diff --git a/README.md b/README.md index 8145faf..134beea 100644 --- a/README.md +++ b/README.md @@ -49,7 +49,7 @@ Jupiter notebook (with matlab or octave kernel) ### Note on Homework and online forms -The Homeworks are graded based upon effort and completeness. The forms are not graded at +The Homeworks are graded based upon effort, correctness, and completeness. The forms are not graded at all, if they are completed you get credit. It is *your* responsibility to make sure your answers are correct. Use the homeworks and forms as a study guide for the exams. In general, I will not post homework solutions. diff --git a/lecture_16/lecture_16.ipynb b/lecture_16/lecture_16.ipynb index be266b3..206e84c 100644 --- a/lecture_16/lecture_16.ipynb +++ b/lecture_16/lecture_16.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 171, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 172, "metadata": { "collapsed": true }, @@ -119,7 +119,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 173, "metadata": { "collapsed": false }, @@ -773,7 +773,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 174, "metadata": { "collapsed": false }, @@ -811,7 +811,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 175, "metadata": { "collapsed": false }, @@ -830,7 +830,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 177, "metadata": { "collapsed": false }, @@ -849,7 +849,7 @@ }, { "cell_type": "code", - "execution_count": 83, + "execution_count": 180, "metadata": { "collapsed": false }, @@ -896,561 +896,539 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.05\n", + "\t\t\n", + "\t\t5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.1\n", + "\t\t\n", + "\t\t10\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.15\n", + "\t\t\n", + "\t\t15\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.2\n", + "\t\t\n", + "\t\t20\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-20\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-15\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-10\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t15\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\n", - "\n", - "\t\n", - "\t\trelative counts\n", - "\t\n", - "\n", - "\n", - "\t\n", - "\t\tx position (cm)\n", - "\t\n", - "\n", + "\t\n", "\n", "\n", "\tgnuplot_plot_1a\n", - "\n", + "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_5a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_6a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_7a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_8a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_9a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_10a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_11a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_12a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_13a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_14a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_15a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_16a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_17a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_18a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_19a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_20a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_21a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_22a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_23a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_24a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_25a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_26a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_27a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_28a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_29a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_30a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_31a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_32a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_33a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_34a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_35a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_36a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_37a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_38a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_39a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_40a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_41a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_42a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_43a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_44a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_45a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_46a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_47a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_48a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_49a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_50a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_51a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_52a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_53a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_54a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_55a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_56a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_57a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_58a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_59a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_60a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_61a\n", "\n", "\n", "\n", - "\t\n", - "\t\n", - "\tgnuplot_plot_62a\n", - "\n", - "\t\n", - "\t\n", - "\tgnuplot_plot_63a\n", - "\n", - "\t\n", - "\t\n", - "\tgnuplot_plot_64a\n", - "\n", - "\t\n", + "\t\n", "\t\n", - "\n", + "\n", "\n", "\n", "\n", @@ -1470,27 +1448,12 @@ } ], "source": [ - "x_vals=linspace(-15,20,30);\n", - "hist(x_darts,x_vals,1);\n", - "[histFreq, histXout] = hist(x_darts, 30);\n", - "binWidth = histXout(2)-histXout(1);\n", - "bar(histXout, histFreq/binWidth/sum(histFreq));\n", - "pdfnorm = @(x) 1/sqrt(2*s_x^2*pi).*exp(-(x-mu_x).^2/2/s_x^2);\n", - "%cdfnorm = @(x) 1/2*(1+erf((x-mu_x)./sqrt(2*s_x^2)));\n", - "%hist(x_darts,x_vals,trapz(x,f))%,cdfnorm(max(x_darts))/2)\n", - "hold on;\n", - "plot(x_vals,pdfnorm(x_vals))\n", - "n=2; % n=1, 68% confidence, n=2, 95% confidence, n=3, 99% conf\n", - " plot([mu_x+n*s_x,mu_x+n*s_x],[0,0.1],'r-')\n", - " plot([mu_x-n*s_x,mu_x-n*s_x],[0,0.1],'r-')\n", - "\n", - "xlabel('x position (cm)')\n", - "ylabel('relative counts')" + "hist(x_darts,30,100)" ] }, { "cell_type": "code", - "execution_count": 84, + "execution_count": 182, "metadata": { "collapsed": false }, @@ -1628,450 +1591,450 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_5a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_6a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_7a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_8a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_9a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_10a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_11a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_12a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_13a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_14a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_15a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_16a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_17a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_18a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_19a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_20a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_21a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_22a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_23a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_24a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_25a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_26a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_27a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_28a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_29a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_30a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_31a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_32a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_33a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_34a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_35a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + 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"\t\n", "\t\n", "\tgnuplot_plot_58a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_59a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_60a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_61a\n", "\n", @@ -2081,15 +2044,15 @@ "\t\n", "\tgnuplot_plot_62a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_63a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_64a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -2111,19 +2074,19 @@ } ], "source": [ - "y_vals=linspace(-15,20,30);\n", - "hist(y_darts,y_vals,1);\n", - "[histFreq, histXout] = hist(y_darts, 30);\n", + "x_vals=linspace(-15,20,30);\n", + "hist(x_darts,x_vals,1);\n", + "[histFreq, histXout] = hist(x_darts, 30);\n", "binWidth = histXout(2)-histXout(1);\n", "bar(histXout, histFreq/binWidth/sum(histFreq));\n", - "pdfnorm = @(x) 1/sqrt(2*s_y^2*pi).*exp(-(x-mu_y).