Nonlinear fit of segmented curve

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Eric
Eric 2012-12-4
评论： Jonathan Gößwein 2022-10-14
How would I go about getting a nonlinear least-squares fit of a segmented curve? In this case, I have a short, linear, lag period followed by a logistic growth phase (typical of bacterial growth in culture).
Thus, for x < T0, y = Y0; for x >= T0, y = Y0 + (Plateau-Y0)*(1 - exp(-K*(X-X0)).
I need least squares estimates for each of the parameters: T0, Y0, Plateau, and K
I've attempted to use a custom function in the curve fitting toolbox, but cannot figure out how to allow for the two curves.
Thanks!

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Teja Muppirala
Teja Muppirala 2012-12-5
It is no problem to fit piecewise curves in MATLAB using the Curve Fitting Toolbox. You can deal with piecewise functions by multiplying each piece by its respective domain. For example:
rng(0); %Just fixing the random number generator for initial conditions
X = (0:0.01:10)';
% True Values
Y0_true = 3;
PLATEAU_true = 5;
K_true = 1;
X0_true = 4;
Y = [Y0_true] * (X <= X0_true) + [Y0_true + (PLATEAU_true-Y0_true)*(1 - exp(-K_true*(X-X0_true)))].* (X > X0_true);
Y = Y + 0.1*randn(size(Y));
plot(X,Y);
ftobj = fittype('[Y0] * (x <= X0) + [Y0 + (PLATEAU-Y0)*(1 - exp(-K*(x-X0)))].* (x > X0)');
cfobj = fit(X,Y,ftobj,'startpoint',rand(4,1))
hold on;
plot(X,cfobj(X),'r','linewidth',2);
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user001
user001 2019-9-26
编辑：user001 2019-9-26
This does not work for me in R2017b. The fit is effectively the line y=4 (K=4.2, plateau=4, X0 = -1.94, Y0 = -1.98). The fit works only if noise is not added to the simulated data.
Jonathan Gößwein
Jonathan Gößwein 2022-10-14
The problem are the startpoints, rand(4,1) does not work indeed, but with an appropriate selection the method works (e.g. the true values).

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John Petersen
John Petersen 2012-12-4
Not sure how you would do that, but you could try using a sigmoid function which will get you close, relatively speaking. Something like, for example,
y2 = Y0 + (Plateau-Y0)./(1 + exp(-K*(X-X0)));
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Eric
Eric 2012-12-5
Thanks much for your rapid response, John! However, in order for this to be acceptable for publication (and more fundamentally, for comparison with other work), I need to be able to fit it to the originally specified equation. The values of those parameters are used to describe the response in a way that's meaningful for the relevant research community. The frustrating part is that this is an amazingly simple operation with other software (GraphPad Prism) with a built-in equation. I just can't figure out how to specify a segmented equation in a way that the Curve Fitting Toolbox will understand. Perhaps there's something analogous with how a discontinuous function would be entered?

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laoya
laoya 2013-5-14
Hi Teja Muppirala,
I am also interested in this topic. Now my problem is: if the express of curves are not expressed explicitly, but should be calculated by functions, how to use this function?
Thanks, Tang Laoya

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