TRACES (Time-series Relationship Analysis with Comprehensive Evaluation Suite) 1. Pearson Correlation Coefficient $r$ ) $$
r = \frac{\sum_{i=1}^n \left(X_i - \bar{X}\right)\left(Y_i - \bar{Y}\right)}{\sqrt{\sum_{i=1}^n \left(X_i - \bar{X}\right)^2} \sqrt{\sum_{i=1}^n \left(Y_i - \bar{Y}\right)^2}}
$$
Where:
$X_i$ and $Y_i$ are individual sample points
$\bar{X}$ and $\bar{Y}$ are the means of the $X$ and $Y$ samples
$n$ is the number of paired samples
2. Spearman's Rank Correlation Coefficient $\rho$ ) $$
\rho = 1 - \frac{6 \sum_{i=1}^n d_i^2}{n\left(n^2 - 1\right)}
$$
Where:
$d_i = rank(X_i) - rank(Y_i)$ is the difference between ranks
$n$ is the number of observations
3. Kendall's Tau $\tau$ ) $$
\tau = \frac{C - D}{\sqrt{(C + D + X)(C + D + Y)}}
$$
Where:
$C$ is the number of concordant pairs
$D$ is the number of discordant pairs
$X$ is the number of pairs tied only in $X$
$Y$ is the number of pairs tied only in $Y$
4. Cross-Correlation Function $$
\text{CCF}(k) = \frac{\sum_{i=1}^{n - k} \left(X_i - \bar{X}\right)\left(Y_{i + k} - \bar{Y}\right)}{\sqrt{\sum_{i=1}^n \left(X_i - \bar{X}\right)^2} \sqrt{\sum_{i=1}^n \left(Y_i - \bar{Y}\right)^2}}
$$
Where:
$k$ is the lag (shift between series)
$n$ is the series length
$\text{CCF}(k)$ is cross-correlation at lag $k$
5. Rolling Window Correlation $r_w$ ) $$
r_w(t) = r\left(X_{[t-w+1:t]}, Y_{[t-w+1:t]}\right)
$$
Where:
$w$ is the window size
$t$ is the current time point
$[t-w+1:t]$ represents the window interval
6. Relationship Confidence Score $C_s$ ) $$
C_s = \frac{S_c}{3} \cdot \max(|r|, |\rho|, |\tau|)
$$
Where:
$S_c$ is the count of significant correlations
$r$ , $\rho$ , $\tau$ are Pearson, Spearman, and Kendall coefficients
7. Lag Impact Ratio $L_r$ ) $$
L_r = \left|\frac{\max_{k}(\text{CCF}(k))}{\text{CCF}(0)}\right|
$$
Where:
$\text{CCF}(k)$ is cross-correlation at lag $k$
$\text{CCF}(0)$ is zero-lag correlation