Christopher Godwin Udomboso [a],*

Transcription

1 Progress in Alied Mathematics Vol. 5, No., 03, DOI: /j.am ISSN 95-5X Print ISSN Online On Some Proerties of a Hetereogeneous Transfer Function Involving Symmetric Saturated Linear SATLINS with Hyerbolic Tangent Sigmoid TANSIG Transfer Functions Christoher Godwin Udomboso a,* a Deartment of Statistics, University of Ibadan, Ibadan, Nigeria. * Corresonding author. Address: Deartment of Statistics, University of Ibadan, Ibadan, Nigeria; cg.udomboso@mail.ui.edu.ng Received: February 0, 03/ Acceted: Aril, 03/ Published: Aril 30, 03 Abstract: Transfer functions mas the inut layer of the statistical neural network model to the outut layer. To do this erfectly, the function must lie within certain bounds. This is a roerty of robability distributions. This aer establishes the heterogeneous transfer function, SATLINS TANSIG, as a Probability Distribution Functions.d.f by showing that it is roer. It also shows the mean and variance. Key words: Statistical neural network; SATLINS; TANSIG; SATLINS - TANSIG; Mean; Variance Udomboso, C. G. 03. On Some Proerties of a Hetereogeneous Transfer Function Involving Symmetric Saturated Linear SATLINS with Hyerbolic Tangent Sigmoid TAN- SIG Transfer Functions. Progress in Alied Mathematics, 5, Available from htt:// DOI: /j.am INTRODUCTION Artificial Neural Networks ANNs are non linear maing structures based on the function of the human brain. They imitate the training of the human brain and can rocess roblems involving non linear and comlex data even if the data are imrecise and noisy. They are owerful tools for modeling, esecially when the underlying data relationshi is unknown, and are ideally suited for modeling linear 40

2 Udomboso, C. G./Progress in Alied Mathematics, 5, 03 and non linear data. This is why their alications to statistical roblems are very interesting. Such networks are called the Statistical Neural Network SNN. This study centres on the Multi-Layer Percetron MLP which haens to be the most commonly used tye of ANN. It has been found to be owerful in terms of model recision in the usage of homogeneous transfer functions TFs, esecially with comlex or large data set. The choice of MLP is because it is the only ANN tye that allows for statistical inference. The modeling ower of an SNN model lies on the transfer function that is used. There are several transfer functions that may be used in a given SNN model. S- elections are made from a fixed ool of different transfer functions, and ossibly using runing techniques to dro functions that are not useful. U till now, known literatures and researches have reorted network analysis using one transfer functions that is, homogeneous models. For examle, 9 used the sigmoid transfer function, while 0 comared logistic and hyerbolic tangent transfer functions, 4 used the tangential transfer function that is, family of tangents functions, as well as used the symmetric saturated linear transfer function. This study endeavours to investigate an analytical derivation of a heterogeneous transfer function using the symmetric saturated linear SATLINS as well as the hyerbolic tangent sigmoid TANSIG transfer functions. It further showed that the derived transfer function is a roer robability density function.d.f. Hence the mean and variance were also derived.. THE STATISTICAL NEURAL NETWORK MODEL Anders 996 roosed a statistical neural network model given as y fx, w u where y is the deendent variable, X x 0, x,..., x I is a vector of indeendent variables, w α, β, γ is the network weight: α is the weight of the inut unit, β is the weight of the hidden unit, and γ is the weight of the outut unit, and u i is the stochastic term that is normally distributed that is, u i N0, σ I n. Basically, fx, w is the artificial neural network function, exressed as H I fx, w αx β h g γ hi x i. h where g. is the transfer function. The roosed convoluted form of the artificial neural network function used in this study is H I I fx, w αx β h g γ hi x i g γ hi x i h and thus, the form of the statistical neural network model roosed is H I I y αx β h g γ hi x i g γ hi x i u i u j h where y is the deendent variable, X x 0, x,..., x I is a vector of indeendent 4

