Hi @sstore23,
In addition to Ali’s suggestions, I would suggest the following:
- It is unusual that the LOO error is so small, and then the validation error is larger than 1. There might be an issue with your validation set. How large is it? Are the training set and the validation set sampled from the same distribution? Your data is two-dimensional, so it is easy to visualize: Plot your training data (
X) and the validation data (Xval) in the same figure and confirm visually that they follow the same distribution. How does the distribution look like? Is there maybe dependence in the X-data? - Make a scatter plot of your data and the corresponding model output by using the Matlab command
scatterwith filled circles that are colored according to the output value. Does the color vary smoothly over the input domain, or are there jumps in the data (or outliers)? Polynomial approximation works best when the model is smooth and varying slowly. - Is your model deterministic - or maybe stochastic?
- You mention
Note that regression-based PCE is a form of polynomial regression, just with a special polynomial family, and in the case of UQLab, by default using a sparsity constraint (LARS). The pictures that you show (the left one from PCE, the right one from “polynomial regression”?) use a very different set of points, which might explain the difference.
Good luck! 