Robust Diffusion Recursive Least M-Estimate Adaptive Filtering and Its Performance Analysis

Abstract

This paper presents a robust distributed adaptive algorithm called diffusion recursive least M-estimate (DRLM), which enhances the robustness of the diffusion recursive least square (DRLS) algorithm against impulsive noise by incorporating the strong resistance to impulsive noise of the modified Huber function (MHF). However, the tracking speed of the proposed DRLM algorithm is not favorable in scenario where the network parameter vector of interest changes abruptly. To overcome this drawback, a new variable forgetting factor (VFF) strategy is devised and the VFF-DRLM algorithm is obtained. Then, the asymptotic unbiasedness and steady-state network mean square deviation (NMSD) performance of the DRLM algorithm are analyzed. Finally, the accuracy of the steady-state analysis result of the DRLM algorithm and the superiority of the proposed algorithms compared to other competing algorithms are demonstrated by extensive numerical simulations.

Appendix A

1.1 Detailed Derivation of the DRLM Algorithm

Similar to [2], we first update the matrix ${\varvec{P}}_{k,i}$ from ${\varvec{P}}_{k,i - 1}$ by using the rank-1 update, i.e.,

$$ {\varvec{P}}_{k,i}^{0} \leftarrow \lambda^{ - 1} {\varvec{P}}_{k,i - 1} $$

From node $l = 1$ to $N$, repeat:

$$ \begin{aligned} {\varvec{P}}_{k,i}^{l} \leftarrow&\, \left[ {\left( {{\varvec{P}}_{k,i}^{l - 1} } \right)^{ - 1} + c_{l,k} q_{l} (i){\varvec{u}}_{l,i}^{T} {\varvec{u}}_{l,i} } \right]^{ - 1} \hfill \\ {\varvec{P}}_{k,i} \leftarrow &\,{\varvec{P}}_{k,i}^{N} \hfill \\ \end{aligned} $$

(A-1)

where the matrix ${\varvec{P}}_{k,i}^{l}$ denotes the intermediate result after every rank-1 update. Actually, we need only consider the neighbors of node k, from (A-1) and using the matrix inversion lemma, we can obtain

$$ {\varvec{P}}_{k,i} \leftarrow \lambda^{ - 1} {\varvec{P}}_{k,i - 1} $$

For node $l \in {\mathcal{N}}_{k}$, repeat:

$$ {\varvec{P}}_{k,i} \leftarrow {\varvec{P}}_{k,i} - \frac{{c_{l,k} q_{l} (i){\varvec{P}}_{k,i} {\varvec{u}}_{l,i}^{T} {\varvec{u}}_{l,i} {\varvec{P}}_{k,i} }}{{1 + c_{l,k} q_{l} (i){\varvec{u}}_{l,i} {\varvec{P}}_{k,i} {\varvec{u}}_{l,i}^{T} }} $$

(A-2)

Next, we define the matrices ${\mathbf{\mathcal{D}}}_{i}^{l}$, ${\mathbf{\mathcal{U}}}_{i}^{l}$, ${\mathbf{\mathcal{W}}}_{k,i}^{l}$ and ${\mathbf{\mathcal{Q}}}_{i}^{l}$ as follows:

$$ {\mathbf{\mathcal{D}}}_{i}^{l} = col\left\{ {d_{l} (i),d_{l - 1} (i),...,d_{1} (i),{\mathbf{\mathcal{D}}}_{i - 1} } \right\}\quad $$

$$ {\mathbf{\mathcal{U}}}_{i}^{l} = col\left\{ {{\varvec{u}}_{l,i} ,{\varvec{u}}_{l - 1,i} ,...,{\varvec{u}}_{1,i} ,{\mathbf{\mathcal{U}}}_{i - 1} } \right\} $$

$$ {\mathbf{\mathcal{W}}}_{k,i}^{l} = diag\left\{ {c_{l,k} ,c_{l - 1,k} ,...,c_{1,k} ,\lambda {\mathbf{\mathcal{W}}}_{k,i - 1} } \right\} $$

$$ {\mathbf{\mathcal{Q}}}_{i}^{l} = diag\left\{ {q_{l} (i),q_{l - 1} (i),...,q_{1} (i),{\mathbf{\mathcal{Q}}}_{i - 1} } \right\} $$

