{ "cells": [ { "cell_type": "code", "execution_count": 171, "metadata": { "collapsed": true }, "outputs": [], "source": [ "setdefaults" ] }, { "cell_type": "code", "execution_count": 172, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%plot --format svg" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "![question 1](q1.png)\n", "\n", "![question 2](q2.png)\n", "\n", "### Project Ideas so far\n", "\n", "- Nothing yet...probably something heat transfer related\n", "\n", "- Modeling Propulsion or Propagation of Sound Waves\n", "\n", "- Low Thrust Orbital Transfer\n", "\n", "- Tracking motion of a satellite entering orbit until impact\n", "\n", "- What ever you think is best.\n", "\n", "- You had heat transfer project as an option; that sounded cool\n", "\n", "- Heat transfer through a pipe\n", "\n", "- I would prefer to do something with beam/plate mechanics or vibrations than a heat transfer or thermo problem\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Questions from you:\n", "\n", "- Is attempting to divide by zero an acceptable project idea?\n", "\n", "- Would it be alright if we worked in a group of 4?\n", "\n", "- What are acceptable project topics?\n", "\n", "- How do the exams look? \n", "\n", "- Is there no pdf for the lecture today?\n", "\n", "- Thank you for making the formatted lectures available!\n", "\n", "- did you do anything cool over spring break?\n", "\n", "- Could we have a group of 4? We don't want to have to choose which one of us is on their own.\n", "\n", "- In HW 5 should there be 4 vectors as an input?\n", "\n", "- Would it be possible for me to join a group of 3? I seem to be the odd man out in two 3 member groups that my friends are in." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Statistics and Curve-Fitting\n", "## Linear Regression\n", "\n", "Often, we have a model with empirical parameters. (e.g. Young's modulus, Poisson's ratio, drag coefficient, coefficient of restitution, spring constant)\n", "\n", "Experimental measurements are prone to a number of stochastic (random) factors:\n", "\n", "- Environmental noise\n", " \n", "- Measurement Uncertainty\n", " \n", "- Factors not accounted in model (e.g. 2D effects of 1D approximation)\n", "\n", "These can lead to **noise** (lack of precision) and **bias** (lack of accuracy)\n", "\n", "Consider a piece of glass being stretched. \n", "\n", "![movie of stretching glass in microtensile machine](sgs_strain.gif)\n", "\n", "It is clear that a straight line is a \"good\" fit, but how good and what line?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Statistics\n", "\n", "How do we describe *precision* and *accuracy*?\n", "\n", "- mean\n", "\n", "- standard deviation\n", "\n", "Take our class dart problem\n" ] }, { "cell_type": "code", "execution_count": 173, "metadata": { "collapsed": false }, "outputs": [ { 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"\t\n", "\n", "\t\n", "\n", "\t\n", "\n", "\t\n", "\t\tx position (cm)\n", "\t\n", "\n", "\n", "\t\n", "\t\ty position (cm)\n", "\t\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "gauss2d = @(x,y) exp(-((x-0).^2/2/1^2+(y-0).^2/2/5^2));\n", "\n", "x=linspace(-20,20,71);\n", "y=linspace(-20,20,31);\n", "scatter3(x_darts, y_darts,ones(length(x_darts),1))\n", "xlabel('x position (cm)')\n", "ylabel('y position (cm)')\n", "hold on\n", "[X,Y]=meshgrid(x,y);\n", "view([1,1,1])\n", "\n", "surf(X,Y,gauss2d(X,Y))" ] }, { "cell_type": "code", "execution_count": 190, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "Gnuplot\n", "Produced by GNUPLOT 5.0 patchlevel 3 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\t\n", "\t \n", 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"\tgnuplot_plot_20a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_21a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "hist(darts(:,2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Monte Carlo Simulations\n", "\n", "Monte Carlo models use random numbers to either understand statistics or generate a solution. \n", "\n", "### Example 1:\n", "#### Calculate $\\pi$ with random numbers. \n", "\n", "Assuming we can actually generate random numbers (a topic of philosophical and heated debates) we can populate a unit square with random points and determine the ratio of points inside and outside of a circle.