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"dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+1" diff --git a/Project.toml b/Project.toml new file mode 100644 index 0000000..8668cce --- /dev/null +++ b/Project.toml @@ -0,0 +1,23 @@ +[deps] +CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b" +DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" +DataFramesMeta = "1313f7d8-7da2-5740-9ea0-a2ca25f37964" +ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" +Franklin = "713c75ef-9fc9-4b05-94a9-213340da978e" +GLM = "38e38edf-8417-5370-95a0-9cbb8c7f171a" +NodeJS = "2bd173c7-0d6d-553b-b6af-13a54713934c" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Pluto = "c3e4b0f8-55cb-11ea-2926-15256bba5781" +PlutoSliderServer = "2fc8631c-6f24-4c5b-bca7-cbb509c42db4" +PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee" +RDatasets = "ce6b1742-4840-55fa-b093-852dadbb1d8b" +StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd" + +[compat] +DataFrames = "1" +DataFramesMeta = "0.10" +Franklin = "0.10" +GLM = "1" +Pluto = "0.18, 0.19" +PlutoSliderServer = "0.3" +StatsPlots = "0.14" diff --git a/assets/favicon.ico b/assets/favicon.ico new file mode 100644 index 0000000..ef91212 Binary files /dev/null and b/assets/favicon.ico differ diff --git a/assets/favicon.png b/assets/favicon.png new file mode 100644 index 0000000..67bce35 Binary files /dev/null and b/assets/favicon.png differ diff --git a/assets/hamburger.svg b/assets/hamburger.svg new file mode 100644 index 0000000..233bf21 --- /dev/null +++ b/assets/hamburger.svg @@ -0,0 +1 @@ + \ No newline at end of file diff --git a/assets/notes/descent-methods/code/_ceval_1.jl b/assets/notes/descent-methods/code/_ceval_1.jl new file mode 100644 index 0000000..1b95b0b --- /dev/null +++ b/assets/notes/descent-methods/code/_ceval_1.jl @@ -0,0 +1,8 @@ +# This file was generated, do not modify it. # hide +using LinearAlgebra +import ForwardDiff:gradient,hessian + +f(x) = (1 - x[1])^2 + 100.0(x[2] - x[1]^2)^2 +∇f(x) = gradient(f, x) +∇²f(x) = hessian(f, x) +;#hide diff --git a/assets/notes/descent-methods/code/_ceval_2.jl b/assets/notes/descent-methods/code/_ceval_2.jl new file mode 100644 index 0000000..031c120 --- /dev/null +++ b/assets/notes/descent-methods/code/_ceval_2.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +x̄ = [1., 1.] +@show norm(∇f(x̄)) +@show isposdef(∇²f(x̄)) diff --git a/assets/notes/descent-methods/code/_ceval_3.jl b/assets/notes/descent-methods/code/_ceval_3.jl new file mode 100644 index 0000000..28f3d4e --- /dev/null +++ b/assets/notes/descent-methods/code/_ceval_3.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +steepestDescent(f, x) = -gradient(f, x) +;#hide diff --git a/assets/notes/descent-methods/code/descent.jl b/assets/notes/descent-methods/code/descent.jl new file mode 100644 index 0000000..0d6e682 --- /dev/null +++ b/assets/notes/descent-methods/code/descent.jl @@ -0,0 +1,8 @@ +# This file was generated, do not modify it. # hide +using Plots; gr() + +x = y = -0.25:0.001:1.5 +z = Surface((x,y)->f([x,y]), x, y) +contourf(x,y,z, levels=collect([10^k for k ∈ -4.0:3.0]), colorbar=:none, leg=false) +scatter!([1.],[1.], c="gold", ms=10, shape=:star5) +savefig(joinpath(@OUTPUT,"surface")) # hide \ No newline at end of file diff --git a/assets/notes/descent-methods/code/output/_ceval_1.out b/assets/notes/descent-methods/code/output/_ceval_1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/descent-methods/code/output/_ceval_1.res b/assets/notes/descent-methods/code/output/_ceval_1.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/descent-methods/code/output/_ceval_1.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/descent-methods/code/output/_ceval_2.out b/assets/notes/descent-methods/code/output/_ceval_2.out new file mode 100644 index 0000000..c901b1d --- /dev/null +++ b/assets/notes/descent-methods/code/output/_ceval_2.out @@ -0,0 +1,2 @@ +norm(∇f(x̄)) = 0.0 +isposdef(∇²f(x̄)) = true diff --git a/assets/notes/descent-methods/code/output/_ceval_2.res b/assets/notes/descent-methods/code/output/_ceval_2.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/descent-methods/code/output/_ceval_2.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/descent-methods/code/output/_ceval_3.out b/assets/notes/descent-methods/code/output/_ceval_3.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/descent-methods/code/output/_ceval_3.res b/assets/notes/descent-methods/code/output/_ceval_3.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/descent-methods/code/output/_ceval_3.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/descent-methods/code/output/descent.out b/assets/notes/descent-methods/code/output/descent.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/descent-methods/code/output/descent.res b/assets/notes/descent-methods/code/output/descent.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/descent-methods/code/output/descent.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/descent-methods/code/output/setup.out b/assets/notes/descent-methods/code/output/setup.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/descent-methods/code/output/setup.res b/assets/notes/descent-methods/code/output/setup.res new file mode 100644 index 0000000..1fcb670 --- /dev/null +++ b/assets/notes/descent-methods/code/output/setup.res @@ -0,0 +1 @@ +Random.TaskLocalRNG() \ No newline at end of file diff --git a/assets/notes/descent-methods/code/output/surface.png b/assets/notes/descent-methods/code/output/surface.png new file mode 100644 index 0000000..7a348e8 Binary files /dev/null and b/assets/notes/descent-methods/code/output/surface.png differ diff --git a/assets/notes/descent-methods/code/setup.jl b/assets/notes/descent-methods/code/setup.jl new file mode 100644 index 0000000..1336505 --- /dev/null +++ b/assets/notes/descent-methods/code/setup.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +# hideall +using Random; Random.seed!(12) \ No newline at end of file diff --git a/assets/notes/gradients/code/_ceval_1.jl b/assets/notes/gradients/code/_ceval_1.jl new file mode 100644 index 0000000..d8bec88 --- /dev/null +++ b/assets/notes/gradients/code/_ceval_1.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +a, b = 1, 100 +f(x) = (a - x[1])^2 + b*(x[2] - x[1]^2)^2 +∇f(x) = [-2(a - x[1]) - 4b*(x[2] - x[1]^2)*x[1] , 2b*(x[2] - x[1]^2) ] +; #hide diff --git a/assets/notes/gradients/code/_ceval_2.jl b/assets/notes/gradients/code/_ceval_2.jl new file mode 100644 index 0000000..fce77bf --- /dev/null +++ b/assets/notes/gradients/code/_ceval_2.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +using ForwardDiff +∇fad(x) = ForwardDiff.gradient(f, x) +; #hide diff --git a/assets/notes/gradients/code/_ceval_3.jl b/assets/notes/gradients/code/_ceval_3.jl new file mode 100644 index 0000000..1f5cf0b --- /dev/null +++ b/assets/notes/gradients/code/_ceval_3.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +x = rand(2) +∇f(x) == ∇fad(x) diff --git a/assets/notes/gradients/code/_ceval_4.jl b/assets/notes/gradients/code/_ceval_4.jl new file mode 100644 index 0000000..7194f75 --- /dev/null +++ b/assets/notes/gradients/code/_ceval_4.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +a = collect(1:5) +ForwardDiff.gradient(x->a'x, rand(5)) +; #hide diff --git a/assets/notes/gradients/code/output/_ceval_1.out b/assets/notes/gradients/code/output/_ceval_1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/gradients/code/output/_ceval_1.res b/assets/notes/gradients/code/output/_ceval_1.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/gradients/code/output/_ceval_1.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/gradients/code/output/_ceval_2.out b/assets/notes/gradients/code/output/_ceval_2.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/gradients/code/output/_ceval_2.res b/assets/notes/gradients/code/output/_ceval_2.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/gradients/code/output/_ceval_2.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/gradients/code/output/_ceval_3.out b/assets/notes/gradients/code/output/_ceval_3.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/gradients/code/output/_ceval_3.res b/assets/notes/gradients/code/output/_ceval_3.res new file mode 100644 index 0000000..f32a580 --- /dev/null +++ b/assets/notes/gradients/code/output/_ceval_3.res @@ -0,0 +1 @@ +true \ No newline at end of file diff --git a/assets/notes/gradients/code/output/_ceval_4.out b/assets/notes/gradients/code/output/_ceval_4.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/gradients/code/output/_ceval_4.res b/assets/notes/gradients/code/output/_ceval_4.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/gradients/code/output/_ceval_4.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/gradients/code/output/setup.out b/assets/notes/gradients/code/output/setup.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/gradients/code/output/setup.res b/assets/notes/gradients/code/output/setup.res new file mode 100644 index 0000000..1fcb670 --- /dev/null +++ b/assets/notes/gradients/code/output/setup.res @@ -0,0 +1 @@ +Random.TaskLocalRNG() \ No newline at end of file diff --git a/assets/notes/gradients/code/setup.jl b/assets/notes/gradients/code/setup.jl new file mode 100644 index 0000000..e616335 --- /dev/null +++ b/assets/notes/gradients/code/setup.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +# hideall +using Random; Random.seed!(1234) \ No newline at end of file diff --git a/assets/notes/gradients/gradient-normal.svg b/assets/notes/gradients/gradient-normal.svg new file mode 100644 index 0000000..a96435a --- /dev/null +++ b/assets/notes/gradients/gradient-normal.svg @@ -0,0 +1,50 @@ + + + + Untitled + Drawing exported from Concepts: Smarter Sketching + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-fit.jl b/assets/notes/least-squares/code/ls-fit.jl new file mode 100644 index 0000000..d66e9f8 --- /dev/null +++ b/assets/notes/least-squares/code/ls-fit.jl @@ -0,0 +1,6 @@ +# This file was generated, do not modify it. # hide +using StatsPlots; pyplot() +x₁, x₄ = coef(ols)[[1,4]] +@df mtcars scatter(:WT, :MPG, xlabel="weight (1000s lbs)", ylabel="MPG", leg=false) +@df mtcars plot!(:WT, x₁ .+ x₄*:WT) +savefig(joinpath(@OUTPUT,"ls-fit")) # hide \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-glm.jl b/assets/notes/least-squares/code/ls-glm.jl new file mode 100644 index 0000000..4a73eb2 --- /dev/null +++ b/assets/notes/least-squares/code/ls-glm.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +using GLM +ols = lm(@formula(MPG ~ Disp + HP + WT), mtcars) \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-n2o-1.jl b/assets/notes/least-squares/code/ls-n2o-1.jl new file mode 100644 index 0000000..84c576a --- /dev/null +++ b/assets/notes/least-squares/code/ls-n2o-1.jl @@ -0,0 +1,9 @@ +# This file was generated, do not modify it. # hide +using CSV, DataFrames, DataFramesMeta, Statistics +df = CSV.read(fname, DataFrame, comment="#", normalizenames=true, delim=" ", ignorerepeated=true) +dfg = @chain df begin + @select(:year = :N2OcatsMLOyr, :month = :N2OcatsMLOmon, :ppb = :N2OcatsMLOm) + @subset((!isnan).(:ppb)) + @by([:year, :month], :avgppb = mean(:ppb)) +end +describe(dfg, :min, :max, cols=[:year, :avgppb]) \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-n2o-2.jl b/assets/notes/least-squares/code/ls-n2o-2.jl new file mode 100644 index 0000000..73d22dc --- /dev/null +++ b/assets/notes/least-squares/code/ls-n2o-2.jl @@ -0,0 +1,6 @@ +# This file was generated, do not modify it. # hide +@df dfg scatter(:avgppb) +p(t) = α + β*t +plot!(p, lw=3) +plot!(leg=false, xlab="Month index from 1999 to 2021", ylab="Monthly average N₂O ppb") +savefig(joinpath(@OUTPUT,"ls-n2o-2")) # hide \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-n2o-3.jl b/assets/notes/least-squares/code/ls-n2o-3.jl new file mode 100644 index 0000000..bb71297 --- /dev/null +++ b/assets/notes/least-squares/code/ls-n2o-3.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +@df dfg scatter(:avgppb - p.(t)) +plot!(leg=false, xlab="Month index from 1999 to 2021", ylab="ppb deviation from mean") # hide +savefig(joinpath(@OUTPUT,"ls-n2o-3")) # hide \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-n2o-4.jl b/assets/notes/least-squares/code/ls-n2o-4.jl new file mode 100644 index 0000000..c3617cc --- /dev/null +++ b/assets/notes/least-squares/code/ls-n2o-4.jl @@ -0,0 +1,2 @@ +# This file was generated, do not modify it. # hide +α, β = A \ b \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-n2o.jl b/assets/notes/least-squares/code/ls-n2o.jl new file mode 100644 index 0000000..1430180 --- /dev/null +++ b/assets/notes/least-squares/code/ls-n2o.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +fname = "mlo_N2O_All.dat" +if !isfile(fname) + download("https://gml.noaa.gov/aftp/data/hats/n2o/insituGCs/CATS/hourly/mlo_N2O_All.dat", fname) +end \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls-size.jl b/assets/notes/least-squares/code/ls-size.jl new file mode 100644 index 0000000..8c0ee16 --- /dev/null +++ b/assets/notes/least-squares/code/ls-size.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +# hideall +print("$(size(mtcars,1))") \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls1-1.jl b/assets/notes/least-squares/code/ls1-1.jl new file mode 100644 index 0000000..37130fd --- /dev/null +++ b/assets/notes/least-squares/code/ls1-1.jl @@ -0,0 +1,2 @@ +# This file was generated, do not modify it. # hide +first(mtcars, 5) \ No newline at end of file diff --git a/assets/notes/least-squares/code/ls1.jl b/assets/notes/least-squares/code/ls1.jl new file mode 100644 index 0000000..b5f3119 --- /dev/null +++ b/assets/notes/least-squares/code/ls1.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +x1 = A \ b +@show norm(x1 - x, Inf) \ No newline at end of file diff --git a/assets/notes/least-squares/code/output/ls-fit.out b/assets/notes/least-squares/code/output/ls-fit.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/least-squares/code/output/ls-fit.png b/assets/notes/least-squares/code/output/ls-fit.png new file mode 100644 index 0000000..566aaaa Binary files /dev/null and b/assets/notes/least-squares/code/output/ls-fit.png differ diff --git a/assets/notes/least-squares/code/output/ls-fit.res b/assets/notes/least-squares/code/output/ls-fit.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/least-squares/code/output/ls-fit.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/least-squares/code/output/ls-glm.out b/assets/notes/least-squares/code/output/ls-glm.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/least-squares/code/output/ls-glm.res b/assets/notes/least-squares/code/output/ls-glm.res new file mode 100644 index 0000000..55682b0 --- /dev/null +++ b/assets/notes/least-squares/code/output/ls-glm.res @@ -0,0 +1,13 @@ +StatsModels.TableRegressionModel{GLM.LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}} + +MPG ~ 1 + Disp + HP + WT + +Coefficients: +─────────────────────────────────────────────────────────────────────────────── + Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95% +─────────────────────────────────────────────────────────────────────────────── +(Intercept) 37.1055 2.11082 17.58 <1e-15 32.7817 41.4293 +Disp -0.000937009 0.0103497 -0.09 0.9285 -0.0221375 0.0202635 +HP -0.0311566 0.0114358 -2.72 0.0110 -0.0545817 -0.00773139 +WT -3.80089 1.06619 -3.56 0.0013 -5.98488 -1.6169 +─────────────────────────────────────────────────────────────────────────────── \ No newline at end of file diff --git a/assets/notes/least-squares/code/output/ls-n2o-1.out b/assets/notes/least-squares/code/output/ls-n2o-1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/least-squares/code/output/ls-n2o-1.res b/assets/notes/least-squares/code/output/ls-n2o-1.res new file mode 100644 index 0000000..4633b13 --- /dev/null +++ b/assets/notes/least-squares/code/output/ls-n2o-1.res @@ -0,0 +1,6 @@ +2×3 DataFrame + Row │ variable min max + │ Symbol Real Real +─────┼───────────────────────────── + 1 │ year 1999 2022 + 2 │ 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diff --git a/assets/notes/least-squares/code/output/ls1-1.out b/assets/notes/least-squares/code/output/ls1-1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/least-squares/code/output/ls1-1.res b/assets/notes/least-squares/code/output/ls1-1.res new file mode 100644 index 0000000..2fb4bc2 --- /dev/null +++ b/assets/notes/least-squares/code/output/ls1-1.res @@ -0,0 +1,9 @@ +5×12 DataFrame + Row │ Model MPG Cyl Disp HP DRat WT QSec VS AM Gear Carb + │ String31 Float64 Int64 Float64 Int64 Float64 Float64 Float64 Int64 Int64 Int64 Int64 +─────┼────────────────────────────────────────────────────────────────────────────────────────────────────────── + 1 │ Mazda RX4 21.0 6 160.0 110 3.9 2.62 16.46 0 1 4 4 + 2 │ Mazda RX4 Wag 21.0 6 160.0 110 3.9 2.875 17.02 0 1 4 4 + 3 │ Datsun 710 22.8 4 108.0 93 3.85 2.32 18.61 1 1 4 1 + 4 │ Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1 + 5 │ Hornet Sportabout 18.7 8 360.0 175 3.15 3.44 17.02 0 0 3 2 \ No newline at end of file diff --git a/assets/notes/least-squares/code/output/ls1.out b/assets/notes/least-squares/code/output/ls1.out new file mode 100644 index 0000000..4590c88 --- /dev/null +++ b/assets/notes/least-squares/code/output/ls1.out @@ -0,0 +1 @@ +norm(x1 - x, Inf) = 4.334310688136611e-13 diff --git a/assets/notes/least-squares/code/output/ls1.res b/assets/notes/least-squares/code/output/ls1.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/least-squares/code/output/ls1.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/least-squares/code/output/mlo_N2O_All.dat b/assets/notes/least-squares/code/output/mlo_N2O_All.dat new file mode 100644 index 0000000..38cf70e --- /dev/null +++ b/assets/notes/least-squares/code/output/mlo_N2O_All.dat @@ -0,0 +1,195480 @@ +# file: mlo_N2O_All.dat +# date: 2023-08-29 +# +# Nitrous Oxide (N2O) data from hourly in situ samples analyzed on a gas chromatograph located +# at Mauna Loa (MLO), Hawaii (19.539 N, 155.578 W, elevation: 3397 m). +# +# Atmospheric measurements from the NOAA/GML Chromatograph for Atmospheric Trace +# Species (CATS) program. This GC replaces the RITS GCs operated during the +# period of 1986 through 2000. This work was funded in part by the Atmospheric Chemistry Project of +# NOAA's Climate and Global Change Program. +# +# Data use policy see https://gml.noaa.gov/hats/hats_datause.html +# +# Citation: +# G.S. Dutton and B.D. Hall (2023), +# Global Atmospheric Nitrous Oxide Dry Air Mole Fractions from the +# NOAA GML Halocarbons in situ Network, 1998-2023, Version: 2023-08-29, +# https://doi.org/10.7289/V5X0659V +# +# For contact information see https://gml.noaa.gov/about/stafflist.html +# +# Calibration scale used: NOAA 2006A +# More information about the calibration scale can be found at: https://gml.noaa.gov/ccl/ +# +# See (http://www.esrl.noaa.gov/gmd/obop/) for station statistics and personnel. +# +# The CATS GCs sample air once an hour. Hourly data are provided in parts-per-billion, ppb. +# Abbreviations used: +# yr=year, mon=month, m=monthly data; gc=type of sample--in situ GC data, sd=standard deviation of samples, +# n=number of samples, brw=Pt. 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b/assets/notes/nonlinear-least-squares/code/_ceval_1.jl new file mode 100644 index 0000000..2c0502c --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_1.jl @@ -0,0 +1,12 @@ +# This file was generated, do not modify it. # hide +function gauss_newton(r, J, x; ε=1e-4, maxits=30) + err = [] + for i = 1:maxits + rk, Jk = r(x), J(x) + push!(err, norm(Jk'rk)) + err[end] ≤ ε && break + x = x - Jk\rk + end + return x, err +end +; # hide diff --git a/assets/notes/nonlinear-least-squares/code/_ceval_2.jl b/assets/notes/nonlinear-least-squares/code/_ceval_2.jl new file mode 100644 index 0000000..390684d --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_2.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +r(x, δ, b) = [ δ[i] - norm(x - b[:,i]) for i in 1:length(δ) ] +; #hide diff --git a/assets/notes/nonlinear-least-squares/code/_ceval_3.jl b/assets/notes/nonlinear-least-squares/code/_ceval_3.jl new file mode 100644 index 0000000..e19c0dc --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_3.jl @@ -0,0 +1,6 @@ +# This file was generated, do not modify it. # hide +using Random +using Statistics +using LinearAlgebra +using ForwardDiff +using Plots diff --git a/assets/notes/nonlinear-least-squares/code/_ceval_4.jl b/assets/notes/nonlinear-least-squares/code/_ceval_4.jl new file mode 100644 index 0000000..bba3bd8 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_4.jl @@ -0,0 +1,8 @@ +# This file was generated, do not modify it. # hide +m = 13 # number of beacons +b = 2rand(2, m) .- 1 # place beacons uniformly in the unit box +x = zeros(2) # true position +x0 = 2rand(2) .- 1 # initial guess of the unknown position +ν = .5*rand(m) +δ = [ norm(x - b[:,i]) + ν[i] for i in 1:m] +; # hide diff --git a/assets/notes/nonlinear-least-squares/code/_ceval_5.jl b/assets/notes/nonlinear-least-squares/code/_ceval_5.jl new file mode 100644 index 0000000..5f958d0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_5.jl @@ -0,0 +1,9 @@ +# This file was generated, do not modify it. # hide +function plotmap(b, x, x0) +scatter(xlim=(-1,+1), ylim=(-1,+1), leg=:outertopright,frame=:box, aspect_ratio=:equal) +scatter!(b[1,:], b[2,:], label="beacons", shape=:square, ms=7) +scatter!(x[1,:], x[2,:], label="true position", shape=:xcross, ms=7) +scatter!(x0[1,:], x0[2,:], label="initial guess", c="yellow", ms=7) +end +plotmap(b, x, x0) +savefig("map"); # hide diff --git a/assets/notes/nonlinear-least-squares/code/_ceval_6.jl b/assets/notes/nonlinear-least-squares/code/_ceval_6.jl new file mode 100644 index 0000000..ef31c74 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_6.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +J(x) = ForwardDiff.jacobian(x->r(x, δ, b), x) +xs, err = gauss_newton(x->r(x, δ, b), J, x0) +plot(err, yaxis=:log) +savefig("convergence"); # hide diff --git a/assets/notes/nonlinear-least-squares/code/_ceval_7.jl b/assets/notes/nonlinear-least-squares/code/_ceval_7.jl new file mode 100644 index 0000000..6c62718 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/_ceval_7.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +plotmap(b, x, x0) +scatter!(xs[1,:], xs[2,:], label="solution", shape=:star7, c="green", ms=7) +savefig("map-soln"); # hide diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_1.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_1.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_1.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_1.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_2.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_2.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_2.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_2.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_2.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_3.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_3.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_3.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_3.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_3.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_4.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_4.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_4.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_4.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_4.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_5.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_5.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_5.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_5.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_5.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_6.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_6.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_6.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_6.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_6.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_7.out b/assets/notes/nonlinear-least-squares/code/output/_ceval_7.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/_ceval_7.res b/assets/notes/nonlinear-least-squares/code/output/_ceval_7.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/_ceval_7.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/output/convergence.png b/assets/notes/nonlinear-least-squares/code/output/convergence.png new file mode 100644 index 0000000..6db40ab Binary files /dev/null and b/assets/notes/nonlinear-least-squares/code/output/convergence.png differ diff --git a/assets/notes/nonlinear-least-squares/code/output/map-soln.png b/assets/notes/nonlinear-least-squares/code/output/map-soln.png new file mode 100644 index 0000000..06a4595 Binary files /dev/null and b/assets/notes/nonlinear-least-squares/code/output/map-soln.png differ diff --git a/assets/notes/nonlinear-least-squares/code/output/map.png b/assets/notes/nonlinear-least-squares/code/output/map.png new file mode 100644 index 0000000..c5f82f3 Binary files /dev/null and b/assets/notes/nonlinear-least-squares/code/output/map.png differ diff --git a/assets/notes/nonlinear-least-squares/code/output/setup.out b/assets/notes/nonlinear-least-squares/code/output/setup.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/nonlinear-least-squares/code/output/setup.res b/assets/notes/nonlinear-least-squares/code/output/setup.res new file mode 100644 index 0000000..1fcb670 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/output/setup.res @@ -0,0 +1 @@ +Random.TaskLocalRNG() \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/code/setup.jl b/assets/notes/nonlinear-least-squares/code/setup.jl new file mode 100644 index 0000000..1336505 --- /dev/null +++ b/assets/notes/nonlinear-least-squares/code/setup.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +# hideall +using Random; Random.seed!(12) \ No newline at end of file diff --git a/assets/notes/nonlinear-least-squares/position-distances.svg b/assets/notes/nonlinear-least-squares/position-distances.svg new file mode 100644 index 0000000..68c39ee --- /dev/null +++ b/assets/notes/nonlinear-least-squares/position-distances.svg @@ -0,0 +1,50 @@ + + + + Untitled + Drawing exported from Concepts: Smarter Sketching + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/_ceval_1.jl b/assets/notes/qr-factorization/code/_ceval_1.jl new file mode 100644 index 0000000..350cf8f --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_1.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +# hideall +import Random; Random.seed!(1234); diff --git a/assets/notes/qr-factorization/code/_ceval_10.jl b/assets/notes/qr-factorization/code/_ceval_10.jl new file mode 100644 index 0000000..7aa0d46 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_10.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +θ = LinRange(0,3,400) +ε = 1e-7 +A = @. [sin(θ)^2 cos(θ+ε)^2 θ^0] +; #hide diff --git a/assets/notes/qr-factorization/code/_ceval_11.jl b/assets/notes/qr-factorization/code/_ceval_11.jl new file mode 100644 index 0000000..1db6c29 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_11.jl @@ -0,0 +1,2 @@ +# This file was generated, do not modify it. # hide +rank(A) == 3 diff --git a/assets/notes/qr-factorization/code/_ceval_12.jl b/assets/notes/qr-factorization/code/_ceval_12.jl new file mode 100644 index 0000000..24c5284 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_12.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +xᵉ = [1., 2., 1.] +b = A*xᵉ +; #hide diff --git a/assets/notes/qr-factorization/code/_ceval_13.jl b/assets/notes/qr-factorization/code/_ceval_13.jl new file mode 100644 index 0000000..f8ab944 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_13.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +xⁿ = A'A \ A'b +eⁿ = norm(xᵉ - xⁿ) / norm(xᵉ) +@show xⁿ +@show eⁿ diff --git a/assets/notes/qr-factorization/code/_ceval_14.jl b/assets/notes/qr-factorization/code/_ceval_14.jl new file mode 100644 index 0000000..808a9a4 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_14.jl @@ -0,0 +1,6 @@ +# This file was generated, do not modify it. # hide +Q, R = qr(A); Q = Matrix(Q) +xʳ = R \ (Q'b) +eʳ = norm(xᵉ - xʳ) / norm(xᵉ) +@show xʳ +@show eʳ diff --git a/assets/notes/qr-factorization/code/_ceval_15.jl b/assets/notes/qr-factorization/code/_ceval_15.jl new file mode 100644 index 0000000..dbe227d --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_15.