03-maximum_likelihood.ipynb

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       "        var id_name = '#vis-ebc1f085-64ec-469b-8bab-a901433cf75f';\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 model = {\"panes\":[{\"diagrams\":[{\"type\":\"line\",\"options\":{\"x\":\"m\",\"y\":\"Test set\",\"title\":\"Test set\",\"color\":\"green\"},\"data\":\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\"},{\"type\":\"line\",\"options\":{\"x\":\"m\",\"y\":\"Training set\",\"title\":\"Training set\",\"color\":\"blue\"},\"data\":\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\"}],\"options\":{\"x_label\":\"Log likelihood for N=10\",\"y_label\":\"\",\"legend\":true,\"zoom\":true,\"width\":800,\"xrange\":[0,8],\"yrange\":[-10.959248334611097,54.78957627143174]}}],\"data\":{\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\":[{\"Test set\":-10.959248334611097,\"Training set\":-10.724659158818707,\"m\":0},{\"Test set\":-8.394954941428635,\"Training set\":-8.35172452595085,\"m\":1},{\"Test set\":-8.405406955434055,\"Training set\":-8.293834872758055,\"m\":2},{\"Test set\":6.22863542433814,\"Training set\":6.69355894304122,\"m\":3},{\"Test set\":6.227504317715638,\"Training set\":6.6936262828487525,\"m\":4},{\"Test set\":3.442481754400797,\"Training set\":9.935165809299908,\"m\":5},{\"Test set\":2.6903231988997343,\"Training set\":12.664068141026764,\"m\":6},{\"Test set\":2.7092083106614098,\"Training set\":12.894157291800312,\"m\":7},{\"Test set\":3.327909256286116,\"Training set\":54.78957627143174,\"m\":8}]},\"extension\":[]}\n",
       "        var id_name = '#vis-89fd7f3f-faaf-4f4e-bb59-4b4fdbecc85c';\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": [
       "#<Nyaplot::Frame:0x007f15ad6d19f0 @properties={:panes=>[#<Nyaplot::Plot:0x007f15ad6b90a8 @properties={:diagrams=>[#<Nyaplot::Diagram:0x007f15ad6fc5b0 @properties={:type=>:line, :options=>{:x=>:m, :y=>:\"Test set\", :title=>\"Test set\", :color=>\"green\"}, :data=>\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\"}, @xrange=[0, 8], @yrange=[-10.959248334611097, 6.22863542433814]>, #<Nyaplot::Diagram:0x007f15ad6e7a48 @properties={:type=>:line, :options=>{:x=>:m, :y=>:\"Training set\", :title=>\"Training set\", :color=>\"blue\"}, :data=>\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\"}, @xrange=[0, 8], @yrange=[-10.724659158818707, 54.78957627143174]>], :options=>{:x_label=>\"Log likelihood for N=10\", :y_label=>\"\", :legend=>true, :zoom=>true, :width=>800, :xrange=>[0, 8], :yrange=>[-10.959248334611097, 54.78957627143174]}}>], :data=>{\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\"=>#<Nyaplot::DataFrame:0x007f15ad6fd5c8 @name=\"109907ad-2330-4a2c-ab16-8723fdf8e7c8\", @rows=[{:\"Test set\"=>-10.959248334611097, :\"Training set\"=>-10.724659158818707, :m=>0}, {:\"Test set\"=>-8.394954941428635, :\"Training set\"=>-8.35172452595085, :m=>1}, {:\"Test set\"=>-8.405406955434055, :\"Training set\"=>-8.293834872758055, :m=>2}, {:\"Test set\"=>6.22863542433814, :\"Training set\"=>6.69355894304122, :m=>3}, {:\"Test set\"=>6.227504317715638, :\"Training set\"=>6.6936262828487525, :m=>4}, {:\"Test set\"=>3.442481754400797, :\"Training set\"=>9.935165809299908, :m=>5}, {:\"Test set\"=>2.6903231988997343, :\"Training set\"=>12.664068141026764, :m=>6}, {:\"Test set\"=>2.7092083106614098, :\"Training set\"=>12.894157291800312, :m=>7}, {:\"Test set\"=>3.327909256286116, :\"Training set\"=>54.78957627143174, :m=>8}]>}, :extension=>[]}>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "require 'daru'\n",
    "require 'nyaplot'\n",
    "\n",
    "N = 10\n",
    "M = [0, 1, 3, 9]\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",
