diff --git a/notebooks/k_means.livemd b/notebooks/k_means.livemd index dd8aa3ff..f95ccb93 100644 --- a/notebooks/k_means.livemd +++ b/notebooks/k_means.livemd @@ -12,7 +12,8 @@ Mix.install([ {:scidata, "~> 0.1.10"}, {:req, "~> 0.3.9"}, {:kino, "~> 0.10.0"}, - {:kino_vega_lite, "~> 0.1.9"} + {:kino_vega_lite, "~> 0.1.9"}, + {:tucan, github: "pnezis/tucan", ref: "6549cdc"} ]) ``` @@ -27,7 +28,6 @@ Mix.install([ The main purpose of this livebook is to introduce the **KMeans** clustering algorithm. We will explore KMeans in three different use cases. ```elixir -alias VegaLite, as: Vl alias Scholar.Cluster.KMeans require Explorer.DataFrame, as: DF Nx.global_default_backend(EXLA.Backend) @@ -36,14 +36,6 @@ key = Nx.Random.key(42) -``` - -02:30:02.231 [info] TfrtCpuClient created. - -``` - - - ``` #Nx.Tensor< u32[2] @@ -143,15 +135,11 @@ DF.summarise( We see that `petal_length` and `petal_width` are the most distinguishing features. Let's explore them a little bit more. ```elixir -Vl.new(title: [text: "Histograms of petal_length column by species", offset: 25, anchor: :middle]) -|> Vl.data_from_values(df) -|> Vl.facet( - [field: "species"], - Vl.new(width: 200, height: 200) - |> Vl.mark(:bar) - |> Vl.encode_field(:x, "petal_length", type: :quantitative, bin: true) - |> Vl.encode(:y, aggregate: :count, scale: [domain: [0, 55]]) -) +Tucan.histogram(df, "petal_length", color_by: "species") +|> Tucan.facet_by(:column, "species") +|> Tucan.Scale.set_y_domain(0, 55) +|> Tucan.set_size(200, 200) +|> Tucan.set_title("Histograms of petal_length column by species", offset: 25, anchor: :middle) ``` @@ -161,19 +149,11 @@ Vl.new(title: [text: "Histograms of petal_length column by species", offset: 25, ``` ```elixir -Vl.new( - width: 300, - height: 300, - title: [ - text: "Scatterplot of data samples projected on plane petal_width x petal_length", - offset: 25 - ] +Tucan.scatter(df, "petal_length", "petal_width", filled: true, color_by: "species") +|> Tucan.set_size(300, 300) +|> Tucan.set_title("Scatterplot of data samples projected on plane petal_width x petal_length", + offset: 25 ) -|> Vl.data_from_values(df) -|> Vl.mark(:circle) -|> Vl.encode_field(:x, "petal_length", type: :quantitative) -|> Vl.encode_field(:y, "petal_width", type: :quantitative) -|> Vl.encode_field(:color, "species") ``` @@ -183,19 +163,11 @@ Vl.new( ``` ```elixir -Vl.new( - title: [ - text: "Scatterplot of data samples projected on plane petal_width x petal_length by species", - offset: 25 - ] -) -|> Vl.data_from_values(df) -|> Vl.facet( - [field: "species"], - Vl.new(width: 200, height: 200) - |> Vl.mark(:point) - |> Vl.encode_field(:x, "petal_length", type: :quantitative) - |> Vl.encode_field(:y, "petal_width", type: :quantitative) +Tucan.scatter(df, "petal_length", "petal_width") +|> Tucan.facet_by(:column, "species") +|> Tucan.set_title( + "Scatterplot of data samples projected on plane petal_width x petal_length by species", + offset: 25 ) ``` @@ -227,15 +199,12 @@ inertias = for model <- models, do: Nx.to_number(model.inertia) ``` ```elixir -Vl.new(width: 600, height: 300, title: "Elbow Plot") -|> Vl.data_from_values(num_clusters: clusterings, inertia: inertias) -|> Vl.mark(:line) -|> Vl.encode_field(:x, "num_clusters", - title: "Number of Clusters", - type: :ordinal, - axis: [label_angle: 0] +Tucan.lineplot([num_clusters: clusterings, inertia: inertias], "num_clusters", "inertia", + x: [type: :nominal, axis: [label_angle: 0]], + title: "Elbow Plot" ) -|> Vl.encode_field(:y, "inertia", title: "Inertia", type: :quantitative) +|> Tucan.Axes.set_xy_titles("Number of Clusters", "Inertia") +|> Tucan.set_size(600, 300) ``` @@ -298,28 +267,19 @@ coords = [ cluster_petal_width: best_model.clusters[[.., 3]] |> Nx.to_flat_list() ] -Vl.new( - width: 300, - height: 300, - title: [ - text: - "Scatterplot of data samples projected on plane petal_width x petal_length with calculated centroids", - offset: 25 - ] -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(df) - |> Vl.mark(:circle) - |> Vl.encode_field(:x, "petal_length", type: :quantitative) - |> Vl.encode_field(:y, "petal_width", type: :quantitative) - |> Vl.encode_field(:color, "species"), - Vl.new() - |> Vl.data_from_values(coords) - |> Vl.mark(:circle, color: :green, size: 100) - |> Vl.encode_field(:x, "cluster_petal_length", type: :quantitative) - |> Vl.encode_field(:y, "cluster_petal_width", type: :quantitative) +Tucan.layers([ + Tucan.scatter(df, "petal_length", "petal_width", color_by: "species", filled: true), + Tucan.scatter(coords, "cluster_petal_length", "cluster_petal_width", + filled: true, + point_size: 100, + point_color: "green" + ) ]) +|> Tucan.set_size(300, 300) +|> Tucan.set_title( + "Scatterplot of data samples projected on plane petal_width x petal_length with calculated centroids", + offset: 25 +) ``` @@ -1203,19 +1163,14 @@ data = [ inertia: for(model <- models, do: Nx.to_number(model.inertia)) ] -Vl.new(width: 600, height: 300, title: "Elbow Plot") -|> Vl.data_from_values(data) -|> Vl.mark(:line) -|> Vl.encode_field(:x, "num_clusters", - title: "Number of Clusters", - type: :ordinal, - axis: [label_angle: 0] -) -|> Vl.encode_field(:y, "inertia", - title: "Inertia", - type: :quantitative, - scale: [domain: [4800, 11500]] +Tucan.lineplot(data, "num_clusters", "inertia", + x: [type: :ordinal, axis: [label_angle: 0]], + width: 600, + height: 300 ) +|> Tucan.Axes.set_xy_titles("Number of Clusters", "Inertia") +|> Tucan.Scale.set_y_domain(4800, 11500) +|> Tucan.set_title("Elbow Plot") ``` @@ -1247,19 +1202,15 @@ silhouette_scores = ```elixir data = [num_clusters: nums_clusters, silhouette_scores: silhouette_scores] -Vl.new(width: 600, height: 300, title: "Silhouette score vs Number of Clusters") -|> Vl.data_from_values(data) -|> Vl.mark(:line, point: [point: true, color: :dark_blue]) -|> Vl.encode_field(:x, "num_clusters", - title: "Number of Clusters", - type: :ordinal, - axis: [label_angle: 0] -) -|> Vl.encode_field(:y, "silhouette_scores", - title: "Silhouette score", - type: :quantitative, - scale: [domain: [0.088, 0.205]] +Tucan.lineplot(data, "num_clusters", "silhouette_scores", + points: true, + point_color: "darkBlue", + x: [type: :ordinal, axis: [label_angle: 0]] ) +|> Tucan.Axes.set_xy_titles("Number of Clusters", "Silhouette score") +|> Tucan.Scale.set_y_domain(0.088, 0.205) +|> Tucan.set_size(600, 300) +|> Tucan.set_title("Silhouette score vs Number of Clusters") ``` diff --git a/notebooks/k_nearest_neighbors.livemd b/notebooks/k_nearest_neighbors.livemd index fe7dc492..10d8ab9f 100644 --- a/notebooks/k_nearest_neighbors.livemd +++ b/notebooks/k_nearest_neighbors.livemd @@ -1,1048 +1,872 @@ - - -# K-Nearest Neighbors - -```elixir -Mix.install([ - {:scholar, "~> 0.2.0"}, - {:explorer, "~> 0.6.1"}, - {:exla, "~> 0.6.0"}, - {:nx, "~> 0.6.0", override: true}, - {:req, "~> 0.3.9"}, - {:kino_vega_lite, "~> 0.1.9"}, - {:kino, "~> 0.10.0"}, - {:scidata, "~> 0.1.10"}, - {:kino_explorer, "~> 0.1.7"} -]) -``` - - - -``` -:ok -``` - -## Setup - -We will extensively use VegaLite, Explorer, and Scholar throughout this guide, so let's define some aliases: - -```elixir -alias VegaLite, as: Vl -require Explorer.DataFrame, as: DF -require Explorer.Series, as: S -require Explorer.Query, as: Q -alias Scholar.Neighbors.KNearestNeighbors, as: KNN -alias Scholar.Metrics.Classification -alias Scholar.Metrics.Regression -``` - - - -``` -Scholar.Metrics.Regression -``` - -And let's configure `EXLA.Backend` as our default across the notebook and all branched sections: - -```elixir -Nx.global_default_backend(EXLA.Backend) -seed = 42 -key = Nx.Random.key(42) -``` - - - -``` - -02:33:33.173 [info] TfrtCpuClient created. - -``` - - - -``` -#Nx.Tensor< - u32[2] - EXLA.Backend - [0, 42] -> -``` - -## Introduction - -This notebook will cover the three primary applications of K-Nearest Neighbors: classification, regression, and -anomaly detection. - - - -## Classification - -The process of classification using KNN is relatively straightforward. We start by working with a dataset where each sample is assigned a label. To predict the label of a new sample, we compute the distance between it and all the other samples in the dataset. Next, we select only the $k$ closest samples based on the distance metric, where $k$ is a user-defined parameter. We then examine the labels of these $k$ samples and choose the label that appears most frequently. This label is assigned as the predicted label for the new sample. - -Let's first grasp how kNN works visually with a small example. - -```elixir -data = - DF.new( - x: [-1, 0.2, -0.5, -2.1, -2.3, -2.2, 0.1, 0.3, 0.4, 0.7, 1.3], - y: [-0.1, 0.4, 0.5, 0.4, 1.1, -1.0, -0.1, 0.2, 1.5, 1.6, 0.9], - label: [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1] - ) - -point_to_predict = DF.new(x: [0.0], y: [0.0]) - -k_eq_3 = DF.new(x: [0.45], y: [0.45], name: ["k = 3"]) -k_eq_5 = DF.new(x: [0.72], y: [0.72], name: ["k = 5"]) - -Vl.new( - title: [ - text: "Scatterplot showing KNN prediction process", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(data) - |> Vl.mark(:point, size: 200, filled: true) - |> Vl.encode_field(:shape, "label", type: :nominal) - |> Vl.encode_field(:x, "x", - type: :quantitative, - scale: [domain: [-2.4, 1.5]], - axis: [grid: false] - ) - |> Vl.encode_field(:y, "y", - type: :quantitative, - scale: [domain: [-1.2, 1.8]], - axis: [grid: false] - ) - |> Vl.encode_field(:color, "label", type: :nominal, scale: [scheme: :plasma]), - Vl.new() - |> Vl.data_from_values(point_to_predict) - |> Vl.mark(:point, color: :green, size: 400, shape: :triangle, filled: true) - |> Vl.encode_field(:x, "x", type: :quantitative) - |> Vl.encode_field(:y, "y", type: :quantitative), - Vl.new() - |> Vl.data_from_values(point_to_predict) - |> Vl.mark(:arc, - outer_radius: 99, - inner_radius: 97, - color: :brown, - stroke_dash: [stroke_dash: [5, 5]] - ) - |> Vl.encode_field(:x, "x", type: :quantitative) - |> Vl.encode_field(:y, "y", type: :quantitative), - Vl.new() - |> Vl.data_from_values(point_to_predict) - |> Vl.mark(:arc, - outer_radius: 170, - inner_radius: 168, - color: :blue - ) - |> Vl.encode_field(:x, "x", type: :quantitative) - |> Vl.encode_field(:y, "y", type: :quantitative), - Vl.new() - |> Vl.data_from_values(k_eq_3) - |> Vl.mark(:text, size: 20, color: :brown) - |> Vl.encode_field(:text, "name", type: :nominal) - |> Vl.encode_field(:x, "x", type: :quantitative) - |> Vl.encode_field(:y, "y", type: :quantitative), - Vl.new() - |> Vl.data_from_values(k_eq_5) - |> Vl.mark(:text, size: 20, color: :blue) - |> Vl.encode_field(:text, "name", type: :nominal) - |> Vl.encode_field(:x, "x", type: :quantitative) - |> Vl.encode_field(:y, "y", type: :quantitative) -]) -``` - - - -```vega-lite -{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","height":630,"layer":[{"data":{"values":[{"label":0,"x":-1.0,"y":-0.1},{"label":0,"x":0.2,"y":0.4},{"label":0,"x":-0.5,"y":0.5},{"label":0,"x":-2.1,"y":0.4},{"label":0,"x":-2.3,"y":1.1},{"label":0,"x":-2.2,"y":-1.0},{"label":1,"x":0.1,"y":-0.1},{"label":1,"x":0.3,"y":0.2},{"label":1,"x":0.4,"y":1.5},{"label":1,"x":0.7,"y":1.6},{"label":1,"x":1.3,"y":0.9}]},"encoding":{"color":{"field":"label","scale":{"scheme":"plasma"},"type":"nominal"},"shape":{"field":"label","type":"nominal"},"x":{"axis":{"grid":false},"field":"x","scale":{"domain":[-2.4,1.5]},"type":"quantitative"},"y":{"axis":{"grid":false},"field":"y","scale":{"domain":[-1.2,1.8]},"type":"quantitative"}},"mark":{"filled":true,"size":200,"type":"point"}},{"data":{"values":[{"x":0.0,"y":0.0}]},"encoding":{"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"green","filled":true,"shape":"triangle","size":400,"type":"point"}},{"data":{"values":[{"x":0.0,"y":0.0}]},"encoding":{"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"brown","innerRadius":97,"outerRadius":99,"type":"arc"}},{"data":{"values":[{"x":0.0,"y":0.0}]},"encoding":{"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"blue","innerRadius":168,"outerRadius":170,"type":"arc"}},{"data":{"values":[{"name":"k = 3","x":0.45,"y":0.45}]},"encoding":{"text":{"field":"name","type":"nominal"},"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"brown","size":20,"type":"text"}},{"data":{"values":[{"name":"k = 5","x":0.72,"y":0.72}]},"encoding":{"text":{"field":"name","type":"nominal"},"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"blue","size":20,"type":"text"}}],"title":{"offset":20,"text":"Scatterplot showing KNN prediction process"},"width":630} -``` - -Note that the final prediction may vary depending on the value of $k$. For example, with $k = 3$, we would predict the green triangle as an orange square (1), whereas with $k = 5$, the purple circle (0) would be the final prediction. - -Let's now test this using Scholar code. First, we will define our data. - -```elixir -x = Nx.stack(DF.discard(data, "label"), axis: 1) -labels = Nx.stack(DF.select(data, "label"), axis: 1) |> Nx.squeeze(axes: [1]) -x_pred = Nx.stack(point_to_predict, axis: 1) -``` - - - -``` -#Nx.Tensor< - f64[1][2] - EXLA.Backend - [ - [0.0, 0.0] - ] -> -``` - -Let's now try with $k = 3$. - -```elixir -model = KNN.fit(x, labels, num_classes: 2, num_neighbors: 3) -KNN.predict(model, x_pred) -``` - - - -``` -#Nx.Tensor< - s64[1] - EXLA.Backend - [1] -> -``` - -And $k = 5$. - -```elixir -model = KNN.fit(x, labels, num_classes: 2, num_neighbors: 5) -KNN.predict(model, x_pred) -``` - - - -``` -#Nx.Tensor< - s64[1] - EXLA.Backend - [0] -> -``` - -As we can see, the predictions match our intuition from analyzing the plot. - -Now, let's try KNN on a more complicated dataset: the Wine dataset. Our task will be to classify the quality of the wine. Before we load the dataset into `Explorer.DataFrame` for more efficient exploration, let's check some more detailed information about the dataset. - -```elixir -info = - Req.get!