project#5.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a07d11fa-87bb-426b-9465-76881f9dcf52",
   "metadata": {},
   "source": [
    "# Math 266 Python Project #5. Ordinary Least Squares"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85adec76-208a-4f3f-94d5-0c320558eeed",
   "metadata": {},
   "source": [
    "Simple linear regression involves finding a linear equation or linear regression model with a single independent variable and a single dependent variable. The parameters of the model are \"learned\" by using oberved data. Afterward the model can be used to predict outputs given new data. We will begin our activity by looking at examples of  ordinary linear regression."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8891ab67-05cf-4310-b5d6-bf424255cb06",
   "metadata": {
    "tags": []
   },
   "source": [
    "Instructions:  Read and execute the cells below and follow the instructions.\n",
    "\n",
    "Execute the cell below to load necessary modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "9b2f8e43-7519-4c92-937c-515b400f9620",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: wget in /Users/rmartin/opt/anaconda3/lib/python3.9/site-packages (3.2)\n",
      "\u001b[33mWARNING: There was an error checking the latest version of pip.\u001b[0m\u001b[33m\n",
      "\u001b[0m"
     ]
    }
   ],
   "source": [
    "! pip install wget\n",
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "import wget\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from numpy.linalg import norm\n",
    "import seaborn as sns\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "586766b3-19d0-4d66-bb9e-efd6637fae85",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">Load the data for our first model. </span></h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ae686b78-1274-4369-a2f5-f0f3a9995533",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100% [..............................................................] 288 / 288"
     ]
    }
   ],
   "source": [
    "file = wget.download(\"https://github.com/rmartin977/Math-266/blob/main/data_1.npy?raw=true\")\n",
    "data_1 = np.load(file)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2a66a18-0927-4e22-b6f1-423c600f79aa",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">View the data </span></h2>\n",
    "The data consist of 10 (x,y) pairs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "41264837-d7de-4777-a058-fb7bc54fcc7c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.5, 3.4],\n",
       "       [0.5, 1.7],\n",
       "       [2.2, 3.9],\n",
       "       [1.9, 3.2],\n",
       "       [3.1, 4. ],\n",
       "       [2.3, 3.7],\n",
       "       [2. , 2.6],\n",
       "       [1. , 2.1],\n",
       "       [1.5, 2.6],\n",
       "       [1.1, 1.9]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc63b4de-b4e6-4f56-83cc-21e33f89bd05",
   "metadata": {},
   "source": [
    "## Extract the x and y vectors from the data matrix and plot the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "799cdd6c-086a-4d26-b0ce-9c6eb9b1d956",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = data_1[:,0]\n",
    "y = data_1[:,1]\n",
    "plt.scatter(x,y)\n",
    "plt.xlim([0,4])\n",
    "plt.ylim([0,5])\n",
    "plt.grid\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d401eb4-170a-4591-84ba-089e3a1688cd",
   "metadata": {},
   "source": [
    "The linear regression model is shown below.  $X$ is the design matrix. The rows of $X$ are our inputs or \"feature vectors\". $\\vec{\\mathbf{\\beta}}$ is the parameter vector, what we are solving for. $\\vec{\\mathbf{y}}$ is the observation vector containing the labels for each input\n",
    "\n",
    "\\begin{align}\n",
    "X  \\vec{\\mathbf{\\beta}}  = \\vec{\\mathbf{y}}   \\\\  \n",
    "\\end{align}\n",
    "\n",
    "We will solve by using the standard normal equations used in least squares.\n",
    "\n",
    "\\begin{align}\n",
    "\\vec{\\mathbf{\\beta}}  =   (X^T X)^{-1} X^T  \\vec{\\mathbf{y}} \\\\   \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42dcf114-bf72-43e8-90fd-855a5c6d2c86",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">Construct design matrix. </span></h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8aaaad80-bd33-4efa-b59e-9aeb4f18aa4d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 2.5],\n",
       "       [1. , 0.5],\n",
       "       [1. , 2.2],\n",
       "       [1. , 1.9],\n",
       "       [1. , 3.1],\n",
       "       [1. , 2.3],\n",
       "       [1. , 2. ],\n",
       "       [1. , 1. ],\n",
       "       [1. , 1.5],\n",
       "       [1. , 1.1]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = np.c_[np.ones_like(x),x]\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a083604-e7ed-43f1-a519-64283970bce4",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">Solve system. </span></h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "6fcc2f67-57e4-45e8-b06c-1614a41fb77f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.10326185, 0.99819787])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta_1 = la.inv(X.T@X)@X.T@y\n",
    "beta_1 # this vector holds the y-intercept and slope value for our model."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba5d3fbf-7b38-4db5-8bd6-04cffb082b43",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">Plot resulting least square line. </span></h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "39ccf644-851b-4e81-976e-06493ed2b2b1",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f90e6462e80>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = data_1[:,0]\n",
    "y = data_1[:,1]\n",
    "plt.scatter(x,y)\n",
    "plt.xlim([0,4])\n",
    "plt.ylim([0,5])\n",
    "plt.grid\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "\n",
    "plt.plot(x,beta_1[0] + beta_1[1]*x,c='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cae5e0ea-0b58-4ce8-8b7f-c7907bfaee7f",
   "metadata": {},
   "source": [
    "## For an input of 3.0 the model predicts and output of 4.1.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "a725257c-1db9-48e6-b344-39864dc22569",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.1"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_predict = beta_1[0] + beta_1[1]*3.0\n",