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +xᵇ = A \ b +eᵇ = norm(xᵉ - xᵇ) / norm(xᵉ) +@show xᵇ +@show eᵇ diff --git a/assets/notes/qr-factorization/code/_ceval_2.jl b/assets/notes/qr-factorization/code/_ceval_2.jl new file mode 100644 index 0000000..37c652a --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_2.jl @@ -0,0 +1,2 @@ +# This file was generated, do not modify it. # hide +using LinearAlgebra diff --git a/assets/notes/qr-factorization/code/_ceval_3.jl b/assets/notes/qr-factorization/code/_ceval_3.jl new file mode 100644 index 0000000..4ad16d9 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_3.jl @@ -0,0 +1,5 @@ +# This file was generated, do not modify it. # hide +e₁ = [1, 0]; e₂ = [0, 1] +@show e₁'e₂ +@show norm(e₁) +@show norm(e₂) diff --git a/assets/notes/qr-factorization/code/_ceval_4.jl b/assets/notes/qr-factorization/code/_ceval_4.jl new file mode 100644 index 0000000..d09128b --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_4.jl @@ -0,0 +1,4 @@ +# This file was generated, do not modify it. # hide +m, n = 4, 3 +A = randn(m,n) +Q, R = qr(A) diff --git a/assets/notes/qr-factorization/code/_ceval_5.jl b/assets/notes/qr-factorization/code/_ceval_5.jl new file mode 100644 index 0000000..83f1825 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_5.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +@show norm(Q'*Q - I) +@show norm(Q*Q' - I) diff --git a/assets/notes/qr-factorization/code/_ceval_6.jl b/assets/notes/qr-factorization/code/_ceval_6.jl new file mode 100644 index 0000000..5e1c855 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_6.jl @@ -0,0 +1,7 @@ +# This file was generated, do not modify it. # hide +F = qr(A) +Q₁ = Matrix(F.Q) +R₁ = F.R +@show size(Q₁) +@show size(R₁) +@show norm(A - Q₁*R₁) diff --git a/assets/notes/qr-factorization/code/_ceval_7.jl b/assets/notes/qr-factorization/code/_ceval_7.jl new file mode 100644 index 0000000..8fb4866 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_7.jl @@ -0,0 +1,8 @@ +# This file was generated, do not modify it. # hide +m, n = 4, 3 +A = randn(m, n) +b = randn(m) +F = qr(A) +Q₁, R₁ = Matrix(F.Q), F.R +x = R₁ \ Q₁'b # <--- do NOT be tempted to use x = inv(R₁)*Q₁'b +; # hide diff --git a/assets/notes/qr-factorization/code/_ceval_8.jl b/assets/notes/qr-factorization/code/_ceval_8.jl new file mode 100644 index 0000000..1b0f106 --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_8.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +r = b - A*x +@show norm(A'r) diff --git a/assets/notes/qr-factorization/code/_ceval_9.jl b/assets/notes/qr-factorization/code/_ceval_9.jl new file mode 100644 index 0000000..29a8bab --- /dev/null +++ b/assets/notes/qr-factorization/code/_ceval_9.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +Q₂ = F.Q[:,n+1:end] +norm(Q₂'b) ≈ norm(r) diff --git a/assets/notes/qr-factorization/code/output/_ceval_1.out b/assets/notes/qr-factorization/code/output/_ceval_1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_1.res b/assets/notes/qr-factorization/code/output/_ceval_1.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_1.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_10.out b/assets/notes/qr-factorization/code/output/_ceval_10.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_10.res b/assets/notes/qr-factorization/code/output/_ceval_10.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_10.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_11.out b/assets/notes/qr-factorization/code/output/_ceval_11.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_11.res b/assets/notes/qr-factorization/code/output/_ceval_11.res new file mode 100644 index 0000000..f32a580 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_11.res @@ -0,0 +1 @@ +true \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_12.out b/assets/notes/qr-factorization/code/output/_ceval_12.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_12.res b/assets/notes/qr-factorization/code/output/_ceval_12.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_12.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_13.out b/assets/notes/qr-factorization/code/output/_ceval_13.out new file mode 100644 index 0000000..61b3028 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_13.out @@ -0,0 +1,2 @@ +xⁿ = [1.0769230768721778, 2.0769230770796003, 0.9230769230769231] +eⁿ = 0.054392829346937996 diff --git a/assets/notes/qr-factorization/code/output/_ceval_13.res b/assets/notes/qr-factorization/code/output/_ceval_13.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_13.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_14.out b/assets/notes/qr-factorization/code/output/_ceval_14.out new file mode 100644 index 0000000..e250d5a --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_14.out @@ -0,0 +1,2 @@ +xʳ = [0.9999999962497133, 1.9999999962497135, 1.000000003750286] +eʳ = 2.65185298142961e-9 diff --git a/assets/notes/qr-factorization/code/output/_ceval_14.res b/assets/notes/qr-factorization/code/output/_ceval_14.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_14.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_15.out b/assets/notes/qr-factorization/code/output/_ceval_15.out new file mode 100644 index 0000000..ab93f1a --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_15.out @@ -0,0 +1,2 @@ +xᵇ = [0.9999999998372602, 1.9999999998372604, 1.0000000001627405] +eᵇ = 1.1507453610876192e-10 diff --git a/assets/notes/qr-factorization/code/output/_ceval_15.res b/assets/notes/qr-factorization/code/output/_ceval_15.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_15.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_2.out b/assets/notes/qr-factorization/code/output/_ceval_2.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_2.res b/assets/notes/qr-factorization/code/output/_ceval_2.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_2.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_3.out b/assets/notes/qr-factorization/code/output/_ceval_3.out new file mode 100644 index 0000000..18df710 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_3.out @@ -0,0 +1,3 @@ +e₁' * e₂ = 0 +norm(e₁) = 1.0 +norm(e₂) = 1.0 diff --git a/assets/notes/qr-factorization/code/output/_ceval_3.res b/assets/notes/qr-factorization/code/output/_ceval_3.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_3.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_4.out b/assets/notes/qr-factorization/code/output/_ceval_4.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_4.res b/assets/notes/qr-factorization/code/output/_ceval_4.res new file mode 100644 index 0000000..f92d9e4 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_4.res @@ -0,0 +1,12 @@ +LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}} +Q factor: +4×4 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}: + -0.589037 -0.609377 0.391674 0.35817 + 0.594233 -0.0982621 0.0108708 0.798194 + -0.547289 0.568937 -0.37926 0.482646 + 0.0199064 0.543429 0.838233 0.0406633 +R factor: +3×3 Matrix{Float64}: + -1.64787 -1.95628 1.26521 + 0.0 2.8769 -0.0479272 + 0.0 0.0 1.30317 \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_5.out b/assets/notes/qr-factorization/code/output/_ceval_5.out new file mode 100644 index 0000000..29824a4 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_5.out @@ -0,0 +1,2 @@ +norm(Q' * Q - I) = 5.928593550334434e-16 +norm(Q * Q' - I) = 4.896838930763021e-16 diff --git a/assets/notes/qr-factorization/code/output/_ceval_5.res b/assets/notes/qr-factorization/code/output/_ceval_5.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_5.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_6.out b/assets/notes/qr-factorization/code/output/_ceval_6.out new file mode 100644 index 0000000..5d9873c --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_6.out @@ -0,0 +1,3 @@ +size(Q₁) = (4, 3) +size(R₁) = (3, 3) +norm(A - Q₁ * R₁) = 7.494005416219807e-16 diff --git a/assets/notes/qr-factorization/code/output/_ceval_6.res b/assets/notes/qr-factorization/code/output/_ceval_6.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_6.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_7.out b/assets/notes/qr-factorization/code/output/_ceval_7.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_7.res b/assets/notes/qr-factorization/code/output/_ceval_7.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_7.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_8.out b/assets/notes/qr-factorization/code/output/_ceval_8.out new file mode 100644 index 0000000..46e48c8 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_8.out @@ -0,0 +1 @@ +norm(A' * r) = 3.820983142562135e-15 diff --git a/assets/notes/qr-factorization/code/output/_ceval_8.res b/assets/notes/qr-factorization/code/output/_ceval_8.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_8.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/qr-factorization/code/output/_ceval_9.out b/assets/notes/qr-factorization/code/output/_ceval_9.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/qr-factorization/code/output/_ceval_9.res b/assets/notes/qr-factorization/code/output/_ceval_9.res new file mode 100644 index 0000000..f32a580 --- /dev/null +++ b/assets/notes/qr-factorization/code/output/_ceval_9.res @@ -0,0 +1 @@ +true \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/log-range.jl b/assets/notes/regularized-least-squares/code/log-range.jl new file mode 100644 index 0000000..9939c6d --- /dev/null +++ b/assets/notes/regularized-least-squares/code/log-range.jl @@ -0,0 +1,2 @@ +# This file was generated, do not modify it. # hide +LogRange(x1, x2, n) = (10^y for y in range(log10(x1), log10(x2), length=n)) \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/ls-reg-noisy1.jl b/assets/notes/regularized-least-squares/code/ls-reg-noisy1.jl new file mode 100644 index 0000000..e1f5b04 --- /dev/null +++ b/assets/notes/regularized-least-squares/code/ls-reg-noisy1.jl @@ -0,0 +1,8 @@ +# This file was generated, do not modify it. # hide +n = 300 +t = LinRange(0, 4, n) +x = @. sin(t) + t*cos(t)^2 +w = 0.1*randn(n) +b = x + w +plot(b, leg=:topleft, label="noisy signal") +savefig(joinpath(@OUTPUT,"ls-reg-noisy1")) # hide \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/ls-reg-noisy2.jl b/assets/notes/regularized-least-squares/code/ls-reg-noisy2.jl new file mode 100644 index 0000000..8a06927 --- /dev/null +++ b/assets/notes/regularized-least-squares/code/ls-reg-noisy2.jl @@ -0,0 +1,2 @@ +# This file was generated, do not modify it. # hide +finiteDiff(4) \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/ls-reg-noisy3.jl b/assets/notes/regularized-least-squares/code/ls-reg-noisy3.jl new file mode 100644 index 0000000..706a71d --- /dev/null +++ b/assets/notes/regularized-least-squares/code/ls-reg-noisy3.jl @@ -0,0 +1,10 @@ +# This file was generated, do not modify it. # hide +D = finiteDiff(n) +b̂ = [b; zeros(n-1)] +plot(t, b, w=1, leg =:topleft, label="noisy data") +for λ in LogRange(1e0, 1e4, 3) + Â = [ I; √λ*D ] + xLS = Â \ b̂ + plot!(t, xLS, w=2, label="regularized solution: λ = $(λ)") +end +savefig(joinpath(@OUTPUT,"ls-reg-noisy3")) # hide \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/output/log-range.out b/assets/notes/regularized-least-squares/code/output/log-range.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/regularized-least-squares/code/output/log-range.res b/assets/notes/regularized-least-squares/code/output/log-range.res new file mode 100644 index 0000000..dd7c95a --- /dev/null +++ b/assets/notes/regularized-least-squares/code/output/log-range.res @@ -0,0 +1 @@ +LogRange (generic function with 1 method) \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.out b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.png b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.png new file mode 100644 index 0000000..9db44bf Binary files /dev/null and b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.png differ diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.res b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy1.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy2.out b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy2.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy2.res b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy2.res new file mode 100644 index 0000000..b68552d --- /dev/null +++ b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy2.res @@ -0,0 +1,4 @@ +3×4 Matrix{Float64}: + 1.0 -1.0 0.0 0.0 + 0.0 1.0 -1.0 0.0 + 0.0 0.0 1.0 -1.0 \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.out b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.png b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.png new file mode 100644 index 0000000..603c4d9 Binary files /dev/null and b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.png differ diff --git a/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.res b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/regularized-least-squares/code/output/ls-reg-noisy3.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/output/setup.out b/assets/notes/regularized-least-squares/code/output/setup.out new file mode 100644 index 0000000..e69de29 diff --git a/assets/notes/regularized-least-squares/code/output/setup.res b/assets/notes/regularized-least-squares/code/output/setup.res new file mode 100644 index 0000000..97896a0 --- /dev/null +++ b/assets/notes/regularized-least-squares/code/output/setup.res @@ -0,0 +1 @@ +nothing \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/code/setup.jl b/assets/notes/regularized-least-squares/code/setup.jl new file mode 100644 index 0000000..a2275ff --- /dev/null +++ b/assets/notes/regularized-least-squares/code/setup.jl @@ -0,0 +1,3 @@ +# This file was generated, do not modify it. # hide +using LinearAlgebra +using Plots \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/pareto-curve-levels.svg b/assets/notes/regularized-least-squares/pareto-curve-levels.svg new file mode 100644 index 0000000..14de39f --- /dev/null +++ b/assets/notes/regularized-least-squares/pareto-curve-levels.svg @@ -0,0 +1,52 @@ + + + + Sketchnoting Objects + Drawing exported from Concepts: Smarter Sketching + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/assets/notes/regularized-least-squares/pareto-curve.svg b/assets/notes/regularized-least-squares/pareto-curve.svg new file mode 100644 index 0000000..e31e735 --- /dev/null +++ b/assets/notes/regularized-least-squares/pareto-curve.svg @@ -0,0 +1,65 @@ + + + + Sketchnoting Objects + Drawing exported from Concepts: Smarter Sketching + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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} + +.katex { font-size: 1em !important; } + +.katex-display .katex { + /* Overwrite KaTeX settings. */ + display: inline-block; + + /* Allow display equations to wrap on small screens. */ + white-space: normal; +} + +.katex-display::after { + counter-increment: eqnum; + content: "(" counter(eqnum) ")"; + position: relative; + float: right; + padding-right: 5px; +} + +/* ================================================================== + CODE & HIGHLIGHT.JS +================================================================== */ + +code { + background-color: var(--block-background); + padding: 0.1em 0.2em; + border-radius: 2px; + font-size: var(--small); +} + +/* .franklin-content code { */ + +/* background-color: rgba(27,31,35,0.05); */ + +/* padding: 0.1em 0.2em; */ + +/* border-radius: 2px; */ + +/* font-size: 90%; } */ + +.hljs { + font-size: var(--small); + line-height: 1.35em; + border-radius: 10px; +} + +.hljs-meta, +.hljs-metas, +.hljs-metap { font-weight: bold; } + +.hljs-meta { color: rgb(25, 179, 51); } + +.hljs-metas { color: red; } + +.hljs-metap { color: rgb(51, 131, 231); } + +.code-output { + background: var(--output-background); + border: 1px dashed #dbdbdb; +} + +/* ================================================================== + BOXES +================================================================== */ + +.franklin-content .colbox-blue { + background-color: #eef3f5; + padding-top: 5px; + padding-right: 10px; + padding-left: 10px; + padding-bottom: 5px; + margin-left: 5px; + margin-top: 5px; + margin-bottom: 5px; + border-radius: 0 10px 10px 0; + border-left: 5px solid #4c9cf1; +} diff --git a/css/jtd.css b/css/jtd.css new file mode 100644 index 0000000..30f0f8c --- /dev/null +++ b/css/jtd.css @@ -0,0 +1,767 @@ +/* +This CSS is adapted from https://pmarsceill.github.io/just-the-docs/ +Copyright Patrick Marsceill, MIT license +https://github.com/pmarsceill/just-the-docs/blob/master/LICENSE.txt +(MIT licensed) +*/ + +html { + font-family: sans-serif; + -ms-text-size-adjust: 100%; + -webkit-text-size-adjust: 100%; + font-size: 15px; +} + +@media (min-width: 31.25rem) { + html { + font-size: 19px; + } +} + +body { + margin: 0; + position: relative; + padding-bottom: 4rem; + font-family: -apple-system, "helvetica neue", helvetica, roboto, noto, "segoe ui", arial, sans-serif; + font-size: inherit; + line-height: 1.5; + color: #5c5962; + background-color: #fff; +} + +@media (min-width: 50rem) { + body { + position: static; + padding-bottom: 0; + } +} + +h1 { + font-size: 2em; + margin: 0.67em 0; +} + +sub, +sup { + font-size: 75%; + line-height: 0; + position: relative; + vertical-align: baseline; +} + +sup { + top: -0.5em; +} + +sub { + bottom: -0.25em; +} + +img { + border: 0; +} + +p, +h1, +h2, +h3, +h4, +h5, +h6, +ol, +ul, +pre, +address, +blockquote, +dl, +div, +fieldset, +form, +hr, +noscript, +table { + margin-top: 0; +} + +h1, +h2, +h3, +h4, +h5, +h6 { + margin-top: 1.2em; + margin-bottom: 0.8em; + font-weight: 500; + line-height: 1.15; + color: #27262b; +} + +p { + margin-bottom: 1em; +} + +a { + color: #0e39b0; + text-decoration: none; +} + +pre { + overflow: auto; + margin: 0; + margin-bottom: 0.8em; +} + +code { + font-family: "SFMono-Regular", Menlo, Consolas, monospace; + font-size: 14px; + line-height: 1.4; + padding: 0.2em 0.15em; + font-weight: 400; + background-color: #f5f6fa; + border: 1px solid #eeebee; + border-radius: 4px; +} + +li:not(:last-child) { + margin-bottom: 5px; +} + +img { + max-width: 100%; + height: auto; +} + +hr { + height: 1px; + padding: 0; + margin: 2rem 0; + background-color: #eeebee; + border: 0; +} + +@media (min-width: 50rem) { + .page-wrap { + position: absolute; + top: 0; + left: 0; + display: flex; + width: 100%; + height: 100%; + overflow-x: hidden; + overflow-y: hidden; + } +} + +/************************************************************ +============================================================= + +SIDE BAR + +============================================================ +************************************************************/ + +.side-bar { + z-index: 100; + display: flex; + flex-wrap: wrap; + background-color: #f5f6fa; +} + +.side-bar .header { + display: flex; + min-height: 60px; + align-items: center; +} + +.side-bar .header .title { + padding-right: 1rem; + padding-left: 1rem; + flex-grow: 1; + display: flex; + height: 100%; + align-items: center; + padding-top: 0.75rem; + padding-bottom: 0.75rem; + color: #27262b; + font-size: 18px !important; + line-height: 1.15em; +} + +.show-menu { + cursor: pointer; + appearance: none; + display: flex; + height: 100%; + line-height: 60px; + padding: 1rem; + align-items: center; + color: #0e39b0; + text-transform: uppercase; + background: transparent; + text-align: right; + padding-right: 2rem; + border: 0; + width: calc(100% - 220px); + background-color: white; +} + +.side-bar .menu { + width: 100%; + padding-right: 1rem; + padding-left: 1rem; +} + +.menu-list { + padding: 0; + margin-top: 0; + margin-bottom: 0; + list-style: none; +} + +.menu-list-child-list { + padding-left: 0.75rem; + list-style: none; +} + +.menu-list-child-list .menu-list-link { + color: #5c5962; +} + +.menu-list-child-list .menu-list-item { + position: relative; +} + +.menu-list-child-list .menu-list-item::before { + position: absolute; + margin-top: 0.3em; + margin-left: -0.8em; + color: rgba(92, 89, 98, 0.3); + content: "- "; +} + +.menu-list-child-list .menu-list-item.active::before { + color: #5c5962; +} + +.menu-list-item { + font-size: 14px !important; + margin: 0; +} +.menu-list-parent .menu-list-child-list{ + display:none +} +.menu-list-parent.active .menu-list-child-list{ + display:block +} +.menu-list-item .menu-list-child-list { + display: none; +} + +.menu-list-item.active .menu-list-child-list { + display: block; +} +.menu-list-item.active a { + font-weight:600; + color:#27262b; + text-decoration:none +} +.menu-list-link { + display: block; + padding-top: 0.25rem; + padding-bottom: 0.25rem; +} + +.menu-list-link.active { + font-weight: 600; + color: #27262b; + text-decoration: none; +} + +.side-bar .footer { + width: 150px; + font-size: 13px; + padding-right: 1rem; + padding-left: 1rem; + position: absolute; + bottom: 0; + left: 0; + padding-top: 1rem; + padding-bottom: 1rem; +} + +/* Hide checkbox */ +input[type=checkbox] { display: none; } + +/* Show menu when invisible checkbox is checked */ +input[type=checkbox]:checked ~ #side-menu { display: block; } + +/* side bar adjustments for narrow mode (bar on top) */ +@media (max-width: 49.99rem) { + /* Toggle menu off */ + .side-bar .menu { + display: none; + padding-left: 35px; + padding-bottom: 15px; + } + + .side-bar { + border-bottom: 1px solid #eeebee; + } + + .side-bar .header { + padding-left: 15px; + width: 150px; + } /* middle space */ + .show-menu { display: block; } +} + +/* side bar adjustments for narrow mode (bar on top) */ +@media (min-width: 31.25rem) { + .side-bar .header .title { + font-size: 24px !important; + } + + .menu-list-item { + font-size: 16px !important; + } +} + +/* side bar adjustments for medium mode (bar on side) */ +@media (min-width: 50rem) { + .side-bar { + flex-wrap: nowrap; + position: absolute; + width: 248px; + height: 100%; + flex-direction: column; + align-items: flex-end; + } + + .side-bar .header { + z-index: 101; + height: 60px; + max-height: 60px; + border-bottom: 1px solid #eeebee; + padding-right: 2rem; + padding-left: 2rem; + } + + .side-bar .header .title { + padding-top: 0.5rem; + padding-bottom: 0.5rem; + } + + .show-menu { + display: none; + } + + .side-bar .menu { + width: 135px; + padding-right: 2rem; + padding-left: 2rem; + padding-top: 3rem; + padding-bottom: 1rem; + overflow-y: auto; + flex: 1 1 auto; + } + + .side-bar .footer { + padding-right: 2rem; + padding-left: 2rem; + position: static; + } +} + +/* side bar adjustments for wide mode (bar on side) */ +@media (min-width: 66.5rem) { + .side-bar { + width: calc((100% - 66.5rem) / 2 + 264px); + min-width: 264px; + } +} + + +/************************************************************ +============================================================= + +MAIN CONTENT + +============================================================ +************************************************************/ + +@media (min-width: 50rem) { + .main-content-wrap { + position: absolute; + top: 0; + left: 0; + width: 100%; + height: 100%; + -webkit-overflow-scrolling: touch; + overflow-x: hidden; + overflow-y: scroll; + } + + .main-content { + position: relative; + max-width: 800px; + margin-left: 248px; + } +} + +@media (min-width: 66.5rem) { + .main-content { + margin-left: calc((100% - 1064px) / 2 + 264px); + } +} + +.main-header { + padding-right: 1rem; + padding-left: 1rem; + display: none; + + /* padding-top:1rem; */ + + /* padding-bottom:1rem; */ + background-color: #f5f6fa; + height: 60px; + line-height: 60px; +} + +@media (min-width: 50rem) { + .main-header { + padding-right: 2rem; + padding-left: 2rem; + } +} + +@media (min-width: 50rem) { + .main-header { + display: flex; + justify-content: flex-end; + height: 60px; + background-color: #fff; + border-bottom: 1px solid #eeebee; + } +} + +.main-footer { + font-size: 14px; + color: darkgray; + border-top: 1px solid #eeebee; + padding-top: 15px; +} + +.franklin-content { + padding-right: 1rem; + padding-left: 1rem; + padding-top: 1rem; + padding-bottom: 1rem; + line-height: 1.5; +} + +@media (min-width: 50rem) { + .franklin-content { + padding-right: 2rem; + padding-left: 2rem; + padding-top: 2rem; + padding-bottom: 2rem; + } +} + +/************************************************************ +============================================================= + +Franklin SPECIFICS + +============================================================ +************************************************************/ + +.franklin-content a { + overflow: hidden; + text-overflow: ellipsis; + white-space: nowrap; +} + +.franklin-content ul, +ol { + line-height: 1em; +} + +.franklin-content li { + margin-top: 0.3em; + line-height: 1.3em; +} + +.franklin-content ul, +.franklin-content ol { + padding-left: 1.5em; +} + +.franklin-content h1:first-of-type { + margin-top: 0.5em; +} + +.franklin-content h1, +h2, +h3, +h4, +h5, +h6 { + position: relative; +} + +td.fndef-backref { + min-width: 20px; +} + +.franklin-content table { + margin-left: auto; + margin-right: auto; + border-collapse: collapse; + text-align: center; + margin-bottom: 1em; +} + +.franklin-content tr, +th, +td { + font-size: 14px; + padding: 10px; + border: 1px solid black; +} + +.franklin-content table tbody tr td { + border: 1px solid black; +} + +.franklin-content blockquote { + background: lemonchiffon; + border-left: 7px solid #a8a8a8; + margin: 1.5em 10px; + padding: 0.5em 10px; + font-style: italic; +} + +.franklin-content blockquote p { + display: inline; +} + +.franklin-content .row { + display: block; +} + +.franklin-content .left { + float: left; + margin-right: 15px; +} + +.franklin-content .right { + float: right; +} + +.franklin-content .container img { + width: auto; + padding-left: 0; + border-radius: 10px; + margin-bottom: 1em; +} + +.franklin-content table.fndef { + margin: 0; + margin-bottom: 1em; + border: 0; +} + +.franklin-content .fndef tr { + padding: 0; + border: 0; + text-align: left; +} + +.franklin-content .fndef td { + padding: 0; + border: 0; +} + +.franklin-content .fndef td.fndef-backref { + vertical-align: top; + font-size: 80%; + padding-left: 5px; +} + +.franklin-content .fndef td.fndef-content { + font-size: 90%; + padding-left: 10px; +} + +.franklin-content .fndef tr { + border-left: 2px solid lightgray; +} + +.franklin-content img { + width: 70%; + text-align: center; + padding-left: 10%; +} + +.franklin-content .img-small img { + width: 50%; + text-align: center; + padding-left: 20%; +} + +.franklin-content h1 > a {white-space: normal;} +.franklin-content h2 > a {white-space: normal;} +.franklin-content h3 > a {white-space: normal;} + +/* ================================================================== + KATEX +================================================================== */ + +body { counter-reset: eqnum; } + +.katex { font-size: 1em !important; } + +.katex-display .katex { + display: inline-block; +} /* overwrite katex settings */ + +.katex-display::after { + counter-increment: eqnum; + content: "(" counter(eqnum) ")"; + position: relative; + float: right; + padding-right: 5px; +} + +.nonumber .katex-display::after { + counter-increment: nothing; + content: ""; +} + +/* ================================================================== + CODE & HIGHLIGHT.JS +================================================================== */ + +.hljs { + font-size: 14px; + line-height: 1.35em; + border-radius: 10px; + padding: 1em; +} + +.hljs-meta, +.hljs-metas, +.hljs-metap { font-weight: bold; } + +.hljs-meta { color: rgb(25, 179, 51); } + +.hljs-metas { color: red; } + +.hljs-metap { color: rgb(51, 131, 231); } + +franklin-toc { + /* Avoid clickable elements being too close together. */ + margin: 0.6rem 0; +} + +.note { + margin-top: 1.5em; + width: 95%; + margin-left:auto; + margin-right:auto; + background-color: aliceblue; + border-radius: 5px; + margin-bottom: 1em; +} +.note .content { + padding: 10px; + padding-left: 12px; +} +.note .title{ + font-size: 105%; + border-top-left-radius: 5px; + border-top-right-radius: 5px; + padding-left: 7px; + padding-top: 2px; + color: white; + background: cornflowerblue; +} + +.warning { + margin-top: 1.5em; + width: 95%; + margin-left:auto; + margin-right:auto; + background-color: #f1b3b1; + border-radius: 5px; + margin-bottom: 1em; +} +.warning .content { + padding: 10px; + padding-left: 12px; +} +.warning .title{ + font-size: 105%; + border-top-left-radius: 5px; + border-top-right-radius: 5px; + padding-left: 7px; + padding-top: 2px; + color: white; + background: crimson; +} + +.definition{ + background-color: aliceblue; + border-left: 5px solid cornflowerblue; + border-radius: 10px; + padding: 10px; + margin-bottom: 1em; +} + +.theorem{ + background-color: aliceblue; + border-left: 5px solid cornflowerblue; + border-radius: 10px; + padding: 10px; + margin-bottom: 1em; +} + +.page-foot { + margin-top: 2em; + border-top: 1px solid #eeebee; + padding-top:20px; + padding-bottom:15px; + color: gray; + font-size: 90%; +} + +/* ----------------- */ +/* Clipboard. See `_/libs/clipboard.js */ +/* ----------------- */ + +pre { + position: relative; + /* padding: 30px; */ +} + +.copy-button { + cursor: pointer; + border: 0; + font-size: 10px; + text-transform: uppercase; + font-weight: 500; + padding: 6px 10px 10px; + color: #795548; + background-color: transparent; + position: absolute; + top: 0; + right: 0; +} + +.copy-button:hover, +.copy-button:focus, +.copy-button:active { + outline: 0; + background-color: #eee9e6; +} \ No newline at end of file diff --git a/grades/index.html b/grades/index.html new file mode 100644 index 0000000..4638238 --- /dev/null +++ b/grades/index.html @@ -0,0 +1,66 @@ + + + + + + + + + + + + + Grades and policies + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Grades and policies

