MATLAB - Exponential Curve Fitting without Toolbox - matlab

Question

I have data points (x, y) that I need to fit an exponential function to,
y = A + B * exp(C * x),
but I can neither use the Curve Fitting Toolbox nor the Optimization Toolbox.
User rayryeng was good enough to help me with working code:
x = [0 0.0036 0.0071 0.0107 0.0143 0.0178 0.0214 0.0250 0.0285 0.0321 0.0357 0.0392 0.0428 0.0464 0.0464];
y = [1.3985 1.3310 1.2741 1.2175 1.1694 1.1213 1.0804 1.0395 1.0043 0.9691 0.9385 0.9080 0.8809 0.7856 0.7856];
M = [ones(numel(x),1), x(:)]; %// Ensure x is a column vector
lny = log(y(:)); %// Ensure y is a column vector and take ln
X = M\lny; %// Solve for parameters
A = exp(X(1)); %// Solve for A
b = X(2); %// Get b
xval = linspace(min(x), max(x));
yval = A*exp(b*xval);
plot(x,y,'r.',xval,yval,'b');
However, this code only fits the equation without offset
y = A * exp(B * x).
How can I extend this code to fit the three-parameter equation?
In another attempt, I managed to fit the function using fminsearch:
function [xval, yval] = curve_fitting_exponential_1_optimized(x,y,xval)
start_point = rand(1, 3);
model = #expfun;
est = fminsearch(model, start_point);
function [sse, FittedCurve] = expfun(params)
A = params(1);
B = params(2);
C = params(3);
FittedCurve = A + B .* exp(-C * x);
ErrorVector = FittedCurve - y;
sse = sum(ErrorVector .^ 2);
end
yval = est(1)+est(2) * exp(-est(3) * xval);
end
The problem here is that the result depends on the starting point which is randomly chosen, so I don't get a stable solution. But since I need the function for automatization, I need something stable. How can I get a stable solution?

Answer 1

How to adapt rayryeng's code for three parameters?
rayryeng used the strategy to linearize a nonlinear equation so that standard regression methods can be applied. See also Jubobs' answer to a similar question.
This strategy does no longer work if there is a non-zero offset A. We can fix the situation by getting a rough estimate of the offset. As rubenvb mentioned in the comments, we could estimate A by min(y), but then the logarithm gets applied to a zero. Instead, we could leave a bit of space between our guess of A and the minimum of the data, say half its range. Then we subtract A from the data and use rayreng's method:
x = x(:); % bring the data into the standard, more
y = y(:); % convenient format of column vectors
Aguess = min(y) - (max(y) - min(y) / 2;
guess = [ones(size(x)), -x] \ log(y - Aguess);
Bguess = exp(guess(1));
Cguess = guess(2);
For the given data, this results in
Aguess = 0.4792
Bguess = 0.9440
Cguess = 21.7609
Other than for the two-parameter situation, we cannot expect this to be a good fit. Its SSE is 0.007331.
How to get a stable solution?
This guess is however useful as a starting point for the nonlinear optimization:
start_point = [Aguess, Bguess, Cguess];
est = fminsearch(#expfun, start_point);
Aest = est(1);
Best = est(2);
Cest = est(3);
Now the optimization arrives at a stable estimate, because the computation is deterministic:
Aest = -0.1266
Best = 1.5106
Cest = 10.2314
The SSE of this estimate is 0.004041.
This is what the data (blue dots) and fitted curves (green: guess, red: optimized) look like:

Answer 2

Here is the whole function in all its glory - special thanks to A. Donda!
function [xval, yval] = curve_fitting_exponential_1_optimized(x,y,xval)
x = x(:); % bring the data into the standard, more convenient format of column vectors
y = y(:);
Aguess = min(y) - (max(y)-min(y)) / 2;
guess = [ones(size(x)), -x] \ log(y - Aguess);
Bguess = exp(guess(1));
Cguess = guess(2);
start_point = [Aguess, Bguess, Cguess];
est = fminsearch(#expfun, start_point);
function [sse, FittedCurve] = expfun(params)
A = params(1);
B = params(2);
C = params(3);
FittedCurve = A + B .* exp(-C * x);
ErrorVector = FittedCurve - y;
sse = sum(ErrorVector .^ 2);
end
yval = est(1)+est(2) * exp(-est(3) * xval);
end