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.