Fit data to lagged custom function

Hello,

I would like to ask if you can advice the correct approach I can follow to estimate the parameters of a custom lagged function

(1) y(t)=c^2*a+y(t-1)*(a-1)

where c is a known constant.

to a time series data (I can use the symbilic function to create (1) )

Thank you.

Best regards

Paolo

[Torsten]

I would like to ask if you can advice the correct approach I can follow to estimate the parameters of a custom lagged function

You mean the parameter "a" ?

[Paolo]

Hi Torsten,

thank you for your feedback. Yes I mean "a"; I forgot to mention that the time series of Y is already available and I know Y(0)= 0.04356

Best regards

Paolo

Answer 1

[Harsh Kumar]

Hope this may help ,

% Assuming you have your y data in a vector called 'y'
% and c is your known constant

% Step 1: Prepare data
y_lag = [NaN; y(1:end-1)];  % Create lagged y, with NaN for the first value
y = y(2:end);  % Remove the first value of y to match dimensions
y_lag = y_lag(2:end);

% Step 2 & 3: Define the objective function
obj_fun = @(a) sum((y - (c^2*a + y_lag*(a-1))).^2);

% Step 4: Use optimization to find the best 'a'
options = optimset('Display', 'iter');
a_est = fminsearch(obj_fun, 0.5, options);  % 0.5 is an initial guess for 'a'

% Print the result
fprintf('Estimated value of a: %f\n', a_est);

% Optional: Plot the results
y_pred = c^2*a_est + y_lag*(a_est-1);
plot(y, 'b-', 'DisplayName', 'Observed');
hold on;
plot(y_pred, 'r--', 'DisplayName', 'Predicted');
legend('show');
title('Observed vs Predicted y(t)');

[Paolo]

thanks!

Paolo