simulation_model9 function

Convenience function for generating functional data

Periodic functions with outliers of different amplitude. The main model is of the form [REMOVE_ME]$X_i(t) = a_{1i}\sin \pi + a_{2i}\cos\pi + e_i(t), [REMOVE_ME_2]$

with contamination model of the form [REMOVE_ME]$X_i(t) = (b_{1i}\sin\pi + b_{2i}\cos\pi)(1-u_i) +(c_{1i}\sin\pi + c_{2i}\cos\pi)u_i + e_i(t), [REMOVE_ME_2]$

where $t\in [0,1]$, $\pi \in [0, 2\pi]$, $a_{1i}$, $a_{2i}$ follows uniform distribution in an interval $[a_1, a_2]$

$b_{1i}$, $b_{i1}$ follows uniform distribution in an interval $[b_1, b_2]$; $c_{1i}$, $c_{i1}$ follows uniform distribution in an interval $[c_1, c_2]$; $u_i$ follows Bernoulli distribution and $e_i(t)$ is a 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_model9(
  n = 100,
  p = 50,
  outlier_rate = 0.05,
  kprob = 0.5,
  ai = c(3, 8),
  bi = c(1.5, 2.5),
  ci = c(9, 10.5),
  cov_alpha = 1,
  cov_beta = 1,
  cov_nu = 1,
  deterministic = TRUE,
  seed = NULL,
  plot = F,
  plot_title = "Simulation Model 9",
  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.

• kprob: The probability $P(u_i = 1)$. Set to $0.5$ by default.

• ai: A vector of two values containing $a_{1i}$ and $a_{2i}$

  in the main model. Set to c(3, 8) by default.

• bi: A vector of 2 values containing $b_{1i}$ and $b_{2i}$ in the contamination model. Set to c(1.5, 2.5) by default.

• ci: A vector of 2 values containing $c_1i$ and $c_2i$ in the contamination model. Set to c(9, 10.5) by default.

• cov_alpha: A value indicating the coefficient of the exponential function of the covariance matrix, i.e., the $\alpha$ in the covariance function. Set to $1$ by default.

• cov_beta: A value indicating the coefficient of the terms inside the exponential function of the covariance matrix, i.e., the $\beta$ in the covariance function. Set to $1$ by default.

• cov_nu: 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. Set to $1$ 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

Periodic functions with outliers of different amplitude. The main model is of the form

with contamination model of the form

where $t\in [0,1]$, $\pi \in [0, 2\pi]$, $a_{1i}$, $a_{2i}$ follows uniform distribution in an interval $[a_1, a_2]$

$b_{1i}$, $b_{i1}$ follows uniform distribution in an interval $[b_1, b_2]$; $c_{1i}$, $c_{i1}$ follows uniform distribution in an interval $[c_1, c_2]$; $u_i$ follows Bernoulli distribution and $e_i(t)$ is a 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_model9(plot = TRUE)
dim(dt$data)
dt$true_outliers