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A value or the magrittr placeholder.

A function call using the magrittr semantics.

A matrix, possibly non-symmetric.

A number, the starting number of digits to round from ifmat is not symmetric

A matrix, possibly singular.

A number, a jitter term to add on the diagonal.

A list, containing the information coming from aMagmaClust model, previously trained using thetrain_magmaclust function.

A vector, containing values the likelihood is evaluated on.

A vector or matrix, specifying the mean parameter.

A matrix, specifying the inverse of covariance parameter.

A logical value, indicating whether we return the log-likelihood.

An interval of values we want to draw uniformly in.

A tibble or data frame. Columns required: ID, Input, Output.Additional columns for covariates can be specified.

A vector, corresponding to the prior mean of the mean GP.

A kernel function, associated with the mean GP.

A kernel function, associated with the individual GPs.

A named vector, tibble or data frame of hyper-parametersassociated withkern_0.

A tibble or data frame of hyper-parametersassociated withkern_i.

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

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing values we want to compute elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

A list of the K mean GPs at union of observed timestamps.

A kernel function used to compute the covariance matrix atcorresponding timestamps.

A List of the K posterior covariance of the mean GP (mu_k).Used to compute correction term (cor_term).

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing the values we want to compute the elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

List of parameters for the K mean GPs.

A kernel function used to compute the covariance matrix atcorresponding timestamps.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing values we want to compute elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

List of parameters for the K mean Gaussian processes.

A kernel function used to compute the covariance matrix atcorresponding timestamps.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or named vector of hyper-parametersfor each clusters.

A tibble, data frame or named vector of hyper-parametersfor each individuals.

A tibble containing values we want to compute elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

Kernel used to compute the covariance matrix of individuals GPsat corresponding inputs.

Kernel used to compute the covariance matrix of the mean GPsat corresponding inputs.

A list of parameters for the variational distributionsof the K mean GPs.

Prior value of the mean parameter of the mean GPs (mu_k).Length = 1 or nrow(db).

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A vector of inputs.

As many vector of covariates as desired. We advise to giveexplicit names when using the function.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing the values we want to compute the logL on.Required columns: Input, Output. Additional covariate columns are allowed.

A vector, specifying the mean of the GP at the reference inputs.

A kernel function.

(optional) A matrix, corresponding to covariance parameter ofthe hyper-posterior. Used to compute the hyper-prior distribution of a newindividual in Magma.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing the values we want to compute the logL on.Required columns: Input, Output. Additional covariate columns are allowed.

A vector, specifying the mean of the GPs at the reference inputs.

A kernel function.

A matrix, covariance parameter of the hyper-posterior.Used to compute the correction term.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble or data frame containing hyper-parameters for allindividuals.

A tibble containing the values we want to compute the logL on.Required columns: ID, Input, Output. Additional covariate columns areallowed.

A vector, specifying the mean of the GPs at the reference inputs.

A kernel function.

A matrix, covariance parameter of the hyper-posterior.Used to compute the correction term.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing the values we want to compute the elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

A list of the k means of the GPs at union of observed timestamps.

A kernel function

A list of the k posterior covariance of the mean GP (mu_k).Used to compute correction term (cor_term)

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing the values we want to compute the elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

List of parameters for the K mean Gaussian processes.

A kernel function.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame or name vector of hyper-parameters.

A tibble containing values we want to compute elbo on.Required columns: Input, Output. Additional covariate columns are allowed.

List of parameters for the K mean Gaussian processes.

A kernel function used to compute the covariance matrix atcorresponding timestamps.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

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

A tibble containing data we want to evaluate the logL on.Required columns: Input, Output. Additional covariate columns are allowed.

A tibble or data frame, indicating the mixture probabilitiesof each cluster for the new individual/task.

A list of hyper-posterior mean parameters for all clusters.

A kernel function.

A list of hyper-posterior covariance parameters for allclusters.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A function, or a character string indicating the chosen type ofkernel among:

• "SE": the Squared Exponential 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

In case of a custom kernel function, the argumentlist_hp has to beprovided as well, for designing a tibble with the correct names ofhyper-parameters.

A vector, associating anID value with each individualfor whom hyper-parameters are generated. If NULL (default) only one set ofhyper-parameters is return without theID column.

A vector of characters, providing the name of eachhyper-parameter, in case wherekern is a custom kernel function.

A logical value, indicating whether a 'noise' hyper-parametershould be included.

A logical value, indicating whether the set ofhyper-parameters is assumed to be common to all individuals.

A list, containing the information coming from aMagma model, previously trained using thetrain_magmafunction. Iftrained_model is not provided, the argumentsdata,hp_0,hp_i,kern_0, andkern_iare all required.

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 asreference for the observations (e.g. time for longitudinal data). The'Output' column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach reference 'Input'. Recovered fromtrained_model if notprovided.

A named vector, tibble or data frame of hyper-parametersassociated withkern_0. Recovered fromtrained_model if notprovided.

A tibble or data frame of hyper-parametersassociated withkern_i. Recovered fromtrained_model if notprovided.

A kernel function, associated with the mean GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not). Recovered fromtrained_model if not provided.

A kernel function, associated with the individual GPs. ("SE","PERIO" and "RQ" are aso available here). Recovered fromtrained_model if not provided.

