Compute the posterior predictive distribution in Magma. Providing data of any new individual/task, its trained hyper-parameters and a previously trained Magma model, the predictive distribution is evaluated on any arbitrary inputs that are specified through the 'grid_inputs' argument.

## Usage

```
pred_magma(
data,
trained_model = NULL,
grid_inputs = NULL,
hp = NULL,
kern = "SE",
hyperpost = NULL,
get_hyperpost = FALSE,
get_full_cov = FALSE,
plot = TRUE,
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'.

- trained_model
A list, containing the information coming from a Magma model, previously trained using the

`train_magma`

function.- grid_inputs
The grid of inputs (reference Input and covariates) values on which the GP should be evaluated. Ideally, this argument should be a tibble or a data frame, providing the same columns as

`data`

, except 'Output'. Nonetheless, in cases where`data`

provides only one 'Input' column, the`grid_inputs`

argument can be NULL (default) or a vector. This vector would be used as reference input for prediction and if NULL, a vector of length 500 is defined, ranging between the min and max Input values of`data`

.- hp
A named vector, tibble or data frame of hyper-parameters associated with

`kern`

. The columns/elements should be named according to the hyper-parameters that are used in`kern`

. The function`train_gp`

can be used to learn maximum-likelihood estimators of the hyper-parameters.- 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 the 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 a previous prediction with`pred_magma`

, with the argument`get_hyperpost`

set to TRUE. The 'mean' element should be a data frame with two columns 'Input' and 'Output'. The 'cov' element should be a covariance matrix with colnames and rownames corresponding to the 'Input' in 'mean'. In all cases, the column 'Input' should contain all the values appearing both in the 'Input' column of`data`

and in`grid_inputs`

.- get_hyperpost
A logical value, indicating whether the hyper-posterior distribution of the mean process should be returned. This can be useful when planning to perform several predictions on the same grid of inputs, since recomputation of the hyper-posterior can be prohibitive for high dimensional grids.

- get_full_cov
A logical value, indicating whether the full posterior covariance matrix should be returned.

- plot
A logical value, indicating whether a plot of the results is automatically displayed.

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

## Value

A tibble, representing Magma predictions as two column 'Mean' and
'Var', evaluated on the `grid_inputs`

. The column 'Input' and
additional covariates columns are associated to each predicted values.
If the `get_full_cov`

or `get_hyperpost`

arguments are TRUE,
the function returns a list, in which the tibble described above is
defined as 'pred_gp' and the full posterior covariance matrix is
defined as 'cov', and the hyper-posterior distribution of the mean process
is defined as 'hyperpost'.