Design of Adaptive Algorithms for Estimation in Single-Agent and Multi-Agent Environments

Abstract

Contribution of adaptive filters in the evolution of modern communications is remarkable. The logic of adaptive filtering establishes a significant part in the tool-set of statistical signal processing. For the problem of adaptive estimation, computational cost, convergence rate, steady-state error, stability and generalization are considered to be the main challenges. In this work several novel adaptive algorithms are developed to address these challenges. This work focus on two different types of environments, namely single-agent environments and multi-agent environments. In singleagent environments, data arrives at a single node for the estimation of a parameter of interest. For such environments we propose a novel idea of q-calculus based adaptive analysis. The qgradient is an extension of the classical gradient vector based on the concept of Jackson’s derivative. The q-derivative takes larger steps in the search direction as it evaluates the secant of the cost function rather than the tangent (as in the case of a conventional derivative). Motivated by this, in this work we developed several algorithms/ideas in which we addressed the short comings of the standard least mean square (LMS) and its variants by using q-calculus, such as, the q-least mean square (q-LMS) algorithm, in which we minimize the LMS cost function by employing the concept of q-derivative instead of the conventional derivative, the q-normalized least mean square (qNLMS), the two-dimensional q-least mean square (2D-qLMS) algorithm and the q-LMS for tracking a non-stationary channel. Consequently, several new explicit closed-form expressions for the meansquare-error (MSE) behavior are derived for the transient and steady-state analysis. On the other hand multi-agent signal processing has attract a number of researchers owing to better statistical inference in wireless networks and is therefore effectively utilized in many applications such as wireless sensor networks, smart grids etc. In multi-agent environments, data arrives at multiple nodes that are distributed over a geographical area and have a common task of estimation of a desired parameter. There are two major estimation techniques used for distributed environments: (1) centralized estimation, and (2) decentralized estimation. In centralized estimation, all the estimations take place at a single processor by sharing data from each node at a centralized unit. Such an estimation requires powerful processor along with massive amount of communication and power. To overcome these problems a decentralized estimation solution is proposed in literature, in which each node has its local estimate which is shared with the neighbors in an explicit manner (such as incremental, diffusion etc). Decentralized estimation techniques suffer from: (1) link failure problem, (2) instability issues, and (3) computational cost (particularly in the context of state-space estimation models). To deal with these problems, a number of innovative methodologies are proposed. Firstly a convex combination-based incremental least mean square (LMS) algorithm is proposed to overcome the problems of link failure between the nodes and instability in case local divergence in incremental mode of cooperation. The proposed algorithm is developed by employing convex combination of two filters. The adaptation of one filter is based on the estimate of adjacent node (incremental type), while the adaptation of the other is based on the estimate of the current local node at the previous time instant. These two filters are then fused together by using a suitable mixing parameter. Secondly to minimize the steady-state error, an optimum error non-linearity based incremental mode of cooperation is proposed. Thirdly, to reduce the computational cost for state-space estimation in distributed environments, a state-space least mean square algorithm for diffusion mode of cooperation is proposed. The proposed algorithm minimizes the computational complexity at each node which intern provides a significant advantage in terms of computational cost of the overall network and hence, can improve the response time of the network. Both the convergence in the mean and the mean square analysis of the proposed algorithm are performed and the transient and steady-state behavior of the proposed algorithm is analyzed.