( - "https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality.names" - ).body - -Kino.Markdown.new(info) -``` - -```elixir -data = - Req.get!( - "https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality-white.csv" - ).body - -df_data = DF.load_csv!(data, delimiter: ";", dtypes: %{"total sulfur dioxide": :float}) -tensor_data = Nx.stack(df_data, axis: 1) -df_data -``` - -As we can see, there are no null values in the dataset. Now, let's check the size of the dataset. - -```elixir -DF.shape(df_data) -``` - - - -``` -{4898, 12} -``` - -Let's check some statistical properties of the dataset. We will start with *skewness*, which measures the asymmetry of the probability distribution of a random variable about its mean. To better understand this concept, please take a look at the picture below. - -| ![Skewness](https://upload.wikimedia.org/wikipedia/commons/c/cc/Relationship_between_mean_and_median_under_different_skewness.png) | -| :--------------------------------------------------------------------------------------------------------------------------------: | -| Figure 1: A general relationship of mean and median under differently skewed unimodal distribution | - -Now, let's check the skewness of our dataset using Scholar.Stats.skew function. - -```elixir -Scholar.Stats.skew(tensor_data) -``` - - - -``` -#Nx.Tensor< - f64[12] - EXLA.Backend - [0.6475530855160632, 1.5764965159574844, 1.2815277799152376, 1.0767638711454446, 5.0217921696710315, 1.4063140718346212, 0.3905901775815236, 0.9774735389046988, 0.45764233925379794, 0.9768943947733456, 0.4871927332763434, 0.15574868141362447] -> -``` - -As we can see, all features have positive skewness, which means that their distributions are more similar to the left plot in the picture above. - -Moving on to another statistical function, let's discuss *kurtosis*. Kurtosis measures how much data is located in the tails of distributions. If the kurtosis is greater than 0, the distribution is said to be "platykurtic", indicating that it has more extreme values than a univariate normal distribution. Similarly, a "leptokurtic" distribution has positive kurtosis and less extreme values, while a "mesokurtic" distribution has the same kurtosis as a normal distribution. Let's check the kurtosis of our dataset. - -| ![Skewness](https://vitalflux.com/wp-content/uploads/2023/01/fourth-moment-kurtosis-640x334.png) | -| :----------------------------------------------------------------------------------------------: | -| Figure 2: Plot showing Platykurtic, Mesokurtic and Leptokurtic distributions | - -```elixir -Scholar.Stats.kurtosis(tensor_data) -``` - - - -``` -#Nx.Tensor< - f64[12] - EXLA.Backend - [2.168736944824719, 5.08520490451785, 6.167374226819426, 3.4650542966046363, 37.52503905008619, 11.453415905047144, 0.5700448984658735, 9.78258726703508, 0.5290085383907339, 1.5880812942840778, -0.6989373013774784, 0.21508011570192975] -> -``` - -Almost all features have positive kurtosis (they are leptokurtic). `alcohol` has negative excess kurtosis, which means it is platykurtic. Below there is Kernel Density Estimate (KDE) to check how exactly this tail look like. - -```elixir -# Increase the sample size (or use 1.0 to plot all data) -sample = DF.sample(df_data, 0.5, seed: seed) - -Vl.new( - title: [ - text: "KDE plots of Features", - anchor: :middle, - offset: 20 - ], - width: 700, - height: 300 -) -|> Vl.data_from_values(sample, only: ~w(alcohol density pH)) -|> Vl.concat([ - Vl.new( - title: [ - text: "KDE plot of alcohol feature", - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.transform(density: "alcohol") - |> Vl.mark(:area, color: "lightblue") - |> Vl.encode_field(:x, "value", type: :quantitative, title: "% of alcohol") - |> Vl.encode_field(:y, "density", type: :quantitative), - Vl.new( - title: [ - text: "KDE plot of pH feature", - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.transform(density: "pH") - |> Vl.mark(:area, color: "lightgreen") - |> Vl.encode_field(:x, "value", type: :quantitative, title: "pH") - |> Vl.encode_field(:y, "density", type: :quantitative) -]) -``` - - - -```vega-lite -{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","concat":[{"encoding":{"x":{"field":"value","title":"% of alcohol","type":"quantitative"},"y":{"field":"density","type":"quantitative"}},"height":300,"mark":{"color":"lightblue","type":"area"},"title":{"offset":20,"text":"KDE plot of alcohol feature"},"transform":[{"density":"alcohol"}],"width":350},{"encoding":{"x":{"field":"value","title":"pH","type":"quantitative"},"y":{"field":"density","type":"quantitative"}},"height":300,"mark":{"color":"lightgreen","type":"area"},"title":{"offset":20,"text":"KDE plot of pH 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plots of Features"},"width":700} -``` - -The previous analysis indicates that the alcohol feature has more extreme values than pH. Now, we will create a correlation heatmap to investigate the relationships between the features in the dataset. - -```elixir -correlation = Scholar.Covariance.correlation_matrix(tensor_data, biased: true) -``` - - - -``` -#Nx.Tensor< - f64[12][12] - EXLA.Backend - [ - [1.0, -0.022697290146647113, 0.2891806976936754, 0.08902070136217163, 0.023085643656347757, -0.049395859081173187, 0.09106975615864087, 0.2653310138391866, -0.42585829099138195, -0.017142985021137338, -0.12088112319453302, -0.11366283071301796], - [-0.022697290146647113, 1.0000000000000007, -0.14947181064857518, 0.06428606009099526, 0.07051157147938472, -0.09701193927796008, 0.0892605035517784, 0.02711384546243434, -0.03191536827348892, -0.03572814690581474, 0.06771794277504414, -0.19472296892113428], - [0.2891806976936754, -0.14947181064857518, 1.0000000000000009, 0.09421162429049387, 0.11436444838681839, 0.09407722101469612, 0.12113079766441823, 0.14950257057002272, -0.16374821140062368, 0.06233094028523288, -0.07572873005717942, -0.009209090883975417], - [0.08902070136217163, 0.06428606009099526, 0.09421162429049387, 1.0000000000000002, 0.0886845359412256, 0.299098353693518, 0.40143931115992487, 0.8389664549045833, -0.19413345399025597, -0.026664365853858334, -0.45063122203172884, -0.09757682889469321], - [0.023085643656347757, 0.07051157147938472, ...], - ... - ] -> -``` - -```elixir -{corr_size, _} = Nx.shape(correlation) -correlation_list = Nx.to_flat_list(correlation) -column_names = DF.names(df_data) - -corr_to_plot = - DF.new( - x: List.flatten(List.duplicate(column_names, corr_size)), - y: List.flatten(for name <- column_names, do: List.duplicate(name, corr_size)), - corr_val: Enum.map(correlation_list, fn x -> Float.round(x, 2) end) - ) - -Vl.new( - title: [ - text: "Correlation Matrix for California Housing", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.data_from_values(corr_to_plot) -|> Vl.config(axis: [grid: false, tickband: :extent]) -|> Vl.layers([ - Vl.new() - |> Vl.mark(:rect) - |> Vl.encode_field(:x, "x", type: :nominal, title: "", sort: false) - |> Vl.encode_field(:y, "y", type: :nominal, title: "", sort: false) - |> Vl.encode_field(:color, "corr_val", type: :quantitative, scale: [scheme: :inferno]), - Vl.new() - |> Vl.mark(:text) - |> Vl.encode_field(:x, "x", type: :nominal, title: "") - |> Vl.encode_field(:y, "y", type: :nominal, title: "") - |> Vl.encode_field(:text, "corr_val", type: :quantitative) - |> Vl.encode_field(:color, "corr_val", - type: :quantitative, - condition: [ - [test: "datum['corr_val'] < 0", value: :white], - [test: "datum['corr_val'] >= 0", value: :black] - ] - ) -]) -``` - - - -```vega-lite -{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"corr_val":1.0,"x":"fixed acidity","y":"fixed acidity"},{"corr_val":-0.02,"x":"volatile acidity","y":"fixed acidity"},{"corr_val":0.29,"x":"citric acid","y":"fixed acidity"},{"corr_val":0.09,"x":"residual sugar","y":"fixed acidity"},{"corr_val":0.02,"x":"chlorides","y":"fixed acidity"},{"corr_val":-0.05,"x":"free sulfur dioxide","y":"fixed acidity"},{"corr_val":0.09,"x":"total sulfur dioxide","y":"fixed acidity"},{"corr_val":0.27,"x":"density","y":"fixed acidity"},{"corr_val":-0.43,"x":"pH","y":"fixed acidity"},{"corr_val":-0.02,"x":"sulphates","y":"fixed acidity"},{"corr_val":-0.12,"x":"alcohol","y":"fixed acidity"},{"corr_val":-0.11,"x":"quality","y":"fixed acidity"},{"corr_val":-0.02,"x":"fixed acidity","y":"volatile acidity"},{"corr_val":1.0,"x":"volatile acidity","y":"volatile acidity"},{"corr_val":-0.15,"x":"citric acid","y":"volatile acidity"},{"corr_val":0.06,"x":"residual sugar","y":"volatile acidity"},{"corr_val":0.07,"x":"chlorides","y":"volatile acidity"},{"corr_val":-0.1,"x":"free sulfur dioxide","y":"volatile acidity"},{"corr_val":0.09,"x":"total sulfur dioxide","y":"volatile acidity"},{"corr_val":0.03,"x":"density","y":"volatile acidity"},{"corr_val":-0.03,"x":"pH","y":"volatile acidity"},{"corr_val":-0.04,"x":"sulphates","y":"volatile acidity"},{"corr_val":0.07,"x":"alcohol","y":"volatile acidity"},{"corr_val":-0.19,"x":"quality","y":"volatile acidity"},{"corr_val":0.29,"x":"fixed acidity","y":"citric acid"},{"corr_val":-0.15,"x":"volatile acidity","y":"citric acid"},{"corr_val":1.0,"x":"citric acid","y":"citric acid"},{"corr_val":0.09,"x":"residual sugar","y":"citric acid"},{"corr_val":0.11,"x":"chlorides","y":"citric acid"},{"corr_val":0.09,"x":"free sulfur dioxide","y":"citric acid"},{"corr_val":0.12,"x":"total sulfur dioxide","y":"citric acid"},{"corr_val":0.15,"x":"density","y":"citric acid"},{"corr_val":-0.16,"x":"pH","y":"citric acid"},{"corr_val":0.06,"x":"sulphates","y":"citric acid"},{"corr_val":-0.08,"x":"alcohol","y":"citric acid"},{"corr_val":-0.01,"x":"quality","y":"citric acid"},{"corr_val":0.09,"x":"fixed acidity","y":"residual sugar"},{"corr_val":0.06,"x":"volatile acidity","y":"residual sugar"},{"corr_val":0.09,"x":"citric acid","y":"residual sugar"},{"corr_val":1.0,"x":"residual sugar","y":"residual sugar"},{"corr_val":0.09,"x":"chlorides","y":"residual sugar"},{"corr_val":0.3,"x":"free sulfur dioxide","y":"residual sugar"},{"corr_val":0.4,"x":"total sulfur dioxide","y":"residual sugar"},{"corr_val":0.84,"x":"density","y":"residual sugar"},{"corr_val":-0.19,"x":"pH","y":"residual sugar"},{"corr_val":-0.03,"x":"sulphates","y":"residual sugar"},{"corr_val":-0.45,"x":"alcohol","y":"residual sugar"},{"corr_val":-0.1,"x":"quality","y":"residual sugar"},{"corr_val":0.02,"x":"fixed acidity","y":"chlorides"},{"corr_val":0.07,"x":"volatile acidity","y":"chlorides"},{"corr_val":0.11,"x":"citric acid","y":"chlorides"},{"corr_val":0.09,"x":"residual sugar","y":"chlorides"},{"corr_val":1.0,"x":"chlorides","y":"chlorides"},{"corr_val":0.1,"x":"free sulfur dioxide","y":"chlorides"},{"corr_val":0.2,"x":"total sulfur dioxide","y":"chlorides"},{"corr_val":0.26,"x":"density","y":"chlorides"},{"corr_val":-0.09,"x":"pH","y":"chlorides"},{"corr_val":0.02,"x":"sulphates","y":"chlorides"},{"corr_val":-0.36,"x":"alcohol","y":"chlorides"},{"corr_val":-0.21,"x":"quality","y":"chlorides"},{"corr_val":-0.05,"x":"fixed acidity","y":"free sulfur dioxide"},{"corr_val":-0.1,"x":"volatile acidity","y":"free sulfur dioxide"},{"corr_val":0.09,"x":"citric acid","y":"free sulfur dioxide"},{"corr_val":0.3,"x":"residual sugar","y":"free sulfur dioxide"},{"corr_val":0.1,"x":"chlorides","y":"free sulfur dioxide"},{"corr_val":1.0,"x":"free sulfur dioxide","y":"free sulfur dioxide"},{"corr_val":0.62,"x":"total sulfur dioxide","y":"free sulfur dioxide"},{"corr_val":0.29,"x":"density","y":"free sulfur dioxide"},{"corr_val":0.0,"x":"pH","y":"free sulfur dioxide"},{"corr_val":0.06,"x":"sulphates","y":"free sulfur dioxide"},{"corr_val":-0.25,"x":"alcohol","y":"free sulfur dioxide"},{"corr_val":0.01,"x":"quality","y":"free sulfur dioxide"},{"corr_val":0.09,"x":"fixed acidity","y":"total sulfur dioxide"},{"corr_val":0.09,"x":"volatile acidity","y":"total sulfur dioxide"},{"corr_val":0.12,"x":"citric acid","y":"total sulfur dioxide"},{"corr_val":0.4,"x":"residual sugar","y":"total sulfur dioxide"},{"corr_val":0.2,"x":"chlorides","y":"total sulfur dioxide"},{"corr_val":0.62,"x":"free sulfur dioxide","y":"total sulfur dioxide"},{"corr_val":1.0,"x":"total sulfur dioxide","y":"total sulfur dioxide"},{"corr_val":0.53,"x":"density","y":"total sulfur dioxide"},{"corr_val":0.0,"x":"pH","y":"total sulfur dioxide"},{"corr_val":0.13,"x":"sulphates","y":"total sulfur dioxide"},{"corr_val":-0.45,"x":"alcohol","y":"total sulfur dioxide"},{"corr_val":-0.17,"x":"quality","y":"total sulfur dioxide"},{"corr_val":0.27,"x":"fixed acidity","y":"density"},{"corr_val":0.03,"x":"volatile acidity","y":"density"},{"corr_val":0.15,"x":"citric acid","y":"density"},{"corr_val":0.84,"x":"residual sugar","y":"density"},{"corr_val":0.26,"x":"chlorides","y":"density"},{"corr_val":0.29,"x":"free sulfur dioxide","y":"density"},{"corr_val":0.53,"x":"total sulfur dioxide","y":"density"},{"corr_val":1.0,"x":"density","y":"density"},{"corr_val":-0.09,"x":"pH","y":"density"},{"corr_val":0.07,"x":"sulphates","y":"density"},{"corr_val":-0.78,"x":"alcohol","y":"density"},{"corr_val":-0.31,"x":"quality","y":"density"},{"corr_val":-0.43,"x":"fixed acidity","y":"pH"},{"corr_val":-0.03,"x":"volatile acidity","y":"pH"},{"corr_val":-0.16,"x":"citric acid","y":"pH"},{"corr_val":-0.19,"x":"residual sugar","y":"pH"},{"corr_val":-0.09,"x":"chlorides","y":"pH"},{"corr_val":0.0,"x":"free