linear_models.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# math \n",
    "import numpy as np\n",
    "\n",
    "# random\n",
    "from numpy.random import rand, randn\n",
    "np.random.seed(6)\n",
    "\n",
    "# dataframes\n",
    "import pandas as pd\n",
    "\n",
    "# plotting\n",
    "import matplotlib.pyplot as plt\n",
    "# linear regression two ways\n",
    "import statsmodels.api as sm\n",
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "# pytorch utils\n",
    "import torch\n",
    "from torch import optim\n",
    "import torch.nn as nn\n",
    "import torch.nn.functional as F\n",
    "\n",
    "torch.manual_seed(324891)\n",
    "\n",
    "from patsy import dmatrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot line y = w*x+b\n",
    "def plotline(w,b, xmin=-100,xmax=100,nsamples=100,label=''):\n",
    "    x = np.linspace(xmin, xmax, nsamples)\n",
    "    y = w * x + b\n",
    "    plt.plot(x, y, color='k', label=label)\n",
    "    plt.legend(loc=\"upper left\")\n",
    "    plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot function y = f(x)\n",
    "def plotfunc(f, xmin=-100,xmax=100,nsamples=100,label=\"\"):\n",
    "    x = np.linspace(xmin, xmax, nsamples)\n",
    "    y = f(x)\n",
    "    plt.plot(x, y, color='k', label=label)\n",
    "    plt.legend(loc=\"upper left\")\n",
    "    plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate and plot data\n",
    "n = 15\n",
    "def sample_data(num_points):\n",
    "    x = rand(num_points)*10\n",
    "    y = .2 + .2*x + .1*np.sin(x) + .1*randn(num_points) - .01*((x-5)/6)**4\n",
    "    return x,y\n",
    "\n",
    "x,y = sample_data(n)\n",
    "\n",
    "\n",
    "def plotdata(x=x,y=y, margin=.05):\n",
    "    plt.scatter(x,y, label=\"data\")\n",
    "    plt.legend(loc=\"upper left\")\n",
    "    plt.xlabel(\"x\") \n",
    "    plt.ylabel(\"y\")     \n",
    "    range_y = np.max(y) - np.min(y)\n",
    "    range_x = np.max(x) - np.min(x)\n",
    "    plt.ylim([np.min(y)-margin*range_y,np.max(y)+margin*range_y])\n",
    "    plt.xlim([np.min(x)-margin*range_x,np.max(x)+margin*range_x])\n",
    "    \n",
    "plotdata()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Approximating with the mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-5-6dd8b0414b56>:3: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n",
      "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n",
      "  w = np.linalg.lstsq(X, y)[0]\n"
     ]
    }
   ],
   "source": [
    "# the mean solves a very simple least squares problem:\n",
    "X = np.ones((n,1))\n",
    "w = np.linalg.lstsq(X, y)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.220446049250313e-16"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check the solution to our least squares problem is the mean\n",
    "np.abs(np.mean(y) - w[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the fit\n",
    "plotdata()\n",
    "plotline(0, w[0], label=\"mean\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is this a good model? Would you trust this model to make predictions on new inputs $x$?\n",
    "\n",
    "* (A) yes \n",
    "* (B) no"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Approximating with a line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[8.92860151 1.        ]\n",
      " [3.31979805 1.        ]\n",
      " [8.21229123 1.        ]\n",
      " [0.41696626 1.        ]\n",
      " [1.0765668  1.        ]\n",
      " [5.95052064 1.        ]\n",
      " [5.29817362 1.        ]\n",
      " [4.18807429 1.        ]\n",
      " [3.35407849 1.        ]\n",
      " [6.22519432 1.        ]\n",
      " [4.38141426 1.        ]\n",
      " [7.35882106 1.        ]\n",
      " [5.18036412 1.        ]\n",
      " [5.788586   1.        ]\n",
      " [6.45355096 1.        ]]\n"
     ]
    }
   ],
   "source": [