^2/2/s_y^2);\n", + "pdfnorm = @(x) 1/sqrt(2*s_x^2*pi).*exp(-(x-mu_x).^2/2/s_x^2);\n", "%cdfnorm = @(x) 1/2*(1+erf((x-mu_x)./sqrt(2*s_x^2)));\n", "%hist(x_darts,x_vals,trapz(x,f))%,cdfnorm(max(x_darts))/2)\n", "hold on;\n", - "plot(y_vals,pdfnorm(y_vals))\n", + "plot(x_vals,pdfnorm(x_vals))\n", "n=2; % n=1, 68% confidence, n=2, 95% confidence, n=3, 99% conf\n", - " plot([mu_y+n*s_y,mu_y+n*s_y],[0,0.1],'r-')\n", - " plot([mu_y-n*s_y,mu_y-n*s_y],[0,0.1],'r-')\n", + " plot([mu_x+n*s_x,mu_x+n*s_x],[0,0.1],'r-')\n", + " plot([mu_x-n*s_x,mu_x-n*s_x],[0,0.1],'r-')\n", "\n", "xlabel('x position (cm)')\n", "ylabel('relative counts')" @@ -2131,7 +2094,7 @@ }, { "cell_type": "code", - "execution_count": 76, + "execution_count": 183, "metadata": { "collapsed": false }, @@ -2178,5238 +2141,17319 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.2\n", + "\t\t\n", + "\t\t0.05\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.4\n", + 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}, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% plot the distribution in x- and y-directions\n", + "gauss2d = @(x,y) exp(-((x-mu_x).^2/2/s_x^2+(y-mu_y).^2/2/s_y^2));\n", + "\n", + "x=linspace(-20,20,31);\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": 187, + "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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"\t\n", - "\n", - "\n", - "\t\t\n", - "\t\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_12a\n", + "\n", "\n", "\n", - "\t\t\n", - "\t\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_13a\n", + "\n", "\n", - "\n", - "\t\t\n", - "\t\t\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_14a\n", + "\n", "\n", "\n", - "\t\t\n", - "\t\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_15a\n", + "\n", "\n", - "\n", - "\t\t\n", - "\t\t\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_16a\n", + "\n", "\n", "\n", - "\t\t\n", - "\t\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_17a\n", + "\n", "\n", - "\n", - "\t\t\n", - "\t\t\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_18a\n", + "\n", "\n", "\n", - "\t\t\n", - "\t\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_19a\n", + "\n", "\n", - "\n", - "\t\t\n", - "\t\t\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", "\t\n", + "\tgnuplot_plot_20a\n", + "\n", "\n", "\n", - "\t\n", - "\n", - "\t\n", - "\n", - "\t\n", - "\n", - "\t\n", - "\t\tx position (cm)\n", + "\t\n", "\t\n", + "\tgnuplot_plot_21a\n", + "\n", "\n", - "\n", - "\t\n", - "\t\ty position (cm)\n", + "\n", + "\t\n", "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", "\n", "\n", "\n", @@ -7425,19 +19469,7 @@ } ], "source": [ - "% plot the distribution in x- and y-directions\n", - "gauss2d = @(x,y) exp(-((x-mu_x).^2/2/s_x^2+(y-mu_y).^2/2/s_y^2));\n", - "\n", - "x=linspace(-20,20,31);\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))" + "hist(darts(:,2))" ] }, { @@ -7473,7 +19505,7 @@ }, { "cell_type": "code", - "execution_count": 96, + "execution_count": 191, "metadata": { "collapsed": false }, @@ -7493,7 +19525,7 @@ }, { "cell_type": "code", - "execution_count": 98, + "execution_count": 194, "metadata": { "collapsed": false }, @@ -7502,15 +19534,15 @@ "name": "stdout", "output_type": "stream", "text": [ - "mean value for pi = 3.132400\n", - "standard deviation is 0.054925\n" + "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(1000);\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))" @@ -7532,7 +19564,7 @@ }, { "cell_type": "code", - "execution_count": 112, + "execution_count": 196, "metadata": { "collapsed": false }, @@ -7695,300 +19727,300 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_5a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_6a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_7a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_8a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_9a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_10a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_11a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_12a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_13a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_14a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_15a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", 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"\tgnuplot_plot_27a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_28a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_29a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_30a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_31a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_32a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_33a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_34a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_35a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_36a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_37a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_38a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_39a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_40a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_41a\n", "\n", @@ -8047,7 +20079,7 @@ }, { "cell_type": "code", - "execution_count": 113, + "execution_count": 197, "metadata": { "collapsed": false }, @@ -8146,52 +20178,27 @@ "\n", "\n", "\t\t\n", - "\t\t1.8\n", - "\t\n", - "\n", - "\n", - "\t\t\n", - "\t\t1.82\n", + "\t\t1.75\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t1.84\n", - "\t\n", - "\n", - "\n", - "\t\t\n", - "\t\t1.86\n", + "\t\t1.8\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t1.88\n", - "\t\n", - "\n", - "\n", - "\t\t\n", - "\t\t1.9\n", + "\t\t1.85\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t1.92\n", - "\t\n", - "\n", - "\n", - "\t\t\n", - "\t\t1.94\n", + "\t\t1.9\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t1.96\n", - "\t\n", - "\n", - "\n", - "\t\t\n", - "\t\t1.98\n", + "\t\t1.95\n", "\t\n", "\n", "\n", @@ -8220,300 +20227,300 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_5a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_6a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_7a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_8a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_9a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_10a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_11a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_12a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_13a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_14a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_15a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_16a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_17a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_18a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_19a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_20a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_21a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_22a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_23a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_24a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_25a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_26a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_27a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_28a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_29a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_30a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_31a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_32a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_33a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_34a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_35a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_36a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_37a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_38a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_39a\n", "\n", "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_40a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_41a\n", "\n", @@ -8637,7 +20644,7 @@ }, { "cell_type": "code", - "execution_count": 123, + "execution_count": 200, "metadata": { "collapsed": false }, @@ -8786,12 +20793,20 @@ "\n", "\n", "\n", - "\tgnuplot_plot_1a\n", - "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tdata\n", + "\n", "\n", "\n", - "\t \n", - "\t\n", + "\t\n", + "\t\tdata\n", + "\t\n", + "\n", + "\n", + "\t\n", "\t\n", "\t\n", "\t\n", @@ -8799,11 +20814,17 @@ "\t\n", "\t\n", "\t\n", + "\t\n", "\n", "\t\n", - "\tgnuplot_plot_2a\n", + "\tbest-fit\n", + "\n", + "\t\n", + "\t\tbest-fit\n", + "\t\n", + "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -8844,6 +20865,7 @@ "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)')" ] diff --git a/lecture_17/.ipynb_checkpoints/lecture_16-checkpoint.ipynb b/lecture_17/.ipynb_checkpoints/lecture_16-checkpoint.ipynb new file mode 100644 index 0000000..37443ae --- /dev/null +++ b/lecture_17/.ipynb_checkpoints/lecture_16-checkpoint.ipynb @@ -0,0 +1,754 @@ +{ + "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": [ + "# 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": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "# General Regression (Linear and nonlinear)" + ] + }, + { + "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 +} diff --git a/lecture_17/.ipynb_checkpoints/lecture_17-checkpoint.ipynb b/lecture_17/.ipynb_checkpoints/lecture_17-checkpoint.ipynb new file mode 100644 index 0000000..2fd6442 --- /dev/null +++ b/lecture_17/.ipynb_checkpoints/lecture_17-checkpoint.