dis_qcd: Constructs a pairwise distance matrix based on the quantile...

In mlmts: Machine Learning Algorithms for Multivariate Time Series

Constructs a pairwise distance matrix based on the quantile cross-spectral density (QCD)

Description

dis_qcd returns a pairwise distance matrix based on the dissimilarity introduced by \insertCitelopez2021quantile;textualmlmts.

Usage

dis_qcd(X, levels = c(0.1, 0.5, 0.9), freq = NULL, features = FALSE, ...)

Arguments

X	A list of MTS (numerical matrices).
levels	The set of probability levels.
freq	Vector of frequencies in which the smoothed CCR-periodograms must be computed. If freq=NULL (default), the set of Fourier frequencies is considered.
features	Logical. If features = FALSE (default), a distance matrix is returned. Otherwise, the function returns a dataset of feature vectors.
...	Additional parameters for the function. See smoothedPG.

Details

Given a collection of MTS, the function returns the pairwise distance matrix, where the distance between two MTS \boldsymbol X_T and \boldsymbol Y_T is defined as

d_{QCF}(\boldsymbol X_T, \boldsymbol Y_T)=\Bigg[\sum_{j_1=1}^{d}\sum_{j_2=1}^{d}\sum_{i=1}^{r} \sum_{i'=1}^{r}\sum_{k=1}^{K}\Big(\Re\big({\widehat G_{j_1,j_2}^{\boldsymbol X_T}(\omega_{k}, \tau_{i}, \tau_{i^ {\prime}})}\big) -\Re\big({\widehat G_{j_1,j_2}^{\boldsymbol Y_T}(\omega_{k}, \tau_{i}, \tau_{i^ {\prime}})\big)}\Big)^2+

\sum_{j_1=1}^{d}\sum_{j_2=1}^{d}\sum_{i=1}^{r}\sum_{i'=1}^{r}\sum_{k=1}^{K}\Big(\Im\big({\widehat G_{j_1,j_2} ^{\boldsymbol X_T}(\omega_{k}, \tau_{i}, \tau_{i^ {\prime}})}\big) -\Im\big({\widehat G_{j_1,j_2}^{\boldsymbol Y_T}(\omega_{k}, \tau_{i}, \tau_{i^ {\prime}})\big)}\Big)^2\Bigg]^{1/2},

where {\widehat G_{j_1,j_2}^{\boldsymbol X_T}(\omega_{k}, \tau_{i}, \tau_{i^ {\prime}})} and {\widehat G_{j_1,j_2}^{\boldsymbol Y_T}(\omega_{k}, \tau_{i}, \tau_{i^ {\prime}})} are estimates of the quantile cross-spectral densities (so-called smoothed CCR-periodograms) with respect to the variables j_1 and j_2 and probability levels \tau_i and \tau_{i^\prime} for series \boldsymbol X_T and \boldsymbol Y_T, respectively, and \Re(\cdot) and \Im(\cdot) denote the real part and imaginary part operators, respectively.

Value

If features = FALSE (default), returns a distance matrix based on the distance d_{QCD}. Otherwise, the function returns a dataset of feature vectors, i.e., each row in the dataset contains the features employed to compute the distance d_{QCF}.

Author(s)

Ángel López-Oriona, José A. Vilar

References

\insertRef

lopez2021quantilemlmts

See Also

dis_qcf

Examples

toy_dataset <- AtrialFibrillation$data[1 : 4] # Selecting the first 4 MTS from the
# dataset AtrialFibrillation
distance_matrix <- dis_qcd(toy_dataset) # Computing the pairwise
# distance matrix based on the distance dis_qcd
distance_matrix <- dis_qcd(toy_dataset, levels = c(0.4, 0.8)) # Changing
# the probability levels to compute the QCD-based estimators
distance_matrix <- dis_qcd(toy_dataset, freq = 0.5) # Considering only
# a single frequency for the computation of d_qcd
feature_dataset <- dis_qcd(toy_dataset, features = TRUE) # Computing
# the corresponding dataset of features

mlmts documentation built on Sept. 11, 2024, 6:41 p.m.