    "X = np.column_stack([x, np.ones(n)])\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How to fit a least squares model\n",
    "There are many ways to fit a least squares model in python!\n",
    "1. QR decomposition + `scipy.linalg.solve`\n",
    "2. `numpy.linalg.lstsq`\n",
    "3. `statsmodels.OLS`\n",
    "4. `sklearn.LinearRegression()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.20462642, 0.21131099])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# method 0: scipy\n",
    "from scipy.linalg import qr, solve\n",
    "Q,R = qr(X, mode='economic')\n",
    "w = solve(R, Q.T @ y)\n",
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-10-4844ed0cbb71>:2: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n",
      "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n",
      "  w = np.linalg.lstsq(X, y)[0]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0.20462642, 0.21131099])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# method 1: numpy\n",
    "w = np.linalg.lstsq(X, y)[0]\n",
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.20462642, 0.21131099])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# method 2: statsmodels\n",
    "model = sm.OLS(y, X).fit()\n",
    "w = model.params\n",
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.20462642, 0.21131099])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# method 3: sklearn\n",
    "model = LinearRegression().fit(x[None,].T,y)\n",
    "w = np.array([model.coef_[0], model.intercept_])\n",
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we have to do some work to reconcile the default types of NumPy and PyTorch. Note that NumPy will default to float64 on a 64-bit system. PyTorch models default to float32 as GPUs can do up to 64x the operations per clock cycle when using float32 instead of float64. \n",
    "\n",
    "[CUDA operations per cycle comparison](https://docs.nvidia.com/cuda/cuda-c-programming-guide/index.html#arithmetic-instructions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.17200051, 0.23789968], dtype=float32)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# method 4: PyTorch via SGD\n",
    "\n",
    "def TorchLinRegModel(n : int = 2):\n",
    "    \"\"\"Create a simple linear regression model in PyTorch. Note that\n",
    "    we're manually including the bias in this demo.\"\"\"\n",
    "    return nn.Linear(in_features=n, out_features=1, bias=False)\n",
    "\n",
    "model = TorchLinRegModel()\n",
    "model.zero_grad()\n",
    "optimizer = optim.SGD(model.parameters(), lr=5e-6)\n",
    "loss_function = nn.MSELoss()\n",
    "\n",
    "n_epochs = 5000\n",
    "batch_size = 8\n",
    "\n",
    "for epoch in range(n_epochs):\n",
    "    # Let's do a random order after converting to a torch array\n",
    "    shuffled_indices = torch.randperm(x.shape[0])\n",
    "    for batch in range(0, x.shape[0], batch_size):\n",
    "        sample_idxs = shuffled_indices[batch:batch+batch_size]\n",
    "        \n",
    "        # Read minibatch data. PyTorch models default to float32  \n",
    "        # so we need to convert from double precision to float32\n",
    "        tensor = torch.from_numpy(X[sample_idxs, :]).float()\n",
    "        output = model(tensor)\n",
    "        \n",
    "        batch_labels = torch.from_numpy(y[sample_idxs]).float()\n",
    "        batch_labels = batch_labels.view(-1, 1)\n",
    "        \n",
    "        # loss.backward() calculates the gradient of the loss with\n",
    "        # respect to the weights\n",
    "        loss = loss_function(output, batch_labels)\n",
    "        loss.backward()\n",
    "        \n",
    "        # optimizer.step() moves in the direction of the gradient\n",
    "        # based on the learning rate (i.e. step size)\n",
    "        optimizer.step()\n",
    "        \n",
