regressionTutorial.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "# See for a better viewing angle and maybe different axes limits in fig2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits import mplot3d\n",
    "import ipywidgets as widgets\n",
    "import warnings"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a beginner's tutorial. If you play with the concepts in this tutorial, you can go from knowing next to nothing about linear algebra to an intuition that will help with multiple linear regression.\n",
    "\n",
    "*M.N. Shadlen* for Cold Spring Harbor 1998 <br>\n",
    "Translated into Python by *Chrysa Papadaniil* 9/21.<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's suppose that you have ordered pairs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1  1]\n",
      " [ 1  1]\n",
      " [ 2  3]]\n"
     ]
    }
   ],
   "source": [
    "data = np.array([[-1, 1], [1, 1], [2, 3]])\n",
    "print(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our job is to find the best fitting line for these three points. Here are the points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(data[:,0], data[:,1], s=80, marker = 'o', edgecolors='g')\n",
    "plt.ylim(-1,4)\n",
    "plt.xlim(-3,3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are going to set this up as a matrix equation.\n",
    "\n",
    "$$\n",
    "Ax = b.\n",
    "$$\n",
    "\n",
    "The term $x$ is somewhat unfortunate because $x$ is not the independent variable, but the constant and slope. $b$ is almost as unfortunate since many of us think of $b$ as the slope of a line.  Here it is the value plotted on the ordinate. Let's start by making $A$.  It is a 3 x 2  matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.ones((3,2))\n",
    "A[:,1] = data[:,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the first column of A is just a column of ones. The second column contains the values plotted on the abscissa -- what we would normally call the $x$-variable: the values -1 1 2. To avoid confusion, let's call the \"$x$-values\" $a$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = data[:,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's call the \"$y$-values\" vector $b$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = data[:,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know $A$ and we know $b$.  We want to find the best solution for $x$. The first thing you need to convince yourself of is that the 2 by 1 column vector, $x$, contains the constant and slope of the best fitting line. This is a good time to remind yourself of how a matrix multiplies a vector."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a wild guess at $x$.  Let's guess that the intercept is .5 and the slope is 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.transpose([.5, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that you get out 3 values from $Ax$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.5  1.5  2.5]\n"
     ]
    }
   ],
   "source": [
    "b_guess = A.dot(x)\n",
    "print(b_guess)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot it \n",
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(a, b, s=80, marker = 'o', edgecolors='g')\n",
    "plt.plot(a, b_guess, 'g+--', markersize=20)\n",
    "plt.ylim(-1,4)\n",
    "plt.xlim(-3,3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a pretty crummy fit, but that's not the point. All you're supposed to see right now is that by going across the rows of $A$ and down the column vector $x$, you get the equation:\n",
    "\n",
    "$$\n",
    "A[0,0]*x[0] + A[0,1]*x[1]\\\\\n",
    "A[1,0]*x[0] + A[1,1]*x[1]\\\\\n",
    "A[2,0]*x[0] + A[2,1]*x[1]\\\\\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you don't understand that last equation, stop.  There's no point in going any further. Ask for help or pick up a book. If you understand the equation, then ask yourself the following. Suppose we made $A$ as a column of 1s, a column like $A[:,1]$ and a third column of the squares:  $A[:,2] = A[:,1]*A[:,1]$. We would be looking for a column vector, $x$, that has 3 components. Do you see that these are the coefficients of the best fitting quadratic? $x0 + x1*a + x2*a^2 = b$. Best to convince yourself of that too before going on.\n",
    "\n",
