total_mixed_c_correlation_2 {otsfeatures}

Computes the total mixed cumulative quantile correlation (TMCQC) between an ordinal and a real-valued time series

Description

total_mixed_c_correlation_2 returns the TMCQC between an ordinal and a real-valued time series

Usage

total_mixed_c_correlation_2(
  o_series,
  n_series,
  lag = 1,
  states,
  features = FALSE
)

Arguments

o_series

An OTS.

n_series

A real-valued time series.

lag

The considered lag (default is 1).

states

A numerical vector containing the corresponding states.

features

Logical. If features = FALSE (default), the value of the TMCLC is returned. Otherwise, the function returns a vector with the individual components of the TMCQC.

Details

Given a OTS of length T with range \mathcal{S}=\{s_0, s_1, \ldots, s_n\}, \overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}, and the cumulative binarized time series, which is defined as \overline{\boldsymbol Y}_t=\{\overline{\boldsymbol Y}_1, \ldots, \overline{\boldsymbol Y}_T\}, with \overline{\boldsymbol Y}_k=(\overline{Y}_{k,0}, \ldots, \overline{Y}_{k,n-1})^\top such that \overline{Y}_{k,i}=1 if \overline{X}_k \leq s_i (k=1,\ldots,T , i=0,\ldots,n-1), the function computes the estimated TMCQC given by

\widehat{\Psi}_2^m(l)=\frac{1}{n}\sum_{i=0}^{n-1}\int_{0}^{1}\widehat{\psi}^\rho_{i}(l)^2d\rho,

where \widehat{\psi}_{i}^\rho(l)=\widehat{Corr}\big(Y_{t,i}, I(Z_{t-l}\leq q_{Z_t}(\rho)) \big), with \overline{Z}_t=\{\overline{Z}_1,\ldots, \overline{Z}_T\} being a T-length real-valued time series, \rho \in (0, 1) a probability level, I(\cdot) the indicator function and q_{Z_t} the quantile function of the corresponding real-valued process. If features = TRUE, the function returns a vector whose components are the quantities \int_{0}^{1}\widehat{\psi}^\rho_{i}(l)^2d\rho, i=0,1, \ldots,n-1.

Value

If features = FALSE (default), returns the value of the TMCQC. Otherwise, the function returns a vector of features, i.e., the vector contains the features employed to compute the TMCLC.

Author(s)

Ángel López-Oriona, José A. Vilar

Examples

tmclc <- total_mixed_c_correlation_2(o_series = SyntheticData1$data[[1]],
n_series = rnorm(600), states = 0 : 5) # Computing the TMCQC
# between the first series in dataset SyntheticData1 and white noise
feature_vector <- total_mixed_c_correlation_2(o_series = SyntheticData1$data[[1]],
n_series = rnorm(600), states = 0 : 5, features = TRUE) # Computing the corresponding
# vector of features