Hyper-prior mean parameter of the mean GP. This argument,can be specified under various formats, such as:

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

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

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

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

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

A vector or a data frame, indicating the grid ofadditional reference inputs on which the mean process' hyper-posteriorshould be evaluated.

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

A list, containing the information coming from aMagma model, previously trained using thetrain_magmafunction. Iftrained_model is not provided, the argumentsdata,mixture,hp_k,hp_i,kern_k, andkern_i are all required.

A tibble or data frame. Required columns:ID,Input,Output. Additional columns for covariates can be specified.TheID column contains the unique names/codes used to identify eachindividual/task (or batch of data).TheInput column should define the variable that is used asreference for the observations (e.g. time for longitudinal data). TheOutput column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach referenceInput. Recovered fromtrained_model if notprovided.

A tibble or data frame, indicating the mixture probabilitiesof each cluster for each individual. Required column:ID.Recovered fromtrained_model if notprovided.

A tibble or data frame of hyper-parametersassociated withkern_k. Recovered fromtrained_model if notprovided.

A tibble or data frame of hyper-parametersassociated withkern_i. Recovered fromtrained_model if notprovided.

A kernel function, associated with the mean GPs.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not). Recovered fromtrained_model if not provided.

A kernel function, associated with the individual GPs. ("SE","LIN", PERIO" and "RQ" are also available here). Recovered fromtrained_model if not provided.

The set of hyper-prior mean parameters (m_k) for the Kmean GPs, one value for each cluster.cluster. This argument can be specified under various formats, such as:

• NULL (default). All hyper-prior means would be set to 0 everywhere.

• A numerical vector of the same length as the number of clusters.Each number is associated with one cluster, and consideredto be the hyper-prior mean parameter of the cluster (i.e. a constantfunction at allInput).

• A list of functions. Each function is associated with one cluster. Thesefunctions are all evaluated at allInput values, to providespecific hyper-prior mean vectors for each cluster.

A vector or a data frame, indicating the grid ofadditional reference inputs on which the mean process' hyper-posteriorshould be evaluated.

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

A tibble containing common Input and associated Output valuesto cluster.

A number of clusters assumed for running the kmeans algorithm.

A number, indicating how many re-starts of kmeans are set.

A boolean, indicating whether we want an outcome summary

A tibble or data frame. Required columns:ID,Input,Output.

A number, indicating the number of clusters.

A vector of characters. Each element should correspond tothe name of one cluster.

A number of restart used in the underlying kmeans algorithm

A vector, matrix, data frame or tibble containing all inputs forone individual. If a vector, the elements are used as reference, otherwise, one column should be named 'Input' to indicate that it represents thereference (e.g. 'Input' would contain the timestamps in time-seriesapplications). The other columns are considered as being covariates. Ifno column is named 'Input', the first one is used by default.

A kernel function. Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A list, data frame or tibble containing the hyper-parameters usedin the kernel. The name of the elements (or columns) should correspondexactly to those used in the kernel definition. Ifhp contains anelement or a column 'Noise', its value will be added on the diagonal ofthe covariance matrix.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simplyreturns the covariance matrix.

(optional) A vector, matrix, data frame or tibble under thesame format asinput. This argument should be used only when thekernel needs to be evaluated between two different sets of inputs,typically resulting in a non-square matrix.

A vector, matrix, data frame or tibble containing all inputs forone individual. If a vector, the elements are used as reference, otherwise,one column should be named 'Input' to indicate that it represents thereference (e.g. 'Input' would contain the timestamps in time-seriesapplications). The other columns are considered as being covariates. Ifno column is named 'Input', the first one is used by default.

A kernel function. Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A list, data frame or tibble containing the hyper-parameters usedin the kernel. The name of the elements (or columns) should correspondexactly to those used in the kernel definition.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simply returnsthe inverse covariance matrix.

A vector (or matrix if vectorized = T) of inputs.

A vector (or matrix if vectorized = T) of inputs.

A tibble, data frame or named vector, containing the kernel'shyperparameters. Required columns: 'lin_slope' and 'lin_offset'.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simplyreturns the evaluation of the kernel.

A logical value, indicating whether the function providesa vectorized version for speeded-up calculations. If TRUE, thexandy arguments should be the vector or matrix containing allinputs for which the kernel is evaluated on all pairs of elements.If FALSE, thex andy arguments are simply two inputs.

A tibble or data frame of input data. Required column: 'ID'.Suggested column: 'Input' (for indicating the reference input).

A kernel function.

A tibble or data frame, containing the hyper-parameters associatedwith each individual.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simply returnsthe list of covariance matrices.

A tibble or data frame of input data. Required column: 'ID'.Suggested column: 'Input' (for indicating the reference input).

A kernel function.

A tibble or data frame, containing the hyper-parameters associatedwith each individual.

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

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simply returnsthe list of covariance matrices.

A tibble, data frame or named vector containing hyper-parameters.

A tibble containing the values we want to compute the logL on.Required columns: Input, Output. Additional covariate columns are allowed.

A vector, specifying the mean of the GP at the reference inputs.

A kernel function.

(optional) A matrix, corresponding to covariance parameter ofthe hyper-posterior. Used to compute the hyper-prior distribution of a newindividual in Magma.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

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

A tibble containing values we want to compute logL on.Required columns: Input, Output. Additional covariate columns are allowed.

A vector, specifying the mean of the GP at the reference inputs.

A kernel function.

