Hello! I’m looking for a bit of general advice or some examples to learn from rather than model triage (at this point). My question relates to best practices for the specification of multiple continuous predictors in a hierarchical Gaussian process. I have been working from the excellent 2017 StanCo…

[image] nerutenbeck: I am thinking of it as analogous to the decadal effect and the annual effect in the birthday problem or the region / state effects in the case study I mentioned in the OP. Oh right, that makes sense. [image] nerutenbeck: but another continuous variable that has a sin…

[image] nerutenbeck: if I can no longer lean on the neatly binned diameter class information to efficiently build the covariance matrix If you can’t do binning, you can check out a basis function representation to keep the dimensions of your linear algebra under control: Practical Hilbert spa…

Thanks, @bbbales2 , this paper looks very interesting and helpful! I will check it out more thoroughly and see if I can come up with an implementation for my application. (By the way, last year you helped me identify some (now obvious) issues with earlier strategies I had for modeling a related prob…

By multiple levels do you mean something like: \mu \sim N(X_\mu \beta_\mu, \sigma_\mu)\\ y \sim N(X_y \beta_y + \mu, \sigma_y) (Assuming there are N data points y and J groups \mu and the indexing above makes sense in some way) Then the alternative would be something like: \mu \sim N(0, \sigma…

[image] bbbales2: By multiple levels do you mean something like: \mu \sim N(X_\mu \beta_\mu, \sigma_\mu)\\ y \sim N(X_y \beta_y + \mu, \sigma_y) (Assuming there are N data points y and J groups \mu and the indexing above makes sense in some way) Then the alternative would be something like: …

[image] nerutenbeck: I am also interested in incorporating a plot-level variable such as time since disturbance that I expect to have an interaction with the size effect in predicting count Can you write out equations for what you mean here? For instance I could imagine an interaction effect…

[image] bbbales2: Can you write out equations for what you mean here? I will give it a shot! My current model is: y \sim \textbf{NegBinom}(e^{\eta}, \phi)\\ \eta = \mu + \textsf{offset}_{plot} + g_{\mu} + g_{plot} + log(exposure)\\ \mu \sim N(0, 0.1)\\ \textsf{offset}_{plot} \sim N(0, \si…

[image] nerutenbeck: g_{\mu} is the mean diameter effect on counts across plots (general trend) and g_{plot} is the plot-level diameter effect on counts So in this case, if the GPs model a function that we’re trying to estimate, then g_{\mu} is a function tree diameter to a log density of tre…

[image] bbbales2: So in this case, if the GPs model a function that we’re trying to estimate, then g_{\mu} is a function tree diameter to a log density of trees (trying to think of what the units are) and g_{plot} is a function from X to log density of trees. What is X here? Is this a spatial t…

Great, I will try the straight two dimensional kernel and look into the HS approximation. I also want to try out automatic relevance determination in this case. Thanks again for your time and advice, @bbbales2 . I really appreciate it.

Oh yeah if it becomes too difficult to model the full function f(x, y), I think the first thing you could do to simplify that is model f(x, y) as two functions f(x) f(y) (before switching to linear models).

Multiple continuous predictors in hierarchical Gaussian process

Modeling

bbbales2 July 15, 2020, 9:51pm 2

If you can’t do binning, you can check out a basis function representation to keep the dimensions of your linear algebra under control: Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming

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Multiple continuous predictors in hierarchical Gaussian process

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