    "        # PyTorch is still saving our past gradient even though we \n",
    "        # changed our model parameters. Clearing them here prevents\n",
    "        # us from accidentally applying them again.\n",
    "        optimizer.zero_grad()\n",
    "        \n",
    "        \n",
    "wpytorch = list(model.parameters())[0][0].detach().numpy()\n",
    "wpytorch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the fit\n",
    "plotdata()\n",
    "plotline(w[0], w[1], label=\"linear fit\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is this a good model? Would you trust this model to make predictions on new inputs $x$?\n",
    "\n",
    "* (A) yes \n",
    "* (B) no"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fbd163e2cd0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot fit on out of sample data\n",
    "plotdata()\n",
    "plotline(w[0], w[1])\n",
    "\n",
    "xtest,ytest = sample_data(20)\n",
    "plt.scatter(xtest,ytest,label=\"test\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Approximating with a polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>order0</th>\n",
       "      <th>order1</th>\n",
       "      <th>order2</th>\n",
       "      <th>order3</th>\n",
       "      <th>order4</th>\n",
       "      <th>order5</th>\n",
       "      <th>order6</th>\n",
       "      <th>order7</th>\n",
       "      <th>order8</th>\n",
       "      <th>order9</th>\n",
       "      <th>order10</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>8.928602</td>\n",
       "      <td>79.719925</td>\n",
       "      <td>711.787443</td>\n",
       "      <td>6355.266442</td>\n",
       "      <td>56743.641581</td>\n",
       "      <td>506641.364154</td>\n",
       "      <td>4.523599e+06</td>\n",
       "      <td>4.038941e+07</td>\n",
       "      <td>3.606210e+08</td>\n",
       "      <td>3.219841e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>3.319798</td>\n",
       "      <td>11.021059</td>\n",
       "      <td>36.587691</td>\n",
       "      <td>121.463744</td>\n",
       "      <td>403.235101</td>\n",
       "      <td>1338.659102</td>\n",
       "      <td>4.444078e+03</td>\n",
       "      <td>1.475344e+04</td>\n",
       "      <td>4.897845e+04</td>\n",
       "      <td>1.625985e+05</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>8.212291</td>\n",
       "      <td>67.441727</td>\n",
       "      <td>553.851105</td>\n",
       "      <td>4548.386575</td>\n",
       "      <td>37352.675185</td>\n",
       "      <td>306751.046862</td>\n",
       "      <td>2.519129e+06</td>\n",
       "      <td>2.068782e+07</td>\n",
       "      <td>1.698944e+08</td>\n",
       "      <td>1.395222e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0.416966</td>\n",
       "      <td>0.173861</td>\n",
       "      <td>0.072494</td>\n",
       "      <td>0.030228</td>\n",
       "      <td>0.012604</td>\n",
       "      <td>0.005255</td>\n",
       "      <td>2.191323e-03</td>\n",
       "      <td>9.137077e-04</td>\n",
       "      <td>3.809853e-04</td>\n",
       "      <td>1.588580e-04</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1.076567</td>\n",
       "      <td>1.158996</td>\n",
       "      <td>1.247737</td>\n",
       "      <td>1.343272</td>\n",
       "      <td>1.446122</td>\n",
       "      <td>1.556847</td>\n",
       "      <td>1.676050e+00</td>\n",
       "      <td>1.804379e+00</td>\n",
       "      <td>1.942535e+00</td>\n",
       "      <td>2.091269e+00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1.0</td>\n",
       "      <td>5.950521</td>\n",
       "      <td>35.408696</td>\n",
       "      <td>210.700176</td>\n",
       "      <td>1253.775746</td>\n",
       "      <td>7460.618458</td>\n",
       "      <td>44394.564137</td>\n",
       "      <td>2.641708e+05</td>\n",
       "      <td>1.571954e+06</td>\n",
