# A Robust Diffusion Estimation Algorithm with Self-Adjusting Step-Size in WSNs

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Minimization Error Entropy Criterion

## 3. Proposed Algorithms

#### 3.1. Diffusion MEE-SAS Algorithm

Algorithm 1: DMEE-SAS Algorithm |

Initialize: ${w}_{k,i}=0$ |

for $i=1:T$ |

for each node k: |

Adaptation |

${\mu}_{k}(i)=2{\mu}_{k}[V(0)\u2012V({e}_{k,i})]$ |

${\phi}_{k,i+1}={w}_{k,i}\u2012{\mu}_{k}(i)\frac{1}{{\sigma}^{2}L}{\displaystyle \sum _{j=i\u2012L+1}^{i}}{G}_{\sigma \sqrt{2}}({e}_{k,i}\u2012{e}_{k,j})({e}_{k,i}\u2012{e}_{k,j})({u}_{k,j}\u2012{u}_{k,i})$ |

Combination |

${w}_{k,i+1}={\displaystyle \sum _{l\in {N}_{k}}}{c}_{lk}{\phi}_{l,i+1}$ |

end for |

#### 3.2. Performance Analysis

#### 3.2.1. Mean Performance

#### 3.2.2. Mean-Square Performance

#### 3.2.3. Instantaneous MSD

#### 3.3. An Improving Scheme for the DMEE-SAS Algorithm

Algorithm 2: Improving DMEE-SAS Algorithm |

Initialize: |

${w}_{k,i}=0$ |

for $i=1:T$ |

for each node k: |

Adaptation |

each node calculates the switching time using Equation (50). |

each node updates intermediate estimate ${\phi}_{k,i}$ according to the first equation of Equation (51). |