A matrix, covariance parameter of the hyper-posterior.Used to compute the correction term.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble, data frame of hyper-parameters.

A tibble containing the values we want to compute the logL on.Required columns: ID, Input, Output. Additional covariate columns areallowed.

A vector, specifying the mean of the GP at the reference inputs.

A kernel function.

A matrix, covariance parameter of the hyper-posterior.Used to compute the correction term.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A named vector, tibble or data frame, containing thehyper-parameters associated with the mean GP.

A tibble or data frame, containing the hyper-parameters with theindividual GPs.

A tibble or data frame. Columns required: ID, Input, Output.Additional columns for covariates can be specified.

A vector, corresponding to the prior mean of the mean GP.

A kernel function, associated with the mean GP.

A kernel function, associated with the individual GPs.

A tibble, coming out of the E step, containing the Input andassociated Output of the hyper-posterior mean parameter.

A matrix, coming out of the E step, being the hyper-posteriorcovariance parameter.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

A tibble or data frame. Columns required: ID, Input, Output.Additional columns for covariates can be specified.

A vector, corresponding to the prior mean of the mean GP.

A kernel function, associated with the mean GP.

A kernel function, associated with the individual GPs.

A named vector, tibble or data frame, containing thehyper-parameters from the previous M-step (or initialisation) associatedwith the mean GP.

A tibble or data frame, containing the hyper-parametersfrom the previous M-step (or initialisation) associated with theindividual GPs.

A tibble, coming out of the E step, containing the Input andassociated Output of the hyper-posterior mean parameter.

A matrix, coming out of the E step, being the hyper-posteriorcovariance parameter.

A logical value, indicating whether the set ofhyper-parameters is assumed to be common to all indiviuals.

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

A vector (or matrix if vectorized = T) of inputs.

A vector (or matrix if vectorized = T) of inputs.

A tibble, data frame or named vector, containing the kernel'shyperparameters. Required columns: 'perio_variance', 'perio_lengthscale',and 'period'.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simply returnsthe evaluation of the kernel.

A logical value, indicating whether the function providesa vectorized version for speeded-up calculations. If TRUE, thexandy arguments should be the vector or matrix containing allinputs for which the kernel is evaluated on all pairs of elements.If FALSE, thex andy arguments are simply two inputs.

A data frame or tibble with format : ID, Input, Output.

A boolean indicating whether data should be coloured bycluster. Requires a column named 'Cluster'.

A boolean indicating whether the legend should be displayed.

A tibble, typically coming from thepred_giffunction. Required columns: 'Input', 'Mean', 'Var' and 'Index'.

A vector of character strings, indicating which input shouldbe displayed. If NULL(default) the 'Input' column is used for the x-axis.If providing a 2-dimensional vector, the corresponding columns are usedfor the x-axis and y-axis.

(Optional) 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 asreference for the observations (e.g. time for longitudinal data). The'Output' column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach reference 'Input'.

(Optional) A tibble or data frame, containing the trainingdata of the Magma model. The data set should have the same format as thedata argument with an additional column 'ID' for identifying thedifferent individuals/tasks. If provided, those data are displayed asbackward colourful points (each colour corresponding to oneindividual/task).

(Optional) A tibble or a data frame, containing the 'Input'and associated 'Output' prior mean parameter of the GP prediction.

A vector, indicating the grid of values on the y-axis for whichprobabilities should be computed for heatmaps of 1-dimensionalpredictions. If NULL (default), a vector of length 50 is defined, rangingbetween the min and max 'Output' values contained inpred_gp.

A logical value indicating whether the GP prediction should berepresented as a heatmap of probabilities for 1-dimensional inputs. IfFALSE (default), the mean curve and associated 95% CI are displayed.

A number between 0 and 1 (default is 0.95), indicating thelevel of the Credible Interval associated with the posterior mean curve.

A number, controlling the size of thedata points.

A number, controlling the size of thedata_train points.

A number, between 0 and 1, controlling transparencyof thedata_train points.

A logical value indicating whether the animation shouldbe exported as a .gif file.

A character string defining the path where the GIF file should beexported.

Any additional parameters that can be passed to the functiontransition_states from thegganimatepackage.

A tibble or data frame, typically coming frompred_magma orpred_gp functions. Requiredcolumns: 'Input', 'Mean', 'Var'. Additional covariate columns may bepresent in case of multi-dimensional inputs.

A vector of character strings, indicating which input shouldbe displayed. If NULL (default) the 'Input' column is used for the x-axis.If providing a 2-dimensional vector, the corresponding columns are usedfor the x-axis and y-axis.

(Optional) A tibble or data frame. Required columns: 'Input','Output'. Additional columns for covariates can be specified. Thisargument corresponds to the raw data on which the prediction has beenperformed.

(Optional) A tibble or data frame, containing the trainingdata of the Magma model. The data set should have the same format as thedata argument with an additional required column 'ID' foridentifying the different individuals/tasks. If provided, those data aredisplayed as backward colourful points (each colour corresponding to oneindividual/task).

(Optional) A tibble or a data frame, containing the 'Input'and associated 'Output' prior mean parameter of the GP prediction.

A vector, indicating the grid of values on the y-axis for whichprobabilities should be computed for heatmaps of 1-dimensionalpredictions. If NULL (default), a vector of length 50 is defined, rangingbetween the min and max 'Output' values contained inpred_gp.

