2.1 Quantile Correlation (QC)

Let \(X\) and \(Y\) be two random variables. For a quantile level \(\tau\in(0,1)\), define the influence function:

\[ \phi_\tau(w) = \tau - I(w < 0), \]

where \(I(\cdot)\) is the indicator function. The quantile covariance of \(Y\) and \(X\) at \(\tau\) is:

\[ \mathrm{qcov}_\tau(Y,X) = \mathrm{Cov}\bigl(\phi_\tau(Y - Q_{\tau,Y}),\,X - \mathbb{E}[X]\bigr), \]

with \(Q_{\tau,Y}\) the \(\tau\)th quantile of \(Y\). The Quantile Correlation (QC) is then:

\[ \mathrm{QC}_\tau(X,Y) = \frac{\mathrm{qcov}_\tau(Y,X)}{\sqrt{\mathrm{Var}(\phi_\tau(Y - Q_{\tau,Y}))\,\mathrm{Var}(X)}}. \]

2.2 Maximal Overlap Discrete Wavelet Transform (MODWT)

To capture scale-dependent dynamics, each series is decomposed into \(J\) levels of detail coefficients via MODWT (Percival & Walden, 2000):

1. Filter: Convolve the series with low-pass (\(h_j\)) and high-pass (\(g_j\)) filters to obtain approximation \(a_j\) and detail \(d_j\) coefficients.
2. Upsample: For level \(j+1\), upsample filters by inserting zeros and repeat filtering on \(a_j\).
3. Collect: The resulting vectors \(d_1,\dots,d_J\) isolate fluctuations at dyadic scales (fine to coarse).

2.3 Wavelet Quantile Correlation (WQC)

For each level \(j=1,\dots,J\) and quantile \(\tau\):

1. Decompose series \(X\) and \(Y\) into detail coefficients \(d_j[X]\) and \(d_j[Y]\).
2. Compute \(\mathrm{QC}_\tau(d_j[X],d_j[Y])\) using the formula above.
3. Simulate \(n_{\mathrm{sim}}\) Gaussian surrogates matching the empirical mean and variance of each \(d_j[\cdot]\) to obtain confidence intervals at the 2.5% and 97.5% percentiles.

The function quantile_correlation_analysis() automates these computations, returning a data frame:

3.1 1. Multi-Series Analysis with apply_quantile_correlation()

library(wqca)
set.seed(123)
# Reference series plus two targets
data <- data.frame(
  x = rnorm(512),
  y = 0.6 * rnorm(512) + 0.4 * rnorm(512),
  z = rnorm(512)
)
quantiles <- c(0.1, 0.5, 0.9)
# Compute WQC up to 3 levels with 300 bootstrap sims
res_df <- apply_quantile_correlation(data, quantiles, J = 3, n_sim = 300)
# View top rows
head(res_df)

3.2 2. Single-Pair Analysis with quantile_correlation_analysis()

library(wqca)
set.seed(456)
x <- rnorm(600)
y <- 0.8 * x + 0.2 * rnorm(600)
quantiles <- c(0.05, 0.5, 0.95)
# Analyze one pair at 4 levels with 200 sims
single_res <- quantile_correlation_analysis(x, y, quantiles, J = 4, n_sim = 200)
# Inspect results
single_res