3 On Some Proerties of a Hetereogeneous Transfer Function Involving Symmetric Saturated Linear SATLINS with Hyerbolic Tangent Sigmoid TANSIG Transfer Functions variables, w α, β, γ is the network weight: α is the weight of the inut unit, β is the weight of the hidden unit, and γ is the weight of the outut unit, u i and u j are the stochastic terms that are normally distributed that is, u i, u j N0, σ I n, and g. and g. are the transfer functions. In this aer, we investigate the distributional roerties of the heterogeneous transfer function arising from the convolution of SATLINS and TANSIG. Let g. Symmetric Saturated Linear function SATLINS, defined as, n < satlins g. f n n, n 3, n > Let g. Hyerbolic Tangent Sigmoid function TANSIG, defined as tansig g. f 3 n 4 en 3. SYMMETRIC SATURATING LINEAR AND HYPERBOL- IC TANGENT SIGMOID MODEL SATLINS TANSIG i. For n <, f n a. This imlies that f n m. Let f 3 m em en e m 5 fn f n f 3 n r r f n mf 3 mdm n r e n e m r dm e m e 4m... dm e r n r 6 ii. For n, f n n. This imlies that f n m n m. f 3 m em em e m f n f 3 n f n mf 3 mdm, n n m e m dm 4 7

4 Udomboso, C. G./Progress in Alied Mathematics, 5, 03 I m emdm decreases raidly for any interval of m. Hence I 0. Therefore, 7 becomes f n f 3 n n e m dm n mdm n 3n n e n n n e iii. For n >, f n a. This imlies that f n m as 8 Therefore, f n f 3 n f n f 3 n f 3 m e m n f n mf 3 mdm e m dm e m e 4m... dm dm e n The summary of the derived function is given as I I g γ hi x i g γ hi x i fn en e er n r, n < n 3n n n en n e, n 9 0 n en e, n > Equation 0 is the derived transfer function for the Symmetric Saturated Linear transfer function and the Hyerbolic Tangent transfer function. 4. DISTRIBUTIONAL PROPERTIES OF THE SATLINS - TANSIG MODEL We now show that the derived transfer function is a robability density function. By definition, the robability density function.d.f of function fx of a random variable X : Ω R is said to be a roer.d.f if for x,, x X, we have that, fxdx, 43 x X

5 On Some Proerties of a Hetereogeneous Transfer Function Involving Symmetric Saturated Linear SATLINS with Hyerbolic Tangent Sigmoid TANSIG Transfer Functions From the derived transfer function f n f 3 n e n n 3n n 7 3 where A e r n 3n n ne n e r n e n n r n 7 3 A e e e n e n r e n n ne n e e. e n ne Hence, for suitable values of such that A 4 3, we conclude that f n f 3 n is a robability density function. We next obtain the mean and variance of the derived transfer function. En n n e n n 3n n n e r e n n r n en e ne r n nr n e n n e 44 n e n 3 3n n

6 Udomboso, C. G./Progress in Alied Mathematics, 5, 03 3 n n e ne n e n Let I udv n e n, where udv uv vdu. Also, let u n and v e n. Then, I n en ne n e e e e e 4 3 e Therefore, mean is given as e En e e e 3 e e 4 3 e Also, En n e n n n 3n n n n e n n n 3n n n e r e n n r i nfty 3 5 n 3 e n n en e n e r n e n 3 e n n e n n n r 3 n 3 e n e Let J udv n3 e n, where udv uv vdu. Also, let u n 3 45

