Simulation Models

Model 1

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) = μt + e_i(t), and the contamination model model is of the form:

X_i(t) = μt + qk_i + e_i(t) where:

• t ∈ [0, 1],
• e_i(t) is a Gaussian process with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν},
• k_i ∈ {−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.

This model can be accessed with the simulation_model1() function in fdaoutlier.

library(fdaoutlier)
dtss <- simulation_model1(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

The returned object is a list containing a matrix of the data and a vector of the indices of the true outliers:

dim(dtss$data)
#> [1] 100  50
dtss$true_outliers
#>  [1] 11 14 20 43 53 70 79 81 83 96

The simulated data can be tuned using additional parameters to simulation_model1(). The following parameters modify the data generated by simulation_model1():

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• q: the shift parameter q in the contamination model which controls how far the outliers are from the mean function.
• kprob: the probability that k_i = 1, i.e., P(k_i = 1) in the contamination model
• cov_alpha: the coefficient α in the covariance function.
• cov_beta: the coefficient β in the covariance function.
• cov_nu: the coefficient ν in the covariance function.

Additional plotting parameters allows for modifying the plot title (plot_title), the font size of the title (title_cex), toggle on/off the display of the legend (show_legend), y-axis label (ylabel) and x-axis label (xlabel).

Model 2

This model generates non-persistent magnitude outliers, i.e., the outliers are magnitude outliers for only a portion of the domain of the functional data. The main model is of the form: X_i(t) = μt + e_i(t), with contamination model of the form: X_i(t) = μt + qk_iI_{Ti ≤ t ≤ Ti + l} + e_i(t) where:

• t ∈ [0, 1],
• e_i(t) is a Gaussian process with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν},
• k_i ∈ {−1, 1} with P(k_i = −1) = P(k_i = 1) = 0.5,
• q is a constant controlling how far the outliers are from the mass of the data,
• I is an indicator function,
• T_i is a uniform random variable between an interval [a, b] ⊂ [0, 1],
• and l is a constant specifying for how much of the domain the outliers are away from the mean function.

A call to simulation_model2() generates data from this model:

dtss <- simulation_model2(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model3() to which arguments can be passed are:

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• q: the shift parameter q in the contamination model which controls how far the outliers are from the mean function.
• kprob: the probability that k_i = 1, i.e., P(k_i = 1) in the contamination model.
• a, b: values specifying the interval [a, b] from which T_i is drawn in the contamination model.
• l: the value of l in the contamination model.
• cov_alpha: the coefficient α in the covariance function.
• cov_beta: the coefficient β in the covariance function.
• cov_nu: the coefficient ν in the covariance function.

Additional plotting parameters listed for simulation_model1() also applies.

Model 3

This model generates outliers that are magnitude outliers for a part of the domain. The main model is of the form: X_i(t) = μt + e_i(t), with contamination model of the form: X_i(t) = μt + qk_iI_Ti ≤ t + e_i(t) where:

• t ∈ [0, 1],
• e_i(t) is a Gaussian process with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν},
• k_i ∈ {−1, 1} with P(k_i = −1) = P(k_i = 1) = 0.5,
• q is a constant controlling how far the outliers are from the mass of the data,
• I is an indicator function,
• and T_i is a uniform random variable between an interval [a, b] ⊂ [0, 1].

A call to simulation_model3() generates data from this model:

dtss <- simulation_model3(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model3() to which arguments can be passed are:

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• q: the shift parameter q in the contamination model which controls how far the outliers are from the mean function.
• kprob: the probability that k_i = 1, i.e., P(k_i = 1) in the contamination model.
• a, b: values specifying the interval [a, b] from which T_i is drawn in the contamination model.
• cov_alpha: the coefficient α in the covariance function.
• cov_beta: the coefficient β in the covariance function.
• cov_nu: the coefficient ν in the covariance function.

Additional plotting parameters listed for simulation_model1() also applies.

Model 4

This models generates outliers defined on the reversed interval of the main model. The main model is of the form: X_i(t) = μt(1 − t)^m + e_i(t), with contamination model of the form: X_i(t) = μ(1 − t)t^m + e_i(t) where:

• t ∈ [0, 1],
• e_i(t) is a Gaussian process with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν},
• and m is a constant.

A call to simulation_model4() generates data from this model:

dtss <- simulation_model4(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model4() to which arguments can be passed are:

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• m: the constant m in the main and contamination models.
• cov_alpha: the coefficient α in the covariance function.
• cov_beta: the coefficient β in the covariance function.
• cov_nu: the coefficient ν in the covariance function.

Additional plotting parameters listed for simulation_model1() also applies.

Model 5

This models generates shape outliers with a different covariance structure from that of the main model. The main model is of the form: X_i(t) = μt + e_i(t), with contamination model of the form: X_i(t) = μt + ẽ_i(t), where:

• t ∈ [0, 1],
• and e_i(t) and ẽ_i(t) are Gaussian processes with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν}

A call to simulation_model5() generates data from this model:

dtss <- simulation_model5(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model5() to which arguments can be passed are:

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• cov_alpha: the coefficient α in the covariance function of e_i(t).
• cov_beta: the coefficient β in the covariance function of e_i(t).
• cov_nu: the coefficient ν in the covariance function of e_i(t).
• cov_alpha2: the coefficient α in the covariance function of ẽ_i(t).
• cov_beta2: the coefficient β in the covariance function of ẽ_i(t).
• cov_nu2: the coefficient ν in the covariance function of ẽ_i(t).

