Robust propagation velocity estimation of gastric electrical activity by least mean p-norm blind channel identification

Abstract

The propagation of the gastric slow wave is one of the most important spatial characteristics of gastric electrical activity (GEA). The time delay estimation (TDE) is an effective approach to quantitatively assessing the propagation velocity of GEA. Traditional TDE analyses are developed under the condition reported that the background noise in GEA analysis is Gaussian distributed. Due to the effects of spikes and/or motion artifacts, the GEA obtained from gastric serosal electrodes often contains sharp transitions. This paper proposes robust time delay estimation based on least mean p-norm blind channel identification (BCILMP) under α-stable noise condition. Compared with the least mean square time delay estimation (LMSTDE), the BCILMP provides better performance in the impulsive noise environments. The robustness of the proposed method is demonstrated through computer simulations in both Gaussian and α-stable noise environments. The results of the propagation velocity of real data obtained from gastric serosal electrodes in gastroparetic patients show that the propagation velocity in gastroparetic patients is slower than in the normal subjects reported in the literature, and the slow-wave propagation is directed proximally to distally from the corpus toward the pylorus but not all the variability of the propagation velocity increases monotonously.

Appendix

The derivation process of Eq. (7) is below:

The cost function is J(n) =  E[|e(n)|^p], 1 ≤ p <  α, where ${e(n) = \frac{{{\bf w}^{\rm T} (n){\bf x}(n)}}{{{\left\| {{\bf w}(n)} \right\|}_{p}}}.}$ The steepest descent algorithm has been derived to minimize the p-norm cost function, has following update equation:

$$ {\mathbf{w}}(n + 1) = {\mathbf{w}}(n) + \mu [ - \nabla_{{\mathbf{w}}} J(n)] = {\mathbf{w}}(n) - \mu [e(n)]^{{\langle p - 1\rangle}} \nabla_{{\mathbf{w}}} e(n) $$

(11)

There is w(n +  1) ≈ w(n) and gradient vector ∇ _w e(n) =  0 after convergence gives. Further, has following equation:

$$ \frac{{{\mathbf{x}}(n){\left\| {{\mathbf{w}}(n)} \right\|}_{p} - {\mathbf{w}}(n)^{\rm T} {\mathbf{x}}(n)({\left\| {{\mathbf{w}}(n)} \right\|}_{p} {)}^{\prime}}}{{{\left\| {{\mathbf{w}}(n)} \right\|}^{2}_{p}}} = {\mathbf{0}} $$

(12)

where

$$ \begin{aligned} ({\left\| {{\mathbf{w}}(n)} \right\|}_{p} {)}^{\prime} &= {\left\{{{\left[ {{\sum\limits_{i = 0}^{M - 1} {{\left[ {{\left| {w_{2} (n - i)} \right|}^{p} + {\left| {- w_{1} (n - i)} \right|}^{p}} \right]}}}} \right]}^{{1/p}}} \right\}}^{\prime} \\&= \left[ {\frac{1}{p}{\left\| {{\mathbf{w}}(n)} \right\|}^{{1 - p}}_{p} p{\left| {w_{2} (n)} \right|}^{{p - 1}} \operatorname{sgn} [w_{2} (n)]}, \right. \ldots, \frac{1}{p}{\left\| {{\mathbf{w}}(n)} \right\|}^{{1 - p}}_{p} p{\left| {w_{2} (n - M + 1)} \right|}^{{p - 1}} \operatorname{sgn} [w_{2} (n - M + 1)], \\ & \quad \frac{1}{p}{\left\| {{\mathbf{w}}(n)} \right\|}^{{1 - p}}_{p} p{\left| {- w_{1} (n)} \right|}^{{p - 1}} \operatorname{sgn} [ - w_{1} (n)], \ldots, \left. {\frac{1}{p}{\left\| {{\mathbf{w}}(n)} \right\|}^{{1 - p}}_{p} p{\left| {- w_{1} (n - M + 1)} \right|}^{{p - 1}} \operatorname{sgn} [ - w_{1} (n - M + 1)]} \right]^{\rm T} \\ & = {\left\| {{\mathbf{w}}(n)} \right\|}^{{1 - p}}_{p} {\left[ {{\left| {w_{2} (n)} \right|}^{{\langle p - 1\rangle}}, \ldots, {\left| {w_{2} (n - M + 1)} \right|}^{{\langle p - 1\rangle}}, {\left| {- w_{1} (n)} \right|}^{{\langle p - 1\rangle}}, \ldots, {\left| {- w_{1} (n - M + 1)} \right|}^{{\langle p - 1\rangle}}} \right]}^{\rm T} \\ & = \frac{{{\mathbf{w}}^{{\langle p - 1\rangle}}}}{{{\left\| {\mathbf{w}} \right\|}^{{p - 1}}_{p}}} \\ \end{aligned} $$

then Eq. (12)can be simplified as follow:

$$ \frac{1}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}}{\left[ {{\mathbf{x}}(n) - e(n)\frac{{{\mathbf{w}}(n)^{{\langle p - 1\rangle}}}}{{{\left\| {{\mathbf{w}}(n)} \right\|}^{{p - 1}}_{p}}}} \right]} = {\mathbf{0}} $$

(13)

thus, ${{\mathbf{x}}(n)=e(n)\frac{{[{\mathbf{w}}(n)]^{{\langle p - 1\rangle}}}}{{{\left\| {{\mathbf{w}}(n)} \right\|}^{{p - 1}}_{p}}}}$ can be obtained. Right multiplying ${[{\mathbf{x}}^{\rm T} (n)]^{{\langle p - 1\rangle}} \frac{{{\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}}}$ both side of this equation and taking expectation:

$$ \begin{aligned} E{\left\{{{\mathbf{x}}(n)[{\mathbf{x}}^{\rm T} (n)]^{{\langle p - 1\rangle}} \frac{{{\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}}} \right\}} &= E{\left\{{e(n)\frac{{{\mathbf{w}}(n)^{{\langle p - 1\rangle}} [{\mathbf{x}}^{\rm T} (n)]^{{\langle p - 1\rangle}} {\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}^{p}_{p}}}} \right\}} \\ R_{{_{c}}} (n)\frac{{{\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}} &= E{\left\{{e(n)\frac{{{\mathbf{w}}(n)[{\mathbf{x}}^{\rm T} (n){\mathbf{w}}(n)]^{{\langle p - 1\rangle}}}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p} {\left\| {{\mathbf{w}}(n)} \right\|}^{{p - 1}}_{p}}}} \right\}} = E[{\left| {e(n)} \right|}^{p} ]\frac{{{\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}} \\ \end{aligned} $$

(14)

where ${{\mathbf{R}}_{c} (n) = E\{[{\mathbf{x}}(n)]^{{\langle p - 1\rangle}} {\mathbf{x}}^{\rm T} (n)\}= E\{{\mathbf{x}}(n)[{\mathbf{x}}^{{{\rm T}}} (n)]^{{\langle p - 1\rangle}} \}.}$ We have ${{\mathbf{R}}_{c}(n)\frac{{{\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}} = E[{\left| {e(n)} \right|}^{p} ]\frac{{{\mathbf{w}}(n)}}{{{\left\| {{\mathbf{w}}(n)} \right\|}_{p}}}.}$