Section 8.1 - The Discrete Least Squares Approximation.ipynb

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   "source": [
    "---\n",
    "# Section 8.1: The Discrete Least Squares Approximation\n",
    "---"
   ]
  },
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    "## Polynomial Least squares\n",
    "\n",
    "#### Motivation:\n",
    "\n",
    "If we have several points $$(1,1.3);\\; (2,3.5); (3,4.2); (4,5.0); (5,7); (6,8.8); (7,10.1); (8,12.5); (9,13.0); (10,15.6)$$ to be interpolated as follows:\n",
    "\n",
    "![title](Fig8.1.png)\n",
    "\n",
    "The following polynomial is clearly a poor predictor of the information. \n",
    "\n",
    "![title](Fig8.2.png)\n",
    "\n",
    "A better approach would be to find the \"best\" approximating line, even if it does not agree precisely with the data at any point. \n",
    "\n",
    "![title](Fig8.3.png)\n",
    "\n",
    "The general problem of approximating a set of data, $\\{(x_i, y_i) \\quad |\\quad  i = 1, 2, \\ldots , m\\}$, with an algebraic polynomial\n",
    "\n",
    "$$P_n(x) = a_nx^n + a_{n−1}x^{n−1} + \\cdots + a_1x + a_0,$$\n",
    "of degree $n < m - 1$, using the least squares procedure is handled similarly. We choose the\n",
    "constants $a_0, a_1, \\ldots, a_n$ to minimize the least squares error $E = E_2(a_0, a_1, \\ldots , a_n)$, where\n",
    "\n",
    "$$E=\\sum_{i=1}^m(y_i-P_n(x_i))^2=\\sum_{i=1}^m\\left(y_i-\\left(a_nx_i^n + a_{n−1}x_i^{n−1} + \\cdots + a_1x_i + a_0\\right)\\right)^2,$$\n",
    "\n",
    "for E to be minimized it is necessary that $\\displaystyle \\frac{\\partial E}{\\partial a_j} = 0$, for each $j = 0,1,\\ldots,n$. Thus, for each $j$, we must have\n",
    "\n",
    "$$0=\\displaystyle \\frac{\\partial E}{\\partial a_j}=-2\\sum_{i=1}^m\\left(y_i-\\left(a_nx_i^n + a_{n−1}x_i^{n−1} + \\cdots + a_1x_i + a_0\\right)\\right)x_i^j.$$\n",
    "\n",
    "This gives the **Normal Equations** with $n+1$ equations and $n+1$ unknowns $a_j$ as follows:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "  \\sum_{i=1}^m x_i^0 & \\sum_{i=1}^m x_i^1 & \\cdots & \\sum_{i=1}^m x_i^n \\\\\n",
    "  \\sum_{i=1}^m x_i^1 & \\sum_{i=1}^m x_i^2 & \\cdots & \\sum_{i=1}^m x_i^{n+1} \\\\\n",
    "  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  \\sum_{i=1}^m x_i^n & \\sum_{i=1}^m x_i^{n+1} & \\cdots & \\sum_{i=1}^m x_i^{n+n}\\\\\n",
    "  \\end{bmatrix}\\begin{bmatrix}a_0\\\\a_1\\\\\\vdots\\\\a_n \\end{bmatrix}=\\begin{bmatrix}\\sum_{i=1}^my_ix_i^0\\\\\\sum_{i=1}^my_ix_i^1\\\\\\vdots\\\\\\sum_{i=1}^my_ix_i^n \\end{bmatrix}.$$\n",
    "  \n",
    "If we define $Y=\\begin{bmatrix}y_1\\\\y_2\\\\\\vdots\\\\y_m \\end{bmatrix}\\in \\mathbb{R}^{m}$ and $A\\in \\mathbb{R}^{m\\times (n+1)}$ as\n",
    "\n",
    "$$A=\\begin{bmatrix}\n",
    "1 & x_1 & \\cdots & x_1^n\\\\\n",
    "1 & x_2 & \\cdots & x_2^n\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "1 & x_m & \\cdots & x_m^n\\\\\n",
    "\\end{bmatrix},$$ then\n",
    "\n",
    "$$A^TAa=A^TY$$ are the **Normal Equations**, where $$a=\\begin{bmatrix}a_0\\\\a_1\\\\\\vdots\\\\a_n \\end{bmatrix}.$$ \n",
    "\n",
    "\n",
    "## Over-determined linear systems\n",
    "\n",
    "Let $A \\in \\mathbb{R}^{m \\times n}$ and $b \\in \\mathbb{R}^m$. \n",
    "\n",
    "If $m > n$, we say the linear system \n",
    "\n",
    "$$\n",
    "Ax = b\n",
    "$$ \n",
    "\n",
    "is **over-determined**.\n",
    "\n",
    "Often over-determined linear systems have no solution, typically due to measurement errors in  $b$."