A logical value indicating whether the GP prediction should berepresented as a heatmap of probabilities for 1-dimensional inputs. IfFALSE (default), the mean curve and associated Credible Interval aredisplayed.

A logical value indicating whether the GP prediction should berepresented as a collection of samples drawn from the posterior. IfFALSE (default), the mean curve and associated Credible Interval aredisplayed.

A number, indicating the number of samples to be drawn fromthe predictive posterior distribution. For two-dimensional graphs, onlyone sample can be displayed.

A logical value, indicating whether the mean predictionshould be displayed on the graph whensamples = TRUE.

A number, controlling transparency of the sample curves.

A number between 0 and 1 (default is 0.95), indicating thelevel of the Credible Interval associated with the posterior mean curve.If this this argument is set to 1, the Credible Interval is not displayed.

A number, controlling the size of thedata points.

A number, controlling the size of thedata_train points.

A number, between 0 and 1, controlling transparencyof thedata_train points.

A list of predictions, typically coming frompred_magmaclust. Required elements:pred,mixture,mixture_pred.

A character string, indicating which cluster to plot from.If 'all' (default) the mixture of GPs prediction is displayed as a meancurve (1-D inputs) or a mean heatmap (2-D inputs). Alternatively, if thename of one cluster is provided, the classic mean curve + credibleinterval is displayed (1-D inputs), or a heatmap with colour gradient forthe mean and transparency gradient for the Credible Interval (2-D inputs).

A vector of character strings, indicating which input shouldbe displayed. If NULL (default) the 'Input' column is used for the x-axis.If providing a 2-dimensional vector, the corresponding columns are usedfor the x-axis and y-axis.

(Optional) A tibble or data frame. Required columns:Input,Output. Additional columns for covariates can be specified. Thisargument corresponds to the raw data on which the prediction has beenperformed.

(Optional) A tibble or data frame, containing the trainingdata of the MagmaClust model. The data set should have the same format asthedata argument with an additional required columnID foridentifying the different individuals/tasks. If provided, those data aredisplayed as backward colourful points (each colour corresponding to oneindividual or a cluster, seecol_clust below).

A boolean indicating whether backward points are colouredaccording to the individuals or to their most probable cluster. If onewants to colour by clusters, a columnCluster shall be present indata_train. We advise to usedata_allocate_clusterfor automatically creating a well-formatted dataset from a trainedMagmaClust model.

(Optional) A list providing, for each cluster, atibble containing prior mean parameters of the prediction. This argumenttypically comes as an outcomehyperpost$mean, available throughthetrain_magmaclust,pred_magmaclustfunctions.

A vector, indicating the grid of values on the y-axis for whichprobabilities should be computed for heatmaps of 1-dimensionalpredictions. If NULL (default), a vector of length 50 is defined, rangingbetween the min and max 'Output' values contained inpred.

A logical value indicating whether the GP mixture should berepresented as a heatmap of probabilities for 1-dimensional inputs. IfFALSE (default), the mean curve (and associated Credible Interval ifavailable) are displayed.

A logical value indicating whether the GP mixture should berepresented as a collection of samples drawn from the posterior. IfFALSE (default), the mean curve (and associated Credible Interval ifavailable) are displayed.

A number, indicating the number of samples to be drawn fromthe predictive posterior distribution. For two-dimensional graphs, onlyone sample can be displayed.

A logical value, indicating whether the mean predictionshould be displayed on the graph whensamples = TRUE.

A number, controlling transparency of the sample curves.

A number between 0 and 1 (default is 0.95), indicating thelevel of the Credible Interval associated with the posterior mean curve.If this this argument is set to 1, the Credible Interval is not displayed.

A number, controlling the size of thedata points.

A number, controlling the size of thedata_train points.

A number, between 0 and 1, controlling transparencyof thedata_train points.

A list, typically coming frompred_gp,pred_magma orpred_magmaclust functions, usingthe argument 'get_full_cov = TRUE'. Required elements:pred,cov. This argument is needed ifsamples is missing.

A tibble or data frame, containing the samples generated froma GP, Magma, or MagmaClust prediction. Required columns:Input,Sample,Output. This argument is needed ifpredis missing.

A number, indicating the number of samples to be drawn fromthe predictive posterior distribution. For two-dimensional graphs, onlyone sample can be displayed.

A vector of character strings, indicating which 'column'should be displayed in the case of multidimensional inputs. IfNULL(default) the Input' column is used for the x-axis. If providing a2-dimensional vector, the corresponding columns are used for the x-axisand the y-axis.

A logical value, indicating whether the mean predictionshould be displayed on the graph.

A number, controlling transparency of the sample curves.

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 asreference for the observations (e.g. time for longitudinal data). The'Output' column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach reference 'Input'.

A list, containing the information coming from aMagma model, previously trained using thetrain_magmafunction.

The grid of inputs (reference Input and covariates) valueson which the GP should be evaluated. Ideally, this argument should be atibble or a data frame, providing the same columns asdata, except'Output'. Nonetheless, in cases wheredata provides only one'Input' column, thegrid_inputs argument can be NULL (default) or avector. This vector would be used as reference input for prediction and ifNULL, a vector of length 500 is defined, ranging between the min and maxInput values ofdata.

