simulation_model1 function

Convenience function for generating functional data

This is a typical magnitude model in which outliers are shifted from the normal' non-outlying observations. The main model is of the form: [REMOVE_ME]$X_i(t) = \mu t + e_i(t), [REMOVE_ME_2]$ and the contamination model model is of the form: [REMOVE_ME]$X_i(t) = \mu t + qk_i + e_i(t) [REMOVE_ME_2]$

where $t\in [0,1]$, $e_i(t)$ is a Gaussian process with zero mean and covariance function of the form: [REMOVE_ME]$\gamma(s,t) = \alpha \exp(-\beta|t-s|^\nu), [REMOVE_ME_2]$ $k_i \in \{-1, 1\}$

(usually with $P(k_i = -1) = P(k_i=1) = 0.5$), and $q$ is a constant controlling how far the outliers are from the mean function of the data, usually, $q = 6$ or $q = 8$. The domain of the generated functions is over the interval $[0, 1]$. Please see the simulation models vignette with vignette("simulation_models", package = "fdaoutlier") for more details.

simulation_model1(
  n = 100,
  p = 50,
  outlier_rate = 0.05,
  mu = 4,
  q = 8,
  kprob = 0.5,
  cov_alpha = 1,
  cov_beta = 1,
  cov_nu = 1,
  deterministic = TRUE,
  seed = NULL,
  plot = F,
  plot_title = "Simulation Model 1",
  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 in the main and contamination model. Set to 4 by default.

• q: A value indicating the shift of the outliers from the mean function, i.e., the $q$

  in the contamination model. Used to control how far the outliers are from the mean function. Set to 8 by default.

• kprob: A value between $0$ and $1$ indicating the probability that an outlier will be above or below the mean function, i.e., $P(k_i = 1)$ in the contamination model. Can be used to control the amount of outliers above or below the mean. Set to $0.5$ 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

This is a typical magnitude model in which outliers are shifted from the normal' non-outlying observations. The main model is of the form:

and the contamination model model is of the form:

where $t\in [0,1]$, $e_i(t)$ is a Gaussian process with zero mean and covariance function of the form:

$k_i \in \{-1, 1\}$

(usually with $P(k_i = -1) = P(k_i=1) = 0.5$), and $q$ is a constant controlling how far the outliers are from the mean function of the data, usually, $q = 6$ or $q = 8$. The domain of the generated functions is over the interval $[0, 1]$. Please see the simulation models vignette with vignette("simulation_models", package = "fdaoutlier") for more details.

Examples

dt <- simulation_model1(n = 50, plot = TRUE)
dim(dt$data)
dt$true_outliers