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Learning from correlation with extreme learning machine

  • Li ZhaoEmail author
  • Jie Zhu
Original Article
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Abstract

A seemingly unrelated regression (SUR) refers to several individual equations among which there is not an explicit connection such as one equation’s observation is another equation’s response, but there exists an implicit relation represented by correlated disturbances of response variables. In this paper, SUR is applied to extreme learning machine (ELM) which is a single hidden layer feed-forward neural network where input weights and hidden layer biases are randomly assigned but the weight parameters between hidden and output layers are least-square solutions of a regression equation. A correlation-based extreme learning machine is built using the auxiliary sample which is related to the main sample which we focus on. Considering the weights between hidden and output layers in ELM as a random vector, we derive an explicit representation for the vector’s covariance matrix. The proof of theorems and simulation process indicate that the stronger correlation between main sample and auxiliary sample is, the higher generalization ability is.

Keywords

Extreme learning machine Seemingly unrelated regression model The least square estimator Two-stage improved estimator 

Notes

Acknowledgements

We would like to express our gratitude to all those who helped me during the writing of this paper. We gratefully acknowledge the help of our supervisor, Prof. XiZhao Wang, who has offered us valuable suggestions to revise and improve this paper. This work was supported in part by the National Natural Science Foundation of China (Grant 61772344, Grant 61732011, and Grant 61811530324), in part by the Natural Science Foundation of SZU (Grant 827-000140, Grant 827-000230, and Grant 2017060), in part by Basic Research Project of Knowledge Innovation Program in ShenZhen (JCYJ20180305125850156).

