[Paper Review] Distributed and Recursive Parameter Estimation in Parametrized Linear State-Space Models

S. Sundhar Ram, Venugopal V. Veeravalli+1 more|ArXiv.org|Apr 10, 2008

Target Tracking and Data Fusion in Sensor Networks14 references4 citations

TL;DR

This paper proposes a distributed and recursive parameter estimation algorithm, the Incremental Recursive Prediction Error (IRPE) algorithm, for parametrized linear state-space models in sensor networks. It enables on-line, energy-efficient estimation by allowing sensors to iteratively update estimates using local data and neighbor information, achieving convergence to the true parameter under general conditions, with application to gas leak source localization in a warehouse.

ABSTRACT

We consider a network of sensors deployed to sense a spatio-temporal field and estimate a parameter of interest. We are interested in the case where the temporal process sensed by each sensor can be modeled as a state-space process that is perturbed by random noise and parametrized by an unknown parameter. To estimate the unknown parameter from the measurements that the sensors sequentially collect, we propose a distributed and recursive estimation algorithm, which we refer to as the incremental recursive prediction error algorithm. This algorithm has the distributed property of incremental gradient algorithms and the on-line property of recursive prediction error algorithms. We study the convergence behavior of the algorithm and provide sufficient conditions for its convergence. Our convergence result is rather general and contains as special cases the known convergence results for the incremental versions of the least-mean square algorithm. Finally, we use the algorithm developed in this paper to identify the source of a gas-leak (diffusing source) in a closed warehouse and also report numerical simulations to verify convergence.

Motivation & Objective

• To develop a distributed and recursive estimation algorithm suitable for real-time, energy-constrained sensor networks.
• To enable on-line parameter estimation in parametrized linear state-space models without centralized data fusion.
• To reduce communication and memory costs by using only summary statistics and incremental updates.
• To provide convergence guarantees for the algorithm under general conditions, extending prior work on incremental and recursive methods.
• To apply the method to a real-world problem: localizing a gas leak in a warehouse using sensor measurements.

Proposed method

• The IRPE algorithm combines incremental gradient updates with recursive prediction error estimation, enabling distributed, on-line parameter updates.
• Sensors form a cycle and update estimates sequentially, using local measurements and the estimate received from the upstream neighbor.
• Each sensor maintains a local estimate and a prediction error, updating recursively using a gradient-like update rule based on prediction error.
• The algorithm uses a fixed-step size and ensures that the parameter estimate converges to the true value under mild assumptions on the model and measurement process.
• A hybrid version is proposed where sensors are grouped into clusters, allowing intra-cluster averaging before inter-cluster incremental updates.
• The method is applied to a gas leak localization problem by deriving a state-space model from the advection-diffusion equation using Green's function technique.

Experimental results

Research questions

• RQ1 Can a distributed and recursive parameter estimation algorithm be designed for linear state-space models that ensures convergence to the true parameter?
• RQ2 How does the performance of the distributed IRPE algorithm compare to centralized and non-incremental distributed methods in terms of convergence and accuracy?
• RQ3 What is the impact of network topology and clustering on the convergence behavior of the IRPE algorithm?
• RQ4 Can the algorithm handle real-world sensor data with noise and spatially distributed observations of a common underlying process?
• RQ5 Under what conditions does the IRPE algorithm avoid local minima in non-convex estimation problems?

Key findings

• The IRPE algorithm converges to the true parameter value under general conditions, including weak assumptions on the model and measurement noise.
• The standard IRPE algorithm fails to converge to the correct source location in simulations, getting trapped in a local minimum due to lack of information sharing across sensors.
• When sensors are clustered and intra-cluster averaging is used, the hybrid IRPE version successfully converges to the true source location (x*=37, y*=48).
• The centralized RPE method performs best, but the hybrid IRPE achieves near-optimal performance with significantly reduced communication and memory usage.
• The algorithm enables real-time estimation with constant memory usage per sensor and minimal communication per time slot, making it suitable for wireless sensor networks.
• The method successfully localizes a gas leak in a warehouse using 27 sensors, demonstrating practical applicability in a non-Bayesian, distributed setting.