Functional Dependence Measures

• Functional dependence measures are statistical tools that quantify the strength of relationships between random objects using values from 0 (independence) to 1 (deterministic dependence).
• They integrate methodologies from information theory, rank-based approaches, and kernel methods, ensuring practical interpretability in high-dimensional and functional data settings.
• Recent advances employ graph-based and matrix entropy techniques to robustly analyze complex data in areas like functional data analysis, machine learning, and time series.

Functional dependence measures quantify the strength and character of the relationship between two random objects, with particular emphasis on functional data, non-monotonic relationships, and generalizations beyond scalar or finite-dimensional settings. Originating from both information-theoretic and rank-based traditions, these measures aim to reflect not just the presence, but the degree and interpretability of dependence—ranging from independence to deterministic functional relationships—with precise mathematical guarantees and domain-specific flexibility. Recent developments address functional data analysis (FDA), equitability, ordering, and high-dimensional (including infinite-dimensional) settings, substantiating their importance for statistical modeling, machine learning, and applied sciences.

1. Conceptual Foundations of Functional Dependence

Functional dependence is formalized via measures or coefficients that map pairs (X, Y)—where X and/or Y may be random variables, vectors, or functions—to a real number or functional object. The guiding principles proposed by Reimherr and Nicolae (Reimherr et al., 2013) are:

• Existence: The measure $D(X;Y)$ must be well-defined for the given data types, including function spaces.
• Range: $D(X;Y) \in [0,1]$, with 0 denoting independence and 1 corresponding to deterministic (functional) relationships.
• Interpretability: Each value of $D$ should admit an application-driven interpretation as the "fraction of relevant information in $Y$ carried by $X$."

Interpretability is often achieved by normalizing an information or loss-based quantity: $D(X ; Y) = \frac{I(X ; Y)}{I(Y ; Y)}$ where $I(X ; Y)$ measures an interpretable form of information (e.g., variance explained, entropy reduction) (Reimherr et al., 2013). This framework subsumes $R^2$, mutual information ratios, Fisher information efficiency, and extends to loss or divergence-based constructs.

For functional data, these measures must respect the structure of the input/output (e.g., curves, time series), and often rely on $L^2$ norms, pointwise variances, or eigen-expansions.

2. Archetypes and Construction Strategies

Prototypical functional dependence measures include:

• Functional Correlation Ratio ($L^2$-based): Quantifies the proportion of variance ("energy") in a functional $Y$ explained by $X$:

$$D_1(X ; Y) = 1 - \frac{\mathbb{E}\|Y - \mathbb{E}[Y | X]\|^2}{\mathbb{E}\|Y - \mathbb{E}[Y]\|^2}$$

with variants such as pointwise variance-weighted and principal component-based reductions (Reimherr et al., 2013).

• Integrated $R^2$: For scalar $Y$ and arbitrary $X$, this fully nonparametric coefficient is:

$$\nu(Y, X) = \int \frac{\operatorname{Var}(\mathbb{E}[1_{Y > t} | X])}{\operatorname{Var}(1_{Y > t})} d\tilde{\mu}(t)$$

where the weighting in $t$ allows sensitivity to central or tail associations. $\nu(Y, X) = 0$ if and only if $Y \perp X$ and $=1$ if and only if $Y$ is a measurable function of $X$ (Azadkia et al., 23 May 2025).

• Azadkia–Chatterjee’s Dependence Coefficient: Defined via the conditional survival function, extended to metric spaces, including functional data:

$$T(X, Y) = 6 \int \operatorname{Var}(P(Y \geq t | X)) \, dF(t)$$

and estimated by nearest-neighbor graphs, with boundary properties $T=0$ iff independence, $T=1$ iff deterministic function (Hörmann et al., 13 May 2024).

• Matrix-based Entropy Functionals: $T_\alpha^*$ and $D_\alpha^*$ are nonparametric dependence measures based on normalized matrix Renyi entropies constructed from kernel Gram matrices of the data, and are suited for multi-dimensional, even infinite-dimensional, features (Yu et al., 2021).
• Generalized (Non-monotonic) Spearman Correlations: Use orthonormal function systems (e.g., Legendre, cosine) to form basis-specific correlations, with sharp bounds and transformation-invariance, revealing complex non-monotonic relationships (McNeil et al., 11 Dec 2025).

3. Theoretical Guarantees and Ordering

A rigorous axiomatization of "strength of functional dependence" is given via conditional-convex (ccx) ordering (Ansari et al., 9 Nov 2025). The salient axioms are:

1. Law-invariance and transformation-invariance: All dependence assessments are invariant under bijective reparameterizations.
2. Extremality: Independence is minimal, and $Y=f(X)$ maximal, for dependence ordering.
3. Information-monotonicity: Adding information (predictors) cannot decrease dependence.
4. Characterization of conditional independence and copula-invariance.

This ordering is realized via convex ordering of conditional survival probabilities $p_v(X) = P[Y \ge q_Y(v) | X]$, and subsumes monotonicity properties of several leading dependence measures (e.g., Chatterjee's $\xi$, functionals of convex generators) (Ansari et al., 9 Nov 2025). For Gaussian, additive noise, and many copula models, ccx-ordering is explicitly computable.

Rearranged dependence measures employ the Hardy–Littlewood rearrangement to enforce the exact detection of independence ($0$) and functional dependence ($1$) for copula-based statistics, reconciling traditional rank and moment-based indices with the requirements of functional relationships (Strothmann et al., 2022).

