1. Physics 114: Lecture 19 Least Squares Fit to 2D Data Dale E. Gary NJIT Physics Department

2. Apr 19, 2010 Nonlinear Least Squares Fitting  We saw that for a function y(x) the chi-square is which can be used to fit a curve to data points y i.  In just the same way, we can easily extend this idea to fitting a function z(x,y), that is, a surface, to a 2D data set of measurements z i (x,y).  You know that it takes three parameters to fit a Gaussian curve  To extend this to a 2D Gaussian is straightforward. It is:  Note that this has 5 parameters, but the most general way to describe a 2D Gaussian actually has 6 parameters, which you will do for homework.

3. Apr 19, 2010 Plotting a 2D Gaussian in MatLAB  Let’s introduce some useful MatLAB commands for creating a 2D Gaussian.  One can use a brute force method such as z = zeros(101,101); a = 2.5; bx = 2.3; by = -1.2; cx = 1.5; cy = 0.8; i = 0; for x = -5:0.1:5 i = i + 1; j = 0; for y = -5:0.1:5 j = j + 1; z(i,j) = a*exp(-((x-bx)/cx)^2 – ((y-by)/cy)^2); end imagesc(-5:0.1:5,-5:0.1:5,z)  Or, we can plot it as a surface to provide x and y coordinates surf(-5:0.1:5,-5:0.1:5,z)

4. Apr 19, 2010 Better Way of Creating 2D Functions  That way of using for loops is a very slow, so here is another way of doing it for “round” Gaussians: x = -5:0.1:5; y = -5:0.1:5; siz = size(x); siz = siz(2); a = 2.5; bx = 2.3; by = -1.2; c = 0.8; dist = zeros(siz,siz); for i=1:siz for j=1:siz dist(i,j) = x(i)^2 + y(j)^2; % Create “distance” array once end % use the next two statements multiple times for different Gaussians sdist = circshift(dist,round([bx/0.1,by/0.1])); % Shifted “distance” array z = a*exp(-sdist/c); imagesc(x,y,z);

5. Apr 19, 2010 Best Way of Creating 2D Functions  The best way, however, is to use the meshgrid() MatLAB function: [x, y] = meshgrid(-5:.1:5, -5:.1:5); a = 2.5; bx = 2.3; by = -1.2; cx = 1.5; cy = 0.8; z = a*exp( - (((x-bx)/cx).^2 + ((y-by)/cy).^2) ; surf(x,y,z);  This is completely general, and allows vector calculations which are very fast.  Here is another example, the sinc() function: z = a*sinc(sqrt(((x-bx)/cx).^2 + ((y-by)/cy).^2));  You can see that the meshgrid() function is very powerful as a shortcut for 2D function calculation.  To add noise, just add sig*randn(size(z)), where sig is the standard deviation of the noise.

6. Apr 19, 2010 Searching Parameter Space in 2D  Just as we did in the 1D case, searching parameter space is just a trial and error exercise. % Initial values around which to search a0 = max(max(z)); [ix iy] = find(z == a0); bx0 = x(ix,iy); by0 = y(ix,iy); cx0 = 1; cy0 = 1; astep = 0.05; bstep = 0.05; cstep = 0.05; chisqmin = 1e20; for a = a0-5*astep:astep:a0+5*astep for bx = bx0-5*bstep:bstep:bx0+5*bstep for by = by0-5*bstep:bstep:by0+5*bstep for cx = cx0-5*cstep:cstep:cx0+5*cstep for cy = cy0-5*cstep:cstep:cy0+5*cstep ztest = a*exp(-(((x-bx)/cx).^2 + ((y-by)/cy).^2)); chisq = sum(sum(((ztest-z)/sig).^2)); if (chisq < chisqmin); chisqmin = chisq; params = [a,bx,by,cx,cy]; end end params = 2.1465 2.3000 -1.2000 1.2500 0.8000

7. Apr 19, 2010 The Result Looks Good, but…  To calculate the result of the fit, just type ztest = params(1)*exp(-(((x-params(2))/params(4)).^2 + ((y-params(3))/params(5)).^2)); imagesc(x(1,:),y(:,1),ztest)  This gives a reasonable result, but compare the fit parameters with the (known) params: a = 2.1; bx = 2.3; by = -1.2; cx = 1.3; cy = 0.8; params = 2.15 2.3 -1.2 1.25 0.8  The value of cx is especially far off, because our search window was limited to 0.75-1.25, so it could not reach 1.3.  Look at the residuals: imagesc(x(1,:),y(:,1),ztest-z)  Can you see a pattern in the noise?  To bring out the pattern better, let’s introduce convolution.

8. 1D Convolution and Smoothing  Let’s create a noisy sine wave: u = -5:.1:5; w = sin(u*pi)+0.5*randn(size(u)); plot(u,w)  We can now smooth the data by convolving it with a vector [1,1,1], which does a 3-point running sum. wsm = conv(w,[1,1,1]); whos wsm Name Size Bytes Class Attributes wsm 1x103 824 double  Notice wsm is now of length 103. That means we cannot plot(u,wsm), but we can plot(u,wsm(2:102)). Now we see another problem.  Try this: plot(u,w) hold on; plot(u,wsm(2:102)/3,’r’) Apr 19, 2010

9. 2D Convolution  To do 2D smoothing, we will use a 2D kernel k = ones(3,3), and use the conv2() function. So to smooth the residuals of our fit, we can use zsm = conv2(ztest-z,k)/9.; imagesc(x,y,zsm(2:102,2:102))  Now we can see the effect of missing the value of cx by 0.05 due to our limited search range.  There are other uses for convolution, such as edge detection. For example, we can convolve with a kernel k = [1,1,1,1,1,1].  Or a kernel k = [1,1,1,1,1,1]’. Or even a kernel k = eye(6). Or k = rot90(eye(6)). Apr 19, 2010