Comparative analysis on hidden neurons estimation in multi layer perceptron neural networks for wind speed forecasting

Abstract

In this paper methodologies are proposed to estimate the number of hidden neurons that are to be placed numbers in the hidden layer of artificial neural networks (ANN) and certain new criteria are evolved for fixing this hidden neuron in multilayer perceptron neural networks. On the computation of the number of hidden neurons, the developed neural network model is applied for wind speed forecasting application. There is a possibility of over fitting or under fitting occurrence due to the random selection of hidden neurons in ANN model and this is addressed in this paper. Contribution is done in developing various 151 different criteria and the evolved criteria are tested for their validity employing various statistical error means. Simulation results prove that the proposed methodology minimized the computational error and enhanced the prediction accuracy. Convergence theorem is employed over the developed criterion to validate its applicability for fixing the number of hidden neurons. To evaluate the effectiveness of the proposed approach simulations were carried out on collected real-time wind data. Simulated results confirm that with minimum errors the presented approach can be utilized for wind speed forecasting. Comparative analysis has been performed for the estimation of the number of hidden neurons in multilayer perceptron neural networks. The presented approach is compact, enhances the accuracy rate with reduced error and faster convergence.

Appendix

Considers the applicability of different criteria with ‘n’ as input parameters. All developed criteria should satisfy the convergence theorem. If the limit of a sequence is finite, the sequence is called a convergent sequence. If the limit of a sequence does not tend to be a finite number, the sequence is called divergent (Dass 2009).

The convergence theorem characteristics are given below.

• 1. A convergent sequence has a finite limit.

• 2. All convergent sequences are bounded sequence.

• 3. All bounded point has a finite limit.

• 4. Convergent sequence needed condition is that it has finite limit and is bounded.

• 5. An oscillatory sequence does not tend to have a unique limit.

In a network there is no change occurring in the state of the network, regardless of the operation is called the stable network. For neural network model most important property is it always converges to a stable state. In real-time optimization problem the convergence plays a major role, the risk of getting stuck at some local minima problem in a network is prevented by the convergence. The convergence of sequence infinite has been established in convergence theorem because of the discontinuities in the model. The real-time neural optimization solvers are designed with the use of convergence properties.

Presenting the convergence of the considered sequence as follows,

$$\begin{aligned} \hbox {Taking the sequence} \,\,u_n =\frac{11\left( {n+1} \right) }{n-3} \end{aligned}$$

(13)

Apply convergence theorem,

$$\begin{aligned} n\mathop {\rightarrow }\limits ^{\lim } \infty u_n =n\mathop \rightarrow \limits ^{\lim } \infty \frac{11\left( {n+1} \right) }{n-3}=n\mathop \rightarrow \limits ^{\lim } \infty \frac{n\left( {11+1/n} \right) }{n\left( {1-3/n} \right) }=11\ne 0,\quad \hbox {it has a finite value.} \end{aligned}$$

(14)

Hence, the terms of a sequence have a finite limit value and are bounded so the considered sequence is convergent sequence.

$$\begin{aligned} \hbox {Take the sequence}\,\,u_{n} =\frac{8n^{2}-2}{n^{2}-15} \end{aligned}$$

(15)

Apply convergence theorem,

$$\begin{aligned} n\mathop \rightarrow \limits ^{\lim } \infty u_n =n\mathop \rightarrow \limits ^{\lim } \infty \frac{8n^{2}-2}{n^{2}-15}=n\mathop \rightarrow \limits ^{\lim } \infty \left( {\frac{n^{2}}{n^{2}}\left[ {\frac{8-2/{n^{2}}}{1-{15}/{n^{2}}}} \right] } \right) =8\ne 0,\quad \hbox {it has a finite value}. \end{aligned}$$

(16)

Hence, the terms of a sequence have a finite limit value and are bounded so the considered sequence is convergent sequence.