Hi everyone,
Are there any suitable tools in UQLab for handling heteroscedastic problems? I have a simple 2D dataset and would like to model how the mean and variance of y changes with x. Any suggestions? Thanks!
Dear @HZhang ,
Thank you for your question. UQLab does not currently offer an “out-of-the-box” solution for directly modeling both the mean and variance in the presence of heteroscedastic noise. However, there are different approaches you can follow.
One option is to use feasible generalized least squares (FGLS). With FGLS, the model mean and variance are estimated in an iterative way: First, an initial model (e.g. PCE) is fitted (e.g., via OLS), and the residuals are used to estimate the heteroscedasticity pattern; then, a weighted least squares regression is performed using these estimates. For more details, see Chapter 8 in [1].
Another option is to treat the problem as a stochastic emulation task and model the entire conditional response distribution y given x. To do this, you can use UQLab’s stochastic polynomial chaos expansion (SPCE) (user manual here, usage examples here) or generalized lambda models (GLaM) modules (manual here, examples here), which allow you to approximate the full distribution y|x. Once the emulator is built, the mean and variance can be extracted via simple post-processing. Please note that while the stochastic emulator approach is more flexible and informative (as it captures the entire response distribution), it typically requires a sufficiently large dataset (often on the order of 100+ points).
I hope this helps!
Best regards,
Katerina
[1] J. M. Wooldridge. Introductory Econometrics: A Modern Approach. Cengage Learning, 5th
edition, 2013
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Dear @Katerina ,
Thank you for the thorough explanations, I truly appreciate it! This is really helpful!
Best,
Hongzhou
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