论文部分内容阅读
Abstract
This researchworkpresents new developmentin thefield ofnatural science, wherecomparisonis made theoretically on the efficiency of both classical regression models and that of artificial neural network models, with various transfer functions without data consideration. The results obtained based on variance estimation indicates that ANN is better which coincides with the results of Authors in the past on the efficiency of ANN over the traditional regression models. The certain conditions required for ANN efficiency over the conventional regression models were noted only that the optimal number of hidden layers and neurons needed to achieve minimum error is still open to further investigation.
Key words
Artificial neural network models; Transfer functions; Hidden layers; Regression models
1.INTRODUCTION
Neural networks are being widely used in manyfields of study. This can be attributed to the fact that these networks are attempts to model the capabilities of human brains. Since the last decade, neural networks have been used as a theoretically sound alternative to traditional regression models.
Although neural networks (NNs) originated in mathematical neurology, the rather simplified practical models currently in use have moved steadily towards thefield of statistics. A number of researchers have illustrated the connection of neural networks to traditional statistical methods. For example Gallinari, Thiria, Badran and Fogelman-Soullie (1991) have presented analytical results that establish a link between discriminant analysis and multilayer perceptions(MLP) used for classification problems. Cheng and Titterington ((1994) made a detailed analysis and comparison of various neural network models with traditional statistical methods.
Neural networks are being used in the areas of prediction and classification, areas where regression models and other related statistical techniques have traditionally been used. Ripley (1994) discusses the statistical aspects of neural networks and classifies neural networks as one of a class offlexible non-linear regression methods.
2.AIM AND OBJECTIVES OF THE STUDY
The main purpose of this study is to compare efficiency of traditional regression model and that of artificial neuralnetworkmodelswith anattemptto recognizethe onewithbetterdiscriminatingandpredictivepower.
For the realization of the above intention, the following measures are the underlined objectives:
(i) To compare theoretically a ANN model with a logistic transfer function and a logistic regression model.
(ii) To compare analytically the ANN model with linear transfer function and linear regression model.
(iii) To estimate the means of the estimates of parameters of various classes of non linear regression and that of the ANN.
(iv) To estimate and comparevariances of the parametersof both ANN and that of traditional regression model.
(v) To identify a better model based on the result obtained from the comparison.
3.SIGNIFICANCE OF THE STUDY
Methodologicaldisputes that arise in practice oftenturn on questions of the natureinterpretationand justifications of methods and models that relieved on to learn from incomplete and often“observational”(or non experimental) data, the methodology of statistical inference and statistical modeling. This research work is of very high significance as it attempts to unravel the truth and settle the scores of methodological disputes in thefield of mathematical statistics, Accounting andfinance, Health and medicine, Engineeringand Manufacturing, Marketing and general body of knowledge with regards to using classical regression and artificial neural modeling.
As theperformanceofa particulartechniquein comparisonto othertechniquedependonvariousfactors like the size of the sample, among others, in this study, attempt is comparing both techniques (ANN and logistic models) analytically without data consideration.
4.METHODOLOGY
The purpose of the study is to have the theoretical explanation of both techniques, which include their variance analysis. The classical regression model includes:
(i) Log-linear model
lnY =α0+α1x + ei
(ii) Linear-log model
Y =α0+α1lnx + ei
(iii) YLR=?α0+?α1(1 + e?x)?1Logistic regression
The ANN class of models
5.COMPARATIVE ANALYSIS OF THEVARIANCES OF TRADITIONAL REGRESSION MODEL AND ANN MODELS
5.1Linear-log Versus ANN with Logistic as Transfer Function In the section, the variances of the model under review will be compared.
Let YLGrepresents a lines-log model and YLANN denotes variable of a ANN with logistic as transfer function.
Var[YLG] = Var[(YLG?YLANN) + YLANN]
Recall that Var(Z+M) = Var(Z) + Var(M) + 2Cov(Z, M)
Let
6.CERTAIN CONDITIONS
From the critical analysis of the variances of the models under review, the following are noted.
(i) ANN model with logistic function out-performed the linear-log if and only if)ln x?%1 + e?(γ0+γ1x)&?1*2
> 0(ii) ANN model with Hyperbolic as transfer function will be better than that of logistic regression iff
7.SUMMARY, CONCLUSION AND RECOMMENDATION
7.1Summary
This research work has presented an analytically the efficiency of both traditional regression model of various classes and the ANN of various transfer function. Analysis of variance of each model was also conducted without data consideration.
REFERENCES
[1]Akaike H. (1973). Information Theory and an Estimation of the Maximum Likelihood Principle. Second International Symposium on Information Theory. Budapest.
[2]Amemiya T. (1980). Selection of Regressors. International Economics Review, 21, 331-354 leisten.
[3]Anders U. (1996). Was Neuronale Netze Wirklich Leisten. Die Bank, 3, 162-165.
[4]Davidson R. Mackinnon J.G. (1993). Estimation and Inference in Econometrics. Oxford University Press.
[5]Efron B., Tibshirani R. (1986).Boostrap Methods for Standard Errors Confidence Intervals and Other Measures of Statistical Accuracy. Statistical Science, 1(1), 54-77.
[6]Hardle W. (1993). Applied NonParametrics Regression. Cambridge University Press.
[7]Hornik K., Stinchcombe M., White H. (1989). Multilayer Feedforward Networks are Universal Approximators. Neural Networks, 2, 356-366.
