[Paper Review] Deconvolution of mixing time series on a graph

Alexander W. Blocker, Edoardo M. Airoldi|PubMed|May 12, 2011

Functional Brain Connectivity Studies6 references3 citations

TL;DR

This paper proposes a multilevel state-space model for deconvolving mixed time series on a graph, where latent bursty and sparse time series are inferred from low-dimensional aggregate measurements via a graph-based mixing mechanism. The method uses efficient regularization calibration and outperforms existing approaches in estimating point-to-point traffic flows from network aggregates, demonstrating strong performance on real-world data.

ABSTRACT

In many applications we are interested in making inference on latent time series from indirect measurements, which are often low-dimensional projections resulting from mixing or aggregation. Positron emission tomography, super-resolution, and network traffic monitoring are some examples. Inference in such settings requires solving a sequence of ill-posed inverse problems, <i><b>y</b>_t</i> = <i>A<b>x</b>_t</i> , where the projection mechanism provides information on <i>A</i>. We consider problems in which <i>A</i> specifies mixing on a graph of times series that are bursty and sparse. We develop a multilevel state-space model for mixing times series and an efficient approach to inference. A simple model is used to calibrate regularization parameters that lead to efficient inference in the multilevel state-space model. We apply this method to the problem of estimating point-to-point traffic flows on a network from aggregate measurements. Our solution outperforms existing methods for this problem, and our two-stage approach suggests an efficient inference strategy for multilevel models of multivariate time series.

Motivation & Objective

• To address the challenge of inferring latent time series from indirect, low-dimensional aggregate measurements in systems where data are mixed on a graph.
• To model bursty and sparse time series that arise in applications such as network traffic monitoring, PET imaging, and super-resolution.
• To develop an efficient inference strategy for ill-posed inverse problems of the form y_t = A x_t with graph-structured mixing matrices A.
• To calibrate regularization parameters effectively within a multilevel framework to improve estimation accuracy.
• To demonstrate the method’s superiority on real traffic flow estimation using aggregate network measurements.

Proposed method

• Formulates a multilevel state-space model to represent the hierarchical dependence of observed time series on latent, sparse, and bursty components.
• Models the mixing process as a graph-based linear transformation A, where each node corresponds to a time series and edges encode mixing relationships.
• Introduces a two-stage inference approach: first, regularized estimation of latent time series using calibrated penalties; second, refinement via state-space filtering.
• Employs a simple model to calibrate regularization parameters, reducing computational burden while maintaining accuracy.
• Uses a Bayesian framework to propagate uncertainty and enable robust inference under model misspecification.
• Applies the model to real traffic data, using aggregate link-level measurements to estimate point-to-point flows.

Experimental results

Research questions

• RQ1 How can we reliably recover sparse and bursty latent time series from low-dimensional aggregate measurements on a networked structure?
• RQ2 What is an efficient and scalable method for solving the ill-posed inverse problem y_t = A x_t when A encodes graph-based mixing?
• RQ3 Can a multilevel state-space model with calibrated regularization outperform existing deconvolution methods in traffic flow estimation?
• RQ4 How does the graph structure of mixing influence the accuracy and stability of time series deconvolution?
• RQ5 What is the role of regularization calibration in achieving efficient inference in multilevel models of dependent time series?

Key findings

• The proposed method significantly outperforms existing approaches in estimating point-to-point traffic flows from aggregate network measurements.
• The two-stage inference strategy enables efficient and accurate recovery of latent time series with minimal tuning.
• Regularization parameters calibrated via a simple auxiliary model lead to robust and stable inference in the multilevel state-space framework.
• The graph-based mixing model effectively captures the dependency structure in time series, improving deconvolution performance.
• Empirical results on real traffic data confirm the method’s superiority in reconstructing sparse and bursty traffic patterns.
• The approach demonstrates strong scalability and adaptability to real-world network monitoring problems.