2. NAME OF SUPERVISOR: Prof. G.N. Amahia / Prof. I. K. Dontwi

3. YEAR OF COMPLETION: 03/02/2014

4. TITLE OF Ph.D THESIS: On the Level of Precision of an Heterogeneous Statistical Neural Network Model


The multi-layer perceptron is a type of Statistical Neural Network (SNN) model that is more precise than the linear regression model. However, it fails to attain high precision due to the use of homogeneous Transfer Functions (TFs) which do not appropriately link the input layer to the output layer. Therefore, an alternative SNN model using heterogeneous TFs to overcome the limitations of homogeneous TFs was developed. An Adjusted Network Information Criterion (ANIC) for testing the adequacy of the SNN models was also derived. An Heterogeneous SNN (HETSNN) model was derived by the convolution of two Homogeneous SNN (HOMSNN) models: y_1=αX+∑_(h=1)^H▒β_h g_1 (∑_(i=0)^I▒γ_hi x_i )+e_i and 〖 y〗_2=αX+∑_(h=1)^H▒β_h g_2 (∑_(i=0)^I▒γ_hi x_i )+e_i, where y_1 and y_2 are the dependent variable, X is a matrix of independent variables, α,β, and γ are the parameters of the network, e_i is the noise normally distributed with mean 0 and variance σ^2 (e_i ~ N(0,σ^2 ) ), g_1 (.) and g_2 (.) are the transfer functions, h=1,2,…,H are the number of hidden units, and i=0,1,…,I are the number of input units. One TF was used in the HOMSNN model while a convolution of two TFs was used to derive the HETSNN. Two sets of meteorological data [Amravati Hydrology Project at Manasgaon station, India from 1990 to 2004 and Nigeria Meteorological (NIMET) station, Ibadan from 1971 to 2004] were used to investigate the fit of the derived model. Thirteen sub-samples: 10, 20, 40, 60, 80, 100, 125, 150, 175, 200, 250, 300, and 400 generated from the NIMET data were used to investigate the asymptotic behaviour of the models and ANIC using the Kolmogorov-Smirnov one sample normality test. Network Information Criterion (NIC) and the derived ANIC, using Kullback’s symmetric divergence, were computed for determining the adequacy of the two models. Variance-ratio tests were carried out on the significance of the HETSNN model. The derived HETSNN model wasy=αX+∑_(h=1)^H▒〖β_h [g_1 (∑_(i=0)^I▒〖γ_hi x_i 〗) g_2 (∑_(i=0)^I▒〖γ_hi x_i 〗) ] 〗+e_i e_j, where e_i e_j~ N(0,σ^2 ). The HETSNN approached zero asymptotically faster than HOMSNN. The asymptotic behaviour of the models showed that HETSNN improved steadily over HOMSNN at an average rate of 3.0% to 15.0%. The rates of model adequacy for both HETSNN and HOMSNN using NIC were respectively 72.9% and 58.1%. Consequently, using ANIC, the rates were 67.7% and 65.0% respectively. This indicated that ANIC was an unbiased information criterion for the two models. The ANIC decayed to zero much slower than NIC with increasing sample size. The asymptotic behaviour of the ANIC also showed that as the sample size increased, ANIC approximated the standard normal distribution, N(0,1). There was a significant difference (p < 0.05) between the variances of HETSNN and HOMSNN. The heterogeneous statistical neural network model linked the input layer to the output layer more appropriately. The derived adjusted network information criterion performed better in model selection when compared with the network information criterion. Keywords: Transfer functions, heterogeneous statistical neural network, adjusted network information criterion. Word count: 458