    "Now this way of setting up the equation jives with your sense of matrix times vector by going along the rows of the matrix and multiplying the elements by those going down the column vector. Another way to put that is that each element in $b$ is a dot-product of the corresponding row in $A$ with the column vector $x$. That's worth thinking about, but it does not lend itself to a particularly lucid view of the regression problem.  For this, it is worth thinking about the matrix times the vector in a different way. Instead of going along the rows of $A$, consider the columns.  This is actually the easier way to think about it. Notice that $b$ is the same dimension as a column in $A$ (3 by 1). Convince yourself that $b$ is just the weighted sum of the columns in $A$. The weights come from the elements of $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.5  1.5  2.5]\n"
     ]
    }
   ],
   "source": [
    "b_guess = x[0]*A[:,0] + x[1]*A[:,1]\n",
    "print(b_guess)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, it's not worth going on if you don't see this version of $A*x$. \n",
    "\n",
    "Now here is where things start getting interesting. Once we recognize that $b$ is the weighted sum of just 2 vectors,  it must be the case that b lies in the plane that these vectors define. No matter how you cut it, you can only find vectors, $b$, that you can get to by adding the two vectors that form the columns of $A$. This plane is called the column-space of $A$. It includes the origin.\n",
    "\n",
    "It helps to look at a picture.  The way I'm going to draw this picture includes a step or two that we're not ready for. Just execute it for the time being. Later, we'll come back to why the matrix algebra works."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can imagine the plane:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.plot_trisurf((0, A[0,0] ,A[0,1]), (0, A[1,0], A[1,1]), (0, A[2,0], A[2,1]),  color = 'blue')\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now here's the part that you need to ignore for a moment. The next line makes a projection matrix. We're going to use it to project the $xy$ plane into the column space of $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = (A.dot(np.linalg.inv(np.transpose(A).dot(A)))).dot(np.transpose(A)) # A * inv(A'*A) * A'\n",
    "q = 2*P.dot(np.array([[1, 1, -2, -2, 1], [1, -2, -2, 1 , 1], [1, 1, 1, 1, 1]]))\n",
    "\n",
    "#Plot it\n",
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "ax.set_xlabel('b1', fontsize=13)\n",
    "ax.set_ylabel('b2', fontsize=13)\n",
    "ax.set_zlabel('b3', fontsize=13)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can try the above in interactive mode with `%matplotlib qt`. This will plot our plane in a pop-up window. Then, you can try rotating the plot to view the plane from different angles. (To go back to non-interactive plots within the notebook, you have to use `%matplotlib inline`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib qt \n",
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "ax.set_xlabel('b1', fontsize=13)\n",
    "ax.set_ylabel('b2', fontsize=13)\n",
    "ax.set_zlabel('b3', fontsize=13)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or you can try a widget to see some different angles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c441d34d0e4a418e954d7c0bd5838a42",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=0, description='angle', max=90, min=-90, step=10), Output()), _dom_class…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline \n",
    "@widgets.interact(angle=widgets.IntSlider(min=-90, max=90, step=10))\n",
    "def plot_plane(angle):\n",
    "    fig = plt.figure(figsize=(12,12))\n",
    "    ax = plt.axes(projection='3d')\n",
    "    ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "    plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "    plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "    ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "    ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "    ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "    ax.view_init(angle,50)\n",
    "    plt.ylim(-4,4)\n",
    "    plt.xlim(-4,4)\n",
    "    ax.set_zlim(-2,3)\n",
    "    ax.set_xlabel('b1', fontsize=13)\n",
    "    ax.set_ylabel('b2', fontsize=13)\n",
    "    ax.set_zlabel('b3', fontsize=13)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Are you convinced that no matter what we guess for our solution, $x$, we will end up with something in the plane? You should have convinced yourself of this earlier. But here is a quick demo.\n",
    "\n",
    "It is important to realize that any guess we make for the answer to our problem will make a vector in the plane spanned by $A$'s column vectors.\n",
