15_linear_regression-checkpoint.ipynb

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    "setdefaults"
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    "%plot --format svg"
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   "source": [
    "# General Linear Regression"
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    "In general, you may want to fit other polynomials besides degree-1 (straight-lines)\n",
    "\n",
    "$y=a_{0}+a_{1}x+a_{2}x^{2}+\\cdots+a_{m}x^{m}+e$\n",
    "\n",
    "Now, the solution for $a_{0},~a_{1},...a_{m}$ is the minimization of m+1-dependent linear equations. \n",
    "\n",
    "Consider the following data:\n",
    "\n",
    "| x | y |\n",
    "|---|---|\n",
    "| 0.00 | 21.50 |\n",
    "| 2.00 | 20.84 |\n",
    "| 4.00 | 23.19 |\n",
    "| 6.00 | 22.69 |\n",
    "| 8.00 | 30.27 |\n",
    "| 10.00 | 40.11 |\n",
    "| 12.00 | 43.31 |\n",
    "| 14.00 | 54.79 |\n",
    "| 16.00 | 70.88 |\n",
    "| 18.00 | 89.48 |\n",
    "\n"
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    "load xy_data.csv\n",
    "x=xy_data(:,1);\n",
    "y=xy_data(:,2);\n",
    "plot(x,y,'o')"
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      "xy_data =\n",
      "\n",
      "    0.00000   21.50000\n",
      "    2.00000   20.84000\n",
      "    4.00000   23.19000\n",
      "    6.00000   22.69000\n",
      "    8.00000   30.27000\n",
      "   10.00000   40.11000\n",
      "   12.00000   43.31000\n",
      "   14.00000   54.79000\n",
      "   16.00000   70.88000\n",
      "   18.00000   89.48000\n",
      "\n"
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    "The model can be rewritten as \n",
    "\n",
    "$y=\\left[Z\\right]a+e$\n",
    "\n",
    "where $a=\\left[\\begin{array}{c}\n",
    "        a_{0}\\\\\n",
    "        a_{1}\\\\\n",
    "        a_{2}\\end{array}\\right]$\n",
    "        \n",
    "$[Z]=\\left[\\begin{array} \n",
    "1 & x_{1} & x_{1}^{2} \\\\\n",
    "1 & x_{2} & x_{2}^{2} \\\\\n",
    "1 & x_{3} & x_{3}^{2} \\\\\n",
    "1 & x_{4} & x_{4}^{2} \\\\\n",
    "1 & x_{5} & x_{5}^{2} \\\\\n",
    "1 & x_{6} & x_{6}^{2} \\\\\n",
    "1 & x_{7} & x_{7}^{2} \\\\\n",
    "1 & x_{8} & x_{8}^{2} \\\\\n",
    "1 & x_{9} & x_{9}^{2} \\\\\n",
    "1 & x_{10} & x_{10}^{2} \\end{array}\\right]$\n",
    "\n",
    "The sum of squares residuals for this model is\n",
    "\n",
    "$S_{r}=\\sum_{i=1}^{n}\\left(y_{i}-\\sum_{j=0}^{m}a_{j}z_{ji}\\right)^2$\n",
    "\n",
    "Minimizing this function results in\n",
    "\n",
    "$y=[Z]a$\n",
    "\n",
    "->**A standard Linear Algebra Problem**\n",
    "\n",
    "*the vector a is unknown, and Z is calculated based upon the assumed function*\n"
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     "text": [
      "Z =\n",
      "\n",
      "     1     0     0\n",
      "     1     2     4\n",
      "     1     4    16\n",
      "     1     6    36\n",
      "     1     8    64\n",
      "     1    10   100\n",
      "     1    12   144\n",
      "     1    14   196\n",
      "     1    16   256\n",
      "     1    18   324\n",
      "\n",
      "a =\n",
      "\n",
      "   22.46691\n",
      "   -1.36269\n",
      "    0.27634\n",
      "\n"
     ]
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   "source": [
    "Z=[x.^0,x,x.^2]\n",
    "\n",
    "a=Z\\y"
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    "x_fcn=linspace(min(x),max(x));\n",
    "plot(x,y,'o',x_fcn,a(1)+a(2)*x_fcn+a(3)*x_fcn.^2)"
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    "### General Coefficient of Determination\n",
    "\n",
    "$r^{2}=\\frac{S_{t}-S_{r}}{S_{t}}=1-\\frac{S_{r}}{S_t}$"
   ]
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     "text": [
      "St =  27.923\n",
      "Sr =  2.6366\n"
     ]
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   "source": [
    "St=std(y)\n",
    "Sr=std(y-a(1)-a(2)*x-a(3)*x.^2)"
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  {
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   "execution_count": 19,
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     "text": [
      "the coefficient of determination for this fit is 0.880485\n",
      "the correlation coefficient this fit is 0.938342\n"
     ]
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    "r2=1-Sr/St;\n",
    "r=sqrt(r2);\n",
    "\n",
    "fprintf('the coefficient of determination for this fit is %f',r2)\n",
