On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-Old Problem.

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally unknown. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance R via post-fit residuals, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

Adaptive Bayesian Learning and Forecasting of Epidemic Evolution-Data Analysis of the COVID-19 Outbreak.

Since the beginning of 2020, the outbreak of a new strain of Coronavirus has caused hundreds of thousands of deaths and put under heavy pressure the world's most advanced healthcare systems. In order to slow down the spread of the disease, known as COVID-19, and reduce the stress on healthcare structures and intensive care units, many governments have taken drastic and unprecedented measures, such as closure of schools, shops and entire industries, and enforced drastic social distancing regulations, including local and national lockdowns. To effectively address such pandemics in a systematic and informed manner in the future, it is of fundamental importance to develop mathematical models and algorithms to predict the evolution of the spread of the disease to support policy and decision making at the governmental level. There is a strong literature describing the application of Bayesian sequential and adaptive dynamic estimation to surveillance (tracking and prediction) of objects such as missiles and ships; and in this article, we transfer some of its key lessons to epidemiology. We show that we can reliably estimate and forecast the evolution of the infections from daily - and possibly uncertain - publicly available information provided by authorities, e.g., daily numbers of infected and recovered individuals. The proposed method is able to estimate infection and recovery parameters, and to track and predict the epidemiological curve with good accuracy when applied to real data from Lombardia region in Italy, and from the USA. In these scenarios, the mean absolute percentage error computed after the lockdown is on average below 5% when the forecast is at 7 days, and below 10% when the forecast horizon is 14 days.

On a multimode test sequencing problem.

Test sequencing is a binary identification problem wherein one needs to develop a minimal expected cost test procedure to determine which one of a finite number of possible failure states, if any, is present. In this paper, we consider a multimode test sequencing (MMTS) problem, in which tests are distributed among multiple modes and additional transition costs will be incurred if a test sequence involves mode changes. The multimode test sequencing problem can be solved optimally via dynamic programming or AND/OR graph search methods. However, for large systems, the associated computation with dynamic programming or AND/OR graph search methods is substantial due to the rapidly increasing number of OR nodes (denoting ambiguity states and current modes) and AND nodes (denoting next modes and tests) in the search graph. In order to overcome the computational explosion, we propose to apply three heuristic algorithms based on information gain: information gain heuristic (IG), mode capability evaluation (MC), and mode capability evaluation with limited exploration of depth and degree of mode Isolation (MCLEI). We also propose to apply rollout strategies, which are guaranteed to improve the performance of heuristics, as long as the heuristics are sequentially improving. We show computational results, which suggest that the information-heuristic based rollout policies are significantly better than traditional information gain heuristic. We also show that among the three information heuristics proposed, MCLEI achieves the best tradeoff between optimality and computational complexity.