Hello, does anyone know how to find the horizontal asymptote of an exponential decay function?
MATLAB: Horizontal asymptote of exponential decay
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If X is a vector of numel(X) = 6, then
Z=A*exp(B*X);if A,B are constants; otherwise it depends on what they are and what is the end result wanted...if they're both also vectors of same shape as X, then
Z=A.*exp(B.*X);for element-wise multiplication. From the Q? it doesn't sound like you want/intend matrix multiplication, but that's doable, too, if that were to the desired result and have commensurately-sized Arrays/Vectors for conformant product dimensions.
You want to fit an exponential envelope function to the curve, essentially a least upper bound function. So the exponential mode would be something of the form...
Y_e = a + b*exp(-c*x)If you know that the lower asymptote is exactly 1, then your model would be
Y_e = 1 + b*exp(-x)We might estimate those coefficients by a simple log transformation, or we could use a nonlinear estimation. I'll show the log transformation. First, I'll create a damped sine wave curve as an example.
x = 0:100;y = 1 + sin(x - pi/3).*exp(-0.2*x);First, use only those points that fall above 1. So assuming vectors of points x and y...
ind = (y > 1);x1 = x(ind);% and transform y by a log transformation
y1 = log(y(ind) - 1);A = [ones(numel(x1),1),x1(:)];% estimate the model coeffs = lsqlin(A,y1(:),-A,-y1(:),[],[],[],[],[],optimset('algorithm','active-set','display','off'));
coeffs = -0.00091842 -0.19999Don't forget that when we logged the model, we transformed the coefficient b. We need to exponentiate
b = exp(coeffs(1))b = 0.99908c = coeffs(2);% now plot the data and the curve
plot(x,y,'ro',x,1 + b*exp(c*x),'b-)
I used lsqlin from the optimization toolbox to do the fit. The fundamental ideas in this solution were:
- Transform the problem to linearity.- Solve the linear least squares problem, such that all residuals had the proper sign.If you don't know the lower asymptote for the curve, then you would need to use a nonlinear optimization tool for the fit. The constraints would be such that the necessary optimizer would be fmincon.
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