\n", "\n", "![Unit circle and unit square](MonteCarloPi.gif)\n", "\n", "![1/4 Unit circle and 1/4 unit square](MonteCarloPi_rand.gif)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The ratio of the area of the circle to the square is:\n", "\n", "$\\frac{\\pi r^{2}}{4r^{2}}=\\frac{\\pi}{4}$\n", "\n", "So if we know the fraction of random points that are within the unit circle, then we can calculate $\\pi$\n", "\n", "(number of points in circle)/(total number of points)=$\\pi/4$" ] }, { "cell_type": "code", "execution_count": 191, "metadata": { "collapsed": false }, "outputs": [], "source": [ "function our_pi=montecarlopi(N)\n", " % Create random x-y-coordinates\n", "\n", " x=rand(N,1);\n", " y=rand(N,1);\n", " R=sqrt(x.^2+y.^2); % compute radius\n", " num_in_circle=sum(R<1);\n", " total_num_pts =length(R);\n", " our_pi = 4*num_in_circle/total_num_pts;\n", "end\n" ] }, { "cell_type": "code", "execution_count": 194, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mean value for pi = 3.138880\n", "standard deviation is 0.009956\n" ] } ], "source": [ "test_pi=zeros(10,1);\n", "for i=1:10\n", " test_pi(i)=montecarlopi(10000);\n", "end\n", "fprintf('mean value for pi = %f\\n',mean(test_pi))\n", "fprintf('standard deviation is %f',std(test_pi))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example 2\n", "#### Determine uncertainty in failure stress based on geometry\n", "\n", "In this example, we know that a steel bar will break under 940 MPa tensile stress. The bar is 1 mm by 2 mm with a tolerance of 1 %. What is the range of tensile loads that can be safely applied to the beam?\n", "\n", "$\\sigma_{UTS}=\\frac{F_{fail}}{wh}$\n", "\n", "$F_{fail}=\\sigma_{UTS}wh$" ] }, { "cell_type": "code", "execution_count": 196, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "Gnuplot\n", "Produced by GNUPLOT 5.0 patchlevel 3 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.01\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.02\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.03\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.04\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.05\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.06\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.07\n", 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"\t\n", "\t\n", "\tgnuplot_plot_35a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_36a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_37a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_38a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_39a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_40a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_41a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N=1000;\n", "r=rand(N,1);\n", "wmean=1; % in mm\n", "wmin=wmean-wmean*0.01;\n", "wmax=wmean+wmean*0.01;\n", "hmean=2; % in mm\n", "hmin=hmean-hmean*0.01;\n", "hmax=hmean+hmean*0.01;\n", "\n", "wrand=wmin+(wmax-wmin)*r;\n", "hrand=hmin+(hmax-hmin)*r;\n", "\n", "uts=940; % in N/mm^2=MPa\n", "\n", "Ffail=uts.*wrand.*hrand*1e-3; % force in kN\n", "hist(Ffail,20,1)\n", "xlabel('failure load (kN)')\n", "ylabel('relative counts')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Normally, the tolerance is not a maximum/minimum specification, but instead a normal distribution that describes the standard deviation, or the 68 % confidence interval.\n", "\n", "So instead, we should generate normally distributed dimensions." ] }, { "cell_type": "code", "execution_count": 197, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "Gnuplot\n", "Produced by GNUPLOT 5.0 patchlevel 3 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.02\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.04\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.06\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.08\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.1\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.12\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.14\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.16\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1.75\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1.8\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1.85\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1.9\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1.95\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t2\n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\n", "\t\n", "\t\trelative counts\n", "\t\n", "\n", "\n", "\t\n", "\t\tfailure load (kN)\n", "\t\n", "\n", "\n", "\n", "\tgnuplot_plot_1a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_5a\n", 