08-bayes_regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<main>:9: warning: already initialized constant BETA\n",
      "<main>:7: warning: previous definition of BETA was here\n",
      "<main>:10: warning: already initialized constant ALPHA\n",
      "<main>:8: warning: previous definition of ALPHA was here\n",
      "<main>:11: warning: already initialized constant ORDER\n",
      "<main>:9: warning: previous definition of ORDER was here\n"
     ]
    },
    {
     "data": {
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       "        var id_name = '#vis-90943fcc-b6a3-4332-bc6a-8fd755c80201';\n",
       "        Nyaplot.core.parse(model, id_name);\n",
       "\n",
       "        require(['downloadable'], function(downloadable){\n",
       "          var svg = d3.select(id_name).select(\"svg\");\n",
       "\t  if(!svg.empty())\n",
       "\t    svg.call(downloadable().filename('fig'));\n",
       "\t});\n",
       "    };\n",
       "    if(window['Nyaplot']==undefined){\n",
       "        window.addEventListener('load_nyaplot', render, false);\n",
       "\treturn;\n",
       "    } else {\n",
       "       render();\n",
       "    }\n",
       "})();\n",
       "</script>\n"
      ],
      "text/plain": [
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       "        var id_name = '#vis-e5bc5a1f-ed55-467c-a31f-6fc1d9599031';\n",
       "        Nyaplot.core.parse(model, id_name);\n",
       "\n",
       "        require(['downloadable'], function(downloadable){\n",
       "          var svg = d3.select(id_name).select(\"svg\");\n",
       "\t  if(!svg.empty())\n",
       "\t    svg.call(downloadable().filename('fig'));\n",
       "\t});\n",
       "    };\n",
       "    if(window['Nyaplot']==undefined){\n",
       "        window.addEventListener('load_nyaplot', render, false);\n",
       "\treturn;\n",
       "    } else {\n",
       "       render();\n",
       "    }\n",
       "})();\n",
       "</script>\n"
      ],
      "text/plain": [
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   "source": [
    "# ベイズ推定による回帰分析\n",
    "\n",
    "require 'daru'\n",
    "require 'numo/narray'\n",
    "require 'open3'\n",
    "require 'nyaplot'\n",
    "\n",
    "DATASET_NUMS = [4, 5, 10, 100]\n",
    "\n",
    "BETA = 1.0 / (0.3) ** 2 # 真の分布の分散\n",
    "ALPHA = 1.0 / 100 ** 2  # 事前分布の分散\n",
    "ORDER = 9               # 多項式の字数\n",
    "\n",
    "def normal_rand(mu = 0, sigma = 1.0)\n",
    "  random = Random.new\n",
    "  (Math.sqrt(-2 * Math.log(random.rand)) * Math.sin(2 * Math::PI * random.rand) * sigma) + mu\n",
    "end\n",
    "\n",
    "# データセット {x_n,y_n} (n=1...N) を用意\n",
    "def create_dataset(num)\n",
    "  dataset = Daru::DataFrame.new({'x': [], 'y': []})  \n",
    "\n",
    "  num.times do |i|\n",
    "    x = i.to_f / (num - 1).to_f\n",
    "    y = Math.sin(2 * Math::PI * x) + normal_rand(0, 0.3)\n",
    "    dataset.add_row(Daru::Vector.new([x, y], index: [:x, :y]))\n",
    "  end\n",
    "  \n",
    "  return dataset\n",
    "end\n",
    "\n",
    "# 事後分布に基づく推定曲線、および、事後分布の平均と分散を計算\n",
    "def resolve(dataset, m)\n",
    "  t = dataset.y\n",
    "\n",
    "  columns = {}\n",
    "  (m+1).times do |i|\n",
    "    columns[\"x**#{i}\"] = dataset.x ** i\n",
    "  end\n",
    "  phis = Daru::DataFrame.new(columns)\n",
    "\n",
    "  phiphi = nil\n",
    "  phis.each_row_with_index do |line, index|\n",
    "    phi = Daru::DataFrame.new(x: line)\n",
    "    if index == 0\n",
    "      phiphi = phi.to_matrix * phi.transpose.to_matrix\n",
    "    else\n",
    "      phiphi += phi.to_matrix * phi.transpose.to_matrix\n",
    "    end\n",
    "  end\n",
    "\n",
    "  s_inv = Matrix[*(ALPHA * Numo::DFloat.eye(m + 1))] + BETA * phiphi\n",
    "  s = s_inv.inv # 事後分布の共分散行列\n",
    "\n",
    "  # 平均 m(x)\n",
    "  mean_fun = lambda {|x0|\n",
    "    phi_x0 = Numo::NArray[*((m + 1).times.map {|i| (x0 ** i).to_a })]\n",
    "    tmp = 0\n",
    "    phis.each_row_with_index do |line, index|\n",
    "      if index == 0\n",
    "        tmp = t[index] * Numo::NArray[*line.to_a]\n",
