A novel criterion to select hidden neuron numbers in improved back propagation networks for wind speed forecasting

Abstract

This paper analyzes various earlier approaches for selection of hidden neuron numbers in artificial neural networks and proposes a novel criterion to select the hidden neuron numbers in improved back propagation networks for wind speed forecasting application. Either over fitting or under fitting problem is caused because of the random selection of hidden neuron numbers in artificial neural networks. This paper presents the solution of either over fitting or under fitting problems. In order to select the hidden neuron numbers, 151 different criteria are tested by means of the statistical errors. The simulation is performed on collected real-time wind data and simulation results prove that proposed approach reduces the error to a minimal value and enhances forecasting accuracy The perfect building of improved back propagation networks employing the fixation criterion is substantiated based on the convergence theorem. Comparative analyses performed prove the selection of hidden neuron numbers in improved back propagation networks is highly effective in nature.

Appendix

Considers different criterion ‘n’ as input parameters number. All considered criteria are satisfied based on the convergence theorem. If the limit of sequence is finite, the sequence is called convergent sequence. If the limit of the sequence does not tend to a finite number, the sequence is called divergent [36].

The convergence theorem characteristics are given below.

• 1. A convergent sequence has a finite limit.

• 2. All convergent sequences are bounded.

• 3. All bounded point has a finite limit.

• 4. Convergent sequence needed condition 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 stable network. In the neural network model most important property is it always converges to a stable state. For real-time optimization problem the convergence play a major role, risk of getting stuck at some local mini ma problem in a network prevented by the convergence. The convergence of sequence infinite has been established in convergence theorem because of the discontinuities in model. The real-time neural optimization solvers are designed using the convergence properties.

Discuss convergence of the considered sequence as follows. Taking the sequence

$$ u_{n} =\frac{5n+2}{n-4} $$

(A.a)

Apply convergence theorem

$$\begin{array}{@{}rcl@{}} n\overset{\lim}{\to} \infty \,u_{n} &=&n\overset{\lim}{\to} \infty \frac{5n+2}{n-4}=n\overset{\lim}{\to} \infty \,\frac{n\left( 5+^{2}{}/_{n}\right)}{n\left( 1+^{4}{}/_{n}\right)}\\ &&=5\ne 0,\,\,\text{it has finite value.} \end{array} $$

(A.b)

Hence, the sequence possesses a finite limit value and bounded so the considered sequence is convergent in nature.

Take the sequence

$$\begin{array}{@{}rcl@{}} u_{n} =\frac{4n^{2}+7}{n^{2}-24} \end{array} $$

(A.c)

Apply convergence theorem

$$\begin{array}{@{}rcl@{}} n\overset{\lim}{\to} \infty \,u_{n}&=&n\overset{\lim}{\to} \infty \frac{4n^{2}+7}{n^{2}-24}=n\overset{\lim}{\to} \infty \left( \frac{n^{2}}{n^{2}}\left[\frac{4+^{7}{}/_{n^{2}}}{1-^{24}{}/_{n^{2}}}\right]\right)\\ &&=4\ne 0,\,\,\text{it has finite value.} \end{array} $$

(A.d)

Hence, the sequence possesses a finite limit value and bounded so the considered sequence is convergent in nature.