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%%\title{\mbox{When an iris flower is normal then are others abnormal?}}
\title{A weighting function approach for neural network 
$~~~~~~$nonlinear time series analysis of satellite remote sensing of rainstorms
}
\author{{\bf Lisheng Xu, Jilie Ding , and Xiaobo Deng} \\
            \normalsize Satellite Remote Sensing and Nonlinear Sciences Lab \\ [\bs]
            \normalsize China Meteorological Administration$-$Atmospheric Sounding Key Laboratory \\ [\bs]
            \normalsize Chengdu University of Information Technology , Chengdu, SC610041, China \\[\bs]}
            


       
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\maketitle

\begin{abstracttext}
\noindent
\textit{One of frequently used neural networks, i.e., a radial-based function network (RBFN) with 
Gaussian activation functions is employed to study the nonlinear time series by carrying out 
the characterization experiments for a GMS-5 satellite 11$\mu$m IR observations of rainstorm process. 
The proposed methodology mainly uses RBFN to approximate the nonlinear time series signal first; 
then the characteristics of its weighting functions changed with time are analyzed. 
The difficulty due to the effects of high noise on the signal processing using neural networks is addressed.
 Thus, finally a more integrated method combining the neural network analysis with wavelet packet
 decomposition is introduced. The preliminary results show that the proposed approach for nonlinear
 time series analysis is efficient and promising.}
\end{abstracttext}


\bigskip
\section{INTRODUCTION}
\noindent
In recent years, extreme weather events, such as severe storms and extreme rainfalls, have
been increased likely worldwide, especially over Europe and Asian monsoon regions [1], which 
have caused, and will still frequently cause great or sudden disasters, with terrible accidents 
of human lives and economic society in the world. The knowledge of formation and evolution of rainstorms, 
and their forecasting are crucial to studies of weather, climate, and environment problems. It is difficult
 to solve the satellite retrieval problems in cloudy atmospheres with rainstorms using the traditional and 
classical optical retrieval approach. Therefore, it is imperative to develop new and modern retrieval theories
 and methods for satellite optical remote sensing of storms.

\bigskip
\noindent
We can imagine that the evolution of storm process is a nonlinear dynamic process, with chaos 
and fractal properties [2]. Thus, to develop a new and modern satellite remote sensing method to reveal 
the occurrence and evolution process of severe storms, we would like to combine satellite remote sensing 
with nonlinear sciences to study the issues. 

\bigskip
\noindent
Chaotic systems are an important class of dynamical systems. Often, the only information we have about
such systems is in the form of a time series. The process of analyzing time series constitutes a field of 
science known as time-series analysis. Its objective is to build a model for the unknown dynamical system
that generated the time series. 

\bigskip
\noindent
Chaotic time-series analysis, or nonlinear time-series analysis, cannot be studied satisfactorily by 
linear time series analysis, which fails to detect any nonlinear correlations present and cannot provide a 
complete characterization of the underlying dynamics and, thus, describe the nonlinear structure in chaotic 
time series. However, uncovering the deterministic structure is important because it allows for construction 
of more realistic and better models and thus improved predictive capabilities.

\bigskip
\noindent
Over the last two decades many nonlinear time series methods have been developed in the theory of nonlinear
 dynamics, commonly known as chaos theory. Since artificial neural networks are a high complex nonlinear dynamical
 system with powerful signal processing capability and have all demonstrated superior performance in many engineering 
applications, in this paper we would like to use neural networks combined with wavelet analysis to study the nonlinear
 time series of satellite remote sensing of rainstorm process. The preliminary results show that the proposed approach 
of neural network weight series combined with wavelet packet decomposition has potential for nonlinear time series 
analysis with high noise effects. 


\section{Neural network nonlinear time series prediction}
\noindent
Neural networks have been applied to many areas of statistics, classification and pattern recognition, and time
series analysis. In many areas of statistics, especially in time series analysis, neural networks play an important role.
The historical development of neural computation is written in some books [3,4]. 

