Partial Least Squares regression - confidence interval of the predicted variable (response)

Hello all,

I am interested in obtaining confidence intervals for the response variable of PLS (Partial Least Squares Regression). Can someone help me on that? Here is my attempt at it:

% https://nl.mathworks.com/help/stats/partial-least-squares-regression-and-principal-components-regression.html

load spectra

X = NIR; % independent variables

y = octane; % dependent variables

PLS_comp = 3; % number of PLS components

[XL,yl,XS,YS,beta,PCTVAR,mse,stats] = plsregress(X,y,PLS_comp); % PLS regression

yfit = [ones(size(X,1),1) X]*beta; % Model fit

residuals = y - yfit; % Ordinary residuals vector

alpha_stat = 0.05; % Significance level

dgf = length(y) - PLS_comp - 1; % Degree of freedom

RMSE_model = sqrt(sum(residuals.^2)/dgf); % Degree of freedom corrected root-mean squared error (standard deviation estimator)

t_Student = tinv((1-alpha_stat/2),dgf); % t-value Student distribution

delta = t_Student*RMSE_model*sqrt(1+stats.T2); % CI boundaries

figure()

set(gcf,'color','white','position',[100 100 500 500])

errorbar(y,yfit,delta,'o')

hold on; grid minor;

hline = refline([1 0]);

hline.Color = 'k';

hline.LineStyle = ':';

xlabel('Measured')

ylabel('Predicted')

Questions are:

• Is there a better (or simpler) way to do it? (maybe even using a MATLAB standard function). I tried to follow the guidelines of this paper here, in case someone is wondering about the degrees of freedom: 10.1016/j.chemolab.2009.11.003
• Is this approach correct? The confidence intervals look too big to be correct
• This T2 statistic from the stats struct is not retrievable for data outside the training data. How do I collect it for a new spectra? (if same approach is used). I cannot get confidence intervals of prediction the way I did it.

Kind regards,

Gustavo

Answer 1

[Torsten]

I did not look into your code in detail, but I think you could use the output structure "gof" from MATLAB's "fit" together with "confint" to compare with your statistical parameters.

[Gustavo Lunardon]

Sounds reasonable. Might try with the predint command too.