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_17/gen_examples.m b/lecture_17/gen_examples.m new file mode 100644 index 0000000..c5d6890 --- /dev/null +++ b/lecture_17/gen_examples.m @@ -0,0 +1,5 @@ +x1=linspace(0,1,1000)'; +y1=0*x1+0.1*rand(length(x1),1); + +x2=linspace(0,1,10)'; +y2=1*x2; diff --git a/lecture_17/in-class_regression.pdf b/lecture_17/in-class_regression.pdf new file mode 100644 index 0000000..628b495 Binary files /dev/null and b/lecture_17/in-class_regression.pdf differ diff --git a/lecture_17/lecture_17.ipynb b/lecture_17/lecture_17.ipynb new file mode 100644 index 0000000..35e61da --- /dev/null +++ b/lecture_17/lecture_17.ipynb @@ -0,0 +1,2710 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "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", + "\n", + "- Modeling Couette flow with a pressure gradient using a discretized form of the Navier-Stokes equation and comparing it to the analytical solution\n", + "\n", + "- Software to instruct a robotic arm to orient itself based on input from a gyroscope" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Questions from you:\n", + "\n", + "How was your spring break\n", + "\n", + "Are grades for hw 3 and 4 going to be posted?\n", + "\n", + "For class occasionally switching to doc cam and working through problems by hand might help get ideas across.\n", + "\n", + "Are you assigning those who do not have groups to groups?\n", + "\n", + "How do you graph a standard distribution in Matlab?\n", + "\n", + "What is the longest code you have written?\n", + "\n", + "how to approach probability for hw 5?\n", + "??\n", + "\n", + "Will you be assigning groups to people who do not currently have one? \n", + "\n", + "what are some basic guidelines we should use to brainstorm project ideas?\n", + "\n", + "Are you a fan of bananas?\n", + "\n", + "Going through code isn't the most helpful, because you can easily lose interest. But I am not sure what else you can do.\n", + "\n", + "Has lecture 15 been posted yet? Looking in the repository I can't seem to find it." + ] + }, + { + "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": 68, + "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": "code", + "execution_count": 72, + "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\t-2000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1000\n", + "\t\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\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\tresiduals (N)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tvelocity (m/s)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tModel does not capture measurements\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", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot(x,y-a(1)-40*x)\n", + "legend('data','best-fit','Location','NorthWest')\n", + "ylabel('residuals (N)')\n", + "xlabel('velocity (m/s)')\n", + "title('Model does not capture measurements')" + ] + }, + { + "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": 73, + "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": 74, + "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": 76, + "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": 77, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a1 =\n", + "\n", + " 0.0482496\n", + " 0.0017062\n", + "\n", + "a2 =\n", + "\n", + " 0\n", + " 1\n", + "\n", + "Sr1 = 0.83505\n", + "St1 = 0.83529\n", + "coefficient of determination in Case 1 is 0.000291\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": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "# General Linear Regression" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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" + ] + }, + { + "cell_type": "code", + "execution_count": 78, + "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\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", + "\t\t\n", + "\t\t90\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "load xy_data.csv\n", + "x=xy_data(1,:)';\n", + "y=xy_data(2,:)';\n", + "plot(x,y,'o')" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "xy_data =\n", + "\n", + " Columns 1 through 7:\n", + "\n", + " 0.00000 2.00000 4.00000 6.00000 8.00000 10.00000 12.00000\n", + " 21.50114 20.84153 23.19201 22.69498 30.26687 40.11075 43.30543\n", + "\n", + " Columns 8 through 11:\n", + "\n", + " 14.00000 16.00000 18.00000 20.00000\n", + " 54.78730 70.88443 89.48368 97.28135\n", + "\n" + ] + } + ], + "source": [ + "xy_data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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", + "$S_{r}=\\sum_{i=1}^{n}\\left(y_{i}-\\sum_{j=0}^{m}a_{j}z_{ji}\\right)$\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" + ] + }, + { + "cell_type": "code", + "execution_count": 79, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "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", + " 1 20 400\n", + "\n", + "a =\n", + "\n", + " 21.40341\n", + " -0.81538\n", + " 0.23935\n", + "\n" + ] + } + ], + "source": [ + "Z=[x.^0,x,x.^2]\n", + "\n", + "a=Z\\y" + ] + }, + { + "cell_type": "code", + "execution_count": 80, + "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\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "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)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### General Coefficient of Determination\n", + "\n", + "$r^{2}=\\frac{S_{t}-S_{r}}{S_{t}}=1-\\frac{S_{r}}{S_t}$" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "St = 27.923\n", + "Sr = 2.6366\n" + ] + } + ], + "source": [ + "St=std(y)\n", + "Sr=std(y-a(1)-a(2)*x-a(3)*x.^2)" + ] + }, + { + "cell_type": "code", + "execution_count": 81, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "the coefficient of determination for this fit is 0.880485\n", + "the correlation coefficient this fit is 0.938342\n" + ] + } + ], + "source": [ + "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": "markdown", + "metadata": {}, + "source": [ + "### Compare this to a straight line fit" + ] + }, + { + "cell_type": "code", + "execution_count": 83, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "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" + ] + }, + { + "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\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "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": 85, + "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\t-15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tresiduals of straight line\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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": [ + "plot(x,y-a_line(1)-a_line(2)*x)\n", + "title('residuals of straight line')" + ] + }, + { + "cell_type": "code", + "execution_count": 86, + "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\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tresiduals of parabola\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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": [ + "plot(x,y-a(1)-a(2)*x-a(3)*x.^2)\n", + "title('residuals of parabola')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Warning \n", + "**Coefficient of determination reduction does not always mean a better fit**\n", + "\n", + "Try the function, \n", + "\n", + "$y(x)=a_0+a_{1}x+a_{2}x^{2}+a_{4}x^{4}+a_{5}x^{5}+a_{5}x^{5}+a_{6}x^{6}+a_{7}x^{7}+a_{8}x^{8}$" + ] + }, + { + "cell_type": "code", + "execution_count": 90, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "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" + ] + } + ], + "source": [ + "Z=[ones(size(x)) x x.^2 x.^3 x.^4 x.^5 x.^6 x.^7 x.^8];\n", + "a_overfit=Z\\y" + ] + }, + { + "cell_type": "code", + "execution_count": 91, + "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\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\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", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tparabola\n", + "\n", + "\t\n", + "\t\tparabola\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tn=8-fit\n", + "\n", + "\t\n", + "\t\tn=8-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": [ + "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')" + ] + }, + { + "cell_type": "code", + "execution_count": 92, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "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": 6, + "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\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + 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[ + "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": 8, + "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\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", 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"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 +} diff --git a/lecture_17/octave-workspace b/lecture_17/octave-workspace new file mode 100644 index 0000000..c9ec2aa Binary files /dev/null and b/lecture_17/octave-workspace differ diff --git a/lecture_17/q1.png b/lecture_17/q1.png new file mode 100644 index 0000000..7c89cbf Binary files /dev/null and b/lecture_17/q1.png differ diff --git a/lecture_17/q2.png b/lecture_17/q2.png new file mode 100644 index 