A list, containing the elements 'mean' and 'cov', theparameters of the hyper-posterior distribution of the mean process.Typically, this argument should from a previous learning usingtrain_magma, or a previous prediction withpred_magma, with the argumentget_hyperpost set toTRUE. The 'mean' element should be a data frame with two columns 'Input'and 'Output'. The 'cov' element should be a covariance matrix withcolnames and rownames corresponding to the 'Input' in 'mean'. In allcases, the column 'Input' should contain all the values appearing both inthe 'Input' column ofdata and ingrid_inputs.

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

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

• A number. The mean would be a constant function.

• A function. This function is defined as the mean.

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

A named vector, tibble or data frame of hyper-parametersassociated withkern. The columns/elements should be namedaccording to the hyper-parameters that are used inkern. Thefunctiontrain_gp can be used to learn maximum-likelihoodestimators of the hyper-parameters,

A kernel function, defining the covariance structure of the GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

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

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 asreference for the observations (e.g. time for longitudinal data). The'Output' column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach reference 'Input'. If NULL, the prior GP is returned.

The grid of inputs (reference Input and covariates) valueson which the GP should be evaluated. Ideally, this argument should be atibble or a data frame, providing the same columns asdata, except'Output'. Nonetheless, in cases wheredata provides only one'Input' column, thegrid_inputs argument can be NULL (default) or avector. This vector would be used as reference input for prediction and ifNULL, a vector of length 500 is defined, ranging between the min and maxInput values ofdata.

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

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

• A number. The mean would be a constant function.

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

A named vector, tibble or data frame of hyper-parametersassociated withkern. The columns/elements should be namedaccording to the hyper-parameters that are used inkern. If NULL(default), the functiontrain_gp is called with randominitial values for learning maximum-likelihood estimators of thehyper-parameters associated withkern.

A kernel function, defining the covariance structure of the GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A logical value, indicating whether the full posteriorcovariance matrix should be returned.

A logical value, indicating whether a plot of the results isautomatically displayed.

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

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 asreference for the observations (e.g. time for longitudinal data). The'Output' column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach reference 'Input'. If NULL, the mean process fromtrained_model is returned as a generic prediction.

A list, containing the information coming from aMagma model, previously trained using thetrain_magmafunction.

The grid of inputs (reference Input and covariates) valueson which the GP should be evaluated. Ideally, this argument should be atibble or a data frame, providing the same columns asdata, except'Output'. Nonetheless, in cases wheredata provides only one'Input' column, thegrid_inputs argument can be NULL (default) or avector. This vector would be used as reference input for prediction and ifNULL, a vector of length 500 is defined, ranging between the min and maxInput values ofdata.

A named vector, tibble or data frame of hyper-parametersassociated withkern. The columns/elements should be namedaccording to the hyper-parameters that are used inkern. Thefunctiontrain_gp can be used to learn maximum-likelihoodestimators of the hyper-parameters.

A kernel function, defining the covariance structure of the GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A list, containing the elements 'mean' and 'cov', theparameters of the hyper-posterior distribution of the mean process.Typically, this argument should come from a previous learning usingtrain_magma, or a previous prediction withpred_magma, with the argumentget_hyperpost set toTRUE. The 'mean' element should be a data frame with two columns 'Input'and 'Output'. The 'cov' element should be a covariance matrix withcolnames and rownames corresponding to the 'Input' in 'mean'. In allcases, the column 'Input' should contain all the values appearing both inthe 'Input' column ofdata and ingrid_inputs.

A logical value, indicating whether the hyper-posteriordistribution of the mean process should be returned. This can be usefulwhen planning to perform several predictions on the same grid of inputs,since recomputation of the hyper-posterior can be prohibitive for highdimensional grids.

A logical value, indicating whether the full posteriorcovariance matrix should be returned.

A logical value, indicating whether a plot of the results isautomatically displayed.

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

A tibble or data frame. Required columns:Input,Output. Additional columns for covariates can be specified.TheInput column should define the variable that is used asreference for the observations (e.g. time for longitudinal data). TheOutput column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach reference 'Input'. If NULL, the mixture of mean processes fromtrained_model is returned as a generic prediction.

A list, containing the information coming from aMagmaClust model, previously trained using thetrain_magmaclust function. Iftrained_model is set toNULL, thehyperpost andprop_mixture arguments are mandatoryto perform required re-computations for the prediction to succeed.

The grid of inputs (reference Input and covariates) valueson which the GP should be evaluated. Ideally, this argument should be atibble or a data frame, providing the same columns asdata, except'Output'. Nonetheless, in cases wheredata provides only one'Input' column, thegrid_inputs argument can be NULL (default) or avector. This vector would be used as reference input for prediction and ifNULL, a vector of length 500 is defined, ranging between the min and maxInput values ofdata.

A tibble or data frame, indicating the mixture probabilitiesof each cluster for the new individual/task.If NULL, thetrain_gp_clust function is used to computethese posterior probabilities according todata.

A named vector, tibble or data frame of hyper-parametersassociated withkern. The columns/elements should be namedaccording to the hyper-parameters that are used inkern. Thetrain_gp_clust function can be used to learnmaximum-likelihood estimators of the hyper-parameters.

A kernel function, defining the covariance structure of the GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A list, containing the elementsmean,cov andmixture the parameters of the hyper-posterior distributions of themean processes. Typically, this argument should come from a previouslearning usingtrain_magmaclust, or a previous predictionwithpred_magmaclust, with the argumentget_hyperpostset to TRUE.

