Recovering lscale in a two-dimensional Gaussian process regression

#1

In order to better work with two-dimensional Gaussian process models in brms I would like to generate data with known parameters. Given that brms only has the exponential quadratic kernel implemented, in particular I’d like to see if I can recover the lscale parameter of this kernel.

In the following code, I generate data on a grid, and attempt to recover lscale = 0.5, and sdx1x2 = 0.25. Despite a sample size of 100 (the highest I want to go due to speed constraints), I am not able to recover a parameter value close to the lscale I used to generate the data, even with the gp(..., scale=FALSE) argument set. The sdx1x2 parameter is also off. The model samples well.

Is this because I am misinterpreting the lscale parameter in some way, or have generated the data incorrectly. Alternatively, is it a sample size issue (i.e. more data are required to ensure generated data converges on intended parameters)?

library(MASS)
library(brms)

## Generate data with a exponential quadratic kernel
## covariance structure in two dimensions

set.seed(186429)
## Specify parameters
N <- 100
lscale <- 0.5
sdx1x2 <- 0.25
b0 <- 0

## Generate exponential quadratic kernel covariance given lscale
exp_quad <- function(D, lscale, eps=sqrt(.Machine$double.eps)) {
 Sigma <- exp(-D/2*lscale^2) + diag(eps, nrow(D))
}
rows <- seq(0, 1, length.out=trunc(sqrt(N)))
cols <- seq(0, 1, length.out=trunc(sqrt(N)))
df <- expand.grid(x1=rows, x2=cols)
Sigma <- exp_quad(as.matrix(dist(cbind(df$x1, df$x2))), lscale)
## Ensure variance reflects sdx1x2
Sigma <- Sigma * sdx1x2^2

## Draw from this Gaussian process
df$gpx1x2 <- mvrnorm(n=1, mu=rep(0, nrow(Sigma)), Sigma=Sigma)

## Generate data y ~ b0 + gpx1x2
df$y <- b0 + df$gpx1x2

## Try to recover parameters using brms
## set gp(..., scale=FALSE) to maximize interpretability
m <- brm(y ~ gp(x1, x2, scale=FALSE), data=df,
         iter=2000, cores=4, refresh=1)

 summary(m)
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: y ~ gp(x1, x2, scale = FALSE) 
   Data: df (Number of observations: 100) 
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup samples = 4000

Gaussian Process Terms: 
               Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sdgp(gpx1x2)       0.06      0.02     0.04     0.10       1043 1.00
lscale(gpx1x2)     0.25      0.03     0.19     0.33        643 1.00

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept     0.38      0.03     0.33     0.43       1502 1.00

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma     0.02      0.00     0.02     0.03       1818 1.00

Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample 
is a crude measure of effective sample size, and Rhat is the potential 
scale reduction factor on split chains (at convergence, Rhat = 1).

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#2

That multiplies by lscale^2. Take a look at @bbbales2’s version:

cov_exp_quad = function(x, y, sigma, l) {
  outer(x, y, function(xi, xj) { sigma^2 * exp(-(xi - xj)^2 / (2 * l^2)) })
}
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#3

This was very helpful @andre.pfeuffer . It demonstrated that the main issue with my code is that I was not squaring the distance, and secondarily I had the brackets in the wrong place.

Replace the exp_quad() function with this code:

exp_quad <- function(D, lscale, eps=sqrt(.Machine$double.eps)) {
  Sigma <- exp(-D^2/(2*lscale^2)) + diag(eps, nrow(D))
}

AND adjust

df$y <- b0 + df$gpx1x2 + rnorm(nrow(Sigma), sd=0.1)

in order to add enough variability that the sampling doesn’t get stuck. It then samples like a charm, and recovers the input parameters perfectly!

I took my formula for exponential quadratic from the brms::gp() help. Here is the rendered version:

image

@paul.buerkner is there a typo in the help, here? Shouldn’t the Euclidean norm be squared? This is what fooled me. I don’t think Euclidean norm notation implies the squared distance, but I should probably know!

Thanks again for making an awesome package that truly does everything.

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#4

You are right, I fixed that.

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