C2M1L1___Least Squares__M.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "-----\n",
    "You (as an intrepid electrical engineer) wish to determine the resistance of an electrical component using Ohm's law. You remember from your high school circuit classes that $$V = RI$$ \n",
    "\n",
    "where $V$ is the voltage in volts, $R$ is resistance in ohms, and $I$ is electrical current in amperes. Using a multimeter, you measure the voltage drop across the resistor at various current values and collect the following data:\n",
    "\n",
    "| Current (A) | Voltage (V) |\n",
    "|-------------|-------------|\n",
    "| 0.2         | 1.23        |\n",
    "| 0.3         | 1.38        |\n",
    "| 0.4         | 2.06        |\n",
    "| 0.5         | 2.47        |\n",
    "| 0.6         | 3.17        |\n",
    "\n",
    "With the data in hand, your goals are to: \n",
    "1. Fit a line through the origin (i.e., determine the parameter $R$ for $y = Rx$) to this data by using the method of least squares. You may assume that all measurements are of equal importance. \n",
    "2. Consider what the best estimate of the resistance is, in ohms, for this component.\n",
    "\n",
    "## Getting Started\n",
    "----\n",
    "\n",
    "The first step is to import the neccesary Python modules and load the current values and voltage measurements into NumPy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy.linalg import inv\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Store the voltage and current data as column vectors.\n",
    "# I = np.array([[0.2, 0.3, 0.4, 0.5, 0.6]]).T\n",
    "# V = np.array([[1.23, 1.38, 2.06, 2.47, 3.17]]).T\n",
    "\n",
    "I = np.mat([0.2, 0.3, 0.4, 0.5, 0.6]).T\n",
    "V = np.mat([1.23, 1.38, 2.06, 2.47, 3.17]).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's always useful to plot the measurements - can you see the linear relationship between current and voltage?"
   ]
  },
  {
   "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": [
    "# plt.scatter(I, V)\n",
    "\n",
    "plt.scatter(np.asarray(I), np.asarray(V))\n",
    "\n",
    "plt.xlabel('Current (A)')\n",
    "plt.ylabel('Voltage (V)')\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimating the Slope Parameter\n",
    "----\n",
    "Let's try to estimate the slope parameter $R$ (i.e., the resistance) using the least squares formulation from Module 1, Lesson 1 - \"The Squared Error Criterion and the Method of Least Squares\":\n",
    "\n",
    "\\begin{align}\n",
    "\\hat{R} = \\left(\\mathbf{H}^T\\mathbf{H}\\right)^{-1}\\mathbf{H}^T\\mathbf{y}\n",
    "\\end{align}\n",
    "\n",
    "If we know that we're looking for the slope parameter $R$, how do we define the matrix $\\mathbf{H}$ (the Jacobian matrix) and vector $\\mathbf{y}$ (containing the stacked voltage measurements)? Hint #1: When reviewing material from Module 1, carefully note the relationship defined by Ohm's law (this problem is slightly different from the example provided in the Module). Hint #2: the Jacobian matrix contains the partial derivatives of the measurement equation with respect to the parameter(s) of interest."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slope parameter of the best-fit line (i.e., the resistance) is:\n",
      "5.134444444444445\n"
     ]
    }
   ],
   "source": [
    "# Define the H matrix - what does it contain?\n",
    "# H = ...\n",
    "\n",
    "H = I\n",
    "\n",
    "H_transpose = H.transpose()\n",
    "inverse = inv(np.dot(H_transpose, H))\n",
    "\n",
    "# Now estimate the resistance parameter.\n",
    "# R = ... \n",
    "\n",
    "y = V\n",
    "R = inverse * H_transpose * y\n",
    "\n",
    "print('The slope parameter of the best-fit line (i.e., the resistance) is:')\n",
    "print(R[0, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.134444444444445\n"
     ]
    }
   ],
   "source": [
    "R = R[0, 0]\n",
    "print(R)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the Results\n",
    "----\n",
    "Now let's plot our result. How do we relate our linear parameter fit to the resistance value in ohms?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": false
   },
   "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": [
    "# I_line = np.arange(0, 0.8, 0.1).reshape(8, 1)\n",
    "I_line = np.arange(0, 0.8, 0.1)\n",
    "V_line = R*I_line\n",
    "\n",
    "# plt.scatter(I, V)\n",
    "plt.scatter(np.asarray(I), np.asarray(V))\n",
    "\n",
    "plt.plot(I_line, V_line)\n",
    "plt.xlabel('Current (A)')\n",
    "plt.ylabel('Voltage (V)')\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you have implemented the steps correctly, the slope parameter $\\hat{R}$ should be very close (within a couple of tenths of ohms) to the actual resistance value of $R = 5~\\Omega$. However, the estimated value will not match the true resistance value exactly, since we have only a limited number of noisy measurements."
   ]
  }
 ],
 "metadata": {
  "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}