Which regressors do the parameter estimates belong to returned by 'nlarx'?

I estimate a model using 'nlarx' from data like this

mdl = nlarx(data,sys);

Then, in the 'mdl' object has a field

mdl.Report.Parameters.ParVector

which lists the parameter estimates in a single column vector. Now, i see that the number of parameters estimates equal the number of 'true's in

sys.RegressorUsage

which is a table, however, I do not find information anywhere, in the Matlab documentation or on the internet, which individual instant of 'true' in sys.RegressorUsage, i.e., which individual regressors, belongs to which entry of mdl.Report.Parameters.ParVector.

Furthermore, i would like to ask where i can find p-values or standard errors for each of these parameter estimates.

Any help would be much appreciated!

[Umar]

Hi Tamas,

Please see my response to your posted comments.

Question: which individual regressors, belongs to which entry of mdl.Report.Parameters.ParVector.

Please bear n mind that the mdl.Report.Parameters.ParVectorparameter contains estimates in a column vector. To associate individual regressors with these estimates, you can refer to the sys.RegressorUsage table. Each 'true' entry in sys.RegressorUsage corresponds to a parameter estimate in ParVector. The order of 'true' entries aligns with the order of parameter estimates.

Question: where i can find p-values or standard errors for each of these parameter estimates.

I am not sure if MATLAB's NLARX model implementation directly provide p-values or standard errors. You may need to calculate these values separately using statistical methods or by employing additional toolboxes in MATLAB for statistical analysis. I will provide a code snippet example which will utilize functions in Matlab to calculate p-values and standard errors for parameter estimates. Additionally, generate plots to visualize the data effectively.

% Generate sample data

x = randn(100, 1);

y = 2*x + randn(100, 1);

% Fit a linear regression model

mdl = fitlm(x, y);

% Display parameter estimates, p-values, and standard errors

disp('Parameter Estimates:');

disp(mdl.Coefficients.Estimate);

disp('P-Values:');

disp(mdl.Coefficients.pValue);

disp('Standard Errors:');

disp(mdl.Coefficients.SE);

% Plot the data and regression line

figure;

scatter(x, y);

hold on;

plot(mdl);

xlabel('X');

ylabel('Y');

title('Linear Regression Model');

legend('Data Points', 'Regression Line');

So, I created random data points for x and generate corresponding y values with some noise. Then,use the fitlm function to fit a linear regression model to the data and display the parameter estimates, p-values, and standard errors for the model coefficients and finally, create a scatter plot of the data points and overlay the regression line to visualize the linear relationship between x and y.

Please see attached plot along with test results.

By following this code snippet, you can efficiently calculate p-values, standard errors, and visualize the regression model in Matlab.

[Tamas]

Hi Umar,

Many thanks for the quick response!

As for the first question, i think i understand your answer. Sorry about being tedious about it, but would you be kind to confirm if this is what you meant? Say, we 1. define the model 2. estimate parameters and 3. extract the parameter estimates like this:

sys = idnlarx(output_name,input_name,orders);
mdl = nlarx(data,sys);
par_est = mdl.Report.Parameters.ParVector;

These are the regressors

myregs = getreg(sys);

that we potentially use in each of the

neqs = length(output_name);

number of (potentially coupled) equations that the system consists of. That is, we have a number

nterms = length(myregs)*neqs;

of terms in the whole system. But we want to knock out some of those terms, which we can do by setting entries of the

regus = sys.RegressorUsage;

table 'false'. This table corresponds to the 2D matrix

ru_mat = table2array(regus);

whose entries we can keep count of by the "column-wise" "linear" indices of it:

myregs_linind = (1:nterms)';

These linear indices of the matrix would correspond to "ordinary" indices of the vector coming from the "colonisation" of the matrix:

ru_vec = ru_mat(:);

Finally, is it correct that the subselected terms (regressor & equation/output variable)

whichregneq = myregs_linind(ru_vec);

correspond entry for entry with par_est?

As for the second question about uncertainty quantification. Sorry, i know 'fitlm' would return those things. But i need 'nlarx' to do that. Do i miss a point of yours?

If not, is it possible that "inside" 'nlarx' those p-values/SEs are calculated but not provided as an output for the user?

[Umar]

Hi Tamas,

Regarding your first query,” correspond entry for entry with par_est?”, the alignment of subselected terms with parameter estimates is correctly established. The 'true' entries in sys.RegressorUsage correspond to parameter estimates in mdl.Report.Parameters.ParVector. By setting entries in the sys.RegressorUsage table to 'false', specific terms can be excluded. The linear indices derived from sys.RegressorUsage matrix help in selecting the desired terms. The whichregneq vector, obtained from these indices, indeed corresponds entry for entry with par_est, ensuring the correct association between regressors, equations, and parameter estimates.

Addressing your last query, “ about uncertainty quantification. Sorry, i know 'fitlm' would return those things. But i need 'nlarx' to do that. Do i miss a point of yours?If not, is it possible that "inside" 'nlarx' those p-values/SEs are calculated but not provided as an output for the user?”

Sorry, if I missed details in my earlier post, 'nlarx' focuses on system identification rather than statistical inference. While 'nlarx' does not directly output p-values or SEs, you can still assess model uncertainty through techniques like bootstrapping or Monte Carlo simulations to derive confidence intervals for model parameters if this s what you are trying to accomplish. Also, you can still consider using 'fitlm' or other statistical tools alongside 'nlarx' for a comprehensive analysis.