Additional plotting parameters listed for simulation_model1() also applies.

Model 6

This models generates shape outliers that have a different shape for a portion of the domain. The main model is of the form: X_i(t) = μt + e_i(t), with contamination model of the form: $$X_i(t) = \mu t + (-1)^u\cdot q + (-1)^{(1-u)}\left(\frac{1}{\sqrt{r\pi}}\right)\exp{(-z(t-v)^w)} + e_i(t)$$ where:

• t ∈ [0, 1],
• e_i(t) is a Gaussian process with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν},
• u follows Bernoulli distribution with probability P(u = 1) = 0.5,
• q, r, z and w are constants,
• v follows a Uniform distribution between an interval [a, b].

A call to simulation_model6() generates data from this model:

dtss <- simulation_model6(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model6() to which arguments can be passed are:

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• q: the constant term q in the contamination model.
• kprob: the probability P(u = 1)
• a, b: values specifying the interval of from which v in the contamination model is drawn.
• pi_coeff: the constant r in the contamination model.
• exp_pow: the constant w in the contamination model.
• exp_coeff: the constant z in the contamination model.
• cov_alpha: the coefficient α in the covariance function.
• cov_beta: the coefficient β in the covariance function.
• cov_nu: the coefficient ν in the covariance function.

Additional plotting parameters listed for simulation_model1() also applies.

Model 7

This model generates pure shape outliers that are periodic. The main model is of the form: X_i(t) = μt + e_i(t), with contamination model of the form: X_i(t) = μt + ksin (rπ(t + θ)) + e_i(t), where:

• t ∈ [0, 1],
• and e_i(t) is a Gaussian processes with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν}
• θ is uniformly distributed in an interval [a, b].
• k, r are constants

A call to simulation_model7() generates data from this model:

dtss <- simulation_model7(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model7() to which arguments can be passed are:

• mu: the coefficient μ in the main and contamination models controlling the mean function.
• cov_alpha: the coefficient α in the covariance function of e_i(t).
• cov_beta: the coefficient β in the covariance function of e_i(t).
• cov_nu: the coefficient ν in the covariance function of e_i(t).
• sin_coeff: the coefficient k in the contamination model.
• pi_coeff: the coefficient r in the contamination model.
• a, b: values specifying the interval of from which θ is to be drawn.

Additional plotting parameters listed for simulation_model1() also applies.

Model 8

This model generates pure shape outliers that are periodic. The main model is of the form: X_i(t) = ksin (rπt) + e_i(t), with contamination model of the form: X_i(t) = ksin (rπt + v) + e_i(t), where:

• t ∈ [0, 1],
• and e_i(t) is a Gaussian processes with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν}
• k, r, v are constants

A call to simulation_model8() generates data from this model:

dtss <- simulation_model8(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model7() to which arguments can be passed are:

• cov_alpha: the coefficient α in the covariance function of e_i(t).
• cov_beta: the coefficient β in the covariance function of e_i(t).
• cov_nu: the coefficient ν in the covariance function of e_i(t).
• sin_coeff: the coefficient k in the main and contamination model.
• pi_coeff: the coefficient r in the main and contamination model.
• constant: the value of the constant v in the contamination model.

Additional plotting parameters listed for simulation_model1() also applies.

Model 9

Periodic functions with outliers of different amplitude. The main model is of the form: X_i(t) = a_1isin π + a_2icos π + e_i(t), with contamination model of the form: X_i(t) = (b_1isin π + b_2icos π)(1 − u_i) + (c_1isin π + c_2icos π)u_i + e_i(t), where:

• t ∈ [0, 1],
• π ∈ [0, 2π]
• a_1i, a_2i follows uniform distribution in an interval [a₁, a₂]
• b_1i, b_2i follows uniform distribution in an interval [b₁, b₂]
• c_1i, c_2i follows uniform distribution in an interval [c₁, c₂]
• u_i follows Bernoulli distribution
• and e_i(t) is a Gaussian processes with zero mean and covariance function of the form: γ(s, t) = αexp {−β|t − s|^ν}

A call to simulation_model9() generates data from this model:

dtss <- simulation_model9(n = 100, p = 50, outlier_rate = .1,
                          seed = 50, plot = F)

Additional parameters of simulation_model9() to which arguments can be passed are:

• kprob the probability P(u_i = 1)
• ai a vector of 2 values containing a₁ and a₂ indicating the interval from which a_1i and a_2i are drawn in the main model.
  
• bi a vector of 2 values containing b₁ and b₂ indicating the interval from which a_1i and a_2i are drawn in the main model.
• ci a vector of 2 values containing c₁ and c₂ indicating the interval from which c_1i and c_2i are drawn in the main model.
• cov_alpha: the coefficient α in the covariance function of e_i(t).
• cov_beta: the coefficient β in the covariance function of e_i(t).
• cov_nu: the coefficient ν in the covariance function of e_i(t).

Additional plotting parameters listed for simulation_model1() also applies.