       "      <td>9.353942e+06</td>\n",
       "      <td>5.566083e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1.0</td>\n",
       "      <td>5.298174</td>\n",
       "      <td>28.070644</td>\n",
       "      <td>148.723144</td>\n",
       "      <td>787.961039</td>\n",
       "      <td>4174.754394</td>\n",
       "      <td>22118.573606</td>\n",
       "      <td>1.171880e+05</td>\n",
       "      <td>6.208826e+05</td>\n",
       "      <td>3.289544e+06</td>\n",
       "      <td>1.742857e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>1.0</td>\n",
       "      <td>4.188074</td>\n",
       "      <td>17.539966</td>\n",
       "      <td>73.458682</td>\n",
       "      <td>307.650415</td>\n",
       "      <td>1288.462792</td>\n",
       "      <td>5396.177888</td>\n",
       "      <td>2.259959e+04</td>\n",
       "      <td>9.464878e+04</td>\n",
       "      <td>3.963961e+05</td>\n",
       "      <td>1.660136e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>1.0</td>\n",
       "      <td>3.354078</td>\n",
       "      <td>11.249843</td>\n",
       "      <td>37.732855</td>\n",
       "      <td>126.558957</td>\n",
       "      <td>424.488676</td>\n",
       "      <td>1423.768339</td>\n",
       "      <td>4.775431e+03</td>\n",
       "      <td>1.601717e+04</td>\n",
       "      <td>5.372284e+04</td>\n",
       "      <td>1.801906e+05</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>1.0</td>\n",
       "      <td>6.225194</td>\n",
       "      <td>38.753044</td>\n",
       "      <td>241.245232</td>\n",
       "      <td>1501.798446</td>\n",
       "      <td>9348.987159</td>\n",
       "      <td>58199.261781</td>\n",
       "      <td>3.623017e+05</td>\n",
       "      <td>2.255399e+06</td>\n",
       "      <td>1.404029e+07</td>\n",
       "      <td>8.740356e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1.0</td>\n",
       "      <td>4.381414</td>\n",
       "      <td>19.196791</td>\n",
       "      <td>84.109094</td>\n",
       "      <td>368.516782</td>\n",
       "      <td>1614.624684</td>\n",
       "      <td>7074.339615</td>\n",
       "      <td>3.099561e+04</td>\n",
       "      <td>1.358046e+05</td>\n",
       "      <td>5.950163e+05</td>\n",
       "      <td>2.607013e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>1.0</td>\n",
       "      <td>7.358821</td>\n",
       "      <td>54.152247</td>\n",
       "      <td>398.496699</td>\n",
       "      <td>2932.465903</td>\n",
       "      <td>21579.491857</td>\n",
       "      <td>158799.619217</td>\n",
       "      <td>1.168578e+06</td>\n",
       "      <td>8.599356e+06</td>\n",
       "      <td>6.328112e+07</td>\n",
       "      <td>4.656745e+08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1.0</td>\n",
       "      <td>5.180364</td>\n",
       "      <td>26.836172</td>\n",
       "      <td>139.021145</td>\n",
       "      <td>720.180150</td>\n",
       "      <td>3730.795409</td>\n",
       "      <td>19326.878674</td>\n",
       "      <td>1.001203e+05</td>\n",
       "      <td>5.186594e+05</td>\n",
       "      <td>2.686845e+06</td>\n",
       "      <td>1.391883e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1.0</td>\n",
       "      <td>5.788586</td>\n",
       "      <td>33.507728</td>\n",
       "      <td>193.962365</td>\n",
       "      <td>1122.767829</td>\n",
       "      <td>6499.238141</td>\n",
       "      <td>37621.398928</td>\n",
       "      <td>2.177747e+05</td>\n",
       "      <td>1.260608e+06</td>\n",
       "      <td>7.297136e+06</td>\n",
       "      <td>4.224010e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1.0</td>\n",
       "      <td>6.453551</td>\n",
       "      <td>41.648320</td>\n",
       "      <td>268.779555</td>\n",
       "      <td>1734.582557</td>\n",
       "      <td>11194.216925</td>\n",
