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"%plot --format svg"
]
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"![question 1](q1.png)\n",
"\n",
"![question 2](q2.png)\n",
"\n",
"### Project Ideas so far\n",
"\n",
"- Nothing yet...probably something heat transfer related\n",
"\n",
"- Modeling Propulsion or Propagation of Sound Waves\n",
"\n",
"- Low Thrust Orbital Transfer\n",
"\n",
"- Tracking motion of a satellite entering orbit until impact\n",
"\n",
"- What ever you think is best.\n",
"\n",
"- You had heat transfer project as an option; that sounded cool\n",
"\n",
"- Heat transfer through a pipe\n",
"\n",
"- I would prefer to do something with beam/plate mechanics or vibrations than a heat transfer or thermo problem\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Questions from you:\n",
"\n",
"- Is attempting to divide by zero an acceptable project idea?\n",
"\n",
"- Would it be alright if we worked in a group of 4?\n",
"\n",
"- What are acceptable project topics?\n",
"\n",
"- How do the exams look? \n",
"\n",
"- Is there no pdf for the lecture today?\n",
"\n",
"- Thank you for making the formatted lectures available!\n",
"\n",
"- did you do anything cool over spring break?\n",
"\n",
"- Could we have a group of 4? We don't want to have to choose which one of us is on their own.\n",
"\n",
"- In HW 5 should there be 4 vectors as an input?\n",
"\n",
"- Would it be possible for me to join a group of 3? I seem to be the odd man out in two 3 member groups that my friends are in."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Statistics and Curve-Fitting\n",
"## Linear Regression\n",
"\n",
"Often, we have a model with empirical parameters. (e.g. Young's modulus, Poisson's ratio, drag coefficient, coefficient of restitution, spring constant)\n",
"\n",
"Experimental measurements are prone to a number of stochastic (random) factors:\n",
"\n",
"- Environmental noise\n",
" \n",
"- Measurement Uncertainty\n",
" \n",
"- Factors not accounted in model (e.g. 2D effects of 1D approximation)\n",
"\n",
"These can lead to **noise** (lack of precision) and **bias** (lack of accuracy)\n",
"\n",
"Consider a piece of glass being stretched. \n",
"\n",
"![movie of stretching glass in microtensile machine](sgs_strain.gif)\n",
"\n",
"It is clear that a straight line is a \"good\" fit, but how good and what line?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Statistics\n",
"\n",
"How do we describe *precision* and *accuracy*?\n",
"\n",
"- mean\n",
"\n",
"- standard deviation\n",
"\n",
"Take our class dart problem\n"
]
},
{
"cell_type": "code",
"execution_count": 173,
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"collapsed": false
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"source": [
"darts=dlmread('compiled_data.csv',',');\n",
"x_darts=darts(:,1).*cosd(darts(:,2));\n",
"y_darts=darts(:,1).*sind(darts(:,2));\n",
"\n",
"colormap(colorcube(length(darts(:,3))))\n",
"\n",
"scatter(x_darts, y_darts, [], darts(:,3))\n",
"xlabel('x position (cm)')\n",
"ylabel('y position (cm)')"
]
},
{
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"name": "stdout",
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"text": [
"mu_x = 0.90447\r\n"
]
}
],
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"mu_x=mean(x_darts)"
]
},
{
"cell_type": "code",
"execution_count": 26,
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{
"name": "stdout",
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"text": [
"s_x = 4.2747\r\n"
]
}
],
"source": [
"s_x=std(x_darts)"
]
},
{
"cell_type": "code",
"execution_count": 175,
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{
"name": "stdout",
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"text": [
"mu_y = 0.88450\r\n"
]
}
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"source": [
"mu_y=mean(y_darts)"
]
},
{
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"execution_count": 177,
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"name": "stdout",
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"s_y = 4.6834\r\n"
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"s_y=std(y_darts)"
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"hist(x_darts,30,100)"
]
},
{
"cell_type": "code",
"execution_count": 182,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"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')"
]
},
{
"cell_type": "code",
"execution_count": 183,
"metadata": {
"collapsed": false
},
"outputs": [
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y_vals=linspace(-15,20,30);\n",
"hist(y_darts,y_vals,1);\n",
"[histFreq, histXout] = hist(y_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",