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    "---"
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    "## Minimizing the error\n",
    "\n",
    "Since we cannot solve $Ax=b$ exactly, we want to find $x$ such that the **residual**\n",
    "\n",
    "$$\n",
    "r = b - Ax\n",
    "$$\n",
    "\n",
    "is as small as possible:\n",
    "\n",
    "$$\n",
    "\\min_x \\|b - Ax\\|\n",
    "$$\n",
    "\n",
    "We can consider many different norms, such as:\n",
    "\n",
    "$$\n",
    "\\|b - Ax\\|_1 = \\sum_{i=1}^m \\big| b_i - (Ax)_i \\big| = \\sum_{i=1}^m \\big| b_i - a_i^Tx \\big|\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\|b - Ax\\|_2 = \\sqrt{\\sum_{i=1}^m \\big( b_i - (Ax)_i \\big)^2} = \\sqrt{\\sum_{i=1}^m \\big( b_i - a_i^Tx \\big)^2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\|b - Ax\\|_\\infty = \\max_{1\\leq i\\leq m} \\big| b_i - (Ax)_i \\big| = \\max_{1\\leq i\\leq m} \\big| b_i - a_i^Tx \\big|\n",
    "$$\n",
    "\n",
    "\n",
    "\n"
   ]
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    "---"
   ]
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    "## Least squares\n",
    "\n",
    "When the error in the entries of $b$ are believed to be **identically and independently [normally distributed](https://en.wikipedia.org/wiki/Normal_distribution)** with **zero mean** and **constant variance**, the best choice is to minimize $\\|b - Ax\\|_2$. \n",
    "\n",
    "In this case, the $x$ that minimizes $\\|b - Ax\\|_2$ is the **maximum likelihood estimator** of the true solution.\n",
    "\n",
    "Minimizing $\\|b - Ax\\|_2$ is equivalent to minimizing\n",
    "\n",
    "$$\n",
    "\\|b - Ax\\|_2^2 = \\sum_{i=1}^m \\big( b_i - (Ax)_i \\big)^2.\n",
    "$$\n",
    "\n",
    "\n",
    "The $x$ that minimizes $\\|b - Ax\\|_2$ (or equivalently $\\|b - Ax\\|_2^2$) is called the **least-squares solution** because it is minimizing the **sum-of-the-squares** of the errors."
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    "![Paranormal distribution](http://m.jech.bmj.com/content/60/1/6/embed/graphic-1.gif \"A visual comparison of normal and paranormal distributions\")\n",
    "\n",
    "<center>\n",
    "[A visual comparison of normal and paranormal distributions](http://jech.bmj.com/content/60/1/6.full) by Matthew Freeman\n",
    "</center>\n"
   ]
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    "---"
   ]
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    "## Solving the least-squares problem\n",
    "\n",
    "In Julia (or MATLAB), we can solve the least-squares problem\n",
    "\n",
    "$$\n",
    "\\min_x \\|b - Ax\\|_2\n",
    "$$\n",
    "\n",
    "by using the same **backslash** function that we used to solve $n \\times n$ linear systems:\n",
    "\n",
    "```julia\n",
    "x = A\\b\n",
    "```\n",
    "\n",
    "Julia will recognize that the linear system is over-determined and will use an algorithm for solving the least-squares problem that is based on the $QR$-factorization of $A$."