Then, the intermediate estimate vector can be given by

$$ \user2{\varphi }_{k,i}^{l} = {\varvec{P}}_{k,i}^{l} ({\mathbf{\mathcal{U}}}_{i}^{l} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{l} {\mathbf{\mathcal{W}}}_{k,i}^{l} {\mathbf{\mathcal{D}}}_{i}^{l} $$

(A-3)

where

$$ {\varvec{P}}_{k,i}^{l} = \left[ {\lambda^{i + 1}{\varvec{\varPi}}+ ({\mathbf{\mathcal{U}}}_{i}^{l} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{l} {\mathbf{\mathcal{W}}}_{k,i}^{l} {\mathbf{\mathcal{U}}}_{i}^{l} } \right]^{ - 1} $$

(A-4)

Since we have $\user2{\varphi }_{k,i - 1} = \user2{\varphi }_{k,i - 1}^{N} \triangleq \user2{\varphi }_{k,i}^{0}$, the $\user2{\varphi }_{k,i}$ can be updated recursively from $\user2{\varphi }_{k,i - 1}$ by instead updating $\user2{\varphi }_{k,i}^{N}$ recursively from $\user2{\varphi }_{k,i}^{0}$. Assuming that $j$ is the minimum index satisfying $c_{j,k} \ne 0$, $\user2{\varphi }_{k,i}^{j}$ can be deduced from $\user2{\varphi }_{k,i}^{0}$ as follows

$$ \begin{aligned} \user2{\varphi }_{k,i}^{j} = & {\varvec{P}}_{k,i}^{j} ({\mathbf{\mathcal{U}}}_{i}^{j} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{j} {\mathbf{\mathcal{W}}}_{k,i}^{j} {\mathbf{\mathcal{D}}}_{i}^{j} \\ & = \underbrace {{{\varvec{P}}_{k,i}^{0} ({\mathbf{\mathcal{U}}}_{i}^{0} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{0} {\mathbf{\mathcal{W}}}_{k,i}^{0} {\mathbf{\mathcal{D}}}_{i}^{0} }}_{{\user2{\varphi }_{k,i - 1} }} + c_{j,k} q_{j} (i){\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} \\ & \left( {1 - \frac{{c_{j,k} q_{j} (i){\varvec{u}}_{j,i} {\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} }}{{1 + c_{j,k} q_{j} (i){\varvec{u}}_{j,i} {\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} }}} \right)d_{j} (i) \\ & - \frac{{c_{j,k} q_{j} (i){\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} {\varvec{u}}_{j,i} }}{{1 + c_{j,k} q_{j} (i){\varvec{u}}_{j,i} {\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} }}\underbrace {{{\varvec{P}}_{k,i}^{0} ({\mathbf{\mathcal{U}}}_{i}^{0} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{0} {\mathbf{\mathcal{W}}}_{k,i}^{0} {\mathbf{\mathcal{D}}}_{i}^{0} }}_{{\user2{\varphi }_{k,i - 1} }} \\ & = \user2{\varphi }_{k,i - 1} + \frac{{c_{j,k} q_{j} (i){\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} }}{{1 + c_{j,k} q_{j} (i){\varvec{u}}_{j,i} {\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} }}(d_{j} (i) - {\varvec{u}}_{j,i} \user2{\varphi }_{k,i - 1} ) \\ \end{aligned} $$