    "fprintf('the correlation coefficient this fit is %f',r)"
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   "source": [
    "### Compare this to a straight line fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
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     "text": [
      "St =  27.923\n",
      "Sr =  9.2520\n",
      "the coefficient of determination for this fit is 0.668655\n",
      "the correlation coefficient this fit is 0.817713\n"
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    "Z=[ones(size(x)) x];\n",
    "a_line=Z\\y;\n",
    "x_fcn=linspace(min(x),max(x));\n",
    "plot(x,y,'o',x_fcn,a(1)+a(2)*x_fcn+a(3)*x_fcn.^2,...\n",
    "x_fcn,a_line(1)+a_line(2)*x_fcn)\n",
    "St=std(y)\n",
    "Sr=std(y-a_line(1)-a_line(2)*x)\n",
    "r2=1-Sr/St;\n",
    "r=sqrt(r2);\n",
    "\n",
    "fprintf('the coefficient of determination for this fit is %f',r2)\n",
    "fprintf('the correlation coefficient this fit is %f',r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
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    "plot(x,y-a(1)-a(2)*x-a(3)*x.^2)\n",
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    "## Warning \n",
    "**Coefficient of determination reduction does not always mean a better fit**\n",
    "\n",
    "Try the function, \n",
    "\n",
    "$y(x)=a_0+a_{1}x+a_{2}x^{2}+a_{4}x^{4}+a_{5}x^{5}+a_{5}x^{5}+a_{6}x^{6}+a_{7}x^{7}+a_{8}x^{8}$"
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      "a_overfit =\n",
      "\n",
      "   2.1487e+01\n",
      "  -1.4264e+01\n",
      "   1.5240e+01\n",
      "  -6.0483e+00\n",
      "   1.1887e+00\n",
      "  -1.2651e-01\n",
      "   7.4379e-03\n",
      "  -2.2702e-04\n",
      "   2.8063e-06\n",
      "\n"
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    "Z=[ones(size(x)) x x.^2 x.^3 x.^4 x.^5 x.^6 x.^7 x.^8];\n",
    "a_overfit=Z\\y"
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    "plot(x,y,'o',x_fcn,a(1)+a(2)*x_fcn+a(3)*x_fcn.^2,...\n",
    "x_fcn,a_overfit(1)+a_overfit(2)*x_fcn+a_overfit(3)*x_fcn.^2+...\n",
    "a_overfit(4)*x_fcn.^3+a_overfit(5)*x_fcn.^4+...\n",
    "a_overfit(6)*x_fcn.^5+a_overfit(7)*x_fcn.^6+...\n",
    "a_overfit(8)*x_fcn.^7+a_overfit(9)*x_fcn.^8)\n",
    "legend('data','parabola','n=8-fit','Location','NorthWest')"
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  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
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   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "St =  27.923\n",
      "Sr =  0.77320\n",
      "the coefficient of determination for this fit is 0.972309\n",
      "the correlation coefficient this fit is 0.986057\n"
     ]
    }
   ],
   "source": [
    "St=std(y)\n",
    "Sr=std(y-polyval(a_overfit(end:-1:1),x))\n",
    "r2=1-Sr/St;\n",
    "r=sqrt(r2);\n",
    "\n",
    "fprintf('the coefficient of determination for this fit is %f',r2)\n",
    "fprintf('the correlation coefficient this fit is %f',r)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linear Regression is only limited by the ability to separate the parameters from the function to achieve\n",
    "\n",
    "$y=[Z]a$\n",
    "\n",
    "$Z$ can be any function of the independent variable(s)\n",
    "\n",
    "**Example**:\n",
    "We assume two functions are added together, sin(t) and sin(3t). What are the amplitudes?\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
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   "outputs": [
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   "source": [
    "load sin_data.csv\n",
    "t=sin_data(1,:)';\n",
    "y=sin_data(2,:)';\n",
    "plot(t,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
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   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "a =\n",
      "\n",
      "   0.99727\n",
      "   0.50251\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Z=[sin(t) sin(3*t)];\n",
    "a=Z\\y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
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      "text/plain": [
       "<IPython.core.display.SVG object>"
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     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "plot(t,y,'.',t,a(1)*sin(t)+a(2)*sin(3*t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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