Mathematical Problems in Engineering

Volume 2015, Article ID 385876, 4 pages

http://dx.doi.org/10.1155/2015/385876

## Wavelet Network Model Based on Multiple Criteria Decision Making for Forecasting Temperature Time Series

School of Environmental Sciences, Beijing Normal University, Beijing 100875, China

Received 24 October 2014; Revised 18 January 2015; Accepted 25 January 2015

Academic Editor: Hector Puebla

Copyright © 2015 Jian Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to nonlinear and multiscale characteristics of temperature time series, a new model called wavelet network model based on multiple criteria decision making (WNMCDM) has been proposed, which combines the advantage of wavelet analysis, multiple criteria decision making, and artificial neural network. One case for forecasting extreme monthly maximum temperature of Miyun Reservoir has been conducted to examine the performance of WNMCDM model. Compared with nearest neighbor bootstrapping regression (NNBR), the probability of relative error smaller than 10% increases from 65.79% to 84.21% (forecast period ) and from 51.35% to 91.89% by WNMCDM model. Similarly, the probability of relative error smaller than 20% increases from 84.21% to 97.37% and from 81.08% to 91.89% by WNMCDM model. Therefore, WNMCDM model is superior to NNBR model in forecasting temperature time series.

#### 1. Introduction

Temperature time series are closely related to human life. The accurate prediction of the temperature time series offers important information for the city planning, land use, the design of civil project, and water resource management. Meteorological system is the result of the comprehensive effect of climate factors and human activity factors [1]. It is difficult for a single method to establish an effective model [2].

In recent years, wavelet analysis has become a research boom. It has huge advances in signal processing, image compress and encoding, tongue encoding, mode identification, and nonlinear science fields. Wavelet analysis has good multitime and scale features, which provides useful decompositions of original time series; so wavelet-transformed data improves other models the forecasting ability by capturing useful information on various resolution levels. The document [3] pointed out the potential applications of wavelet analysis to analyze temperature series. Jones and Moberg [4] studied the multiscale characteristics of temperature. Therefore, wavelet analysis has made great progress in the analysis of temperature time series.

Artificial neural network (ANN) has shown great ability in modeling and forecasting nonlinear and nonstationary time series in meteorology and water resources engineering due to its adaptive, self-organizing, self-learning ability. Campolo et al. [5] reported that their ANN model had better prediction accuracy and flexibility than statistical regression and simple conceptual models. Fan and Fu [6] presented an improved BP algorithm to optimize weights of neural network and achieved great prediction effect. In conclusion, ANN is a good method to predict temperature series.

Wavelet neural network model for predicting time series has become a hot study area since Zhang and Benveniste firstly proposed concepts and algorithms of wavelet neural network (WNN) and applied it for chaotic time series prediction [7]. The wavelet network model achieved good results in studying and predicting chaotic time series. Lv and Zhao [8] indicated that wavelet network method is more accurate than neural network from simulation results and can be effectively used in the prediction of nonlinear time series. Wang and Ding [9] revealed that wavelet network method could increase the forecasted accuracy and prolong the length time of prediction. But the hidden nodes are difficult to decide, and multiple criteria decision making can solve this problem. Wavelet network model based on multiple criteria decision making (WNMCDM) is firstly proposed in this paper. We will introduce the theory of WNMCDM model and prove the feasibility and accuracy by a practical case.

#### 2. WNMCDM Model

Each part of wavelet network model based on multiple criteria decision making (WNMCDM) plays an importance role in forecasting time series. Wavelet analysis is used to determine cycle of temperature series and obtain high and low frequency components. Then artificial neural network is applied to predict future temperature by using above high and low frequency components. In the meantime, multiple criteria decision making is critical in determining the hidden nodes of ANN. The specific steps are shown in following context.

*Step 1 (wavelet analysis). *Discrete wavelet transform is selected to decompose and reconstruct the time series because observed time series in the real world are usually discrete, such as monthly runoff series and monthly temperature series [10, 11]. We adopt common discrete wavelet transform a trous in this paper. Complex time sequence is decomposed into different frequency blocks by [12]
where is the discrete low-pass filter; in this paper, spline defined as is used [13]; , are background information (low frequency) and detail information (high frequency) and ; is the scale which generally takes the natural logarithm of . are called discrete wavelet transform with the resolution level .

*Step 2 (artificial neural network). *Recent studies have shown that three-layer ANN network model can depict any complex nonlinear function, which basically solves the forecasting and simulation work. So three-layer neural network is also suitable for predicting temperature series. But input data needs to be standardized firstly by limiting to the range . The input of BP network is , and the number of nodes is . The output is , and the number of nodes is 1. Figure 1 shows the structure of ANN. Hidden layer nodes are determined by multiple criteria decision making by Step 3. Conjugate gradient momentum BP algorithm has been adopted in this paper.