sulfur dioxide","y":"pH"},{"corr_val":0.0,"x":"total sulfur dioxide","y":"pH"},{"corr_val":-0.09,"x":"density","y":"pH"},{"corr_val":1.0,"x":"pH","y":"pH"},{"corr_val":0.16,"x":"sulphates","y":"pH"},{"corr_val":0.12,"x":"alcohol","y":"pH"},{"corr_val":0.1,"x":"quality","y":"pH"},{"corr_val":-0.02,"x":"fixed acidity","y":"sulphates"},{"corr_val":-0.04,"x":"volatile acidity","y":"sulphates"},{"corr_val":0.06,"x":"citric acid","y":"sulphates"},{"corr_val":-0.03,"x":"residual sugar","y":"sulphates"},{"corr_val":0.02,"x":"chlorides","y":"sulphates"},{"corr_val":0.06,"x":"free sulfur dioxide","y":"sulphates"},{"corr_val":0.13,"x":"total sulfur dioxide","y":"sulphates"},{"corr_val":0.07,"x":"density","y":"sulphates"},{"corr_val":0.16,"x":"pH","y":"sulphates"},{"corr_val":1.0,"x":"sulphates","y":"sulphates"},{"corr_val":-0.02,"x":"alcohol","y":"sulphates"},{"corr_val":0.05,"x":"quality","y":"sulphates"},{"corr_val":-0.12,"x":"fixed acidity","y":"alcohol"},{"corr_val":0.07,"x":"volatile acidity","y":"alcohol"},{"corr_val":-0.08,"x":"citric acid","y":"alcohol"},{"corr_val":-0.45,"x":"residual sugar","y":"alcohol"},{"corr_val":-0.36,"x":"chlorides","y":"alcohol"},{"corr_val":-0.25,"x":"free sulfur dioxide","y":"alcohol"},{"corr_val":-0.45,"x":"total sulfur dioxide","y":"alcohol"},{"corr_val":-0.78,"x":"density","y":"alcohol"},{"corr_val":0.12,"x":"pH","y":"alcohol"},{"corr_val":-0.02,"x":"sulphates","y":"alcohol"},{"corr_val":1.0,"x":"alcohol","y":"alcohol"},{"corr_val":0.44,"x":"quality","y":"alcohol"},{"corr_val":-0.11,"x":"fixed acidity","y":"quality"},{"corr_val":-0.19,"x":"volatile acidity","y":"quality"},{"corr_val":-0.01,"x":"citric acid","y":"quality"},{"corr_val":-0.1,"x":"residual sugar","y":"quality"},{"corr_val":-0.21,"x":"chlorides","y":"quality"},{"corr_val":0.01,"x":"free sulfur dioxide","y":"quality"},{"corr_val":-0.17,"x":"total sulfur dioxide","y":"quality"},{"corr_val":-0.31,"x":"density","y":"quality"},{"corr_val":0.1,"x":"pH","y":"quality"},{"corr_val":0.05,"x":"sulphates","y":"quality"},{"corr_val":0.44,"x":"alcohol","y":"quality"},{"corr_val":1.0,"x":"quality","y":"quality"}]},"height":630,"layer":[{"encoding":{"color":{"field":"corr_val","scale":{"scheme":"viridis"},"type":"quantitative"},"x":{"field":"x","type":"nominal"},"y":{"field":"y","type":"nominal"}},"mark":"rect"},{"encoding":{"text":{"field":"corr_val","type":"quantitative"},"x":{"field":"x","type":"nominal"},"y":{"field":"y","type":"nominal"}},"mark":"text"}],"title":{"offset":20,"text":"Correlation Matrix for California Housing"},"width":630} -``` - -We can observe from the correlation heatmap that there is no strong correlation between the quality of wine and any single feature. However, we can see that there are two strong correlations: density is proportional to the amount of residual sugar and inversely proportional to the amount of alcohol. - -Given that citric acid, free sulfur dioxide, and sulphates are not strongly correlated with the quality of wine or any other features, we can drop them from the dataset to simplify our analysis. - -Before dropping the features, it is important to consider their potential impact on the final prediction. Even if they are not strongly correlated with the target variable (quality), they may still provide useful information for the KNN algorithm. It is recommended to experiment with different feature subsets to find the optimal combination for the given task. You can try running the notebook again without dropping these features. - -```elixir -tensor_data = - DF.discard(df_data, ["citric acid", "free sulfur dioxide", "sulphates"]) |> Nx.stack(axis: 1) -``` - - - -``` -#Nx.Tensor< - f64[4898][9] - EXLA.Backend - [ - [7.0, 0.27, 20.7, 0.045, 170.0, 1.001, 3.0, 8.8, 6.0], - [6.3, 0.3, 1.6, 0.049, 132.0, 0.994, 3.3, 9.5, 6.0], - [8.1, 0.28, 6.9, 0.05, 97.0, 0.9951, 3.26, 10.1, 6.0], - [7.2, 0.23, 8.5, 0.058, 186.0, 0.9956, 3.19, 9.9, 6.0], - [7.2, 0.23, 8.5, 0.058, 186.0, 0.9956, 3.19, 9.9, 6.0], - [8.1, 0.28, 6.9, 0.05, 97.0, ...], - ... - ] -> -``` - -The next step will be splitting the dataset into features and labels. To make the classification task easier, we will convert the labels into three categories: "poor," "medium," and "excellent". Additionally, we will normalize the features using the standard scaler. - -```elixir -x = tensor_data[[.., 0..-2//1]] -y = tensor_data[[.., -1]] - -# convert quality into labels quality: (0, 4) poor => 0, (5, 6) medium => 1, (7, 10) excelent => 2 - -poor_and_better_mask = Nx.greater(y, 4) -medium_and_better_mask = Nx.greater(y, 6) - -y = Nx.select(poor_and_better_mask, Nx.select(medium_and_better_mask, 2, 1), 0) -x = Scholar.Preprocessing.standard_scale(x) - -{x, y} -``` - - - -``` -{#Nx.Tensor< - f64[4898][8] - EXLA.Backend - [ - [-0.29385297541380284, -0.4368649169854585, -0.0027291121847267076, -0.44164614831586296, 3.1698834995014824, -0.4213312276408997, -0.37885264350988346, -0.2556031247705665], - [-0.30872791733061694, -0.43622741947473787, -0.40860252734351166, -0.4415611486477669, 2.3623866525887167, -0.4214799770600679, -0.3724776684026774, -0.24072818285375244], - [-0.27047806668738067, -0.43665241781521824, -0.29597796711620483, -0.4415398987307429, 1.6186395567480112, -0.42145660215134145, -0.37332766508363824, -0.22797823263934036], - [-0.2896029920089988, -0.4377149136664193, -0.2619780998777726, -0.44136989939455074, 3.509882171885805, -0.42144597719282945, -0.3748151592753196, -0.23222821604414437], - [-0.2896029920089988, -0.4377149136664193, -0.2619780998777726, -0.44136989939455074, 3.509882171885805, -0.42144597719282945, -0.3748151592753196, -0.23222821604414437], - [-0.27047806668738067, -0.43665241781521824, -0.29597796711620483, -0.4415398987307429, 1.6186395567480112, -0.42145660215134145, -0.37332766508363824, -0.22797823263934036], - [-0.310852909033019, ...], - ... - ] - >, - #Nx.Tensor< - s64[4898] - EXLA.Backend - [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, ...] - >} -``` - -After performing EDA on our wine dataset, we can now train our KNN model and make predictions. However, the question arises as to what value of neighbors should we choose for the KNN model. To answer this question, we will use *K-Fold Cross-Validation* (specifically, Nested Cross-Validation). K-Fold is a procedure performed on the training dataset where we split the data into k smaller subsets (folds). For each tested parameter value, we run k separate trainings and testings, where one fold is the validation (test) dataset, and the other k-1 folds are the training dataset. To make this process clear, there is a GIF below that explains it visually. - -| ![KFold](https://upload.wikimedia.org/wikipedia/commons/4/4b/KfoldCV.gif) | -| :----------------------------------------------------------------------------------------------------------------------------------------------------------: | -| Figure 3: Illustration of KFold Cross-Validation when n = 12 observations and k = 3. After data is shuffled, a total of 3 models will be trained and tested. | - -```elixir -defmodule Comb do - def combinations([]), do: [[]] - - def combinations([{name, values} | opts]) do - for subcombination <- combinations(opts), value <- values do - [{name, value} | subcombination] - end - end -end - -Comb.combinations(task: [:classification], num_classes: [3], num_neighbors: Enum.to_list(1..30)) -``` - - - -``` -[ - [task: :classification, num_classes: 3, num_neighbors: 1], - [task: :classification, num_classes: 3, num_neighbors: 2], - [task: :classification, num_classes: 3, num_neighbors: 3], - [task: :classification, num_classes: 3, num_neighbors: 4], - [task: :classification, num_classes: 3, num_neighbors: 5], - [task: :classification, num_classes: 3, num_neighbors: 6], - [task: :classification, num_classes: 3, num_neighbors: 7], - [task: :classification, num_classes: 3, num_neighbors: 8], - [task: :classification, num_classes: 3, num_neighbors: 9], - [task: :classification, num_classes: 3, num_neighbors: 10], - [task: :classification, num_classes: 3, num_neighbors: 11], - [task: :classification, num_classes: 3, num_neighbors: 12], - [task: :classification, num_classes: 3, num_neighbors: 13], - [task: :classification, num_classes: 3, num_neighbors: 14], - [task: :classification, num_classes: 3, num_neighbors: 15], - [task: :classification, num_classes: 3, num_neighbors: 16], - [task: :classification, num_classes: 3, num_neighbors: 17], - [task: :classification, num_classes: 3, num_neighbors: 18], - [task: :classification, num_classes: 3, num_neighbors: 19], - [task: :classification, num_classes: 3, num_neighbors: 20], - [task: :classification, num_classes: 3, num_neighbors: 21], - [task: :classification, num_classes: 3, num_neighbors: 22], - [task: :classification, num_classes: 3, num_neighbors: 23], - [task: :classification, num_classes: 3, num_neighbors: 24], - [task: :classification, num_classes: 3, num_neighbors: 25], - [task: :classification, num_classes: 3, num_neighbors: 26], - [task: :classification, num_classes: 3, num_neighbors: 27], - [task: :classification, num_classes: 3, num_neighbors: 28], - [task: :classification, num_classes: 3, num_neighbors: 29], - [task: :classification, num_classes: 3, num_neighbors: 30] -] -``` - -The code implementing KFold cross-validation is located at the beginning of the notebook. - -```elixir -min_num_neighbors = 1 -param_space_size = 30 - -# K in KFold -k = 10 -train_size = 4000 - -{indices, _new_key} = Nx.Random.shuffle(key, Nx.iota(Nx.shape(y))) -x = Nx.take(x, indices) -y = Nx.take(y, indices) -{train_data, test_data} = Nx.split(x, train_size) -{train_labels, test_labels} = Nx.split(y, train_size) - -folding_fun = fn x -> Scholar.ModelSelection.k_fold_split(x, k) end - -scoring_fun = fn x, y, opts -> - {x_train, x_test} = x - {y_train, y_test} = y - - model = - KNN.fit(x_train, y_train, - num_classes: opts[:num_classes], - num_neighbors: opts[:num_neighbors], - task: opts[:task] - ) - - prediction = KNN.predict(model, x_test) - - accuracy = Classification.accuracy(y_test, prediction) - [accuracy] -end - -res = - Scholar.ModelSelection.grid_search(train_data, train_labels, folding_fun, scoring_fun, - task: [:classification], - num_classes: [3], - num_neighbors: Enum.to_list(min_num_neighbors..param_space_size) - ) - -res = for(sub_res <- res, do: sub_res[:score]) |> Nx.concatenate() |> Nx.as_type(:f64) - -results = - DF.new(accuracy: res, num_neighbors: Nx.add(Nx.iota({param_space_size}), min_num_neighbors)) -``` - -To potentially improve our KNN model's performance, we can try setting the weights parameter to `:distance` and use weighted version of grid search. Next, we can visualize the results of KFold using a plot. - -```elixir -Vl.new( - title: [ - text: "Accuracy vs Number of Neighbors", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.data_from_values(results) -|> Vl.mark(:line, color: :gray, stroke_dash: [5, 5], point: true) -|> Vl.encode_field(:x, "num_neighbors", type: :quantitative, scale: [domain: [1, 31]]) -|> Vl.encode_field(:y, "accuracy", type: :quantitative, scale: [domain: [0.7, 0.85]]) -``` - - - -```vega-lite -{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"accuracy":0.7682499885559082,"num_neighbors":1},{"accuracy":0.734250009059906,"num_neighbors":2},{"accuracy":0.7425000071525574,"num_neighbors":3},{"accuracy":0.7512499690055847,"num_neighbors":4},{"accuracy":0.7512499690055847,"num_neighbors":5},{"accuracy":0.749750018119812,"num_neighbors":6},{"accuracy":0.7477499842643738,"num_neighbors":7},{"accuracy":0.7487500309944153,"num_neighbors":8},{"accuracy":0.7487500309944153,"num_neighbors":9},{"accuracy":0.749250054359436,"num_neighbors":10},{"accuracy":0.7489999532699585,"num_neighbors":11},{"accuracy":0.7477499842643738,"num_neighbors":12},{"accuracy":0.7475000023841858,"num_neighbors":13},{"accuracy":0.7419999837875366,"num_neighbors":14},{"accuracy":0.7487500309944153,"num_neighbors":15},{"accuracy":0.747249960899353,"num_neighbors":16},{"accuracy":0.7505000233650208,"num_neighbors":17},{"accuracy":0.7475000023841858,"num_neighbors":18},{"accuracy":0.7475000023841858,"num_neighbors":19},{"accuracy":0.7477499842643738,"num_neighbors":20},{"accuracy":0.7480000257492065,"num_neighbors":21},{"accuracy":0.7477499842643738,"num_neighbors":22},{"accuracy":0.7459999918937683,"num_neighbors":23},{"accuracy":0.7442499995231628,"num_neighbors":24},{"accuracy":0.7442499995231628,"num_neighbors":25},{"accuracy":0.7437499761581421,"num_neighbors":26},{"accuracy":0.7434999942779541,"num_neighbors":27},{"accuracy":0.7464999556541443,"num_neighbors":28},{"accuracy":0.7430000305175781,"num_neighbors":29},{"accuracy":0.747249960899353,"num_neighbors":30}]},"encoding":{"x":{"field":"num_neighbors","scale":{"domain":[1,31]},"type":"quantitative"},"y":{"field":"accuracy","scale":{"domain":[0.7,0.8]},"type":"quantitative"}},"height":630,"mark":{"color":"gray","point":true,"strokeDash":[5,5],"type":"line"},"title":{"offset":20,"text":"Accuracy vs Number of Neighbors"},"width":630} -``` - -```elixir -best_neighbors = Nx.add(Nx.argmax(res), min_num_neighbors) |> Nx.to_number() -``` - - - -``` -1 -``` - -Ok, let's test our best model on test data. - -```elixir -best_model = KNN.fit(train_data, train_labels, num_neighbors: best_neighbors, num_classes: 3) - -final_prediction = KNN.predict(best_model, test_data) -Classification.accuracy(test_labels, final_prediction) -``` - - - -``` -#Nx.Tensor< - f32 - EXLA.Backend - 0.7672605514526367 -> -``` - -Pretty well, now let's move on to a regression task using KNN. - - - -## Regression - -KNNs can also be used in regression tasks. We will explore this on a dataset about Airfoil Self-Noise. Here's a brief description of the dataset: - -**Data Set Information**: - -The NASA data set comprises different size NACA 0012 airfoils at various wind tunnel speeds and angles of attack. The span of the airfoil and the observer position were the same in all of the experiments. - -**Attribute Information**: - -This problem has the following inputs: - -1. Frequency, in Hertzs. -2. Angle of attack, in degrees. -3. Chord length, in meters. -4. Free-stream velocity, in meters per second. -5. Suction side displacement thickness, in meters. - -The only output is: - -1. Scaled sound pressure level, in decibels. - -Dua, D. and Graff, C. (2019). UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science. - -```elixir -dataset = - Req.get!