Correlation Coefficient: Inter-Series Dependencies

Pearson’s correlation coefficient (r) is a measure of the linear statistical dependence between two random variables X and Y. Formally: r = Cov(X, Y) / (σ_X · σ_Y) = (E[XY] − E[X]·E[Y]) / (√(D[X])·√(D[Y])), where r ∈ [−1, 1]. A value of r = 0 indicates the absence of a linear relationship (but not necessarily independence), while |r| = 1 indicates a functional linear relationship. The sample correlation coefficient r̂ = Σ(xᵢ − x̄)(yᵢ − ȳ) / √(Σ(xᵢ − x̄)²·Σ(yᵢ − ȳ)²) is a consistent estimator of the theoretical ρ and is asymptotically normal.

Spearman’s rank correlation coefficient (ρ_s) measures monotonic (not necessarily linear) dependence between variables. The calculation is based on the ranks of observations: ρ_s = 1 − 6·Σdᵢ² / (n·(n² − 1)), where dᵢ = rg(xᵢ) − rg(yᵢ) is the difference in ranks. Spearman’s advantage over Pearson lies in its robustness to outliers and applicability to non-linear monotonic relationships. To test the significance of ρ_s, Student’s t-statistic is used: t = ρ_s·√((n − 2)/(1 − ρ_s²)) with (n − 2) degrees of freedom, provided n ≥ 10.

The Autocorrelation Function (ACF) is a tool for detecting dependencies between elements of a single time series at different lags (shifts). The ACF is defined as r(k) = Cov(Xₜ, Xₜ₊ₖ) / D[X] = (E[Xₜ·Xₜ₊ₖ] − μ²) / σ², where k is the lag. For a truly random sequence, r(k) = 0 for k ≠ 0. Statistically significant non-zero autocorrelations indicate the presence of patterns or generator defects. The significance limits are defined as ±z_{α/2}/√n (Bartlett’s formula), where n is the sequence length. A correlogram — the graphical representation of the ACF for k = 1, 2, ..., K — is a standard diagnostic tool.

Lag analysis allows for revealing periodic structures and hidden dependencies in the outcome sequence. The Partial Autocorrelation Function (PACF) eliminates the influence of intermediate lags: PACF(k) = Corr(Xₜ, Xₜ₊ₖ | Xₜ₊₁, ..., Xₜ₊ₖ₋₁), which allows for determining the order of the autoregressive AR(p) model. The Ljung-Box test checks the global hypothesis of the absence of autocorrelations at lags 1, ..., m: Q = n·(n + 2)·Σₖ₌₁ᵐ r²(k)/(n − k), where Q ~ χ²(m) under H₀. The detection of significant autocorrelations in the outcome sequence of a certified generator is a critical indicator of a violation of cryptographic strength.