simulation_model5 function

Convenience function for generating functional data

This models generates shape outliers with a different covariance structure from that of the main model. The main model is of the form: [REMOVE_ME]$X_i(t) = \mu t + e_i(t), [REMOVE_ME_2]$ contamination model of the form: [REMOVE_ME]$X_i(t) = \mu t + \tilde{e}_i(t), [REMOVE_ME_2]$ where $t\in [0,1]$, and $e_i(t)$

and $\tilde{e}_i(t)$ are Gaussian processes with zero mean and covariance function of the form: [REMOVE_ME]$\gamma(s,t) = \alpha\exp(-\beta|t-s|^\nu) [REMOVE_ME_2]$

Please see the simulation models vignette with vignette("simulation_models", package = "fdaoutlier") for more details.

simulation_model5(
  n = 100,
  p = 50,
  outlier_rate = 0.05,
  mu = 4,
  cov_alpha = 1,
  cov_beta = 1,
  cov_nu = 1,
  cov_alpha2 = 5,
  cov_beta2 = 2,
  cov_nu2 = 0.5,
  deterministic = TRUE,
  seed = NULL,
  plot = F,
  plot_title = "Simulation Model 5",
  title_cex = 1.5,
  show_legend = T,
  ylabel = "",
  xlabel = "gridpoints"
)

Arguments

• n: The number of curves to generate. Set to $100$ by default.

• p: The number of evaluation points of the curves. Curves are usually generated over the interval $[0, 1]$. Set to $50$ by default.

• outlier_rate: A value between $[0, 1]$ indicating the percentage of outliers. A value of $0.06$ indicates about $6\%$ of the observations will be outliers depending on whether the parameter deterministic is TRUE or not. Set to $0.05$ by default.

• mu: The mean value of the functions. Set to 4 by default.

• cov_alpha, cov_alpha2: A value indicating the coefficient of the exponential function of the covariance matrix, i.e., the $\alpha$ in the covariance function. cov_alpha is for the main model while cov_alpha2 is for the covariance function of the contamination model. cov_alpha is set to $1$ by default while cov_alpha2 is set to $5$ by default.

• cov_beta, cov_beta2: A value indicating the coefficient of the terms inside the exponential function of the covariance matrix, i.e., the $\beta$ in the covariance function. cov_beta

  is for the main model while cov_beta2 is for the covariance function of the contamination model. cov_beta is set to $1$ by default while cov_beta2 is set to $2$ by default.

• cov_nu, cov_nu2: A value indicating the power to which to raise the terms inside the exponential function of the covariance matrix, i.e., the $\nu$ in the covariance function. cov_nu is for the main model while cov_nu2 is for the covariance function of the contamination model. cov_nu is set to $1$ by default while cov_nu2 is set to $0.5$ by default.

• deterministic: A logical value. If TRUE, the function will always return round(n*outlier_rate) outliers and consequently the number of outliers is always constant. If FALSE, the number of outliers are determined using n Bernoulli trials with probability outlier_rate, and consequently the number of outliers returned is random. TRUE by default.

• seed: A seed to set for reproducibility. NULL by default in which case a seed is not set.

• plot: A logical value indicating whether to plot data.

• plot_title: Title of plot if plot is TRUE

• title_cex: Numerical value indicating the size of the plot title relative to the device default. Set to 1.5 by default. Ignored if plot = FALSE.

• show_legend: A logical indicating whether to add legend to plot if plot = TRUE.

• ylabel: The label of the y-axis. Set to "" by default.

• xlabel: The label of the x-axis if plot = TRUE. Set to "gridpoints" by default.

Returns

A list containing: - data: a matrix of size n by p containing the simulated data set

• true_outliers: a vector of integers indicating the row index of the outliers in the generated data.

Description

This models generates shape outliers with a different covariance structure from that of the main model. The main model is of the form:

contamination model of the form:

where $t\in [0,1]$, and $e_i(t)$

and $\tilde{e}_i(t)$ are Gaussian processes with zero mean and covariance function of the form:

Please see the simulation models vignette with vignette("simulation_models", package = "fdaoutlier") for more details.

Examples

dt <- simulation_model5(plot = TRUE)
dt$true_outliers
dim(dt$data)