0000000..7eea25c Binary files /dev/null and b/lecture_17/q2.png differ diff --git a/lecture_17/sin_data.csv b/lecture_17/sin_data.csv new file mode 100644 index 0000000..a5ff272 --- /dev/null +++ b/lecture_17/sin_data.csv @@ -0,0 +1,2 @@ + 0.00000000e+00 1.26933037e-01 2.53866073e-01 3.80799110e-01 5.07732146e-01 6.34665183e-01 7.61598219e-01 8.88531256e-01 1.01546429e+00 1.14239733e+00 1.26933037e+00 1.39626340e+00 1.52319644e+00 1.65012947e+00 1.77706251e+00 1.90399555e+00 2.03092858e+00 2.15786162e+00 2.28479466e+00 2.41172769e+00 2.53866073e+00 2.66559377e+00 2.79252680e+00 2.91945984e+00 3.04639288e+00 3.17332591e+00 3.30025895e+00 3.42719199e+00 3.55412502e+00 3.68105806e+00 3.80799110e+00 3.93492413e+00 4.06185717e+00 4.18879020e+00 4.31572324e+00 4.44265628e+00 4.56958931e+00 4.69652235e+00 4.82345539e+00 4.95038842e+00 5.07732146e+00 5.20425450e+00 5.33118753e+00 5.45812057e+00 5.58505361e+00 5.71198664e+00 5.83891968e+00 5.96585272e+00 6.09278575e+00 6.21971879e+00 6.34665183e+00 6.47358486e+00 6.60051790e+00 6.72745093e+00 6.85438397e+00 6.98131701e+00 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7.77225453e-01 5.92618124e-01 3.08822183e-01 -1.07884730e-03 -3.46563271e-01 -5.64836023e-01 -8.11931510e-01 -1.05925186e+00 -1.13323611e+00 -1.11986890e+00 -8.88336727e-01 -9.54113139e-01 -6.81378679e-01 -6.02369117e-01 -4.78684439e-01 -5.88160325e-01 -4.93580777e-01 -5.68747320e-01 -7.51641934e-01 -8.14672884e-01 -9.53191554e-01 -9.55337518e-01 -9.85995556e-01 -9.63373597e-01 -1.01511061e+00 -7.56467517e-01 -4.17379564e-01 -1.22340361e-01 2.16273929e-01 5.16909714e-01 7.77031694e-01 1.00653798e+00 9.35718089e-01 1.00660116e+00 1.11177057e+00 9.85485116e-01 8.54344900e-01 6.26444042e-01 6.28124048e-01 4.27764254e-01 5.93991751e-01 4.79248018e-01 7.17522492e-01 7.35927848e-01 9.08802925e-01 9.38646871e-01 1.13125860e+00 1.07247935e+00 1.05198782e+00 9.41647332e-01 6.98801244e-01 4.03193543e-01 1.37009682e-01 -1.43203880e-01 -4.64369445e-01 -6.94978252e-01 -1.03483196e+00 -1.10261288e+00 -1.12892727e+00 -1.03902484e+00 -8.53573083e-01 -7.01815315e-01 -6.84745997e-01 -6.14189417e-01 -4.70090797e-01 -5.95052432e-01 -5.96497000e-01 -5.66861911e-01 -7.18239679e-01 -9.52873043e-01 -9.37512847e-01 -1.15782985e+00 -1.03858206e+00 -1.03182712e+00 -8.45121554e-01 -5.61821980e-01 -2.83427014e-01 -8.27056140e-02 diff --git a/lecture_17/xy_data.csv b/lecture_17/xy_data.csv new file mode 100644 index 0000000..15b919d --- /dev/null +++ b/lecture_17/xy_data.csv @@ -0,0 +1,2 @@ + 0.00000000e+00 2.00000000e+00 4.00000000e+00 6.00000000e+00 8.00000000e+00 1.00000000e+01 1.20000000e+01 1.40000000e+01 1.60000000e+01 1.80000000e+01 2.00000000e+01 + 2.15011412e+01 2.08415256e+01 2.31920098e+01 2.26949773e+01 3.02668745e+01 4.01107461e+01 4.33054255e+01 5.47873036e+01 7.08844291e+01 8.94836828e+01 9.72813533e+01 diff --git a/lecture_16/.compiled_data.csv.swp b/lecture_18/.Newtint.m.swp similarity index 70% rename from lecture_16/.compiled_data.csv.swp rename to lecture_18/.Newtint.m.swp index 70ab7de..769fe2b 100644 Binary files a/lecture_16/.compiled_data.csv.swp and b/lecture_18/.Newtint.m.swp differ diff --git a/lecture_18/.ipynb_checkpoints/lecture_18-checkpoint.ipynb b/lecture_18/.ipynb_checkpoints/lecture_18-checkpoint.ipynb new file mode 100644 index 0000000..2fd6442 --- /dev/null +++ b/lecture_18/.ipynb_checkpoints/lecture_18-checkpoint.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_18/Newtint.m b/lecture_18/Newtint.m new file mode 100644 index 0000000..e4c6c83 --- /dev/null +++ b/lecture_18/Newtint.m @@ -0,0 +1,34 @@ +function yint = Newtint(x,y,xx) +% Newtint: Newton interpolating polynomial +% yint = Newtint(x,y,xx): Uses an (n - 1)-order Newton +% interpolating polynomial based on n data points (x, y) +% to determine a value of the dependent variable (yint) +% at a given value of the independent variable, xx. +% input: +% x = independent variable +% y = dependent variable +% xx = value of independent variable at which +% interpolation is calculated +% output: +% yint = interpolated value of dependent variable + +% compute the finite divided differences in the form of a +% difference table +n = length(x); +if length(y)~=n, error('x and y must be same length'); end +b = zeros(n,n); +% assign dependent variables to the first column of b. +b(:,1) = y(:); % the (:) ensures that y is a column vector. +for j = 2:n + for i = 1:n-j+1 + b(i,j) = (b(i+1,j-1)-b(i,j-1))/(x(i+j-1)-x(i)); + end +end +%b +% use the finite divided differences to interpolate +xt = 1; +yint = b(1,1); +for j = 1:n-1 + xt = xt*(xx-x(j)); + yint = yint+b(1,j+1)*xt; +end diff --git a/lecture_18/challenger_oring.csv b/lecture_18/challenger_oring.csv new file mode 100644 index 0000000..11d647e --- /dev/null +++ b/lecture_18/challenger_oring.csv @@ -0,0 +1,24 @@ +Flight#,Temp,O-Ring Problem +1,53,1 +2,57,1 +3,58,1 +4,63,1 +5,66,0 +6,66.8,0 +7,67,0 +8,67.2,0 +9,68,0 +10,69,0 +11,69.8,1 +12,69.8,0 +13,70.2,1 +14,70.2,0 +15,72,0 +16,73,0 +17,75,0 +18,75,1 +19,75.8,0 +20,76.2,0 +21,78,0 +22,79,0 +23,81,0 diff --git a/lecture_18/lecture_18.ipynb b/lecture_18/lecture_18.ipynb new file mode 100644 index 0000000..bf7731c --- /dev/null +++ b/lecture_18/lecture_18.ipynb @@ -0,0 +1,2193 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 146, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 107, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 115, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 0.081014\n", + " 0.317493\n", + " 0.690279\n", + " 1.169170\n", + " 1.715370\n", + " 2.284630\n", + " 2.830830\n", + " 3.309721\n", + " 3.682507\n", + " 3.918986\n", + "\n" + ] + } + ], + "source": [ + "N=10;\n", + "A_beam=diag(ones(N,1))*2+diag(ones(N-1,1)*-1,-1)+diag(ones(N-1,1)*-1,1);\n", + "[v,e]=eig(A_beam);\n", + "diag(e)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Nonlinear Regression\n", + "\n", + "We can define any function and minimize the sum of squares error even if the constants cannot be separated.\n", + "\n", + "$S_{r}=\\left[y-f(z_{1},z_{2},...)\\right]^{2}$\n", + "\n", + "Consider the function, \n", + "\n", + "$f(x) = a_{0}(1-e^{a_{1}x})$\n", + "\n", + "We can define the sum of squares error as a function of $a_{0}$ and $a_{1}$:\n", + "\n", + "$f_{SSE}(a_{0},a_{1})=\\sum_{i=1}^{n}\\left[y- a_{0}(1-e^{a_{1}x})\\right]^{2}$" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "function [SSE,yhat] = sse_nonlin_exp(a,x,y)\n", + " % This is a sum of squares error function based on \n", + " % the two input constants a0 and a1 where a=[a0,a1]\n", + " % and the data is x (independent), y (dependent)\n", + " % and yhat is the model with the given a0 and a1 values\n", + " a0=a(1);\n", + " a1=a(2);\n", + " yhat=a0*(1-exp(a1*x));\n", + " SSE=sum((y-a0*(1-exp(a1*x))).^2);\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Where the data we are fitting is:\n", + "\n", + "| x | y |\n", + "|---|---|\n", + " | 0.0 | 0.41213|\n", + " | 1.0 | -2.65190|\n", + " | 2.0 | 15.04049|\n", + " | 3.0 | 5.19368|\n", + " | 4.0 | -0.71086|\n", + " | 5.0 | 12.69008|\n", + " | 6.0 | 29.20309|\n", + " | 7.0 | 58.68879|\n", + " | 8.0 | 91.61117|\n", + " | 9.0 | 173.75492|\n", + " | 10.0 | 259.04083|" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "data=[\n", + " 0.00000 0.41213\n", + " 1.00000 -2.65190\n", + " 2.00000 15.04049\n", + " 3.00000 5.19368\n", + " 4.00000 -0.71086\n", + " 5.00000 12.69008\n", + " 6.00000 29.20309\n", + " 7.00000 58.68879\n", + " 8.00000 91.61117\n", + " 9.00000 173.75492\n", + " 10.00000 259.04083\n", + "\n", + "];\n" + ] + }, + { + "cell_type": "code", + "execution_count": 116, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "the sum of squares for a0=-2.00 and a1=0.20 is 98118.4\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-50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t150\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t200\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t250\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t300\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "[SSE,yhat]=sse_nonlin_exp([-2,0.2],data(:,1),data(:,2));\n", + "fprintf('the sum of squares for a0=%1.2f and a1=%1.2f is %1.1f',...\n", + "-2,0.2,SSE)\n", + "plot(data(:,1),data(:,2),'o',data(:,1),yhat)" + ] + }, + { + "cell_type": "code", + "execution_count": 127, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a =\n", + "\n", + " -1.71891 0.50449\n", + "\n", + "fsse = 633.70\n" + ] + } + ], + "source": [ + "[a,fsse]=fminsearch(@(a) sse_nonlin_exp(a,data(:,1),data(:,2)),[-2,0.2])" + ] + }, + { + "cell_type": "code", + "execution_count": 128, + "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\t-50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t150\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t200\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t250\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t300\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "[sse,yhat]=sse_nonlin_exp(a,data(:,1),data(:,2));\n", + "plot(data(:,1),data(:,2),'o',data(:,1),yhat)" + ] + }, + { + "cell_type": "code", + "execution_count": 130, + "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\t-15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tresiduals of function\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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": [ + "[sse,yhat]=sse_nonlin_exp(a,data(:,1),data(:,2));\n", + "plot(data(:,1),data(:,2)-yhat)\n", + "title('residuals of function')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Case Study: Logistic Regression\n", + "\n", + "Many times the variable you predict is a binary (or discrete) value, such as pass/fail, broken/not-broken, etc. \n", + "\n", + "One method to fit this type of data is called [**logistic regression**](https://en.wikipedia.org/wiki/Logistic_regression).