Compute the posterior predictive distribution in MagmaClust. Providing data from any new individual/task, its trained hyper-parameters and a previously trained MagmaClust model, the multi-task posterior distribution is evaluated on any arbitrary inputs that are specified through the 'grid_inputs' argument. Due to the nature of the model, the prediction is defined as a mixture of Gaussian distributions. Therefore the present function computes the parameters of the predictive distribution associated with each cluster, as well as the posterior mixture probabilities for this new individual/task.

## Usage

```
pred_magmaclust(
data,
trained_model = NULL,
grid_inputs = NULL,
mixture = NULL,
hp = NULL,
kern = "SE",
hyperpost = NULL,
prop_mixture = 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 MagmaClust model, previously trained using the

`train_magmaclust`

function. If`trained_model`

is set to NULL, the`hyperpost`

and`prop_mixture`

arguments are mandatory to perform required re-computations for the prediction to succeed.- 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`

.- mixture
A tibble or data frame, indicating the mixture probabilities of each cluster for the new individual/task. If NULL, the

`train_gp_clust`

function is used to compute these posterior probabilities according to`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`train_gp_clust`

function 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`

,`cov`

and`mixture`

the parameters of the hyper-posterior distributions of the mean processes. Typically, this argument should come from a previous learning using`train_magmaclust`

, or a previous prediction with`pred_magmaclust`

, with the argument`get_hyperpost`

set to TRUE.- prop_mixture
A tibble or a named vector of the mixture proportions. Each name of column or element should refer to a cluster. The value associated with each cluster is a number between 0 and 1. If both

`mixture`

and`trained_model`

are set to NULL, this argument allows to recompute mixture probabilities, thanks to the`hyperpost`

argument and the`train_gp_clust`

function.- get_hyperpost
A logical value, indicating whether the hyper-posterior distributions of the mean processes 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 matrices 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 list of GP prediction results composed of:

pred: As sub-list containing, for each cluster:

pred_gp: A tibble, representing the GP predictions as two column

`Mean`

and`Var`

, evaluated on the`grid_inputs`

. The column`Input`

and additional covariates columns are associated with each predicted values.proba: A number, the posterior probability associated with this cluster.

cov (if

`get_full_cov`

= TRUE): A matrix, the full posterior covariance matrix associated with this cluster.

mixture: A tibble, indicating the mixture probabilities of each cluster for the predicted individual/task.

hyperpost (if

`get_hyperpost`

= TRUE): A list, containing the hyper-posterior distributions information useful for visualisation purposes.