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"\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_20a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_21a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_22a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_23a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_24a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_25a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_26a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_27a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_28a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_29a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_30a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_31a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_32a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_33a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_34a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_35a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_36a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_37a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_38a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_39a\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\tgnuplot_plot_40a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\tgnuplot_plot_41a\n", "\n", "\n", "\n", "\t\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N=1000;\n", "wmean=1; % in mm\n", "wstd=wmean*0.01; % standard deviation in mm\n", "hmean=2; % in mm\n", "hstd=hmean*0.01; % standard deviation in mm\n", "\n", "\n", "wrand=normrnd(wmean,wstd,[N,1]);\n", "hrand=normrnd(hmean,hstd,[N,1]);\n", "uts=940; % in N/mm^2=MPa\n", "\n", "Ffail=uts.*wrand.*hrand*1e-3; % force in kN\n", "hist(Ffail,20,1)\n", "xlabel('failure load (kN)')\n", "ylabel('relative counts')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear Least Squares Regression\n", "\n", "When approximating a set of data as a polynomial, there will always be error introduced (except in 2 cases). \n", "\n", "For a straight line, the actual data points, $y_{i}$ as a function of the independent variable, $x_{i}$, is:\n", "\n", "$y_{i}=a_{0}+a_{1}x_{i}+e_{i}$\n", "\n", "where $a_{0}$ and $a_{1}$ are the intercept and slope of the line and $e_{i}$ is the error between the approximate function and the recorded data point. \n", "\n", "We make the following assumptions in this analysis:\n", "\n", "1. Each x has a fixed value; it is not random and is known without error.\n", "\n", "2. The y values are independent random variables and all have the same variance.\n", "\n", "3. The y values for a given x must be normally distributed.\n", "\n", "The total error is \n", "\n", "$\\sum_{i=1}^{n}e_{i}=\\sum_{i=1}^{n}(y_{i}-a_{0}-a_{1}x_{i})$\n", "\n", "we don't care about the sign though. One approach has been demonstrated to provide a unique solution is minimizing the sum of squares error or\n", "\n", "$S_{r}=\\sum_{i=1}^{n}e_{i}^{2}=\\sum_{i=1}^{n}(y_{i}-a_{0}-a_{1}x_{i})^{2}$\n", "\n", "Where, $S_{r}$ is the sum of squares error (SSE). \n", "\n", "$\\frac{\\partial S_{r}}{\\partial a_{0}}=-2\\sum(y_{i}-a_{0}-a_{1}x_{i})$\n", "\n", "$\\frac{\\partial S_{r}}{\\partial a_{1}}=-2\\sum(y_{i}-a_{0}-a_{1}x_{i})x_{i}$\n", "\n", "The minimum $S_{r}$ occurrs when the partial derivatives are 0. \n", "\n", "$\\sum y_{i}= \\sum a_{0}+\\sum a_{1}x_{i}$\n", "\n", "$\\sum x_{i}y_{i}= \\sum a_{0}x_{i}+\\sum a_{1}x_{i}^{2}$\n", "\n", "$\\left[\\begin{array}{c}\n", "\\sum y_{i}\\\\\n", "\\sum x_{i}y_{i}\\end{array}\\right]=\n", "\\left[\\begin{array}{cc}\n", "n & \\sum x_{i}\\\\\n", "\\sum x_{i} & \\sum x_{i}^{2}\\end{array}\\right]\n", "\\left[\\begin{array}{c}\n", "a_{0}\\\\\n", "a_{1}\\end{array}\\right]$\n", "\n", "\n", "$b=Ax$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example \n", "\n", "Find drag coefficient with best-fit line to experimental data\n", "\n", "|i | v (m/s) | F (N) |\n", "|---|---|---|\n", "|1 | 10 | 25 |\n", "|2 | 20 | 70 |\n", "|3 | 30 | 380|\n", "|4 | 40 | 550|\n", "|5 | 50 | 610|\n", "|6 | 60 | 1220|\n", "|7 | 70 | 830 |\n", "|8 |80 | 1450|" ] }, { "cell_type": "code", "execution_count": 200, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a =\n", "\n", " -234.286\n", " 19.470\n", "\n" ] }, { "data": { "image/svg+xml": [ "\n", "\n", "Gnuplot\n", "Produced by GNUPLOT 5.0 patchlevel 3 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", "\t\t\n", "\t\t-500\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t500\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1000\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1500\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t2000\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t10\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t20\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t30\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t40\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t50\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t60\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t70\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t80\n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\n", "\t\n", "\t\tvelocity (m/s)\n", "\t\n", "\n", "\n", "\t\n", "\t\tForce (N)\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\tdata\n", "\n", "\n", "\n", "\t\n", "\t\tdata\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\n", "\t\n", "\tbest-fit\n", "\n", "\t\n", "\t\tbest-fit\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "drag_data=[...