"%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",
"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",
"\n",
"xlabel('x position (cm)')\n",
"ylabel('relative counts')"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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"\n",
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"\t \n",
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"\n",
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"text/plain": [
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x_exp=empirical_cdf(x_vals,x_darts);\n",
"plot(x_vals,x_exp)\n",
"hold on;\n",
"plot(x_vals,normcdf(x_vals,mu_x,s_x),'k-')\n",
"legend('experimental CDF','Normal CDF','Location','SouthEast')"
]
},
{
"cell_type": "code",
"execution_count": 185,
"metadata": {
"collapsed": false
},
"outputs": [
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"text/plain": [
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},
"metadata": {},
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],
"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": [
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"gauss2d = @(x,y) exp(-((x-0).^2/2/1^2+(y-0).^2/2/5^2));\n",
"\n",
"x=linspace(-20,20,71);\n",
"y=linspace(-20,20,31);\n",
"scatter3(x_darts, y_darts,ones(length(x_darts),1))\n",
"xlabel('x position (cm)')\n",
"ylabel('y position (cm)')\n",
"hold on\n",
"[X,Y]=meshgrid(x,y);\n",
"view([1,1,1])\n",
"\n",
"surf(X,Y,gauss2d(X,Y))"
]
},
{
"cell_type": "code",
"execution_count": 190,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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"source": [
"hist(darts(:,2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Monte Carlo Simulations\n",
"\n",
"Monte Carlo models use random numbers to either understand statistics or generate a solution. \n",
"\n",
"### Example 1:\n",
"#### Calculate $\\pi$ with random numbers. \n",
"\n",
"Assuming we can actually generate random numbers (a topic of philosophical and heated debates) we can populate a unit square with random points and determine the ratio of points inside and outside of a circle.\n",
"\n",
"![Unit circle and unit square](MonteCarloPi.gif)\n",
"\n",
"![1/4 Unit circle and 1/4 unit square](MonteCarloPi_rand.gif)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The ratio of the area of the circle to the square is:\n",
"\n",
"$\\frac{\\pi r^{2}}{4r^{2}}=\\frac{\\pi}{4}$\n",
"\n",
"So if we know the fraction of random points that are within the unit circle, then we can calculate $\\pi$\n",
"\n",
"(number of points in circle)/(total number of points)=$\\pi/4$"
]
},
{
"cell_type": "code",
"execution_count": 191,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"function our_pi=montecarlopi(N)\n",
" % Create random x-y-coordinates\n",
"\n",
" x=rand(N,1);\n",
" y=rand(N,1);\n",
" R=sqrt(x.^2+y.^2); % compute radius\n",
" num_in_circle=sum(R<1);\n",
" total_num_pts =length(R);\n",
" our_pi = 4*num_in_circle/total_num_pts;\n",
"end\n"
]
},
{
"cell_type": "code",
"execution_count": 194,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"mean value for pi = 3.138880\n",
"standard deviation is 0.009956\n"
]
}
],
"source": [
"test_pi=zeros(10,1);\n",
"for i=1:10\n",
" test_pi(i)=montecarlopi(10000);\n",
"end\n",
"fprintf('mean value for pi = %f\\n',mean(test_pi))\n",
"fprintf('standard deviation is %f',std(test_pi))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 2\n",
"#### Determine uncertainty in failure stress based on geometry\n",
"\n",
"In this example, we know that a steel bar will break under 940 MPa tensile stress. The bar is 1 mm by 2 mm with a tolerance of 1 %. What is the range of tensile loads that can be safely applied to the beam?\n",
"\n",
"$\\sigma_{UTS}=\\frac{F_{fail}}{wh}$\n",
"\n",
"$F_{fail}=\\sigma_{UTS}wh$"
]
},
{
"cell_type": "code",
"execution_count": 196,
"metadata": {
"collapsed": false
},
"outputs": [
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"source": [
"N=1000;\n",
"r=rand(N,1);\n",
"wmean=1; % in mm\n",
"wmin=wmean-wmean*0.01;\n",
"wmax=wmean+wmean*0.01;\n",
"hmean=2; % in mm\n",
"hmin=hmean-hmean*0.01;\n",
"hmax=hmean+hmean*0.01;\n",
"\n",
"wrand=wmin+(wmax-wmin)*r;\n",
"hrand=hmin+(hmax-hmin)*r;\n",
"\n",
"uts=940; % in N/mm^2=MPa\n",
"\n",
"Ffail=uts.*wrand.*hrand*1e-3; % force in kN\n",
"hist(Ffail,20,1)\n",
"xlabel('failure load (kN)')\n",
"ylabel('relative counts')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Normally, the tolerance is not a maximum/minimum specification, but instead a normal distribution that describes the standard deviation, or the 68 % confidence interval.\n",
"\n",
"So instead, we should generate normally distributed dimensions."
