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

deviationerrorMATLABpolyfitpolyvalstandard

I would like to know why POLYVAL produces error bounds with at least 50% of the predictions in MATLAB 6.5 (R13).

Best Answer

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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