Combination |

${w}_{k,i+1}={\displaystyle \sum _{l\in {N}_{k}}}{c}_{lk}{\phi}_{l,i+1}$ |

end for |

## 4. Simulation Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Wang, G.; Li, C.; Dong, L. Noise estimation using mean square cross prediction error for speech enhancement. IEEE Trans. Circuits Syst. I Regul. Pap.
**2010**, 57, 1489–1499. [Google Scholar] [CrossRef] - Xia, Y.; Jahanchahi, C.; Nitta, T. Performance bounds of quaternion estimators. IEEE Trans. Neural Netw. Learn. Syst.
**2015**, 26, 3287–3292. [Google Scholar] [PubMed] - Xia, Y.; Douglas, S.C.; Mandic, D.P. Adaptive frequency estimation in smart grid applications: Exploiting noncircularity and widely linear adaptive estimators. IEEE Signal Process. Mag.
**2012**, 29, 44–54. [Google Scholar] [CrossRef] - Liu, Y.; Li, C. Complex-valued Bayesian parameter estimation via Markov chain Monte Carlo. Inf. Sci.
**2016**, 326, 334–349. [Google Scholar] [CrossRef] - Xia, Y.; Mandic, D.P. Widely linear adaptive frequency estimation of unbalanced three-phase power systems. IEEE Trans. Instrum. Meas.
**2012**, 61, 74–83. [Google Scholar] [CrossRef] - Xia, Y.; Blazic, Z.; Mandic, D.P. Complex-valued least squares frequency estimation for unbalanced power systems. IEEE Trans. Instrum. Meas.
**2015**, 64, 638–648. [Google Scholar] [CrossRef] - Lopes, C.G.; Sayed, A.H. Incremental Adaptive Strategies over Distributed Networks. IEEE Trans. Signal Process.
**2007**, 55, 4064–4077. [Google Scholar] [CrossRef] - Rabbat, M.G.; Nowak, R.D. Quantized incremental algorithms for distributed optimization. IEEE J. Sel. Areas Commun.
**2006**, 23, 798–808. [Google Scholar] [CrossRef] - Nedic, A.; Bertsekas, D.P. Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim.
**2001**, 12, 109–138. [Google Scholar] [CrossRef] - Kar, S.; Moura, J.M.F. Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise. IEEE Trans. Signal Process.
**2009**, 57, 355–369. [Google Scholar] [CrossRef] - Carli, R.; Chiuso, A.; Schenato, L. Distributed Kalman filtering based on consensus strategies. IEEE J. Sel. Areas Commun.
**2008**, 26, 622–633. [Google Scholar] [CrossRef] - Cattivelli, F.S.; Sayed, A.H. Diffusion LMS Strategies for Distributed Estimation. IEEE Trans. Signal Process.
**2010**, 58, 1035–1048. [Google Scholar] [CrossRef] - Chen, J.; Sayed, A.H. Diffusion adaptation strategies for distributed optimization and learning over networks. IEEE Trans. Signal Process.
**2012**, 60, 4289–4305. [Google Scholar] [CrossRef] - Chen, F.; Shao, X. Broken-motifs Diffusion LMS Algorithm for Reducing Communication Load. Signal Process.
**2017**, 133, 213–218. [Google Scholar] [CrossRef] - Xia, Y.; Mandic, D.P.; Sayed, A.H. An Adaptive Diffusion Augmented CLMS Algorithm for Distributed Filtering of Noncircular Complex Signals. IEEE Signal Process. Lett.
**2011**, 18, 659–662. [Google Scholar] - Kanna, S.; Mandic, D.P. Steady-State Behavior of General Complex-Valued Diffusion LMS Strategies. IEEE Signal Process. Lett.
**2016**, 23, 722–726. [Google Scholar] [CrossRef] - Kanna, S.; Talebi, S.P.; Mandic, D.P. Diffusion widely linear adaptive estimation of system frequency in distributed power grids. In Proceedings of the IEEE International Energy Conference, Leuven, Belgium, 8 April 2014; pp. 772–778. [Google Scholar]
- Jahanchahi, C.; Mandic, D.P. An adaptive diffusion quaternion LMS algorithm for distributed networks of 3D and 4D vector sensors. In Proceedings of the IEEE Signal Processing Conference, Napa, CA, USA, 11 August 2013; pp. 1–5. [Google Scholar]
- Abdolee, R.; Champagne, B.; Sayed, A.H. A diffusion LMS strategy for parameter estimation in noisy regressor applications. In Proceedings of the Signal Processing Conference, Ann Arbor, MI, USA, 5 August 2012; pp. 749–753. [Google Scholar]
- Abdolee, R.; Champagne, B. Diffusion LMS Strategies in Sensor Networks With Noisy Input Data. IEEE/ACM Trans. Netw.
**2016**, 24, 3–14. [Google Scholar] [CrossRef] - Abdolee, R.; Champagne, B.; Sayed, A. Diffusion Adaptation over Multi-Agent Networks with Wireless Link Impairments. IEEE Trans. Mob. Comput.
**2016**, 15, 1362–1376. [Google Scholar] [CrossRef] - Kanna, S.; Dini, D.H.; Xia, Y. Distributed Widely Linear Kalman Filtering for Frequency Estimation in Power Networks. IEEE Trans. Signal Inf. Process. Netw.
**2015**, 1, 45–57. [Google Scholar] [CrossRef] - Middleton, D. Non-Gaussian noise models in signal processing for telecommunications: New methods an results for class A and class B noise models. IEEE Trans. Inf. Theory
**1999**, 45, 1129–1149. [Google Scholar] [CrossRef] - Zoubir, A.M.; Koivunen, V.; Chakhchoukh, Y.; Muma, M. Robust estimation in signal processing: A tutorial-style treatment of fundamental concepts. IEEE Signal Process. Mag.
**2012**, 29, 61–80. [Google Scholar] [CrossRef] - Zoubir, A.M.; Brcich, R.F. Multiuser detection in heavy tailed noise. Digit. Signal Process.
**2002**, 12, 262–273. [Google Scholar] [CrossRef] - Wen, F. Diffusion least-mean P-power algorithms for distributed estimation in alpha-stable noise environments. Electron. Lett.