( - "https://archive.ics.uci.edu/ml/machine-learning-databases/00291/airfoil_self_noise.dat" - ).body - -column_names = [ - "frequency", - "angle_of_attack", - "chord_length", - "free_stream_velocity", - "suction_side_displacement_thickness", - "scaled_sound_pressure_level" -] - -dataset_df = DF.load_csv!(dataset, header: false, delimiter: "\t") |> DF.rename(column_names) -labels = DF.select(dataset_df, "scaled_sound_pressure_level") -data = DF.discard(dataset_df, "scaled_sound_pressure_level") -``` - -Below are the KDE plots of the dataset's features. We will analyze them in the next step. - -```elixir -Vl.new( - title: [ - text: "KDE plots of Features", - anchor: :middle, - offset: 20 - ] -) -|> Vl.data_from_values(data) -|> Vl.concat( - [ - Vl.new() - |> Vl.concat( - [ - Vl.new( - title: [ - text: "KDE plot of Frequency feature", - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.transform(density: "frequency") - |> Vl.mark(:area, color: "lightblue") - |> Vl.encode_field(:x, "value", type: :quantitative, title: "Frequency (Hz)") - |> Vl.encode_field(:y, "density", type: :quantitative), - Vl.new( - title: [ - text: "KDE plot of Angle of Attack feature", - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.transform(density: "angle_of_attack") - |> Vl.mark(:area, color: "lightgreen") - |> Vl.encode_field(:x, "value", type: :quantitative, title: "Angle of attack") - |> Vl.encode_field(:y, "density", type: :quantitative), - Vl.new( - title: [ - text: "KDE plot of Chord length feature", - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.transform(density: "chord_length") - |> Vl.mark(:area, color: "pink") - |> Vl.encode_field(:x, "value", type: :quantitative, title: "Chord length") - |> Vl.encode_field(:y, "density", type: :quantitative) - ], - :vertical - ), - Vl.new() - |> Vl.concat( - [ - Vl.new( - title: [ - text: "KDE plot of Free Stream Velocity feature", - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.data_from_values(data) - |> Vl.transform(density: "free_stream_velocity") - |> Vl.mark(:area, color: "purple") - |> Vl.encode_field(:x, "value", type: :quantitative, title: "Free-stream velocity") - |> Vl.encode_field(:y, "density", type: :quantitative), - Vl.new( - title: [ - text: ["KDE plot of Suction Side Displacement", "Thickness feature"], - offset: 20 - ], - width: 350, - height: 300 - ) - |> Vl.data_from_values(data) - |> Vl.transform(density: "suction_side_displacement_thickness") - |> Vl.mark(:area, color: "red") - |> Vl.encode_field(:x, "value", - type: :quantitative, - title: "Suction Side Displacement Thickness" - ) - |> Vl.encode_field(:y, "density", type: :quantitative) - ], - :vertical - ) - ], - :horizontal -) -``` - - - -```vega-lite 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Side Displacement Thickness","type":"quantitative"},"y":{"field":"density","type":"quantitative"}},"height":300,"mark":{"color":"red","type":"area"},"title":{"offset":20,"text":["KDE plot of Suction Side Displacement","Thickness feature"]},"transform":[{"density":"suction_side_displacement_thickness"}],"width":350}]}],"title":{"anchor":"middle","offset":20,"text":"KDE plots of Features"}} -``` - -The Suction Side Displacement Thickness and Frequency features have asymmetric distributions with many observations having small values. To make these distributions less skewed, we will use the logarithms of these two features. - -```elixir -data = - DF.mutate(data, - frequency_log: log(frequency), - suction_side_displacement_thickness_log: log(suction_side_displacement_thickness) - ) - -data = DF.discard(data, ["frequency", "suction_side_displacement_thickness"]) -``` - -Scale data and convert them to tensors. - -```elixir -x = Nx.stack(data, axis: 1) |> Scholar.Preprocessing.standard_scale() -y = Nx.stack(labels, axis: 1) |> Nx.squeeze(axes: [1]) -{x, y} -``` - - - -``` -{#Nx.Tensor< - f64[1503][5] - EXLA.Backend - [ - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.24959453243691035, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.2391306895323175, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.22866684662772468, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.21709086757891163, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.2066270246743188, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.19616318176972597, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.18532568847301975, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.17412335981632007, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.1636595169117272, -0.8410438169469526], - [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.15282202361502104, ...], - ... - ] - >, - #Nx.Tensor< - f64[1503] - EXLA.Backend - [126.201, 125.201, 125.951, 127.591, 127.461, 125.571, 125.201, 123.061, 121.301, 119.541, 117.151, 115.391, 112.241, 108.721, 126.416, 127.696, 128.086, 126.966, 126.086, 126.986, 126.616, 124.106, 123.236, 121.106, 119.606, 117.976, 116.476, 113.076, 111.076, 118.129, 119.319, 122.779, 124.809, 126.959, 128.629, 129.099, 127.899, 125.499, 124.049, 123.689, 121.399, 120.319, 119.229, 117.789, 116.229, 114.779, 112.139, 109.619, ...] - >} -``` - -In this section, we will apply a similar KFold procedure as in the classification section to find the best value for the `:num_neighbors` parameter in the KNN regression model. - -```elixir -min_num_neighbors = 1 -param_space_size = 30 - -# K in KFold -k = 10 -train_size = 1100 - -{indices, _new_key} = Nx.Random.shuffle(key, Nx.iota(Nx.shape(y))) -x_shuffled = Nx.take(x, indices) -y_shuffled = Nx.take(y, indices) -{train_data, test_data} = Nx.split(x_shuffled, train_size) -{train_labels, test_labels} = Nx.split(y_shuffled, train_size) - -folding_fun = fn x -> Scholar.ModelSelection.k_fold_split(x, k) end - -scoring_fun = fn x, y, opts -> - {x_train, x_test} = x - {y_train, y_test} = y - - model = - KNN.fit(x_train, y_train, - num_classes: opts[:num_classes], - num_neighbors: opts[:num_neighbors], - task: opts[:task] - ) - - prediction = KNN.predict(model, x_test) - - rmse = Regression.mean_square_error(y_test, prediction) |> Nx.sqrt() - [rmse] -end - -res = - Scholar.ModelSelection.grid_search(train_data, train_labels, folding_fun, scoring_fun, - task: [:regression], - # dummy, for regression it doesn't matter - num_classes: [3], - num_neighbors: Enum.to_list(min_num_neighbors..param_space_size) - ) - -res = for(sub_res <- res, do: sub_res[:score]) |> Nx.concatenate() |> Nx.as_type(:f64) - -results = DF.new(RMSE: res, num_neighbors: Nx.add(Nx.iota({param_space_size}), min_num_neighbors)) -``` - -```elixir -Vl.new( - title: [ - text: "RMSE vs. Number of Neighbors", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.data_from_values(results) -|> Vl.mark(:line, color: :gray, stroke_dash: [5, 5], point: true) -|> Vl.encode_field(:x, "num_neighbors", type: :quantitative, scale: [domain: [1, 31]]) -|> Vl.encode_field(:y, "RMSE", type: :quantitative, scale: [domain: [2.6, 6.0]]) -``` - - - -```vega-lite -{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"RMSE":2.954160435119645,"num_neighbors":1},{"RMSE":2.705136057088256,"num_neighbors":2},{"RMSE":2.9964430952256733,"num_neighbors":3},{"RMSE":3.2905484274001795,"num_neighbors":4},{"RMSE":3.4905466912822405,"num_neighbors":5},{"RMSE":3.6884594409820104,"num_neighbors":6},{"RMSE":3.8750230152441363,"num_neighbors":7},{"RMSE":4.047035866627379,"num_neighbors":8},{"RMSE":4.189331570582891,"num_neighbors":9},{"RMSE":4.324340953175694,"num_neighbors":10},{"RMSE":4.439531671658643,"num_neighbors":11},{"RMSE":4.571325241080606,"num_neighbors":12},{"RMSE":4.672862066067107,"num_neighbors":13},{"RMSE":4.787292392293998,"num_neighbors":14},{"RMSE":4.879060063659762,"num_neighbors":15},{"RMSE":4.978485008641893,"num_neighbors":16},{"RMSE":5.065103101842437,"num_neighbors":17},{"RMSE":5.160077780567819,"num_neighbors":18},{"RMSE":5.23622641273977,"num_neighbors":19},{"RMSE":5.284432293863683,"num_neighbors":20},{"RMSE":5.3405242911271005,"num_neighbors":21},{"RMSE":5.3922427790028795,"num_neighbors":22},{"RMSE":5.44295673910584,"num_neighbors":23},{"RMSE":5.47931644940631,"num_neighbors":24},{"RMSE":5.523618789257428,"num_neighbors":25},{"RMSE":5.567065728104048,"num_neighbors":26},{"RMSE":5.602044586338607,"num_neighbors":27},{"RMSE":5.641831179764594,"num_neighbors":28},{"RMSE":5.675274254551784,"num_neighbors":29},{"RMSE":5.708321768081241,"num_neighbors":30}]},"encoding":{"x":{"field":"num_neighbors","scale":{"domain":[1,31]},"type":"quantitative"},"y":{"field":"RMSE","scale":{"domain":[2.6,6.0]},"type":"quantitative"}},"height":630,"mark":{"color":"gray","point":true,"strokeDash":[5,5],"type":"line"},"title":{"offset":20,"text":"RMSE vs. Number of Neighbors"},"width":630} -``` - -```elixir -best_neighbors = Nx.add(Nx.argmin(res), min_num_neighbors) |> Nx.to_number() -``` - - - -``` -2 -``` - -The best value is 2. Now try this value to a final prediction. - -```elixir -best_model = - KNN.fit(train_data, train_labels, - num_neighbors: best_neighbors, - num_classes: 3, - task: :regression - ) - -final_prediction = KNN.predict(best_model, test_data) -Regression.mean_square_error(test_labels, final_prediction) |> Nx.sqrt() -``` - - - -``` -#Nx.Tensor< - f64 - EXLA.Backend - 2.2414312639566227 -> -``` - -### Outliers prediction - - - -KNNs can be used for outlier detection. In this task, we will also use the Airfoil Self-Noise dataset. We will calculate for each sample a mean value of distances to the three closest points from dataset. - -```elixir -model = KNN.fit(x, y, num_neighbors: 3, task: :regression) -{num_samples, _} = Nx.shape(x) -{distances, indices} = KNN.k_neighbors(model, x) - -mean_distances = Nx.mean(distances, axes: [1], keep_axes: true) - -samples_mean_distances = - Nx.concatenate([mean_distances, Nx.iota({num_samples, 1})], axis: 1) - |> DF.new() - |> DF.rename(["mean_distances", "index"]) -``` - -Now, plot the mean values of distances. - -```elixir -Vl.new( - title: [ - text: "Plot to check anomalies", - offset: 20 - ], - width: 800, - height: 630 -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(samples_mean_distances) - |> Vl.mark(:line, color: :jet) - |> Vl.encode_field(:x, "index", type: :quantitative, scale: [domain: [0, 1510]]) - |> Vl.encode_field(:y, "mean_distances", - type: :quantitative, - title: "Mean distance to clusters", - scale: [domain: [0.0065, 0.012]] - ), - Vl.new() - |> Vl.data_from_values(DF.new(y: [0.009])) - |> Vl.mark(:rule, color: :red, stroke_dash: [5, 5]) - |> Vl.encode_field(:y, "y", type: :quantitative, title: "") -]) -``` - - - -```vega-lite 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distance to clusters","type":"quantitative"}},"mark":{"color":"jet","type":"line"}},{"data":{"values":[{"y":0.009}]},"encoding":{"y":{"field":"y","type":"quantitative"}},"mark":{"color":"red","strokeDash":[5,5],"type":"rule"}}],"title":{"offset":20,"text":"Plot to check anomalies"},"width":800} -``` - -We can assume, that samples with average distance bigger than 0.0105 can be treated as outliers. - -```elixir -data = DF.put(data, :is_outlier, Nx.greater(mean_distances, 0.0105)) -``` - -```elixir -Vl.new( - title: [ - text: "Logarithm of Frequency vs Logarithm of Suction Side Displacement Thickness", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.data_from_values(data) -|> Vl.layers([ - Vl.new() - |> Vl.param("brush", select: :interval) - |> Vl.mark(:circle) - |> Vl.encode_field(:x, "frequency_log", type: :quantitative, scale: [domain: [4.5, 10.5]]) - |> Vl.encode_field(:y, "suction_side_displacement_thickness_log", - type: :quantitative, - scale: [domain: [-8.0, -2.5]] - ) - |> Vl.encode(:color, - condition: [param: "brush", field: "is_outlier", type: :nominal], - value: :gray - ), - Vl.new() - |> Vl.transform(filter: [param: "brush", field: "is_outlier", equal: true]) - |> Vl.mark(:text) - |> Vl.encode(:x, value: 11) - |> Vl.encode(:y, value: -3.5) - |> Vl.encode_field(:text, "filter", - type: :quantitative, - aggregate: :count - ) -]) -``` - - - -```vega-lite 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of Frequency vs Logarithm of Suction Side Displacement Thickness"},"width":630} -``` + + +# K-Nearest Neighbors + +```elixir +Mix.install([ + {:scholar, "~> 0.2.0"}, + {:explorer, "~> 0.6.1"}, + {:exla, "~> 0.6.0"}, + {:nx, "~> 0.6.0", override: true}, + {:req, "~> 0.3.9"}, + {:kino_vega_lite, "~> 0.1.9"}, + {:kino, "~> 0.10.0"}, + {:scidata, "~> 0.1.10"}, + {:kino_explorer, "~> 0.1.7"}, + {:tucan, github: "pnezis/tucan", ref: "6549cdc"} +]) +``` + + + +``` +:ok +``` + +## Setup + +We will extensively use VegaLite, Explorer, and Scholar throughout this guide, so let's