\bigskip
\noindent
There are two kinds of time series, i.e., linear time series and nonlinear time series, or chaotic time series.
For time series analysis, the very first object of the study is forecasting, which is one of controversial domain and 
the subject of a tremendous effort in research and development. Time series forecasting can be studied using 
well-established statistical models, which, however, have some drawbacks as pointed out by Hill et al. [5] and Kajitani
et al. [6], thus, are not suitable for nonlinear time series forecasting. Some approaches for nonlinear time series 
forecasting are commonly used, for example, chaos model, neural network model, and random work model, etc.. Especially, 
since the last decade in 20 century, a considerable attention has been devoted to dynamical system theory for the study 
of nonlinear time series. It is worth pointing out that the starting point of dynamical system time series analysis is 
the determination of past series information to be used for model building. Once this is done, models can be constructed 
through several ways, and neural networks are natural candidates for this task because of their universal approximation 
properties. Neural networks, for example, the feed forward neural network (FFNN) model or the radial based function neural 
network (RBFN) model, have been found to work as well or better than many competing models mentioned, especially 
outperformed the classical random walk model [6-9].

\bigskip
\noindent
Golovko et al. [10] discuss a synthetic use of neural networks to chaotic signal processing, including nonlinear time 
series analysis, the identification of chaotic behavior, forecasting and dynamical reconstruction. Pavlidis et al. [11] 
also propose a time series forecasting methodology that draws from the disciplines of chaotic time series analysis, 
clustering, and neural networks, and apply it to perform multi-step-ahead prediction.

\bigskip
\noindent
For the purpose of nonlinear time series forecasting, in the present study, we shall alternatively use 
NN weighting function �Cbased approach to characterize the nonlinear time series, first.
\\

\bigskip
\section{Radial basis function neural network}
\noindent
In the literature a large variety of neural networks has been proposed for modeling the dynamic behavior of a system.
Three types of neural networks are frequently used, i.e., the FFNN, the RBFN, and the Elman neural network [12]. 

\bigskip
\noindent
Jayawardena et al.[9] found that RBFN is similar to FFNN. The RBFN method has the advantage that it does not require
a long calculation time and does not suffer with the overtraining problem. More neurons may be required for RBFN than 
standard feed-forward back propagation (BP) networks, but often they can be designed in a fraction of the time it takes
to train standard feed-forward networks. They work best when many training vectors are available [13].

\bigskip
\noindent
It is commonly known that linearity in parameters in RBFN allows the use of least squares error based updating 
schemes that have faster convergence than the gradient-descent methods used to update the nonlinear parameters of 
multi-layer BPNN.

\bigskip
\noindent
Furthermore, Gaussian-like radial based functions (RBFs) are local (give a significant response only in a neighbourhood 
near the centre) and are more commonly used than multiquadric-type RBFs which have a global response. Its expression is 
simple and the analyticity is good. Thus, in this paper, RBFN with Gaussian activation functions is employed.

\bigskip
\section{Results and discussions}
\noindent
A severe rainstorm process with sudden occurrence and great floods and disasters occurred in Wuhan area located in the 
mid basin of Yangtze River in China on 21-27 July 1998. The corresponding observation time series of Geostationary 
Meteorological Satellite (GMS)-5 11$\mu$m IR channel brightness temperatures (BT) are used in this study since GMS-5 
observations have higher time resolution than those of the polar-orbiting satellites. 