This researchworkpresents new developmentin thefield ofnatural science, wherecomparisonis made theoretically on the efficiency of both classical regression models and that of artificial neural network models, with various transfer functions without data consideration. The results obtained based on variance estimation indicates that ANN is better which coincides with the results of Authors in the past on the efficiency of ANN over the traditional regression models. The certain conditions required for ANN efficiency over the conventional regression models were noted only that the optimal number of hidden layers and neurons needed to achieve minimum error is still open to further investigation.
Key words
Artificial neural network models; Transfer functions; Hidden layers; Regression models
1.INTRODUCTION
Neural networks are being widely used in manyfields of study. This can be attributed to the fact that these networks are attempts to model the capabilities of human brains. Since the last decade, neural networks have been used as a theoretically sound alternative to traditional regression models.
Although neural networks (NNs) originated in mathematical neurology, the rather simplified practical models currently in use have moved steadily towards thefield of statistics. A number of researchers have illustrated the connection of neural networks to traditional statistical methods. For example Gallinari, Thiria, Badran and Fogelman-Soullie (1991) have presented analytical results that establish a link between discriminant analysis and multilayer perceptions(MLP) used for classification problems. Cheng and Titterington ((1994) made a detailed analysis and comparison of various neural network models with traditional statistical methods.
Neural networks are being used in the areas of prediction and classification, areas where regression models and other related statistical techniques have traditionally been used. Ripley (1994) discusses the statistical aspects of neural networks and classifies neural networks as one of a class offlexible non-linear regression methods.
2.AIM AND OBJECTIVES OF THE STUDY
The main purpose of this study is to compare efficiency of traditional regression model and that of artificial neuralnetworkmodelswith anattemptto recognizethe onewithbetterdiscriminatingandpredictivepower.
For the realization of the above intention, the following measures are the underlined objectives:
(i) To compare theoretically a ANN model with a logistic transfer function and a logistic regression model.
(ii) To compare analytically the ANN model with linear transfer function and linear regression model.
(iii) To estimate the means of the estimates of parameters of various classes of non linear regression and that of the ANN.
(iv) To estimate and comparevariances of the parametersof both ANN and that of traditional regression model.
(v) To identify a better model based on the result obtained from the comparison.
3.SIGNIFICANCE OF THE STUDY
Methodologicaldisputes that arise in practice oftenturn on questions of the natureinterpretationand justifications of methods and models that relieved on to learn from incomplete and often“observational”(or non experimental) data, the methodology of statistical inference and statistical modeling. This research work is of very high significance as it attempts to unravel the truth and settle the scores of methodological disputes in thefield of mathematical statistics, Accounting andfinance, Health and medicine, Engineeringand Manufacturing, Marketing and general body of knowledge with regards to using classical regression and artificial neural modeling.
As theperformanceofa particulartechniquein comparisonto othertechniquedependonvariousfactors like the size of the sample, among others, in this study, attempt is comparing both techniques (ANN and logistic models) analytically without data consideration.
4.METHODOLOGY
The purpose of the study is to have the theoretical explanation of both techniques, which include their variance analysis. The classical regression model includes:
(i) Log-linear model
lnY =α0+α1x + ei
(ii) Linear-log model
Y =α0+α1lnx + ei
(iii) YLR=?α0+?α1(1 + e?x)?1Logistic regression
The ANN class of models
5.COMPARATIVE ANALYSIS OF THEVARIANCES OF TRADITIONAL REGRESSION MODEL AND ANN MODELS
5.1Linear-log Versus ANN with Logistic as Transfer Function In the section, the variances of the model under review will be compared.
Let YLGrepresents a lines-log model and YLANN denotes variable of a ANN with logistic as transfer function.
Var[YLG] = Var[(YLG?YLANN) + YLANN]
Recall that Var(Z+M) = Var(Z) + Var(M) + 2Cov(Z, M)
Let
6.CERTAIN CONDITIONS
From the critical analysis of the variances of the models under review, the following are noted.
(i) ANN model with logistic function out-performed the linear-log if and only if)ln x?%1 + e?(γ0+γ1x)&?1*2
> 0(ii) ANN model with Hyperbolic as transfer function will be better than that of logistic regression iff
7.SUMMARY, CONCLUSION AND RECOMMENDATION
7.1Summary
This research work has presented an analytically the efficiency of both traditional regression model of various classes and the ANN of various transfer function. Analysis of variance of each model was also conducted without data consideration.
REFERENCES
[1]Akaike H. (1973). Information Theory and an Estimation of the Maximum Likelihood Principle. Second International Symposium on Information Theory. Budapest.
[2]Amemiya T. (1980). Selection of Regressors. International Economics Review, 21, 331-354 leisten.
[3]Anders U. (1996). Was Neuronale Netze Wirklich Leisten. Die Bank, 3, 162-165.
[4]Davidson R. Mackinnon J.G. (1993). Estimation and Inference in Econometrics. Oxford University Press.
[5]Efron B., Tibshirani R. (1986).Boostrap Methods for Standard Errors Confidence Intervals and Other Measures of Statistical Accuracy. Statistical Science, 1(1), 54-77.
[6]Hardle W. (1993). Applied NonParametrics Regression. Cambridge University Press.
[7]Hornik K., Stinchcombe M., White H. (1989). Multilayer Feedforward Networks are Universal Approximators. Neural Networks, 2, 356-366.