brms (version 2.1.0)

gp: Set up Gaussian process terms in brms

Description

Function used to set up a Gaussian process term in brms. The function does not evaluate its arguments -- it exists purely to help set up a model with Gaussian process terms.

Usage

gp(..., by = NA, cov = "exp_quad", scale = TRUE)

Arguments

...

One or more predictors for the Gaussian process.

by

A numeric or factor variable of the same length as each predictor. In the numeric vector case, the elements multiply the values returned by the Gaussian process. In the factor variable case, a separate Gaussian process is fitted for each factor level.

cov

Name of the covariance kernel. By default, the exponentiated-quadratic kernel "exp_quad" is used.

scale

Logical; If TRUE (the default), predictors are scaled so that the maximum Euclidean distance between two points is 1. Since the default prior on lscale expects scaled predictors, it is recommended to manually specify priors on lscale, if scale is set to FALSE.

Value

An object of class 'gpterm', which is a list of arguments to be interpreted by the formula parsing functions of brms.

Details

A Gaussian process is a stochastic process, which describes the relation between one or more predictors \(x = (x_1, ..., x_d)\) and a response \(f(x)\), where \(d\) is the number of predictors. A Gaussian process is the generalization of the multivariate normal distribution to an infinite number of dimensions. Thus, it can be interpreted as a prior over functions. Any finite sample realized from this stochastic process is jointly multivariate normal, with a covariance matrix defined by the covariance kernel \(k_p(x)\), where \(p\) is the vector of parameters of the Gaussian process: $$f(x) ~ MVN(0, k_p(x))$$ The smoothness and general behavior of the function \(f\) depends only on the choice of covariance kernel. For a more detailed introduction to Gaussian processes, see https://en.wikipedia.org/wiki/Gaussian_process.

Below, we describe the currently supported covariance kernels:

• "exp_quad": The exponentiated-quadratic kernel is defined as \(k(x_i, x_j) = sdgp^2 exp(- || x_i - x_j || / (2 lscale^2)\), where \(|| . ||\) is the Euclidean norm, \(sdgp\) is a standard deviation parameter, and \(lscale\) is characteristic length-scale parameter. The latter practically measures how close two points \(x_i\) and \(x_j\) have to be to influence each other substantially.

In the current implementation, "exp_quad" is the only supported covariance kernel. More options will follow in the future.

See Also

brmsformula

Examples

# NOT RUN {
# simulate data using the mgcv package
dat <- mgcv::gamSim(1, n = 30, scale = 2)

# fit a simple gaussian process model
fit1 <- brm(y ~ gp(x2), dat, chains = 2)
summary(fit1)
me1 <- marginal_effects(fit1, nsamples = 200, spaghetti = TRUE)
plot(me1, ask = FALSE, points = TRUE)

# fit a more complicated gaussian process model
fit2 <- brm(y ~ gp(x0) + x1 + gp(x2) + x3, dat, chains = 2)
summary(fit2)
me2 <- marginal_effects(fit2, nsamples = 200, spaghetti = TRUE)
plot(me2, ask = FALSE, points = TRUE)

# fit a multivariate gaussian process model
fit3 <- brm(y ~ gp(x1, x2), dat, chains = 2)
summary(fit3)
me3 <- marginal_effects(fit3, nsamples = 200, spaghetti = TRUE)
plot(me3, ask = FALSE, points = TRUE)

# compare model fit
LOO(fit1, fit2, fit3)

# simulate data with a factor covariate
dat2 <- mgcv::gamSim(4, n = 90, scale = 2)

# fit separate gaussian processes for different levels of 'fac'
fit4 <- brm(y ~ gp(x2, by = fac), dat2, chains = 2)
summary(fit4)
plot(marginal_effects(fit4), points = TRUE)
# }
# NOT RUN {
# }