4. Estimation, Computation, and Practical Implementation

Estimation strategies vary by measure, but for high-dimensional and functional settings, two broad trends dominate:

• Nearest neighbor or graph-based statistics: Used in nonparametric settings, such as Azadkia–Chatterjee's coefficient and Integrated $R^2$, allow estimation without tuning, even when $X$ is functional or lies in a general metric space. In infinite dimensions, self-normalization schemes are critical to accommodate the unbounded degree growth in dependency graphs (Azadkia et al., 23 May 2025, Hörmann et al., 13 May 2024).
• Kernel and matrix-functional approaches: Matrix-based dependence measures rely on (Hadamard product) kernel Gram matrices and their spectra to compute entropy proxies and their differences for multivariate or functional data, avoiding explicit density estimation (Yu et al., 2021).
• Rank- and copula-based estimators: Generalized Spearman coefficients, rearranged measures, and function-valued dependence maps (e.g., $q_H(x,y)$ on the unit square) are typically empirically computed via rank statistics and piecewise estimators of empirical copulas (Ledwina, 2014, McNeil et al., 11 Dec 2025).

Most approaches are designed for computational practicality—$O(n \log n)$ for nearest-neighbors, $O(n^2)$ or $O(n^3)$ for kernel methods, and efficient when grid or copula-based.

5. Equitability, Power, and Comparative Performance

Equitability is a core criterion: a dependence measure should ascribe similar values to relationships with identical noise level but different functional forms. Power-equitability ("weak equitability") demands only equal probability of detection under identical signal-to-noise settings, even if raw scores differ (Jiang et al., 2015). Simulations demonstrate that functions like HHG, Copula Dependence Coefficient, and Integrated $R^2$ attain high power-equitability, outperforming several traditional measures (e.g., MIC, distance correlation) when evaluated on a diverse set of functional forms and noise models (Rainio, 2021, Azadkia et al., 23 May 2025).

Matrix-based entropy functionals maintain high test power in detecting nonlinear dependencies, even in high dimension (Yu et al., 2021). MICe achieves the best practical equitability among leading measures, but at the cost of power retention as noise increases, while TICe and SDDP offer state-of-the-art power at the potential expense of precise equitability (Reshef et al., 2015). Weak-equitability is provably impossible in the strongest sense for all function classes simultaneously; all known measures trade statistical properties across function types (Jiang et al., 2015).

6. Functional-Dependence in Time Series and Functional Data Analysis

Temporal and functional data introduce serial or spatial dependence structures requiring specialized measures:

• L^p–m-approximability: For functional time series, weak dependence is quantified via moment-decay under coupled $m$-dependent approximations. This allows uniform convergence and central limit theorems for function-valued sequences under non-mixing innovations, and supports reliable inference for principal components, regression, and change-point analysis (Hörmann et al., 2010).
• Functional dependence measures in empirical process theory: Quantified by $\delta_\nu(k)$, the $L^\nu$-norm of the effect of changing the $k$-th innovation, these measures govern maximal inequalities and functional central limit theorems under polynomial or geometric decay, typically requiring weaker assumptions than classical mixing (Phandoidaen et al., 2021).

A functional dependence measure must suit the structure and dimensionality of the data, providing both theoretical guarantees for inference and computational viability.

7. Visualization, Non-Monotonicity, and High-Resolution Diagnosis

Function-valued dependence measures like $q_H(x, y)$ (Ledwina, 2014) allow detailed, local visualization of dependence across the support, revealing tail effects, asymmetries, and patterns (e.g., checkerboard, cross patterns) missed by scalar indices. Generalized Spearman correlations constructed from orthonormal systems provide a matrix of "basis correlations," facilitating exploration, function elicitation, and the construction of copula models matching observed non-monotonic dependencies (McNeil et al., 11 Dec 2025). Stochastic inversion techniques for uniform-distribution-preserving transforms further enable the construction of extremal copulas attaining prescribed margins or dependence scores.

Summary Table: Illustrative Functional Dependence Measures

Measure / Concept	Key Property / Setting	Reference
Functional Correlation Ratio ($L^2$)	Fraction of $L^2$-energy explained	(Reimherr et al., 2013)
Integrated $R^2$ ($\nu(Y, X)$)	Model-free, interpretable, 0 iff $\perp$, 1 iff functional	(Azadkia et al., 23 May 2025)
Azadkia–Chatterjee's $\xi$	Tuning-free, NN-based, extends to functional data	(Hörmann et al., 13 May 2024)
Rearranged $\rho$, $\tau$	Attain 0 iff $\perp$, 1 iff functional	(Strothmann et al., 2022)
Matrix-based $T_\alpha^*, D_\alpha^*$	Multivariate, nonparametric, entropy-based	(Yu et al., 2021)
ccx-Ordering	Full ordering by conditional convex order	(Ansari et al., 9 Nov 2025)
Generalized Spearman / Basis Correlations	Orthonormal expansions, non-monotonicity, copula models	(McNeil et al., 11 Dec 2025)
L^p–m-approximability ($\nu_p(\cdot)$)	Moment-based decay, functional time series	(Hörmann et al., 2010)

Functional dependence measures thus comprise a rich ecosystem of interpretative, flexible, and theoretically robust tools bridging classical statistics, information theory, FDA, and modern high-dimensional inference. Their continued development and comparative study are central to robust analysis of complex data structures in contemporary research.