]
},
{
"cell_type": "code",
"execution_count": 197,
"metadata": {
"collapsed": false
},
"outputs": [
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"\n",
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" \n",
" \n",
" "
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"N=1000;\n",
"wmean=1; % in mm\n",
"wstd=wmean*0.01; % standard deviation in mm\n",
"hmean=2; % in mm\n",
"hstd=hmean*0.01; % standard deviation in mm\n",
"\n",
"\n",
"wrand=normrnd(wmean,wstd,[N,1]);\n",
"hrand=normrnd(hmean,hstd,[N,1]);\n",
"uts=940; % in N/mm^2=MPa\n",
"\n",
"Ffail=uts.*wrand.*hrand*1e-3; % force in kN\n",
"hist(Ffail,20,1)\n",
"xlabel('failure load (kN)')\n",
"ylabel('relative counts')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Linear Least Squares Regression\n",
"\n",
"When approximating a set of data as a polynomial, there will always be error introduced (except in 2 cases). \n",
"\n",
"For a straight line, the actual data points, $y_{i}$ as a function of the independent variable, $x_{i}$, is:\n",
"\n",
"$y_{i}=a_{0}+a_{1}x_{i}+e_{i}$\n",
"\n",
"where $a_{0}$ and $a_{1}$ are the intercept and slope of the line and $e_{i}$ is the error between the approximate function and the recorded data point. \n",
"\n",
"We make the following assumptions in this analysis:\n",
"\n",
"1. Each x has a fixed value; it is not random and is known without error.\n",
"\n",
"2. The y values are independent random variables and all have the same variance.\n",
"\n",
"3. The y values for a given x must be normally distributed.\n",
"\n",
"The total error is \n",
"\n",
"$\\sum_{i=1}^{n}e_{i}=\\sum_{i=1}^{n}(y_{i}-a_{0}-a_{1}x_{i})$\n",
"\n",
"we don't care about the sign though. One approach has been demonstrated to provide a unique solution is minimizing the sum of squares error or\n",
"\n",
"$S_{r}=\\sum_{i=1}^{n}e_{i}^{2}=\\sum_{i=1}^{n}(y_{i}-a_{0}-a_{1}x_{i})^{2}$\n",
"\n",
"Where, $S_{r}$ is the sum of squares error (SSE). \n",
"\n",
"$\\frac{\\partial S_{r}}{\\partial a_{0}}=-2\\sum(y_{i}-a_{0}-a_{1}x_{i})$\n",
"\n",
"$\\frac{\\partial S_{r}}{\\partial a_{1}}=-2\\sum(y_{i}-a_{0}-a_{1}x_{i})x_{i}$\n",
"\n",
"The minimum $S_{r}$ occurrs when the partial derivatives are 0. \n",
"\n",
"$\\sum y_{i}= \\sum a_{0}+\\sum a_{1}x_{i}$\n",
"\n",
"$\\sum x_{i}y_{i}= \\sum a_{0}x_{i}+\\sum a_{1}x_{i}^{2}$\n",
"\n",
"$\\left[\\begin{array}{c}\n",
"\\sum y_{i}\\\\\n",
"\\sum x_{i}y_{i}\\end{array}\\right]=\n",
"\\left[\\begin{array}{cc}\n",
"n & \\sum x_{i}\\\\\n",
"\\sum x_{i} & \\sum x_{i}^{2}\\end{array}\\right]\n",
"\\left[\\begin{array}{c}\n",
"a_{0}\\\\\n",
"a_{1}\\end{array}\\right]$\n",
"\n",
"\n",
"$b=Ax$\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example \n",
"\n",
"Find drag coefficient with best-fit line to experimental data\n",
"\n",
"|i | v (m/s) | F (N) |\n",
"|---|---|---|\n",