References

  1. 1.
    Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57(298):348–368MathSciNetzbMATHGoogle Scholar
  2. 2.
    Hubert M, Verdonck T, Yorulmaz z (2016) Fast robust sur with economical and actuarial applications. Stat Anal Data Min ASA Data Sci J 10(2):77–88MathSciNetGoogle Scholar
  3. 3.
    Wang H (2010) Sparse seemingly unrelated regression modelling: applications in finance and econometrics. Comput Stat Data Anal 54(11):2866–2877MathSciNetzbMATHGoogle Scholar
  4. 4.
    Foschi P, Kontoghiorghes EJ (2004) A computationally efficient method for solving sur models with orthogonal regressors. Linear Algebra Appl 388(1):193–200MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fraser D, Rekkas M, Wong A (2005) Highly accurate likelihood analysis for the seemingly unrelated regression problem. J Econom 127(1):17–33MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dufour J-M, Khalaf L (2002) Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions. J Econom 106(1):143–170MathSciNetzbMATHGoogle Scholar
  7. 7.
    Zellner A, Ando T (2010) A direct monte carlo approach for bayesian analysis of the seemingly unrelated regression model. J Econom 159(1):33–45MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zellner A, Huang DS (1962) Further properties of efficient estimators for seemingly unrelated regression equations. Int Econ Rev 3(3):300–313zbMATHGoogle Scholar
  9. 9.
    Magnus JR (1978) Maximum likelihood estimation of the gls model with unknown parameters in the disturbance covariance matrix. J Econom 7(3):281–312MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kakwani NC (1967) The unbiasedness of Zellner’s seemingly unrelated regression equations estimators. Publ Am Stat Assoc 62(317):141–142MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zellner A (1963) Estimators for seemingly unrelated regression equations: some exact finite sample results. J Am Stat Assoc 58(304):977–992zbMATHGoogle Scholar
  12. 12.
    Revankar NS (1974) Some finite sample results in the context of two seemingly unrelated regression equations. J Am Stat Assoc 69(345):187–190MathSciNetzbMATHGoogle Scholar
  13. 13.
    Revankar NS (1976) Use of restricted residuals in sur systems: some finite sample results. J Am Stat Assoc 71(353):183–188MathSciNetzbMATHGoogle Scholar
  14. 14.
    Liu A (2002) Efficient estimation of two seemingly unrelated regression equations. J Multivar Anal 82(2):445–456MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ma T, Ye R (2010) Efficient improved estimation of the parameters in two seemingly unrelated regression models. J Stat Plan Inference 140(9):2749–2754MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wang L, Lian H, Singh RS (2011) On efficient estimators of two seemingly unrelated regressions. Stat Probab Lett 81(5):563–570MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhao L, Xu X (2017) Generalized canonical correlation variables improved estimation in high dimensional seemingly unrelated regression models. Stat Probab Lett 126:119–126MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kurata H, Kariya T (1996) Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model. Ann Stat 24(4):1547–1559MathSciNetzbMATHGoogle Scholar
  19. 19.
    Chauvin Y, Rumelhart DE (1995) Back-propagation: theory, architecture, and applications. Lawrence Erlbaum Associates Inc., HillsdaleGoogle Scholar
  20. 20.
    Huang G-B, Zhu Q-Y, Siew CK (2006) Extreme learning machine: theory and applications. Neurocomputing 70(1–3):489–501Google Scholar
  21. 21.
    Huang G-B, Zhou HM, Ding XJ, Zhang R (2012) Extreme learning machine for regression and multi-class classification. IEEE Trans Syst Man Cybern Part B Cybern 42(2):513–529Google Scholar
  22. 22.
    Huang Z, Yu Y, Gu J (2015) A novel method for traffic sign recognition based on extreme learning machine. In: Intelligent control and automation, pp 1451–1456Google Scholar
  23. 23.
    Zhang L, Wang X, Huang GB, Liu T, Tan X (2018) Taste recognition in e-tongue using local discriminant preservation projection. IEEE Trans Cybern PP(99):1–14Google Scholar
  24. 24.
    Wang J, Zhang L, Cao J-J, Han D (2018) Nbwelm: naive bayesian based weighted extreme learning machine. Int J Mach Learn Cybern 9(1):21–35Google Scholar
  25. 25.
    Wang R, Chen D, Kwong S (2014) Fuzzy rough set based active learning. IEEE Trans Fuzzy Syst 22(6):1699–1704Google Scholar
  26. 26.
    Wang R, Wang X-Z, Kwong S, Chen X (2017) Incorporating diversity and informativeness in multiple-instance active learning. IEEE Trans Fuzzy Syst 25(6):1460–1475Google Scholar
  27. 27.
    Srivastava DG (1987) Seemingly unrelated regression models. Dekker, New YorkzbMATHGoogle Scholar
  28. 28.
    Wang X-Z, Wang R, Chen X (2018) Discovering the relationship between generalization and uncertainty by incorporating complexity of classification. IEEE Trans Cybern 48(2):703–715Google Scholar
  29. 29.
    Cao J, Zhang K, Luo M, Yin C, Lai X (2016) Extreme learning machine and adaptive sparse representation for image classification. Neural Netw 81(C):91–102Google Scholar
  30. 30.
    Zhao H, Guo X, Wang M, Li T, Pang C, Georgakopoulos D (2018) Analyze EEG signals with extreme learning machine based on PMIS feature selection. Int J Mach Learn Cybern 9(2):243–249Google Scholar
  31. 31.
    Wang R, Chow C-Y, Kwong S (2016) Ambiguity based multiclass active learning. IEEE Trans Fuzzy Syst 24(1):242–248Google Scholar
  32. 32.
    Luo X, Yang X, Jiang C, Ban X (2018) Timeliness online regularized extreme learning machine. Int J Mach Learn Cybern 9(3):465–476Google Scholar
  33. 33.
    Zhao X, Cao W, Zhu H, Ming Z, Ashfaq RAR (2018) An initial study on the rank of input matrix for extreme learning machine. Int J Mach Learn Cybern 9(5):867–879Google Scholar
  34. 34.
    Zhao L, Yan L, Xu X (2018) High correlated residuals improved estimation in the high dimensional SUR model. Commun Stat Simul Comput 47(7):1583–1605.  https://doi.org/10.1080/03610918.2017.1309429 MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The College of Computer Science and Software EngineeringShenzhen UniversityShenzhenChina
  2. 2.Guangdong Key Laboratory of Intelligent Information ProcessingShenzhen UniversityShenzhenChina
  3. 3.Department of Information ManagementThe National Police University for Criminal JusticeBeijingChina

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