BIOE 198MI Biomedical Data Analysis. Spring Semester Lab 6: Least Squares Curve Fitting Methods

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1 BIOE 98MI Biomedical Data Analysis. Spring Semester 09. Lab 6: Least Squares Curve Fitting Methods A. Modeling Data You have a pair of night vision goggles for seeing IR fluorescent photons emerging from the skin of patients whose lymphatic vessels were injected with a fluorescent dye (see Fig ). These goggles are necessary because the detector can see light levels below the sensitivity of the human eye. The manufacturer says the output voltage signal is directly proportional to the input light levels with slope one. Great! However, there is additive noise. Before noise suppression, the output signal from the goggles y relative to deterministic input x is modeled simply as: y x + n, where for each x the noise sample n is a time-independent sample drawn from the normal pdf, (0,0). (In reality, x is a Poisson random variable, which we ignore in this example.) Assume the units of all three terms is micro candela per square meter (cd/m ). Fig.. Images from the lab of Dr. Eva Sevick-Muraca, University of Texas Health Science Center at Houston, TX. Publication: Journal of Vascular Surgery: Venous and Lymphatic Disorders, Volume 4, Issue, January 06, Pages 9-7. Example. You obtain one of these devices and decide to experiment in the lab before seeing patients. The experiment is to input a known quantity of light x into the device and measure its response y. The experiment begins with input flux x() 0 μcd/m and is indexed in steps of 0 cd/m, i.e., 0, 0,, 00 cd/m while output y (t) is recorded. The results are: Experimental Data: x y

2 Plotting the x,y pairs of recorded measurements, you find the red circles in Fig. Compared to the noise free model provided by the manufacturer, y x (solid black line), the results are not very impressive because of the noise. The device assumes a linear input-output relationship, so these data can be fit to a linear regression line of the form z P x P using the method of least squares: MSSD argmin d i (x) argmin (y i P x i P ). P,P P,P i This equation tells us to find parameters P and P that yield the minimum sum squared differences (MSSD) between each of the 0 measurements and a straight line. The differences (see Fig below) are defined as i Fig.. (Above) Graphical representation of data in Example. Least-squares regression line is z while modeled output are labeled y and measured data y. (Below) The distances d between y and z at each x are displayed along with z equation, GOF and R values. d i y i z i y i (P x i P ), for i. For statistical optimization reasons, we use d i instead of d i. otice that in this example, the least-squares regression line z is not the same as the model function y. To apply the method of least squares, we must assume y is linearly related to x or a transformation of x deviations from the regression line (residuals d i ) are normally distributed random variables (d i, σ i ) all variances i are equal. My code for generating these results is (note the new plotting function at the end!) close all x0:0:00;rng('default');yx+0*randn(size(x)); yx+n where n~(0,0) Ppolyfit(x,y,); Apply the method of least squares: P(),P() zpolyval(p,x); Compute the regression line zp*x + P myplot(x,x,':b');hold on;myplot(x,y,'or'),myplot(x,z,'-k') xlabel('x (input)');ylabel('output'); legend('x','y','z','location','southeast') qgoodnessoffit(y,z,'mse'); str['gof between z and y is ' numstr(q)]; disp(str); qgoodnessoffit(y,x,'mse'); str['gof between x and y is ' numstr(q)]; disp(str); The following correlation coefficient relates x to y. Acorrcoef(x,y);str['R^ for x vs y is ' numstr(a(,))]; disp(str); str4['regression line: y ' numstr(p()) 'x + ' numstr(p())]; disp(str4); function myplot(x,y,z) I adapted myplot to modify line/point types plot(x,y,numstr(z),'linewidth',) ax gca; current axes ax.fontsize 0; end

3 ote how I adapted myplot from Lab b, I called it myplot, to add the ability to control line and point properties. The Pearson correlation coefficient R is found from either of the two off-diagonal terms obtained from the correlation matrix generated between x and y by the Matlab function corrcoef(x,y); R square of sample covariance product of two sample variances s xy s x s ( i (x i x ) (y i y )) y i(x i x ) (y i y ) [ (x i y i x i j y j y i x j + j x i j j j y j )] [ (x i x i j x j + ( j x i j) )] [ (y i y i j y j + ( y j [ i x i y i i x i [ i x i ( i x i)( j x j ) + ( x j i i j ) )] j y j i y i j x j + ( j x j y j j ) ] [ i y i i j )] ( y i j ) i )( j y j ) + ( y j i ] [ i x i y i x i i i y i i y i i x i + x i i y i i ) [ i x i ( i x i)( i x i ) + ( x i [ i x i y i ( i x i )( i y i )] [ x i ( j x j ) i ] [ y i ( j y j ) i ] Whew! To follow the math, note that y similarly for x. ] [ i y i ( y i i ) i )( i y i ) + ( y i [ i x iy i x y ] [ x i x i ] i ] [ y i y i y i j y j. Also, ( i j ) y y j i y and ] ] Pearson s correlation coefficient R describes the strength of the linear relationship between two functions or arrays of data (Fig 3). It estimates the fraction of sample variance s y explained by changes in x. While 0 R, we also have R. For example, when R ~, (see S4 in Fig 3 and in Example above where R ~ 0.96), we see that variations in x mostly explain variation in y ; increasing one variable by unit increases the other by approximately one unit and we say the two variables are strongly positively correlated. However, when R 0 (see S and S3 in Fig 3), there is no relationship between the two variables. The downside of correlation is that it doesn t tell us which variable is dependent. We will come back to R and R in a minute. First, another test statistic of interest is the Goodness of Fit measure, GOF. Using the Matlab function goodnessoffit, I separately tested how well x and z in Fig each represent y by selecting mean-square error as the GOF metric: MSE i (y i z i ). Since Fig. 3. -D scatter plots for two variables with different means, variances, and correlation coefficients. S: R -0.40, sy < sx. S: R 0, sy > sx. S3: R 0, sy sx. S4: R 0.98, sy sx. The code is given at the end of this lab write up. 3