\n", + "\n", + "[Logistic Regression link 2](http://www.holehouse.org/mlclass/06_Logistic_Regression.html)\n", + "\n", + "We use a function that varies from 0 to 1 called a logistic function:\n", + "\n", + "$\\sigma(t)=\\frac{1}{1+e^{-t}}$\n", + "\n", + "We can use this function to describe the likelihood of failure (1) or success (0). When t=0, the probability of failure is 50%. " + ] + }, + { + "cell_type": "code", + "execution_count": 131, + "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\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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": [ + "t=linspace(-10,10);\n", + "sigma=@(t) 1./(1+exp(-t));\n", + "plot(t,sigma(t))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we make the assumption that we can predict the boundary between the pass-fail criteria with a function of our independent variable e.g.\n", + "\n", + "$y=\\left\\{\\begin{array}{cc} \n", + "1 & a_{0}+a_{1}x +\\epsilon >0 \\\\\n", + "0 & else \\end{array} \\right\\}$\n", + "\n", + "so the logistic function is now:\n", + "\n", + "$\\sigma(x)=\\frac{1}{1+e^{-(a_{0}+a_{1}x)}}$\n", + "\n", + "Here, there is not a direct sum of squares error, so we minimize a cost function: \n", + "\n", + "$J(a_{0},a_{1})=\\sum_{i=1}^{n}\\left[-y_{i}\\log(\\sigma(x_{i}))-(1-y_{i})\\log((1-\\sigma(x_{i})))\\right]$\n", + "\n", + "y=0,1 \n", + "\n", + "So the cost function either sums the " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example: Challenger O-ring failures\n", + "\n", + "The O-rings on the Challenger shuttles had problems when temperatures became low. We can look at the conditions when damage was observed to determine the likelihood of failure. \n", + "\n", + "[Challenger O-ring data powerpoint](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjZvL7jkP3SAhUp04MKHXkXDkMQFggcMAA&url=http%3A%2F%2Fwww.stat.ufl.edu%2F~winner%2Fcases%2Fchallenger.ppt&usg=AFQjCNFyjwT7NmRthDkDEgch75Fc5dc66w&sig2=_qeteX6-ZEBwPW8SZN1mIA)" + ] + }, + { + "cell_type": "code", + "execution_count": 132, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "oring =\n", + "\n", + " 1.00000 53.00000 1.00000\n", + " 2.00000 57.00000 1.00000\n", + " 3.00000 58.00000 1.00000\n", + " 4.00000 63.00000 1.00000\n", + " 5.00000 66.00000 0.00000\n", + " 6.00000 66.80000 0.00000\n", + " 7.00000 67.00000 0.00000\n", + " 8.00000 67.20000 0.00000\n", + " 9.00000 68.00000 0.00000\n", + " 10.00000 69.00000 0.00000\n", + " 11.00000 69.80000 1.00000\n", + " 12.00000 69.80000 0.00000\n", + " 13.00000 70.20000 1.00000\n", + " 14.00000 70.20000 0.00000\n", + " 15.00000 72.00000 0.00000\n", + " 16.00000 73.00000 0.00000\n", + " 17.00000 75.00000 0.00000\n", + " 18.00000 75.00000 1.00000\n", + " 19.00000 75.80000 0.00000\n", + " 20.00000 76.20000 0.00000\n", + " 21.00000 78.00000 0.00000\n", + " 22.00000 79.00000 0.00000\n", + " 23.00000 81.00000 0.00000\n", + "\n" + ] + } + ], + "source": [ + "% read data from csv file \n", + "% col 1 = index\n", + "% col 2 = temperature\n", + "% col 3 = 1 if damaged, 0 if undamaged\n", + "oring=dlmread('challenger_oring.csv',',',1,0)" + ] + }, + { + "cell_type": "code", + "execution_count": 134, + "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\t50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t55\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t65\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t70\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t75\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t85\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tfailure (1)/ pass (0)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tTemp (F)\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot(oring(:,2),oring(:,3),'o')\n", + "xlabel('Temp (F)')\n", + "ylabel('failure (1)/ pass (0)')" + ] + }, + { + "cell_type": "code", + "execution_count": 135, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "function J=sse_logistic(a,x,y)\n", + " % Create function to calculate cost of logistic function\n", + " % t = a0+a1*x\n", + " % sigma(t) = 1./(1+e^(-t))\n", + " sigma=@(t) 1./(1+exp(-t));\n", + " a0=a(1);\n", + " a1=a(2);\n", + " t=a0+a1*x;\n", + " J = 1/length(x)*sum(-y.*log(sigma(t))-(1-y).*log(1-sigma(t)));\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 142, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "J = 0.88822\n", + "a =\n", + "\n", + " 15.03501 -0.23205\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\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\t1.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t55\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t65\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t70\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t75\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t85\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "J=sse_logistic([10,-0.2],oring(:,2),oring(:,3))\n", + "a=fminsearch(@(a) sse_logistic(a,oring(:,2),oring(:,3)),[0,-3])\n", + "\n", + "T=linspace(50,85);\n", + "plot(oring(:,2),oring(:,3),'o',T,sigma(a(1)+a(2)*T),T,a(1)+a(2)*T)\n", + "axis([50,85,-0.1,1.2])" + ] + }, + { + "cell_type": "code", + "execution_count": 139, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "probability of failure when 70 degrees is 23.00% \n", + "probability of failure when 60 degrees is 75.25%\n", + "probability of failure when 36 degrees is 99.87%\n" + ] + } + ], + "source": [ + "fprintf('probability of failure when 70 degrees is %1.2f%% ',100*sigma(a(1)+a(2)*70))\n", + "fprintf('probability of failure when 60 degrees is %1.2f%%',100*sigma(a(1)+a(2)*60))\n", + "fprintf('probability of failure when 36 degrees is %1.2f%%',100*sigma(a(1)+a(2)*36))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Interpolation\n", + "\n", + "Using regression (linear and nonlinear) you are faced with the problem, that you have lots of noisy data and you want to fit a physical model to it. \n", + "\n", + "You can use interpolation to solve the opposite problem, you have a little data with very little noise.\n", + "\n", + "## Linear interpolation\n", + "\n", + "If you are trying to find the value of f(x) for x between $x_{1}$ and $x_{2}$, then you can match the slopes\n", + "\n", + "$\\frac{f(x)-f(x_{1})}{x-x_{1}}=\\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$\n", + "\n", + "or\n", + "\n", + "$f(x)=f(x_{1})+(x-x_{1})\\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$\n", + "\n", + "### Example: Logarithms\n", + "\n", + "Engineers used to have to use interpolation in logarithm tables for calculations. Find ln(2) from \n", + "\n", + "a. ln(1) and ln(6)\n", + "\n", + "b. ln(1) and ln(4)\n", + "\n", + "c. just calculate it as ln(2)" + ] + }, + { + "cell_type": "code", + "execution_count": 149, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ln(2)~0.358352\n", + "ln(2)~0.462098\n", + "ln(2)~0.549306\n", + "ln(2)=0.693147\n" + ] + } + ], + "source": [ + "ln2_16=log(1)+(log(6)-log(1))/(6-1)*(2-1);\n", + "fprintf('ln(2)~%f\\n',ln2_16)\n", + "ln2_14=log(1)+(log(4)-log(1))/(4-1)*(2-1);\n", + "ln2_13=log(1)+(log(3)-log(1))/(3-1)*(2-1);\n", + "fprintf('ln(2)~%f\\n',ln2_14)\n", + "fprintf('ln(2)~%f\\n',ln2_13)\n", + "ln2=log(2);\n", + "fprintf('ln(2)=%f\\n',ln2)" + ] + }, + { + "cell_type": "code", + "execution_count": 147, + "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\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tln(x)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tx\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t\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": [ + "x=linspace(1,6);\n", + "plot(x,log(x),2,log(2),'*',...\n", + "[1,2,6],[log(1),ln2_16,log(6)],'o-',...\n", + "[1,2,4],[log(1),ln2_14,log(4)],'s-')\n", + "ylabel('ln(x)')\n", + "xlabel('x')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Quadratic interpolation (intro curvature)\n", + "\n", + "Assume function is parabola between 3 points. The function is can be written as:\n", + "\n", + "$f_{2}(x)=b_{1}+b_{2}(x-x_{1})+b_{3}(x-x_{1})(x-x_{2})$\n", + "\n", + "When $x=x_{1}$\n", + "\n", + "$f(x_{1})=b_{1}$\n", + "\n", + "when $x=x_{2}$\n", + "\n", + "$b_{2}=\\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$\n", + "\n", + "when $x=x_{3}$\n", + "\n", + "$b_{3}=\\frac{\\frac{f(x_{3})-f(x_{2})}{x_{3}-x_{2}}\n", + "-\\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}{x_{3}-x_{1}}$\n", + "\n", + "#### Reexamining the ln(2) with ln(1), ln(4), and ln(6):" + ] + }, + { + "cell_type": "code", + "execution_count": 154, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Z =\n", + "\n", + " 1 1 1\n", + " 1 4 16\n", + " 1 600 360000\n", + "\n", + "ans = 5.1766e+05\n", + "ans =\n", + "\n", + " -4.6513e-01\n", + " 4.6589e-01\n", + " -7.5741e-04\n", + "\n" + ] + } + ], + "source": [ + "x=[1,4,600]';\n", + "Z=[x.^0,x.^1,x.