VAR estimators using binary measurements

TL;DR

This paper introduces two new algorithms for estimating Gaussian VAR models from 1-bit measurements, addressing the challenges of quantisation and variance ratio estimation in the vector case.

Contribution

The paper presents novel algorithms that extend scalar quantisation methods to vector VAR models, including one that uses asymmetric thresholds and another that employs pairwise symmetric quantisation.

Findings

Both algorithms demonstrate high efficiency in numerical simulations.

The asymmetric method requires prior variance knowledge for threshold setting.

The symmetric method avoids threshold selection by using pairwise differences.

Abstract

In this paper, two novel algorithms to estimate a Gaussian Vector Autoregressive (VAR) model from 1-bit measurements are introduced. They are based on the Yule-Walker scheme modified to account for quantisation. The scalar case has been studied before. The main difficulty when going from the scalar to the vector case is how to estimate the ratios of the variances of pairwise components of the VAR model. The first method overcomes this difficulty by requiring the quantisation to be non-symmetric: each component of the VAR model output is replaced by a binary "zero" or a binary "one" depending on whether its value is greater than a strictly positive threshold. Different components of the VAR model can have different thresholds. As the choice of these thresholds has a strong influence on the performance, this first method is best suited for applications where the variance of each time series is approximately known prior to choosing the corresponding threshold. The second method relies instead on symmetric quantisations of not only each component of the VAR model but also on the pairwise differences of the components. These additional measurements are equivalent to a ranking of the instantaneous VAR model output, from the smallest component to the largest component. This avoids the need for choosing thresholds but requires additional hardware for quantising the components in pairs. Numerical simulations show the efficiency of both schemes.