4 regression line z is the MSSD result, it make sense that z will fit data y even better than the true underlying model y. Summary: Correlation R is useful for quantifying the existence of relationships between variables, but it cannot establish causal relationships. That is, there is no way to tell if one variable is causing the response observed by the other; i.e., is y (x) true or is x(y ) true? We prefer R over R when correlation is used to explain variance in the data. However, we use R to describe scatter plots (Fig 3) because it tells us if the correlation between variables is negative or positive (compare S and S4 in Fig 3). While R describes correlation between x and y, GOF describe how well y is described by linear regression line z. Exercise. Look at the line of code below. It generates random data in three columns. The first two columns are uncorrelated random data, R ~ 0, while the last two are perfectly correlated random data, R. If the correlation coefficient between columns i and j is R ij, the resulting correlation matrix from R R R 3 Matlab has elements given by ( R R 3 R R 3 R 3 ). What fraction of each random variable can be R 33 explained by the others? Why is the matrix symmetric about the diagonal? clear all;yzeros(000,3);yrandn(000,);y(:,3)3*y(:,);ccorrcoef(y) C Assignment : Obtain the file lab6.mat that contains three variables. t is a time axis while y and y are data arrays such that y (t) and y (t). I suggest you approach this assignment in stages. (a) Plot y and y versus t. Adapt the code from Example above to propose linear models for both y and y. ote that y is one data recording without signal averaging, while y a data recording resulting after averaging 0 waveforms. The linear models will not fit well, so now try higher-order polynomials. (b) Change the code to enter a range of m in Ppolyfit(t,y,m) from the keyboard, where 0 m 8 is the order of polynomial generated to fit to the data. P is a vector of values where P()*t.^m + P()*t.^(m-) P(m)*t + P(m+). Use goodnessoffit to compute the MSE values between z and y and between z and y as a function of m. (c) Create code that allows the operator to view the MSE values versus m from (b) and then input a value of m from the keyboard to see plots of both y and y. Explore plots of model fits for various values of m and tell me what you consider achieves the best fit. At m0, for example, the model is a constant, and hence MSE should be large. At m8, the model is clearly fitting noise. Hence, you need to tell me how you would best select m based on your observations of the analysis of y and y. ote that you do not know the true model function, so your answer needs to be related from your analysis. This discussion needs to appear in the DISCUSSIO section. Grading: pts for introduction of the problem. pts for methods adopted to decide on optimal model. 3 pts for visualization of results that will lead to a convincing decision given in Discussion & Conclusions ( pts). pt for overall appearance of the report. 4

5 Code that generated Figure 3 scatter plots. ot an assignment! The following script simulates four flow cytometry data sets that are each bivariate normal. Parameter vectors vary between data sets. clear all; close all; rho-0.4;s[00 40];m[500 80];50; First data set zrandn(); ^ is the number of data points simulated Xs()*z+m(); X and Y convert from standard normal pdf Ys()*(rho*z+sqrt(-rho^)*randn())+m(); plot(x,y,'k.');axis([ ]);axis square; plot on fixed axis text(730,70,'s');hold on label the first group S rho0;s[40 00];m[ ];0; Second data set zrandn(); Xs()*z+m(); Ys()*(rho*z+sqrt(-rho^)*randn())+m(); plot(x,y,'ro');text(60,880,'s') rho0;s3[30 30];m3[00 700];330; Third data set zrandn(3); X3s3()*z+m3(); Y3s3()*(rho*z+sqrt(-rho^)*randn(3))+m3(); plot(x3,y3,'bx') text(80,850,'s3') rho0.98;s4[40 40];m4[ ];450; Fourth data set zrandn(4); X4s4()*z+m4(); Y4s4()*(rho*z+sqrt(-rho^)*randn(4))+m4(); plot(x4,y4,'bx');hold off title('bivariant ormal Flow Cytometry Data'); xlabel('side Scatter Intensity');ylabel('Forward Scatter Intensity') text(850,580,'s4');hold off save('f','x','y','x','y','x3','y3') use these for the assignment load('f.mat'); then use whos command to see what is loaded [rho]corr(x,y);rtrace(rho)/; Here we estimate Pearson s str ['rho_ ' numstr(r)];disp(str) correlation and average for [rho]corr(x,y);rtrace(rho)/; all points along diagonal str ['rho_ ' numstr(r)];disp(str) [rho3]corr(x3,y3);r3trace(rho3)/3; str ['rho_3 ' numstr(r3)];disp(str) [rho4]corr(x4,y4);r4trace(rho4)/4; str ['rho_4 ' numstr(r4)];disp(str) 5