7 On Some Proerties of a Hetereogeneous Transfer Function Involving Symmetric Saturated Linear SATLINS with Hyerbolic Tangent Sigmoid TANSIG Transfer Functions and v e n. Then, J Thus, En 3 5 Hence, n 3 en 3 n e n { e e 3 e e e e e 4 3 e } { e e 3 varn En En 3 5 { e e 3 { 5. CONCLUSION e e e e } e 4 3 e e e e e e 4 3 e } e e e e 3 e e 4 3 e This study derived a heterogeneous transfer function involving the symmetric saturated linear and hyerbolic tangent sigmoid transfer functions. It went further to show that the derived transfer function is a roer robability distribution function.d.f, having mean and variance. REFERENCES Reso, J. P A comarison of artificial neural networks and statistical regression with biological resources alications. M.S. Thesis. University of Maryland, College Park, USA. Adeoju, G. A., Ogunjuyigbe, S. O. A., & Alawode, K. O Alication of neural network to load forecasting in Nigerian electrical ower system. The Pacific Journal of Science and Technology, 8, Adewole, A. P., Akinwale, A. T., & Akintomide, A. B. 0. Artificial neural network model for forecasting foreign exchange rate. World of Comuter Science and Information Technology Journal, 3, Adeyiga, J. A., Ezike, J. O. J., Omotosho, A., & Amakulor, W. 0. A neural network based model for detecting irregularities in e-banking transactions. African Journal of Comuter and ICTs, 4, Akaike, H A new look at the statistical model identification. IEEE Transactions on Automatic Control, 9, Akinwale, A. T., Arogundade, O. T., & Adekoya, A. F Translated Nigeria stock market rices using artificial neural network for effective rediction. Journal of Theoretical and Alied Information Technology, }

8 Udomboso, C. G./Progress in Alied Mathematics, 5, 03 7 Anders, U Statistical model building for neural networks. Nunberg, Germany: AFIR Colloqium. 8 Anderson, J. A An introduction to neural networks. Prentice Hall. 9 Ashigwuike, C. E. 0. Estimation of solar ower generation in some Nigerian cities using artificial neural network. Journal of Chemical, Biological and Physical Sciences,, Battiti, R. 99. First- and second-order methods for learning: Between steeest descent and Newtons method. Neural Comutation, 4, Carling, A. 99. Introducing neural networks. England: Sigma Press. Falode, A. O., & Udomboso, C. G. 0. Predictive modeling of gas roduction, utilization and flaring in Nigeria using TSRM and TSNN: A comarative aroach. International Journal of Environmental Engineering. Acceted 3 Foresee, F. D., & Hagan, M. T Gauss-Newton aroximation to Bayesian regularization. In IEEE International Conference on Neural Networks, 3, New York: IEEE. 4 Gan, C., Limsombunchai, V., Clemes, M., & Weng, A Consumer choice rediction artificial neural networks versus logistic models. Journal of Social Sciences, 4, Golden, R. M Mathematical methods for neural network analysis and design. Cambridge: MIT Press. 6 Ibeh, G. F., Agbo, G. A., Rabia, S., & Chikwenze, A. R. 0. Comarison of emirical and artificial neural network models for the correlation of monthly average global solar radiation with sunshine hours in Minna, Niger State, Nigeria. International Journal of Physical Sciences, 7 8, Lawrence, J Introduction to neural networks: design, theory, and alications. Nevada City, CA: California Scientific Software Press. 8 Maren, A., Harston, C., & Pa, R Handbook of neural comuting alications. San Diego, CA: Academic Press. 9 Nelson, M. M., & Illingworth, W. T. 99. A ractical guide to neural nets. USA: Addison-Wesley Publishing Comany, Inc. 0 Omole, O., Falode, O.A., & Deng, A. D Prediction of Nigerian crude oil viscosity using artificial neural network. Petroleum & Coal, 5 3, Smith, M Neural networks for statistical modeling. New York: Van Nostrand Reinhold. Taylor, J. G Neural networks and their alications. Wiley. 3 Tayfur G. 00. Artificial neural networks for sheet sediment transort. Hydrol. Sci. J., 47 6, Torak, F., & Cigizoglu, H. K Predicting longitudinal Disersion coefficient in natural streams by artificial neural networks. Hydrol. Processes, 0, Udomboso, C. G., & Amahia, G. N. 0. Comarative analysis of rainfall rediction using statistical neural network and classical linear regression model. Journal of Modern Mathematics and Statistics, 5 3, Warner, B., & Misra, M Understanding neural networks as statistical tools. The American Statistician, 50 4,