MATLAB: Is the POLYCONF function computing the confidence interval or the prediction interval in the Statistics Toolbox 5.0.1 (R14SP1)

boundconfidenceintervalMATLABpolyconfpolyvalpredictionstatisticsStatistics and Machine Learning Toolbox

When I type "doc polyconf" at the MATLAB command prompt, the documentation states that POLYCONF computes a confidence interval. However, when I type "help polyconf" at the MATLAB command prompt, the file help indicates that POLYCONF computes a prediction interval.

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This enhancement has been incorporated in Release 14 Service Pack 3 (R14SP3). For previous product releases, read below for any possible workarounds:

This has been verified as an error within the documentation for POLYCONF within the Statistics Toolbox 5.0.1 (R14SP1). The POLYCONF function computes the prediction interval, not the confidence interval.

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MATLAB: How to calculate the standard error related to the confidence interval output by the NLPARCI function in the Statistics Toolbox 5.0.1(R14SP1)

This enhancement has been incorporated in Release 2006a (R2006a). For previous product releases, read below for any possible workarounds:

There is a well-defined relation between the width of the confidence interval in standard errors and the percentage of observations falling within the confidence interval. To calculate the standard error, first you will need to find the confidence interval. Then you can use the TINV function to calculate the number of standard errors spanned by the confidence interval. Finally, divide the confidence interval by the number of standard errors spanned by the confidence interval to find the standard error.

Here is an illustration of how to compute the standard error:

1. Fit a model using the NLINFIT function, and get the parameters.

load hald 
y = heat; 
x = ingredients(:,4); 
[b,r,J] = nlinfit(y,x,@(b,x) b(1)+b(2)*x, 1:2);

2. Estimate the confidence interval using the NLPARCI function:

alpha = 0.05
ci = nlparci(b,r,J,alpha);   % 95% confidence intervals

3. Compute the standard error from the confidence interval using the TINV function:

   t = tinv(1-alpha/2,length(y)-length(b));
   nlinfit_se = (ci(:,2)-ci(:,1)) ./ (2*t)  % Standard Error

MATLAB: Does POLYVAL produce error bounds with at least 50% of the predictions in MATLAB 6.5 (R13)

The bounds produced by POLYVAL surround an interval that is one standard error below and above the estimate. This standard error measures both the uncertainty in estimating the polynomial, and the additional uncertainty in predicting a new observation.

You can use the POLYCONF function in the Statistics Toolbox to compute a confidence interval for predicing a new observation. These intervals have the form

 p_hat(x) +/- T * SE

where p_hat (read this as "P Hat") is the estimated polynomial, SE is the standard error, and T is a quantile from the t distribution.

In the POLYVAL function we effectively set T=1. The confidence level of the resulting intervals depends on the number of error degrees of freedom (d.f.). It has a low value equal to 50% when there is 1 d.f., and it rises to 68% as the d.f. get larger. Here's an illustration using a range of d.f. values:

errordf = [1 2 3 10 50 10000];
tcdf(1,errordf) - tcdf(-1,errordf)
 ans =
     0.5000    0.5774    0.6090    0.6591    0.6779    0.6827

This is the reason for the claim in the file help.

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Related Solutions

MATLAB: How to calculate the standard error related to the confidence interval output by the NLPARCI function in the Statistics Toolbox 5.0.1(R14SP1)

MATLAB: Does POLYVAL produce error bounds with at least 50% of the predictions in MATLAB 6.5 (R13)

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