Regression of stochastic processes with exponentially correleted noise. secn-PR is a MATLAB package designed to estimate a stochastic differential equation from multiple data sources. The regression is flexable enough to account for white or exponentially correlated noise. It employs a block bootstrap algorithm to estimate uncertainties.

To be added

This example follows the script exampleScript.m in the ./examples/ directory. Further information on the functions/classes/options can be found in the headers of each file.

Take some time-series observations that are vectors, and use the buildObservation() function to form an observationData object. The observations must have the same time-step.

This data set was calculated from a prescribed SDE, so the function estimates can be compared to the true values later.

%% Make observation object
observationDataAB = buildObservation(trueModel.dt,...
    observationVectorA,observationVectorB);
figure,observationDataAB.plotObservation;

From the observations, conditional moments can be calculated using the buildMoments() function. Options for the calculations are in the MomentOptionsClass class definition.

% Setting options (timeShiftSamplePoints,nEvalPoints,evalLims,bandwidth)
momentOptions = MomentOptionsClass(1:60,20,[-1,1],0.1);

% Making moments
momentDataAB = buildMoments(observationDataAB,momentOptions);
figure,momentDataAB.plotMoment1()

Adjustments to the options can be made and the . The choice of maximum lag should be as small as possible whilst still retaining the shape of the moments.

% Making adjustments to moment calculations
momentOptions.timeShiftSamplePoints = 1:15;
momentDataAB = buildMoments(observationDataAB,momentOptions);
figure,momentDataAB.plotMoment1()

From the conditional moments, the correlation time and drift and noise functions of the stochastic process can be estimated using the estimateSPmodel() function. Options for the calculations are in the FitOptionsClass class definition. For example, fixThetaValue allows for setting the correlation time to a particular value rather than solving for it.

% Setting options (no arguments => default)
fitOptions = FitOptionsClass();

% Estimate stochastic process model
SPmodelAB = estimateSPmodel(momentDataAB,fitOptions);

% Review autocorrelation and moments functional fit
figure,SPmodelAB.plotAutocorrelation();
figure,SPmodelAB.plotMoment1();

The drift and noise functions are plotted here with the true input functions.

% View estimated functions
figure
subplot(1,2,1)
h = SPmodelAB.plotDrift();
h.model = plot(SPmodelAB.momentData.evalPoints,...
    trueModel.drift(SPmodelAB.momentData.evalPoints),'r-','LineWidth',2);
legend([h.scatter,h.model],{'Data','True model'},'Location','NorthEast')
subplot(1,2,2)
h = SPmodelAB.plotNoise();
h.model = plot(SPmodelAB.momentData.evalPoints,...
    trueModel.noise(SPmodelAB.momentData.evalPoints),'r-','LineWidth',2);
legend([h.scatter,h.model],{'Data','True model'},'Location','NorthWest')

Uncertainties can be estimated through a block bootstrap algorithm, using the estimateBootstrapModel() function. Options for the calculations are in the BootstrapOptionsClass class definition.

% Setting options (nBootstrapSamples)
bootstrapOptions = BootstrapOptionsClass(20);

% Estimate uncertainties
SPbootstrapAB = estimateBootstrapModel(SPmodelAB,bootstrapOptions);

% View estimated uncertainties
figure
subplot(1,2,1)
h = SPbootstrapAB.plotDrift();
h.model = plot(SPmodelAB.momentData.evalPoints,...
    trueModel.drift(SPmodelAB.momentData.evalPoints),'r-','LineWidth',2);
legend([h.scatter,h.fill,h.model],'Best estimate',...
    '95% confidence interval','True model')
subplot(1,2,2)
h = SPbootstrapAB.plotNoise();
h.model = plot(SPmodelAB.momentData.evalPoints,...
    trueModel.noise(SPmodelAB.momentData.evalPoints),'r-','LineWidth',2);
legend([h.scatter,h.fill,h.model],'Best estimate',...
    '95% confidence interval','True model')

The drift and noise functions and their uncertainties are plotted here with the true input functions.

This workflow is streamlined in the fullSPestimate() function, see exampleScript2.m for more details.

• Implement arbitrary timeShiftSamplePoints vector
• Implement arbitrary spatial evaluation points
• Implement option of forcing a particular correlation time
• Add compatibility for combining observations of different time-steps
• Make a plotting function for the distribution of correlation times
• Add more kernel functions to the library

• Version 1.0 - First official release
• Version 0.9 - Introduced version I used for my research.

• Lehle, B., & Peinke, J. (2018). Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise. Physical Review E, 97(1), 012113.
• Lamouroux, D., & Lehnertz, K. (2009). Kernel-based regression of drift and diffusion coefficients of stochastic processes. Physics Letters A, 373(39), 3507-3512.
• Kunsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. The annals of Statistics, 1217-1241.