    "        next\n",
    "      end\n",
    "\n",
    "      tmp += t[index] * Numo::NArray[*line.to_a]\n",
    "    end\n",
    "    BETA * phi_x0.transpose.dot(Numo::NArray[*s.to_a]).dot(tmp)\n",
    "  }\n",
    "\n",
    "  # 標準偏差 s(x)\n",
    "  deviation_fun = lambda {|x0|\n",
    "    phi_x0 = Numo::NArray[*((m + 1).times.map {|i| (x0 ** i).to_a })]\n",
    "    deviation = (1.0 / BETA + phi_x0.transpose.dot(Numo::NArray[*s.to_a]).dot(phi_x0)).map {|v| v < 0 ? Float::NAN : Math.sqrt(v) }\n",
    "    deviation.diagonal\n",
    "  }\n",
    "\n",
    "  tmp = nil\n",
    "  phis.each_row_with_index do |line, index|\n",
    "    if index == 0\n",
    "      tmp = t[index] * Numo::NArray[*line.to_a]\n",
    "      next\n",
    "    end\n",
    "\n",
    "    tmp += t[index] * Numo::NArray[*line.to_a]\n",
    "  end\n",
    "  mean = BETA * Numo::NArray[*s.to_a].dot(tmp).flatten\n",
    "\n",
    "  return mean_fun, deviation_fun, mean, s\n",
    "end\n",
    "\n",
    "fig1 = Nyaplot::Frame.new\n",
    "fig2 = Nyaplot::Frame.new\n",
    "\n",
    "DATASET_NUMS.each do |num|\n",
    "  train_set = create_dataset(num)\n",
    "  mean_fun, deviation_fun, mean, sigma = resolve(train_set, ORDER)\n",
    "  command = \"python -c 'import numpy; print numpy.random.multivariate_normal(#{mean.to_a.inspect}, #{sigma.to_a.inspect}, 4).tolist()'\"\n",
    "  output, std_error, status = Open3.capture3(command)\n",
    "  ws_samples = Daru::DataFrame.rows(eval(output))\n",
    "  \n",
    "  # トレーニングセットを表示\n",
    "  plot1 = Nyaplot::Plot.new\n",
    "  scatter1 = plot1.add(:scatter, train_set.x.to_a, train_set.y.to_a)\n",
    "  scatter1.color('blue')\n",
    "  scatter1.title('train_set')\n",
    "  \n",
    "  plot1.configure do\n",
    "    x_label(\"N=#{num}\")\n",
    "    y_label('')\n",
    "    xrange([-0.05, 1.05])\n",
    "    yrange([-2, 2])\n",
    "    legend(true)\n",
    "    height(300)\n",
    "    width(490)\n",
    "  end\n",
    "  \n",
    "  plot2 = Nyaplot::Plot.new\n",
    "  scatter2 = plot2.add(:scatter, train_set.x.to_a, train_set.y.to_a)\n",
    "  scatter2.color('blue')\n",
    "  scatter2.title('train_set')\n",
    "\n",
    "  plot2.configure do\n",
    "    x_label(\"N=#{num}\")\n",
    "    y_label('')\n",
    "    xrange([-0.05, 1.05])\n",
    "    yrange([-2, 2])\n",
    "    legend(true)\n",
    "    height(300)\n",
    "    width(490)\n",
    "  end\n",
    "  \n",
    "  linex = Numo::NArray[*(0..1).step(0.01).to_a]\n",
    "\n",
    "  # 真の曲線を表示\n",
    "  liney = (2 * Math::PI * linex).map {|v| Math.sin(v) }\n",
    "  collect_line = plot1.add(:line, linex, liney)\n",
    "  collect_line.color('green')\n",
    "  collect_line.title('collect')\n",
    "\n",
    "  # 平均と標準偏差の曲線を表示\n",
    "  m = mean_fun.call(linex)\n",
    "  d = deviation_fun.call(linex)\n",
    "  mean_line = plot1.add(:line, linex, m)\n",
    "  mean_line.color('red')\n",
    "  mean_line.title('mean')\n",
    "  lower_std_line = plot1.add(:line, linex, m - d)\n",
    "  lower_std_line.color('black')\n",
    "  lower_std_line.title('')\n",
    "  upper_std_line = plot1.add(:line, linex, m + d)\n",
    "  upper_std_line.color('black')\n",
    "  upper_std_line.title('')\n",
    "\n",
    "  # 多項式のサンプルを表示\n",
    "  liney = m\n",
    "  mean_line = plot2.add(:line, linex, liney)\n",
    "  mean_line.color('red')\n",
    "  mean_line.title('mean')\n",
    "  \n",
    "  f = lambda {|x, ws|\n",
    "    y = 0\n",
    "    ws.each_with_index do |w, i|\n",
    "      y += w * (x ** i.to_i)\n",
    "    end\n",
    "    y\n",
    "  }\n",
    "\n",
    "  ws_samples.each_row do |ws|\n",
    "    liney = f.call(linex, ws)\n",
    "    sample_line = plot2.add(:line, linex, liney)\n",
    "    sample_line.color('pink')\n",
    "    sample_line.title('sample')\n",
    "  end\n",
    "  \n",
    "  fig1.add(plot1)\n",
    "  fig2.add(plot2)\n",
    "end\n",
    "\n",
    "fig1.show\n",
    "fig2.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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