       "      <td>72242.449371</td>\n",
       "      <td>4.662203e+05</td>\n",
       "      <td>3.008777e+06</td>\n",
       "      <td>1.941729e+07</td>\n",
       "      <td>1.253105e+08</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    order0    order1     order2      order3       order4        order5  \\\n",
       "0      1.0  8.928602  79.719925  711.787443  6355.266442  56743.641581   \n",
       "1      1.0  3.319798  11.021059   36.587691   121.463744    403.235101   \n",
       "2      1.0  8.212291  67.441727  553.851105  4548.386575  37352.675185   \n",
       "3      1.0  0.416966   0.173861    0.072494     0.030228      0.012604   \n",
       "4      1.0  1.076567   1.158996    1.247737     1.343272      1.446122   \n",
       "5      1.0  5.950521  35.408696  210.700176  1253.775746   7460.618458   \n",
       "6      1.0  5.298174  28.070644  148.723144   787.961039   4174.754394   \n",
       "7      1.0  4.188074  17.539966   73.458682   307.650415   1288.462792   \n",
       "8      1.0  3.354078  11.249843   37.732855   126.558957    424.488676   \n",
       "9      1.0  6.225194  38.753044  241.245232  1501.798446   9348.987159   \n",
       "10     1.0  4.381414  19.196791   84.109094   368.516782   1614.624684   \n",
       "11     1.0  7.358821  54.152247  398.496699  2932.465903  21579.491857   \n",
       "12     1.0  5.180364  26.836172  139.021145   720.180150   3730.795409   \n",
       "13     1.0  5.788586  33.507728  193.962365  1122.767829   6499.238141   \n",
       "14     1.0  6.453551  41.648320  268.779555  1734.582557  11194.216925   \n",
       "\n",
       "           order6        order7        order8        order9       order10  \n",
       "0   506641.364154  4.523599e+06  4.038941e+07  3.606210e+08  3.219841e+09  \n",
       "1     1338.659102  4.444078e+03  1.475344e+04  4.897845e+04  1.625985e+05  \n",
       "2   306751.046862  2.519129e+06  2.068782e+07  1.698944e+08  1.395222e+09  \n",
       "3        0.005255  2.191323e-03  9.137077e-04  3.809853e-04  1.588580e-04  \n",
       "4        1.556847  1.676050e+00  1.804379e+00  1.942535e+00  2.091269e+00  \n",
       "5    44394.564137  2.641708e+05  1.571954e+06  9.353942e+06  5.566083e+07  \n",
       "6    22118.573606  1.171880e+05  6.208826e+05  3.289544e+06  1.742857e+07  \n",
       "7     5396.177888  2.259959e+04  9.464878e+04  3.963961e+05  1.660136e+06  \n",
       "8     1423.768339  4.775431e+03  1.601717e+04  5.372284e+04  1.801906e+05  \n",
       "9    58199.261781  3.623017e+05  2.255399e+06  1.404029e+07  8.740356e+07  \n",
       "10    7074.339615  3.099561e+04  1.358046e+05  5.950163e+05  2.607013e+06  \n",
       "11  158799.619217  1.168578e+06  8.599356e+06  6.328112e+07  4.656745e+08  \n",
       "12   19326.878674  1.001203e+05  5.186594e+05  2.686845e+06  1.391883e+07  \n",
       "13   37621.398928  2.177747e+05  1.260608e+06  7.297136e+06  4.224010e+07  \n",
       "14   72242.449371  4.662203e+05  3.008777e+06  1.941729e+07  1.253105e+08  "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# first, construct a Vandermonde matrix\n",
    "n = 15\n",
    "max_order = 10\n",
    "X = np.zeros((n, max_order+1))\n",
    "\n",
    "# method 1: use the default Vandermonde function from numpy\n",
    "X = np.vander(x, max_order+1, increasing = True)\n",
    "# method 2: use a for loop\n",
    "# for k in range(max_order+1):\n",
    "#     X[:,k] = x**(k)\n",
    "    \n",
    "# create a pandas dataframe with data = Vandermonde\n",
    "col_list = [\"order\"+str(i) for i in range(max_order+1)]   \n",
    "df1 = pd.DataFrame(data = X, columns=col_list)\n",
    "df1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# helper function: create a Vandermonde matrix in pandas.dataframe\n",
    "def Vandermonde_df(x, order = max_order):\n",
    "    X = np.vander(x, order+1, increasing = True)\n",
    "    col_list = [\"order\"+str(i) for i in range(order+1)]   \n",
    "    df = pd.DataFrame(data = X, columns=col_list)\n",
    "    return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "order0     121.634847\n",
      "order1    -580.750926\n",
      "order2     971.307923\n",
      "order3    -804.409277\n",
      "order4     387.304320\n",
      "order5    -117.184775\n",
      "order6      23.031717\n",
      "order7      -2.942067\n",
      "order8       0.235877\n",
      "order9      -0.010787\n",
      "order10      0.000215\n",
      "dtype: float64\n"
     ]
    }
   ],
   "source": [
    "# solve least squares problem\n",
    "model = sm.OLS(y, df1).fit()\n",
    "w = model.params\n",
    "print(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0     2.130451\n",
       "1     0.847466\n",
       "2     2.022922\n",
       "3     0.383902\n",
       "4     0.591226\n",
       "5     1.472116\n",
       "6     1.193113\n",
       "7     0.980092\n",
       "8     0.811906\n",
       "9     1.526781\n",
       "10    1.037409\n",
       "11    1.667587\n",
       "12    1.154493\n",
       "13    1.410077\n",
       "14    1.517587\n",
       "dtype: float64"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# output the prediction given our polynomial fit\n",
    "# using X*w\n",
    "df1.dot(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0     2.130451\n",
       "1     0.847466\n",
       "2     2.022922\n",
       "3     0.383902\n",
       "4     0.591226\n",
       "5     1.472116\n",
       "6     1.193113\n",
       "7     0.980092\n",
       "8     0.811906\n",
       "9     1.526781\n",
       "10    1.037409\n",
       "11    1.667587\n",
       "12    1.154493\n",
       "13    1.410077\n",
       "14    1.517587\n",
       "dtype: float64"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# output the prediction given our polynomial fit\n",
    "# using statsmodels predict function\n",
    "model.predict(df1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "#computes our polynomial fit evaluated at x\n",
    "def p(x, order = max_order, w = w):\n",
    "    df = Vandermonde_df(x, order)\n",
    "    y = df.dot(w)\n",
    "    return np.array(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot fit\n",
    "plotdata()\n",
    "plotfunc(p, xmin=0, xmax=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is this a good model? Would you trust this model to make predictions on new inputs $x$?\n",
    "\n",
    "* (A) yes \n",
    "* (B) no"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fbd16615d60>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot fit on out of sample data\n",
    "plotdata()\n",
    "plotfunc(p, xmin=0, xmax=10)\n",
    "\n",
    "xtest,ytest = sample_data(100)\n",
    "plt.scatter(xtest,ytest,label=\"test\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Choosing the best model order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.53558514,  0.09219372,  0.11267527,  0.10313493,  0.07633348,\n",
       "        0.11427176,  0.09231839,  0.22500731,  0.96274495,  1.11348529,\n",
       "       16.46716875])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max_model_order = 10\n",
    "rmse = np.zeros(max_model_order+1)       # array to store root mean square model errors\n",
    "rmse_train = np.zeros(max_model_order+1) # array to store root mean square model errors on training set\n",
    "xtest,ytest = sample_data(50)   # generate test set\n",
    "\n",
    "for model_order in range(max_model_order+1):\n",
    "    # form Vandermonde matrix\n",
    "    df = Vandermonde_df(x, model_order)\n",
    "    \n",
    "    # solve least squares problem\n",
    "    model = sm.OLS(y, df).fit()\n",
    "    w = model.params\n",
    "    ytrain = df.dot(w)\n",
    "    \n",