+

Grade distribution

+ +

Collaboration

+

Most homeworks involve programming tasks. You may collaborate and consult with other students in the course, but you must hand in your own assignments. If you have collaborated or consulted with someone while working on your assignment, you must acknowledge this explicitly in your submitted assignment. If you are unsure about any of these rules, consult with one of the teaching staff or visit the departmental webpage on Collaboration and Plagiarism.

+

Policies

+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + diff --git a/homework/hw1.jl b/homework/hw1.jl new file mode 100644 index 0000000..9f4caea --- /dev/null +++ b/homework/hw1.jl @@ -0,0 +1,1037 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ a89095b2-ff36-48ba-b65c-27d4659d1e7d +using Plots + +# ╔═╡ caf735c8-d31e-4be6-b34b-52bff9cae3ad +using LinearAlgebra + +# ╔═╡ 30de9262-41a9-481d-b2c3-edb1193434cc +# Fill in these fields, and execute the field either by hiting "Shift-Enter" or clicking the play button at the bottom right of this cell. +student = (name = "Harry Potter", ubcid = "123456789") + +# ╔═╡ 1718256d-d4c7-4eea-a931-dd8d5f12bc50 +md"""Student: **$(student.name)** (ID# $(student.ubcid))""" + +# ╔═╡ 0690f1b6-71de-11ec-27b8-f36e49769002 +md"""# CPSC 406 — Homework 1""" + +# ╔═╡ ca2da5ca-0127-436d-a8d8-9454af111117 +md""" +This first homework is modeled after first assignment for the really neat MIT course on [Introduction to Computational Thinking](https://computationalthinking.mit.edu/Spring21/). You'll find lots of information on those course pages, including helpful information on [Installing Julia and Pluto](https://computationalthinking.mit.edu/Spring21/installation/). + +All homework assignments for CPSC 406 will be completed using a [Pluto notebook](https://github.com/fonsp/Pluto.jl). You will submit your notebook using Canvas. + +!!! deadline + This homework is due **Tuesday, January 17**. + + +""" + +# ╔═╡ 0f229544-3b79-4136-86e8-6f68bdee4ad3 +md""" +## Exercise 1 + +Hide this cell by clicking the "eye" button on the top-left of the cell. +""" + +# ╔═╡ 1ef5542f-5ccf-497f-962c-8af518f7a68e +md""" +## Exercise 2 — Plotting + +This [notebook on plotting](https://github.com/fonsp/Pluto.jl/blob/main/sample/Plots.jl.jl) is a great way to get started with creating plots in Pluto. + +The function $f(x) = \sin(x^2)$ looks like this on the interval $[-\pi,+\pi]$. +""" + +# ╔═╡ 3efb76fb-ca72-4756-abd1-f66742ae6e66 +# Each cell allows for only a single command. Get around this by wrapping multiple commands between `begin` and `end` +begin + f(x) = sin(x^2) + plot(f, -π, π) +end + +# ╔═╡ f1504490-3d56-4422-9ced-086066a52f42 +md"""Plot the function $g(x) = cos(x^2)$.""" + +# ╔═╡ 27574dd3-b96b-43ab-95ca-d4632cfa5096 +# Place your code in this cell. + +# ╔═╡ 23205a93-e7a8-44ac-8abf-c3de190d4e3f +md"""Now use the `plot` and `plot!` functions to create a plot that overlays the graphs of $f$ and $g$.""" + +# ╔═╡ 6308459e-d538-4f31-b360-4e271635031c +begin + plot(f, label="f") + #plot!( ..., label="g") # fill in the blank +end + +# ╔═╡ 5981a538-c243-4241-9634-51f42663288f +md""" +## Exercise 3 +The "speed of convergence" measures tells us how fast a sequence converges to its limit (if one exists). + +A sequence $\{x_k\}$ converging to $x^*$ is said to converge _linearly_ with rate $\mu$ if + +$$\lim_{k\to\infty} \frac{|x_{k+1}-x^*|}{|x_k-x^*|}=\mu\in(0,1).$$ + +The sequence is said to converge _quadratically_ with rate $\mu$ if + +$$\lim_{k\to\infty} \frac{|x_{k+1}-x^*|}{|x_k-x^*|^2}=\mu\in(0,\infty).$$ + +The sequence is said to converge _superlinearly_ if + +$$\lim_{k\to\infty} \frac{|x_{k+1}-x^*|}{|x_k-x^*|}=0.$$ + +### Part a + +Derive the convergence order and the convergence rate and order for each of the sequences $\{1/2^k\}$, $\{2^{-2^k}\}$, and $\{3^{-k^2}\}$. + +### Part b + +Provide two plots of the values of the first 50 elements of these sequences: the first plots the sequence value vs $k$; the second plots the log of the sequence value vs $k$. + +For much more on the speed of sequence convergence, see [How Many Steps Still Left to x*](https://doi.org/10.1137/19M1244858), by Emil Cătinaş, and [Wikipedia](https://en.wikipedia.org/wiki/Rate_of_convergence). +""" + +# ╔═╡ 91d9170f-164a-49e4-8ea5-be2e573abb78 +md""" +## Exercise 4: Eigenvalues + +Prove that the eigenvalues of an orthogonal matrix $Q$ must all lie on the unit circle, i.e., every (possibly complex-valued) eigenvalue $\lambda$ of $Q$ has magnitude $|\lambda|=1$. + +The code below illustrates this property. +""" + +# ╔═╡ 53469720-8f6f-46d8-b1a2-f553fd0df35c +begin + Q = qr(randn(50,50)).Q # generate a random orthogonal matrix + scatter(eigvals(Q), aspect_ratio=:equal) +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" + +[compat] +Plots = "~1.25.4" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.8.5" +manifest_format = "2.0" +project_hash = "a303196bd18d5207cbda8be2fa596e24d456ea3e" + +[[deps.Adapt]] +deps = ["LinearAlgebra"] +git-tree-sha1 = "af92965fb30777147966f58acb05da51c5616b5f" +uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" +version = "3.3.3" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.Artifacts]] 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╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/homework/hw1/index.html b/homework/hw1/index.html new file mode 100644 index 0000000..ab39ad6 --- /dev/null +++ b/homework/hw1/index.html @@ -0,0 +1,16 @@ + + + + + + +
\ No newline at end of file diff --git a/homework/hw2.jl b/homework/hw2.jl new file mode 100644 index 0000000..61b5963 --- /dev/null +++ b/homework/hw2.jl @@ -0,0 +1,1194 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ 5d2515b8-dc83-4a62-a3c7-38480a3246e1 +using FFTW, Random, LinearAlgebra, Plots + +# ╔═╡ ad9518bb-471d-48f9-abc1-3e3ecac3bf0c +md"""# Homework 2""" + +# ╔═╡ f93a3136-7812-11ec-2c52-e30ce2ff34ca +# Fill in your name and student ID +student = (name = "Harry Potter", ubcid = "123456789"); + +# ╔═╡ 4ef808f0-13fc-44a8-acdf-f01e0c7b5a10 +md""" +**Student:** $(student.name) (ID# $(student.ubcid)) +""" + +# ╔═╡ 38ee44b9-4642-4a5f-a581-2ce5e8187875 +md"""Useful packages are already loaded at the end of this notebook. Feel free to add any other packages you might need.""" + +# ╔═╡ 011a67fe-f829-42b9-bc26-54db0d3905c1 +md"""----""" + +# ╔═╡ cfa9eca1-4a2c-40d1-8963-0fa0d92eaf9c +md"""## Q1: Constant model +Derive an expression for the least-squares solution that corresponds to the matrix and right-hand side +```math +A = \begin{bmatrix} 1\\ \vdots\\1 \end{bmatrix} +\quad\text{and}\quad +b = \begin{bmatrix} b_1 \\ \vdots \\ b_m \end{bmatrix}. +``` +""" + +# ╔═╡ 6ff25cfb-48f9-46e1-9ce6-47de70bfb458 +md"""## Q2: Quadratic fit to data +Consider the NO₂ [trendline example](https://friedlander.io/ubc-cpsc-406/notes/least-squares/#example_trendline) from the class notes. +- Compute the coefficients $\alpha_1,\ldots\alpha_3$ that cause the quadratic model + ```math + q(t) = \alpha_1 + \alpha_2 t + \alpha_3 t^2 + ``` + to fit the detrended data (see the last plot of that example) in the least-squares sense. +- Provide a plot of the detrended data and overlay the fitted function $q(t)$. +""" + +# ╔═╡ 4f53eb39-34d9-4a7c-b122-bcca27d859fe +md""" +Suppose that we believe that the underlying signal `s` (ie, the signal without noise) is very uniform, and is in fact the superposition of a few sinusoidal-like elements, such as are contained in the columns of the [Discrete Cosine Transform](https://en.wikipedia.org/wiki/Discrete_cosine_transform). + +Here are the first 5 columns of the DCT, to give you the idea: +""" + +# ╔═╡ 27d6b83b-cce4-4821-b89f-5e996a37a445 +md""" +A standard least-squares obtained via `x = D\b` doesn't give us a recovered signal `D*x` that's close to the true solution `s`: +""" + +# ╔═╡ 9b8106ca-5e69-4de1-b523-c77fbec4076d +md"""The question, then, is how to choose the right columns of $D$ in order to obtain a good guess of the underlying signal? In other words, is there a sparse vector `x` (i.e., one with just a few nonzeros) such that +```math +Dx \approx s? +``` + +We'll try to solve this problem by building up a solution `x` element by element. + +The following function, which you'll need to complete, solves a sequence of least-squares problems of increasing dimension. First it solves a least-squares problem with only a single column to obtain a single DCT coefficient; then it solves a least-squares problem with two columsn to obtain two DCT coefficients, and so on. + +How do we choose the columns? Here's one approach. Because the least-squares problem +```math +\min_x\ \tfrac12\|Dx-b\|^2 +``` +has the optimality condition +```math +D^T r = 0, \quad r = b - Dx, +``` +we start with a trivial solution $x_0 = 0$ and associated residual $r_0 = b$, and then find the index of +```math +z_0 := D^T r_0 +``` +largest in magnitude, i.e., choose the element +```math + j_0 = \arg\max_j\ |z_0[j]|. +``` +We then pull that column from $D$ and solve a new LS problem +```math +x_1 := \arg\min\ \tfrac12\|Sx-b\|^2, \quad\text{with}\quad S = D[:, j_0]. +``` +We then repeat the iteration by updating the residual as $r_1=b - Sx_1$ and finding the largest element of $|z_1|$, where $z_1:=D^T r_1$, and so on. +""" + +# ╔═╡ d532f991-33a2-4560-a510-1522df1bdfdc +md""" +!!! do + Fix the next cell by completing the function. The lines marked `fill in` are placeholders to help guide you, but feel free to modify the function as you wish. +(The function throws an error because of the `# fill in` lines.) +""" + +# ╔═╡ ca578974-6695-4b5a-a14d-85faa04ac4a1 +"Solve a sequence of least-squares problems. Returns the indices `idx` and corresponding coefficients" +function lsup(D, b; its=4) + r = b + x = []; idx = []; S = zeros(length(b), 0) + for i in 1:its + z = # fill in + j = # fill in + push!(idx, j) + S = # fill in + x = # fill in + r = # fill in + end + return idx, x +end + +# ╔═╡ db870974-0ecb-43d8-b8e3-82671c09dd05 +md""" +!!! do + This next cell should work once you get `lsup` defined. +""" + +# ╔═╡ 635d79b5-989f-492d-b101-250f1c5c25b6 +md""" +--- +!!! note + The hidden cells below contain helper code. +""" + +# ╔═╡ a7f3b7fe-c50c-4f5d-9a6d-5d14dfd7949a +# Create an n-dimensional explicit Discrete Cosine Transform matrix. +function DCT(n::Integer) + D = zeros(n, n) + for i in 1:n + e = zeros(n); e[i] = 1 + D[:,i] = FFTW.dct(e) + end + return D +end; + +# ╔═╡ 3502dd92-178e-4d26-9c5c-a047183c3823 +# Generate a signal using 4 random elements from the DCT basis. +function signal(n=200, k=4, ϵ=0.1) + Random.seed!(19) + x = zeros(n) + p = randperm(n)[1:k] + x[p] .= rand(k) + noise = ϵ*randn(n) + s = FFTW.dct(x) # true signal + b = s + noise # noisy signal + return s, b +end; + +# ╔═╡ b34d0256-fb12-4148-a29d-c562c635f8c7 +s, b = signal(); # for Q3 + +# ╔═╡ ad1f8c5b-347e-47cd-8f0c-369d1fc2ebff +md"""## Q3: Iterative regularization + +Suppose that you observe the following noisy signal (of length $(length(b))), which is stored in the variable `b`. The true (and "unknown") signal `s` is displayed in blue. +""" + +# ╔═╡ f6caf981-69b6-41b9-9ab4-fd4c331bc8da +plot([s b], w = [2 1], label=["true signal `s`" "observed noisy signal `b`"]) + +# ╔═╡ 6d0c3d84-1d4d-46eb-9599-b6c8c0ac7475 +D = DCT(length(b)); # generates an explicit DCT matrix + +# ╔═╡ 7c75d286-f549-49aa-8a14-36315474d314 +plot(D[:,1:5]) + +# ╔═╡ 5d52aca1-df32-4cf4-a334-f2b9cf255a60 +let + x = D \ b + s_LS = D*x + plot([s s_LS], w=[2 1], label=["true signal `s`" "recovered signal via least-squares"]) +end + +# ╔═╡ 185e88f6-f3cb-4b68-9686-69cb6a1eaf8b +# Plot the recovered signal +let + idx, xs = lsup(D, b, its=4) + s_up = D[:,idx]*xs + plot([s s_up], w=[2 1], label=["true signal `s`" "recovered signal"]) +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Random = 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"1.1.4+4" + +[[deps.Xorg_libXrandr_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"] +git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631" +uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484" +version = "1.5.2+4" + +[[deps.Xorg_libXrender_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] +git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" +uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" +version = "0.9.10+4" + +[[deps.Xorg_libpthread_stubs_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" +uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" +version = "0.1.0+3" + +[[deps.Xorg_libxcb_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] +git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6" +uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" +version = "1.13.0+3" + +[[deps.Xorg_libxkbfile_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] +git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2" +uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a" +version = "1.1.0+4" + +[[deps.Xorg_xcb_util_image_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] +git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97" +uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b" +version = "0.4.0+1" + +[[deps.Xorg_xcb_util_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"] +git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1" +uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5" +version = "0.4.0+1" + +[[deps.Xorg_xcb_util_keysyms_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] +git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00" +uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7" +version = "0.4.0+1" + +[[deps.Xorg_xcb_util_renderutil_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] +git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" +uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" +version = "0.3.9+1" + +[[deps.Xorg_xcb_util_wm_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] +git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" +uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" +version = "0.4.1+1" + +[[deps.Xorg_xkbcomp_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] +git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" +uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" +version = "1.4.2+4" + +[[deps.Xorg_xkeyboard_config_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] +git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" +uuid = "33bec58e-1273-512f-9401-5d533626f822" +version = "2.27.0+4" + +[[deps.Xorg_xtrans_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" +uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" +version = "1.4.0+3" + +[[deps.Zlib_jll]] +deps = ["Libdl"] +uuid = "83775a58-1f1d-513f-b197-d71354ab007a" +version = "1.2.12+3" + +[[deps.Zstd_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927" +uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" +version = "1.5.2+0" + +[[deps.fzf_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56" +uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09" +version = "0.29.0+0" + +[[deps.libaom_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" +uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" +version = "3.4.0+0" + +[[deps.libass_jll]] +deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] +git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" +uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" +version = "0.15.1+0" + +[[deps.libblastrampoline_jll]] +deps = ["Artifacts", "Libdl", "OpenBLAS_jll"] +uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" +version = "5.1.1+0" + +[[deps.libfdk_aac_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" +uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" +version = "2.0.2+0" + +[[deps.libpng_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] +git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" +uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" +version = "1.6.38+0" + +[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] +git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.7+1" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.48.0+0" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+0" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "2021.5.5+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+0" +""" + +# ╔═╡ Cell order: +# ╟─4ef808f0-13fc-44a8-acdf-f01e0c7b5a10 +# ╟─ad9518bb-471d-48f9-abc1-3e3ecac3bf0c +# ╠═f93a3136-7812-11ec-2c52-e30ce2ff34ca +# ╟─38ee44b9-4642-4a5f-a581-2ce5e8187875 +# ╟─011a67fe-f829-42b9-bc26-54db0d3905c1 +# ╟─cfa9eca1-4a2c-40d1-8963-0fa0d92eaf9c +# ╟─6ff25cfb-48f9-46e1-9ce6-47de70bfb458 +# ╟─ad1f8c5b-347e-47cd-8f0c-369d1fc2ebff +# ╠═f6caf981-69b6-41b9-9ab4-fd4c331bc8da +# ╟─4f53eb39-34d9-4a7c-b122-bcca27d859fe +# ╠═6d0c3d84-1d4d-46eb-9599-b6c8c0ac7475 +# ╠═7c75d286-f549-49aa-8a14-36315474d314 +# ╟─27d6b83b-cce4-4821-b89f-5e996a37a445 +# ╠═5d52aca1-df32-4cf4-a334-f2b9cf255a60 +# ╟─9b8106ca-5e69-4de1-b523-c77fbec4076d +# ╟─d532f991-33a2-4560-a510-1522df1bdfdc +# ╠═ca578974-6695-4b5a-a14d-85faa04ac4a1 +# ╟─db870974-0ecb-43d8-b8e3-82671c09dd05 +# ╠═185e88f6-f3cb-4b68-9686-69cb6a1eaf8b +# ╟─635d79b5-989f-492d-b101-250f1c5c25b6 +# ╟─5d2515b8-dc83-4a62-a3c7-38480a3246e1 +# ╟─b34d0256-fb12-4148-a29d-c562c635f8c7 +# ╟─a7f3b7fe-c50c-4f5d-9a6d-5d14dfd7949a +# ╟─3502dd92-178e-4d26-9c5c-a047183c3823 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/homework/hw2/index.html b/homework/hw2/index.html new file mode 100644 index 0000000..37f1ab4 --- /dev/null +++ b/homework/hw2/index.html @@ -0,0 +1,16 @@ + + + + + + +
\ No newline at end of file diff --git a/homework/hw3/index.html b/homework/hw3/index.html new file mode 100644 index 0000000..4dbb6aa --- /dev/null +++ b/homework/hw3/index.html @@ -0,0 +1,65 @@ + + + + + + + + + + + + + + Homework 3 + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Homework 3

+

Q1. Quadratic functions

+ +

Q2. Homogeneity of the directional derivative

+

(a) Prove that the directional derivative (when it exists) of a function f:ℜn→ℜf:\Re^n\to\Re is positively homogeneous of degree one, that is, show that for all α≥0\alpha\ge0,

+f′(x;αd)=αf′(x;d). +f'(x;\alpha d) = \alpha f'(x;d). + +

Use this fact to argue that the directions dd in which the directional derivative is evaluated can always be normalized, so that ∥d∥2=1\|d\|_2=1.

