07. Regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ScikitLearn.CrossValidation \n",
    "using Plots \n",
    "using Statistics\n",
    "using PyCall \n",
    "using DataFrames \n",
    "using GLM  # Generalized Linear Models in Julia \n",
    "using CSV\n",
    "using MLBase\n",
    "using LsqFit\n",
    "using RDatasets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PyObject <module 'numpy' from '/home/pabvald/.venv_py37/lib/python3.8/site-packages/numpy/__init__.py'>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#np = PyCall.pyimport(\"numpy\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data preparation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DataFrame"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset = CSV.read(\"data/real-state.csv\");\n",
    "typeof(dataset)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{String,1}:\n",
       " \"No\"\n",
       " \"X1 transaction date\"\n",
       " \"X2 house age\"\n",
       " \"X3 distance to the nearest MRT station\"\n",
       " \"X4 number of convenience stores\"\n",
       " \"X5 latitude\"\n",
       " \"X6 longitude\"\n",
       " \"Y house price of unit area\""
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset_names = names(dataset)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We rename the dataset columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{String,1}:\n",
       " \"No\"\n",
       " \"X1\"\n",
       " \"X2\"\n",
       " \"X3\"\n",
       " \"X4\"\n",
       " \"X5\"\n",
       " \"X6\"\n",
       " \"Y\""
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rename!(dataset, Symbol(\"X1 transaction date\") => :X1,\n",
    "                 Symbol(\"X2 house age\") => :X2,\n",
    "                 Symbol(\"X3 distance to the nearest MRT station\")=> :X3,\n",
    "                 Symbol(\"X4 number of convenience stores\") => :X4,\n",
    "                 Symbol(\"X5 latitude\") => :X5,\n",
    "                 Symbol(\"X6 longitude\") => :X6,\n",
    "                 Symbol(\"Y house price of unit area\") => :Y                 \n",
    "                )\n",
    "names(dataset)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = Matrix(dataset[:,[:X1,:X2, :X3, :X4, :X5, :X6]]); # house features\n",
    "Y = Vector(dataset[:,:Y]); # house price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(414, 6)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(414,)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size(Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since all the features have positive values, I scale the features so that they have the same weigth."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "scale! (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function standarize!(X)\n",
    "    for col in Array(1:size(X,2))\n",
    "        X[:,col] = (X[:, col] .- mean(X[:,col])) ./ std(X[:, col])\n",
    "    end\n",
    "end\n",
    "\n",
    "function scale!(X)    \n",
    "    for col in Array(1:size(X,2))\n",
    "        minimum = min(X[:,col]...)\n",
    "        maximum = max(X[:, col]...)\n",
    "        X[:,col] = (X[:, col] .- minimum) ./ (maximum -minimum)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "scale!(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot some of the features vs the house's price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(X[:,2], Y, label=nothing, title=\"Age vs Price\", markercolor=:red)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "lm(X, y, allowrankdeficient::Bool=false; wts=similar(y, 0))\n",
       "\\end{verbatim}\n",
       "An alias for \\texttt{fit(LinearModel, X, y, allowrankdeficient)}\n",
       "\n",
       "The arguments \\texttt{X} and \\texttt{y} can be a \\texttt{Matrix} and a \\texttt{Vector} or a \\texttt{Formula} and a \\texttt{DataFrame}.\n",
       "\n",
       "The keyword argument \\texttt{wts} can be a \\texttt{Vector} specifying frequency weights for observations. Such weights are equivalent to repeating each observation a number of times equal to its weight. Do note that this interpretation gives equal point estimates but different standard errors from analytical (a.k.a. inverse variance) weights and from probability (a.k.a. sampling) weights which are the default in some other software.\n",
       "\n"
      ],
      "text/markdown": [
       "```\n",
       "lm(X, y, allowrankdeficient::Bool=false; wts=similar(y, 0))\n",
       "```\n",
       "\n",
       "An alias for `fit(LinearModel, X, y, allowrankdeficient)`\n",
       "\n",
       "The arguments `X` and `y` can be a `Matrix` and a `Vector` or a `Formula` and a `DataFrame`.\n",
       "\n",
       "The keyword argument `wts` can be a `Vector` specifying frequency weights for observations. Such weights are equivalent to repeating each observation a number of times equal to its weight. Do note that this interpretation gives equal point estimates but different standard errors from analytical (a.k.a. inverse variance) weights and from probability (a.k.a. sampling) weights which are the default in some other software.\n"
      ],
      "text/plain": [
       "\u001b[36m  lm(X, y, allowrankdeficient::Bool=false; wts=similar(y, 0))\u001b[39m\n",
       "\n",
       "  An alias for \u001b[36mfit(LinearModel, X, y, allowrankdeficient)\u001b[39m\n",
       "\n",
       "  The arguments \u001b[36mX\u001b[39m and \u001b[36my\u001b[39m can be a \u001b[36mMatrix\u001b[39m and a \u001b[36mVector\u001b[39m or a \u001b[36mFormula\u001b[39m and a\n",
       "  \u001b[36mDataFrame\u001b[39m.\n",
       "\n",
       "  The keyword argument \u001b[36mwts\u001b[39m can be a \u001b[36mVector\u001b[39m specifying frequency weights for\n",
       "  observations. Such weights are equivalent to repeating each observation a\n",
       "  number of times equal to its weight. Do note that this interpretation gives\n",
       "  equal point estimates but different standard errors from analytical (a.k.a.\n",
       "  inverse variance) weights and from probability (a.k.a. sampling) weights\n",
       "  which are the default in some other software."
