Chapter - 3. ANN Approach for Efficient Computation of Logarithm and Antilogarithm of Decimal Numbers

Transcription

1 Chapter - 3 ANN Approach for Efficient Computation of Logarithm and Antilogarithm of Decimal Numbers

2 Chapter - 3 ANN Approach for Efficient Computation of Logarithm and Antilogarithm of Decimal Numbers 3.1 Introduction The artificial neurons are inherently associated with nonlinearities. Hence an ANN structure can be conveniently used to model different linear and nonlinear systems. To assess the modeling potentiality o f ANN structure, we have addressed a simple problem o f logarithm and antilogarithm in this chapter. Computation o f logarithm o f a number and its inverse computation are frequently encountered in many applications. The logarithm o f a number x to the base 10 is defined by y a= log io (x) To get back the original number, we have to compute the antilog o f ya which is defined as * = antilogio (y j The objective o f this chapter is to develop a simple neural structure for the computation o f log o f any number and another simple structure to compute the original num ber from its antilog. 3.2 Modeling o f an ANN structure to compute logarithm For developing a logarithmic processor, the Neural Network (NN) is used in an adaptive manner as shown in Fig In this figure x refers to an input number whose log value is to be computed.

3 ya Fig. 3.1 An ANN Scheme The output o f the NN processor computes an estimate o f the log value. In the beginning the parameters o f the neural processor are initialized to some random values. The desired log value o f the input number is known to us which is compared with the estimated value y a. The difference of these two numbers is denoted as the error e. Knowledge o f this error e is used to compute the change in the connecting weights o f the neural model. To completely train the network, a set o f input patterns lying between 1 to 10 is first generated which is used to train the network. Using the standard log table, the log values o f these numbers are also obtained to serve as the desired values w hile training. Since computation o f logarithm is relatively a simple application, we have considered a single layer network to serve as a logarithmic processor. This is shown in Fig Fig. 3,2. Scheme for logjo computation using a single layer, single neuron ANN

4 The input x is weighted by a single weight w and a bias weight Wb to produce the linear output. This linear output is passed through a nonlinear sigmoid function defined in Section o f chapter 2. The estimated output y a is compared with corresponding log value ya to produce error e. Initially this error will be large but as the training o f the weights continues, on the application o f different patterns, mean square error (M SE) obtained from e gradually diminishes to a very small value. The number o f patterns used for training was 80. The procedure for adaptation o f the neural model is known as pattern matching method. Procedure for learning o f this method is outlined in Section o f Chapter 2. The MSE is recorded for each iteration as long as training continues. The relation between MSE and iterations has been plotted and shown in Fig The curve is known as a learning characteristics of the neural model. Each o f the values o f \x and a are set to 0.1 for obtaining the best convergence. The steady state connecting weights w and Wb obtained from simulation are given in Table 3.1. These two weights and the single neuron represent the neural log model which is very simple and can predict the logarithm o f any number lying between 1 to 10. To assess the potentiality o f the simulated log model, various numbers between 1 to 10 were used as input to the model and the computed outputs were compared with the true log values. The test results are presented in Table 3.2. It is observed that the percentage o f error is less than 1.75% in all cases. The accuracy can further be improved by incorporating one more neuron in the second layer or by increasing number o f inputs by functional expansion method. However in these two

5 cases the complexity would increase compared to the accuracy achieved, o, , o c U) co S ' o o o o O o O o o CM CO Iterations Fig. 3.3 Learning Characteristics for log computation Table 3.1 Weights obtained from simulation for log computation a = 0.1, = 0.1 w bias weight \vb Table 3.2 Comparison between simulated and true log results. R andom Decimal N um ber x Desired Result ya Estim ated Result Percentage E rro r y«

6 3.3 Modeling of an ANN structure to compute Antiiogarithm Computation o f antilogarithm is an inverse process which is basically a nonlinear operation. In an antilog processor, the input will be the log o f a number and output will be the original number. Such an inverse processor can be conveniently designed by a neural network. A multilayer ANN structure can be used to model an antilog processor but would involve more complexity and cost as well. In this chapter we have proposed an economical single layer ANN structure using functional expansion as is shown in Fig Fig. 3.4: ANN Schem e for an tilo g y com putation using functional expansion A single input ya which is the logio o f x is functionally expanded to nine values such as y^ cos7iya, co s2 7 ty a,co s8 7 ty a The purpose o f this expansion is to introduce some nonlinearity to the input so that the requirement o f number o f multilayers is reduced to one. These nine points are weighted by a set o f nine random weights and linearly added in the ANN. This sum is added with a bias weight w b and is passed through a nonlinear sigmoid function to produce the estimate o f the desired number as *. Then x is compared with true value x to produce error e. In the