    "We'll make 5 random guesses for $g$ and see what we get out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "//anaconda3/lib/python3.7/site-packages/numpy/lib/stride_tricks.py:341: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.\n",
      "  array = np.array(array, copy=False, subok=subok)\n",
      "//anaconda3/lib/python3.7/site-packages/numpy/core/_asarray.py:171: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.\n",
      "  return array(a, dtype, copy=False, order=order, subok=True)\n",
      "//anaconda3/lib/python3.7/site-packages/numpy/core/_asarray.py:102: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    }
   ],
   "source": [
    "%matplotlib qt \n",
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "ax.set_xlabel('b1', fontsize=13)\n",
    "ax.set_ylabel('b2', fontsize=13)\n",
    "ax.set_zlabel('b3', fontsize=13)\n",
    "\n",
    "for i in range(1,5):\n",
    "    g = 2*(np.random.rand(2,1) - 0.5)\n",
    "    res = A.dot(g)\n",
    "    ax.plot3D(res[0],res[1],res[2],'c*', markersize =9)\n",
    "    plt.plot((0, res[0]), (0, res[1]), (0, res[2]), 'c', lw = 3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that you have convinced yourself that no matter what we choose for $x$, we are going to get out (after multiplication by $A$) a  vector that lies in the plane. That is unfortunate because the perfect solution would be the vector $b$, and it is usually the case that $b$ does not lie in our plane! What a bummer. But then again, if it did, we would have a perfect solution -- and that's not what we're dealing with in least squares.\n",
    "\n",
    "Let's look at the vector $b$ (rotate the plot to see it better).  Remember this is the list of ordinate values for the points in our first plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "ax.set_xlabel('b1', fontsize=13)\n",
    "ax.set_ylabel('b2', fontsize=13)\n",
    "ax.set_zlabel('b3', fontsize=13)\n",
    "\n",
    "ax.plot3D([b[0]], [b[1]] , [b[2]] ,'g*', markersize =9) #Latest addition\n",
    "plt.plot((0, b[0]), (0, b[1]), (0, b[2]), 'g', lw=3)  #Latest addition\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So now what? Clearly $b$ is not in the column space of $A$. Yet no matter what we choose for $x$, we will get a set of predicted values that form a vector that is in the column-space. Of course we want to find x so that the vector we get out is the closest one to the real solution.  \n",
    "\n",
    "Let's call the solution $x\\_bar$.  After all, there is no vector $x$, such that $A*x = b$.\n",
    "\n",
    "\n",
    "Now from the picture in front of you, it ought to be obvious that the closest we can get to $b$ is a vector that ends in the projection of $b$ on the blue plane. I'm going to drop the end of vector $b$ onto the plane. Try to ignore the math for the moment and grasp the geometry.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "b_proj = ((A.dot(np.linalg.inv(np.transpose(A).dot(A)))).dot(np.transpose(A))).dot(b)\n",
    "\n",
    "#Plot it\n",
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "ax.set_xlabel('b1', fontsize=13)\n",
    "ax.set_ylabel('b2', fontsize=13)\n",
    "ax.set_zlabel('b3', fontsize=13)\n",
    "\n",
    "ax.plot3D([b[0]], [b[1]] , [b[2]] ,'g*', markersize =9)\n",
    "plt.plot((0, b[0]), (0, b[1]), (0, b[2]), 'g', lw=3) \n",
    "plt.plot((0, b_proj[0]), (0, b_proj[1]), (0, b_proj[2]), 'g--', lw=5) #Latest addition\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The thick broken line is the projection of b onto the column-space of $A$.  Let's call this vector $b\\_proj$. It is the predicted value that we would get out of $A*x\\_bar = b\\_proj$. One way to see this is to return to the first plot and look at where $b\\_proj$ lies. These are the green asterisks connected by the green broken line.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline \n",
    "\n",
    "#Plot it \n",
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(data[:,0], data[:,1], s=80, marker = 'o', edgecolors='g')\n",
    "plt.plot(a, b_proj, 'g*--', markersize=20, lw=4)\n",
    "plt.ylim(-1,4)\n",
    "plt.xlim(-3,3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our $b\\_proj$ is the column vector containing the ordinate values of these three points. You can see that these are the predicted values that lie on the best fitting line. The way that we would say that this is the best fitting line is that the distance from the asterisks to the data values (vertically) is minimized.\n",
    "\n",
