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Learning Multi-Frequency Partial Correlation Graphs

Gabriele D'Acunto, Paolo Di Lorenzo, Francesco Bonchi, Stefania Sardellitti, Sergio Barbarossa

TL;DR

The paper tackles learning conditional dependencies among multiple time series across different frequency bands by introducing a multi-frequency, block-sparse partial-correlation graph ($K$-PCG). It develops two nonconvex methods: a closed-form CF approach that requires per-band sparsity and an iterative IA approach that jointly learns the CSD and its inverse via a successive convex approximation framework with inner ADMM. CF recovers per-band arc sets by selecting the top $s_m$ columns of a flattened inverse CSD, while IA solves a sequence of strongly convex surrogates to converge to stationary points. Across synthetic experiments, IA and especially CF with full sparsity knowledge outperform baselines, and a financial case study demonstrates that partial correlations concentrate in a few frequency bands, underscoring the practical value of frequency-aware learning. Public JAX code is provided to enable results reproduction and broader adoption across finance, climate, industrial, and biomedical domains.

Abstract

Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.

Learning Multi-Frequency Partial Correlation Graphs

TL;DR

The paper tackles learning conditional dependencies among multiple time series across different frequency bands by introducing a multi-frequency, block-sparse partial-correlation graph (-PCG). It develops two nonconvex methods: a closed-form CF approach that requires per-band sparsity and an iterative IA approach that jointly learns the CSD and its inverse via a successive convex approximation framework with inner ADMM. CF recovers per-band arc sets by selecting the top columns of a flattened inverse CSD, while IA solves a sequence of strongly convex surrogates to converge to stationary points. Across synthetic experiments, IA and especially CF with full sparsity knowledge outperform baselines, and a financial case study demonstrates that partial correlations concentrate in a few frequency bands, underscoring the practical value of frequency-aware learning. Public JAX code is provided to enable results reproduction and broader adoption across finance, climate, industrial, and biomedical domains.

Abstract

Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.
Paper Structure (25 sections, 2 theorems, 74 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 25 sections, 2 theorems, 74 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Conditional linear independence over frequency bands
  3. Estimation of Inverse CSD Tensor
  4. Problem statement
  5. The closed-form (CF) method
  6. The iterative approximation (IA) method
  7. Building the strongly convex surrogate problem
  8. Subproblem in TEXT
  9. Solution for TEXT
  10. Solution for TEXT
  11. Solution for TEXT
  12. Solution for TEXT
  13. Subproblem in TEXT
  14. Solution for TEXT
  15. Solution for TEXT
  16. ...and 10 more sections

Key Result

Lemma 1

The update $\underline{\widehat{\mathbf{p}}}_k^{t+1}$ in eq:upP is equal to with $\boldsymbol{\Gamma}_1$ defined as in eq:Gamma1.

Figures (4)

  • Figure 1: Inverse CSD tensor and its components. Nonzero off-diagonal entries are given in blue, white otherwise. Diagonal entries are shaded. The block-sparsity feature refers to the occurrence of partial correlations only over some $\mathcal{K}_m$. In this example, time series $i=2$ and $j=3$ are not partially correlated at the second block of frequencies $\mathcal{K}_2$, where $m \in [3]$ and each block is made of two frequencies.
  • Figure 2: The figure depicts the underlying multiscale causal structure together with the behavior of the strictly lower triangular part of $\mathbf{P}_k$ along frequencies. Different colors are associated with interactions that (i) are not present in the causal structure (blue), (ii) exist within the causal graph related to time scale 2 (green), and (iii) show up in both time scales (magenta). Finally, dashed lines indicate the splitting points of the frequency bands associated with the relevant time scales.
  • Figure 3: Left: performance in terms of SHD (lower better) for all methods, along $\bar{N}_{\mathcal{Y}}$. Right: difference of SHD between IA-bs and TS-GLASSO baseline, and CF-fk. Points (line plot) and bars height (bar plot) represent median values obtained over $50$ runs. Error bars (bar plot) represent $95\%$ interval. For readability, the line plot on the left has been cut at SHD$=50$, while in the bar plot on the right, we omit the naive baseline.
  • Figure 4: The 4 layers of the block-sparse multi-frequency partial correlation graph ($4$-PCG) among the time series of $17$ industry portfolios, where each layer corresponds to a different frequency band, provided by \ref{['subfig:exp_fin_ours']}) the IA method and \ref{['subfig:exp_fin_naive']}) the naive baseline. The lower triangular part of each matrix reports the $2018-2019$ period, while the upper triangular part the $2020-2021$ period.

Theorems & Definitions (5)

  • Definition 1: $K$-frequency partial correlation graph
  • Lemma 1
  • proof
  • Lemma 2
  • proof