    "y_predict.round(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4992a24-3294-44fc-909a-bfaa7cfec856",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">Load the data for our second model. </span></h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "1b17a36d-fe50-4f80-b992-5bc6624a6306",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100% [..............................................................] 288 / 288"
     ]
    }
   ],
   "source": [
    "file2 = wget.download(\"https://github.com/rmartin977/Math-266/blob/main/data_2.npy?raw=true\")\n",
    "data_2 = np.load(file2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "17f30ae8-29ca-4cc4-926c-d752af214857",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.5, 2.4],\n",
       "       [0.5, 0.7],\n",
       "       [2.2, 2.9],\n",
       "       [1.9, 2.2],\n",
       "       [3.1, 3. ],\n",
       "       [2.3, 2.7],\n",
       "       [2. , 1.6],\n",
       "       [1. , 1.1],\n",
       "       [1.5, 1.6],\n",
       "       [1.1, 0.9]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c5c1672-21b3-464e-bddf-63ec8608bd2e",
   "metadata": {},
   "source": [
    "## Now build a linear model for this second data set by repeating the above steps.  No need to graph the data.  Use your model to predict the output when the input is 3.0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "077f2459-4bf2-4d81-bc93-09489259b97e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# your code here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3f16f329-c099-4637-b9ab-c39596f9ba03",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5c946af1-4d23-4fcb-8487-fe7e7045b27e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8270b7c2-71cd-441b-83d8-780c9d893c40",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "63611fe2-348e-4f7a-93a3-5675061dbd5c",
   "metadata": {},
   "source": [
    "# Multiple Regression."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df5d0ec0-5568-463d-a277-23b8be7b6f96",
   "metadata": {
    "tags": []
   },
   "source": [
    "A multiple regression model is one that attempts to predict a dependent variable  based on the value of multiple independent variables. Below  you will predict the selling price of a house based on the number of bedrooms and square footage of the house. This is an example of two independent variables. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e0298f4-a16c-41a8-845e-e7a244647c5b",
   "metadata": {},
   "source": [
    "<h2><span style=\"color:blue\">Load the housing data. </span></h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "981d696d-25d5-4c2a-a796-b0f7ef15a901",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100% [................................................................................] 1232 / 1232"
     ]
    }
   ],
   "source": [
    "file3 = wget.download(\"https://github.com/rmartin977/Math-266/blob/main/housing_data.npy?raw=true\")                  \n",
    "data_3 = np.load(file3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "17d69123-677d-42da-b6be-a99751560f7e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(46, 3)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_3.shape  # there are 46 data points in this set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "1253f757-4435-4196-97d3-15a0b2e6275a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1600,      3, 329900],\n",
       "       [  2400,      3, 369000],\n",
       "       [  1416,      2, 232000],\n",
       "       [  3000,      4, 539900],\n",
       "       [  1985,      4, 299900]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_3[:5] # view the first 5 rows"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5281df3f-a2ac-4572-8ead-399a73242284",
   "metadata": {},
   "source": [
    "## Above we see the first house is 1600 sqr ft., 3 bedrooms and sold for *$329,900.00*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10359a14-4380-4a11-8b74-889b1b2d889b",
   "metadata": {},
   "source": [
    "The first house has a feature vector = $\\begin{bmatrix}1600  \\\\3  \\end{bmatrix}$. Our model has the form: $$y =\\beta_0 + \\beta_1x_1+\\beta_2x_2 $$  See chapter 5 in the text for reference.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99f56ded-68cb-4ff3-820e-d77f1a6f3598",
   "metadata": {},
   "source": [
    "## Now follow the steps above to construct a model for this data. You need to:\n",
    "1. Construct design Matrix X.\n",
    "2. Use normal equations to solve for  the paramter vector $\\vec{\\mathbf{\\beta}} =\\begin{bmatrix}\\beta_0  \\\\ \\beta_1 \\\\ \\beta_2 \\end{bmatrix}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "2d301039-55d6-4209-8562-7c70d27a6204",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = # enter necessary command to construct design design matrix\n",
    "y = # enter necessary command to extract the third column from data_3 matrix.  Note zero based indexing so it is column #2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a194361a-15e4-4eb6-8caa-529485ce6d7d",
   "metadata": {},
   "outputs": [],
   "source": [
    "beta_3 = # enter command to solve for parameter vector.  Note there are three entries in this vector\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a90bbf5-b05a-4a1f-9081-20d0f35ad2d4",
   "metadata": {},
   "source": [
    "## Now use your mode to predict the price of a house with 3000 sq ft and 3 bedrooms. Record you answer.\n",
    " We give you the command below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7a3fc0d8-92b3-4285-83f8-86fc92ed6a0f",
   "metadata": {},
   "outputs": [],
   "source": [
    "y_predict=beta_3[0]+ beta_3[1]*3000 + beta_3[2]*3\n",
    "y_predict.round(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8540fd8-1f42-4c30-b972-495c842cdd02",
   "metadata": {},
   "source": [
    "## Your turn...\n",
    "Answer the following 4 questions.  Go to gradescope project #5 and enter your results.\n",
    "\n",
    "1. What output does the second model give with an input of 3.0?\n",
    "2. What price does you model  predict for a 3000 sqr ft house with 3 bedrooms.\n",
    "3. We note that each house has 2 features.  If we were going to build a model to predict digit values for mnist images\n",
    "   what size would the feature vector be?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1cdc9856-34f1-4764-a12e-d7eedcf59ab1",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}