\bigskip
\noindent  
Fig. 1 depicts the GMS-5 observation time series $S(t)$ in Wuhan in July 1998(solid line),  and its approximation 
time series $S(t)_{NN}$ by RBFN (dashed line). Fig. 2 shows the relative error of the approximation changes with time t, 
demonstrating that the approximation of RBFN to the signal of the GMS-5 observation time series $S(t)$ is very good. 
The weights of $S(t)_{NN}$ changing with time are plotted in Fig. 3. It is seen from Fig. 3 that at $t{\approx}$300 hours, i.e., 
on 12-13 of July, there is a dramatic change in the weight time series, which indicates that before about one week 
there is a remarkable sign in the weight time series $S(t)_{NN}$ prophetic of the occurrence of severe rainstorm process.
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\bigskip
\noindent
In our further study of the same rainstorm process using the new quantities defined by inter-discipline of fractal and 
wavelet packet, similar results are obtained, which will be published elsewhere. 

\bigskip
\noindent
The effects of noise on the weight time series $S(t)_{NN}$are shown in Fig.4, assuming that the time series $S(t)$ is 
contaminated by the noise with the ratio of noise to signal being 0.01, 0.05 and 0.1, respectively. 

\bigskip
\noindent 
It is obvious by comparison of Fig.3 and Fig.4 that with the increase of noise, the characteristic with the remarkable
sign in the weight time series $S(t)_{NN}$ mentioned above is disappeared gradually. So, signal processing using neural network 
time series analysis with high noise is also a challenging problem.

\bigskip
\noindent
To overcome the difficult, we propose a method combined wavelet packet analysis with RBFN technique. The objective 
of the method is to remove the effect of noise. Define a new quantity as:
\begin{eqnarray}
S^*(t) = S(t)+ {\delta} S_m {\eta}
\end{eqnarray}
where $S_m = <S(t)>$, i.e., the average of $S(t)$, $\delta$ denotes a parameter which represents noise level, $\eta$ denotes 
a random number changing from 0 to 1. The second term in the right hand side of Eq. (1) represents a random noise. 

\bigskip
\noindent
The wavelet transform of a signal evolving in time provides a tool for time-frequency localization [14]. It is known that 
the wavelet packets method is a generalization of wavelet decomposition, which can be used for numerous expansions of a 
given signal, and thus, offers a richer signal analysis. Thus, for analyzing the time series $S^*(t)$, the discrete wavelet 
packet transform (DWPT) [14,15] is used. Fig.5 shows the DWPT decomposition with level $j$ = 1 for the time series $S^*(t)$ and 
the 2 elements in the first level in the decomposition. It is obvious that the element $s_{10}$ is mainly composed of low 
frequencies, and the element $s_{11}$ is composed of high frequency including the noise. 

\bigskip
\noindent
Motivated by the results of the wavelet packet decomposition, we perform the RBFN time series analysis for the element 
$s_{10}$. The weights of the signal $s_{10NN}$ approximated by RBFN to the time series $s_{10}$ changing with time $t$ are plotted in 
Fig.6, which depicts the effects of the noise with $\delta$= 0.1 on the weight time series. Obviously, the dramatic change 
of the weight series at $t{\approx}$300 hours still appears in Fig.6. This indicates that the approach of combination of
wavelet packet with RBFN is effective for the signal processing with higher noise.  

\bigskip
\section{CONCLUSION}
\noindent
Forecasting of extreme weather and climate events using satellite remote sensing time series is important and of signal
processing problems which is challenging due to small sample sizes, non-linearity, and complexity of the system. Neural 
networks have been very successful in a number of signal processing applications. 

\bigskip
\noindent
This paper presents an approach for a nonlinear time series analysis and applies it to characterize the GMS-5 satellite
11$\mu$m IR channel BT observation time series of rainstorm process and the noise effects are addressed. The proposed approach 
draws from the disciplines of the neural network and wavelet analyses. First, the RBFN is used to approximate the nonlinear 
time series. Then, the weight series is analyzed for characterizing the time series for its further forecasting study. 
Finally, since the effects of noise on the signal processing often occur, the DWPT decomposition is employed for overcoming 
the difficulty. The preliminary results demonstrate that this approach for characterization of nonlinear time series with 
higher noise is efficient and promising and, thus, needs further effort in research and development.



\bigskip
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