    "\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",
    "def log_likelihood(dataset, f)\n",
    "  dev = 0.0\n",
    "  n = dataset.size.to_f\n",
    "  dataset.each_row do |line|\n",
    "    x = line.x\n",
    "    y = line.y\n",
    "    dev += (y - f.call(x))**2\n",
    "  end\n",
    "  err = dev * 0.5\n",
    "  beta = n / dev\n",
    "  -beta * err + 0.5 * n * Math.log(0.5 * beta / Math::PI)\n",
    "end\n",
    "\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",
    "  phi = Daru::DataFrame.new(columns)\n",
    "\n",
    "  tmp = (phi.transpose.to_matrix * phi.to_matrix).inv\n",
    "  ws = (tmp * phi.transpose.to_matrix) * Vector.elements(t.to_a)\n",
    "\n",
    "  f = lambda {|x|\n",
    "    y = 0\n",
    "    ws.each_with_index do |w, i|\n",
    "      y += w * (x ** i)\n",
    "    end\n",
    "\n",
    "    y\n",
    "  }\n",
    "\n",
    "  sigma2 = 0.0\n",
    "  dataset.each_row do |line|\n",
    "    sigma2 += (f.call(line.x) - line.y)**2\n",
    "  end\n",
    "  sigma2 /= dataset.size\n",
    "\n",
    "  return f, ws, Math.sqrt(sigma2)\n",
    "end\n",
    "\n",
    "train_set = create_dataset(N)\n",
    "test_set = create_dataset(N)\n",
    "df_ws = {}\n",
    "\n",
    "fig = Nyaplot::Frame.new\n",
    "\n",
    "M.each_with_index do |m, c|\n",
    "  f, ws, sigma = resolve(train_set, m)\n",
    "  df_ws[\"M=#{m}\"] = Daru::Vector.new(ws, name: \"M=#{m}\")\n",
    "  \n",
    "  plot = Nyaplot::Plot.new\n",
    "  sc = plot.add_with_df(train_set.to_nyaplotdf, :scatter, :x, :y)\n",
    "  sc.title(\"train_set\")\n",
    "  sc.color('blue')\n",
    "  \n",
    "  linex = (0..1).step(1.0 / (101 - 1)).to_a\n",
    "  liney = linex.map do |x|\n",
    "    Math.sin(2 * Math::PI * x)\n",
    "  end\n",
    "  line_answer = plot.add(:line, linex, liney)\n",
    "  line_answer.title('answer')\n",
    "  line_answer.color('green')\n",
    "  \n",
    "  liney = linex.map do |x|\n",
    "    f.call(x)\n",
    "  end\n",
    "  line_middle = plot.add(:line, linex, liney)\n",
    "  line_middle.title(\"Sigma=#{sprintf(\"%.2f\", sigma)}\")\n",
    "  line_middle.color('red')\n",
    "  line_upper = plot.add(:line, linex, liney.map {|y| y + sigma })\n",
    "  line_upper.color('red')\n",
    "  line_lower = plot.add(:line, linex, liney.map {|y| y - sigma } )\n",
    "  line_lower.color('red')\n",
    "\n",
    "  plot.configure do\n",
    "    x_label(\"M=#{m}\")\n",
    "    y_label('')\n",
    "    xrange([-0.05, 1.05])\n",
    "    yrange([-1.5, 1.5])\n",
    "    legend(true)\n",
    "    height(300)\n",
    "    width(490)\n",
    "  end\n",
    "  \n",
    "  fig.add(plot)\n",
    "end\n",
    "\n",
    "fig.show\n",
    "\n",
    "df = Daru::DataFrame.new({m: [], 'Test set': [], 'Training set': []})  \n",
    "9.times do |m|\n",
    "  f, ws, sigma = resolve(train_set, m)\n",
    "  train_mlh = log_likelihood(train_set, f)\n",
    "  test_mlh = log_likelihood(test_set, f)\n",
    "  df.add_row(Daru::Vector.new([m, test_mlh, train_mlh], index: [:m, 'Test set'.to_sym, 'Training set'.to_sym]))\n",
    "end\n",
    "\n",
    "df.plot(type: [:line, :line], x: [:m, :m], y: ['Test set'.to_sym, 'Training set'.to_sym]) do |plot, diagrams|\n",
    "  test_set_diagram = diagrams[0]\n",
    "  train_set_diagram = diagrams[1]\n",
    "  \n",
    "  train_set_diagram.title('Training set')\n",
    "  train_set_diagram.color('blue')\n",
    "  test_set_diagram.title('Test set')\n",
    "  test_set_diagram.color('green')\n",
    "  \n",
    "  plot.x_label(\"Log likelihood for N=#{N}\")\n",
    "  plot.y_label('')\n",
    "  plot.legend(true)\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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