7 beginning the magnitude o f this error is high as the connecting weights are not properly adjusted. To develop the antilog processor, we consider a set o f log values and the original numbers to serve as input and desired output respectively for training the model. Each tim e one value o f ya (logjo(x) ) is applied and the corresponding antilog value x is computed and the magnitude o f the difference between x and x is calculated. The pattern matching o f learning described in Section o f Chapter 2 is used to adapt the nine weights and one bias weight Wb so that the average MSE progressively decreases. The training is continued until the MSE attains the possible minimum value. The selection o f p. and a is very crucial as they control the rate o f convergence as well as minimum MSE. In this simulation, [i = 0.1 and a = 0.7 were used. The learning characteristics for developing the antilog computation model is shown in Fig The steady state MSE is set at around -40 db after about 1500 iterations. When the training is complete, the learning is withdrawn and the final weights achieved correspond to steady state weights o f the antilog neural model. Table 3.3 represents the magnitudes o f the steady state weights obtained from the computer simulation o f the antilog model. Determination o f nine connecting weights and Wb constitute the development o f neural network. To test the efficiency o f this model, random decimal values were generated and their logio values were used as the inputs to predict the desired results x. The results have been compared with the true values o f x and the percentage o f error are computed. These results are shown in Table 3.4. It is observed from the table that percentage o f error is less than 1.75% in each case. This error can further be minimized by increasing the num ber o f functional expansion points at the input.

8 -60 I o o o Iterations Fig. 3.5 Learning Characteristics for antilog computation Table 3.3 Weights obtained from simulation of antilog computation a = 0.1, ^ = 0.7 Wo W l w w W W W w7-0, w bias weight Wb

9 Table 3.4. C om parison between sim ulated an d tru e antilog results log(x) Desired Result y* Estimated Result y a Percentage E rro r Conclusions Since the artificial neuron contains nonlinearity, an A NN structure is quite suitable to model nonlinear systems. Keeping this in view, investigation has been made in this chapter to develop simple ANN model to compute logarithm and antilogarithm values o f decimal numbers. The problem chosen in this chapter is though quite simple, its study reveals the potentiality o f nonlinearity modeling o f neural networks. In both log and antilog computation, single neuron has been used to yield simple structure. In log computation a single weight and a bias weigh have been evaluated by simulation to act as a model for log computation. On testing the efficiency o f this model, it is observed that the percentage o f error lies less than 1.75% which is quite acceptable. The hardware cost o f this method is very cheap. Similarly a functional expansion based antilog model has been developed through simulation. The performance o f this model has been tested by supplying unknown data and the predicted results o f this model are observed to lie between 2% less than the true results. Thus it can be concluded that neural structure is a potential tool for nonlinear modeling which has been verified by developing log and antilog models and testing their perform ance.

10 References [l.jr u m elh art, D.E., H int on, G.E. and W illiam s, R.J. 1986, Learning internal representation by error propagation, in R um elhart, D.E. an d M cclelland, J.L.; (Ems), Parallel Distributed Processing, Cambridge, M.A. FIT Press Chap 8, pp [2.] Jam es A. F reem an and D avid M. S kapura, Neural Networks, Algorithms, Applications, and Programming Technique, Addison Wesley Publishing Company, [3.JY. Pao,Adaptive Pattern Recognition and Neural Networks, Reading, M.A. : Addison W esley Publishers, [4.]G.Panda and A.K.Saxena, An Effecient Data Communication Scheme using Artificial Neural Network, pp proceeding o f COMNET-94 at Patna. [5.]G.Panda,R.K.Singh and A.K.Saxena, Artificial Neural Network Approach for Efficient Computation o f Logarithm and Antilogarithm o f Decimal Numbers, pp 69-73, proceeding o f COM NET-97 at Patna.