^2]\n", + "cond(Z)\n", + "Z\\log(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 155, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "b1 = 0\n", + "b2 = 0.46210\n", + "b3 = -0.051873\n" + ] + } + ], + "source": [ + "x1=1;\n", + "x2=4;\n", + "x3=6;\n", + "f1=log(x1);\n", + "f2=log(x2);\n", + "f3=log(x3);\n", + "\n", + "b1=f1\n", + "b2=(f2-b1)/(x2-x1)\n", + "b3=(f3-f2)/(x3-x2)-b2;\n", + "b3=b3/(x3-x1)" + ] + }, + { + "cell_type": "code", + "execution_count": 157, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 0.56584\r\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\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\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": [ + "x=linspace(1,6);\n", + "f=@(x) b1+b2*(x-x1)+b3*(x-x1).*(x-x2);\n", + "plot(x,log(x),2,log(2),'*',...\n", + "[1,4,6],[log(1),log(4),log(6)],'ro',...\n", + "x,f(x),'r-',2,f(2),'s')\n", + "f(2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Newton's Interpolating Polynomials\n", + "\n", + "For n-data points, we can fit an (n-1)th-polynomial\n", + "\n", + "$f_{n-1}(x)=b_{1}+b_{2}(x-x_{1})+\\cdots+b_{n}(x-x_{1})(x-x_{2})\\cdots(x-x_{n})$\n", + "\n", + "where \n", + "\n", + "$b_{1}=f(x_{1})$\n", + "\n", + "$b_{2}=\\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$\n", + "\n", + "$b_{3}=\\frac{\\frac{f(x_{3})-f(x_{2})}{x_{3}-x_{2}}\n", + "-b_{2}}{x_{3}-x_{1}}$\n", + "\n", + "$\\vdots$\n", + "\n", + "$b_{n}=f(x_{n},x_{n-1},...,x_{2},x_{1})\n", + "=\\frac{f(x_{n},x_{n-1},...x_{2})-f(x_{n-1},x_{n-2},...,x_{1})}{x_{n}-x_{1}}$\n", + "\n", + "**e.g. for 4 data points:**\n", + "\n", + "![Newton Interpolation Iterations](newton_interpolation.png)" + ] + }, + { + "cell_type": "code", + "execution_count": 160, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "b =\n", + "\n", + " 0.00000 0.54931 -0.08721 0.01178\n", + " 1.09861 0.28768 -0.02832 0.00000\n", + " 1.38629 0.20273 0.00000 0.00000\n", + " 1.79176 0.00000 0.00000 0.00000\n", + "\n", + "ans = 0.66007\n", + "ans =\n", + "\n", + " 0.00000\n", + " 1.09861\n", + " 1.38629\n", + " 1.79176\n", + "\n" + ] + } + ], + "source": [ + "Newtint([1,3,4,6],log([1,3,4,6]),2)\n", + "log([1,3,4,6]')" + ] + }, + { + "cell_type": "code", + "execution_count": 162, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ln(2)=0.693147\n", + "ln(2)~0.366349\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-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\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", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "\n", + "x=[0.2,3,10,20,50,60]; % define independent var's\n", + "y=log(x); % define dependent var's\n", + "xx=linspace(min(x),max(x));\n", + "yy=zeros(size(xx));\n", + "for i=1:length(xx)\n", + " yy(i)=Newtint(x,y,xx(i));\n", + "end\n", + "plot(xx,log(xx),2,log(2),'*',...\n", + "x,y,'ro',...\n", + "xx,yy,'r-')\n", + "\n", + "fprintf('ln(2)=%f',log(2))\n", + "fprintf('ln(2)~%f',Newtint(x,y,2))" + ] + }, + { + "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 +} diff --git a/lecture_18/newton_interpolation.png b/lecture_18/newton_interpolation.png new file mode 100644 index 0000000..5990cb5 Binary files /dev/null and b/lecture_18/newton_interpolation.png differ diff --git a/lecture_18/octave-workspace b/lecture_18/octave-workspace new file mode 100644 index 0000000..c651696 Binary files /dev/null and b/lecture_18/octave-workspace differ diff --git a/lecture_19/.ipynb_checkpoints/lecture 19-checkpoint.ipynb b/lecture_19/.ipynb_checkpoints/lecture 19-checkpoint.ipynb new file mode 100644 index 0000000..2911efe --- /dev/null +++ b/lecture_19/.ipynb_checkpoints/lecture 19-checkpoint.ipynb @@ -0,0 +1,497 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Splines (Brief introduction before next section)\n", + "\n", + "Following interpolation discussion, instead of estimating 9 data points with an eighth-order polynomial, it makes more sense to fit sections of the curve to lower-order polynomials:\n", + "\n", + "0. zeroth-order (nearest neighbor)\n", + "\n", + "1. first-order (linear interpolation)\n", + "\n", + "2. third-order (cubic interpolation)\n", + "\n", + "Matlab and Octave have built-in functions for 1D and 2D interpolation:\n", + "\n", + "`interp1`\n", + "\n", + "`interp2`" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp1' is a function from the file /usr/share/octave/4.0.0/m/general/interp1.m\n", + "\n", + " -- Function File: YI = interp1 (X, Y, XI)\n", + " -- Function File: YI = interp1 (Y, XI)\n", + " -- Function File: YI = interp1 (..., METHOD)\n", + " -- Function File: YI = interp1 (..., EXTRAP)\n", + " -- Function File: YI = interp1 (..., \"left\")\n", + " -- Function File: YI = interp1 (..., \"right\")\n", + " -- Function File: PP = interp1 (..., \"pp\")\n", + "\n", + " One-dimensional interpolation.\n", + "\n", + " Interpolate input data to determine the value of YI at the points\n", + " XI. If not specified, X is taken to be the indices of Y ('1:length\n", + " (Y)'). If Y is a matrix or an N-dimensional array, the\n", + " interpolation is performed on each column of Y.\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"previous\"\n", + " Return the previous neighbor.\n", + "\n", + " \"next\"\n", + " Return the next neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " Adding '*' to the start of any method above forces 'interp1' to\n", + " assume that X is uniformly spaced, and only 'X(1)' and 'X(2)' are\n", + " referenced. This is usually faster, and is never slower. The\n", + " default method is \"linear\".\n", + "\n", + " If EXTRAP is the string \"extrap\", then extrapolate values beyond\n", + " the endpoints using the current METHOD. If EXTRAP is a number,\n", + " then replace values beyond the endpoints with that number. When\n", + " unspecified, EXTRAP defaults to 'NA'.\n", + "\n", + " If the string argument \"pp\" is specified, then XI should not be\n", + " supplied and 'interp1' returns a piecewise polynomial object. This\n", + " object can later be used with 'ppval' to evaluate the\n", + " interpolation. There is an equivalence, such that 'ppval (interp1\n", + " (X, Y, METHOD, \"pp\"), XI) == interp1 (X, Y, XI, METHOD, \"extrap\")'.\n", + "\n", + " Duplicate points in X specify a discontinuous interpolant. There\n", + " may be at most 2 consecutive points with the same value. If X is\n", + " increasing, the default discontinuous interpolant is\n", + " right-continuous. If X is decreasing, the default discontinuous\n", + " interpolant is left-continuous. The continuity condition of the\n", + " interpolant may be specified by using the options \"left\" or \"right\"\n", + " to select a left-continuous or right-continuous interpolant,\n", + " respectively. Discontinuous interpolation is only allowed for\n", + " \"nearest\" and \"linear\" methods; in all other cases, the X-values\n", + " must be unique.\n", + "\n", + " An example of the use of 'interp1' is\n", + "\n", + " xf = [0:0.05:10];\n", + " yf = sin (2*pi*xf/5);\n", + " xp = [0:10];\n", + " yp = sin (2*pi*xp/5);\n", + " lin = interp1 (xp, yp, xf);\n", + " near = interp1 (xp, yp, xf, \"nearest\");\n", + " pch = interp1 (xp, yp, xf, \"pchip\");\n", + " spl = interp1 (xp, yp, xf, \"spline\");\n", + " plot (xf,yf,\"r\", xf,near,\"g\", xf,lin,\"b\", xf,pch,\"c\", xf,spl,\"m\",\n", + " xp,yp,\"r*\");\n", + " legend (\"original\", \"nearest\", \"linear\", \"pchip\", \"spline\");\n", + "\n", + " See also: pchip, spline, interpft, interp2, interp3, interpn.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp1" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp2' is a function from the file /usr/share/octave/4.0.0/m/general/interp2.m\n", + "\n", + " -- Function File: ZI = interp2 (X, Y, Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, N)\n", + " -- Function File: ZI = interp2 (Z)\n", + " -- Function File: ZI = interp2 (..., METHOD)\n", + " -- Function File: ZI = interp2 (..., METHOD, EXTRAP)\n", + "\n", + " Two-dimensional interpolation.\n", + "\n", + " Interpolate reference data X, Y, Z to determine ZI at the\n", + " coordinates XI, YI. The reference data X, Y can be matrices, as\n", + " returned by 'meshgrid', in which case the sizes of X, Y, and Z must\n", + " be equal. If X, Y are vectors describing a grid then 'length (X)\n", + " == columns (Z)' and 'length (Y) == rows (Z)'. In either case the\n", + " input data must be strictly monotonic.\n", + "\n", + " If called without X, Y, and just a single reference data matrix Z,\n", + " the 2-D region 'X = 1:columns (Z), Y = 1:rows (Z)' is assumed.\n", + " This saves memory if the grid is regular and the distance between\n", + " points is not important.\n", + "\n", + " If called with a single reference data matrix Z and a refinement\n", + " value N, then perform interpolation over a grid where each original\n", + " interval has been recursively subdivided N times. This results in\n", + " '2^N-1' additional points for every interval in the original grid.\n", + " If N is omitted a value of 1 is used. As an example, the interval\n", + " [0,1] with 'N==2' results in a refined interval with points at [0,\n", + " 1/4, 1/2, 3/4, 1].\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " EXTRAP is a scalar number. It replaces values beyond the endpoints\n", + " with EXTRAP. Note that if EXTRAPVAL is used, METHOD must be\n", + " specified as well. If EXTRAP is omitted and the METHOD is\n", + " \"spline\", then the extrapolated values of the \"spline\" are used.\n", + " Otherwise the default EXTRAP value for any other METHOD is \"NA\".\n", + "\n", + " See also: interp1, interp3, interpn, meshgrid.