\n", "1 , 10 , 25 \n", "2 , 20 , 70 \n", "3 , 30 , 380\n", "4 , 40 , 550\n", "5 , 50 , 610\n", "6 , 60 , 1220\n", "7 , 70 , 830 \n", "8 ,80 , 1450];\n", "x=drag_data(:,2);\n", "y=drag_data(:,3);\n", "\n", "b=[sum(y);sum(x.*y)];\n", "A=[length(x),sum(x);\n", " sum(x), sum(x.^2)];\n", " \n", "a=A\\b\n", "\n", "plot(x,y,'o',x,a(1)+a(2)*x)\n", "legend('data','best-fit','Location','NorthWest')\n", "xlabel('Force (N)')\n", "ylabel('velocity (m/s)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How do we know its a \"good\" fit? \n", "\n", "Can compare the sum of squares error to the total sum of squares of the dependent variable (here F). \n", "\n", "$S_{r}=\\sum(y_{i}-a_{0}-a_{1}x_{i})^{2}$\n", "\n", "$S_{t}=\\sum(y_{i}-\\bar{y})^{2}$\n", "\n", "Then, we can calculate the *coefficient of determination*, $r^{2}$ or *correlation coefficient*, r. \n", "\n", "$r^{2}=\\frac{S_{t}-S_{r}}{S_{t}}$\n", "\n", "This represents the relative improvement of assuming that y is a function of x (if the x-values are not random and the y-values are random)\n", "\n", "For further information regarding statistical work on regression, look at \n", "[NIST Statistics Handbook](http://www.itl.nist.gov/div898/handbook/pmd/section4/pmd44.htm)" ] }, { "cell_type": "code", "execution_count": 128, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "St = 1.8083e+06\n", "St = 1.8083e+06\n" ] } ], "source": [ "Sr=sum((y-a(1)-a(2)*x).^2);\n", "St=std(y)^2*(length(y)-1)\n", "St=sum((y-mean(y)).^2)" ] }, { "cell_type": "code", "execution_count": 130, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "r2 = 0.88049\n", "r = 0.93834\n" ] } ], "source": [ "r2=(St-Sr)/St\n", "r=sqrt(r2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Limiting cases \n", "\n", "#### $r^{2}=0$ $S_{r}=S{t}$\n", "\n", "#### $r^{2}=1$ $S_{r}=0$" ] }, { "cell_type": "code", "execution_count": 152, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "Gnuplot\n", "Produced by GNUPLOT 5.0 patchlevel 3 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.2\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.4\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.6\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.8\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.2\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.4\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.6\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t0.8\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t1\n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\tCase 1\n", "\n", "\n", "\n", "\t\n", "\t\tCase 1\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\n", "\t\n", "\tCase 2\n", "\n", "\t\n", "\t\tCase 2\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "gen_examples\n", "plot(x1(1:50:end),y1(1:50:end),'s',x2,y2,'o')\n", "legend('Case 1','Case 2','Location','NorthWest')" ] }, { "cell_type": "code", "execution_count": 157, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a1 =\n", "\n", " 0.0497269\n", " 0.0016818\n", "\n", "a2 =\n", "\n", " 0\n", " 1\n", "\n", "Sr1 = 0.82607\n", "St1 = 0.82631\n", "coefficient of determination in Case 1 is 0.000286\n", "Sr2 = 0\n", "St2 = 1.0185\n", "coefficient of determination in Case 2 is 1.000000\n" ] } ], "source": [ "b1=[sum(y1);sum(x1.*y1)];\n", "A1=[length(x1),sum(x1);\n", " sum(x1), sum(x1.^2)];\n", " \n", "a1=A1\\b1\n", "\n", "b2=[sum(y2);sum(x2.*y2)];\n", "A2=[length(x2),sum(x2);\n", " sum(x2), sum(x2.^2)];\n", " \n", "a2=A2\\b2\n", "\n", "Sr1=sum((y1-a1(1)-a1(2)*x1).^2)\n", "St1=sum((y1-mean(y1)).^2)\n", "fprintf('coefficient of determination in Case 1 is %f\\n',1-Sr1/St1)\n", "\n", "Sr2=sum((y2-a2(1)-a2(2)*x2).^2)\n", "St2=sum((y2-mean(y2)).^2)\n", "\n", "fprintf('coefficient of determination in Case 2 is %f\\n',1-Sr2/St2)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Octave", "language": "octave", "name": "octave" }, "language_info": { "file_extension": ".m", "help_links": [ { "text": "MetaKernel Magics", "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" } ], "mimetype": "text/x-octave", "name": "octave", "version": "0.19.14" } }, "nbformat": 4, "nbformat_minor": 2 }