A tibble or a named vector of the mixture proportions.Each name of column or element should refer to a cluster. The valueassociated with each cluster is a number between 0 and 1. If bothmixture andtrained_model are set to NULL, this argumentallows to recompute mixture probabilities, thanks to thehyperpostargument and thetrain_gp_clust function.

A logical value, indicating whether the hyper-posteriordistributions of the mean processes should be returned. This can be usefulwhen planning to perform several predictions on the same grid of inputs,since recomputation of the hyper-posterior can be prohibitive for highdimensional grids.

A logical value, indicating whether the full posteriorcovariance matrices should be returned.

A logical value, indicating whether a plot of the results isautomatically displayed.

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

A tibble or data frame containing mixture probabilities.

A tibble or data frame. Required columns:ID,InputOutput. TheID column contains the unique names/codes usedto identify each individual/task (or batch of data). TheInputcolumn corresponds to observed locations (an explanatory variable).TheOutput column specifies the associated observed values (theresponse variable). The data frame can also provide as many additionalinputs as desired, with no constraints on the column names.

An integer, which indicates the number of equispaced pointseach column must contain. Each original input value will be collapsed tothe closest point of the new regular grid, and the associated outputs areaveraged using the 'summarise_fct' function. This argument is used when'grid_inputs' is left to 'NULL'. Default value is 30.

A data frame, corresponding to a pre-defined grid ofinputs according to which we want to regularise a dataset. Column namesmust be similar to those appearing indata. IfNULL (default), a default grid of inputs is defined: for eachinput column indata, a regular sequence is created from the minto the max values, with a number of equispaced points being equal to the'size_grid' argument.

A character string or a function. If several similarinputs are associated with different outputs, the user can choose thesummarising function for the output among the following: min, max, mean,median. A custom function can be defined if necessary. Default is "mean".

A vector (or matrix if vectorized = T) of inputs.

A vector (or matrix if vectorized = T) of inputs.

A tibble, data frame or named vector, containing the kernel'shyperparameters. Required columns: 'rq_variance', 'rq_lengthscale', and'rq_scale'.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simply returnsthe evaluation of the kernel.

A logical value, indicating whether the function providesa vectorized version for speeded-up calculations. If TRUE, thexandy arguments should be the vector or matrix containing allinputs for which the kernel is evaluated on all pairs of elements.If FALSE, thex andy arguments are simply two inputs.

A list, typically coming frompred_magma orpred_gp functions, with argument'get_full_cov = TRUE'. Required elements:pred,cov.

A number, indicating the number of samples to be drawn fromthe predictive posterior distribution. For two-dimensional graphs, onlyone sample can be displayed.

A list, typically coming frompred_magmaclust, with argument get_full_cov = TRUE'.Required elements:pred,cov,mixture.

A number, indicating the number of samples to be drawn fromthe predictive posterior distribution. For two-dimensional graphs, onlyone sample can be displayed.

A vector (or matrix if vectorized = T) of inputs.

A vector (or matrix if vectorized = T) of inputs.

A tibble, data frame or named vector, containing the kernel'shyperparameters. Required columns: 'se_variance', 'se_lengthscale'.

A character, indicating according to which hyper-parameter thederivative should be computed. If NULL (default), the function simplyreturns the evaluation of the kernel.

A logical value, indicating whether the function providesa vectorized version for speeded-up calculations. If TRUE, thexandy arguments should be the vector or matrix containing allinputs for which the kernel is evaluated on all pairs of elements.If FALSE, thex andy arguments are simply two inputs.

A tibble or data frame. Columns required:ID,Input,Output. Additional columns for covariates can be specified.TheID column contains the unique names/codes used to identify eachindividual/task (or batch of data).TheInput column should define the variable that is used asreference for the observations (e.g. time for longitudinal data). TheOutput column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach referenceInput.

A boolean, indicating whether a fast approximation shouldbe used for selecting the number of clusters. If TRUE, each Magma orMagmaClust model will perform only one E-step of the training, usingthe same fixed values for the hyper-parameters (ini_hp_k andini_hp_i, or random values if not provided) in all models. Theresulting models should not be considered as trained, but this approachprovides an convenient heuristic to avoid a cumbersome model selectionprocedure.

A vector of integer, corresponding to grid of valuesthat will be tested for the number of clusters.

A tibble or data frame of hyper-parameters associated withkern_k. Thehp function can be used to drawcustom hyper-parameters with the correct format.

A tibble or data frame of hyper-parameters associated withkern_i. Thehp function can be used to drawcustom hyper-parameters with the correct format.db

A kernel function associated to the mean processes.

A kernel function associated to the individuals/tasks.

A boolean indicating whether the plot of V-BIC values for allnumbers of clusters should displayed.

Any additional argument that could be passed totrain_magmaclust.

An integer. The number of individual per cluster.

An integer. The number of observations per individual.

An integer. The number of underlying clusters.

A logical value indicating whether the dataset shouldinclude an additional input covariate named 'Covariate'.

A vector of numbers defining a grid of observations(i.e. the reference inputs).

A vector of numbers defining a grid of observations(i.e. the covariate reference inputs).

A logical value indicating whether the reference inputsare common to all individual.

A logical value indicating whether the hyper-parameters arecommon to all individual. If TRUE and K>1, the hyper-parameters remaindifferent between the clusters.

A logical value indicating whether the values ofhyper-parameters should be added as columns in the dataset.

A logical value indicating whether the name of theclusters should be added as a column in the dataset.

A vector of 2 numbers, defining an interval of admissiblevalues for the variance hyper-parameter of the mean process' kernel.

A vector of 2 numbers, defining an interval of admissiblevalues for the lengthscale hyper-parameter of the mean process' kernel.

A vector of 2 numbers, defining an interval of admissiblevalues for the variance hyper-parameter of the individual process' kernel.

A vector of 2 numbers, defining an interval of admissiblevalues for the lengthscale hyper-parameter of the individual process'kernel.

A vector of 2 numbers, defining an interval of admissiblevalues for the noise hyper-parameter.

A vector of 2 numbers, defining an interval of admissiblevalues for the lambda parameter of the 2D exponential.

A vector of 2 numbers, defining an interval of admissiblevalues for the mean of the 2D exponential.

A vector of 2 numbers, defining an interval ofadmissible values for the lengthscale parameter of the 2D exponential.

A vector of 2 numbers, defining an interval of admissiblevalues for the slope of m0.

A vector of 2 numbers, defining an interval of admissiblevalues for the intercept of m0.

An identification code, whether numeric or character.

A vector of numbers. The input variable that is used as'reference' for input and outputs.

A vector of numbers. Prior mean values of the GP.

A number. The variance hyper-parameter of the SE kernel.

A number. The lengthscale hyper-parameter of the SE kernel.

A number. The noise hyper-parameter.

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

A tibble containing data we want to evaluate the logL on.Required columns: Input, Output. Additional covariate columns are allowed.

A tibble or data frame, indicating the mixture probabilitiesof each cluster for the new individual/task.

A list of hyper-posterior mean parameters for all clusters.

A kernel function.

A list of hyper-posterior covariance parameters for allclusters.

A tibble or a named vector. Each name of column orelement should refer to a cluster. The value associated with each clusteris a number between 0 and 1, corresponding to the mixtureproportions.

A jitter term that is added to the covariance matrix to avoidnumerical issues when inverting, in cases of nearly singular matrices.

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

Mean parameter of the GP. This argument can bespecified 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 thedata argument. This vector would be considered as the evaluationof 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 Inputvalues should include at least the same values as in thedataargument.

A named vector, tibble or data frame of hyper-parametersassociated with thekern of the new individual/task.The columns should be named according to the hyper-parameters that areused inkern. 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. Thehpfunction can be used to draw custom hyper-parameters with the correctformat.

A kernel function, defining the covariance structure of the GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As the²elements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

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 usingtrain_magma, or from thehyperposteriorfunction. Ifhyperpost is provided, the likelihood that ismaximised is the one involved during Magma's prediction step, and theprior_mean argument is ignored. For classic GP training, leavehyperpost to NULL.

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

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

A tibble or a named vector. Each name of column orelement should refer to a cluster. The value associated with each clusteris a number between 0 and 1, corresponding to the mixtureproportions.

A tibble or data frame of hyper-parametersassociated withkern, the individual process kernel. Thehp function can be used to draw custom hyper-parameters withthe correct format.

A kernel function, defining the covariance structure of the GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A list, containing the elementsmean,cov andmixture the parameters of the hyper-posterior distributions of themean processes. Typically, this argument should come from a previouslearning usingtrain_magmaclust, or a previous predictionwithpred_magmaclust, with the argumentget_hyperpostset to TRUE.

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

A number, indicating the maximum number of iterations ofthe EM algorithm to proceed while not reaching convergence.

A number, indicating the threshold of the likelihood gainunder which the EM algorithm will stop.

A tibble or data frame. Required columns:ID,Input,Output. Additional columns for covariates can be specified.TheID column contains the unique names/codes used to identify eachindividual/task (or batch of data).TheInput column should define the variable that is used asreference for the observations (e.g. time for longitudinal data). TheOutput column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach referenceInput.

Hyper-prior mean parameter (m_0) of the mean GP. Thisargument can be specified under various formats, such as:

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

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

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

• A function. This function is defined as the hyper_prior mean.

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

A named vector, tibble or data frame of hyper-parametersassociated withkern_0, the mean process' kernel. Thecolumns/elements should be named according to the hyper-parametersthat are used inkern_0. If NULL (default), random values are usedas initialisation. Thehp function can be used to drawcustom hyper-parameters with the correct format.

A tibble or data frame of hyper-parametersassociated withkern_i, the individual processes' kernel.Required column :ID. TheID column contains the uniquenames/codes used to identify each individual/task. The other columnsshould be named according to the hyper-parameters that are used inkern_i. Compared toini_hp_0 should contain an additional'noise' column to initialise the noise hyper-parameter of the model. IfNULL (default), random values are used as initialisation. Thehp function can be used to draw custom hyper-parameterswith the correct format.

A kernel function, associated with the mean GP.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A kernel function, associated with the individual GPs. ("SE","PERIO" and "RQ" are also available here).

A logical value, indicating whether the set ofhyper-parameters is assumed to be common to all individuals.

A vector, indicating the grid of additional referenceinputs on which the mean process' hyper-posterior should be evaluated.

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

A number, indicating the maximum number of iterations ofthe EM algorithm to proceed while not reaching convergence.