define some aliases: + +```elixir +require Explorer.DataFrame, as: DF +require Explorer.Series, as: S +require Explorer.Query, as: Q +alias Scholar.Neighbors.KNearestNeighbors, as: KNN +alias Scholar.Metrics.Classification +alias Scholar.Metrics.Regression +``` + + + +``` +Scholar.Metrics.Regression +``` + +And let's configure `EXLA.Backend` as our default across the notebook and all branched sections: + +```elixir +Nx.global_default_backend(EXLA.Backend) +seed = 42 +key = Nx.Random.key(42) +``` + + + +``` + +02:33:33.173 [info] TfrtCpuClient created. + +``` + + + +``` +#Nx.Tensor< + u32[2] + EXLA.Backend + [0, 42] +> +``` + +## Introduction + +This notebook will cover the three primary applications of K-Nearest Neighbors: classification, regression, and +anomaly detection. + + + +## Classification + +The process of classification using KNN is relatively straightforward. We start by working with a dataset where each sample is assigned a label. To predict the label of a new sample, we compute the distance between it and all the other samples in the dataset. Next, we select only the $k$ closest samples based on the distance metric, where $k$ is a user-defined parameter. We then examine the labels of these $k$ samples and choose the label that appears most frequently. This label is assigned as the predicted label for the new sample. + +Let's first grasp how kNN works visually with a small example. + +```elixir +data = + DF.new( + x: [-1, 0.2, -0.5, -2.1, -2.3, -2.2, 0.1, 0.3, 0.4, 0.7, 1.3], + y: [-0.1, 0.4, 0.5, 0.4, 1.1, -1.0, -0.1, 0.2, 1.5, 1.6, 0.9], + label: [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1] + ) + +point_to_predict = DF.new(x: [0.0], y: [0.0]) + +k_eq_3 = DF.new(x: [0.45], y: [0.45], name: ["k = 3"]) +k_eq_5 = DF.new(x: [0.72], y: [0.72], name: ["k = 5"]) + +radius = fn x, y -> :math.sqrt(x ** 2 + y ** 2) end + +Tucan.layers([ + Tucan.scatter(data, "x", "y", + point_size: 200, + filled: true, + shape_by: "label", + color_by: "label" + ) + |> Tucan.Scale.set_color_scheme(:plasma), + Tucan.scatter(point_to_predict, "x", "y", + point_size: 400, + point_color: "green", + point_shape: "triangle-up", + filled: true + ), + Tucan.circle(Tucan.new(), {0, 0}, radius.(0.2, 0.4), line_color: "brown", stroke_width: 2), + Tucan.circle(Tucan.new(), {0, 0}, radius.(-1, 0.1), line_color: "blue", stroke_width: 2), + Tucan.annotate(Tucan.new(), 0.42, 0.42, "k = 3", color: "brown", size: 20), + Tucan.annotate(Tucan.new(), 0.81, 0.81, "k = 5", color: "blue", size: 20) +]) +|> Tucan.Scale.set_xy_domain(-2.4, 1.7) +|> Tucan.Grid.set_enabled(false) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Scatterplot showing KNN prediction process", offset: 20) +``` + + + +```vega-lite +{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","height":630,"layer":[{"data":{"values":[{"label":0,"x":-1.0,"y":-0.1},{"label":0,"x":0.2,"y":0.4},{"label":0,"x":-0.5,"y":0.5},{"label":0,"x":-2.1,"y":0.4},{"label":0,"x":-2.3,"y":1.1},{"label":0,"x":-2.2,"y":-1.0},{"label":1,"x":0.1,"y":-0.1},{"label":1,"x":0.3,"y":0.2},{"label":1,"x":0.4,"y":1.5},{"label":1,"x":0.7,"y":1.6},{"label":1,"x":1.3,"y":0.9}]},"encoding":{"color":{"field":"label","scale":{"scheme":"plasma"},"type":"nominal"},"shape":{"field":"label","type":"nominal"},"x":{"axis":{"grid":false},"field":"x","scale":{"domain":[-2.4,1.5]},"type":"quantitative"},"y":{"axis":{"grid":false},"field":"y","scale":{"domain":[-1.2,1.8]},"type":"quantitative"}},"mark":{"filled":true,"size":200,"type":"point"}},{"data":{"values":[{"x":0.0,"y":0.0}]},"encoding":{"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"green","filled":true,"shape":"triangle","size":400,"type":"point"}},{"data":{"values":[{"x":0.0,"y":0.0}]},"encoding":{"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"brown","innerRadius":97,"outerRadius":99,"type":"arc"}},{"data":{"values":[{"x":0.0,"y":0.0}]},"encoding":{"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"blue","innerRadius":168,"outerRadius":170,"type":"arc"}},{"data":{"values":[{"name":"k = 3","x":0.45,"y":0.45}]},"encoding":{"text":{"field":"name","type":"nominal"},"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"brown","size":20,"type":"text"}},{"data":{"values":[{"name":"k = 5","x":0.72,"y":0.72}]},"encoding":{"text":{"field":"name","type":"nominal"},"x":{"field":"x","type":"quantitative"},"y":{"field":"y","type":"quantitative"}},"mark":{"color":"blue","size":20,"type":"text"}}],"title":{"offset":20,"text":"Scatterplot showing KNN prediction process"},"width":630} +``` + +Note that the final prediction may vary depending on the value of $k$. For example, with $k = 3$, we would predict the green triangle as an orange square (1), whereas with $k = 5$, the purple circle (0) would be the final prediction. + +Let's now test this using Scholar code. First, we will define our data. + +```elixir +x = Nx.stack(DF.discard(data, "label"), axis: 1) +labels = Nx.stack(DF.select(data, "label"), axis: 1) |> Nx.squeeze(axes: [1]) +x_pred = Nx.stack(point_to_predict, axis: 1) +``` + + + +``` +#Nx.Tensor< + f64[1][2] + EXLA.Backend + [ + [0.0, 0.0] + ] +> +``` + +Let's now try with $k = 3$. + +```elixir +model = KNN.fit(x, labels, num_classes: 2, num_neighbors: 3) +KNN.predict(model, x_pred) +``` + + + +``` +#Nx.Tensor< + s64[1] + EXLA.Backend + [1] +> +``` + +And $k = 5$. + +```elixir +model = KNN.fit(x, labels, num_classes: 2, num_neighbors: 5) +KNN.predict(model, x_pred) +``` + + + +``` +#Nx.Tensor< + s64[1] + EXLA.Backend + [0] +> +``` + +As we can see, the predictions match our intuition from analyzing the plot. + +Now, let's try KNN on a more complicated dataset: the Wine dataset. Our task will be to classify the quality of the wine. Before we load the dataset into `Explorer.DataFrame` for more efficient exploration, let's check some more detailed information about the dataset. + +```elixir +info = + Req.get!( + "https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality.names" + ).body + +Kino.Markdown.new(info) +``` + +```elixir +data = + Req.get!( + "https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality-white.csv" + ).body + +df_data = DF.load_csv!(data, delimiter: ";", dtypes: %{"total sulfur dioxide": :float}) +tensor_data = Nx.stack(df_data, axis: 1) +df_data +``` + +As we can see, there are no null values in the dataset. Now, let's check the size of the dataset. + +```elixir +DF.shape(df_data) +``` + + + +``` +{4898, 12} +``` + +Let's check some statistical properties of the dataset. We will start with *skewness*, which measures the asymmetry of the probability distribution of a random variable about its mean. To better understand this concept, please take a look at the picture below. + +| ![Skewness](https://upload.wikimedia.org/wikipedia/commons/c/cc/Relationship_between_mean_and_median_under_different_skewness.png) | +| :--------------------------------------------------------------------------------------------------------------------------------: | +| Figure 1: A general relationship of mean and median under differently skewed unimodal distribution | + +Now, let's check the skewness of our dataset using Scholar.Stats.skew function. + +```elixir +Scholar.Stats.skew(tensor_data) +``` + + + +``` +#Nx.Tensor< + f64[12] + EXLA.Backend + [0.6475530855160632, 1.5764965159574844, 1.2815277799152376, 1.0767638711454446, 5.0217921696710315, 1.4063140718346212, 0.3905901775815236, 0.9774735389046988, 0.45764233925379794, 0.9768943947733456, 0.4871927332763434, 0.15574868141362447] +> +``` + +As we can see, all features have positive skewness, which means that their distributions are more similar to the left plot in the picture above. + +Moving on to another statistical function, let's discuss *kurtosis*. Kurtosis measures how much data is located in the tails of distributions. If the kurtosis is greater than 0, the distribution is said to be "platykurtic", indicating that it has more extreme values than a univariate normal distribution. Similarly, a "leptokurtic" distribution has positive kurtosis and less extreme values, while a "mesokurtic" distribution has the same kurtosis as a normal distribution. Let's check the kurtosis of our dataset. + +| ![Skewness](https://vitalflux.com/wp-content/uploads/2023/01/fourth-moment-kurtosis-640x334.png) | +| :----------------------------------------------------------------------------------------------: | +| Figure 2: Plot showing Platykurtic, Mesokurtic and Leptokurtic distributions | + +```elixir +Scholar.Stats.kurtosis(tensor_data) +``` + + + +``` +#Nx.Tensor< + f64[12] + EXLA.Backend + [2.168736944824719, 5.08520490451785, 6.167374226819426, 3.4650542966046363, 37.52503905008619, 11.453415905047144, 0.5700448984658735, 9.78258726703508, 0.5290085383907339, 1.5880812942840778, -0.6989373013774784, 0.21508011570192975] +> +``` + +Almost all features have positive kurtosis (they are leptokurtic). `alcohol` has negative excess kurtosis, which means it is platykurtic. Below there is Kernel Density Estimate (KDE) to check how exactly this tail look like. + +```elixir +# Increase the sample size (or use 1.0 to plot all data) +sample = DF.sample(df_data, 0.5, seed: seed) + +Tucan.hconcat([ + Tucan.density(sample, "alcohol", only: ["alcohol"], area_color: "lightblue") + |> Tucan.Axes.set_x_title("% of alcohol") + |> Tucan.set_size(350, 300) + |> Tucan.set_title("KDE plot of alcohol feature", offset: 20), + Tucan.density(sample, "pH", only: ["pH"], area_color: "lightgreen") + |> Tucan.set_size(350, 300) + |> Tucan.set_title("KDE plot of pH feature", offset: 20) +]) +|> Tucan.set_title("KDE plots of Features", anchor: :middle, offset: 20) +``` + + + +```vega-lite +{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","concat":[{"encoding":{"x":{"field":"value","title":"% of alcohol","type":"quantitative"},"y":{"field":"density","type":"quantitative"}},"height":300,"mark":{"color":"lightblue","type":"area"},"title":{"offset":20,"text":"KDE plot of 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plots of Features"},"width":700} +``` + +The previous analysis indicates that the alcohol feature has more extreme values than pH. Now, we will create a correlation heatmap to investigate the relationships between the features in the dataset. + +```elixir +correlation = Scholar.Covariance.correlation_matrix(tensor_data, biased: true) +``` + + + +``` +#Nx.Tensor< + f64[12][12] + EXLA.Backend + [ + [1.0, -0.022697290146647113, 0.2891806976936754, 0.08902070136217163, 0.023085643656347757, -0.049395859081173187, 0.09106975615864087, 0.2653310138391866, -0.42585829099138195, -0.017142985021137338, -0.12088112319453302, -0.11366283071301796], + [-0.022697290146647113, 1.0000000000000007, -0.14947181064857518, 0.06428606009099526, 0.07051157147938472, -0.09701193927796008, 0.0892605035517784, 0.02711384546243434, -0.03191536827348892, -0.03572814690581474, 0.06771794277504414, -0.19472296892113428], + [0.2891806976936754, -0.14947181064857518, 1.0000000000000009, 0.09421162429049387, 0.11436444838681839, 0.09407722101469612, 0.12113079766441823, 0.14950257057002272, -0.16374821140062368, 0.06233094028523288, -0.07572873005717942, -0.009209090883975417], + [0.08902070136217163, 0.06428606009099526, 0.09421162429049387, 1.0000000000000002, 0.0886845359412256, 0.299098353693518, 0.40143931115992487, 0.8389664549045833, -0.19413345399025597, -0.026664365853858334, -0.45063122203172884, -0.09757682889469321], + [0.023085643656347757, 0.07051157147938472, ...], + ... + ] +> +``` + +```elixir +{corr_size, _} = Nx.shape(correlation) +correlation_list = Nx.to_flat_list(correlation) +column_names = DF.names(df_data) + +corr_to_plot = + DF.new( + x: List.flatten(List.duplicate(column_names, corr_size)), + y: List.flatten(for name <- column_names, do: List.duplicate(name, corr_size)), + corr_val: Enum.map(correlation_list, fn x -> Float.round(x, 2) end) + ) + +Tucan.heatmap(corr_to_plot, "x", "y", "corr_val", annotate: true, text_color: [{nil, 0, "white"}]) +|> Tucan.Scale.set_color_scheme(:inferno) +|> Tucan.Axes.set_title(:color, "Correlation") +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Correlation Matrix for California Housing", offset: 20) +``` + + + +```vega-lite +{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"corr_val":1.0,"x":"fixed acidity","y":"fixed acidity"},{"corr_val":-0.02,"x":"volatile acidity","y":"fixed acidity"},{"corr_val":0.29,"x":"citric acid","y":"fixed acidity"},{"corr_val":0.09,"x":"residual sugar","y":"fixed acidity"},{"corr_val":0.02,"x":"chlorides","y":"fixed acidity"},{"corr_val":-0.05,"x":"free sulfur dioxide","y":"fixed acidity"},{"corr_val":0.09,"x":"total sulfur dioxide","y":"fixed acidity"},{"corr_val":0.27,"x":"density","y":"fixed acidity"},{"corr_val":-0.43,"x":"pH","y":"fixed acidity"},{"corr_val":-0.02,"x":"sulphates","y":"fixed acidity"},{"corr_val":-0.12,"x":"alcohol","y":"fixed acidity"},{"corr_val":-0.11,"x":"quality","y":"fixed acidity"},{"corr_val":-0.02,"x":"fixed acidity","y":"volatile acidity"},{"corr_val":1.0,"x":"volatile acidity","y":"volatile acidity"},{"corr_val":-0.15,"x":"citric acid","y":"volatile acidity"},{"corr_val":0.06,"x":"residual sugar","y":"volatile acidity"},{"corr_val":0.07,"x":"chlorides","y":"volatile acidity"},{"corr_val":-0.1,"x":"free sulfur dioxide","y":"volatile acidity"},{"corr_val":0.09,"x":"total sulfur dioxide","y":"volatile acidity"},{"corr_val":0.03,"x":"density","y":"volatile acidity"},{"corr_val":-0.03,"x":"pH","y":"volatile acidity"},{"corr_val":-0.04,"x":"sulphates","y":"volatile