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?GLM.lm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### a) Extracting the features and the Y in matrixes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is necessary to add a columns of 'ones' so that the LinearModel computes the intercept."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×2 Array{Float64,2}:\n",
       " 1.0  1.0\n",
       " 1.0  0.9\n",
       " 1.0  0.5\n",
       " 1.0  0.5"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1 = hcat(ones(size(X,1)),X[:,4]);\n",
    "X1[1:4,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearModel{GLM.LmResp{Array{Float64,1}},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}}:\n",
       "\n",
       "Coefficients:\n",
       "──────────────────────────────────────────────────────────────\n",
       "      Coef.  Std. Error      t  Pr(>|t|)  Lower 95%  Upper 95%\n",
       "──────────────────────────────────────────────────────────────\n",
       "x1  27.1811     0.94192  28.86    <1e-99    25.3295    29.0327\n",
       "x2  26.3765     1.86829  14.12    <1e-36    22.704     30.0491\n",
       "──────────────────────────────────────────────────────────────\n"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = lm(X1,Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "x1 is the intercept"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b) Using the @formula macro"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Array{Float64,1}},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}},Array{Float64,2}}\n",
       "\n",
       "Y ~ 1 + X\n",
       "\n",
       "Coefficients:\n",
       "───────────────────────────────────────────────────────────────────────\n",
       "               Coef.  Std. Error      t  Pr(>|t|)  Lower 95%  Upper 95%\n",
       "───────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  27.1811     0.94192  28.86    <1e-99    25.3295    29.0327\n",
       "X            26.3765     1.86829  14.12    <1e-36    22.704     30.0491\n",
       "───────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(X=X[:,4], Y=Y)\n",
    "model = lm(@formula(Y ~ X), data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot!(X[:,4], predict(model))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "standarize! (generic function with 1 method)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function standarize!(X)\n",
    "    for j = 1:size(X,2)\n",
    "        X[:,j] = (X[:,j] .- mean(X[:,j])) ./ std(X[:,j])\n",
    "    end\n",
    "    nothing    \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are cases when the predictions should be binary (0 or 1). In such cases, we can use logistic regression so that every predicted value is in the interval [0,1]. Once we have the predictions, we can set a threshold to determine which instances are classified as 0 and which ones are classified as 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(X=[1,2,3,4,5,6,7], Y=[1,0,1,1,1,1,1])\n",
    "linear_reg = lm(@formula(Y ~ X), data)\n",
    "scatter(data.X, data.Y, legend=false, size=(400,300))\n",
    "plot!(data.X, predict(linear_reg))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To apply Logistic Regression, we will use a dataset contained in the RDataset.jl package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nunique</th><th>nmissing</th><th>eltype</th></tr><tr><th></th><th>Symbol</th><th>Union…</th><th>Any</th><th>Union…</th><th>Any</th><th>Union…</th><th>Nothing</th><th>DataType</th></tr></thead><tbody><p>3 rows × 8 columns</p><tr><th>1</th><td>Sex</td><td></td><td>F</td><td></td><td>M</td><td>2</td><td></td><td>CategoricalValue{String,UInt8}</td></tr><tr><th>2</th><td>BWt</td><td>-6.96973e-16</td><td>-1.49104</td><td>-0.0486519</td><td>2.42401</td><td></td><td></td><td>Float64</td></tr><tr><th>3</th><td>HWt</td><td>3.51571e-16</td><td>-1.77873</td><td>-0.21792</td><td>4.05377</td><td></td><td></td><td>Float64</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccccc}\n",
       "\t& variable & mean & min & median & max & nunique & nmissing & eltype\\\\\n",
       "\t\\hline\n",
       "\t& Symbol & Union… & Any & Union… & Any & Union… & Nothing & DataType\\\\\n",
       "\t\\hline\n",
       "\t1 & Sex &  & F &  & M & 2 &  & CategoricalValue\\{String,UInt8\\} \\\\\n",
       "\t2 & BWt & -6.96973e-16 & -1.49104 & -0.0486519 & 2.42401 &  &  & Float64 \\\\\n",
       "\t3 & HWt & 3.51571e-16 & -1.77873 & -0.21792 & 4.05377 &  &  & Float64 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "3×8 DataFrame. Omitted printing of 2 columns\n",
