total_mixed_c_correlation_2 function

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

total_mixed_c_correlation_2 returns the TMCQC between an ordinal and a real-valued time series UTF-8

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.

Returns

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.

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 $\overline{X}_k \leq s_i$ (c("$k=1,\\ldots,T\n$", "$, i=0,\\ldots,n-1$")), the function computes the estimated TMCQC given by

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 $\int_{0}^{1}\widehat{\psi}^\rho_{i}(l)^2d\rho$, $i=0,1, \ldots,n-1$.

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

Author(s)

Ángel López-Oriona, José A. Vilar