    "This distance is the difference between $b$ and $b\\_proj$.<br>\n",
    "Let's return to the vector diagram and convince ourselves.\n",
    "\n",
    "Visualizing this error vector is the key to understanding least squares.\n",
    "\n",
    "It helps to construct a line that represents the error between $b$ and $b\\_proj$.  The error, $err$, is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3,)\n"
     ]
    }
   ],
   "source": [
    "err = b - b_proj\n",
    "print(err.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's add a second dimension and make it a column vector to help us with the math later on:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3, 1)\n"
     ]
    }
   ],
   "source": [
    "err = np.expand_dims(err, axis=1)\n",
    "print(err.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rather than displaying this vector emanating from the origin, we will add it to the end of $b\\_proj$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot3D(A[0,:],A[1,:],A[2,:], 'ro', markersize =10)\n",
    "plt.plot((0, A[0,0]), (0, A[1,0]), (0, A[2,0]), 'r', lw = 3)\n",
    "plt.plot((0, A[0,1]), (0, A[1,1]), (0, A[2,1]), 'r', lw = 3)\n",
    "ax.plot_trisurf(q[0,:], q[1,:], q[2,:],  color = 'b', alpha = 0.7)\n",
    "ax.text(1, 1, 1, '(1,1,1)', size = 'large')\n",
    "ax.text(-1, 1, 2, '(-1,1,2)', size = 'large')\n",
    "ax.view_init(0,50)\n",
    "plt.ylim(-4,4)\n",
    "plt.xlim(-4,4)\n",
    "ax.set_zlim(-2,3)\n",
    "ax.set_xlabel('b1', fontsize=13)\n",
    "ax.set_ylabel('b2', fontsize=13)\n",
    "ax.set_zlabel('b3', fontsize=13)\n",
    "\n",
    "ax.plot3D([b[0]], [b[1]] , [b[2]] ,'g*', markersize =9)\n",
    "plt.plot((0, b[0]), (0, b[1]), (0, b[2]), 'g', lw=3) \n",
    "plt.plot((0, b_proj[0]), (0, b_proj[1]), (0, b_proj[2]), 'g--', lw=5) \n",
    "plt.plot((b_proj[0], b[0]), (b_proj[1], b[1]), (b_proj[2], b[2]), 'y--', lw=5) #Latest addition\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do you appreciate that this $err$ vector is orthogonal to the plane? There are lots of ways to talk about this, but you have to see and believe it first. Feel free to make the above plot interactive if it helps you see it better. Don't go on unless you are absolutely comfortable with the idea that $err$ is orthogonal to the plane spanned by the columns of $A$.\n",
    "\n",
    "O.K., you made it. Do you see that regardless of the dimension of the problem, the least squares solution will be a projection of some high dimensional vector, $b$, on some lower dimensional column space of $A$. Sticking with the 2-dimensional case, if we were fitting a line to 4 points instead of 3, we would be projecting a 4-dimensional vector, $b$, into the 2-dimensional plane spanned by the 2 4-dimensional column vectors of $A$.  One of these would be [1 1 1 1]'.   \n",
    "\n",
    "Let's return to the matter of $err$ and how to talk about it. We have already said that it is orthogonal to the columns of $A$. That means that the dot product of $err$ with any column of $A$ should be zero. Since $err$ is a column vector, we either have to take its transpose and multiply with $A$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[8.8817842e-16]\n",
      "[1.77635684e-15]\n"
     ]
    }
   ],
   "source": [
    "print(np.transpose(err).dot(A[:,0]))     # 1st column \n",
    "print(np.transpose(err).dot(A[:,1]))     # 2nd column "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or take the transpose of the column vectors and multiply these with the $err$ column vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[8.8817842e-16]\n",
      "[1.77635684e-15]\n"
     ]
    }
   ],
   "source": [
    "print(np.transpose(A[:,0]).dot(err))     # 1st column \n",
    "print(np.transpose(A[:,1]).dot(err))     # 2nd column "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I'm assuming you know about dot products.  If you don't, you'll have to pick up a book. You might just recall that it is the product  of the lengths times the cosine of the angle between the vectors. So $z'*z$ is the square of the length of $z$.  You can easily verify that the dot product of two orthogonal vectors is 0 <br>e.g., visualize and compute"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "0\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "print(np.array([1,0]).dot(np.transpose([0, 1]))) # [1 0] * [0 1]'\n",
    "print(np.array([1,1]).dot(np.transpose([-1, 1])))\n",
    "print(np.array([1,1]).dot(np.transpose([-2.7, 2.7]))) # [1 0] * [0 1]'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "...and so forth.\n",