+

(b) Which unit-norm direction minimizes the directional derivative, i.e., which direction provides the largest instantaneous decrease in ff at the point xx?

+

Q3. Nonlinear least squares (Beck, Exercise 4.6)

+

(a) Derive the gradient ∇f(x)\nabla f(x). (Note that this is a slightly different objective than used in the formulation discribed in lecture. Yes, you could use AD to compute this, but here we're asking you to derive it analytically.)

+

(b) Part (ii) of this problem.

+
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + diff --git a/homework/hw4.jl b/homework/hw4.jl new file mode 100644 index 0000000..bc9e4fc --- /dev/null +++ b/homework/hw4.jl @@ -0,0 +1,74 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ c5054d49-06cb-4682-b070-71c3a397077f +md"""# Homework 4 +!!! deadline + Thursday, 16 February, 11:59pm +""" + +# ╔═╡ e0626ba4-87ae-45df-82c7-c7e0c767843d +md"Fill in your name and student ID:" + +# ╔═╡ fe59b1e5-a4b2-45b1-91a8-c3e5fde92f5c +student = (name = "Harry Potter", ubcid = "123456789"); + +# ╔═╡ a70c0b8a-5d35-4c5c-bb79-262a83327717 +md""" +**Student:** $(student.name) (ID# $(student.ubcid)) +""" + +# ╔═╡ f9527e44-3eed-42c4-8be5-47254955d498 +md""" +## Q1. Beck, Exercise 5.2 +""" + +# ╔═╡ 99df8118-7c73-11ec-201b-c75cb0f58794 +md""" +## Q2. Cholesky factorization +Partition the $n$-by-$n$ positive definite matrix $A$ as +```math +A = \begin{bmatrix} a_{11} & w^T \\ w & K \end{bmatrix}, +``` +where, respectively, $a_{11}$ and $w$ are the top-left entry and the remaining first-column of $A$. The first iteration of the Cholesky process decomposes $A$ as +```math +\begin{align} + A = \begin{bmatrix} \alpha & 0 \\ w/\alpha & I \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & K-ww^T/a_{11} \end{bmatrix} + \begin{bmatrix} \alpha & w^T/\alpha \\ 0 & I \end{bmatrix} = R_1^TA_1R_1, +\end{align} +``` +where $\alpha:=\sqrt{a_{11}}$, and $R_1$ and $A_1$ are defined by the matrices in the factorization. +1. This first step is valid because $a_{11}$ is positive. Why? +The next iterations of the factorization proceed inductively on the submatrix $K-ww^T/a_{11}$. But again, that's only possible if its first entry is positive. +2. Prove that this must be the case. +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.8.5" +manifest_format = "2.0" +project_hash = "da39a3ee5e6b4b0d3255bfef95601890afd80709" + +[deps] +""" + +# ╔═╡ Cell order: +# ╟─a70c0b8a-5d35-4c5c-bb79-262a83327717 +# ╟─c5054d49-06cb-4682-b070-71c3a397077f +# ╟─e0626ba4-87ae-45df-82c7-c7e0c767843d +# ╠═fe59b1e5-a4b2-45b1-91a8-c3e5fde92f5c +# ╟─f9527e44-3eed-42c4-8be5-47254955d498 +# ╟─99df8118-7c73-11ec-201b-c75cb0f58794 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/homework/hw4/index.html b/homework/hw4/index.html new file mode 100644 index 0000000..59b3cf0 --- /dev/null +++ b/homework/hw4/index.html @@ -0,0 +1,16 @@ + + + + + + +
\ No newline at end of file diff --git a/homework/hw5/index.html b/homework/hw5/index.html new file mode 100644 index 0000000..cbcc1d9 --- /dev/null +++ b/homework/hw5/index.html @@ -0,0 +1,69 @@ + + + + + + + + + + + + + + Homework 5 + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Homework 5

+

The first six exercises are from Beck, Introduction to Nonlinear Optimization.

+
    +

  1. (Convexity of set operations) Exercise 6.1

    +
    2. +

  2. (Convex hull) Exercise 6.4

    +
    3. +

  3. (Normal cone) Exercise 6.10

    +
    4. +

  4. (Convex functions) Exercise 7.1 (i–ii)

    +
    5. +

  5. (Affine functions) Exercise 7.3

    +
    6. +

  6. (Strict convexity) Exercise 7.7

    +
    7. +

  7. (Entropy) Show that f(x)=−∑i=1nxilog⁡xif(x) = -\sum_{i=1}^n x_i\log x_i is concave over the probability simplex Δˉn={x∈R+n∣∑i=1nxi=1}\bar\Delta_n=\{ x\in \mathbb R^n_+\mid \sum_{i=1}^n x_i = 1 \}.

    +
    8. +

  8. (Log-Sum-Exp) Show that f(x)=log⁡(∑i=1meaiTx)f(x) = \log\left(\sum_{i=1}^m e^{a_i^T x}\right) is convex in Rn\mathbb R^n, where a1,…,am∈Rna_1,\ldots,a_m\in\mathbb R^n.

    +
    9. +
+
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + diff --git a/homework/hw6/index.html b/homework/hw6/index.html new file mode 100644 index 0000000..8669f7b --- /dev/null +++ b/homework/hw6/index.html @@ -0,0 +1,15 @@ + + + + + +
\ No newline at end of file diff --git a/homework/linear-programming.jl b/homework/linear-programming.jl new file mode 100644 index 0000000..b387ec5 --- /dev/null +++ b/homework/linear-programming.jl @@ -0,0 +1,1162 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ d56ebfcd-2c5f-4e21-a4c9-44c73be68d00 +using LinearAlgebra, Random, JuMP, HiGHS, Statistics, Plots + +# ╔═╡ e4bcc583-0af4-45c6-9824-86a548d79112 +# Fill in these fields, and execute the field either by hiting "Shift-Enter" or clicking the play button at the bottom right of this cell. +student = (name = "Harry Potter", ubcid = "123456789") + +# ╔═╡ 261a62de-d8f0-11ed-324e-0fa7b81588a0 +md"""Student: **$(student.name)** (ID# $(student.ubcid))""" + +# ╔═╡ b49c68b0-f3bc-495b-a8d3-7d4ee098f0d5 +md""" +# Homework 8: Linear Programming +""" + +# ╔═╡ 5c713a05-1e2c-4206-b8a1-fde429313be0 +md""" +## Exercise 1: Simplex + +1. Convert this problem into standard form and construct a basis that corresponds to the point $x=(0,0)$: + +$$\begin{align} + \min_{x_1,x_2} &\quad {-}2x_1-x_2\\ + \text{subj to} &\quad x_1+x_2 \le 6,\ x\ge0. +\end{align}$$ + +2. Carry out the steps of the simplex method, starting with the basic feasible solution above. At each iteration, show the basic index set, the reduced costs, and which variables enter and leave the basis. + +3. Draw a graphical representation of the problem in terms of the original variables $x_1,x_2$, and indicate the path taken by the simplex algorithm. +""" + +# ╔═╡ ed780b9c-049e-4672-9fab-c7e6cef5f132 +# answer here + +# ╔═╡ 049a884e-0aa1-4568-8098-7e381d3f68c9 +md""" +## Exercise 2: Robust optimization + +You are the owner of a factory that makes widgets and gadgets. For each widget and gadget, there is a set of feature measurements, e.g., the weight, dimensions, material composition, etc., represented by a vector $x$. For maximum profit, you want the combinations of these features to exactly match a set of standard specifications, e.g., $Ax = b$, where $A$ dictates how the features are combined, and $b$ is the standard. However, realistically, it is too expensive to exactly satisfy $Ax=b$, so you have to make some concessions. + +Instead, you can solve one of three optimization problems: + +$$\begin{equation} + \min_{x}\;\|Ax-b\|_1 = \sum_{i=1,\ldots,m}|a_i^T x-b_i| +\end{equation}$$ + +$$\begin{equation} + \min_{x}\;\|Ax-b\|_\infty = \max_{i=1,\ldots,n}|a_i^T x-b_i| +\end{equation}$$ + +$$\begin{equation} + \min_{x}\;\|Ax-b\|_2 =\sqrt{ \sum_{i=1,\ldots,n}|a_i^T x-b_i|^2} +\end{equation}$$ +""" + +# ╔═╡ 88aa630e-e6cc-455a-b4a9-73112dda0b5a +md""" +**Part (a).** Write the one-norm problem as a linear program of the form + +$$\begin{equation}\label{eq:linprog-form} + \min_{z}\ c^T z \text{ subject to } \hat{A} z \leq \hat{b}, +\end{equation}$$ + +for some $c$, $\hat{A}$, and $\hat{b}$. +""" + +# ╔═╡ 2a372f65-88f9-48ac-8b7b-7cfe404facdd +# answer here + +# ╔═╡ b0bb5d61-6afc-4c03-8c95-6dd14c1e3e3f +md""" +**Part (b)**. Similarly, write the one-norm problem as a linear program. +""" + +# ╔═╡ 5304c4e2-2184-446d-b955-ec53d8112700 +# answer here + +# ╔═╡ 18ee4b68-7953-4977-9fb1-fae480672b4f +md""" +**Part (c).** For some randomly generated data (below), plot histograms of the residuals obtained by each of the three formulations. + +Here's the data: +""" + +# ╔═╡ 7ed3b8dc-af00-43e5-984b-75b74b555358 +begin + Random.seed!(1) + m = 400 + n = 200 + A = randn(m,n) + b = randn(m) +end; + +# ╔═╡ 67689ed7-f5ed-45d4-b574-751817fef2fe +md"The histogram for the least-squares problem." + +# ╔═╡ 2cd8325e-414c-4e23-a39c-21dd5ed143e5 +begin + x_ls = A \ b + r_ls = A*x_ls - b + histogram(r_ls, bins=-3:.1:3) +end + +# ╔═╡ ec8f86dd-7c70-4e88-b980-40118a14823a +md"Produce the historgrams for the one- and inf-norm formulations, using the same data above." + +# ╔═╡ 0d22c3c8-b55a-41b6-973e-a98d4cea2f41 +# answer here + +# ╔═╡ 8a3c4bec-d6f9-40bb-9710-dbef88756aae +md"## Helper code" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b" +JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" + +[compat] +HiGHS = "~1.5.0" +JuMP = "~1.10.0" +Plots = "~1.38.9" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.8.5" +manifest_format = "2.0" +project_hash = "af77fcb5c339bb5f62a0e8a0be1ea94705e73a37" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.BenchmarkTools]] +deps = ["JSON", "Logging", 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+[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "2021.5.5+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+0" +""" + +# ╔═╡ Cell order: +# ╟─261a62de-d8f0-11ed-324e-0fa7b81588a0 +# ╠═e4bcc583-0af4-45c6-9824-86a548d79112 +# ╠═b49c68b0-f3bc-495b-a8d3-7d4ee098f0d5 +# ╟─5c713a05-1e2c-4206-b8a1-fde429313be0 +# ╠═ed780b9c-049e-4672-9fab-c7e6cef5f132 +# ╟─049a884e-0aa1-4568-8098-7e381d3f68c9 +# 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\ No newline at end of file diff --git a/homework/projection-hw7.jl b/homework/projection-hw7.jl new file mode 100644 index 0000000..171edb5 --- /dev/null +++ b/homework/projection-hw7.jl @@ -0,0 +1,1345 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ 03b47ffa-ee46-4c1f-be21-3f43c211882b +using LinearAlgebra, Random, CSV, DataFrames, Statistics, Plots + +# ╔═╡ fd475ac2-b151-495c-992e-c8a594d6bd90 +# Fill in these fields, and execute the field either by hiting "Shift-Enter" or clicking the play button at the bottom right of this cell. +student = (name = "Harry Potter", ubcid = "123456789") + +# ╔═╡ 8f78e770-dda4-4cd6-b8c1-f7993b0eafae +md"""Student: **$(student.name)** (ID# $(student.ubcid))""" + +# ╔═╡ 19085aff-f3aa-4ece-915b-4e88e16de849 +md""" +# CPSC 406 — Homework 7 +""" + +# ╔═╡ 64666a1a-cebc-453a-8388-83ad8ee4d81d +md""" +## Exercise 1. Projection onto a halfspace +A nonzero n-vector $a$ and a scalar $\beta$ define the halfspace + +$$H^{-}_{a,β}=\{x∈\Re^n\mid ⟨a,x⟩≤β\}.$$ + +**(Part a)**. Derive an explicit expression for the projection of an arbitrary n-vector b onto $H^-_{a,β}$. + +$$\mbox{proj}_{H^-_{a,β}}(b) = +\begin{cases} +blah & \mbox{if $...$}\\ +blah & \mbox{if $...$} +\end{cases}$$ +""" + +# ╔═╡ 1ab340af-eecd-4533-ae04-9dbea4e0f352 +# Answer here + +# ╔═╡ c3a4cfa1-65b5-4910-bd7e-a0028f20f054 +md""" +**(Part b.)** Implement your solution above by completing the following function. +""" + +# ╔═╡ d59ceae2-5bbb-4005-9092-45b01277c13a +# Answer here +""" +Project the n-vector `b` onto the halfspace {x ∣ ⟨a,x⟩ ≤ β} defined by the nonzero vector `a` and the scalar `β`. +""" +function proj_halfspace(b, a, β) + # complete ... +end; + +# ╔═╡ 89cd142d-263e-4ad1-9034-2347cc5576c0 +md"**(Part c.)** Verify numerically with a few examples that your projection code is correct." + +# ╔═╡ 2845cbb3-233b-4bc1-b522-a91db00f166e +# code here... + +# ╔═╡ c9cc4ffa-5a21-4a5c-b532-ca2ad53177c3 +md""" +## Exercise 2. Alternating projection method + +The function `projsplx`, defined in the _Helper Function Section_ below, computes the projection of any n-vector onto the probability simplex $\{x\in\Re^n\mid \sum_j^nx_j=1,\ x≥0\}$. The cell below shows the result of the projection of a vector `b` onto the simplex. +""" + +# ╔═╡ 7cb77d3b-10c0-4a2e-bcd3-8df99d6f1025 +md""" +Now suppose that we wish to compute the projection onto the **intersection** of the probability simplex and an arbitrary halfspace. Can you do this using the projection function `proj_halfspace`, which you implemented above, and `proj_simplex`? + +Yes, you can, using the **[Projection Onto Convex Sets](https://en.wikipedia.org/wiki/Projections_onto_convex_sets)** algorithm, which iteratively projects onto each convex set. The algorithm is simple, and involves the following iteration: + +$$x_{k+1}=P_1(P_2(x_k)),$$ + +where $P_1$ and $P_2$ are projection operators onto the first and second sets. Implement this algorithm for generic projection operators $P_1$ and $P_2$. Exit the loop when $\|x_{k+1}-x_k\|≤ϵ\|x_0\|$ for some positive tolerance $\epsilon$. + +Note that this method can in some situations take many iterations. +""" + +# ╔═╡ 6e8bcee5-1ad2-4e62-93ef-038337b7d4d2 +md"**(Part a.)** Complete the following function." + +# ╔═╡ c9ea95eb-df0d-45eb-ae4a-bc008dd26418 +""" +Project a vector `x` onto the intersection of two convex sets. Takes as input two functions `p1` and `p2` that each take a vector and return the projection of `b0` onto the corresponding sets. +""" +function proj_intersection(x::Vector, p1::Function, p2::Function; maxits=100, ϵ=1e-7) + err = [] + x0 = copy(x) + ϵ *= norm(x0, Inf) + for i ∈ 1:maxits + # complete.... + push!(err, norm(x-x0, Inf)) + err[end] ≤ ϵ && break + x0 = copy(x) + end + return x, err +end; + +# ╔═╡ b4bbd6cd-c622-44fb-abaf-4f9ee725c73d +md""" +**(Part b).** Test your implementationo f `proj_intersection` on a small problem that aims to find the projection of a vector onto the intersection of the unit probability simplex and a halfspace. Verify that the answer is correct. + +!!! warning + Make sure that your test problem is **feasible** -- i.e., that the intersection isn't empty. +""" + +# ╔═╡ 053ffb2b-632a-4939-964e-19ea1e4e71bb +# Answer here + +# ╔═╡ 91afe5bf-bfdb-470b-8025-47fc1aece514 +md""" +## Exercise 3. Projected gradient method. +""" + +# ╔═╡ 3dedec75-d6f0-48ee-a9c6-00624a36842c +md""" +**(Part a).** The function below is meant to minimize a simple quadratic function $\frac12 x^T Qx$ over a convex set $C$. Complete the implementation. Use a fixed step size equal to $1/L$, where $L$ is the Lipschitz constant of the gradient. +""" + +# ╔═╡ 32a209db-2ba5-4d71-9c77-71d62d55a13b +""" +Projected gradient method for the quadratic optimization problem + + minimize_{x} 1/2 x'Qx subj to x ∈ C. + +The function `proj(b)` compute the projection of the vector `b` onto the convex set 𝒞. +""" +function pgrad(Q, proj::Function; maxits=100, ϵ=1e-5) + err = [] + # complete... + return x, err +end; + +# ╔═╡ ca5d15cb-9ad6-4adc-9ad5-391c41f53cfe +md""" +**(Part b).** Use the function `pgrad` to obtain the minimum-risk portfolio from a set of assets with returns `r` and covariances `Σ`, computed here: +""" + +# ╔═╡ 2b4d2f39-b943-4180-ac4c-6474bee5f596 +# Complete this function +function efficient_portfolio(r, Σ, μ) + n = length(r) + p1 = # ... + p2 = # ... + p = b -> proj_intersection(b, p1, p2, maxits=10000, ϵ=1e-8)[1] + x, err = pgrad(Σ, p, maxits=1000) +end; + +# ╔═╡ bf4ff882-046e-4c10-a36e-276075ff7667 +md""" +The plot below shows the error in the optimality of the iterations, as measured by + +$$\|x_{k+1}-x_k\|.$$ +""" + +# ╔═╡ 73ec2bfb-471a-45b4-af52-4a1f0478fd58 +md""" +**(Part c).** Use the function `efficient_portfolio` to compute an optimal portfolio with expected return of 10%. +""" + +# ╔═╡ 1ed17bbc-befc-490b-aa31-cae3c5f03668 +xp1, xp1_itn_errors = efficient_portfolio(...) + +# ╔═╡ a54f1c79-6cb8-4639-87b4-054ad2a0e6a4 +plot(xp1_itn_errors, yscale=:log10, xlab="iterations", ylab="error", leg=false) + +# ╔═╡ 0cc4c836-1a41-47f7-9ca3-80f5a47be30f +md""" +## Appendix: Helper functions +""" + +# ╔═╡ 69e31180-d264-11ed-06e3-9534fbee9c37 +""" +Projection onto the unit simplex {x | sum(x) = τ, x ≥ 0}. + + projsplx!(b, τ) + +In-place variant of `projsplx`. +""" +function projsplx!(b::Vector{T}, τ::T) where T + + n = length(b) + bget = false + + idx = sortperm(b, rev=true) + tsum = zero(T) + + @inbounds for i = 1:n-1 + tsum += b[idx[i]] + tmax = (tsum - τ)/i + if tmax ≥ b[idx[i+1]] + bget = true + break + end + end + + if !bget + tmax = (tsum + b[idx[n]] - τ) / n + end + + @inbounds for i = 1:n + b[i] = max(b[i] - tmax, 0) + end +end; + +# ╔═╡ 069bb7e7-a956-4d7b-bc00-f64bc818cf7e +""" +Projection onto the unit simplex. + + projsplx(b) -> x + +Variant of `projsplx`. +""" +function proj_simplex(b::Vector; τ=1.0) + x = copy(b) + projsplx!(x, τ) + return x +end; + +# ╔═╡ 6d77a73a-3003-4b66-85ea-ecd04da99d62 +let + b = [12.0, -23.0, 16.0] + x = proj_simplex(b) +end + +# ╔═╡ f4eef581-17f2-4ef8-92d1-de7cd5594e4f +stocks = + ["AAPL", "META", "GOOG", "MSFT", "NVDA", # technology + "AMZN", "COST", "EBAY", "TGT", "WMT", # services + "BMO", "BNS", "HBCP", "RY", "TD", # finance + "BP", "CVX", "IMO", "TTE", "XOM" # energy + ]; + +# ╔═╡ e07c50f8-ca5d-47f2-9d08-7001d30e4893 +tmpdir = mktempdir(); + +# ╔═╡ 6dbf55f0-d962-410f-ab90-d6630f4b3c37 +for s in stocks + url = "https://query1.finance.yahoo.com/v7/finance/download/$(s)?period1=1647815267&period2=1679351267&interval=1d&events=history&includeAdjustedClose=true" + Base.download(url, joinpath(tmpdir,"$(s).csv")) +end + +# ╔═╡ 4a175806-229d-4c07-8f71-e544dc87b873 +df = map(stocks) do stock + f = joinpath(tmpdir,"$(stock).csv") + dfs = DataFrame(CSV.File(f, select=[1,6])) # keep only date and closing price + rename!(dfs, "Adj Close" => stock) + select!(dfs, :Date, "$stock" .=> p->(p.-p[1])/p[1]) + rename!(dfs, "$(stock)_function" => "$stock") +end |> z->innerjoin(z..., on=:Date); + +# ╔═╡ 5179d26b-d8a8-4fec-a7ff-153117fd520d +function meancov(df) + r = mean.(eachcol(df[:,2:end])) + Σ = cov(Matrix(df[:,2:end])) + return r, Σ +end; + +# ╔═╡ 079e34d6-312e-42ea-9623-038d7b83a941 +r, Σ = meancov(df); # `meancov` is implemented in the Helper Functions section. + +# ╔═╡ 19f58193-0442-44f4-8aa5-98313c7b8b9f +min_risk_portfolio, min_risk_itn_errors = efficient_portfolio(r, Σ, 0) + +# ╔═╡ 40651b43-9541-4b36-a2f9-8f5a205089d3 +plot(min_risk_itn_errors, yscale=:log10, xlab="iterations", ylab="error", leg=false) + +# ╔═╡ 00000000-0000-0000-0000-000000000001 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\ No newline at end of file diff --git a/index.html b/index.html new file mode 100644 index 0000000..229c92d --- /dev/null +++ b/index.html @@ -0,0 +1,105 @@ + + + + + + + + + + + + + Computational Optimization + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Computational Optimization

+

CPSC 406 ∙ Term 2 of 2022/23 ∙ Department of Computer Science ∙ UBC

+ + This course covers the main algorithms for continuous optimization, including unconstrained and constrained problems, and duality theory and sensitivity. Because linear algebra is central to the subject, the course also covers the main numerical linear algebra operations needed in optimization. + +

  1. Lectures
  2. Teaching staff
  3. Course requirements
  4. Required reading
  5. Discussion board and announcements
  6. Grade distribution
  7. Collaboration
  8. Policies

+

Lectures

+ + +

Teaching staff

+ +

Course requirements

+

One of the following courses are required: CPSC 302 (Numerical Computation for Algebraic Problems), CPSC 303 (Numerical Approximation and Discretization), or MATH 307 (Applied Linear Algebra). You should be comfortable with an the main topics in applied linear algebra, multivariate calculus, and have experience in a numerical computing using a language such as Julia, Matlab, or Python.

+

This courses uses Julia exclusively.

+

Required reading

+

The course is based on a variety of sources, including lecture notes provided on these pages, which also provide links to supplementary reading materials.

+

Discussion board and announcements

+

A student discussion board is hosted on Piazza. All course announcements, including upcoming exams and homework deadlines, will appear on Piazza. Set your Piazza notifications appropriately.

+
⚠ Note
+
Register for the Piazza board.
+

Grade distribution

+ +

Collaboration

+

Most homeworks involve programming tasks. You may collaborate and consult with other students in the course, but you must hand in your own assignments. If you have collaborated or consulted with someone while working on your assignment, you must acknowledge this explicitly in your submitted assignment. If you are unsure about any of these rules, consult with one of the teaching staff or visit the departmental webpage on Collaboration and Plagiarism.

+

Policies

+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
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a=this.fetch();if(-1!==e.endOfExpression.indexOf(a.text))break;if(r&&a.text===r)break;if(t&&En[a.text]&&En[a.text].infix)break;var i=this.parseAtom(r);if(!i)break;"internal"!==i.type&&n.push(i)}return"text"===this.mode&&this.formLigatures(n),this.handleInfixNodes(n)},t.handleInfixNodes=function(e){for(var t,r=-1,a=0;a=0&&this.settings.reportNonstrict("unicodeTextInMathMode",'Latin-1/Unicode text character "'+t[0]+'" used in math mode',e);var s,l=X[this.mode][t].group,h=Hn.range(e);if(U.hasOwnProperty(l)){var m=l;s={type:"atom",mode:this.mode,family:m,loc:h,text:t}}else s={type:l,mode:this.mode,loc:h,text:t};i=s}else{if(!(t.charCodeAt(0)>=128))return null;this.settings.strict&&(w(t.charCodeAt(0))?"math"===this.mode&&this.settings.reportNonstrict("unicodeTextInMathMode",'Unicode text character "'+t[0]+'" used in math mode',e):this.settings.reportNonstrict("unknownSymbol",'Unrecognized Unicode character "'+t[0]+'" ('+t.charCodeAt(0)+")",e)),i={type:"textord",mode:"text",loc:Hn.range(e),text:t}}if(this.consume(),o)for(var c=0;c + + + + + + + + + + + + + + + + + Descent Methods + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +
+

Descent Methods

+ + Descent methods for unconstrained optimization follow at each iteration a search direction related to the current gradient. There are a variety of methods for generating valid descent directions, including steepest descent, scaled descent, coordinate descent, and variations of Newton descent. + +

The methods we consider here minimize a smooth function f:Rn→Rf:\mathbb R^n\to\mathbb R by generating a sequence of iterates

+xk+1=xk+αkdk +x_{k+1} = x_k + α_k d_k + +

that monotonically reduces the function value, i.e.,

+f(xk+1)<f(xk)for allk∈N. +f(x_{k+1}) < f(x_k) \quad\text{for all}\quad k\in\N. + +

The positive scalars αk\alpha_k are steplengths and the vectors dk∈Rnd_k\in\mathbb R^n are descent directions. Our aim is to ensure that the generated sequence converges to a stationary point:

+lim⁡k∈N∥∇f(xk)∥=0. +\lim_{k\in\N}\|\nabla f(x_k)\| = 0. + +

Below we describe methods for computing the descent directions and the steplengths.

