ApplyingMachineLearningLab.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0aeda375",
   "metadata": {},
   "source": [
    "# Practice Lab: Advice for Applying Machine Learning\n",
    "[ 1 - Packages ](#1)\n",
    "<a name=\"1\"></a>\n",
    "## 1 - Packages \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f9de1df5",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.linear_model import LinearRegression, Ridge\n",
    "from sklearn.preprocessing import StandardScaler, PolynomialFeatures\n",
    "from sklearn.model_selection import train_test_split\n",
    "from sklearn.metrics import mean_squared_error\n",
    "import tensorflow as tf\n",
    "from tensorflow.keras.models import Sequential\n",
    "from tensorflow.keras.layers import Dense\n",
    "from tensorflow.keras.activations import relu,linear\n",
    "from tensorflow.keras.losses import SparseCategoricalCrossentropy\n",
    "from tensorflow.keras.optimizers import Adam\n",
    "\n",
    "import logging\n",
    "logging.getLogger(\"tensorflow\").setLevel(logging.ERROR)\n",
    "\n",
    "tf.keras.backend.set_floatx('float64')\n",
    "\n",
    "tf.autograph.set_verbosity(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "efef31da",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X.shape (23,) y.shape (23,)\n",
      "X_train.shape (15,) y_train.shape (15,)\n",
      "X_test.shape (8,) y_test.shape (8,)\n"
     ]
    }
   ],
   "source": [
    "# Generate some data\n",
    "X = np.array([1, 2, 5, 8, 9, 12, 17, 19, 22, 34, 25, 33, 38, 41, 42, 46, 47, 52, 54, 58, 60, 62, 65])\n",
    "y = np.array([220, 250, 340, 390, 400, 450, 465, 489, 501, 523, 544, 567, 588, 600, 643, 654, 666, 678, 679, 690, 695, 699, 705])\n",
    "x_ideal = [1, 65]\n",
    "y_ideal= [220,745]\n",
    "print(\"X.shape\", X.shape, \"y.shape\", y.shape)\n",
    "\n",
    "#split the data using sklearn routine \n",
    "X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.33, random_state=1)\n",
    "print(\"X_train.shape\", X_train.shape, \"y_train.shape\", y_train.shape)\n",
    "print(\"X_test.shape\", X_test.shape, \"y_test.shape\", y_test.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "2c8c468a",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1,figsize=(4,4))\n",
    "ax.plot(x_ideal, y_ideal, \"--\", color = \"orangered\", label=\"y_ideal\", lw=1)\n",
    "ax.set_title(\"Training, Test\",fontsize = 14)\n",
    "ax.set_xlabel(\"x\")\n",
    "ax.set_ylabel(\"y\")\n",
    "\n",
    "ax.scatter(X_train, y_train, color = \"red\",           label=\"train\")\n",
    "ax.scatter(X_test, y_test,   color = \"blue\",   label=\"test\")\n",
    "ax.legend(loc='upper left')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7ca09da",
   "metadata": {},
   "source": [
    "### 2.2 Error calculation for model evaluation, linear regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "588be980",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0050000000000000305"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def eval_mse(y, yhat):\n",
    "    \"\"\" \n",
    "    Calculate the mean squared error on a data set.\n",
    "    Args:\n",
    "      y    : (ndarray  Shape (m,) or (m,1))  target value of each example\n",
    "      yhat : (ndarray  Shape (m,) or (m,1))  predicted value of each example\n",
    "    Returns:\n",
    "      err: (scalar)             \n",
    "    \"\"\"\n",
    "    m = len(y)\n",
    "    err = 0.0\n",
    "    for i in range(m):\n",
    "    ### START CODE HERE ### \n",
    "       err += (y[i] - yhat[i])**2\n",
    "        \n",
    "    err = err / (2 * m)\n",
    "        \n",
    "    \n",
    "    ### END CODE HERE ### \n",
    "    \n",
    "    return(err)\n",
    "\n",
    "y_hat = np.array([2.4, 4.2])\n",
    "y_tmp = np.array([2.3, 4.1])\n",
    "eval_mse(y_hat, y_tmp)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a9dfcaa",
   "metadata": {},
   "source": [
    "### 2.3 Compare performance on training and test data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fe19f691",
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a model in sklearn, train on training data\n",
    "degree = 10\n",
    "lmodel = lin_model(degree)\n",
    "lmodel.fit(X_train, y_train)\n",
    "\n",
    "# predict on training data, find training error\n",
    "yhat = lmodel.predict(X_train)\n",
    "err_train = lmodel.mse(y_train, yhat)\n",
    "\n",
    "# predict on test data, find error\n",
    "yhat = lmodel.predict(X_test)\n",
    "err_test = lmodel.mse(y_test, yhat)\n",
    "\n",
    "max_degree = 9\n",
    "err_train = np.zeros(max_degree)    \n",
    "err_cv = np.zeros(max_degree)      \n",
    "x = np.linspace(0,int(X.max()),100)  \n",
    "y_pred = np.zeros((100,max_degree))  #columns are lines to plot\n",
    "\n",
    "for degree in range(max_degree):\n",
    "    lmodel = lin_model(degree+1)\n",
    "    lmodel.fit(X_train, y_train)\n",
    "    yhat = lmodel.predict(X_train)\n",
    "    err_train[degree] = lmodel.mse(y_train, yhat)\n",
    "    yhat = lmodel.predict(X_cv)\n",
    "    err_cv[degree] = lmodel.mse(y_cv, yhat)\n",
    "    y_pred[:,degree] = lmodel.predict(x)\n",
    "    \n",
    "optimal_degree = np.argmin(err_cv)+1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b25be441",
   "metadata": {},
   "source": [
    "### 3.3 Tuning Regularization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d9e6f418",
   "metadata": {},
   "outputs": [],
   "source": [
    "lambda_range = np.array([0.0, 1e-6, 1e-5, 1e-4,1e-3,1e-2, 1e-1,1,10,100])\n",
    "num_steps = len(lambda_range)\n",
    "degree = 10\n",
    "err_train = np.zeros(num_steps)    \n",
    "err_cv = np.zeros(num_steps)       \n",
    "x = np.linspace(0,int(X.max()),100) \n",
    "y_pred = np.zeros((100,num_steps))  #columns are lines to plot\n",
    "\n",
    "for i in range(num_steps):\n",
    "    lambda_= lambda_range[i]\n",
    "    lmodel = lin_model(degree, regularization=True, lambda_=lambda_)\n",
    "    lmodel.fit(X_train, y_train)\n",
    "    yhat = lmodel.predict(X_train)\n",
    "    err_train[i] = lmodel.mse(y_train, yhat)\n",
    "    yhat = lmodel.predict(X_cv)\n",
    "    err_cv[i] = lmodel.mse(y_cv, yhat)\n",
    "    y_pred[:,i] = lmodel.predict(x)\n",
    "    \n",
    "optimal_reg_idx = np.argmin(err_cv) "
   ]
  }
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