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Nonlinear Poisson regression using neural networks: a simulation study


We describe a novel extension of the Poisson regression model to be based on a multi-layer perceptron, a type of neural network. This relaxes the assumptions of the traditional Poisson regression model, while including it as a special case. In this paper, we describe neural network regression models with six different schemes and compare their performances in three simulated data sets, namely one linear and two nonlinear cases. From the simulation study it is found that the Poisson regression models work well when the linearity assumption is correct, but the neural network models can largely improve the prediction in nonlinear situations.

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Fig. 1


  1. 1.

    McCulloch W, Pitts W (1943) A logical calculus of ideas immanent in nervous activity. Bull Math Biophys 5:115–133. doi:10.1007/BF02478259

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    Ripley BD (1996) Pattern recognition and neural networks. Cambridge University Press, London

    MATH  Google Scholar 

  3. 3.

    Bishop CM (2006) Pattern recognition and machine learning. Springer, New York

    MATH  Google Scholar 

  4. 4.

    Ripley RM, Harris AL, Tarassenko L (2004) Non-linear survival analysis using neural networks. Stat Med 23:825–842. doi:10.1002/sim.1655

    Article  Google Scholar 

  5. 5.

    Faraggi D, Simon R (1995) The maximum likelihood neural network as a statistical classification model. J Statist Plann Inference 46:93–104. doi:10.1016/0378-3758(95)99068-2

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Mathieson MJ (1996) Ordinal models for neural networks. Neural networks in financial engineering. In: Refences A-PN, Abu-Mostafa Y, Moody J, Weigend A (eds), Proceedings of the third international conference on neural networks in the capital markets, pp 23–536

  7. 7.

    Nejadgholi I, Seyyedsalehi SA (2009) Nonlinear normalization of input patterns to speaker variability in speech recognition neural networks. Neural Comput Appl 18:45–55. doi:10.1007/s00521-007-0151-5

    Article  Google Scholar 

  8. 8.

    Baxt WG (1990) Use of an artificial neural network for data analysis in clinical decision-making: the diagnosis of acute coronary occlusion. Neural Comput 2:480–489. doi:10.1162/neco.1990.2.4.480

    Article  Google Scholar 

  9. 9.

    Huang YL, Wang KL, Chen DR (2006) Diagnosis of breast tumors with ultrasonic texture analysis using support vector machines. Neural Comput Appl 15:164–169. doi:10.1007/s00521-005-0019-5

    Article  Google Scholar 

  10. 10.

    Akin M, Kurt MB, Sezgin N, Bayram M (2008) Estimating vigilance level by using EEG and EMG signals. Neural Comput Appl 17:227–236. doi:10.1007/s00521-007-0117-7

    Article  Google Scholar 

  11. 11.

    Bailey TC, Everson RM, Fieldsend JF, Krzanowski WJ, Partridge D, Schetinin V (2007) Representing classifier confidence in the safety critical domain: an illustration from mortality prediction in trauma cases. Neural Comput Appl 16:1–10. doi:10.1007/s00521-006-0053-y

    Google Scholar 

  12. 12.

    Eftekhar B, Mohammad K, Eftekhar H, Ghodsi M, Ketabchi E (2005) Comparison of artificial neural network and logistic regression models for prediction of mortality in head trauma based on initial clinical data. BMC Med Inform Decis Mak 5:3. doi:10.1186/1472-6947-5-3

    Article  Google Scholar 

  13. 13.

    Sadat-Hashemi SM, Kazemnejad A, Lucas C, Badie K (2005) Predicting the type of pregnancy using artificial neural networks and multinomial logistic regression: a comparison study. Neural Comput Appl 14:198–202. doi:10.1007/s00521-004-0454-8

    Article  Google Scholar 

  14. 14.

    Leondes CT (1998) Neural network systems techniques and applications. Academic Press, San Diego

    Google Scholar 

  15. 15.

    Shafi I, Ahmad J, Shah SI, Kashif FM (2008) Computing de-blurred time frequency distributions using artificial neural networks. Circuits Syst Signal Process 27:277–294. doi:10.1007/s00034-008-9027-x

    Article  Google Scholar 

  16. 16.

    Shafi I, Ahmad J, Shah SI, Kashif FM (2007) Evolutionary time-frequency distributions using Bayesian regularised neural network model. IET Signal Process 1:97–106. doi:10.1049/iet-spr:20060311

    Article  Google Scholar 

  17. 17.

    Funahashi K (1989) On the approximate realization of continuous mapping by neural networks. Neural Netw 2:183–192. doi:10.1016/0893-6080(89)90003-8

    Article  Google Scholar 

  18. 18.

    Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2:359–366. doi:10.1016/0893-6080(89)90020-8

    Article  Google Scholar 

  19. 19.

    Pearlmutter BA (1994) Fast exact multiplication by the Hessian. Neural Comput 6:147–160. doi:10.1162/neco.1994.6.1.147

    Article  Google Scholar 

  20. 20.

    Nabney I (2001) Netlab algorithms for pattern recognition. Springer London Ltd, UK

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This study was sponsored by Tehran University of Medical Sciences. Most of this work was carried out during a sabbatical year (student visitor period) of Nader Fallah at the Department of Mathematics and Statistics in Dalhousie University. Authors wish to thank R. Ripley and I. Nabney for their help on R and Matlab Codes. The authors thank Catherine Pretty and Janet Brush for editing this manuscript.

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Correspondence to Kazem Mohammad.

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Fallah, N., Gu, H., Mohammad, K. et al. Nonlinear Poisson regression using neural networks: a simulation study. Neural Comput & Applic 18, 939 (2009).

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  • Generalized linear model
  • Poisson regression
  • Nonlinear regression
  • Multilayer perceptron
  • Simulation