This is a typical magnitude model in which outliers are shifted from the
normal' non-outlying observations. The main model is of the form:
$$X_i(t) = \mu t + e_i(t),$$ and the contamination model model is of the
form: $$X_i(t) = \mu t + qk_i + e_i(t)$$
where \(t\in [0,1]\), \(e_i(t)\) is a Gaussian process with zero mean and
covariance function of the form:
$$\gamma(s,t) = \alpha \exp(-\beta|t-s|^\nu),$$ \(k_i \in \{-1, 1\}\)
(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"
)A list containing:
a matrix of size n by p containing the simulated data set
a vector of integers indicating the row index of the outliers in the generated data.
The number of curves to generate. Set to \(100\) by default.
The number of evaluation points of the curves. Curves are usually generated over the interval \([0, 1]\). Set to \(50\) by default.
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.
The mean value of the functions in the main and contamination model.
Set to 4 by default.
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.
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.
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.
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.
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.
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.
A seed to set for reproducibility. NULL by default in which case a seed
is not set.
A logical value indicating whether to plot data.
Title of plot if plot is TRUE
Numerical value indicating the size of the plot title relative to the device default.
Set to 1.5 by default. Ignored if plot = FALSE.
A logical indicating whether to add legend to plot if plot = TRUE.
The label of the y-axis. Set to "" by default.
The label of the x-axis if plot = TRUE. Set to
"gridpoints" by default.
dt <- simulation_model1(n = 50, plot = TRUE)
dim(dt$data)
dt$true_outliers
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