How to model a time series data with a custom function?

Question

Ana María Malagón Leal 2022-12-21

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https://ww2.mathworks.cn/matlabcentral/answers/1882872-how-to-model-a-time-series-data-with-a-custom-function

回答： Prateek 2023-1-9

I'm trying to fit an ex-gaussian function to a time series of data. I tried to use the fit function of MATLAB. I don't know what I'm doing wrong, I'm not finding the best fitting parameters of my function.

Here is my code and an image of the results (plot on the right).

% Ex-Gaussian model
  exGaussian = @(mu,sigma,tau,a0,a1,time) a0 + a1/2 .* exp((1./(2*tau)) .* (2.*mu+(sigma.^2./tau)-(2.*time))).*...
     erfc((mu+(sigma.^2./tau)-time)./sqrt(2*sigma));
Fs = 20; %sampling frecuency
t = 0:1/Fs:5; %timestamp in seconds
%Parameters for the example data to fit
mu = 2; 
sigma = .1; 
tau = 1;
a0 = 1;
a1 = 19;
%Example data
y  = exGaussian(mu,sigma,tau,a0,a1,t);
figure(1),clf
subplot(1,2,1)
plot(t, y), grid off
title('Example of data')
% Select the options to fit the model
opts = fitoptions('Method','NonlinearLeastSquares',...
     'Lower',[0 0 0 5 10],'Upper',[10 2 10 30 50],...
     'Startpoint',[1 0.1 0.1 1 1]);
 
exGfit = fittype('exGaussian(mu,sigma,tau,a0,a1,t)',...
    'coefficients',{'mu','sigma','tau','a0','a1'},...
    'independent','t','options',opts);
subplot(1,2,2)
[fitResult,  fitStats, fitInfo] = fit(t',y',exGfit);
plot(fitResult,'-r',t,y,'.-b');

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Walter Roberson 2022-12-22

在 MATLAB Online 中打开

Guassians are difficult to fit -- some of the parameters need to be pretty close to the correct value or else you are very likely to get results that are no-where near correct.

That said, in this case it doesn't even do a great job when given the actual parameters as the starting point :(

% Ex-Gaussian model

exGaussian = @(mu,sigma,tau,a0,a1,time) a0 + a1/2 .* exp((1./(2*tau)) .* (2.*mu+(sigma.^2./tau)-(2.*time))).*...

erfc((mu+(sigma.^2./tau)-time)./sqrt(2*sigma));

Fs = 20; %sampling frecuency

t = 0:1/Fs:5; %timestamp in seconds

%Parameters for the example data to fit

mu = 2;

sigma = .1;

tau = 1;

a0 = 1;

a1 = 19;

%Example data

y = exGaussian(mu,sigma,tau,a0,a1,t);

figure(1),clf

subplot(1,2,1)

plot(t, y), grid off

title('Example of data')

% Select the options to fit the model

opts = fitoptions('Method','NonlinearLeastSquares',...

'Lower',[0 0 0 5 10],'Upper',[10 2 10 30 50],...

'Startpoint',[mu, sigma, tau, a0, a1]);

exGfit = fittype(exGaussian,...

'coefficients',{'mu','sigma','tau','a0','a1'},...

'independent','time','options',opts);

subplot(1,2,2)

[fitResult, fitStats, fitInfo] = fit(t',y',exGfit)

fitResult =

General model: fitResult(time) = a0+a1/2.*exp((1./(2*tau)).*(2.*mu+(sigma.^2./tau)-(2.*time) )).*erfc((mu+(sigma.^2./tau)-time)./sqrt(2*sigma)) Coefficients (with 95% confidence bounds): mu = 2.358 (1.745, 2.97) sigma = 0.07661 (-0.009826, 0.163) tau = 0.3223 (-0.0268, 0.6714) a0 = 5 (fixed at bound) a1 = 16.62 (13.08, 20.17)

fitStats = struct with fields:

sse: 607.8905 rsquare: 0.5742 dfe: 97 adjrsquare: 0.5611 rmse: 2.5034

fitInfo = struct with fields:

numobs: 101 numparam: 5 residuals: [101×1 double] Jacobian: [101×5 double] exitflag: 3 firstorderopt: 1.4248 iterations: 12 funcCount: 78 cgiterations: 0 algorithm: 'trust-region-reflective' stepsize: 0.0030 message: 'Success, but fitting stopped because change in residuals less than tolerance (TolFun).'

plot(fitResult,'-r',t,y,'.-b');

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Answer 1

Mathieu NOE 2022-12-24

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https://ww2.mathworks.cn/matlabcentral/answers/1882872-how-to-model-a-time-series-data-with-a-custom-function#answer_1135592

在 MATLAB Online 中打开

hello

a quick and dirty trial with the poor's man solution ( with fminsearch)

the only trick to avoid an error (because with negative sigma, sqrt(sigma) is complex), I modified the function :

sqrt(2*abs(sigma)))

not 100% scientific maybe regarding optimizer usage, but it's seems to work not too bad

made a few trials with different IC, not all will give a good fit but with a liitle effort we can probably refine the IC range

% Ex-Gaussian model
  exGaussian = @(mu,sigma,tau,a0,a1,time) a0 + a1/2 .* exp((1./(2*tau)) .* (2.*mu+(sigma.^2./tau)-(2.*time))).*...
     erfc((mu+(sigma.^2./tau)-time)./sqrt(2*abs(sigma)));
Fs = 20; %sampling frecuency
t = 0:1/Fs:5; %timestamp in seconds
%Parameters for the example data to fit
mu = 2; 
sigma = .1; 
tau = 1;
a0 = 1;
a1 = 19;
%Example data
y  = exGaussian(mu,sigma,tau,a0,a1,t);
figure(1),clf
plot(t, y), grid off
title('Example of data')
% curve fit using fminsearch
obj_fun = @(p) norm(exGaussian(p(1),p(2),p(3),p(4),p(5),t)-y);
p0 = [0.5 0.01 0.5 y(1) max(y)]; % IC guess for mu,sigma,tau,a0,a1
sol = fminsearch(obj_fun, p0);
yfit = exGaussian(sol(1),sol(2),sol(3),sol(4),sol(5), t);
plot(t,yfit,'-',t,y,'r .','MarkerSize', 20);
title('Data', 'FontSize', 20)
ylabel('Amplitude', 'FontSize', 20)
xlabel('t', 'FontSize', 20)
legend('curve fit','raw data');

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Ana María Malagón Leal 2023-1-3

Thanks! That simple change in the equation will help me a lot.

Mathieu NOE 2023-1-4

My pleasure !

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Answer 2

Prateek 2023-1-9

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1882872-how-to-model-a-time-series-data-with-a-custom-function#answer_1144572

Hi Ana,

A good way to fit ex-Gaussian function is to use the "interpolant" option in the curve fitting toolbox. I was able to obtain a good fit for the data providede by you: