# generate_gauss_fdata: Generation of gaussian univariate functional data In roahd: Robust Analysis of High Dimensional Data

## Description

`generate_gauss_fdata` generates a dataset of univariate functional data with a desired mean and covariance function.

## Usage

 `1` ```generate_gauss_fdata(N, centerline, Cov = NULL, CholCov = NULL) ```

## Arguments

 `N` the number of distinct functional observations to generate. `centerline` the centerline of the distribution, represented as a one- dimensional data structure of length P containing the measurement of the centerline on grid points. `Cov` the covariance operator (provided in form of a P x P matrix) that has to be used in the generation of ε(t). At least one argument between `Cov` and `CholCov` should be different from `NULL`. `CholCov` the Cholesky factor of the covariance operator (provided in form of a P x P matrix) that has to be used in the generation of observations from the process ε(t). At least one argument between `Cov` and `CholCov` should be different from `NULL`.

## Details

In particular, the following model is considered for the generation of data:

X(t) = m( t ) + ε(t), for all t in I = [a, b]

where m(t) is the center and ε(t) is a centered gaussian process with covariance function C_i. That is to say:

Cov( ε(s), ε(t) ) = C( s, t ), with s, t in I

All the functions are supposed to be observed on an evenly-spaced, one- dimensional grid of P points: [a = t_0, t_1, …, t_{P-1} = b] \subset I .

## Value

The function returns a matrix containing the discretized values of the generated observations (in form of an N x P matrix).

`exp_cov_function`, `fData`, `generate_gauss_mfdata`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```N = 30 P = 1e2 t0 = 0 tP = 1 time_grid = seq( t0, tP, length.out = P ) C = exp_cov_function( time_grid, alpha = 0.1, beta = 0.2 ) CholC = chol( C ) centerline = sin( 2 * pi * time_grid ) generate_gauss_fdata( N, centerline, Cov = C ) generate_gauss_fdata( N, centerline, CholCov = CholC ) ```