   ]
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   "source": [
    "---\n",
    "## Example: "
   ]
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    "Approximate the points $(x_i,y_i)$ with $i=1,2,\\ldots,10$:\n",
    "\n",
    "$$(1,1.3);\\; (2,3.5);\\; (3,4.2);\\; (4,5.0);\\; (5,7);\\; (6,8.8);\\; (7,10.1);\\; (8,12.5);\\; (9,13.0);\\; (10,15.6)$$\n",
    "\n",
    "### The approximate quadratic polynomial\n",
    "\n",
    "We let our approximation be given by\n",
    "\n",
    "$$\n",
    "p(x) = a_0 + a_1 x + a_2 x^2\n",
    "$$\n",
    "\n",
    "and we want to find the coefficients $a_0, a_1, a_2$.\n",
    "\n",
    "We want to minimize\n",
    "\n",
    "$$\n",
    "\\sum_{i=1}^{m} \\big(y_i - p(x_i)\\big)^2.\n",
    "$$\n",
    "\n",
    "The residual $y_i - p(x_i)$ can be written as\n",
    "\n",
    "$\n",
    "y_i - \\left(a_0 + a_1 x_i + a_2 x^2_i\\right) = y_i - \\begin{bmatrix} 1 & x_i & x^2_i \\end{bmatrix} \\begin{bmatrix} a_0 \\\\ a_1 \\\\ a_2 \\end{bmatrix}.\n",
    "$$"
   ]
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    "Therefore, the $i^\\mathrm{th}$-row of the matrix $A$ is $\\begin{bmatrix} 1 & x_i & x^2_i \\end{bmatrix}$:\n",
    "\n",
    "$$\n",
    "A = \n",
    "\\begin{bmatrix} \n",
    "1 & x_1 & x^2_1 \\\\\n",
    "1 & x_2 & x^2_2 \\\\\n",
    "\\vdots&\\vdots&\\vdots\\\\\n",
    "1 & x_m & x^2_m \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "and $b \\in \\mathbb{R}^m$ such that $b_i = y_i$ for $i = 1,2,\\ldots,m$. \n",
    "\n",
    "Then \n",
    "\n",
    "$$\n",
    "\\sum_{i=1}^m \\big(y_i - p(x_i)\\big)^2 = \\|b - Ax\\|_2^2.\n",
    "$$"
   ]
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     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       "  10    55    385\n",
       "  55   385   3025\n",
       " 385  3025  25333"
      ]
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   "source": [
    "A = [1 1 1; 1 2 2^2; 1 3 3^2; 1 4 4^2;1 5 5^2;\n",
    "    1 6 6^2;1 7 7^2;1 8 8^2;1 9 9^2;1 10 10^2]\n",
    "\n",
    "b = [1.3;3.5;4.2;5.0;7;8.8;10.1;12.5;13.0;15.6]\n",
    "\n",
    "## Solve the least squares problem:  min norm(b - A*x) or equivalently solve A^TA*x=A^T*b\n",
    "x = A\\b\n",
    "\n",
    "A = [1 1 1; 1 2 2^2; 1 3 3^2; 1 4 4^2;1 5 5^2;\n",
    "    1 6 6^2;1 7 7^2;1 8 8^2;1 9 9^2;1 10 10^2]\n",
    "\n",
    "A'*A"
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",
      "text/plain": [
       "PyPlot.Figure(PyObject <matplotlib.figure.Figure object at 0x12b9bae90>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using PyPlot\n",
    "p(x) =  0.406667 + 1.15485*x + 0.0348485*x.^2\n",
    "\n",
    "m = 10\n",
    "xx = (1,2,3,4,5,6,7,8,9,10)\n",
    "tt=linspace(1, 10, m)\n",
    "\n",
    "\n",
    "yy = (1.3,3.5,4.2,5.0,7,8.8,10.1,12.5,13.0,15.6)\n",
    "plot(tt, p(tt), label=\"Least Square Solution\")\n",
    "plot(xx, yy, \"r.\", label=\"Data\")\n",
    "legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The approximating line \n",
    "\n",
    "We let our approximation be given by\n",
    "\n",
    "$$\n",
    "p(x) = a_0 + a_1 x \n",
    "$$\n",
    "\n",
    "and we want to find the coefficients $a_0, a_1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.36   \n",
       "  1.53818"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1 1; 1 2; 1 3; 1 4;1 5;\n",
    "    1 6;1 7;1 8;1 9;1 10]\n",
    "\n",
    "b = [1.3;3.5;4.2;5.0;7;8.8;10.1;12.5;13.0;15.6]\n",
    "\n",
    "## Solve the least squares problem:  min norm(b - A*x) or equivalently solve A^TA*x=A^T*b\n",
    "x = A\\b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.36   \n",
       "  1.53818"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1 1; 1 2; 1 3; 1 4;1 5;\n",
    "    1 6;1 7;1 8;1 9;1 10]\n",
    "b = [1.3;3.5;4.2;5.0;7;8.8;10.1;12.5;13.0;15.6]\n",
    "A'*A\n",
    "\n",
    "c=A'*b\n",
    "#A^TA*x=A^T*b Normal Equations\n",
    "A'*A\\c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "PyPlot.Figure(PyObject <matplotlib.figure.Figure object at 0x1317351d0>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using PyPlot\n",
    "p(x) =  -0.36 + 1.53818*x \n",
    "\n",
    "m = 10\n",
    "xx = (1,2,3,4,5,6,7,8,9,10)\n",
    "tt=linspace(1, 10, m)\n",
    "\n",
    "\n",
    "yy = (1.3,3.5,4.2,5.0,7,8.8,10.1,12.5,13.0,15.6)\n",
    "plot(tt, p(tt), label=\"Least Square Solution\")\n",
    "plot(xx, yy, \"r.\", label=\"Data\")\n",
    "legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Julia 0.6.0",
   "language": "julia",
   "name": "julia-0.6"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "0.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}