time series fitting to statistical moments

Hello,

I have a process for which I know the conditional moments:

mean = exp(-a * t)*(x-mu)

variance = ((1-exp(-2* t* a))* sigma)/2a

where a and sigma are unknown parameters.

By using the fact that conditional moments are linear, I would like to estimate the 2 unknown parameters through a linear regression of X(t+dt) and X^2(t+dt) on X(t) where X(t) is a known time series that I have, and dt is the time interval used in the time series.

Any idea about how to implement this in matlab code, would be really appreciated.

Thanks

Best regards

Paolo

[Sam Chak]

Hi @Paolo, could you at least provide the data for visualization in MATLAB?

[Paolo]

Hi Sam, here the data attached.

thanks for help

Paolo

Answer 1

[Star Strider]

‘where a and sigma are unknown parameters’

Beyond that, I am clueless as to any appropriate way to do this parameter estimation, since there is a significant amount of missing information.

Using my fertile imagination to fill those gaps, try something like tthis —

LD = load('cds.mat');
X = LD.spread;                                      % The ‘X’ Part
X = X(:);
Size_X = size(X)

Size_X = 1×2

2219 1

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t = 0:numel(X)-1;                                   % Undefined, So A Guess
t = t(:);
mu = mean(X)                                        % Undefined, So A Guess

mu = 0.0425

dt = randi([2 10])

dt = 4

Xmm = movmean(X,dt);                                % Use ‘movmean’ To Provide The ‘(X(t+dt)’ Mean
Xmv = movvar(X,dt);                                 % Use ‘movvar’ To Provide The ‘(X(t+dt)’ Variance

fcn = @(b,x) [exp(-b(1) .* t).*(x-mu), ((1-exp(-2 * t .* b(1)) .* b(2))./(2*b(1)))]       % b(1) = a,  b(2) = sigma

fcn = function_handle with value:
    @(b,x)[exp(-b(1).*t).*(x-mu),((1-exp(-2*t.*b(1)).*b(2))./(2*b(1)))]

[B, fv] = fminsearch(@(b) norm([Xmm(:) Xmv(:)] - fcn(b,X(:))), rand(2,1) );
FinalValue = fv

FinalValue = 2.1505

fprintf('\n\nParameters:\n\ta \t= %10.3f\n\tsigma \t= %10.3f\n\n', B)

Parameters:
	a 	=   2782.231
	sigma 	=   2162.584

figure
plot(t, X, DisplayName="X(t)")
hold on
plot(t, Xmm, DisplayName="movmean(X)")
hold off
grid
xlabel("Time")
ylabel("Value")
legend(Location='best')

X = X.^2;                                           % The 'X²' Part
Xmm = movmean(X,dt);                                % Use ‘movmean’ To Provide The ‘(X(t+dt)’ Mean
Xmv = movvar(X,dt);                                 % Use ‘movvar’ To Provide The ‘(X(t+dt)’ Variance

fcn = @(b,x) [exp(-b(1) .* t).*(x-mu), ((1-exp(-2 * t .* b(1)) .* b(2))./(2*b(1)))]       % b(1) = a,  b(2) = sigma

fcn = function_handle with value:
    @(b,x)[exp(-b(1).*t).*(x-mu),((1-exp(-2*t.*b(1)).*b(2))./(2*b(1)))]

[B, fv] = fminsearch(@(b) norm([Xmm(:) Xmv(:)] - fcn(b,X(:))), rand(2,1) );
FinalValue = fv

FinalValue = 0.1420

fprintf('\n\nParameters:\n\ta \t= %10.3f\n\tsigma \t= %10.3f\n\n', B)

Parameters:
	a 	=    525.128
	sigma 	=    109.864

figure
plot(t, X, DisplayName="X(t)^2")
hold on
plot(t, Xmm, DisplayName="movmean(X^2)")
hold off
grid
xlabel("Time")
ylabel("Value")
legend(Location='best')

.