Linear_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic linear regression with synthetic data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.7804851303758737 [1.78908082]\n",
      "R-square is: 0.8571356016801874\n",
      "Training error is: 0.01066999943533126\n",
      "Testing error is: 0.13213357697947492\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np \n",
    "import random as rd\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "x = np.arange(1, 2, 0.1)\n",
    "\n",
    "# Convert x-data into 2D column vectors \n",
    "\n",
    "x_train = x[0:int(len(x)/2)].reshape(-1,1)\n",
    "x_test = x[int(len(x)/2):len(x)].reshape(-1, 1)\n",
    "\n",
    "m = 2*rd.random()\n",
    "c = 10*rd.random()\n",
    "\n",
    "# y-data as 1D row vectors\n",
    "y = m*x+c \n",
    "y_prime = [0]*len(x)\n",
    "\n",
    "for i in range(len(x)):\n",
    "    y_prime[i] = y[i] + rd.random()\n",
    "\n",
    "y_prime = np.array(y_prime)\n",
    "\n",
    "y_train = y_prime[0:int(len(y)/2)]\n",
    "y_test = y_prime[int(len(y)/2):len(y)]\n",
    "\n",
    "\n",
    "## Linear regression model.\n",
    "\n",
    "Model = LinearRegression()\n",
    "Model = LinearRegression().fit(x_train, y_train)\n",
    "r_sq = Model.score(x_train, y_train)\n",
    "\n",
    "plt.scatter(x_train,y_train)\n",
    "\n",
    "Model_c = Model.intercept_\n",
    "Model_m = Model.coef_\n",
    "\n",
    "print(Model.intercept_,Model.coef_)\n",
    "\n",
    "y_trained = Model_m*x_train + Model_c\n",
    "\n",
    "plt.plot(x_train,y_trained)\n",
    "\n",
    "training_error = sum((y_trained.reshape(-1) - y_train)**2)/len(y_train)\n",
    "\n",
    "print('R-square is:',r_sq)\n",
    "print('Training error is:',training_error)\n",
    "\n",
    "# Predicted response\n",
    "\n",
    "y_predicted = Model_m*x_test + Model_c\n",
    "\n",
    "testing_error = sum((y_predicted.reshape(-1) - y_test)**2)/len(y_test)\n",
    "\n",
    "plt.scatter(x_test,y_test)\n",
    "plt.plot(x_test,y_predicted)\n",
    "\n",
    "print('Testing error is:',testing_error)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.32724944, 0.57007948],\n",
       "       [0.78488514, 0.80954067]])"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rng = np.random.default_rng()\n",
    "rng.random([2,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multivariate linear regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.5         0.33333333  0.5       ]\n",
      " [ 0.25       -1.          0.5       ]\n",
      " [ 0.25        0.66666667 -1.        ]\n",
      " [ 1.          1.          1.        ]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0.45714286, 0.25714286, 0.28571429])"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "x1 = np.array([[-1/2,1/4,1/4,1]])\n",
    "x2 = np.array([[1/3,-1,2/3,1]])\n",
    "x3 = np.array([[1/2,1/2,-1,1]])\n",
    "\n",
    "# To solve a multivariate linear regression, one has to provide the data in terms of a matrix with each column representing all the data points in first dimension. \n",
    "# Hence, an N-dimensional problem, with M features, can be solved by using a linear model with an N x M matrix representing the data.\n",
    "\n",
    "Mat = np.transpose(np.concatenate((x1,x2,x3)))\n",
    "\n",
    "Mat[:,0]\n",
    "\n",
    "print(Mat)\n",
    "\n",
    "y = np.array([0,0,0,1])\n",
    "\n",
    "Model = LinearRegression().fit(Mat, y)\n",
    "\n",
    "Model_c = Model.intercept_\n",
    "Model_pi = Model.coef_\n",
    "\n",
    "Model_pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 0. , 0. ],\n",
       "       [0.5, 0. , 0.5],\n",
       "       [0. , 0. , 1. ]])"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = np.array([[1,0,0],[1/3,1/3,1/3],[0,0,1]])\n",
    "\n",
    "for i in range(10):\n",
    "    P = np.matmul(P,P)\n",
    "    \n",
    "P"
   ]
  }
 ],
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