A number, indicating the threshold of the likelihood gainunder which the EM algorithm will stop. The convergence condition isdefined as the difference of likelihoods between two consecutive steps,divided by the absolute value of the last one((LL_n - LL_n-1) / |LL_n| ).

A boolean, indicating whether the EM algorithm shouldstop after only one iteration of the E-step. This advanced feature ismainly used to provide a faster approximation of the model selectionprocedure, by preventing any optimisation over the hyper-parameters.

A tibble or data frame. Columns required:ID,Input,Output. Additional columns for covariates can be specified.TheID column contains the unique names/codes used to identify eachindividual/task (or batch of data).TheInput column should define the variable that is used asreference for the observations (e.g. time for longitudinal data). TheOutput column specifies the observed values (the responsevariable). The data frame can also provide as many covariates as desired,with no constraints on the column names. These covariates are additionalinputs (explanatory variables) of the models that are also observed ateach referenceInput.

A number, indicating the number of clusters ofindividuals/tasks that are assumed to exist among the dataset.

The set of hyper-prior mean parameters (m_k) for the Kmean GPs, one value for each cluster.cluster. This argument can be specified under various formats, such as:

• NULL (default). All hyper-prior means would be set to 0 everywhere.

• A numerical vector of the same length as the number of clusters.Each number is associated with one cluster, and consideredto be the hyper-prior mean parameter of the cluster (i.e. a constantfunction at allInput).

• A list of functions. Each function is associated with one cluster. Thesefunctions are all evaluated at allInput values, to providespecific hyper-prior mean vectors for each cluster.

A tibble or data frame of hyper-parametersassociated withkern_k, the mean process' kernel.Required column :ID. TheID column contains the uniquenames/codes used to identify each cluster. The other columnsshould be named according to the hyper-parameters that are used inkern_k. Thehp function can be used to drawcustom hyper-parameters with the correct format.

A tibble or data frame of hyper-parametersassociated withkern_i, the individual processes' kernel.Required column :ID. TheID column contains the uniquenames/codes used to identify each individual/task. The other columnsshould be named according to the hyper-parameters that are used inkern_i. Thehp function can be used to drawcustom hyper-parameters with the correct format.

A kernel function, associated with the mean GPs.Several popular kernels(seeThe KernelCookbook) are already implemented and can be selected within thefollowing list:

• "SE": (default value) the Squared Exponential Kernel (also calledRadial 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 stringwhere elements are separated by whitespaces (e.g. "SE + PERIO"). As theelements are treated sequentially from the left to the right, the productoperator '*' shall always be used before the '+' operators (e.g.'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

A kernel function, associated with the individual GPs. (Seedetails above inkern_k).

Initial values of the probability to belong to eachcluster for each individual (ini_mixture can be used fora k-means initialisation. Used by default if NULL).

A boolean indicating whether hyper-parameters are commonamong the mean GPs.

A boolean indicating whether hyper-parameters are commonamong the individual GPs.

A vector, indicating the grid of additional referenceinputs on which the mean processes' hyper-posteriors should be evaluated.

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

A number, indicating the maximum number of iterations ofthe VEM algorithm to proceed while not reaching convergence.

A number, indicating the threshold of the likelihood gainunder which the VEM algorithm will stop. The convergence condition isdefined as the difference of elbo between two consecutive steps,divided by the absolute value of the last one((ELBO_n - ELBO_{n-1}) / |ELBO_n| ).

A boolean, indicating whether the VEM algorithm shouldstop after only one iteration of the VE-step. This advanced feature ismainly used to provide a faster approximation of the model selectionprocedure, by preventing any optimisation over the hyper-parameters.

A tibble or data frame. Columns required:ID,Input,Output. Additional columns for covariates can bespecified.

A list of the K hyper-posterior mean parameters.

A list of the K hyper-posterior covariance matrices.

A named vector, tibble or data frame of hyper-parametersassociated withkern, the individual process' kernel. Thecolumns/elements should be named according to the hyper-parametersthat are used inkern.

A kernel function, defining the covariance structure ofthe individual GPs.

A tibble containing the hyper-parameters associatedwith each individual, indicating in which cluster it belongs.

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

A tibble or data frame. Columns required: ID, Input, Output.Additional columns for covariates can be specified.

A named list of vectors, corresponding to the prior meanparameters of the K mean GPs.

A kernel function, associated with the K mean GPs.

A kernel function, associated with the M individual GPs.

A named vector, tibble or data frame of hyper-parametersassociated withkern_k.

A named vector, tibble or data frame of hyper-parametersassociated withkern_i.

A list of mixture values from the previous iteration.

A number, indicating the current iteration of the VEM algorithm.

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

A tibble or data frame. Columns required: ID, Input, Output.Additional columns for covariates can be specified.

A named vector, tibble or data frame, containing thehyper-parameters from the previous M-step (or initialisation) associatedwith the mean GPs.

A named vector, tibble or data frame, containing thehyper-parameters from the previous M-step (or initialisation) associatedwith the individual GPs.

List of parameters of the K mean GPs.

A kernel used to compute the covariance matrix of the mean GPat corresponding timestamps.

A kernel used to compute the covariance matrix of individualsGP at corresponding timestamps.

A named list of prior mean parameters for the K mean GPs.Length = 1 or nrow(unique(db$Input))

A boolean indicating whether hp are common amongmean GPs (for each mu_k)

A boolean indicating whether hp are common amongindividual GPs (for each y_i)

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