Time-series forecasting using GA-tuned radial basis functions
Introduction
Time-series analysis is an important tool for forecasting the future in terms of past history. A time-series is a sequence of values measured over time, in discrete or continuous time units. By studying many related variables together, a better understanding is often obtained. Robust forecasting must rely on how well the time-series is designed.
Many techniques for time-series analysis have been developed assuming linear relationships among the series variables. Unfortunately, many real-world applications involve nonlinearities between environmental variables. Assuming simple relationships among time-series variables can produce poor results regarding the ability to predict the future. In many cases, such inaccuracies can produce major problems. For example, forecasting river flows or business and economic trends needs to be handled very carefully to produce accurate results. In [1], [2], a description of some of the drawbacks of linear modeling for time-series analysis was introduced. This includes for example, their inability to explain the sudden bursts of very large amplitudes at irregular time intervals. It seems necessary, therefore, that nonlinear models be used for the analysis of the real-world temporal data. In [3], modeling for the multi-variable time-series using statistical nonlinear models was addressed.
Time-series analysis and design have attracted a considerable number of interest from the evolutionary computation community to study, model and forecast their behavior. In [4], a GA-based approach was developed for system identification and time-series prediction problems. This approach is called STROGANOFF (STructured Representation On Genetic Algorithms for NOn-linear Function Fitting). ESs have been used in the identification of nonlinear communication channels based on an evolutionary Volterra time-series in [5]. GP was applied to the task of chaotic and time-series prediction problem in [6].
In this paper we develop a nonlinear auto-regressive (NAR) time-series model structure for forecasting applications. GAs are used to tune the RBF parameters: the centers and widths of the radial functions, and their weights such that a global optimal solution for the forecasting problem can be reached. An advantage of this approach is that no assumptions are made about the RBF structure in contrast to more traditional methods [7]. A method for efficient automatic selection of all of the RBF parameters represents a potentially significant advance in the state of the art in time-series analysis and design. We present an initial evaluation of this approach via a comparison between a traditional AR model with parameters tuned using least-squares estimation (LSE) and the proposed AR-RBF model tuned using GAs, both applied to the time-series data representing the weekly averaged exchange rates between the British pound and the US dollar over a four-year period.
Section snippets
Structure of RBF networks
The problem of interpolation of real multivariable time-series can be expressed as follows. Let us consider a set of N data points in the input space Rd, together with their associated desired output values in R:
This data set can be used to characterize a function with one-dimensional output values; multi-dimensional interpolation can be done by generalizing the following equations and algorithms, while considering separately each component of the output vectors.
Traditional AR model
One of the most famous models used in forecasting applications is the auto-regressive model. This model has been studied in many applications [8]. The AR model can be described in the following form:where y(k−τi) is the past system output. The value of n is referred to as the “order” of the model. Some of the disadvantages of this model structure are:
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this model is a linear model and in many cases it is unable to track abrupt changes in measured observations;
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even when we go
AR-RBF model with LSE-tuning
The proposed structure of the AR model is described by adding nonlinearity to the model dynamics as follows:where h(·) is an RBF with the format presented in Eq. (4), wi represents the weight of the RBF, n is the order of the model, and k is a selected constant gain for the time-series.
Our goal is to provide a time-series which will produce an estimated response that best matches the actual response (i.e. the stock market response) and be able to forecast the future. To
Proposed AR-RBF model with GA-tuning
Tuning RBFs means finding the optimal values of the parameters ci, ri and wi used in , . These parameters need to be adjusted to get the best approximation of the function f. GAs are used to find the global optimal tuning parameters for the AR-RBF time-series model.
The proposed representation for GAs to solve this problem is based on coding all parameters cj, rj and wj as genes in a string representation. Candidate solutions (individuals) in this case are just i-dimensional vectors of
Data set
A set of real data is employed in the investigation of forecasting problem. The data represents the weekly averaged exchange rates between the British pound and the US dollar during the period 31 December 1979 to 26 December 1983 [9]. There are 209 observations. The available data were split into two sets, one for training and the other for testing.
AR-RBF tuning using LSE
Using LSE, we estimated the parameters of the time-series model using 100 observations. The values of the model parameters a's, in Eq. (6), were
Conclusions and future work
In this paper we introduced an AR-RBF time-series model which has been successfully used for forecasting applications. The model parameters were automatically tuned using GAs with promising results. A comparison between LSE and GAs for time-series modeling was provided. Our future work will concern the application of the GA-based approach to forecast nonlinear time-series with abrupt changes in measured observations which traditional AR models are unable to model well.
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