    " \n",
    "If you don't understand dot products (sometimes called inner products) don't go on. \n",
    "\n",
    "If you understand the notion of inner products and the fact that $err$ is orthogonal to the columnspace of $A$, then we can move on. The first thing to say is that we can combine the dot products above into a single matrix operation.<br> The  following"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[8.88178420e-16 1.77635684e-15]]\n"
     ]
    }
   ],
   "source": [
    "print(np.transpose(err).dot(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "should be a row vector with two 0s. (You get very tiny numbers)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or we can transpose $A$ and multiply it times the $err$ column vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[8.88178420e-16]\n",
      " [1.77635684e-15]]\n"
     ]
    }
   ],
   "source": [
    "print(np.transpose(A).dot(err))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and get a column vector with 2 zeros."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a very imporant idea here.  The error vector, which is, after all, the residuals between the observed data and the fit, is orthogonal to the columnspace of $A$.  It lies in the left nullspace of $A$.  The reason  this is called the left nullspace is because we have to left-multiply  $err'*A$ to get our zeros.\n",
    "\n",
    "Once we see that this is so, we can derive the NORMAL EQUATIONS in matrix form.  We are looking for the best approximation to $x$ in the equation \n",
    "\n",
    "$$A*x = b. $$\n",
    "\n",
    "We have already admitted that there is no vector $x$ that will work because $A*x$ always lies in the columnspace of $A$, whereas $b$ does not.  So, instead, we'll solve for $x\\_bar$. We want the best solution to \n",
    "\n",
    "$$A*x\\_bar = b.$$\n",
    "\n",
    "Again, we admit we can't achieve equality. So we say that what we really want is to minimize th error between our fit, $A*x\\_bar$, and the observed values, $b$. In least squares, we want to minimize the sum of the $(b - A*x\\_bar)^{2}$. What you should be absolutely convinced of is that $b - A*x\\_bar$ is just the vector err, and that this vector is perpendicular to the columnspace of $A$.   \n",
    "\n",
    "Start with $err$.  We have already convinced ourselves of the fact that \n",
    "\n",
    "$$A' * err = 0$$. \n",
    "\n",
    "If you don't remember this, go back to the last code you executed.<br>\n",
    "But $err$ is just $b - A*x\\_bar$ <br>\n",
    "Substituting for $err$, we get\n",
    "\n",
    "$$\n",
    " \t\tA' * (b - A*x\\_bar) = 0\n",
    "$$\n",
    "\n",
    "Rearranging, we get \n",
    "\n",
    "$$\n",
    " \t\tA'*b = A'*A*x\\_bar\n",
    "$$\n",
    "\n",
    "Remember, we are trying to solve for $x\\_bar$. So we multiply both sides of the equation by the inverse of $(A'*A)$\n",
    "\n",
    "$$\n",
    "\t\t\\text{inv}(A'*A) * A'*b \t= \\text{inv}(A'*A) * (A'*A) * x\\_bar \\\\\n",
    "\t\t\\text{inv}(A'*A) * A'*b \t= x\\_bar \n",
    "$$\n",
    "\n",
    "or in other words,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.28571429 0.57142857]\n"
     ]
    }
   ],
   "source": [
    "x_bar = ((np.linalg.inv(np.transpose(A).dot(A)).dot(np.transpose(A)))).dot(b) #normal equation\n",
    "print(x_bar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The column vector describes the best fitting line to the data. <br>\n",
    "1.2857 is the intercept and 0.5714 is the slope. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot it \n",
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(a, b, s=80, marker = 'o', edgecolors='g')\n",
    "plt.plot(a, b_proj, 'g*', markersize=20, lw=4)\n",
    "\n",
    "#Here we'll create the line function\n",
    "axes = plt.gca()\n",
    "x_vals = np.linspace(-3, 3, 100)  \n",
    "y_vals = 1.2857 + 0.5714 * x_vals\n",
    "plt.plot(x_vals, y_vals, 'g', lw=3)\n",
    "\n",
    "plt.ylim(-1,4)\n",
    "plt.xlim(-3,3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now there is one thing left to appreciate before we extend this to multiple regression. I promised that I would explain the projection matrix we used to make our pretty picture. It would be nice to be able to project a vector onto the columnspace of $A$. Well, we already have one example:  $A * x\\_bar$ is the projection of $b$ onto the columnspace.  Now look at what you get when you left-multiply the equation for $x\\_bar$ by $A$.\n",
    "\n",
    "$$\n",
