Graphical methods to help get initial guesses for multivariate nonlinear regression
fit the model f(x1,x2; a,b) = a*x1 + x2^b to the data given below. This model has two independent variables, and two paramters.
Note it is not easy to visualize this data in 2D, but we can see the function in 3D.
x1 = [1 2 3 4 5 6]'; x2 = [.2 .4 .8 .9 1.1 2.1]'; X = [x1 x2]; % independent variables f = [ 3.3079 6.6358 10.3143 13.6492 17.2755 23.6271]'; plot3(x1,x2,f) xlabel('x1') ylabel('x2') zlabel('f(x1,x2)')
we want to do a nonlinear fit to find a and b that minimize the summed squared errors between the model predictions and the data. With only two variables, we can graph how the summed squared error varies with the parameters, which may help us get initial guesses . Let's assume the parameters lie in a range, here we choose 0 to 5. In other problems you would adjust this as needed.
arange = linspace(0,5); brange = linspace(0,5);
[A,B] = meshgrid(arange, brange);
we use the arrayfun to evaluate the error function for every pair of a,b from the A,B matrices
SSE = arrayfun(@errfunc,A,B);
we use a contour plot because it is easy to see where minima are. Here the colorbar shows us that dark blue is where the minimum values of the contours are. We can see the minimum is near a=3.2, and b = 2.1 by using the data exploration tools in the graph window.
contourf(A,B,SSE,50) colorbar xlabel('a') ylabel('b') hold on plot(3.2, 2.1, 'ro') text(3.4,2.2,'Minimum near here','color','r')
Now the nonlinear fit with our guesses
guesses = [3.18,2.02]; [pars residuals J] = nlinfit(X,f,@model, guesses) parci = nlparci(pars,residuals,'jacobian',J,'alpha',0.05) % show where the best fit is on the contour plot. plot(pars(1),pars(2),'r*') text(pars(1)+0.1,pars(2),'Actual minimum','color','r')
pars = 3.2169 1.9728 residuals = 0.0492 0.0379 0.0196 -0.0309 -0.0161 0.0034 J = 1.0000 -0.0673 2.0000 -0.1503 3.0000 -0.1437 4.0000 -0.0856 5.0000 0.1150 6.0000 3.2067 parci = 3.2034 3.2305 1.9326 2.0130
figure hold all plot3(x1,x2,f,'ko ') plot3(x1,x2,model(pars,[x1 x2]),'r-') xlabel('x1') ylabel('x2') zlabel('f(x1,x2)') legend('data','fit') view(-12,20) % adjust viewing angle to see the curve better.
It can be difficult to figure out initial guesses for nonlinear fitting problems. For one and two dimensional systems, graphical techniques may be useful to visualize how the summed squared error between the model and data depends on the parameters.
function f = model(pars,X) % Nested function for the model x1 = X(:,1); x2 = X(:,2); a = pars(1); b = pars(2); f = a*x1 + x2.^b; end function sse = errfunc(a,b) % Nested function for the summed squared error fit = model([a b],X); sse = sum((fit - f).^2); end
end % categories: data analysis, plotting