acidity"},{"corr_val":0.07,"x":"alcohol","y":"volatile acidity"},{"corr_val":-0.19,"x":"quality","y":"volatile acidity"},{"corr_val":0.29,"x":"fixed acidity","y":"citric acid"},{"corr_val":-0.15,"x":"volatile acidity","y":"citric acid"},{"corr_val":1.0,"x":"citric acid","y":"citric acid"},{"corr_val":0.09,"x":"residual sugar","y":"citric acid"},{"corr_val":0.11,"x":"chlorides","y":"citric acid"},{"corr_val":0.09,"x":"free sulfur dioxide","y":"citric acid"},{"corr_val":0.12,"x":"total sulfur dioxide","y":"citric acid"},{"corr_val":0.15,"x":"density","y":"citric acid"},{"corr_val":-0.16,"x":"pH","y":"citric acid"},{"corr_val":0.06,"x":"sulphates","y":"citric acid"},{"corr_val":-0.08,"x":"alcohol","y":"citric acid"},{"corr_val":-0.01,"x":"quality","y":"citric acid"},{"corr_val":0.09,"x":"fixed acidity","y":"residual sugar"},{"corr_val":0.06,"x":"volatile acidity","y":"residual sugar"},{"corr_val":0.09,"x":"citric acid","y":"residual sugar"},{"corr_val":1.0,"x":"residual sugar","y":"residual sugar"},{"corr_val":0.09,"x":"chlorides","y":"residual sugar"},{"corr_val":0.3,"x":"free sulfur dioxide","y":"residual sugar"},{"corr_val":0.4,"x":"total sulfur dioxide","y":"residual sugar"},{"corr_val":0.84,"x":"density","y":"residual sugar"},{"corr_val":-0.19,"x":"pH","y":"residual sugar"},{"corr_val":-0.03,"x":"sulphates","y":"residual sugar"},{"corr_val":-0.45,"x":"alcohol","y":"residual sugar"},{"corr_val":-0.1,"x":"quality","y":"residual sugar"},{"corr_val":0.02,"x":"fixed acidity","y":"chlorides"},{"corr_val":0.07,"x":"volatile acidity","y":"chlorides"},{"corr_val":0.11,"x":"citric acid","y":"chlorides"},{"corr_val":0.09,"x":"residual sugar","y":"chlorides"},{"corr_val":1.0,"x":"chlorides","y":"chlorides"},{"corr_val":0.1,"x":"free sulfur dioxide","y":"chlorides"},{"corr_val":0.2,"x":"total sulfur dioxide","y":"chlorides"},{"corr_val":0.26,"x":"density","y":"chlorides"},{"corr_val":-0.09,"x":"pH","y":"chlorides"},{"corr_val":0.02,"x":"sulphates","y":"chlorides"},{"corr_val":-0.36,"x":"alcohol","y":"chlorides"},{"corr_val":-0.21,"x":"quality","y":"chlorides"},{"corr_val":-0.05,"x":"fixed acidity","y":"free sulfur dioxide"},{"corr_val":-0.1,"x":"volatile acidity","y":"free sulfur dioxide"},{"corr_val":0.09,"x":"citric acid","y":"free sulfur dioxide"},{"corr_val":0.3,"x":"residual sugar","y":"free sulfur dioxide"},{"corr_val":0.1,"x":"chlorides","y":"free sulfur dioxide"},{"corr_val":1.0,"x":"free sulfur dioxide","y":"free sulfur dioxide"},{"corr_val":0.62,"x":"total sulfur dioxide","y":"free sulfur dioxide"},{"corr_val":0.29,"x":"density","y":"free sulfur dioxide"},{"corr_val":0.0,"x":"pH","y":"free sulfur dioxide"},{"corr_val":0.06,"x":"sulphates","y":"free sulfur dioxide"},{"corr_val":-0.25,"x":"alcohol","y":"free sulfur dioxide"},{"corr_val":0.01,"x":"quality","y":"free sulfur dioxide"},{"corr_val":0.09,"x":"fixed acidity","y":"total sulfur dioxide"},{"corr_val":0.09,"x":"volatile acidity","y":"total sulfur dioxide"},{"corr_val":0.12,"x":"citric acid","y":"total sulfur dioxide"},{"corr_val":0.4,"x":"residual sugar","y":"total sulfur dioxide"},{"corr_val":0.2,"x":"chlorides","y":"total sulfur dioxide"},{"corr_val":0.62,"x":"free sulfur dioxide","y":"total sulfur dioxide"},{"corr_val":1.0,"x":"total sulfur dioxide","y":"total sulfur dioxide"},{"corr_val":0.53,"x":"density","y":"total sulfur dioxide"},{"corr_val":0.0,"x":"pH","y":"total sulfur dioxide"},{"corr_val":0.13,"x":"sulphates","y":"total sulfur dioxide"},{"corr_val":-0.45,"x":"alcohol","y":"total sulfur dioxide"},{"corr_val":-0.17,"x":"quality","y":"total sulfur dioxide"},{"corr_val":0.27,"x":"fixed acidity","y":"density"},{"corr_val":0.03,"x":"volatile acidity","y":"density"},{"corr_val":0.15,"x":"citric acid","y":"density"},{"corr_val":0.84,"x":"residual sugar","y":"density"},{"corr_val":0.26,"x":"chlorides","y":"density"},{"corr_val":0.29,"x":"free sulfur dioxide","y":"density"},{"corr_val":0.53,"x":"total sulfur dioxide","y":"density"},{"corr_val":1.0,"x":"density","y":"density"},{"corr_val":-0.09,"x":"pH","y":"density"},{"corr_val":0.07,"x":"sulphates","y":"density"},{"corr_val":-0.78,"x":"alcohol","y":"density"},{"corr_val":-0.31,"x":"quality","y":"density"},{"corr_val":-0.43,"x":"fixed acidity","y":"pH"},{"corr_val":-0.03,"x":"volatile acidity","y":"pH"},{"corr_val":-0.16,"x":"citric acid","y":"pH"},{"corr_val":-0.19,"x":"residual sugar","y":"pH"},{"corr_val":-0.09,"x":"chlorides","y":"pH"},{"corr_val":0.0,"x":"free sulfur dioxide","y":"pH"},{"corr_val":0.0,"x":"total sulfur dioxide","y":"pH"},{"corr_val":-0.09,"x":"density","y":"pH"},{"corr_val":1.0,"x":"pH","y":"pH"},{"corr_val":0.16,"x":"sulphates","y":"pH"},{"corr_val":0.12,"x":"alcohol","y":"pH"},{"corr_val":0.1,"x":"quality","y":"pH"},{"corr_val":-0.02,"x":"fixed acidity","y":"sulphates"},{"corr_val":-0.04,"x":"volatile acidity","y":"sulphates"},{"corr_val":0.06,"x":"citric acid","y":"sulphates"},{"corr_val":-0.03,"x":"residual sugar","y":"sulphates"},{"corr_val":0.02,"x":"chlorides","y":"sulphates"},{"corr_val":0.06,"x":"free sulfur dioxide","y":"sulphates"},{"corr_val":0.13,"x":"total sulfur dioxide","y":"sulphates"},{"corr_val":0.07,"x":"density","y":"sulphates"},{"corr_val":0.16,"x":"pH","y":"sulphates"},{"corr_val":1.0,"x":"sulphates","y":"sulphates"},{"corr_val":-0.02,"x":"alcohol","y":"sulphates"},{"corr_val":0.05,"x":"quality","y":"sulphates"},{"corr_val":-0.12,"x":"fixed acidity","y":"alcohol"},{"corr_val":0.07,"x":"volatile acidity","y":"alcohol"},{"corr_val":-0.08,"x":"citric acid","y":"alcohol"},{"corr_val":-0.45,"x":"residual sugar","y":"alcohol"},{"corr_val":-0.36,"x":"chlorides","y":"alcohol"},{"corr_val":-0.25,"x":"free sulfur dioxide","y":"alcohol"},{"corr_val":-0.45,"x":"total sulfur dioxide","y":"alcohol"},{"corr_val":-0.78,"x":"density","y":"alcohol"},{"corr_val":0.12,"x":"pH","y":"alcohol"},{"corr_val":-0.02,"x":"sulphates","y":"alcohol"},{"corr_val":1.0,"x":"alcohol","y":"alcohol"},{"corr_val":0.44,"x":"quality","y":"alcohol"},{"corr_val":-0.11,"x":"fixed acidity","y":"quality"},{"corr_val":-0.19,"x":"volatile acidity","y":"quality"},{"corr_val":-0.01,"x":"citric acid","y":"quality"},{"corr_val":-0.1,"x":"residual sugar","y":"quality"},{"corr_val":-0.21,"x":"chlorides","y":"quality"},{"corr_val":0.01,"x":"free sulfur dioxide","y":"quality"},{"corr_val":-0.17,"x":"total sulfur dioxide","y":"quality"},{"corr_val":-0.31,"x":"density","y":"quality"},{"corr_val":0.1,"x":"pH","y":"quality"},{"corr_val":0.05,"x":"sulphates","y":"quality"},{"corr_val":0.44,"x":"alcohol","y":"quality"},{"corr_val":1.0,"x":"quality","y":"quality"}]},"height":630,"layer":[{"encoding":{"color":{"field":"corr_val","scale":{"scheme":"viridis"},"type":"quantitative"},"x":{"field":"x","type":"nominal"},"y":{"field":"y","type":"nominal"}},"mark":"rect"},{"encoding":{"text":{"field":"corr_val","type":"quantitative"},"x":{"field":"x","type":"nominal"},"y":{"field":"y","type":"nominal"}},"mark":"text"}],"title":{"offset":20,"text":"Correlation Matrix for California Housing"},"width":630} +``` + +We can observe from the correlation heatmap that there is no strong correlation between the quality of wine and any single feature. However, we can see that there are two strong correlations: density is proportional to the amount of residual sugar and inversely proportional to the amount of alcohol. + +Given that citric acid, free sulfur dioxide, and sulphates are not strongly correlated with the quality of wine or any other features, we can drop them from the dataset to simplify our analysis. + +Before dropping the features, it is important to consider their potential impact on the final prediction. Even if they are not strongly correlated with the target variable (quality), they may still provide useful information for the KNN algorithm. It is recommended to experiment with different feature subsets to find the optimal combination for the given task. You can try running the notebook again without dropping these features. + +```elixir +tensor_data = + DF.discard(df_data, ["citric acid", "free sulfur dioxide", "sulphates"]) |> Nx.stack(axis: 1) +``` + + + +``` +#Nx.Tensor< + f64[4898][9] + EXLA.Backend + [ + [7.0, 0.27, 20.7, 0.045, 170.0, 1.001, 3.0, 8.8, 6.0], + [6.3, 0.3, 1.6, 0.049, 132.0, 0.994, 3.3, 9.5, 6.0], + [8.1, 0.28, 6.9, 0.05, 97.0, 0.9951, 3.26, 10.1, 6.0], + [7.2, 0.23, 8.5, 0.058, 186.0, 0.9956, 3.19, 9.9, 6.0], + [7.2, 0.23, 8.5, 0.058, 186.0, 0.9956, 3.19, 9.9, 6.0], + [8.1, 0.28, 6.9, 0.05, 97.0, ...], + ... + ] +> +``` + +The next step will be splitting the dataset into features and labels. To make the classification task easier, we will convert the labels into three categories: "poor," "medium," and "excellent". Additionally, we will normalize the features using the standard scaler. + +```elixir +x = tensor_data[[.., 0..-2//1]] +y = tensor_data[[.., -1]] + +# convert quality into labels quality: (0, 4) poor => 0, (5, 6) medium => 1, (7, 10) excelent => 2 + +poor_and_better_mask = Nx.greater(y, 4) +medium_and_better_mask = Nx.greater(y, 6) + +y = Nx.select(poor_and_better_mask, Nx.select(medium_and_better_mask, 2, 1), 0) +x = Scholar.Preprocessing.standard_scale(x) + +{x, y} +``` + + + +``` +{#Nx.Tensor< + f64[4898][8] + EXLA.Backend + [ + [-0.29385297541380284, -0.4368649169854585, -0.0027291121847267076, -0.44164614831586296, 3.1698834995014824, -0.4213312276408997, -0.37885264350988346, -0.2556031247705665], + [-0.30872791733061694, -0.43622741947473787, -0.40860252734351166, -0.4415611486477669, 2.3623866525887167, -0.4214799770600679, -0.3724776684026774, -0.24072818285375244], + [-0.27047806668738067, -0.43665241781521824, -0.29597796711620483, -0.4415398987307429, 1.6186395567480112, -0.42145660215134145, -0.37332766508363824, -0.22797823263934036], + [-0.2896029920089988, -0.4377149136664193, -0.2619780998777726, -0.44136989939455074, 3.509882171885805, -0.42144597719282945, -0.3748151592753196, -0.23222821604414437], + [-0.2896029920089988, -0.4377149136664193, -0.2619780998777726, -0.44136989939455074, 3.509882171885805, -0.42144597719282945, -0.3748151592753196, -0.23222821604414437], + [-0.27047806668738067, -0.43665241781521824, -0.29597796711620483, -0.4415398987307429, 1.6186395567480112, -0.42145660215134145, -0.37332766508363824, -0.22797823263934036], + [-0.310852909033019, ...], + ... + ] + >, + #Nx.Tensor< + s64[4898] + EXLA.Backend + [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, ...] + >} +``` + +After performing EDA on our wine dataset, we can now train our KNN model and make predictions. However, the question arises as to what value of neighbors should we choose for the KNN model. To answer this question, we will use *K-Fold Cross-Validation* (specifically, Nested Cross-Validation). K-Fold is a procedure performed on the training dataset where we split the data into k smaller subsets (folds). For each tested parameter value, we run k separate trainings and testings, where one fold is the validation (test) dataset, and the other k-1 folds are the training dataset. To make this process clear, there is a GIF below that explains it visually. + +| ![KFold](https://upload.wikimedia.org/wikipedia/commons/4/4b/KfoldCV.gif) | +| :----------------------------------------------------------------------------------------------------------------------------------------------------------: | +| Figure 3: Illustration of KFold Cross-Validation when n = 12 observations and k = 3. After data is shuffled, a total of 3 models will be trained and tested. | + +```elixir +defmodule Comb do + def combinations([]), do: [[]] + + def combinations([{name, values} | opts]) do + for subcombination <- combinations(opts), value <- values do + [{name, value} | subcombination] + end + end +end + +Comb.combinations(task: [:classification], num_classes: [3], num_neighbors: Enum.to_list(1..30)) +``` + + + +``` +[ + [task: :classification, num_classes: 3, num_neighbors: 1], + [task: :classification, num_classes: 3, num_neighbors: 2], + [task: :classification, num_classes: 3, num_neighbors: 3], + [task: :classification, num_classes: 3, num_neighbors: 4], + [task: :classification, num_classes: 3, num_neighbors: 5], + [task: :classification, num_classes: 3, num_neighbors: 6], + [task: :classification, num_classes: 3, num_neighbors: 7], + [task: :classification, num_classes: 3, num_neighbors: 8], + [task: :classification, num_classes: 3, num_neighbors: 9], + [task: :classification, num_classes: 3, num_neighbors: 10], + [task: :classification, num_classes: 3, num_neighbors: 11], + [task: :classification, num_classes: 3, num_neighbors: 12], + [task: :classification, num_classes: 3, num_neighbors: 13], + [task: :classification, num_classes: 3, num_neighbors: 14], + [task: :classification, num_classes: 3, num_neighbors: 15], + [task: :classification, num_classes: 3, num_neighbors: 16], + [task: :classification, num_classes: 3, num_neighbors: 17], + [task: :classification, num_classes: 3, num_neighbors: 18], + [task: :classification, num_classes: 3, num_neighbors: 19], + [task: :classification, num_classes: 3, num_neighbors: 20], + [task: :classification, num_classes: 3, num_neighbors: 21], + [task: :classification, num_classes: 3, num_neighbors: 22], + [task: :classification, num_classes: 3, num_neighbors: 23], + [task: :classification, num_classes: 3, num_neighbors: 24], + [task: :classification, num_classes: 3, num_neighbors: 25], + [task: :classification, num_classes: 3, num_neighbors: 26], + [task: :classification, num_classes: 3, num_neighbors: 27], + [task: :classification, num_classes: 3, num_neighbors: 28], + [task: :classification, num_classes: 3, num_neighbors: 29], + [task: :classification, num_classes: 3, num_neighbors: 30] +] +``` + +The code implementing KFold cross-validation is located at the beginning of the notebook. + +```elixir +min_num_neighbors = 1 +param_space_size = 30 + +# K in KFold +k = 10 +train_size = 4000 + +{indices, _new_key} = Nx.Random.shuffle(key, Nx.iota(Nx.shape(y))) +x = Nx.take(x, indices) +y = Nx.take(y, indices) +{train_data, test_data} = Nx.split(x, train_size) +{train_labels, test_labels} = Nx.split(y, train_size) + +folding_fun = fn x -> Scholar.ModelSelection.k_fold_split(x, k) end + +scoring_fun = fn x, y, opts -> + {x_train, x_test} = x + {y_train, y_test} = y + + model = + KNN.fit(x_train, y_train, + num_classes: opts[:num_classes], + num_neighbors: opts[:num_neighbors], + task: opts[:task] + ) + + prediction = KNN.predict(model, x_test) + + accuracy = Classification.accuracy(y_test, prediction) + [accuracy] +end + +res = + Scholar.ModelSelection.grid_search(train_data, train_labels, folding_fun, scoring_fun, + task: [:classification], + num_classes: [3], + num_neighbors: Enum.to_list(min_num_neighbors..param_space_size) + ) + +res = for(sub_res <- res, do: sub_res[:score]) |> Nx.concatenate() |> Nx.as_type(:f64) + +results = + DF.new(accuracy: res, num_neighbors: Nx.add(Nx.iota({param_space_size}), min_num_neighbors)) +``` + +To potentially improve our KNN model's performance, we can try setting the weights parameter to `:distance` and use weighted version of grid search. Next, we can visualize the results of KFold using a plot. + +```elixir +Tucan.lineplot(results, "num_neighbors", "accuracy", + line_color: "gray", + points: true, + stroke_dash: [5, 5] +) +|> Tucan.Scale.set_x_domain(1, 31) +|> Tucan.Scale.set_y_domain(0.7, 0.85) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Accuracy vs Number of Neighbors", offset: 20) +``` + + + +```vega-lite +{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"accuracy":0.7682499885559082,"num_neighbors":1},{"accuracy":0.734250009059906,"num_neighbors":2},{"accuracy":0.7425000071525574,"num_neighbors":3},{"accuracy":0.7512499690055847,"num_neighbors":4},{"accuracy":0.7512499690055847,"num_neighbors":5},{"accuracy":0.749750018119812,"num_neighbors":6},{"accuracy":0.7477499842643738,"num_neighbors":7},{"accuracy":0.7487500309944153,"num_neighbors":8},{"accuracy":0.7487500309944153,"num_neighbors":9},{"accuracy":0.749250054359436,"num_neighbors":10},{"accuracy":0.7489999532699585,"num_neighbors":11},{"accuracy":0.7477499842643738,"num_neighbors":12},{"accuracy":0.7475000023841858,"num_neighbors":13},{"accuracy":0.7419999837875366,"num_neighbors":14},{"accuracy":0.7487500309944153,"num_neighbors":15},{"accuracy":0.747249960899353,"num_neighbors":16},{"accuracy":0.7505000233650208,"num_neighbors":17},{"accuracy":0.7475000023841858,"num_neighbors":18},{"accuracy":0.7475000023841858,"num_neighbors":19},{"accuracy":0.7477499842643738,"num_neighbors":20},{"accuracy":0.7480000257492065,"num_neighbors":21},{"accuracy":0.7477499842643738,"num_neighbors":22},{"accuracy":0.7459999918937683,"num_neighbors":23},{"accuracy":0.7442499995231628,"num_neighbors":24},{"accuracy":0.7442499995231628,"num_neighbors":25},{"accuracy":0.7437499761581421,"num_neighbors":26},{"accuracy":0.7434999942779541,"num_neighbors":27},{"accuracy":0.7464999556541443,"num_neighbors":28},{"accuracy":0.7430000305175781,"num_neighbors":29},{"accuracy":0.747249960899353,"num_neighbors":30}]},"encoding":{"x":{"field":"num_neighbors","scale":{"domain":[1,31]},"type":"quantitative"},"y":{"field":"accuracy","scale":{"domain":[0.7,0.8]},"type":"quantitative"}},"height":630,"mark":{"color":"gray","point":true,"strokeDash":[5,5],"type":"line"},"title":{"offset":20,"text":"Accuracy vs Number of Neighbors"},"width":630} +``` + +```elixir +best_neighbors = Nx.add(Nx.argmax(res), min_num_neighbors) |> Nx.to_number() +``` + + + +``` +1 +``` + +Ok, let's test our best model on test data. + +```elixir +best_model = KNN.fit(train_data, train_labels, num_neighbors: best_neighbors, num_classes: 3) + +final_prediction = KNN.predict(best_model, test_data) +Classification.accuracy(test_labels, final_prediction) +``` + + + +``` +#Nx.Tensor< + f32 + EXLA.Backend + 0.7672605514526367 +> +``` + +Pretty well, now let's move on to a regression task using KNN. + + + +## Regression + +KNNs can also be used in regression tasks. We will explore this on a dataset about Airfoil Self-Noise. Here's a brief description of the dataset: + +**Data Set Information**: + +The NASA data set comprises different size NACA 0012 airfoils at various wind tunnel speeds and angles of attack. The span of the airfoil and the observer position were the same in all of the experiments. + +**Attribute Information**: + +This problem has the following inputs: + +1. Frequency, in Hertzs. +2. Angle of attack, in degrees. +3. Chord length, in meters. +4. Free-stream velocity, in meters per second. +5. Suction side displacement thickness, in meters. + +The only output is: + +1. Scaled sound pressure level, in decibels. + +Dua, D. and Graff, C. (2019). UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science. + +```elixir +dataset = + Req.get!( + "https://archive.ics.uci.edu/ml/machine-learning-databases/00291/airfoil_self_noise.dat" + ).body + +column_names = [ + "frequency", + "angle_of_attack", + "chord_length", + "free_stream_velocity", + "suction_side_displacement_thickness", + "scaled_sound_pressure_level" +] + +dataset_df = DF.load_csv!(dataset, header: false, delimiter: "\t") |> DF.rename(column_names) +labels = DF.select(dataset_df, "scaled_sound_pressure_level") +data = DF.discard(dataset_df, "scaled_sound_pressure_level") +``` + +Below are the KDE plots of the dataset's features. We will analyze them in the next step. + +```elixir +Tucan.new(data, columns: 2) +|> Tucan.concat([ + Tucan.density(Tucan.new(), "frequency", area_color: "lightblue") + |> Tucan.Axes.set_x_title("Frequency (Hz)") + |> Tucan.set_size(350, 300) + |> Tucan.set_title("KDE plot of Frequency feature", offset: 20), + Tucan.density(Tucan.new(), "free_stream_velocity", area_color: "purple") + |> Tucan.Axes.set_x_title("Angle of attack") + |> Tucan.set_size(350, 300) + |> Tucan.set_title("KDE plot of Angle of Attack feature", offset: 20), + Tucan.density(Tucan.new(), "angle_of_attack", area_color: "lightgreen") + |> Tucan.Axes.set_x_title("Angle of attack") + |> Tucan.set_size(350, 300) + |> Tucan.set_title("KDE plot of Angle of Attack feature", offset: 20), + Tucan.density(Tucan.new(), "suction_side_displacement_thickness", area_color: "red") + |> Tucan.Axes.set_x_title("Suction Side Displacement Thickness") + |> Tucan.set_size(350, 300) + |> Tucan.set_title("KDE plot of Suction Side Displacement\nThickness feature", offset: 20) +]) +|> Tucan.set_title("KDE plots of Features", anchor: :middle, offset: 20) +``` + + + + +```vega-lite 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Side Displacement Thickness","type":"quantitative"},"y":{"field":"density","type":"quantitative"}},"height":300,"mark":{"color":"red","type":"area"},"title":{"offset":20,"text":["KDE plot of Suction Side Displacement","Thickness feature"]},"transform":[{"density":"suction_side_displacement_thickness"}],"width":350}]}],"title":{"anchor":"middle","offset":20,"text":"KDE plots of Features"}} +``` + +The Suction Side Displacement Thickness and Frequency features have asymmetric distributions with many observations having small values. To make these distributions less skewed, we will use the logarithms of these two features. + +```elixir +data = + DF.mutate(data, + frequency_log: log(frequency), + suction_side_displacement_thickness_log: log(suction_side_displacement_thickness) + ) + +data = DF.discard(data, ["frequency", "suction_side_displacement_thickness"]) +``` + +Scale data and convert them to tensors. + +```elixir +x = Nx.stack(data, axis: 1) |> Scholar.Preprocessing.standard_scale() +y = Nx.stack(labels, axis: 1) |> Nx.squeeze(axes: [1]) +{x, y} +``` + + + +``` +{#Nx.Tensor< + f64[1503][5] + EXLA.Backend + [ + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.24959453243691035, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.2391306895323175, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.22866684662772468, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.21709086757891163, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.2066270246743188, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.19616318176972597, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.18532568847301975, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.17412335981632007, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.1636595169117272, -0.8410438169469526], + [-0.5630552019680845, -0.5487622533323925, 2.780406864058286, -0.15282202361502104, ...], + ... + ] + >, + #Nx.Tensor< + f64[1503] + EXLA.Backend + [126.201, 125.201, 125.951, 127.591, 127.461, 125.571, 125.201, 123.061, 121.301, 119.541, 117.151, 115.391, 112.241, 108.721, 126.416, 127.696, 128.086, 126.966, 126.086, 126.986, 126.616, 124.106, 123.236, 121.106, 119.606, 117.976, 116.476, 113.076, 111.076, 118.129, 119.319, 122.779, 124.809, 126.959, 128.629, 129.099, 127.899, 125.499, 124.049, 123.689, 121.399, 120.319, 119.229, 117.789, 116.229, 114.779, 112.139, 109.619, ...] + >} +``` + +In this section, we will apply a similar KFold procedure as in the classification section to find the best value for the `:num_neighbors` parameter in the KNN regression model. + +```elixir +min_num_neighbors = 1 +param_space_size = 30 + +# K in KFold +k = 10 +train_size = 1100 + +{indices, _new_key} = Nx.Random.shuffle(key, Nx.iota(Nx.shape(y))) +x_shuffled = Nx.take(x, indices) +y_shuffled = Nx.take(y, indices) +{train_data, test_data} = Nx.split(x_shuffled, train_size) +{train_labels, test_labels} = Nx.split(y_shuffled, train_size) + +folding_fun = fn x -> Scholar.ModelSelection.k_fold_split(x, k) end + +scoring_fun = fn x, y, opts -> + {x_train, x_test} = x + {y_train, y_test} = y + + model = + KNN.fit(x_train, y_train, + num_classes: opts[:num_classes], + num_neighbors: opts[:num_neighbors], + task: opts[:task] + ) + + prediction = KNN.predict(model, x_test) + + rmse = Regression.mean_square_error(y_test, prediction) |> Nx.sqrt() + [rmse] +end + +res = + Scholar.ModelSelection.grid_search(train_data, train_labels, folding_fun, scoring_fun, + task: [:regression], + # dummy, for regression it doesn't matter + num_classes: [3], + num_neighbors: Enum.to_list(min_num_neighbors..param_space_size) + ) + +res = for(sub_res <- res, do: sub_res[:score]) |> Nx.concatenate() |> Nx.as_type(:f64) + +results = DF.new(RMSE: res, num_neighbors: Nx.add(Nx.iota({param_space_size}), min_num_neighbors)) +``` + +```elixir +Tucan.lineplot(results, "num_neighbors", "RMSE", + points: true, + stroke_dash: [5, 5], + line_color: "gray" +) +|> Tucan.Scale.set_x_domain(1, 31) +|> Tucan.Scale.set_y_domain(2.6, 6.0) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("RMSE vs. Number of Neighbors", offset: 20) +``` + + + +```vega-lite +{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"RMSE":2.954160435119645,"num_neighbors":1},{"RMSE":2.705136057088256,"num_neighbors":2},{"RMSE":2.9964430952256733,"num_neighbors":3},{"RMSE":3.2905484274001795,"num_neighbors":4},{"RMSE":3.4905466912822405,"num_neighbors":5},{"RMSE":3.6884594409820104,"num_neighbors":6},{"RMSE":3.8750230152441363,"num_neighbors":7},{"RMSE":4.047035866627379,"num_neighbors":8},{"RMSE":4.189331570582891,"num_neighbors":9},{"RMSE":4.324340953175694,"num_neighbors":10},{"RMSE":4.439531671658643,"num_neighbors":11},{"RMSE":4.571325241080606,"num_neighbors":12},{"RMSE":4.672862066067107,"num_neighbors":13},{"RMSE":4.787292392293998,"num_neighbors":14},{"RMSE":4.879060063659762,"num_neighbors":15},{"RMSE":4.978485008641893,"num_neighbors":16},{"RMSE":5.065103101842437,"num_neighbors":17},{"RMSE":5.160077780567819,"num_neighbors":18},{"RMSE":5.23622641273977,"num_neighbors":19},{"RMSE":5.284432293863683,"num_neighbors":20},{"RMSE":5.3405242911271005,"num_neighbors":21},{"RMSE":5.3922427790028795,"num_neighbors":22},{"RMSE":5.44295673910584,"num_neighbors":23},{"RMSE":5.47931644940631,"num_neighbors":24},{"RMSE":5.523618789257428,"num_neighbors":25},{"RMSE":5.567065728104048,"num_neighbors":26},{"RMSE":5.602044586338607,"num_neighbors":27},{"RMSE":5.641831179764594,"num_neighbors":28},{"RMSE":5.675274254551784,"num_neighbors":29},{"RMSE":5.708321768081241,"num_neighbors":30}]},"encoding":{"x":{"field":"num_neighbors","scale":{"domain":[1,31]},"type":"quantitative"},"y":{"field":"RMSE","scale":{"domain":[2.6,6.0]},"type":"quantitative"}},"height":630,"mark":{"color":"gray","point":true,"strokeDash":[5,5],"type":"line"},"title":{"offset":20,"text":"RMSE vs. Number of Neighbors"},"width":630} +``` + +```elixir +best_neighbors = Nx.add(Nx.argmin(res), min_num_neighbors) |> Nx.to_number() +``` + + + +``` +2 +``` + +The best value is 2. Now try this value to a final prediction. + +```elixir +best_model = + KNN.fit(train_data, train_labels, + num_neighbors: best_neighbors, + num_classes: 3, + task: :regression + ) + +final_prediction = KNN.predict(best_model, test_data) +Regression.mean_square_error(test_labels, final_prediction) |> Nx.sqrt() +``` + + + +``` +#Nx.Tensor< + f64 + EXLA.Backend + 2.2414312639566227 +> +``` + +### Outliers prediction + + + +KNNs can be used for outlier detection. In this task, we will also use the Airfoil Self-Noise dataset. We will calculate for each sample a mean value of