Learning hyper-parameters of any new individual/task in Magma is required in the prediction procedure. This function can also be used to learn hyper-parameters of a simple GP (just let the hyperpost argument set to NULL, and use prior_mean instead). When using within Magma, by providing data for the new individual/task, the hyper-posterior mean and covariance parameters, and initialisation values for the hyper-parameters, the function computes maximum likelihood estimates of the hyper-parameters.

## Usage

train_gp(
data,
prior_mean = NULL,
ini_hp = NULL,
kern = "SE",
hyperpost = NULL,
pen_diag = 1e-10
)

## Arguments

data

A tibble or data frame. Required columns: Input, Output. Additional columns for covariates can be specified. The Input column should define the variable that is used as reference for the observations (e.g. time for longitudinal data). The Output column specifies the observed values (the response variable). The data frame can also provide as many covariates as desired, with no constraints on the column names. These covariates are additional inputs (explanatory variables) of the models that are also observed at each reference Input.

prior_mean

Mean parameter of the GP. This argument can be specified under various formats, such as:

• NULL (default). The hyper-posterior mean would be set to 0 everywhere.

• A number. The hyper-posterior mean would be a constant function.

• A vector of the same length as all the distinct Input values in the data argument. This vector would be considered as the evaluation of the hyper-posterior mean function at the training Inputs.

• A function. This function is defined as the hyper-posterior mean.

• A tibble or data frame. Required columns: Input, Output. The Input values should include at least the same values as in the data argument.

ini_hp

A named vector, tibble or data frame of hyper-parameters associated with the kern of the new individual/task. The columns should be named according to the hyper-parameters that are used in kern. In cases where the model includes a noise term, ini_hp should contain an additional 'noise' column. If NULL (default), random values are used as initialisation.

kern

A kernel function, defining the covariance structure of the GP. Several popular kernels (see The Kernel Cookbook) are already implemented and can be selected within the following list:

• "SE": (default value) the Squared Exponential Kernel (also called Radial Basis Function or Gaussian kernel),

• "LIN": the Linear kernel,

• "PERIO": the Periodic kernel,

• "RQ": the Rational Quadratic kernel. Compound kernels can be created as sums or products of the above kernels. For combining kernels, simply provide a formula as a character string where elements are separated by whitespaces (e.g. "SE + PERIO"). As the² elements are treated sequentially from the left to the right, the product operator '*' shall always be used before the '+' operators (e.g. 'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

hyperpost

A list, containing the elements 'mean' and 'cov', the parameters of the hyper-posterior distribution of the mean process. Typically, this argument should come from a previous learning using train_magma, or from the hyperposterior function. If hyperpost is provided, the likelihood that is maximised is the one involved during Magma's prediction step, and the prior_mean argument is ignored. For classic GP training, leave hyperpost to NULL.

pen_diag

A number. A jitter term, added on the diagonal to prevent numerical issues when inverting nearly singular matrices.

## Value

A tibble, containing the trained hyper-parameters for the kernel of the new individual/task.

## Examples

TRUE
#> [1] TRUE