    "    # compute training error \n",
    "    rmse_train[model_order] = np.mean(np.abs(y - np.array(ytrain))) # sqrt(mean((y - X*w).^2))\n",
    "    \n",
    "    # compute test error\n",
    "    ptest = p(xtest, order=model_order, w=w)\n",
    "    rmse[model_order] = np.mean(np.abs(ytest - np.array(ptest))) # sqrt(mean((ytest - ptest).^2))\n",
    "rmse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'rmse')"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(range(max_model_order+1),rmse, label=\"test error\")\n",
    "plt.plot(range(max_model_order+1),rmse_train, label=\"train error\")\n",
    "plt.xticks(np.arange(0,10,step=1.0))\n",
    "plt.xlabel(\"model order\")\n",
    "plt.ylabel(\"rmse\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Polynomial models for classification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fbd1683afd0>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "n = 20\n",
    "xs = 10*(rand(n) - .5)\n",
    "ys = (xs)**2\n",
    "pos = (xs - 1)**2 >= 7\n",
    "zn=np.zeros(n)\n",
    "pt = plt.scatter(xs[pos],zn[pos],color=\"blue\", label=\"positive\")\n",
    "plt.scatter(xs[~pos],zn[~pos],color=\"red\", label=\"negative\")\n",
    "plt.plot([-2,-2], [-1,1], color=\"black\", label=\"classification boundary\")#, ticks=:none, border=:none, legend=false)\n",
    "plt.ylim([-1,1])\n",
    "plt.grid()\n",
    "# savefig(\"poly-class-1d.pdf\")\n",
    "pt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fbd16908ac0>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pt = plt.scatter(xs[pos],ys[pos],color=\"blue\", label=\"positive\")\n",
    "plt.scatter(xs[~pos],ys[~pos],color=\"red\", label=\"negative\")\n",
    "plt.plot(xs, 1.5*xs + 7, color=\"black\", label=\"classification boundary\")\n",
    "plt.grid()\n",
    "# savefig(\"poly-class-2d.pdf\")\n",
    "pt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Example 1: multivariate polynomial classification \n",
    "n = 1000\n",
    "scale = 50\n",
    "x1 = scale*rand(n) - scale/2\n",
    "x2 = scale*rand(n) - scale/2\n",
    "f = lambda x1, x2: -30 - 9*x1 + 2*x2 + x1**2 - 0*x1*x2 + x2**2\n",
    "y = f(x1,x2)\n",
    "pos = y>0\n",
    "neg = y<0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fbd16940f70>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x1[pos], x2[pos], color=\"blue\", s=4, label='y1')\n",
    "plt.scatter(x1[neg], x2[neg], color=\"red\", s=4, label='y2')\n",
    "plt.legend(loc=\"upper right\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fbd16b4f130>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Example 2: multivariate polynomial classification \n",
    "n = 1000\n",
    "scale = 10\n",
    "x1 = scale*rand(n) - scale/2\n",
    "x2 = scale*rand(n) - scale/2\n",
    "f = lambda x1, x2: -5 - 3*x1 + 2*x2 + x1**2 - x1*x2 + 5*x2**2\n",
    "y = f(x1,x2)\n",
    "pos = y>0\n",
    "neg = y<0\n",
    "\n",
    "plt.scatter(x1[pos], x2[pos], s=4, color=\"blue\", label=\"y1\")\n",
    "plt.scatter(x1[neg], x2[neg], s=4, color=\"red\", label=\"y2\")\n",
    "plt.legend(loc=\"upper right\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fbd16873640>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Example 3: multivariate polynomial classification \n",
    "n = 1000\n",
    "scale = 20\n",
    "x1 = scale*rand(n) - scale/2\n",
    "x2 = scale*rand(n) - scale/2\n",
    "f = lambda x1, x2: -5 - 3*x1 + 2*x2 + x1**2 - x1*x2 - 2*x2**2\n",
    "y = f(x1,x2)\n",
    "pos = y>0\n",
    "neg = y<0\n",
    "\n",
    "plt.scatter(x1[pos], x2[pos], color=\"blue\", s=4, label=\"y1\")\n",
    "plt.scatter(x1[neg], x2[neg], color=\"red\", s=4, label=\"y2\")\n",
    "plt.legend(loc=\"upper right\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "@webio": {
   "lastCommId": null,
   "lastKernelId": null
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}