distances to the three closest points from dataset. + +```elixir +model = KNN.fit(x, y, num_neighbors: 3, task: :regression) +{num_samples, _} = Nx.shape(x) +{distances, indices} = KNN.k_neighbors(model, x) + +mean_distances = Nx.mean(distances, axes: [1], keep_axes: true) + +samples_mean_distances = + Nx.concatenate([mean_distances, Nx.iota({num_samples, 1})], axis: 1) + |> DF.new() + |> DF.rename(["mean_distances", "index"]) +``` + +Now, plot the mean values of distances. + +```elixir +Tucan.lineplot(samples_mean_distances, "index", "mean_distances", line_color: "jet") +|> Tucan.hruler(0.009, line_color: "red", stroke_dash: [5, 5]) +|> Tucan.Scale.set_x_domain(0, 1510) +|> Tucan.Scale.set_y_domain(0.0065, 0.012) +|> Tucan.set_size(800, 630) +|> Tucan.set_title("Plot to check anomalies", offset: 20) +``` + + + +```vega-lite 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distance to clusters","type":"quantitative"}},"mark":{"color":"jet","type":"line"}},{"data":{"values":[{"y":0.009}]},"encoding":{"y":{"field":"y","type":"quantitative"}},"mark":{"color":"red","strokeDash":[5,5],"type":"rule"}}],"title":{"offset":20,"text":"Plot to check anomalies"},"width":800} +``` + +We can assume, that samples with average distance bigger than 0.0105 can be treated as outliers. + +```elixir +data = DF.put(data, :is_outlier, Nx.greater(mean_distances, 0.0105)) +``` + +```elixir +alias VegaLite, as: Vl + +Vl.new( + title: [ + text: "Logarithm of Frequency vs Logarithm of Suction Side Displacement Thickness", + offset: 20 + ], + width: 630, + height: 630 +) +|> Vl.data_from_values(data) +|> Vl.layers([ + Vl.new() + |> Vl.param("brush", select: :interval) + |> Vl.mark(:circle) + |> Vl.encode_field(:x, "frequency_log", type: :quantitative, scale: [domain: [4.5, 10.5]]) + |> Vl.encode_field(:y, "suction_side_displacement_thickness_log", + type: :quantitative, + scale: [domain: [-8.0, -2.5]] + ) + |> Vl.encode(:color, + condition: [param: "brush", field: "is_outlier", type: :nominal], + value: :gray + ), + Vl.new() + |> Vl.transform(filter: [param: "brush", field: "is_outlier", equal: true]) + |> Vl.mark(:text) + |> Vl.encode(:x, value: 11) + |> Vl.encode(:y, value: -3.5) + |> Vl.encode_field(:text, "filter", + type: :quantitative, + aggregate: :count + ) +]) +``` + + + +```vega-lite 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of Frequency vs Logarithm of Suction Side Displacement Thickness"},"width":630} +``` diff --git a/notebooks/linear_regression.livemd b/notebooks/linear_regression.livemd index 20320ea0..b4fb0a35 100644 --- a/notebooks/linear_regression.livemd +++ b/notebooks/linear_regression.livemd @@ -11,7 +11,8 @@ Mix.install([ {:req, "~> 0.3.9"}, {:kino_vega_lite, "~> 0.1.9"}, {:kino, "~> 0.10.0"}, - {:kino_explorer, "~> 0.1.7"} + {:kino_explorer, "~> 0.1.7"}, + {:tucan, github: "pnezis/tucan", ref: "6549cdc"} ]) ``` @@ -26,7 +27,6 @@ Mix.install([ In the livebook, we will cover the typical use cases of linear regression on practical examples. ```elixir -alias VegaLite, as: Vl require Explorer.DataFrame, as: DF require Explorer.Series, as: S alias Scholar.Linear.LinearRegression, as: LR @@ -93,26 +93,11 @@ size = 100 df = DF.new(x: x, y: y) -Vl.new( - title: [ - text: "Scatterplot of Generated Data", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.data_from_values(df) -|> Vl.mark(:circle) -|> Vl.encode_field(:x, "x", - type: :quantitative, - scale: [domain: [-0.05, 2.05]], - axis: [grid: false] -) -|> Vl.encode_field(:y, "y", - type: :quantitative, - scale: [domain: [2.5, 12]], - axis: [grid: false] -) +Tucan.scatter(df, "x", "y", width: 630, height: 630, filled: true) +|> Tucan.Scale.set_x_domain(-0.05, 2.05) +|> Tucan.Scale.set_y_domain(2.5, 12) +|> Tucan.Grid.set_enabled(false) +|> Tucan.set_title("Scatterplot of Generated Data", offset: 20) ``` @@ -161,38 +146,15 @@ line = %{ y: [x_1 * coefficients + intercept, x_2 * coefficients + intercept] } -# Graph scales -x_scale = [domain: [-0.05, 2.05]] -y_scale = [domain: [2.5, 12]] - -Vl.new( - title: [ - text: "Scatterplot of Generated Data", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(df) - |> Vl.mark(:circle) - |> Vl.encode_field(:x, "x", - type: :quantitative, - scale: x_scale, - axis: [grid: false] - ) - |> Vl.encode_field(:y, "y", - type: :quantitative, - scale: y_scale, - axis: [grid: false] - ), - Vl.new() - |> Vl.data_from_values(line) - |> Vl.mark(:line, color: :green) - |> Vl.encode_field(:x, "x", type: :quantitative, scale: x_scale) - |> Vl.encode_field(:y, "y", type: :quantitative, scale: y_scale) +Tucan.layers([ + Tucan.scatter(df, "x", "y", filled: true), + Tucan.lineplot(line, "x", "y", line_color: "green") ]) +|> Tucan.Grid.set_enabled(false) +|> Tucan.Scale.set_x_domain(-0.05, 2.05) +|> Tucan.Scale.set_y_domain(2.5, 12) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Scatterplot of Generated Data", offset: 20) ``` @@ -229,39 +191,16 @@ prediction = %{ y: [y_prediction] } -Vl.new( - title: [ - text: "Scatterplot of Generated Data", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(df) - |> Vl.mark(:circle) - |> Vl.encode_field(:x, "x", - type: :quantitative, - scale: x_scale, - axis: [grid: false] - ) - |> Vl.encode_field(:y, "y", - type: :quantitative, - scale: y_scale, - axis: [grid: false] - ), - Vl.new() - |> Vl.data_from_values(line) - |> Vl.mark(:line, color: :green) - |> Vl.encode_field(:x, "x", type: :quantitative, scale: x_scale) - |> Vl.encode_field(:y, "y", type: :quantitative, scale: y_scale), - Vl.new() - |> Vl.data_from_values(prediction) - |> Vl.mark(:circle, color: :red, size: 80) - |> Vl.encode_field(:x, "x", type: :quantitative) - |> Vl.encode_field(:y, "y", type: :quantitative) +Tucan.layers([ + Tucan.scatter(df, "x", "y", filled: true), + Tucan.lineplot(line, "x", "y", line_color: "green"), + Tucan.scatter(prediction, "x", "y", point_color: "red", point_size: 80, filled: true) ]) +|> Tucan.Grid.set_enabled(false) +|> Tucan.Scale.set_x_domain(-0.05, 2.05) +|> Tucan.Scale.set_y_domain(2.5, 12) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Scatterplot of Generated Data", offset: 20) ``` @@ -332,30 +271,12 @@ Now let's plot the generated data: {x, y} = PolynomialData.data() df = DF.new(x: Nx.to_flat_list(x), y: Nx.to_flat_list(y)) -# Graph scales -x_scale = [domain: [-2.05, 2.05]] -y_scale = [domain: [2.5, 20]] - -Vl.new( - title: [ - text: "Scatterplot of Generated Data", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.data_from_values(df) -|> Vl.mark(:circle) -|> Vl.encode_field(:x, "x", - type: :quantitative, - scale: x_scale, - axis: [grid: false] -) -|> Vl.encode_field(:y, "y", - type: :quantitative, - scale: y_scale, - axis: [grid: false] -) +Tucan.scatter(df, "x", "y", filled: true) +|> Tucan.Scale.set_x_domain(-2.05, 2.05) +|> Tucan.Scale.set_y_domain(2.5, 20) +|> Tucan.Grid.set_enabled(false) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Scatterplot of Generated Data", offset: 20) ``` @@ -404,52 +325,14 @@ df_results = y: y_values |> Nx.to_flat_list() ) -# Plot the predicted values -Vl.new( - title: [ - text: "Scatterplot of Predictions", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(df) - |> Vl.mark(:circle) - |> Vl.encode_field(:x, "x", - type: :quantitative, - scale: x_scale, - axis: [grid: false] - ) - |> Vl.encode_field(:y, "y", - type: :quantitative, - scale: y_scale, - axis: [grid: false] - ), - Vl.new() - |> Vl.data_from_values(df_results) - |> Vl.mark(:line, color: :green) - |> Vl.encode_field(:x, "x", - type: :quantitative, - scale: x_scale - ) - |> Vl.encode_field(:y, "y", - type: :quantitative, - scale: y_scale - ), - Vl.new() - |> Vl.data_from_values(df_linear_results) - |> Vl.mark(:line, color: :red) - |> Vl.encode_field(:x, "x", - type: :quantitative, - scale: x_scale - ) - |> Vl.encode_field(:y, "y", - type: :quantitative, - scale: y_scale - ) +Tucan.layers([ + Tucan.scatter(df, "x", "y", filled: true), + Tucan.lineplot(df_results, "x", "y", line_color: "green"), + Tucan.lineplot(df_linear_results, "x", "y", line_color: "red", clip: true) ]) +|> Tucan.Grid.set_enabled(false) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Scatterplot of Generated Data", offset: 20) ``` @@ -534,31 +417,17 @@ df_prediction = y: y_predictions ) -Vl.new( - title: [ - text: "Plot of Generated Data and Predictions", - offset: 20 - ], - width: 630, - height: 630 -) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values(df) - |> Vl.mark(:circle) - |> Vl.encode_field(:x, "x", type: :quantitative, scale: x_scale, axis: [grid: false]) - |> Vl.encode_field(:y, "y", type: :quantitative, scale: y_scale, axis: [grid: false]), - Vl.new() - |> Vl.data_from_values(df_results) - |> Vl.mark(:line, color: :green) - |> Vl.encode_field(:x, "x", type: :quantitative, scale: x_scale) - |> Vl.encode_field(:y, "y", type: :quantitative, scale: y_scale), - Vl.new() - |> Vl.data_from_values(df_prediction) - |> Vl.mark(:circle, color: :red, size: 80) - |> Vl.encode_field(:x, "x", type: :quantitative, scale: x_scale) - |> Vl.encode_field(:y, "y", type: :quantitative, scale: y_scale) +x_scale = [domain: [-2.05, 2.05]] +y_scale = [domain: [2.5, 20]] + +Tucan.layers([ + Tucan.scatter(df, "x", "y", filled: true), + Tucan.lineplot(df_results, "x", "y", line_color: "green"), + Tucan.scatter(df_prediction, "x", "y", point_color: "red", point_size: 80, filled: true) ]) +|> Tucan.Grid.set_enabled(false) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Plot of Generated Data and Predictions", offset: 20) ``` @@ -609,24 +478,17 @@ Now, we will plot univariate histograms for each feature of the data set. # Increase the sample size (or use 1.0 to plot all data) sample = DF.sample(df, 0.2, seed: seed) -Vl.new( - title: [ - text: "Univariate Histograms of all features", - anchor: :middle - ], - width: 500, - height: 500, - columns: 3 -) -|> Vl.data_from_values(sample) -|> Vl.concat( +histograms = for name <- List.delete(df.names, "ocean_proximity") do - Vl.new() - |> Vl.mark(:bar) - |> Vl.encode_field(:x, name, bin: [bin: true, maxbins: 50], axis: [ticks: false]) - |> Vl.encode_field(:y, "value count", aggregate: :count) + Tucan.new() + |> Tucan.histogram(name, maxbins: 50, fill_opacity: 1.0) + |> Tucan.Axes.put_options(:x, ticks: false) end -) + +Tucan.new(sample, columns: 3) +|> Tucan.concat(histograms) +|> Tucan.set_title("Univariate Histograms of all features", anchor: :middle) +|> Tucan.set_size(500, 500) ``` @@ -642,44 +504,22 @@ From histograms, we can spot that *median_income* and *median_house_values* have Now let's render the houses as a scatter plot using the latitude and longitude, overlayed on top of California's map. Let's also use color to encode the house prices and use the circle size to indicate the population of districts. ```elixir -Vl.new( - title: [ - text: "Scatterplot of Generated Data", - offset: 20 - ], - width: 630, - height: 630 +Tucan.scatter(df, "longitude", "latitude", + filled: true, + tooltip: true, + only: ~w(latitude longitude median_house_value population) ) -|> Vl.layers([ - Vl.new() - |> Vl.data_from_values([ - %{ - url: - "https://raw.githubusercontent.com/ageron/handson-ml2/master/images/end_to_end_project/california.png" - } - ]) - |> Vl.mark(:image, - width: 630, - aspect: false, - align: :right - ) - |> Vl.encode_field(:url, "url", type: :nominal), - Vl.new() - |> Vl.data_from_values(df, only: ~w(latitude longitude median_house_value population)) - |> Vl.mark(:circle, opacity: 0.4) - |> Vl.encode_field(:x, "longitude", - type: :quantitative, - axis: [grid: false], - scale: [domain: [-124.55, -113.80]] - ) - |> Vl.encode_field(:y, "latitude", - type: :quantitative, - axis: [grid: false], - scale: [domain: [32.45, 42.05]] - ) - |> Vl.encode_field(:color, "median_house_value", type: :quantitative, scale: [scheme: :viridis]) - |> Vl.encode_field(:size, "population", type: :quantitative) -]) +|> Tucan.color_by("median_house_value", type: :quantitative) +|> Tucan.size_by("population") +|> Tucan.Scale.set_x_domain(-124.55, -113.80) +|> Tucan.Scale.set_y_domain(32.45, 42.05) +|> Tucan.Grid.set_enabled(false) +|> Tucan.Scale.set_color_scheme(:viridis) +|> Tucan.background_image( + "https://raw.githubusercontent.com/ageron/handson-ml2/master/images/end_to_end_project/california.png" +) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Scatterplot of Generated Data", offset: 20) ``` @@ -942,34 +782,13 @@ corr_to_plot = corr_val: Enum.map(correlation_list, fn x -> Float.round(x, 2) end) ) -Vl.new( - title: [ - text: "Correlation Matrix for California Housing", - offset: 20 - ], - width: 630, - height: 630 +Tucan.heatmap(corr_to_plot, "x", "y", "corr_val", + annotate: true, + text_color: [{nil, 0, "white"}, {0, nil, "black"}] ) -|> Vl.data_from_values(corr_to_plot) -|> Vl.layers([ - Vl.new() - |> Vl.mark(:rect) - |> Vl.encode_field(:x, "x", type: :nominal, title: "", sort: false) - |> Vl.encode_field(:y, "y", type: :nominal, title: "", sort: false) - |> Vl.encode_field(:color, "corr_val", type: :quantitative, scale: [scheme: :viridis]), - Vl.new() - |> Vl.mark(:text) - |> Vl.encode_field(:x, "x", type: :nominal, title: "") - |> Vl.encode_field(:y, "y", type: :nominal, title: "") - |> Vl.encode_field(:text, "corr_val", type: :quantitative) - |> Vl.encode_field(:color, "corr_val", - type: :quantitative, - condition: [ - [test: "datum['corr_val'] < 0", value: :white], - [test: "datum['corr_val'] >= 0", value: :black] - ] - ) -]) +|> Tucan.Scale.set_color_scheme(:viridis) +|> Tucan.set_size(630, 630) +|> Tucan.set_title("Correlation Matrix for California Housing", offset: 20) ```