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp2" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "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\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\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", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\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": [ + "x=linspace(-pi,pi,9);\n", + "xi=linspace(-pi,pi,100);\n", + "y=sin(x);\n", + "yi_lin=interp1(x,y,xi,'linear');\n", + "yi_spline=interp1(x,y,xi,'spline'); \n", + "yi_cubic=interp1(x,y,xi,'cubic');\n", + "plot(x,y,'o',xi,yi_lin,xi,yi_spline,xi,yi_cubic)\n", + "axis([-pi,pi,-1.5,1.5])\n", + "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example: Accelerate then hold velocity\n", + "\n", + "Here the time is given as vector t in seconds and the " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "t=[0 2 40 56 68 80 84 96 104 110]';\n", + "v=[0 20 20 38 80 80 100 100 125 125]';\n" + ] + } + ], + "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 +} diff --git a/lecture_19/Newtint.m b/lecture_19/Newtint.m new file mode 100644 index 0000000..e4c6c83 --- /dev/null +++ b/lecture_19/Newtint.m @@ -0,0 +1,34 @@ +function yint = Newtint(x,y,xx) +% Newtint: Newton interpolating polynomial +% yint = Newtint(x,y,xx): Uses an (n - 1)-order Newton +% interpolating polynomial based on n data points (x, y) +% to determine a value of the dependent variable (yint) +% at a given value of the independent variable, xx. +% input: +% x = independent variable +% y = dependent variable +% xx = value of independent variable at which +% interpolation is calculated +% output: +% yint = interpolated value of dependent variable + +% compute the finite divided differences in the form of a +% difference table +n = length(x); +if length(y)~=n, error('x and y must be same length'); end +b = zeros(n,n); +% assign dependent variables to the first column of b. +b(:,1) = y(:); % the (:) ensures that y is a column vector. +for j = 2:n + for i = 1:n-j+1 + b(i,j) = (b(i+1,j-1)-b(i,j-1))/(x(i+j-1)-x(i)); + end +end +%b +% use the finite divided differences to interpolate +xt = 1; +yint = b(1,1); +for j = 1:n-1 + xt = xt*(xx-x(j)); + yint = yint+b(1,j+1)*xt; +end diff --git a/lecture_19/lecture 19.ipynb b/lecture_19/lecture 19.ipynb new file mode 100644 index 0000000..e6f9f76 --- /dev/null +++ b/lecture_19/lecture 19.ipynb @@ -0,0 +1,1488 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Questions from last class\n", + "\n", + "![q1](q1.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![q2](q2.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![q3](q3.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![q4](q4.png)\n", + "\n", + "#### Other:\n", + "\n", + "Twice the amount of points needed\n", + "\n", + "depends on what order polynomial it is and how far the data needs to be extrapolated \n", + "\n", + "As man you as possible \n", + "\n", + "Never extrapolate unless linear interpolation.\n", + "\n", + "You shouldn't. 2 if the linear is a good fit for the region, and you absolutely have to.\n", + "\n", + "Wait can you do that?\n", + "\n", + "Don't use extrapolation\n", + "\n", + "do not extrapolate\n", + "\n", + "As many data points as you have\n", + "\n", + "the more the better so that the best polynomial can be made through the data points\n", + "\n", + "Twice the amount of points needed" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Questions from you\n", + "\n", + "- when will the project assignment be finalized? Also do you pronounce it \"jiff\" or \"gif\"?\n", + "\n", + "- If blue is red and red is blue, then what is purple? \n", + "\n", + "- How do we open the .ipynb lecture files? Or will the lectures continue to be also saved in pdf (last few have not).\n", + "\n", + "- When will we be put on teams for the final project?\n", + "\n", + "- What is the grading rubric for the project?\n", + "\n", + "- How to sync repository with files from laptop like hw without using Github desktop \n", + "\n", + "- Are there any upcoming deadlines for the project?\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Splines (Brief introduction before next section)\n", + "\n", + "Following interpolation discussion, instead of estimating 9 data points with an eighth-order polynomial, it makes more sense to fit sections of the curve to lower-order polynomials:\n", + "\n", + "0. zeroth-order (nearest neighbor)\n", + "\n", + "1. first-order (linear interpolation)\n", + "\n", + "2. third-order (cubic interpolation)\n", + "\n", + "Matlab and Octave have built-in functions for 1D and 2D interpolation:\n", + "\n", + "`interp1`\n", + "\n", + "`interp2`" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp1' is a function from the file /usr/share/octave/4.0.0/m/general/interp1.m\n", + "\n", + " -- Function File: YI = interp1 (X, Y, XI)\n", + " -- Function File: YI = interp1 (Y, XI)\n", + " -- Function File: YI = interp1 (..., METHOD)\n", + " -- Function File: YI = interp1 (..., EXTRAP)\n", + " -- Function File: YI = interp1 (..., \"left\")\n", + " -- Function File: YI = interp1 (..., \"right\")\n", + " -- Function File: PP = interp1 (..., \"pp\")\n", + "\n", + " One-dimensional interpolation.\n", + "\n", + " Interpolate input data to determine the value of YI at the points\n", + " XI. If not specified, X is taken to be the indices of Y ('1:length\n", + " (Y)'). If Y is a matrix or an N-dimensional array, the\n", + " interpolation is performed on each column of Y.\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"previous\"\n", + " Return the previous neighbor.\n", + "\n", + " \"next\"\n", + " Return the next neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " Adding '*' to the start of any method above forces 'interp1' to\n", + " assume that X is uniformly spaced, and only 'X(1)' and 'X(2)' are\n", + " referenced. This is usually faster, and is never slower. The\n", + " default method is \"linear\".\n", + "\n", + " If EXTRAP is the string \"extrap\", then extrapolate values beyond\n", + " the endpoints using the current METHOD. If EXTRAP is a number,\n", + " then replace values beyond the endpoints with that number. When\n", + " unspecified, EXTRAP defaults to 'NA'.\n", + "\n", + " If the string argument \"pp\" is specified, then XI should not be\n", + " supplied and 'interp1' returns a piecewise polynomial object. This\n", + " object can later be used with 'ppval' to evaluate the\n", + " interpolation. There is an equivalence, such that 'ppval (interp1\n", + " (X, Y, METHOD, \"pp\"), XI) == interp1 (X, Y, XI, METHOD, \"extrap\")'.\n", + "\n", + " Duplicate points in X specify a discontinuous interpolant. There\n", + " may be at most 2 consecutive points with the same value. If X is\n", + " increasing, the default discontinuous interpolant is\n", + " right-continuous. If X is decreasing, the default discontinuous\n", + " interpolant is left-continuous. The continuity condition of the\n", + " interpolant may be specified by using the options \"left\" or \"right\"\n", + " to select a left-continuous or right-continuous interpolant,\n", + " respectively. Discontinuous interpolation is only allowed for\n", + " \"nearest\" and \"linear\" methods; in all other cases, the X-values\n", + " must be unique.\n", + "\n", + " An example of the use of 'interp1' is\n", + "\n", + " xf = [0:0.05:10];\n", + " yf = sin (2*pi*xf/5);\n", + " xp = [0:10];\n", + " yp = sin (2*pi*xp/5);\n", + " lin = interp1 (xp, yp, xf);\n", + " near = interp1 (xp, yp, xf, \"nearest\");\n", + " pch = interp1 (xp, yp, xf, \"pchip\");\n", + " spl = interp1 (xp, yp, xf, \"spline\");\n", + " plot (xf,yf,\"r\", xf,near,\"g\", xf,lin,\"b\", xf,pch,\"c\", xf,spl,\"m\",\n", + " xp,yp,\"r*\");\n", + " legend (\"original\", \"nearest\", \"linear\", \"pchip\", \"spline\");\n", + "\n", + " See also: pchip, spline, interpft, interp2, interp3, interpn.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp1" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp2' is a function from the file /usr/share/octave/4.0.0/m/general/interp2.m\n", + "\n", + " -- Function File: ZI = interp2 (X, Y, Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, N)\n", + " -- Function File: ZI = interp2 (Z)\n", + " -- Function File: ZI = interp2 (..., METHOD)\n", + " -- Function File: ZI = interp2 (..., METHOD, EXTRAP)\n", + "\n", + " Two-dimensional interpolation.\n", + "\n", + " Interpolate reference data X, Y, Z to determine ZI at the\n", + " coordinates XI, YI. The reference data X, Y can be matrices, as\n", + " returned by 'meshgrid', in which case the sizes of X, Y, and Z must\n", + " be equal. If X, Y are vectors describing a grid then 'length (X)\n", + " == columns (Z)' and 'length (Y) == rows (Z)'. In either case the\n", + " input data must be strictly monotonic.\n", + "\n", + " If called without X, Y, and just a single reference data matrix Z,\n", + " the 2-D region 'X = 1:columns (Z), Y = 1:rows (Z)' is assumed.\n", + " This saves memory if the grid is regular and the distance between\n", + " points is not important.\n", + "\n", + " If called with a single reference data matrix Z and a refinement\n", + " value N, then perform interpolation over a grid where each original\n", + " interval has been recursively subdivided N times. This results in\n", + " '2^N-1' additional points for every interval in the original grid.\n", + " If N is omitted a value of 1 is used. As an example, the interval\n", + " [0,1] with 'N==2' results in a refined interval with points at [0,\n", + " 1/4, 1/2, 3/4, 1].