+

Running example

+

In our discussion we'll often use the Rosenbrock test function as a convenient example. Here's the function definition and its gradient and Hessian:

+
using LinearAlgebra
+import ForwardDiff:gradient,hessian
+
+f(x) = (1 - x[1])^2 + 100.0(x[2] - x[1]^2)^2
+∇f(x) = gradient(f, x)
+∇²f(x) = hessian(f, x)
+

The minimizer is x∗=(1,1)x^*=(1,1), which we verify by checking that the gradient is zero and the Hessian is positive definite at that point:

+
x̄ = [1., 1.]
+@show norm(∇f(x̄))
+@show isposdef(∇²f(x̄))
norm(∇f(x̄)) = 0.0
+isposdef(∇²f(x̄)) = true
+
+

Here's a visualization of the function and the minimizer marked with a star:

+
using Plots; gr()
+
+x = y = -0.25:0.001:1.5
+z = Surface((x,y)->f([x,y]), x, y)
+contourf(x,y,z, levels=collect([10^k for k ∈ -4.0:3.0]), colorbar=:none, leg=false)
+scatter!([1.],[1.], c="gold", ms=10, shape=:star5)
+ + +

Observe that for any sequence of positive-definite matrices {Bk}k∈N\{B_k\}_{k\in\N}, the vectors

+dk=−Bk∇f(xk) +d_k = -B_k\nabla f(x_k) + +

are descent directions for ff at points xkx_k because the directional derivative

+f′(xk;dk)=∇f(xk)Tdk=−∇f(xk)T ⁣Bk∇f(xk)<0 +f'(x_k;d_k) = ∇f(x_k)^T d_k = -∇f(x_k)^T\! B_k ∇f(x_k)<0 + +

whenever ff is not stationary at xkx_k, i.e., ∇f(xk)≠0\nabla f(x_k)\ne0. This fact means that we can generate a whole family of descent directions through different choices the scaling matrices BkB_k.

+

Stronger conditions are needed, however, to ensure that the directional derivative doesn't get arbitrarily close to zero. In particular, we need additional conditions to ensure that

+lim sup⁡k∈Nf′(xk;dk)<0. +\limsup_{k\in\N}f'(x_k;d_k)<0. + +

One approach is to require that the eigenvalues of the matrices BkB_k are uniformly positive and bounded. Thus, we assume that there exist positive constants μ1\mu_1 and μ2\mu_2 such that

+0<μ1<λmin(Bk)≤λmax(Bk)<μ2∀k∈N, +0 < \mu_1 < \lambda_{\text{min}}(B_k) \le \lambda_{\text{max}}(B_k) < \mu_2 \quad \forall k\in\N, + +

where λmin(Bk)\lambda_{\text{min}}(B_k) and λmax(Bk)\lambda_{\text{max}}(B_k), respectively, are the minimum and maximum eigenvalues of BkB_k. This condition implies that there are bounds on the directional gradient and the norm of the direction directly related to the gradient:

+f′(xk;dk)=−∇f(xk)TBk∇f(xk)<−μ1∥∇f(xk)∥2∥dk∥=∥−Bk∇f(xk)∥≤∥Bk∥⋅∥∇f(xk)∥<μ2∥∇f(xk)∥.\begin{aligned} + f'(x_k;d_k) &= -∇f(x_k)^T B_k∇f(x_k) < -μ_1∥∇f(x_k)∥^2 \\ + ∥d_k∥ &= ∥{-}B_k∇f(x_k)∥≤∥B_k∥⋅∥∇f(x_k)∥<μ_2∥∇f(x_k)∥. +\end{aligned} +

Thus, the directional derivative of a gradient-related direction vanishes only when at stationary point, and the norm of a gradient-related direction is bounded by a multiple of the norm of the gradient.

+

Below we discuss the most-used search directions.

+

Steepest descent direction

+

The negative gradient d:=−∇f(x)d:=-∇f(x) is evidently a descent direction, because

+f′(xk;−∇f(xk))=−∇f(xk)T∇f(xk)=−∥∇f(xk)∥2<0 +f'(x_k;-∇f(x_k)) = -∇f(x_k)^T ∇f(x_k) = -\|∇f(x_k)\|^2 < 0 + +

whenever ∇f(xk)≠0∇f(x_k)\ne0. The negative gradient is often called the steepest descent direction because it's aligned with the unit-norm direction that makes the directional derivative most negative. In particular, use the Cauchy-Scharz inequality to deduce the bound

+f′(xk;v)=∇f(xk)Tv≥−∥∇f(xk)∥2⋅∥v∥2. + f'(x_k;v) = ∇f(x_k)^T v \ge -∥∇f(x_k)∥_2⋅∥v∥_2. + +

for any direction vv. Then observe that the direction v=−∇f(xk)v=-∇f(x_k) achieves the lower bound.

+

The package ForwardDiff.jl implements the gradient function, which we negate and assign to steepestDescent:

+
steepestDescent(f, x) = -gradient(f, x)
+

Newton direction

+

The Newton direction

+
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + + + + + + + + + diff --git a/notes/gradient-descent.jl b/notes/gradient-descent.jl new file mode 100644 index 0000000..89928c2 --- /dev/null +++ b/notes/gradient-descent.jl @@ -0,0 +1,1700 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═╡ 676c40f2-5b44-4949-b79c-fbda43d8e091 +# Packages +begin + using Colors + import PlutoUI + import Zygote + using CairoMakie + using LinearAlgebra, Random, Printf + Random.seed!(123) +end; + +# ╔═╡ c9cd0266-8255-11ec-163a-559553a6a1df +md""" +**CPSC 406: Computational Optimization** + +# Gradient Descent 📉 +""" + +# ╔═╡ 06062930-7acc-49f8-aae4-240d2f5fcc3b +md""" +## Quadratic functions + +Quadratic functions have the form +```math +q(x) = \tfrac12 x^T A x + b^T x, +``` +where we assume without loss of generality that $A$ is symmetric. The gradient and Hessian are +```math +\nabla q(x) = Ax + b \quad\text{and}\quad \nabla^2 q(x) = A. +``` +""" + +# ╔═╡ 948685b3-ab68-4e0a-8dba-7348f7c68c33 +md"""### Gradient descent with exact linesearch""" + +# ╔═╡ 269fa176-fb18-40df-823c-69ceecd69f67 +md""" +It's possible to use an exact linesearch for quadratic functions. +""" + +# ╔═╡ 87f1b9ad-b455-4a51-9ff1-5ad2a70e6b31 +md""" +## Gradient descent with constant stepsize +""" + +# ╔═╡ 6e7bf402-ed91-4e2a-b15b-386b2308255b +md"""## Gradient with backtracking""" + +# ╔═╡ ba0a8bbc-583e-4124-97b6-1f57409064dc +md""" +## Color matching +""" + +# ╔═╡ 9d93bb5d-0b36-453b-85d6-ae6df1ff92c5 +md"Create a target color:" + +# ╔═╡ b4f57fb9-eb17-4798-a27f-109f41b09960 +color_target = colorant"sandybrown" + +# ╔═╡ 632311a0-7fe3-4c42-baa7-588bd8df6dd1 +md""" +Try to manipuluate these sliders to recreate the target color. The guess and the target are shown next to each other below. +""" + +# ╔═╡ d006d038-f167-4e74-a0c4-f4ccb4ebe1ce +md""" + $(@bind red PlutoUI.Slider(0:.01:1, show_value=true)) red + + $(@bind green PlutoUI.Slider(0:.01:1, show_value=true)) green + + $(@bind blue PlutoUI.Slider(0:.01:1, show_value=true)) blue + +""" + +# ╔═╡ bd9448d4-ccac-4eef-bb71-33d315a25074 +begin + color_guess = RGB(red, green, blue) + [color_guess color_target] +end + +# ╔═╡ 5152c3cc-d002-4af7-b711-1dab5cd16d91 +md""" +Use `colordiff`, which measures the difference between two colors, as the objective function. +""" + +# ╔═╡ 5042ff27-11bf-4c0e-814b-223b5ec74a70 +md""" +colordiff = **$(colordiff( color_guess, color_target ))** +""" + +# ╔═╡ c91c94eb-ca44-461a-882d-1ea8823af581 +obj(x) = colordiff(RGB(x...), color_target)^2 + +# ╔═╡ 9d8e5a4c-3cdb-459a-8d2b-16c750af452d +md"Use Zygote to differentiate the objective and obtain a gradient function. Without further investigation, we can't be certain that `colordiff` is in fact differentiable, though Zygote may still give us something that like a gradient. +" + +# ╔═╡ 9784e3b8-ad97-469a-aa2a-53f09994bf14 +grad(x) = Zygote.gradient(obj, x)[1] + +# ╔═╡ 4150cb58-e04a-4884-a8c1-1a5b3e7924ba +grad([red,green,blue]) + +# ╔═╡ d10d1900-fff1-4267-846e-1b40f2eef944 +md""" +## Helper utilities +""" + +# ╔═╡ dec864c0-8c16-4a74-a193-f6b8710cc445 +begin + struct State + x::Vector + f + norm∇f + end + State(x::Vector, f::Real, ∇f::Vector) = State(x, f, norm(∇f)) +end; + +# ╔═╡ 271d208c-bbc1-4688-800e-00e56d7d9dbf +function gradient_descent_quadratic(A, b, x; ϵ=1e-6) + q = (1/2)x'A*x + b'x; ∇q = A*x + b + trace = [State(x, q, ∇q)] + while norm(∇q) > ϵ + α = dot(∇q, ∇q) / dot(∇q, A*∇q) + x = x - α*∇q + q = (1/2)x'A*x + b'x; ∇q = A*x + b + push!(trace, State(x, q, ∇q)) + end + return x, trace +end; + +# ╔═╡ c989351d-983d-40f2-b7b3-9a1610ed0b9c +q, qtrace = let + A = [1. 0; 0 2.] + b = [0., 0] + x0 = [2., 1] + + x, trace = gradient_descent_quadratic(A, b, x0) + + q(x) = (1/2)x'A*x + b'x + q, trace +end + +# ╔═╡ bfa5474f-8dbf-4d04-83ef-bca468fad0a1 +function gradient_descent_quadratic_conststep(A, b, x, α; ϵ=1e-6, maxits=100) + q = (1/2)x'A*x + b'x; ∇q = A*x + b + trace = [State(x, q, ∇q)] + for k ∈ 1:maxits + norm(∇q) ≤ ϵ && break + x = x - α*∇q + q = (1/2)x'A*x + b'x; ∇q = A*x + b + push!(trace, State(x, q, ∇q)) + end + return x, trace +end + +# ╔═╡ 93416c0d-7b58-4bff-989d-627b49f1233f +qc, qctrace = let + α = 2 # "optimal" + A = [1. 0; 0 2] # L = 2 => α ∈ (0, 2/L) == (0, 1) + b = [0., 0] + x0 = [2., 1] + + x, trace = gradient_descent_quadratic_conststep(A, b, x0, α, maxits=1000) + + q(x) = (1/2)x'A*x + b'x + q, trace +end; + +# ╔═╡ 44a8e5dd-3500-4eda-acf2-359b01a0d6ae +# Gradient method with backtracking +function grad_method_backtracking(f, g, x0; ϵ = 1e-6, μ = 1e-4, maxits = length(x0)) + x = copy(x0) + fk = f(x); ∇fk = g(x) + k = 0 + trace = [State(x, fk, ∇fk)] + while norm(∇fk) > ϵ && k < maxits + α = 1.0 + while ((fk - f(x-α*∇fk)) < μ*α*dot(∇fk,∇fk) ) + α /= 2 + end + x = x - α*∇fk + fk = f(x); ∇fk = g(x) + k += 1 + push!(trace, State(x, fk, ∇fk)) + end + return x, trace +end + +# ╔═╡ 5a0cd0c0-6ea3-410d-bd1b-73ff8d5f85a9 +begin + x0 = rand(3) # random starting vector + xsol, state = grad_method_backtracking(obj, grad, [red, green, blue], maxits=1000) +end + +# ╔═╡ 2cf23da3-4f03-4957-8b9f-47c6bd32999d +RGB(xsol...) # == RDB( x[1], x[2], x[3]) + +# ╔═╡ c3c74290-19c0-40ea-823a-11940ff59c12 +function log(states::Vector{State}) + io = IOBuffer() + write(io, "| k | f | ∇f | \n") + write(io, "|---|---|---| \n") + for (k, s) in enumerate(states) + f = @sprintf("%10.2e", s.f) + nrmf = @sprintf("%10.2e", s.norm∇f) + write(io, "| $(k-1) | $f | $nrmf |\n") + end + Markdown.parse(seekstart(io)) +end + +# ╔═╡ 45b05280-eb7a-4057-aab5-81be10f33d4d +log(qtrace) + +# ╔═╡ 517d402f-19af-428f-9552-fbfc4561df43 +log(qctrace) + +# ╔═╡ ee382544-49e1-4e50-8db2-b890d0fb8e29 +collectx(states::Vector{State}) = reduce(hcat, s.x for s in states) + +# ╔═╡ a20f9c65-1818-4c21-8180-16c71c511e7d +collectf(states::Vector{State}) = [s.f for s in states] + +# ╔═╡ 834d91b0-6285-4054-868d-bd23b7999c5e +collectg(states::Vector{State}) = [s.norm∇f for s in states] + +# ╔═╡ 1e0fd984-40e0-426c-8359-f883ad01d846 +let + f = Figure(); Axis(f[1,1], yscale=log10) + lines!(collectg(state).+eps()) + f +end + +# ╔═╡ 19ed7273-66a3-414e-985b-840ed05eb3de +function plot_iterates(f, trace, x, y) + z = [ f([x,y]) for x in x, y in y ] + fig = Figure(); Axis(fig[1,1], aspect=1) + contour!(x, y, z, levels=20) + xhist = collectx(trace)[1,:]; yhist = collectx(trace)[2,:] + scatterlines!(xhist, yhist, markersize=10) + fig +end + +# ╔═╡ 14cfe2cf-fc19-43f2-ac68-4d9ff74fd0fb +plot_iterates(q, qtrace, -2:.05:2, -2:.05:2) + +# ╔═╡ 15d9d597-f06a-424c-ba2d-0a3d47c1ffd0 +plot_iterates(qc, qctrace, -2:.05:2, -2:.05:2) + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ 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╠═4150cb58-e04a-4884-a8c1-1a5b3e7924ba +# ╠═5a0cd0c0-6ea3-410d-bd1b-73ff8d5f85a9 +# ╠═2cf23da3-4f03-4957-8b9f-47c6bd32999d +# ╠═1e0fd984-40e0-426c-8359-f883ad01d846 +# ╟─d10d1900-fff1-4267-846e-1b40f2eef944 +# ╠═676c40f2-5b44-4949-b79c-fbda43d8e091 +# ╠═dec864c0-8c16-4a74-a193-f6b8710cc445 +# ╠═c3c74290-19c0-40ea-823a-11940ff59c12 +# ╠═ee382544-49e1-4e50-8db2-b890d0fb8e29 +# ╠═a20f9c65-1818-4c21-8180-16c71c511e7d +# ╠═834d91b0-6285-4054-868d-bd23b7999c5e +# ╠═19ed7273-66a3-414e-985b-840ed05eb3de +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/notes/gradient-descent.pdf b/notes/gradient-descent.pdf new file mode 100644 index 0000000..f4bca15 Binary files /dev/null and b/notes/gradient-descent.pdf differ diff --git a/notes/gradient-descent/index.html b/notes/gradient-descent/index.html new file mode 100644 index 0000000..752b100 --- /dev/null +++ b/notes/gradient-descent/index.html @@ -0,0 +1,16 @@ + + + + + + +
\ No newline at end of file diff --git a/notes/gradients/index.html b/notes/gradients/index.html new file mode 100644 index 0000000..2448a35 --- /dev/null +++ b/notes/gradients/index.html @@ -0,0 +1,233 @@ + + + + + + + + + + + + + + + + + + Gradients + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Gradients

+ + + Gradients provide information on a function's sensitivity to perturbations in the input. + +

Directional derivatives

+

The behavior of a function f:Rn→Rf:\mathbb R^n\to\mathbb R along the ray {x+αd∣α∈R+}\{x+αd\mid α\in\mathbb R_+\}, where xx and dd are nn-vectors, is given by the univariate function

+
ϕ(α)=f(x+αd). + \phi(\alpha) = f(x+αd). +
+

From standard calculus, the derivative of ϕ\phi at the origin, when it exists, is the limit

+
ϕ′(0)=lim⁡α→0+ϕ(α)−ϕ(0)α. + \phi'(0) = \lim_{α\to0^+}\frac{\phi(α)-\phi(0)}{α}. +
+

We thus arrive at the following definition.

Definition: (directional derivative) The directional derivative of a function f:Rn→Rf:\mathbb R^n\to\mathbb R at a point x∈Rnx\in\mathbb R^n, along a direction d∈Rnd\in\mathbb R^n, is the limit
f′(x;d)=lim⁡α→0+f(x+αd)−f(x)α. + f'(x;d) = \lim_{α\to0^+}\frac{f(x+αd)-f(x)}{α}. +

+

It follows immediately from this definition that the partial derivatives of ff are simply the directional derivatives of ff along the each of the canonical unit direction e1,…,ene_1,\ldots,e_n, i.e.,

+∂f∂xi(x)≡f′(x;ei). + \frac{\partial f}{\partial x_i}(x) \equiv f'(x;e_i). + +

Descent directions

+

A nonzero vector dd is a descent direction of ff at xx if the directional derivative is negative:

+f′(x;d)<0. + f'(x;d) < 0. + +

It follows directly from the definition of the directional derivative that f(x+αd)<f(x)f(x+αd) < f(x) for all positive α\alpha small enough.

+

Gradient vector

+

The gradient of the function ff is the collection of all the partial derivatives:

+
∇f(x)=[∂f∂x1(x)⋮∂f∂xn(x)]. + \nabla f(x) = \begin{bmatrix} \frac{\partial f}{\partial x_1}(x) + \\\vdots + \\\frac{\partial f}{\partial x_n}(x)\end{bmatrix}. +
+

The gradient and directional derivative are related via the formula

+
f′(x;d)=∇f(x)T ⁣d. + f'(x;d) = \nabla f(x)^T\! d. +
+

If, for example, the direction dd to be the canonical unit direction eie_i, then this formula reduces to

+
f′(x;ei)=∇f(x)T ⁣ei=[∇f(x))]i=∂f∂xi(x), + f'(x;e_i) = \nabla f(x)^T\! e_i = [\nabla f(x))]_i = \frac{\partial f}{\partial x_i}(x), +
+

which confirms the identity (1).

+

Linear approximation

+

The gradient of a continuously differentiable function ff (i.e., ff is differentiable at all xx and ∇f\nabla f is continuous) provides a local linear approximation of ff in the following sense:

+
f(x+d)=f(x)+∇f(x)T ⁣d+ο(∥d∥), + f(x+d) = f(x) + \nabla f(x)^T\! d + \omicron(\| d\|), +
+

The residual ο:R+→R\omicron:\mathbb R_+\to\mathbb R of the approximation decays faster than ∥d∥\| d\|, i.e.,

+
lim⁡α→0+ο(α)/α=0.\lim_{α\to0+}\omicron(α)/α=0.
+

Example

+

Fortunately, there are good computational tools that automatically produce reliable gradients. Consider the 2-dimensional Rosenbrock function and its gradient:

+
f(x)=(a−x1)2+b(x2−x12)2∇f(x)=[−2(a−x1)−4b(x2−x12)x12b(x2−x12)]\begin{aligned} + f(x) &= (a - x_1)^2 + b(x_2 - x_1^2)^2 + \\ \nabla f(x) &= \begin{bmatrix} -2(a-x_1)-4b(x_2-x_1^2)x_1 \\ 2b(x_2-x_1^2)\end{bmatrix} +\end{aligned}
+

Here is the code for ff and its gradient:

+
a, b = 1, 100
+f(x) = (a - x[1])^2 + b*(x[2] - x[1]^2)^2
+∇f(x) = [-2(a - x[1]) - 4b*(x[2] - x[1]^2)*x[1] , 2b*(x[2] - x[1]^2) ]
+

Instead of computing gradients by hand, as we did above, we can use automatic differentiation, such as implemented in the package ForwardDiff, to compute these.

+
using ForwardDiff
+∇fad(x) = ForwardDiff.gradient(f, x)
+

The function ∇fad returns the value of the gradient at x. Let's compare the hand-computed gradient ∇f against that automatically-computed gradient ∇fad at a random point:

+
x = rand(2) 
+∇f(x) == ∇fad(x)
true
+

Calculus rules

+

We derive calculus rules for linear and quadratic functions, which appear often in optimization.

+

Linear functions

+

Let a∈Rna\in\mathbb R^n. The linear function

+
f(x)=aT ⁣x=∑inaixi + f(x) = a^T\! x = \sum_i^n a_i x_i +
+

has the gradient ∇f(x)=a\nabla f(x) = a, and so the gradient is constant. Here's a small example:

+
a = collect(1:5)
+ForwardDiff.gradient(x->a'x, rand(5))
+

Quadratic functions

+

Let A∈Rn×nA\in\mathbb R^{n\times n} be a square matrix. Consider the quadratic function

+f(x)=12xT ⁣Ax. + f(x) = \tfrac12 x^T\! A x. + +

One way to derive the gradient of this function is to write out the quadratic function making explicit all of the coefficients in AA. Here's another approach that uses the product rule:

+
∇f(x)=12∇(xT ⁣a)+12∇(bT ⁣x),\begin{aligned} + ∇f(x) = \tfrac12 ∇(x^T\! a) + \tfrac12 ∇(b^T\! x), +\end{aligned}
+

where a=Axa = Ax and b:=AT ⁣xb:=A^T\! x are held fixed when applying the gradient. Because each of the functions in the right-hand side of this sum is a linear function, we can apply the calculus rule for linear functions to deduce that

+
∇f(x)=12Ax+12AT ⁣x=12(A+AT ⁣)x. + ∇f(x) = \tfrac12 Ax + \tfrac12 A^T\! x = \tfrac12 (A+A^T\!)x. +
+

(Recall that AA is square.) The matrix 12(A+AT ⁣)\tfrac12(A+A^T\!) is the symmetric part of AA. If AA is symmetric, i.e., A=AT ⁣A = A^T\!, then the gradient reduces to

+
∇f(x)=Ax. + ∇f(x) = Ax. +
+

But in optimization, we almost always assume that the matrix that defines the quadratic in (3) is symmetric because always xT ⁣Ax=12xT ⁣(A+AT ⁣)xx^T\! Ax = \tfrac12 x^T\!(A+A^T\!)x, and therefore we can instead work with the symmetric part of AA.

+

Example (2-norm). Consider the two functions

+
f1(x)=∥x∥2andf2(x)=12∥x∥22. +f_1(x) = \| x\|_2 \quad\text{and}\quad f_2(x) = \tfrac12\| x\|_2^2. +
+

The function f2f_2 is of the form (3) with A=IA=I, and so ∇f2(x)=x\nabla f_2(x) = x. Use the chain rule to obtain the gradient of the f1f_1:

+
∇f1(x)=∇(xT ⁣x)12=12(xT ⁣x)−12∇(xT ⁣x)=x∥x∥2, + \nabla f_1(x) = \nabla (x^T\! x)^\tfrac12 = \tfrac12 (x^T\! x)^{-\tfrac12}\nabla (x^T\! x) = \frac{x}{\| x\|_2}, +
+

which isn't differentiable at the origin.

+

Visualizing gradients

+

Gradients can be understood geometrically in relation to the level-set of the function. The αα-level set of a function f:Rn→Rf:\mathbb R^n\to\mathbb R is the set of points that have an equal or lower value at xx:

+
[f≤α]={x∈Rn∣f(x)≤α}. + [f≤α] = \{x\in\mathbb R^n\mid f(x)≤α\}. +
+

Fix any xx and consider the level set [f≤f(x)][f≤f(x)]. For any direction dd that's either a descent direction for ff or a tangent direction for [f≤f(x)][f≤f(x)],

+
f′(x;d)=∇f(x)T ⁣d≤0, + f'(x;d) = ∇f(x)^T\! d ≤ 0, +
+

which implies that the gradient ∇f(x)\nabla f(x) is the outward normal to the level set.