**2013**, 49, 1355–1356. [Google Scholar] [CrossRef] - Ma, W.; Chen, B.; Duan, J.; Zhao, H. Diffusion maximum correntropy criterion algorithms for robust distributed estimation. Digit. Signal Process.
**2016**, 58, 10–19. [Google Scholar] [CrossRef] - Chen, B.; Xing, L.; Zhao, H.; Zheng, N.; Principe, J.C. Generalized correntropy for robust adaptive filtering. IEEE Trans. Signal Process.
**2016**, 64, 3376–3387. [Google Scholar] [CrossRef] - Ma, W.; Qu, H.; Gui, G.; Xu, L.; Zhao, J.; Chen, B. Maximum correntropy criterion based sparse adaptive filtering algorithms for robust channel estimation under non-Gaussian environments. J. Frankl. Inst.
**2015**, 352, 2708–2727. [Google Scholar] [CrossRef] - Chen, B.; Xing, L.; Liang, J.; Zheng, N.; Principe, J.C. Steady-state Mean-square Error Analysis for Adaptive Filtering under the Maximum Correntropy Criterion. IEEE Signal Process. Lett.
**2014**, 21, 880–884. [Google Scholar] - Li, C.; Shen, P.; Liu, Y.; Zhang, Z. Diffusion information theoretic learning for distributed estimation over network. IEEE Trans. Signal Process.
**2013**, 61, 4011–4024. [Google Scholar] [CrossRef] - Chen, B.; Zhu, Y.; Hu, J. Mean-Square Convergence Analysis of ADALINE Training with Minimum Error Entropy Criterion. IEEE Trans. Neural Netw.
**2010**, 21, 1168–1179. [Google Scholar] [CrossRef] [PubMed] - Chen, B.; Hu, H.; Pu, L.; Sun, Z. Stochastic gradient algorithm under (h, phi)-entropy criterion. Circuits Syst. Signal Process.
**2007**, 26, 941–960. [Google Scholar] [CrossRef] - Chen, B.; Zhu, P.; Principe, J.C. Survival information potential: A new criterion for adaptive system training. IEEE Trans. Signal Process.
**2012**, 60, 1184–1194. [Google Scholar] [CrossRef] - Chen, B.; Principe, J.C. Some further results on the minimum error entropy estimation. Entropy
**2012**, 14, 966–977. [Google Scholar] [CrossRef] - Chen, B.; Yuan, Z.; Zheng, N.; Principe, J.C. Kernel minimum error entropy algorithm. Neurocomputing
**2013**, 121, 160–169. [Google Scholar] [CrossRef] - Chen, B.; Xing, L.; Xu, B.; Zhao, H.; Principe, J.C. Insights into the Robustness of Minimum Error Entropy Estimation. IEEE Trans. Neural Netw. Learn. Syst.
**2016**, 1–7. [Google Scholar] [CrossRef] [PubMed] - Lee, H.S.; Kim, S.E.; Lee, J.W. A Variable Step-Size Diffusion LMS Algorithm for Distributed Estimation. IEEE Trans. Signal Process.
**2015**, 63, 1808–1820. [Google Scholar] [CrossRef] - Ghazanfari-Rad, S.; Labeau, F. Optimal variable step-size diffusion LMS algorithms. IEEE Stat. Signal Process. Workshop
**2014**, 464–467. [Google Scholar] [CrossRef] - Ni, J.; Yang, J. Variable step-size diffusion least mean fourth algorithm for distributed estimation. Signal Process.
**2016**, 122, 93–97. [Google Scholar] [CrossRef] - Abdolee, R.; Vakilian, V.; Champagne, B. Tracking Performance and Optimal Adaptation Step-Sizes of Diffusion-LMS Networks. IEEE Trans. Control Netw. Syst.
**2016**. [Google Scholar] [CrossRef] - Han, S.; Rao, S.; Erdogmus, D.; Jeong, K.H.; Principe, J. A minimum-error entropy criterion with self-adjusting step-size (MEE-SAS). Signal Process.
**2007**, 87, 2733–2745. [Google Scholar] [CrossRef] - Erdogmus, D.; Principe, J.C. Convergence properties and data efficiency of the minimum error entropy criterion in ADALINE training. IEEE Trans. Signal Process.
**2003**, 51, 1966–1978. [Google Scholar] [CrossRef] - Shen, P.; Li, C. Minimum Total Error Entropy Method for Parameter Estimation. IEEE Trans. Signal Process.
**2015**, 63, 4079–4090. [Google Scholar] [CrossRef] - Arablouei, R.; Werner, S. Analysis of the gradient-descent total least-squares adaptive filtering algorithm. IEEE Trans. Signal Process.
**2014**, 62, 1256–1264. [Google Scholar] [CrossRef] - Kelley, C.T. Iterative Methods for Optimization; SIAM: Philadelphia, PA, USA, 1999; Volume 9, p. 878. [Google Scholar]
- Abadir, K.M.; Magnus, J.R. Matrix Algebra; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Sayed, A.H. Adaptive Filters; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Khalil, H.K. Nonlinear Systems; Macmillan: New York, NY, USA, 1992. [Google Scholar]
- Li, H.; Li, X.; Anderson, M. A class of adaptive algorithms based on entropy estimation achieving CRLB for linear non-Gaussian filtering. IEEE Trans. Signal Process.
**2012**, 60, 2049–2055. [Google Scholar] [CrossRef] - Zhao, X.; Tu, S.; Sayed, A.H. Diffusion adaptation over networks under imperfect information exchange and non-stationary data. IEEE Trans. Signal Process.
**2012**, 60, 3460–3475. [Google Scholar] [CrossRef]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shao, X.; Chen, F.; Ye, Q.; Duan, S.
A Robust Diffusion Estimation Algorithm with Self-Adjusting Step-Size in WSNs. *Sensors* **2017**, *17*, 824.
https://doi.org/10.3390/s17040824

**AMA Style**

Shao X, Chen F, Ye Q, Duan S.
A Robust Diffusion Estimation Algorithm with Self-Adjusting Step-Size in WSNs. *Sensors*. 2017; 17(4):824.
https://doi.org/10.3390/s17040824

**Chicago/Turabian Style**

Shao, Xiaodan, Feng Chen, Qing Ye, and Shukai Duan.
2017. "A Robust Diffusion Estimation Algorithm with Self-Adjusting Step-Size in WSNs" *Sensors* 17, no. 4: 824.
https://doi.org/10.3390/s17040824