\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " EXTRAP is a scalar number. It replaces values beyond the endpoints\n", + " with EXTRAP. Note that if EXTRAPVAL is used, METHOD must be\n", + " specified as well. If EXTRAP is omitted and the METHOD is\n", + " \"spline\", then the extrapolated values of the \"spline\" are used.\n", + " Otherwise the default EXTRAP value for any other METHOD is \"NA\".\n", + "\n", + " See also: interp1, interp3, interpn, meshgrid.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp2" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "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\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\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", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\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": [ + "x=linspace(-pi,pi,9);\n", + "xi=linspace(-pi,pi,100);\n", + "y=sin(x);\n", + "yi_lin=interp1(x,y,xi,'linear');\n", + "yi_spline=interp1(x,y,xi,'spline'); \n", + "yi_cubic=interp1(x,y,xi,'cubic');\n", + "plot(x,y,'o',xi,yi_lin,xi,yi_spline,xi,yi_cubic)\n", + "axis([-pi,pi,-1.5,1.5])\n", + "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example: Accelerate then hold velocity\n", + "\n", + "Here the time is given as vector t in seconds and the velocity is in mph. " + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "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\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t140\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tv (mph)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tt (s)\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", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\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": [ + "t=[0 20 40 56 68 80 84 96 104 110]';\n", + "v=[0 20 20 38 80 80 100 100 125 125]';\n", + "tt=linspace(0,110)';\n", + "v_lin=interp1(t,v,tt);\n", + "v_spl=interp1(t,v,tt,'spline');\n", + "v_cub=interp1(t,v,tt,'cubic');\n", + "\n", + "plot(t,v,'o',tt,v_lin,tt,v_spl,tt,v_cub)\n", + "xlabel('t (s)')\n", + "ylabel('v (mph)')\n", + "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "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\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tdv/dt (mph/s)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tt (s)\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\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": [ + "t=[0 20 40 56 68 80 84 96 104 110]';\n", + "v=[0 20 20 38 80 80 100 100 125 125]';\n", + "tt=linspace(0,110)';\n", + "v_lin=interp1(t,v,tt);\n", + "v_spl=interp1(t,v,tt,'spline');\n", + "v_cub=interp1(t,v,tt,'cubic');\n", + "\n", + "\n", + "plot(tt(2:end),diff(v_lin)./diff(tt),tt(2:end),diff(v_spl)./diff(tt),tt(2:end),diff(v_cub)./diff(tt))\n", + "xlabel('t (s)')\n", + "ylabel('dv/dt (mph/s)')\n", + "legend('linear','cubic spline','piecewise cubic','Location','NorthWest')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Choose spline wisely\n", + "\n", + "For example of sin(x), not very important\n", + "\n", + "For stop-and-hold examples, the $C^{2}$-continuity should not be preserved. You don't need smooth curves.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Integration\n", + "\n", + "A definite integral is defined by \n", + "\n", + "$I=\\int_{a}^{b}f(x)dx$\n", + "\n", + "To determine the mass of an object with varying density, you can perform a summation\n", + "\n", + "mass=$\\sum_{i=1}^{n}\\rho_{i}\\Delta V_{i}$\n", + "\n", + "or taking the limit as $\\Delta V \\rightarrow dV=dxdydz$\n", + "\n", + "mass=$\\int_{0}^{h}\\int_{0}^{w}\\int_{0}^{l}\\rho(x,y,z)dxdydz$\n", + "\n", + "## Newton-Cotes Formulas\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx=\\int_{a}^{b}f_{n}(x)dx$\n", + "\n", + "where $f_{n}$ is an n$^{th}$-order polynomial approximation of f(x)\n", + "\n", + "## First-Order: Trapezoidal Rule\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx\\approx \\int_{a}^{b}\\left(f(a)+\\frac{f(b)-f(a)}{b-a}(x-a)\\right)dx$\n", + "\n", + "$I\\approx(b-a)\\frac{f(a)+f(b)}{2}$" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "I_trap = 0.78540\n", + "I_act = 1.00000\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\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\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(0,pi)';\n", + "plot(x,sin(x),[0,pi/2],sin([0,pi/2]))\n", + "I_trap=mean(sin(([0,pi/2]))*(diff([0,pi/2])))\n", + "I_act = -(cos(pi/2)-cos(0))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Improve estimate with more points\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx=\\int_{a}^{a+\\Delta x}f(x)dx+\\int_{a+\\Delta x}^{a+2\\Delta x}f(x)dx+ \\cdots \\int_{b-\\Delta x}^{b}f(x)dx$\n", + "\n", + "$I\\approx\\Delta x\\frac{f(a)+f(a+\\Delta x)}{2}+\\Delta x\\frac{f(a+\\Delta x)+f(a+2\\Delta x)}{2}\n", + "+\\cdots \\Delta x\\frac{f(b-\\Delta x)+f(b)}{2}$\n", + "\n", + "$I\\approx \\frac{\\Delta x}{2}\\left(f(a)+2\\sum_{i=1}^{n-1}f(a+i\\Delta x) +f(b)\\right)$" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "For 5 steps\n", + "trapezoid approximation of integral is 0.79 \n", + " actual integral is 1.00\n" + ] + } + ], + "source": [ + "N=5;\n", + "I_trap=trap(@(x) sin(x),0,pi/2,N);\n", + "fprintf('For %i steps\\ntrapezoid approximation of integral is %1.2f \\n actual integral is %1.2f',N,I_trap,I_act)" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "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\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\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": [ + "\n", + "plot(x,sin(x),linspace(0,pi/2,N),sin(linspace(0,pi/2,N)),'o')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Increase accuracy = Increase polynomial order\n", + "\n", + "### Simpson's Rules\n", + "\n", + "When integrating f(x) and using a second order polynomial, this is known as **Simpson's 1/3 Rule**\n", + "\n", + "$I=\\frac{h}{3}(f(x_{0})+4f(x_{1})+f(x_{2}))$\n", + "\n", + "where a=$x_{0}$, b=$x_{2}$, and $x_{1}=\\frac{a+b}{2}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This can be used with n=3 or multiples of 2 intervals\n", + "\n", + "$I=\\int_{x_{0}}^{x_{2}}f(x)dx+\\int_{x_{2}}^{x_{4}}f(x)dx+\\cdots +\\int_{x_{n-2}}^{x_{n}}f(x)dx$\n", + "\n", + "$I=(b-a)\\frac{f(x_{0})+4\\sum_{i=1,3,5}^{n-1}f(x_{i})+2\\sum_{i=2,4,6}^{n-2}f(x_{i})+f(x_{n})}{3n}$" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 1.6235\n", + "Is_1_3 = 1.0023\n" + ] + } + ], + "source": [ + "f=@(x) 0.2+25*x-200*x.^2+675*x.^3-900*x.^4+400*x.^5;\n", + "simpson3(f,0,0.8,4)\n", + "Is_1_3=simpson3(@(x) sin(x),0,pi/2,2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## General Newton-Cotes formulae\n", + "\n", + "![Newton-Cotes Table](newton_cotes.png)" + ] + }, + { + "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 +} diff --git a/lecture_19/newton_cotes.png b/lecture_19/newton_cotes.png new file mode 100644 index 0000000..b734d11 Binary files /dev/null and b/lecture_19/newton_cotes.png differ diff --git a/lecture_19/q1.png b/lecture_19/q1.png new file mode 100644 index 0000000..44ec9da Binary files /dev/null and b/lecture_19/q1.png differ diff --git a/lecture_19/q2.png b/lecture_19/q2.png new file mode 100644 index 0000000..fb27a54 Binary files /dev/null and b/lecture_19/q2.png differ diff --git a/lecture_19/q3.png b/lecture_19/q3.png new file mode 100644 index 0000000..62c4402 Binary files /dev/null and b/lecture_19/q3.png differ diff --git a/lecture_19/q4.png b/lecture_19/q4.png new file mode 100644 index 0000000..5457483 Binary files /dev/null and b/lecture_19/q4.png differ diff --git a/lecture_19/simpson3.m b/lecture_19/simpson3.m new file mode 100644 index 0000000..2ae0c81 --- /dev/null +++ b/lecture_19/simpson3.m @@ -0,0 +1,23 @@ +function I = simpson3(func,a,b,n,varargin) +% simpson3: composite simpson's 1/3 rule +% I = simpson3(func,a,b,n,pl,p2,...): +% composite trapezoidal rule +% input: +% func = name of function to be integrated +% a, b = integration limits +% n = number of segments (default = 100) +% pl,p2,... = additional parameters used by func +% output: +% I = integral estimate +if nargin<3,error('at least 3 input arguments required'),end +if ~(b>a),error('upper bound must be greater than lower'),end +if nargin<4|isempty(n),n=100;end +x = a; h = (b - a)/n; + +xvals=linspace(a,b,n+1); +fvals=func(xvals,varargin{:}); +s=fvals(1); +s = s + 4*sum(fvals(2:2:end-1)); +s = s + 2*sum(fvals(3:2:end-2)); +s = s + fvals(end); +I = (b - a) * s/(3*n); diff --git a/lecture_19/trap.m b/lecture_19/trap.m new file mode 100644 index 0000000..85b8685 --- /dev/null +++ b/lecture_19/trap.m @@ -0,0 +1,22 @@ +function I = trap(func,a,b,n,varargin) +% trap: composite trapezoidal rule quadrature +% I = trap(func,a,b,n,pl,p2,...): +% composite trapezoidal rule +% input: +% func = name of function to be integrated +% a, b = integration limits +% n = number of segments (default = 100) +% pl,p2,... = additional parameters used by func +% output: +% I = integral estimate +if nargin<3,error('at least 3 input arguments required'),end +if ~(b>a),error('upper bound must be greater than lower'),end +if nargin<4|isempty(n),n=100;end + +x = a; h = (b - a)/n; +xvals=linspace(a,b,n); +fvals=func(xvals,varargin{:}); +s=func(a,varargin{:}); +s = s + 2*sum(fvals(2:n-1)); +s = s + func(b,varargin{:}); +I = (b - a) * s/(2*n);