+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + + + + + + + + + diff --git a/notes/least-squares/index.html b/notes/least-squares/index.html new file mode 100644 index 0000000..1fefd9c --- /dev/null +++ b/notes/least-squares/index.html @@ -0,0 +1,314 @@ + + + + + + + + + + + + + + + + + + Least Squares + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Least Squares

+ + Linear least-squares, also known as ordinary linear regression, is a basic optimization problem used for data fitting. It's also a fundamental building block for algorithms meant for more general problems. + +

Example: Multilinear regression

+

We begin with a simple example and use linear regression to build a model that relates the fuel efficiency of a car with its weight, engine displacement, and engine horsepower. Load the mtcars dataset provided in the Rdatasets.jl package:

+
using RDatasets
+mtcars = RDatasets.dataset("datasets", "mtcars")
+ +

Here are the first five rows of the dataset, which contains 32 rows in total:

+
first(mtcars, 5)
+
5×12 DataFrame
+ Row │ Model              MPG      Cyl    Disp     HP     DRat     WT       QSec     VS     AM     Gear   Carb
+     │ String31           Float64  Int64  Float64  Int64  Float64  Float64  Float64  Int64  Int64  Int64  Int64
+─────┼──────────────────────────────────────────────────────────────────────────────────────────────────────────
+   1 │ Mazda RX4             21.0      6    160.0    110     3.9     2.62     16.46      0      1      4      4
+   2 │ Mazda RX4 Wag         21.0      6    160.0    110     3.9     2.875    17.02      0      1      4      4
+   3 │ Datsun 710            22.8      4    108.0     93     3.85    2.32     18.61      1      1      4      1
+   4 │ Hornet 4 Drive        21.4      6    258.0    110     3.08    3.215    19.44      1      0      3      1
+   5 │ Hornet Sportabout     18.7      8    360.0    175     3.15    3.44     17.02      0      0      3      2
+

The model that we wish to build aims to find coefficients x1,…,x4∈Rx_1,\ldots,x_4\in\mathbb R such that for each particular car model i=1,2,…,32i=1,2,\ldots,32,

+
MPGi≈x1+x2⋅Dispi+x3⋅HPi+x4⋅WTi. +\text{MPG}_i \approx x_1 + x_2\cdot\text{Disp}_i + x_3\cdot\text{HP}_i + x_4\cdot\text{WT}_i. +
+

Of course, we shouldn't expect an exact match for all car models ii, and instead the coefficients x1,…,x4x_1,\ldots,x_4 computed by ordinary linear regression minimizes the sum of least-squares deviations:

+min⁡x1,…,x4 ∑i=132(x1+x2⋅Dispi+x3⋅HPi+x4⋅WTi−MPGi)2 +\min_{x_1,\ldots,x_4}\ \sum_{i=1}^{32} (x_1 + x_2\cdot\text{Disp}_i + x_3\cdot\text{HP}_i + x_4\cdot\text{WT}_i - \text{MPG}_i)^2 + +

Because this is such an important problem in general, there exists excellent software packages that make formulating and solving these problems easy. Here's how to do it using the generalized linear models package GLM.jl:

+
using GLM
+ols = lm(@formula(MPG ~ Disp + HP + WT), mtcars)
+
StatsModels.TableRegressionModel{GLM.LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}
+
+MPG ~ 1 + Disp + HP + WT
+
+Coefficients:
+───────────────────────────────────────────────────────────────────────────────
+                    Coef.  Std. Error      t  Pr(>|t|)   Lower 95%    Upper 95%
+───────────────────────────────────────────────────────────────────────────────
+(Intercept)  37.1055        2.11082    17.58    <1e-15  32.7817     41.4293
+Disp         -0.000937009   0.0103497  -0.09    0.9285  -0.0221375   0.0202635
+HP           -0.0311566     0.0114358  -2.72    0.0110  -0.0545817  -0.00773139
+WT           -3.80089       1.06619    -3.56    0.0013  -5.98488    -1.6169
+───────────────────────────────────────────────────────────────────────────────
+

The column marked Coef. shows the values for the 4 computed coefficients x1,…,x4x_1,\ldots,x_4; the remaining columns show the results for various statistical tests that indicate the reliability of each coefficient It appears that the car's weight has the biggest impact on fuel economy. This plot shows the projection of the data and the regression line onto the plane spanned by WT and MPG:

+
using StatsPlots; pyplot()
+x₁, x₄ = coef(ols)[[1,4]]
+@df mtcars scatter(:WT, :MPG, xlabel="weight (1000s lbs)", ylabel="MPG", leg=false)
+@df mtcars plot!(:WT, x₁ .+ x₄*:WT)
+ +

Matrix formulation

+

The objective (1) can be written using matrix notation, as follows: define the matrix and vector

+
A=[1WT1Disp1HP1⋮⋮⋮⋮1WTmDispmHPm],b=[MPG1⋮MPGm],x=[x1x2x3x4], +A = \begin{bmatrix} + 1 & \text{WT}_1 & \text{Disp}_1 & \text{HP}_1 +\\\vdots & \vdots & \vdots & \vdots +\\1 & \text{WT}_m & \text{Disp}_m & \text{HP}_m +\end{bmatrix}, +\quad +b = \begin{bmatrix} \text{MPG}_1 \\ \vdots \\ \text{MPG}_m \end{bmatrix}, +\quad +x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}, +
+

where m=32m=32. Thus, the least-squares soluition x∗x^* minimizes the 2-norm squared ∥r∥2\|r\|^2 of the residual r=b−Axr = b - Ax.

+
Notation
+

The 2-norm of a vector z∈Rnz\in\mathbb R^n is the function

∥z∥2=z12+⋯+zn2. +\|z\|_2 = \sqrt{z_1^2+\cdots+z_n^2}. +
This norm has the property that

+
zT ⁣z=[z1⋮zn]T ⁣[z1⋮zn]=∑i=1nzi2=∥z∥22. + z^T\! z = \begin{bmatrix} z_1\\\vdots\\ z_n\end{bmatrix}^T\! + \begin{bmatrix} z_1\\\vdots\\ z_n\end{bmatrix} + = \sum_{i=1}^n z_i^2 = \|z\|_2^2. +
+

We often leave out the subscript and simply write ∥z∥\|z\| for the 2-norm, unless warned otherwise.

+

General least-squares formulation

+

The general linear-least squares problem is formulated as

+min⁡x∈Rn 12∥r∥2,r=b−Ax, +\min_{x\in\mathbb R^n}\ \tfrac12\|r\|^2, \quad r = b - Ax, + +

where AA is an mm-by-nn matrix, bb is an mm-vector, xx is an nn-vector, and rr is an mm-vector of residuals. Typically, there are many more observations than mm than variables nn, and so m≫nm\gg n. But there are many important application that require least-squares problems where the opposite is true – these problems are considered under determined because there are necessarily infinitely many least-squares solutions xLS.x_\mathrm{\tiny LS}.

+

The following fundamental result describes several equivalent conditions that any least-squares solution must satisfy.

+

Theorem: (Least-squares optimality) The vector xLSx_\mathrm{\tiny LS} solves the least-squares problem (2) if and only if it satisfies the following equivalent conditions:

+
    +

  1. ATr=0A^T r = 0, where r=b−Axr=b-Ax,

    +
    2. +

  2. AT ⁣Ax=ATbA^T\! A x = A^T b,

    +
    3. +

  3. y:=Axy:=Ax is the (unique) orthogonal projection of bb onto range(A)\hbox{range}(A).

    +
    4. +
+

Moreover, xLSx_\mathrm{\tiny LS} is the unique least-squares solution if and only if AA has full rank.

+

This result relies on a fundamental property that for all matrices AA, regardless of their shape, the subspaces

+
range(A)={y∣y=Ax for some x}null(AT ⁣ )={z∣AT ⁣z=0}\begin{aligned} + \hbox{range}(A) &= \{y \mid y = A x \text{ for some } x \} + \\\text{null}(A^T\!\,) &= \{z \mid A^T\! z = 0 \} +\end{aligned}
+

are orthogonal complements, i.e.,

+
range(A)⊕null(A⊺)=Rm. + \hbox{range}(A) \oplus \hbox{null}(A^\intercal) = \mathbb R^m. +
+

The equivalence of the three properties is then straightforward, and it's sufficient to show that the third condition is equivalent to the optimality of the least-squares problem. Orthogonally decompose the residual r=b−yr = b-y as

+
b−y=zn+zr,zr∈range(A),zn∈null(AT ⁣). + b - y = z_n + z_r, \quad z_r\in\hbox{range}(A), \quad z_n\in\hbox{null}(A^T\!). +
+

If yy was not the orthogonal projection of bb, then there must exist some component of b−yb-y in the range of AA. Thus, zr≠0z_r\ne0 and

+
∥r∥2=∥b−y∥2=∥zn+zr∥2=∥zn∥2+∥zr∥2>∥zn∥2=∥b−(y+zr)∥2,\begin{aligned} + \|r\|^2 &= \|b-y\|^2 +\\ &= \|z_n+z_r\|^2 +\\ &= \|z_n\|^2+\|z_r\|^2 +\\ &> \|z_n\|^2=\|b-(y+z_r)\|^2, +\end{aligned}
+

which means we found a new point (y+zr)∈range(A)(y+z_r)\in\hbox{range}(A) with a smaller residual, which contradicts the optimality of xLSx_\mathrm{\tiny LS}, and also the uniqueness of the projection yy.

+

Example: Multilinear regression (continued)

+

For the multilinear regression example, we can obtain the least-squares solution by solving the normal equations as follows:

+
using LinearAlgebra # gives `norm` function
+A = [ones(size(mtcars,1)) mtcars[:,:Disp]  mtcars[:,:HP] mtcars[:,:WT]]
+b = mtcars[:,:MPG]
+x = A'A \ A'b
+@show(norm(x-coef(ols)))
+

norm(x - coef(ols)) = 2.1871395004271944e-14
+
The coefficients computed using the normal equations are almost the same as those computed using by GLM, which likely solves the normal equations using a different method.

+

The least-squares solution should also verify the first condition of the above theorem:

+
r = b - A*x
+@show(norm(A'r))
+
norm(A' * r) = 6.115154620218894e-11
+
+

The backslash operator

+

In this last example, we solved for the least-squares solution by explicitly solving the normal equations AT ⁣Ax=AT ⁣bA^T\! Ax=A^T\! b via the Julia command x = A'A \ A'b. As we'll see in a later lecture, this isn't the best way to solve the least-squares problem. Without getting into the details just yet, it's enough for now to know that Julia allows for the short-hand x = A \ b: Julia recognizes that this is an overdetermined system, and solves for the least-squares solution using a mathematically equivalent approach. Let's verify:

+
x1 = A \ b
+@show norm(x1 - x, Inf)
+
norm(x1 - x, Inf) = 4.334310688136611e-13
+
+

Close enough!

+

Example: Trendline

+

This example uses least-squares to illustrate the increasing concentrations of nitrous oxide (N₂O), which is a greenhouse gas. We'll use a dataset from the Global Monitoring Laboratory:

+
fname = "mlo_N2O_All.dat"
+if !isfile(fname)
+	download("https://gml.noaa.gov/aftp/data/hats/n2o/insituGCs/CATS/hourly/mlo_N2O_All.dat", fname)
+end
+

This dataset provides hourly measurements of N₂O levels in parts per billion (ppb) from 1999 to 2021 from gas sampled at the Manua Loa Observatory in Hawaii. The following code removes rows with missing concentration levels ppb, and creates a new table with average monthly measurements:

+
using CSV, DataFrames, DataFramesMeta, Statistics
+df = CSV.read(fname, DataFrame, comment="#", normalizenames=true, delim=" ", ignorerepeated=true)
+dfg = @chain df begin
+    @select(:year = :N2OcatsMLOyr, :month = :N2OcatsMLOmon, :ppb = :N2OcatsMLOm)
+    @subset((!isnan).(:ppb))
+    @by([:year, :month], :avgppb = mean(:ppb))
+end 
+describe(dfg, :min, :max, cols=[:year, :avgppb])
+
2×3 DataFrame
+ Row │ variable  min      max
+     │ Symbol    Real     Real
+─────┼─────────────────────────────
+   1 │ year      1999     2022
+   2 │ avgppb     315.06   337.484
+

A trend line is defined by the univariate function

+
p(t)=α+βt, + p(t) = \alpha + \beta t, +
+

where α\alpha is the intercept, β\beta is the slope, and tt is time. Here, we'll just index the months by the integers t=1,2,…,Tt=1,2,\ldots,T. We determine the parameters α\alpha and β\beta by solving the least-squares problem

+
min⁡α, β ∑i=1T(p(ti)−ppbi)2, + \min_{\alpha,\ \beta}\ \sum_{i=1}^{T}(p(t_i) - \text{ppb}_i)^2, +
+

which corresponds to the general least-squares problem with

+
A=[1t11t2⋮⋮1T],b=[ppb1ppb2⋮ppbT], andx=[αβ]. + A = \begin{bmatrix} 1 & t_1 \\ 1 & t_2 \\ \vdots & \vdots \\ 1 & T \end{bmatrix}, + \quad + b = \begin{bmatrix} \text{ppb}_1 \\ \text{ppb}_2 \\ \vdots \\ \text{ppb}_T \end{bmatrix}, + \quad\text{ and}\quad + x = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}. +
+

In Julia,

+
T = nrow(dfg)
+t = collect(1:T)
+A = [ones(T)  t]
+b = dfg[!, :avgppb]
+

The least-squares solution is then

+
α, β = A \ b
+

and we can plot the data together with the trend line:

+
@df dfg scatter(:avgppb) 
+p(t) = α + β*t
+plot!(p, lw=3)
+plot!(leg=false, xlab="Month index from 1999 to 2021", ylab="Monthly average N₂O ppb")
+ +

But perhaps most interestingly, we can now "detrend" the data to understand how the concentrations vary around the overall 22-year trend:

+
@df dfg scatter(:avgppb - p.(t))
+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + + + + + + + + + diff --git a/notes/lin-prog-apps.pdf b/notes/lin-prog-apps.pdf new file mode 100644 index 0000000..0235bef Binary files /dev/null and b/notes/lin-prog-apps.pdf differ diff --git a/notes/linear-constraints.pdf b/notes/linear-constraints.pdf new file mode 100644 index 0000000..a40fcf8 Binary files /dev/null and b/notes/linear-constraints.pdf differ diff --git a/notes/lp-duality.jl b/notes/lp-duality.jl new file mode 100644 index 0000000..d4da58a --- /dev/null +++ b/notes/lp-duality.jl @@ -0,0 +1,1148 @@ +### A Pluto.jl notebook ### +# v0.18.4 + +using Markdown +using InteractiveUtils + +# ╔═╡ e88781de-aed3-11ec-33ec-a5846d05897b +using JuMP, Ipopt, Plots, HiGHS + +# ╔═╡ 7c9b81ef-67ba-45cc-bd10-a360bd057628 +md""" +# LP Duality +""" + +# ╔═╡ 2b478e92-8e5d-4203-940f-31a8e48f3333 +md""" +A simple quadratic optimization problem: + +$$\min_{x_1,x_2}\{x_1^2 + x_2^2 \mid x_1+x_2=1\}$$ + +""" + +# ╔═╡ 9f78c416-1c07-4687-900a-ed1a1db581e7 +begin + model = Model(Ipopt.Optimizer) + set_silent(model) + @variable(model, x1) + @variable(model, x2) + @constraint(model, x1 + x2 == 1) + @NLobjective(model, Min, x1^2 + x2^2) + optimize!(model) + value.([x1, x2]), objective_value(model) +end + +# ╔═╡ a4d1b48a-3529-4f58-81a1-f7392c55d609 +md"Define dual function $g$:" + +# ╔═╡ 1123df22-126e-4225-b480-d1e311cc3ccc +function g(y) + model = Model(Ipopt.Optimizer) + set_silent(model) + @variable(model, x1) + @variable(model, x2) + @NLobjective(model, Min, x1^2 + x2^2 + y*(1 - x2 - x1)) + optimize!(model) + objective_value(model) +end + +# ╔═╡ 8abf300c-6f46-4129-a73d-55376af2d3b7 +plot(g, -1, 3) + +# ╔═╡ b136d9f6-5499-4121-921b-699508e34022 +begin + A = [-2 1 1 0 0 + -1 2 0 1 0 + 1 0 0 0 1] + b = [2, 7, 3] + c = [-1, -2, 0, 0, 0] + m, n = size(A) +end + +# ╔═╡ 85c995a9-7b7d-4021-9ec4-8765ced6ba78 +md""" +## Primal problem +""" + +# ╔═╡ 01036d30-c3fb-4069-95ac-fba08c405097 +x, yy = let + model = Model(HiGHS.Optimizer) + @variable(model, x[1:n] ≥ 0) + @objective(model, Min, c'x) + @constraint(model, con, A*x .== b) + set_silent(model) + optimize!(model) + value.(x), dual.(con) +end + +# ╔═╡ d87246f1-eab7-4fb3-b067-937806dd5ff8 +p = c'x + +# ╔═╡ 4fb2b360-68fa-4e1a-be23-10a8dd9b6062 +md""" +## Dual problem +""" + +# ╔═╡ d17c22ef-1236-4468-abed-79ed4df93625 +y = let + model = Model(HiGHS.Optimizer) + @variable(model, y[1:m]) + @objective(model, Max, b'y) + @constraint(model, con, c - A'y .≥ 0) + set_silent(model) + optimize!(model) + value.(y) +end + +# ╔═╡ 5f3b3891-90fe-468c-badc-61386c126859 +d = b'y + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b" +Ipopt = "b6b21f68-93f8-5de0-b562-5493be1d77c9" +JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" + +[compat] +HiGHS = "~1.1.1" +Ipopt = "~1.0.2" +JuMP = "~1.0.0" +Plots = "~1.27.3" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 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\ No newline at end of file diff --git a/notes/lp-games.pdf b/notes/lp-games.pdf new file mode 100644 index 0000000..0788936 Binary files /dev/null and b/notes/lp-games.pdf differ diff --git a/notes/lp-geometry.pdf b/notes/lp-geometry.pdf new file mode 100644 index 0000000..870e880 Binary files /dev/null and b/notes/lp-geometry.pdf differ diff --git a/notes/lp-jump.jl b/notes/lp-jump.jl new file mode 100644 index 0000000..66e5e13 --- /dev/null +++ b/notes/lp-jump.jl @@ -0,0 +1,626 @@ +### A Pluto.jl notebook ### +# v0.18.4 + +using Markdown +using InteractiveUtils + +# ╔═╡ 91e80254-ab22-11ec-10aa-0fcd92bacb88 +using JuMP, PlutoUI + +# ╔═╡ b202edea-3b24-406c-9a70-1db2cbb37e99 +md""" +# Linear Programming with JuMP +""" + +# ╔═╡ f156cea2-603b-4507-b262-14cb465f2fa9 +import Clp, HiGHS + +# ╔═╡ 9e23a531-2209-4092-a1de-68d4edcba8fb +begin + A = [-2 1 1 0 0 + -1 2 0 1 0 + 1 0 0 0 1] + b = [2, 7, 3] + c = [-1, -2, 0, 0, 0] + m, n = size(A) +end + +# ╔═╡ 13dd4455-704d-4df4-b151-7452591f9c51 +begin + model = Model(HiGHS.Optimizer) + @variable(model, x[1:n] ≥ 0) + @objective(model, Min, c'x) + @constraint(model, c1, A*x .== b) + + # set_optimizer_attribute(model, "LogLevel", 4) + # set_optimizer_attribute(model, "PresolveType", 1) + # set_optimizer_attribute(model, "SolveType", 1) + + ## HiGHS options + set_optimizer_attribute(model, "highs_debug_level", 1) + set_optimizer_attribute(model, "log_dev_level", 3) + set_optimizer_attribute(model, "presolve", "off") + set_optimizer_attribute(model, "solver", "simplex") + set_optimizer_attribute(model, "simplex_strategy", "4") + + with_terminal() do + optimize!(model) + end + +end + +# ╔═╡ 96d97354-e899-4dd9-9bd3-92fcc121dafb +solution_summary(model, verbose=true) + +# ╔═╡ e5149742-094e-4cc0-acb2-43b50ebf5b89 +md""" +## Convex modeling language +""" + +# ╔═╡ fe0b5915-09ad-4991-86c2-0379f6870acc +import Convex, SCS, LinearAlgebra, GLPK + +# ╔═╡ 568f5f4a-64fa-4b02-991b-cbfc5501f9b2 +xv = let + x = Convex.Variable(n) + p = Convex.minimize( Convex.norm(A*x-b, Inf) ) + Convex.solve!(p, HiGHS.Optimizer, verbose=true, silent_solver=false) + with_terminal() do + @show p.optval + @show x.value + end + x.value +end + +# ╔═╡ ae48cae2-1f87-439f-b7a2-91f7cd7919df +A*xv - b + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Clp = "e2554f3b-3117-50c0-817c-e040a3ddf72d" +Convex = "f65535da-76fb-5f13-bab9-19810c17039a" +GLPK = "60bf3e95-4087-53dc-ae20-288a0d20c6a6" +HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b" +JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13" + +[compat] +Clp = "~1.0.0" +Convex = "~0.15.0" +GLPK = "~1.0.1" +HiGHS = "~1.1.1" +JuMP = "~0.23.2" +PlutoUI = "~0.7.37" +SCS = "~1.1.1" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it 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+git-tree-sha1 = "ba5c0d3b23220d3598d2877b4cf913e3fcf8add3" +uuid = "f4f2fc5b-1d94-523c-97ea-2ab488bedf4b" +version = "3.2.0+0" + +[[deps.SHA]] +uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" + +[[deps.Serialization]] +uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" + +[[deps.SharedArrays]] +deps = ["Distributed", "Mmap", "Random", "Serialization"] +uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383" + +[[deps.Sockets]] +uuid = "6462fe0b-24de-5631-8697-dd941f90decc" + +[[deps.SparseArrays]] +deps = ["LinearAlgebra", "Random"] +uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" + +[[deps.SpecialFunctions]] +deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"] +git-tree-sha1 = "5ba658aeecaaf96923dce0da9e703bd1fe7666f9" +uuid = "276daf66-3868-5448-9aa4-cd146d93841b" +version = "2.1.4" + +[[deps.StaticArrays]] +deps = ["LinearAlgebra", "Random", "Statistics"] +git-tree-sha1 = "6976fab022fea2ffea3d945159317556e5dad87c" +uuid = 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"8e850b90-86db-534c-a0d3-1478176c7d93" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +""" + +# ╔═╡ Cell order: +# ╟─b202edea-3b24-406c-9a70-1db2cbb37e99 +# ╠═91e80254-ab22-11ec-10aa-0fcd92bacb88 +# ╠═f156cea2-603b-4507-b262-14cb465f2fa9 +# ╠═9e23a531-2209-4092-a1de-68d4edcba8fb +# ╠═13dd4455-704d-4df4-b151-7452591f9c51 +# ╠═96d97354-e899-4dd9-9bd3-92fcc121dafb +# ╟─e5149742-094e-4cc0-acb2-43b50ebf5b89 +# ╠═fe0b5915-09ad-4991-86c2-0379f6870acc +# ╠═568f5f4a-64fa-4b02-991b-cbfc5501f9b2 +# ╠═ae48cae2-1f87-439f-b7a2-91f7cd7919df +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/notes/lp-jump/index.html b/notes/lp-jump/index.html new file mode 100644 index 0000000..e28fe81 --- /dev/null +++ b/notes/lp-jump/index.html @@ -0,0 +1,16 @@ + + + + + + +
\ No newline at end of file diff --git a/notes/lp-qp-barrier.pdf b/notes/lp-qp-barrier.pdf new file mode 100644 index 0000000..8766326 Binary files /dev/null and b/notes/lp-qp-barrier.pdf differ diff --git a/notes/lp-simplex-bnds-2phase.pdf b/notes/lp-simplex-bnds-2phase.pdf new file mode 100644 index 0000000..ef93728 Binary files /dev/null and b/notes/lp-simplex-bnds-2phase.pdf differ diff --git a/notes/lp-simplex-example.pdf b/notes/lp-simplex-example.pdf new file mode 100644 index 0000000..d70d8be Binary files /dev/null and b/notes/lp-simplex-example.pdf differ diff --git a/notes/lp-simplex.pdf b/notes/lp-simplex.pdf new file mode 100644 index 0000000..730af9b Binary files /dev/null and b/notes/lp-simplex.pdf differ diff --git a/notes/lp-standard-form.pdf b/notes/lp-standard-form.pdf new file mode 100644 index 0000000..addffe1 Binary files /dev/null and b/notes/lp-standard-form.pdf differ diff --git a/notes/matrix-games.jl b/notes/matrix-games.jl new file mode 100644 index 0000000..1e23de4 --- /dev/null +++ b/notes/matrix-games.jl @@ -0,0 +1,479 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ a8fab1fa-c21a-468b-b2c4-7c4227fb979a +using JuMP, HiGHS, LinearAlgebra + +# ╔═╡ 176faf8e-da21-11ed-086a-2f1962857849 +md"# Matrix Games" + +# ╔═╡ a8b2f40f-d654-408c-8d17-cf5f7322e562 +md""" +## Rock, Paper, Scissors +""" + +# ╔═╡ 1280a06b-bfbb-4d37-8f87-1650dea12895 +md"Payoff matrix $A$:" + +# ╔═╡ 6a3702e3-bd49-4e15-b148-128fff258e24 +# rock paper scissors +A = [ 0 +1 -1 # rock + -1 0 +1 # paper + +1 -1 0 # scissors +]; + +# ╔═╡ 813750cf-73d9-43b8-8335-252e964532b3 +md""" +### Player X's problem +""" + +# ╔═╡ ad5a76ae-17d6-40b7-ab5d-78a3ead928dc +function playerX(A::Matrix) + m, n = size(A) + model = Model(HiGHS.Optimizer) + set_silent(model) + @variable(model, x[1:n]≥0) + @variable(model, λ) + @constraint(model, sum(x) == 1) + @constraint(model, xcon, λ .≤ A*x) + @objective(model, Max, λ) + optimize!(model) + return value.(x), objective_value(model) +end + +# ╔═╡ 3f3aa6a9-96a9-450c-89f6-f7673a9e86d1 +md"### Player Y's problem" + +# ╔═╡ a9492a83-0547-4cc9-8deb-9c3cdd3d7602 +function playerY(A::Matrix) + m, n = size(A) + model = Model(HiGHS.Optimizer) + set_silent(model) + @variable(model, y[1:m]≥0) + @variable(model, ν) + @constraint(model, sum(y) == 1) + @constraint(model, ycon, ν .≥ A'y) + @objective(model, Min, ν) + optimize!(model) + return value.(y), objective_value(model) +end + +# ╔═╡ 8b5d9c0c-47ca-42c0-8666-7d3eef52bf59 +playerX(A) + +# ╔═╡ 86116587-c804-4680-9a2f-5456c2e21d68 +playerY(A) + +# ╔═╡ 37258aad-9b23-4ccc-8253-d10805d99f46 +md"## Helper functions" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b" +JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" + +[compat] +HiGHS = "~1.5.0" +JuMP = "~1.10.0" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 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\ No newline at end of file diff --git a/notes/max-flow-min-cut.pdf b/notes/max-flow-min-cut.pdf new file mode 100644 index 0000000..7c5c574 Binary files /dev/null and b/notes/max-flow-min-cut.pdf differ diff --git a/notes/max-flow-min-cut_lecture.pdf b/notes/max-flow-min-cut_lecture.pdf new file mode 100644 index 0000000..fc0be32 Binary files /dev/null and b/notes/max-flow-min-cut_lecture.pdf differ diff --git a/notes/nonlinear-least-squares/index.html b/notes/nonlinear-least-squares/index.html new file mode 100644 index 0000000..3bb3a34 --- /dev/null +++ b/notes/nonlinear-least-squares/index.html @@ -0,0 +1,214 @@ + + + + + + + + + + + + + + + + + + Nonlinear Least-Squares + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Nonlinear Least-Squares

+ + + The nonlinear least-squares problem generalizes the linear least-squares problem to include nonlinear residual functions. The Gauss-Newton solves this problem as a sequence of least-squares subproblems. + +

Nonlinear residuals

+

The nonlinear least squares problem seeks to minimize the sum of squares

+min⁡x∈Rn 12∥r(x)∥2, + \min_{x\in\mathbb R^n}\ \tfrac12\|r(x)\|^2, + +

where the mm-vector of residuals

+
r(x)=[r1(x)⋮rm(x)] + r(x)=\begin{bmatrix} r_1(x)\\\vdots\\r_m(x)\end{bmatrix} +
+

is composed of differentiable nonlinear functions ri:Rn→Rr_{i}:\mathbb R^n→\mathbb R. The non-linear least squares problem reduces to linear least-squares when rr is affine, i.e., r(x)=Ax−br(x) = Ax-b for some mm-by-nn matrix AA and mm-vector of observations bb.

