### Abstract

Multi-layered perceptron, a kind of neural network, is suitable for modeling complex phenomena in a noisy and restricted data environment. In spatial interaction modeling, Openshaw (1993) and Fisher and Gopal (1994) verified better performance of perception than classical models. Openshaw wrote “If the dependent variable is binomial or Poisson then the net will deduce this from the training data, there is no need to be explict”. However, both residual patterns of perceptrons and a log-linear gravity model were similar in the results of Fisher and Gopal (1994). They performed log-transform for output data, and trained networks by the least square criterion, which is normal for most perceptrons. If we regard each output of perceptrons as probabilistic expectation, then both of their trained perceptrons and log-linear gravity models would be assumed as log-normal distributions for error term. The similarity of residual patterns (under-prediction in large flows) could be caused by the same assumption of error distribution. To overcome such well-known features of log-normal models, we might use the Poisson regression technique (Flowerdew and Aitkin, 1982). In this paper, I specified PR-perceptron, which is a special type of two-layered perceptrons (Fig. 1). Its features are as follows. 1) By sigmoid hidden-units, PR-perceptrons can approximate any continuous mapping. 2) An exponential output-unit produces a positive real number with no upper limits, and 3) it also brings formal similarities between PR-perceptron and Poisson regression models. 4) Its training criteria is the maximum likelihood in the Poisson distribution. The PR-perceptron shares not simply the estimation process of parameters but also theories of undergoing process of spatial interaction with the Poisson regression, the family of maximum-entropy models, and discrete choice models, e. g. the multinominal logit model.

Original language | English |
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Pages (from-to) | 521-540 |

Number of pages | 20 |

Journal | Japanese Journal of Human Geography |

Volume | 47 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1995 Jan 1 |

Externally published | Yes |

### Keywords

- Poisson regression
- artificial neural network
- migration in Japan
- perceptron
- spatial interaction model

### ASJC Scopus subject areas

- Geography, Planning and Development