Nonlinear Regression

Problem 15.11.

The following model is used to represent the effect of solar radiation on the photosynthesis rate of aquatic plants:

where P = the photosynthesis rate (mg m^-3d^-1), P_m = the maximum photosynthesis rate (mg m^-3d^-1), I = solar radiation (µE m^-2s^-1), and I_sat = optimal solar radiation (µE m^-2s^-1). Use nonlinear regression to evaluate P_m and I_sat based on the following data:

Solution:

First, a M-file function must be created to compute the sum of squares: 

function f = fSSR(a,Im,Pm)

Pp = a(1)*Im/a(2).*exp(-Im/a(2)+1);

% a(1) = Pm, a(2) = Isat

f = sum((Pm-Pp).^2);

In command mode, the data can be entered as:

>> I = [50 80 130 200 250 350 450 550 700];

>> P = [99 177 202 248 229 219 173 142 72];

Employ initial guesses of 200 for the coefficients. The minimization of the function is then implemented by:

>> a = fminsearch(@fSSR, [200, 200], [], I, P)

a =

  238.7124  221.8239

The best-fit model is therefore

The fit along with the data can be displayed graphically as:

>> Ip = linspace(min(I),max(I));

>> Pp = a(1)*Ip/a(2).*exp(-Ip/a(2)+1);

>> plot(I,P,'o',Ip,Pp)

>> xlabel('I'),ylabel('P')

>> title('Effect of solar radiation on photosynthesis rate')