+

Linearizing the residual

+

We can solve non-linear least squares problem (1) by solving a sequence of linear least-squares problem, which result from linearization of rr at the current estimate of the ground truth xx.

+

The linear approximation of rr at a point xk∈Rnx^k \in \mathbb R^n is defined by

+
rk(d):=[r1(xk)+∇r1(xk)T ⁣d⋮rm(xk)+∇rm(xk)T ⁣d]=r(xk)+J(xk)d\begin{aligned} +r^k(d) + &:= \begin{bmatrix} r_1(x^k)+\nabla r_1(x^k)^T\! d \\ \vdots \\ r_m(x^k)+\nabla r_m(x^k)^T\! d\end{bmatrix} +\\ &= r(x^k) + J(x^k)d +\end{aligned}
+

where

+J(x)=[∇r1(x)T⋮∇rm(x)T] +J(x) += \begin{bmatrix} \nabla r_1(x)^T\\ \vdots \\ \nabla r_m(x)^T\end{bmatrix} + +

is the Jacobian of rr evaluated at xx

+

Algorithm: (Gauss-Newton method)
Choose a starting x(0)x^{(0)}
for k=0,1,2,…k=0,1,2,\ldots

+
    +

  1. linearize rr at xkx^{k} to define rk(d):=r(xk)+J(xk)dr^k(d):=r(x^k)+J(x^k)d

    +
    2. +

  2. solve linear least-squares problem

    dk=arg min⁡d 12∥rk(d)∥2 + d^k = \argmin_d\ \tfrac12\| r^k(d)\|^2 +

    +
    3. +

  3. update iterate: xk+1=xk+αdkx^{k+1} = x^k + αd^k for some α∈(0,1]α∈(0,1]

    +
    4. +
If the linesearch parameter α\alpha is held fixed at 1, then this is a "pure" Gauss-Newton iteration. We'll later discuss options for when it's advisable to use smaller steps.

+

Here's a basic version of the method that takes as arguments the residual and Jacobian functions r and J, and a starting vector x.

+
function gauss_newton(r, J, x; ε=1e-4, maxits=30)
+  err = []
+  for i = 1:maxits
+      rk, Jk = r(x), J(x)
+      push!(err, norm(Jk'rk))
+      err[end] ≤ ε && break
+      x = x - Jk\rk 
+    end
+    return x, err
+end
+

The condition err[end] ≤ ε causes the iterations to terminate when the latest residual rk is nearly orthogonal to all the gradients of the residual, i.e., the residual is orthogonal to the tangent space of the level-set for the nonlinear least-squares objective 12∥r(x)∥2\tfrac12\|r(x)\|^2. This is exactly analogous to the optimality condition for linear least-squares.

+

Example: Position estimation from ranges

+ +

Let x∈R2x \in \mathbb R^2 represent the unknown position of an object. Our aim is to find the position of this object relative to a set of mm beacons placed in fixed known positions bi∈R2b_{i} \in \mathbb R^2, i=1,…,mi = 1,\dots,m. The only data available are range measurements δiδ_i that give an estimate of the distance between each beacon bib_{i} and the object at xx, i.e.,

+δi:=∥x−bi∥+νi +δ_i := \| x-b_i\| + ν_i + +

where each scalar νiν_i represents the error between the true (and unkown) distance ∥x−bi∥\| x-b_i\| and the measurement δiδ_i.

+

We can obtain an estimate of the object's position xx by solving the nonlinear least-squares problem (1) where we define the iith residual between the reported distance δiδ_i and the distance between the position bib_i of the iith beacon and xx:

+
ri(x):=δi−∥x−bi∥. + r_i(x) := δ_{i} - \|x-b_i\|. +
+

Here's the residual function, which takes a vector of ranges δδ and a vector of positions b:

+
r(x, δ, b) = [ δ[i] - norm(x - b[:,i]) for i in 1:length(δ) ]
+

Use the following Julia packages for this example.

+
using Random
+using Statistics
+using LinearAlgebra
+using ForwardDiff
+using Plots
+

We simulate data by placing m beacons in random locations:

+
m = 13                 # number of beacons
+b = 2rand(2, m) .- 1   # place beacons uniformly in the unit box 
+x = zeros(2)           # true position 
+x0 = 2rand(2) .- 1     # initial guess of the unknown position
+ν = .5*rand(m)
+δ = [ norm(x - b[:,i]) + ν[i] for i in 1:m]
+

Place these items on a map (we'll wrap this in a function so that we can reuse it below):

+
function plotmap(b, x, x0)
+scatter(xlim=(-1,+1), ylim=(-1,+1), leg=:outertopright,frame=:box, aspect_ratio=:equal)
+scatter!(b[1,:], b[2,:], label="beacons", shape=:square, ms=7)
+scatter!(x[1,:], x[2,:], label="true position", shape=:xcross, ms=7)
+scatter!(x0[1,:], x0[2,:], label="initial guess", c="yellow", ms=7)
+end
+plotmap(b, x, x0)
+ +
J(x) = ForwardDiff.jacobian(x->r(x, δ, b), x)
+xs, err = gauss_newton(x->r(x, δ, b), J, x0)
+plot(err, yaxis=:log)
+ +

Plot the original map and overlay the obtained solution:

+
plotmap(b, x, x0)
+scatter!(xs[1,:], xs[2,:], label="solution", shape=:star7, c="green", ms=7)
+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + + + + + + + + + diff --git a/notes/normal-cone-optimality.pdf b/notes/normal-cone-optimality.pdf new file mode 100644 index 0000000..d31e698 Binary files /dev/null and b/notes/normal-cone-optimality.pdf differ diff --git a/notes/projection.pdf b/notes/projection.pdf new file mode 100644 index 0000000..b121f56 Binary files /dev/null and b/notes/projection.pdf differ diff --git a/notes/qr-factorization/index.html b/notes/qr-factorization/index.html new file mode 100644 index 0000000..5fffc98 --- /dev/null +++ b/notes/qr-factorization/index.html @@ -0,0 +1,366 @@ + + + + + + + + + + + + + + + + + + QR Factorization + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

QR Factorization

+ + The QR factorization of a matrix constructs an orthgonal basis for its columnspace. It's also one of the best computational tools for solving least-squares problems. + + +

Required Reading

+ +
⚠ Note
+

This page uses the LinearAlgebra package, which contains norm and qr.

+
using LinearAlgebra
+

Orthogonal and orthonormal vectors

+

For any two nn-vectors xx and yy, the cosine identity

+
xT ⁣y=∥x∥∥y∥cos⁡(θ) + x^T\! y = \|x\|\|y\|\cos(\theta) +
+

relates the inner product of the vectors to the the angle between them. Thus, the vectors xx and yy are orthogonal when

+
xT ⁣y=0,or equivalently,cos⁡(θ)=0. + x^T\! y = 0, \quad\text{or equivalently,}\quad \cos(\theta) = 0. +
+

Orthogonal vectors are orthonormal when they're also normalized:

+
xT ⁣y=0,∥x∥=1,∥y∥=1. + x^T\! y = 0, \quad \| x\|=1, \quad \| y\|=1. +
+

These definitions extend readily to a collection of vectors: the set of kk vectors {x1,…,xk}\{x_1,\ldots,x_k\} in Rn\mathbb R^n is orthogonal (and orthonormal) whenever

+
xiT ⁣xj=0for alli≠j(and ∥xi∥=1) + x_i^T\! x_j = 0 \quad\text{for all}\quad i\ne j +\qquad (\text{and}\ \| x_i\| = 1) +
+

The 2-dimensional canonical unit vectors e1=(1,0)e_1 = (1,0) and e2=(0,1)e_2 = (0,1) are orthonormal:

+
e₁ = [1, 0];  e₂ = [0, 1]
+@show e₁'e₂
+@show norm(e₁)
+@show norm(e₂)
e₁' * e₂ = 0
+norm(e₁) = 1.0
+norm(e₂) = 1.0
+
+

Orthogonal matrices

+

A square nn-by-nn matrix QQ is orthogonal when its columns are orthonormal:

+
Q=[q1q2⋯qn]andQT ⁣Q=QQT ⁣=I. + Q = \begin{bmatrix} q_1 & q_2 & \cdots & q_n \end{bmatrix} + \quad\text{and}\quad + Q^T\! Q = QQ^T\! = I. +
+

Because a matrix BB is the inverse of a matrix AA if and only if BA=AB=IBA =AB = I, the inverse of an orthogonal matrix is its transpose:

+
Q−1=QT ⁣. +Q^{-1} = Q^T\!. +
+

An orthogonal matrix rotates a vector, but doesn't change its length. Thus inner products are invariant under orthogonal transformations, i.e., for any vectors xx and yy,

+
(Qx)T ⁣(Qy)=xT ⁣QT ⁣Qy=xT ⁣y, +(Qx)^T\!(Qy) = x^T\! Q^T\! Qy = x^T\! y, +
+

and so

+
∥Qx∥2=(Qx)T ⁣(Qx)=xT ⁣x=∥x∥2. + \| Q x \|_2 = \sqrt{(Qx)^T\! (Qx)} = \sqrt{x^T\! x} = \| x \|_2. +
+

Another property of orthogonal matrices is that their determinant is either 11 or −1-1. This can be observed from the fact that for any compatible matrices A A and B B , det⁡(AB)=det⁡(A)det⁡(B)\det( A B ) = \det( A )\det( B ) and det⁡(A)=det⁡(AT)\det( A )= \det( A^T ). Hence,

+
det⁡(QT ⁣Q)=det⁡(I)  ⟺  det⁡(Q)2=1  ⟺  det⁡(Q)=±1. +\det( Q ^T\! Q ) = \det( I ) \iff \det( Q )^2 = 1 \iff \det( Q ) = ±1. +
+

QR Factorization

+

Every mm-by-nn matrix AA has a QR factorization, which means it can be decomposed as the product

+
A=QR, + A = Q R, +
+

where

+
Q=[q1⋯qm] andR=[r11r12⋯r1n0r22⋯r2n⋮⋱⋮00⋯rmn], + Q = \begin{bmatrix} q_1 & \cdots & q_m \end{bmatrix} + \quad\text{ and}\quad + R = \begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1n} + \\ 0 & r_{22} & \cdots & r_{2n} + \\ \vdots & & \ddots & \vdots + \\ 0 & 0 & \cdots & r_{mn} + \end{bmatrix}, +
+

respectively, are an mm-by-mm orthogonal matrix and an mm-by-nn upper-triangular matrix. This factorization isn't unique, unless we impose additional conditions on the factors QQ and RR.

+

Householder reflectors and Givens rotations are the preferred methods for computing the QR factorization. These methods require O(m2n)\Omicron(m^2n) flops to complete.

+

The columns of the orthogonal matrix QQ reveal an orthogonal basis for the range of A,A, and so triangularity of RR gives a simple "recipe" for reconstructing the columns of A=[a1,…,an]A=[a_1,\ldots,a_n] from those of QQ:

+
a1=r11q1a2=r12q1+r22q2a3=r13q1+r23q2+r33q3  ⋮an=r1nq1+r2nq2+⋯+rnnqn.\begin{aligned} + a_1 &= r_{11} q_1 +\\ a_2 &= r_{12} q_1 + r_{22} q_2 +\\ a_3 &= r_{13} q_1 + r_{23} q_2 + r_{33} q_3 +\\ &\ \ \vdots +\\ a_n &= r_{1n} q_1 + r_{2n} q_2 + \cdots + r_{nn} q_n. +\end{aligned}
+

From these formulas we deduce a key property of this factorization: for any index k<nk < n, the span of the leading kk columns of QQ contain the span of the leading kk columns of AA, i.e.,

+
span{a1,…,ak}⊆span{q1,…,qk}. + \hbox{span}\{a_1,\ldots,a_k\} \subseteq \hbox{span}\{q_1,\ldots,q_k\}. +
+

If we further assume that AA has full column rank, this inclusion tightens to an equality, which tells us that leading kk columns of QQ provide a tight basis for the corresponding columns.

+
Theorem: (QR factorization for full column rank matrices) If A=QRA=QR is the QR factorization of an mm-by-nn matrix AA with full column rank, then the first nn columns of QQ form a basis for the range of AA and the remaining m−nm-n columns form a basis for its orthogonal complement:
range(A)=range(Q1)whereQ1=[q1⋯qn],range(A)⊥=null(AT)=range(Q2)whereQ2=[qn+1⋯qm],\begin{aligned} + \hbox{range}(A) &= \hbox{range}(Q_1) \quad \text{where}\quad Q_1 = \begin{bmatrix}q_1 & \cdots & q_n\end{bmatrix}, +\\\hbox{range}(A)^\perp=\hbox{null}(A^T) &= \hbox{range}(Q_2) \quad \text{where}\quad Q_2 = \begin{bmatrix}q_{n+1} & \cdots & q_m\end{bmatrix}, +\end{aligned}
and for k=1,…,nk=1,\ldots,n,
span{a1,…,ak}=span{q1,…,qk} + \hbox{span}\{a_1,\ldots,a_k\} = \hbox{span}\{q_1,\ldots,q_k\} +
where the diagonal elements rkk≠0.r_{kk}\ne 0.
+

This theorem tells us that a full column rank matrix can then be decomposed as

+A=QR=[Q1Q2][R10]=Q1R1, + A = Q R + = \begin{bmatrix}Q_1 & Q_2\end{bmatrix} + \begin{bmatrix}R_1 \\ 0 \end{bmatrix} + = Q_1 R_1, + +

where the triangular submatrix R1=R[1:n,1:n]R_1 = R[1{:}n,1{:}n] has no zeros on its diagonal. The factorization A=Q1R1A = Q_1 R_1 is the thin or economy QR.

+

In Julia, we can compute the full QR decomposition of a matrix using via

+
m, n = 4, 3
+A = randn(m,n)
+Q, R = qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
+Q factor:
+4×4 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
+ -0.589037   -0.609377    0.391674   0.35817
+  0.594233   -0.0982621   0.0108708  0.798194
+ -0.547289    0.568937   -0.37926    0.482646
+  0.0199064   0.543429    0.838233   0.0406633
+R factor:
+3×3 Matrix{Float64}:
+ -1.64787  -1.95628   1.26521
+  0.0       2.8769   -0.0479272
+  0.0       0.0       1.30317
+

and we can verify the orthogonality of Q:

+
@show norm(Q'*Q - I)
+@show norm(Q*Q' - I)
norm(Q' * Q - I) = 5.928593550334434e-16
+norm(Q * Q' - I) = 4.896838930763021e-16
+
+

Julia stores the factor Q as a series of Householder reflectors.

+

Thin QR

+

To extract the "thin" QR factorization from qr, use the Matrix conversion function:

+
F = qr(A)
+Q₁ = Matrix(F.Q)
+R₁ = F.R
+@show size(Q₁)
+@show size(R₁)
+@show norm(A - Q₁*R₁)
size(Q₁) = (4, 3)
+size(R₁) = (3, 3)
+norm(A - Q₁ * R₁) = 7.494005416219807e-16
+
+

Solving least squares via QR

+

The QR factorization can be used to solve the least squares problem

+min⁡x∈Rn ∥Ax−b∥2. + \min_{x\in\mathbb R^n}\ \| A x -b\|^2. + +

Assume throughout this section that AA has full columns rank, and hence m≥nm\geq n. (The QR factorization also can be used to solve the more general problem, but we don't consider that case here.) Let A=QRA = QR be the QR factorization shown in (1). Because the 2-norm is invariant under orthogonal rotation,

+∥Ax−b∥2=∥QT ⁣(Ax−b)∥2=∥[R10]x−[Q1TQ2T]b∥2=∥R1x−Q1Tb∥2+∥Q2Tb∥2.\begin{aligned} +\|Ax -b\|^2 +& = \| Q ^T\! ( Ax -b)\|^2\\ +&=\left\|\begin{bmatrix} R_1\\0\end{bmatrix} x -\begin{bmatrix} Q_1^T\\Q_2^T\end{bmatrix}b\right\|^2\\ +& = \|R_1x - Q_1^T b\|^2+\|Q_2^T b\|^2. +\end{aligned} +

Hence, minimizing ∥R1x−Q1Tb∥\|R_1 x - Q_1^T b\| also minimizes ∥Ax−b∥\|Ax-b\|. Because A A is full rank, R1R_1 is nonsingular, and the least-squares solution is obtained as the unique solution of the system

+R1x=Q1Tb. + R_1 x = Q_1^T b. + +

The procedure for solving least-squares problems via the QR factorization can then be summarized as follows:

+

Algorithm: (Least-squares via QR)

+
    +

  • compute the thin QR factorization A=Q1R1A = Q_1 R_1

    +
    • +

  • backsolve the triangular linear system R1x=Q1TbR_1 x = Q_1^T b

    +
    • +
+

Here's a simple random example in Julia:

+
m, n = 4, 3
+A = randn(m, n)
+b = randn(m)
+F = qr(A)
+Q₁, R₁ = Matrix(F.Q), F.R
+x = R₁ \ Q₁'b   # <--- do NOT be tempted to use x = inv(R₁)*Q₁'b
+

and we can verify that x is the least-squares solution by verifying that the residual r=b−Axr=b-Ax is orthogonal to the columns of AA:

+
r = b - A*x
+@show norm(A'r)
norm(A' * r) = 3.820983142562135e-15
+
+

The last line of the derivation shown in (3) asserts that the norm of the residual is equal to the norm of Q2TbQ_2^T b. Let's check:

+
Q₂ = F.Q[:,n+1:end]
+norm(Q₂'b) ≈ norm(r)
true
+

Accuracy of QR versus normal equations

+

A solution x∗x^* of a full-rank least-squares problem (2) can be obtained as the solution of the normal equations:

+
AT ⁣Ax=AT ⁣b. + A^T\! A x = A^T\! b. +
+

This approach, however, can in some cases suffer from numerical instability, which means that floating-point computations used by most computers yield solutions with unacceptable errors. The QR approach, on the other hand, often allows us to solve a wider-range of problems. The next example, borrowed from the course on Fundamentals of Numerical Computation, illustrates this point.

+

The Pythagorean identity asserts that

+
sin⁡2(θ)+cos⁡2(θ)=1 +\sin^2(θ) + \cos^2(θ) = 1 +
+

for all values of θ\theta. This implies that the matrix

+
[sin⁡2(θ1)cos⁡2(θ1)1⋮⋮⋮sin⁡2(θm)cos⁡2(θm)1], + \begin{bmatrix} \sin^2(\theta_1) & \cos^2(\theta_1) & 1 + \\ \vdots & \vdots & \vdots + \\\sin^2(\theta_m) & \cos^2(\theta_m) & 1\end{bmatrix}, +
+

for any values θ1,…,θm\theta_1,\ldots,\theta_m, doesn't have full column rank. Let's define a slightly perturbed version of this matrix that does have full column rank:

+
θ = LinRange(0,3,400)
+ε = 1e-7
+A = @. [sin(θ)^2   cos(θ+ε)^2   θ^0]
+

We can check that this matrix has full column rank:

+
rank(A) == 3
true
+

Now create a right-hand side that corresponds to a known solution xᵉ:

+
xᵉ = [1., 2., 1.]
+b  = A*xᵉ
+

Solution via the normal equations

+

Here's the solution obtained via the normal equations, and its corresponding relative error:

+
xⁿ = A'A \ A'b
+eⁿ = norm(xᵉ - xⁿ) / norm(xᵉ)
+@show xⁿ
+@show eⁿ
xⁿ = [1.0769230768721778, 2.0769230770796003, 0.9230769230769231]
+eⁿ = 0.054392829346937996
+
+

This solution has only about 1 digit of accuracy.

+

Solution via QR factorization

+

Here's the solution obtained via the QR factorization, and its corresponding relative error:

+
Q, R = qr(A); Q = Matrix(Q)
+xʳ = R \ (Q'b)
+eʳ = norm(xᵉ - xʳ) / norm(xᵉ)
+@show xʳ
+@show eʳ
xʳ = [0.9999999962497133, 1.9999999962497135, 1.000000003750286]
+eʳ = 2.65185298142961e-9
+
+

Clearly, this solution is much more accurate than that obtained via the normal equations. The cost is essentially the same, so there's no real downside to using QR. In general, we can use QR implicitly via the backslash operator, which organizes its computations in a slightly different way, and obtains even better accuracy:

+
xᵇ = A \ b
+eᵇ = norm(xᵉ - xᵇ) / norm(xᵉ)
+@show xᵇ
+@show eᵇ
xᵇ = [0.9999999998372602, 1.9999999998372604, 1.0000000001627405]
+eᵇ = 1.1507453610876192e-10
+
+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + + + + + + + + + diff --git a/notes/regularized-least-squares/index.html b/notes/regularized-least-squares/index.html new file mode 100644 index 0000000..1af2632 --- /dev/null +++ b/notes/regularized-least-squares/index.html @@ -0,0 +1,308 @@ + + + + + + + + + + + + + + + + + + Regularized Least Squares + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

Regularized Least Squares

+ + + Regularization introduces to an optimization problem prior information or assumptions about properties of an optimal solution. A regularization parameter governs the tradeoff between the fidelity to the unmodified problem and the desired solution properties. + +

The Pareto frontier

+

The standard least-squares problem can be interpreted as a method for recovering some underlying object (say, a signal or image) encoded in the vector x0x_0 using noisy measurements of the form

+b:=Ax0+w1, + b := Ax_0 + w_1, + +

where ww is random noise and the matrix AA describes the measurement process. The least-squares problem

+
min⁡x∈Rn ∥Ax−b∥2. + \min_{x\in\mathbb R^n}\ \|Ax-b\|^2. +
+

seeks a solution xx under the implicit assumption that the noise term w1w_1 is small.