"|1 | 10 | 25 |\n",
"|2 | 20 | 70 |\n",
"|3 | 30 | 380|\n",
"|4 | 40 | 550|\n",
"|5 | 50 | 610|\n",
"|6 | 60 | 1220|\n",
"|7 | 70 | 830 |\n",
"|8 |80 | 1450|"
]
},
{
"cell_type": "code",
"execution_count": 200,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a =\n",
"\n",
" -234.286\n",
" 19.470\n",
"\n"
]
},
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"Gnuplot \n",
"Produced by GNUPLOT 5.0 patchlevel 3 \n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
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"\t\n",
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"\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": [
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"Gnuplot \n",
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"\n",
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"\n",
"\t\n",
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"\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",
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"\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",
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"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
" \n",
"\t \n",
"\tCase 2 \n",
"\n",
"\t\n",
"\t\tCase 2 \n",
"\t \n",
" \n",
"\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
"\t\n",
" \n",
"\t \n",
"\n",
" \n",
"\n",
" \n",
"\n",
" \n",
"\n",
" \n",
" \n",
" "
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"gen_examples\n",
"plot(x1(1:50:end),y1(1:50:end),'s',x2,y2,'o')\n",
"legend('Case 1','Case 2','Location','NorthWest')"
]
},
{
"cell_type": "code",
"execution_count": 157,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a1 =\n",
"\n",
" 0.0497269\n",
" 0.0016818\n",
"\n",
"a2 =\n",
"\n",
" 0\n",
" 1\n",
"\n",
"Sr1 = 0.82607\n",
"St1 = 0.82631\n",
"coefficient of determination in Case 1 is 0.000286\n",
"Sr2 = 0\n",
"St2 = 1.0185\n",
"coefficient of determination in Case 2 is 1.000000\n"
]
}
],
"source": [
"b1=[sum(y1);sum(x1.*y1)];\n",
"A1=[length(x1),sum(x1);\n",
" sum(x1), sum(x1.^2)];\n",
" \n",
"a1=A1\\b1\n",
"\n",
"b2=[sum(y2);sum(x2.*y2)];\n",
"A2=[length(x2),sum(x2);\n",
" sum(x2), sum(x2.^2)];\n",
" \n",
"a2=A2\\b2\n",
"\n",
"Sr1=sum((y1-a1(1)-a1(2)*x1).^2)\n",
"St1=sum((y1-mean(y1)).^2)\n",
"fprintf('coefficient of determination in Case 1 is %f\\n',1-Sr1/St1)\n",
"\n",
"Sr2=sum((y2-a2(1)-a2(2)*x2).^2)\n",
"St2=sum((y2-mean(y2)).^2)\n",
"\n",
"fprintf('coefficient of determination in Case 2 is %f\\n',1-Sr2/St2)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Octave",
"language": "octave",
"name": "octave"
},
"language_info": {
"file_extension": ".m",
"help_links": [
{
"text": "MetaKernel Magics",
"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
}
],
"mimetype": "text/x-octave",
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