       "│ Row │ variable │ mean         │ min      │ median     │ max     │ nunique │\n",
       "│     │ \u001b[90mSymbol\u001b[39m   │ \u001b[90mUnion…\u001b[39m       │ \u001b[90mAny\u001b[39m      │ \u001b[90mUnion…\u001b[39m     │ \u001b[90mAny\u001b[39m     │ \u001b[90mUnion…\u001b[39m  │\n",
       "├─────┼──────────┼──────────────┼──────────┼────────────┼─────────┼─────────┤\n",
       "│ 1   │ Sex      │              │ F        │            │ M       │ 2       │\n",
       "│ 2   │ BWt      │ -6.96973e-16 │ -1.49104 │ -0.0486519 │ 2.42401 │         │\n",
       "│ 3   │ HWt      │ 3.51571e-16  │ -1.77873 │ -0.21792   │ 4.05377 │         │"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cats = RDatasets.dataset(\"MASS\", \"cats\");\n",
    "standarize!(cats.BWt)\n",
    "standarize!(cats.HWt)\n",
    "describe(cats)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LabelMap (with 2 labels):\n",
       "[1] F\n",
       "[2] M\n"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lmap = labelmap(cats.Sex)\n",
    "le = labelencode(lmap, cats.Sex);\n",
    "lmap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Female cats (blue) seem to be more present in the lower left corner and Males (orange) seem be present in the top right corner"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(cats.BWt, cats.HWt, color=le, legend=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The *GLM* package allows to apply logistic regression by using the *Binomial()* distribution and the *LogitLink()* link function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StatsModels.TableRegressionModel{GeneralizedLinearModel{GLM.GlmResp{Array{Float64,1},Binomial{Float64},LogitLink},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}},Array{Float64,2}}\n",
       "\n",
       "Y ~ 1 + X1\n",
       "\n",
       "Coefficients:\n",
       "──────────────────────────────────────────────────────────────────────\n",
       "               Coef.  Std. Error     z  Pr(>|z|)  Lower 95%  Upper 95%\n",
       "──────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  1.22128    0.267214  4.57     <1e-5   0.697547    1.74501\n",
       "X1           1.76416    0.327353  5.39     <1e-7   1.12256     2.40576\n",
       "──────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(X1=cats.BWt, X2=cats.HWt, Y=le.-1)\n",
    "logistic_reg = glm(@formula(Y ~ X1), data, Binomial(), LogitLink())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(data.X1, data.Y, label=\"gt gender\", color=6)\n",
    "scatter!(data.X1, predict(logistic_reg), label=\"predicted gender\", color=7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Non Linear Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes we might want to fit a non-linear function to a set of points. In such a case we can use the *LsqFit* package."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can used some artificial data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xvals = 0:0.05:10\n",
    "yvals = 1*exp.(-xvals*2) + 2*sin.(0.8*pi*xvals) + 0.15 * randn(length(xvals));\n",
    "scatter(xvals,yvals,legend=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nlmodel (generic function with 1 method)"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@. nlmodel(x,p) = p[1]*exp(-x*p[2]) + p[3]*sin(0.8*pi*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LsqFit.LsqFitResult{Array{Float64,1},Array{Float64,1},Array{Float64,2},Array{Float64,1}}([0.8757429665112211, 1.698487702493823, 1.9893619091483141], [-0.005597428649230407, 0.009275341845013774, -0.16687780107961814, 0.03552489758210631, 0.3238014673875884, -0.11619549704915921, 0.07464205868326146, -0.0895017545312724, 0.06469204343231238, -0.07864541373529033  …  -0.08978652588567759, -0.054338281086858675, -0.10588190351995896, 0.2233601639237075, 0.03769580373912995, 0.06902053887174942, -0.11207515944391455, -0.0938471284375218, -0.07401530339848988, -0.156991297152993], [1.0000000000052196 0.0 0.0; 0.9185817402187264 -0.040222074908445496 0.12533323355936565; … ; 4.575774015793474e-8 -3.9869757475265434e-7 -0.12533323356282172; 4.203130093641807e-8 -3.6807772838632534e-7 0.0], true, Float64[])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p0 = [0.5, 0.5, 0.5]\n",
    "nlfit = curve_fit(nlmodel, xvals, yvals, p0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = nlfit.param\n",
    "findyvals = p[1]*exp.(-xvals*p[2]) + p[3]*sin.(0.8*pi*xvals)\n",
    "scatter(xvals,yvals,legend=false)\n",
    "plot!(xvals,findyvals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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