+

But what if xx also needs to satisfy some other competing objective? Suppose, for example, that we have also available other set of measurements of x0x_0 of the form d=Cx0+w2d = Cx_0+w_2, where w2w_2 is also random noise. The "best" choice for xx is not necessarily (1), and instead we wish to choose xx to balance the two objective values

+
f1(x)=∥Ax−b∥2andf2(x)=∥Cx−d∥2. +f_1(x) = \|Ax - b\|^2 \quad\text{and}\quad f_2(x) = \|Cx - d\|^2. +
+

Generally, we can make f1(x)f_1(x) or f2(x)f_2(x) small, but not both. The figure below sketches the relationship between the pair {f1(x),f2(x)}\{f_1(x), f_2(x)\} for all values of xx.

+ +

The objective pairs on the boundary of two regions is the Pareto frontier. We can compute these Pareto optimal solutions xx by minimizing the weighted sum objective

+f1(x)+γf2(x)=∥Ax−g∥2+λ∥Cx−d∥2, +f_1(x)+\gamma f_2(x) = \|Ax-g\|^2+ λ\|Cx-d\|^2, + +

where the positive parameter λ\lambda provides the relative weight between the objectives.

+

For fixed scalars λλ and αα, the set

+ℓ(λ,α)={ (f1(x),f2(x))∣f1(x)+λf2(x)=α, x∈Rn } +\ell(\lambda,\alpha) = \{\ (f_1(x),f_2(x)) \mid f_1(x) + \lambda f_2(x) = \alpha,\ x \in \mathbb R^n\ \} + +

forms the graph of a line with slope of −λ-\lambda. We may visualize the optimization problem (2) as the problem of finding the smallest value α\alpha such that the line is tangent to the Pareto frontier.

+ +

Tikhonov regularization

+

A particularly common example of regularized least-squares is Tikhonov, which has the form

+
min⁡x12∥Ax−b∥2+λ12∥Dx∥2, +\min_{x} \tfrac12\|Ax-b\|^2 + \lambda\tfrac12\|Dx\|^2, +
+

for which the objective can be expressed as

+∥Ax−b∥2+λ∥Dx∥2=∥[AλD]x−[b0]∥2. +\|Ax-b\|^2+\lambda\|Dx\|^2 + = \bigg\|\begin{bmatrix} A\\\sqrt{\lambda}D\end{bmatrix} x - \begin{bmatrix} b\\ 0\end{bmatrix}\bigg\|^2. + +

If DD has full column rank, then the stacked matrix

+[AλD] +\begin{bmatrix} A\\\sqrt{\lambda}D\end{bmatrix} + +

necessarily also has full rank for any positive λ\lambda, which implies that the regularized problem always has a well-defined unique solution.

+

Example: Signal denoising

+
⚠ Note
+

This example uses the following packages:

+
using LinearAlgebra
+using Plots
+

Consider a noisy measurement

+b:=x♮+w +b := x^\natural + w + +

of a signal x♮x^\natural, where the vector ww represents unknown noise. Here's a simple 1-dimensional noisy signal:

+
n = 300
+t = LinRange(0, 4, n)
+x = @. sin(t) + t*cos(t)^2
+w = 0.1*randn(n)
+b = x + w
+plot(b, leg=:topleft, label="noisy signal")
+ +

The obvious least-squares problem

+min⁡x 12∥x−b∥2 +\min_{x}\ \tfrac12\|x-b\|^2 + +

isn't useful because the optimal solution is simply the noisy measurements bb, and so it doesn't yield any new information. But suppose we believe that the signal is "smooth" in the sense that the consecutive elements of

+x=(x1,…,xi,xi+1,…,xn) x=(x_1,\ldots,x_i,x_{i+1},\ldots,x_n) +

change relatively little, i.e., the difference ∣xi−xi+1∣|x_i-x_{i+1}| is small relative to xx. In this case, we might balance the least-squares fit against the smoothness of the solution by instead solving the regularized least-squares problem

+min⁡x 12∑i=1∥x−b∥2⏟f1(x)+λ12∑i=1n−1(xi−xi+1)2⏟f2(x). + \min_{x}\ \tfrac12\underbrace{\vphantom{\sum_{i=1}}\|x-b\|^2}_{f_1(x)} + λ\tfrac12 \underbrace{\sum_{i=1}^{n-1}(x_i - x_{i+1})^2}_{f_2(x)}. + +

The role of the regularizer f2(x)f_2(x) is to promote smooth changes in the elements of xx.

+

Matrix notation

+

We can alternatively write the above minimization program in matrix notation. Define the (n−1)(n-1)-by-nn finite difference matrix

+D=[1−1+0⋯+0+00+1−10⋯+0⋱⋱⋱⋮+01−1+00⋯0+1−1], +D = \begin{bmatrix} 1 & -1 & \phantom+0 & \cdots & \phantom+0 & \phantom+0\\ + 0 & \phantom+1 & -1 & 0 & \cdots & \phantom+0\\ + &\ddots & \ddots & \ddots & & \\ + \vdots & & \phantom+0 & 1 & -1 & \phantom+0\\ + 0 & \cdots & & 0 & \phantom+1 & -1\end{bmatrix}, + +

which when applied to a vector xx, yields a vector of the differences:

+Dx=[x1−x2x2−x3⋮xn−1−xn]. +Dx = \begin{bmatrix} x_1 - x_2 \\ x_2 - x_3 \\ \vdots \\ x_{n-1} - x_n\end{bmatrix}. + +

Then we can rephrase the regularization objective as

+f2(x)=∑i=1n−1(xi−xi+1)2=∥Dx∥2. f_2(x) = \sum_{i=1}^{n-1}(x_i - x_{i+1})^2 = \|Dx\|^2. +

This allows for a reformulation of the weighted leas squares objective into a familiar least squares objective:

+∥x−b∥2+λ∥Dx∥2=∥[IλD]⏟A^x−[b0]⏟b^∥2. +\|x-b\|^2+\lambda\|Dx\|^2 = \bigg\|\underbrace{\begin{bmatrix} I\\\sqrt{\lambda}D\end{bmatrix}}_{\hat{A}}x - +\underbrace{\begin{bmatrix} b\\ 0\end{bmatrix}}_{\hat{b}}\bigg\|^2. + +

So the solution to the weighted least squares minimization program (9) satisfies the normal equation A^T ⁣A^x=A^T ⁣b^\hat{A}^T\!\hat{A}x = \hat{A}^T\!\hat{b}, which simplifies to

+(I+λDT ⁣D)x=b. +(I + \lambda D^T\! D)x = b. + +

The following function generates the required finite-difference matrix:

+
finiteDiff(n) = diagm(0 => ones(n), +1 => -ones(n-1))[1:n-1,:]
+

Here's an example of a small finite-difference matrix:

+
finiteDiff(4)
+
3×4 Matrix{Float64}:
+ 1.0  -1.0   0.0   0.0
+ 0.0   1.0  -1.0   0.0
+ 0.0   0.0   1.0  -1.0
+

The function below returns a generator, which produces nn logarithmically-spaced numbers in the interval [x1,x2][x_1,x_2]:

+
LogRange(x1, x2, n) = (10^y for y in range(log10(x1), log10(x2), length=n))
+

Now solve the regularized least-squares problem for several values of λ∈[1,104]\lambda\in[1,10^4]:

+
D = finiteDiff(n)
+b̂ = [b; zeros(n-1)]
+plot(t, b, w=1, leg =:topleft, label="noisy data")
+for λ in LogRange(1e0, 1e4, 3) 
+    Â = [ I; √λ*D ]
+    xLS = Â \ b̂
+    plot!(t, xLS, w=2, label="regularized solution: λ = $(λ)")
+end
+

Note that the expression  = [ I; √λ*D ] contains the UniformScaling object, which represents an identity matrix of any size.

+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
+
+
+ + + + + + + + + + + + + + + + + + + + diff --git a/notes/scaled-and-newton-descent.jl b/notes/scaled-and-newton-descent.jl new file mode 100644 index 0000000..bf442b1 --- /dev/null +++ b/notes/scaled-and-newton-descent.jl @@ -0,0 +1,1130 @@ +### A Pluto.jl notebook ### +# v0.17.7 + +using Markdown +using InteractiveUtils + +# ╔═╡ c76e91d8-ae68-4f31-a461-f7a75522eebc +begin + using ForwardDiff + using Plots + using LinearAlgebra + using Optim +end + +# ╔═╡ ac0e09f2-50a2-46a4-92c2-2796f06961ca +md""" +# Newton's method +""" + +# ╔═╡ b20108ba-29af-4424-bb01-a0839ca413a6 + + +# ╔═╡ 49ed0371-910f-488a-82fd-e9eb86c35af1 + + +# ╔═╡ 6d0e8aec-6c68-4e08-ad22-acb1fb594a6c + + +# ╔═╡ e13046a5-9b57-4523-8b7c-aacbb2b8b15b + + +# ╔═╡ 4378d633-becd-4596-9cbd-4cace22fb397 + + +# ╔═╡ 526e272d-8aa5-4018-85f3-735cf16d1ac7 +md""" +# Appendix +--- +""" + +# ╔═╡ 94025fff-a6c9-434c-a5a0-96e9cf2db30c +begin + struct State + x::Vector + f + norm∇f + end + State(x::Vector, f::Real, ∇f::Vector) = State(x, f, norm(∇f)) +end + +# ╔═╡ 50eb6025-cfe3-47b8-8e4d-475e7c552971 +# Gradient method with backtracking +function grad_method_backtracking(f, g, x0; ϵ = 1e-6, μ = 1e-4, maxits = length(x0)) + x = copy(x0) + fk = f(x); ∇fk = g(x) + k = 0 + trace = [State(x, fk, ∇fk)] + while norm(∇fk) > ϵ && k < maxits + α = 1.0 + while ((fk - f(x-α*∇fk)) < μ*α*dot(∇fk,∇fk) ) + α /= 2 + end + x = x - α*∇fk + fk = f(x); ∇fk = g(x) + k += 1 + push!(trace, State(x, fk, ∇fk)) + end + return x, trace +end + +# ╔═╡ eed36606-8602-11ec-360b-f7b6701b25b8 +# Newton's method with backtracking +function newton_backtracking(f, g, H, x0; ϵ = 1e-6, μ = 1e-4, maxits = length(x0)) + x = copy(x0) + fk = f(x); ∇fk = g(x) + k = 0 + trace = [State(x, fk, ∇fk)] + while norm(∇fk) > ϵ && k < maxits + α = 1.0 + Hk = H(x) + if isposdef(Hk) + d = -Hk\∇fk + else + d = -∇fk + end + while ((fk - f(x+α*d)) < μ*α*dot(∇fk,d) ) + α /= 2 + end + x = x + α*d + fk = f(x); ∇fk = g(x) + k += 1 + push!(trace, State(x, fk, ∇fk)) + end + return x, trace +end + +# ╔═╡ d3f596c1-c5f3-417b-9e55-2b5778ca8764 +begin + #Apply gradient method with backtracking to Rosenbrock function + f(x) = 100(x[2]-x[1]^2)^2+(1-x[1])^2 + ∇f(x) = ForwardDiff.gradient(f,x) + ∇2f(x) = ForwardDiff.hessian(f,x) + x0 = [2.0, 5.0] + nx, nxtrace = newton_backtracking(f, ∇f, ∇2f, x0, μ = 1e-4, maxits = 1000); + gx, gxtrace = grad_method_backtracking(f, ∇f, x0, μ = 1e-4, maxits = 1000); +end + +# ╔═╡ 26c8527f-baed-4e21-a1f5-479301644745 +optimize(f, x0, LBFGS(); autodiff = :forward) + +# ╔═╡ 18e749d4-e9f4-4f6f-b5be-49012cf719bd +collectf(states::Vector{State}) = [s.f for s in states] + +# ╔═╡ 6d39f8bf-adf3-442b-ad6d-c6beb0617678 +begin + plot(collectf(nxtrace).+eps(), yscale=:log10, label="Newton") + #plot!(collectf(gxtrace).+eps(), yscale=:log10, label="gradient descent") +end + +# ╔═╡ 7e741d3e-0e71-4889-8380-47d0d76e5b87 +collectg(states::Vector{State}) = [s.norm∇f for s in states] + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Optim = "429524aa-4258-5aef-a3af-852621145aeb" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" + +[compat] +ForwardDiff = "~0.10.25" +Optim = "~1.6.1" +Plots = "~1.25.7" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.7.1" +manifest_format = "2.0" + +[[deps.Adapt]] +deps = ["LinearAlgebra"] +git-tree-sha1 = "af92965fb30777147966f58acb05da51c5616b5f" +uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" +version = "3.3.3" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" + +[[deps.ArrayInterface]] +deps = ["Compat", "IfElse", "LinearAlgebra", "Requires", "SparseArrays", "Static"] +git-tree-sha1 = "745233d77146ad221629590b6d82fe7f1ddb478f" +uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9" +version = "4.0.3" + +[[deps.Artifacts]] 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https://friedlander.io/ubc-cpsc-406-22-23/homework/hw4/ + 2024-01-09 + monthly + 0.5 + + diff --git a/slides/introduction.html b/slides/introduction.html new file mode 100644 index 0000000..7d94daf --- /dev/null +++ b/slides/introduction.html @@ -0,0 +1,180 @@ +
+

Computational Optimization

+

CPSC 406, Department of Computer Science

+

Professor Michael P. Friedlander

+ +
+
+

Course goals and emphasis

+
    +
  • recognize and formulate the main optimization problem classes
    • +
  • understand how to apply standard algorithms for each class
    • +
  • recognize that an algorithm has succeeded or failed
    • +
  • hands-on experience with mathematical software
    • +
+ +
+
+

Role of optimization

+
    +
  • fitting a statistical model to data (machine learning)
    • +
  • logistics, economics, finance, risk management
    • +
  • theory of games and competition
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  • theory of computer science and algorithms
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  • geometry and analysis
    • +
+ +
+
+

Learning models

+

a model is a simplified abstraction of process

+
    +
  • model parameters
    • +
  • prediction
    • +
+

least-error principle: the optimal parameters minimizes the distance between the model and the observation

+ +
+
+

Mathematical Optimization

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    +
  • Objective function:
    • +
  • feasible set (eg, "constraints"):
    • +
  • decision variables:
    • +
+ +
+
+

Abstract problem

+
    +
  • find such that is minimal, eg,
    • +
+

    +
  • optimal solution set
    • +
+

+
+
+

Gradients wanted

+

Assume the objective is differentiable on the nonempty interior of the feasible set

+

    +
  • measures the objective's sensitivity to feasible perturbations
    • +
  • usually sufficient to devise tractable and implementable algorithms
    • +
+ +
+
+

Example: linear models

+

    +
  • least-squares approximation: minimizes

    • +
  • least absolute-sum approximation: no closed-form solution for minimizing

    • +
+ +
+
+

Example: Scheduling

+

Minimize number nurses needed to meet weekly staffing demands

+

Constraints

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    +
  • each nurse works 5 straight days with 2 days off
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  • nurses required on nights
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+

First attempt

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    +
  • nurses work on night
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  • minimize subject to with
    • +
  • doesn't respect days off constraints
    • +
+ +
+
+

Scheduling: second attempt

+

let be number of nurses starting their 5-day shift on day :

+

    +
  • note the constraint structure. This is almost always true of practical LPs
    • +
  • we may want to restrict to be integer. That's a much harder problem!
    • +
+ +
+
+

Coursework and evaluation

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    +
  • 8 homework assignments (30%) +
      +
    • programming and mathematical deriviations
      • +
    • typeset submissions, correctness, and writing quality graded
      • +
    +
    • +
  • midterm exam (30%): ✏️️ and 🗞️, short mathematical problems
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  • final exam (40%): multiple choice
    • +
+ +
+
+

Programming in Julia

+
    +
  • common programming languages in optimization: Python, Matlab, R; C++
    • +
  • we'll use Julia only +
      +
    • Matlab-like syntax, but free and fast
      • +
    • lots of tutorials available
      • +
    +
    • +
  • Pluto ❤️ or Jupyter or ️notebooks highly recommended for assignments
    • +
+ +
+
+

Assignments

+
    +
  • work alone or collaborate in pairs 👯 +
      +
    • hand in your own assignment and list collaborators
      • +
    +
    • +
  • 3 late days allowed (no permission required) +
      +
    • but no more than 2 late days for a particular assignment
      • +
    +
    • +
  • no solutions posted online +
      +
    • come to office hours to see solutions
      • +
    +
    • +
  • submit solutions to Crowdmark
    • +
+ +
+
+

Homework 1

+
    +
  • no late days, no collaborators
    • +
  • hints at required background
    • +
  • due next week
    • +
+ +
+
+

Resources

+
    +
  • see the course home page for schedule
    • +
  • announcements and discussions on Piazza
    • +
  • TA and instructor office hours start week 2
    • +
+ +
+
\ No newline at end of file diff --git a/slides/least-squares.html b/slides/least-squares.html new file mode 100644 index 0000000..400ac1c --- /dev/null +++ b/slides/least-squares.html @@ -0,0 +1,60 @@ +
+

Linear Least Squares

+ +
    +
  • formulate linear least-squares problems
    • +
  • derive optimality conditions
    • +
  • interpret as a projection
    • +
  • solution via QR factorization
    • +
+ +
+
+

Fitting a line

+
    +
  • pairs of data points
    • +
  • affine model of the data generation process:

    • +
  • choose parameters (intercept) and (slope) that minimize least-squares errors ("residuals"):

    with

    • +
+ +
+
+

Matrix formulation

+
    +
  • observations, model variables
    • +
  • and
    • +
  • least-squares formulation
    • +
+

    +
  • overdetermined ()
    • +
+ +
+
+

Orthogonal projection

+

The orthogonal projection of a vector onto a linear subspace :

+

Let be a least-squares solution to

+

    +
  • is the orthogonal projection of onto
    • +
  • is the orthogonal projection of onto
    • +
+ +
+
+

Projection and normal equations

+ +
+
\ No newline at end of file diff --git a/syllabus.csv b/syllabus.csv new file mode 100644 index 0000000..78d154e --- /dev/null +++ b/syllabus.csv @@ -0,0 +1,30 @@ +" Date "," Lecture Topic "," Readings "," Homework " +"Tue, Jan-10"," [Introduction](/slides/introduction.html) "," ","[HW1](/homework/hw1) out" +"Thu, Jan-12"," [Linear least-squares](/notes/least-squares) ","Sections 3.1--3.2 of [Beck][BeckLink]; [Least-squares notes](https://ubcmath.github.io/MATH307/orthogonality/least_squares.html) of UBC Math 307"," " +"Tue, Jan-17"," [QR Factorization](/notes/qr-factorization) "," ","[HW1](/homework/hw1) due" +"Thu, Jan-19"," [Regularized least-squares](/notes/regularized-least-squares) "," ","[HW2](/homework/hw2) out" +"Tue, Jan-24"," [Gradients](/notes/gradients) ","Section 1.5.1 of [Beck][BeckLink]; Chapter 2 of [Kochenderfer and Wheeler][KochenderferLink]; [Automatic differentiation from scratch, in 10 lines of Julia](https://vladium.com/tutorials/study_julia_with_me/multiple_dispatch/)"," " +"Thu, Jan-26"," [Nonlinear least-squares](/notes/nonlinear-least-squares) ","Sections 3.5 and 4.5 of [Beck][BeckLink]","[HW2](/homework/hw2) due" +"Tue, Jan-31"," Unconstrained optimization ","Sections 4.1, 4.2, 4.4 of [Beck][BeckLink]","[HW3](/homework/hw3) out" +"Thu, Feb-2"," Unconstrained optimization (cont) "," "," " +"Tue, Feb-7"," [Descent methods](/notes/gradient-descent.pdf), [demo](/notes/gradient-descent) ",,"[HW3](/homework/hw3) due" +"Thu, Feb-9"," [Scaled descent](/notes/scaled-and-newton-descent.pdf) ","Section 4.4 of [Beck][BeckLink]","[HW4](/homework/hw4) out" +"Tue, Feb-14","Descent, continued","Chapter 5 of [Beck][BeckLink]"," " +"Thu, Feb-16"," Newton's method and Cholesky"," ","[HW4](/homework/hw4) due" +"Tue, Feb-21"," Midterm break ",, +"Thu, Feb-23"," Midterm break ",, +"Tue, Feb-28"," [Linear constraints and reduced gradient](/notes/linear-constraints.pdf) "," "," " +"Thu, Mar-2"," Midterm exam (in class) ",, +"Tue, Mar-7"," [Convex sets](/notes/convex-sets.pdf) ","Sections 6.1—6.4 of [Beck][BeckLink]","[HW5](/homework/hw5) out" +"Thu, Mar-9"," [Convex functions](/notes/convex-functions.pdf) ","Sections 7.1—7.6 of [Beck][BeckLink]", +"Tue, Mar-14"," [Convex optimality](/notes/normal-cone-optimality.pdf) "," ","[HW5](/homework/hw5) due" +"Thu, Mar-16"," [Projected and proximal gradient](/notes/projection.pdf) "," ", +"Tue, Mar-21"," [Convergence gradient descent](/notes/convergence-gradient-descent.pdf) "," ","[HW6](/homework/hw6) out" +"Thu, Mar-23"," [Stochastic gradient descent](/notes/stochastic-gradient-descent.pdf) ","Sections 6.4 and 8.2 of [Beck][BeckLink]", +"Tue, Mar-28"," [LP intro](/notes/lin-prog-apps.pdf) ",,"HW6 due" +"Thu, Mar-30"," [LP geometry](/notes/lp-geometry.pdf) "," ", +"Tue, Apr-4 "," [LP standard form](/notes/lp-standard-form.pdf) "," ","[HW7](/homework/projection-hw7) out" +"Thu, Apr-6 "," [LP simplex](/notes/lp-simplex.pdf), [LP simplex example](/notes/lp-simplex-example.pdf), [LP 2-sided bounds and phase 1](/notes/lp-simplex-bnds-2phase.pdf), [Demo](/notes/lp-jump)","Section 12.3.1 of [Beck][BeckLink]", +"Tue, Apr-11"," [LP duality](/notes/lp-duality.pdf), [Demo](/notes/lp-duality) [LP & QP interior method](/notes/lp-qp-barrier.pdf) "," ","HW7 due, [HW8](/homework/linear-programming) out (not due)" +"Thu, Apr-13"," [Matrix games](/notes/lp-games.pdf) "," ", +"Fri, Apr-21","Final exam, 7pm"," Location: [PHRM 1201](https://learningspaces.ubc.ca/classrooms/PHRM-1201)", \ No newline at end of file diff --git a/syllabus/index.html b/syllabus/index.html new file mode 100644 index 0000000..2a0275e --- /dev/null +++ b/syllabus/index.html @@ -0,0 +1,49 @@ + + + + + + + + + + + + + Syllabus + + +
+
UBC CPSC 406 2022-T2
+ +
+ + + + +

+

Syllabus

+
⚠ Note
+
This tentative schedule corresponds to the semester January to April, 2023.
+
DateLecture TopicReadingsHomework
Tue, Jan-10IntroductionHW1 out
Thu, Jan-12Linear least-squaresSections 3.1–3.2 of Beck; Least-squares notes of UBC Math 307
Tue, Jan-17QR FactorizationHW1 due
Thu, Jan-19Regularized least-squaresHW2 out
Tue, Jan-24GradientsSection 1.5.1 of Beck; Chapter 2 of Kochenderfer and Wheeler; Automatic differentiation from scratch, in 10 lines of Julia
Thu, Jan-26Nonlinear least-squaresSections 3.5 and 4.5 of BeckHW2 due
Tue, Jan-31Unconstrained optimizationSections 4.1, 4.2, 4.4 of BeckHW3 out
Thu, Feb-2Unconstrained optimization (cont)
Tue, Feb-7Descent methods, demoHW3 due
Thu, Feb-9Scaled descentSection 4.4 of BeckHW4 out
Tue, Feb-14Descent, continuedChapter 5 of Beck
Thu, Feb-16Newton's method and CholeskyHW4 due
Tue, Feb-21Midterm break
Thu, Feb-23Midterm break
Tue, Feb-28Linear constraints and reduced gradient
Thu, Mar-2Midterm exam (in class)
Tue, Mar-7Convex setsSections 6.1—6.4 of BeckHW5 out
Thu, Mar-9Convex functionsSections 7.1—7.6 of Beck
Tue, Mar-14Convex optimalityHW5 due
Thu, Mar-16Projected and proximal gradient
Tue, Mar-21Convergence gradient descentHW6 out
Thu, Mar-23Stochastic gradient descentSections 6.4 and 8.2 of Beck
Tue, Mar-28LP introHW6 due
Thu, Mar-30LP geometry
Tue, Apr-4LP standard formHW7 out
Thu, Apr-6LP simplex, LP simplex example, LP 2-sided bounds and phase 1, DemoSection 12.3.1 of Beck
Tue, Apr-11LP duality, Demo LP & QP interior methodHW7 due, HW8 out (not due)
Thu, Apr-13Matrix games
Fri, Apr-21Final exam, 7pmLocation: PHRM 1201
+ +
+
+ © Michael P. Friedlander | Last modified: January 09, 2024. +
Website built with Franklin.jl and the Julia programming language. +
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