(A-5)

where

$$ ({\mathbf{\mathcal{U}}}_{i}^{j} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{j} {\mathbf{\mathcal{W}}}_{k,i}^{j} {\mathbf{\mathcal{D}}}_{i}^{j} = \left[ {({\mathbf{\mathcal{U}}}_{i}^{0} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{0} {\mathbf{\mathcal{W}}}_{k,i}^{0} {\mathbf{\mathcal{D}}}_{i}^{0} + c_{j,k} q_{j} (i){\varvec{u}}_{j,i}^{T} d_{j} (i)} \right] $$

and

$$ {\varvec{P}}_{k,i}^{j} = \left[ {{\varvec{P}}_{k,i}^{0} - \frac{{c_{j,k} q_{j} (i){\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} {\varvec{u}}_{j,i} {\varvec{P}}_{k,i}^{0} }}{{1 + c_{j,k} q_{j} (i){\varvec{u}}_{j,i} {\varvec{P}}_{k,i}^{0} {\varvec{u}}_{j,i}^{T} }}} \right] $$

(A-6)

Similarly, $\user2{\varphi }_{k,i}^{l}$ can be updated from $\user2{\varphi }_{k,i}^{l - 1}$ recursively, for $l > j$, repeat:

$$ \begin{aligned} \user2{\varphi }_{k,i}^{l} =&\, {\varvec{P}}_{k,i}^{l} \left[ {({\mathbf{\mathcal{U}}}_{i}^{l - 1} )^{T} {\mathbf{\mathcal{Q}}}_{i}^{l - 1} {\mathbf{\mathcal{W}}}_{k,i}^{l - 1} {\mathbf{\mathcal{D}}}_{i}^{l - 1} + c_{l,k} q_{l} (i){\varvec{u}}_{l,i}^{T} d_{l} (i)} \right] \\ =&\, \user2{\varphi }_{k,i}^{l - 1} + \frac{{c_{l,k} q_{l} (i){\varvec{P}}_{k,i}^{l - 1} {\varvec{u}}_{l,i}^{T} (d_{l} (i) - {\varvec{u}}_{l,i} \user2{\varphi }_{k,i}^{l - 1} )}}{{1 + c_{l,k} q_{l} (i){\varvec{u}}_{l,i} {\varvec{P}}_{k,i}^{l - 1} {\varvec{u}}_{l,i}^{T} }} \\ \end{aligned} $$

(A-7)

In ight of (A-2), (A-5) and (A-7), we can summary the increment update process as follows

$$ \user2{\varphi }_{k,i}^{{}} \leftarrow {\varvec{w}}_{k,i - 1} $$

$$ {\varvec{P}}_{k,i}^{{}} \leftarrow \lambda^{ - 1} {\varvec{P}}_{k,i - 1} $$

for $l = 1$ to $N$, repeat

$$ \user2{\varphi }_{k,i}^{{}} \leftarrow \user2{\varphi }_{k,i}^{{}} + \frac{{c_{l,k} q_{l} (i){\varvec{P}}_{k,i}^{{}} {\varvec{u}}_{l,i}^{T} \left[ {d_{l} (i) - {\varvec{u}}_{l,i} \user2{\varphi }_{k,i}^{{}} } \right]}}{{1 + c_{l,k} q_{l} (i){\varvec{u}}_{l,i} {\varvec{P}}_{k,i}^{{}} {\varvec{u}}_{l,i}^{T} }} $$

(A-8)

$$ {\varvec{P}}_{k,i}^{{}} \leftarrow {\varvec{P}}_{k,i}^{{}} - \frac{{c_{l,k} q_{l} (i){\varvec{P}}_{k,i}^{{}} {\varvec{u}}_{l,i}^{T} {\varvec{u}}_{l,i} {\varvec{P}}_{k,i}^{{}} }}{{1 + c_{l,k} q_{l} (i){\varvec{u}}_{l,i} {\varvec{P}}_{k,i}^{{}} {\varvec{u}}_{l,i}^{T} }} $$

(A-9)

Finally, the above obtained intermediate weight vectors are weighted and combined, and the weight vector ${\varvec{w}}_{k,i}$ is given by

$$ {\varvec{w}}_{k,i} = \sum\limits_{{l \in {\mathcal{N}}_{k} }} {a_{l,k} } \user2{\varphi }_{l,i} $$

(A-10)