    "A * x\\_bar = A * \\text{inv}(A'*A) * A' * b\n",
    "$$\n",
    "\n",
    "This leads to a simple idea.  The seemingly messy string of matrices, $A * \\text{inv}(A'*A) * A'$, projects $b$ onto the plane in our figure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.92857143  0.21428571 -0.14285714]\n",
      " [ 0.21428571  0.35714286  0.42857143]\n",
      " [-0.14285714  0.42857143  0.71428571]]\n"
     ]
    }
   ],
   "source": [
    "P = (A.dot(np.linalg.inv(np.transpose(A).dot(A)))).dot(np.transpose(A)) # A * inv(A'*A) * A'\n",
    "print(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Earlier in the tutorial, we used $P$ to project the vertices of a square onto the columnspace of $A$. That's how we made the blue plane.\n",
    "\n",
    "So why is this so cool?  Because it gives us a very easy way to do multiple regression.  Suppose we have the ordered pairs,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[   0    1]\n",
      " [   1   53]\n",
      " [   2   35]\n",
      " [   3   -7]\n",
      " [   4   87]\n",
      " [   5  106]\n",
      " [   6  250]\n",
      " [   7  301]\n",
      " [   8  346]\n",
      " [   9  337]\n",
      " [  10  256]\n",
      " [  11  -25]\n",
      " [  12 -421]]\n"
     ]
    }
   ],
   "source": [
    "data = np.array([\n",
    "[0,1],\n",
    "[1,53],\n",
    "[2,35],\n",
    "[3,-7],\n",
    "[4,87],\n",
    "[5,106],\n",
    "[6,250],\n",
    "[7,301],\n",
    "[8,346],\n",
    "[9,337],\n",
    "[10,256],\n",
    "[11,-25],\n",
    "[12,-421]\n",
    "                ])\n",
    "    \n",
    "print(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot it \n",
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(data[:,0], data[:,1], s=80, marker = 'o', edgecolors='g')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want to fit a 4th order polynomial. The $A$ matrix has 5 columns:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1.0000e+00 0.0000e+00 0.0000e+00 0.0000e+00 0.0000e+00]\n",
      " [1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00]\n",
      " [1.0000e+00 2.0000e+00 4.0000e+00 8.0000e+00 1.6000e+01]\n",
      " [1.0000e+00 3.0000e+00 9.0000e+00 2.7000e+01 8.1000e+01]\n",
      " [1.0000e+00 4.0000e+00 1.6000e+01 6.4000e+01 2.5600e+02]\n",
      " [1.0000e+00 5.0000e+00 2.5000e+01 1.2500e+02 6.2500e+02]\n",
      " [1.0000e+00 6.0000e+00 3.6000e+01 2.1600e+02 1.2960e+03]\n",
      " [1.0000e+00 7.0000e+00 4.9000e+01 3.4300e+02 2.4010e+03]\n",
      " [1.0000e+00 8.0000e+00 6.4000e+01 5.1200e+02 4.0960e+03]\n",
      " [1.0000e+00 9.0000e+00 8.1000e+01 7.2900e+02 6.5610e+03]\n",
      " [1.0000e+00 1.0000e+01 1.0000e+02 1.0000e+03 1.0000e+04]\n",
      " [1.0000e+00 1.1000e+01 1.2100e+02 1.3310e+03 1.4641e+04]\n",
      " [1.0000e+00 1.2000e+01 1.4400e+02 1.7280e+03 2.0736e+04]]\n"
     ]
    }
   ],
   "source": [
    "A = np.ones((13,5))\n",
    "a = data[:,0]\n",
    "b = data[:,1]\n",
    "\n",
    "A[:,1] = a\n",
    "A[:,2] = a**2\n",
    "A[:,3] = a**3\n",
    "A[:,4] = a**4\n",
    "\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want the best solution for $x\\_bar$ in the equation $A*x\\_bar = b$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 20.24531351   4.48909179 -10.09446558   4.39042575  -0.32028143]\n"
     ]
    }
   ],
   "source": [
    "x_bar = ((np.linalg.inv(np.transpose(A).dot(A)).dot(np.transpose(A)))).dot(b) \n",
    "print(x_bar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's that simple.  We can get the fit back with"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pred_vals = A.dot(x_bar)\n",
    "\n",
    "#Plot it \n",
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(a, b, s=80, marker = 'o', edgecolors='g')\n",
    "plt.scatter(a, pred_vals, s=80, marker = '*', edgecolors='r')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can get a predicted curve by increasing the sampling density:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a_dens = np.transpose(np.linspace(min(a),max(a),100))\n",
    "\n",
    "A_dens = np.ones((100,5))\n",
    "A_dens[:,1] = a_dens\n",
    "A_dens[:,2] = a_dens**2\n",
    "A_dens[:,3] = a_dens**3\n",
    "A_dens[:,4] = a_dens**4\n",
    "\n",
    "pred_vals_dens = A_dens.dot(x_bar)\n",
    "\n",
    "#Plot it \n",
    "plt.figure(figsize=(12,5))\n",
    "plt.scatter(a, b, s=80, marker = 'o', edgecolors='g')\n",
    "plt.scatter(a, pred_vals, s=80, marker = '*', edgecolors='r')\n",
    "plt. plot(a_dens,pred_vals_dens,'g:')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}