A Morphological-Rank-Linear evolutionary method for stock market prediction
Introduction
Financial time series forecasting is considered a rather difficult problem, due to the many complex features frequently present in time series, such as irregularities, volatility, trends and noise. For such, a widely number of linear and nonlinear statistical models have been proposed in order to predict future tendencies of financial phenomena based on present and past historical data [4], [32], [26], [31], [33], [8].
Alternatively, approaches based on Artificial Neural Networks (ANNs) have been successfully proposed for nonlinear modeling of time series in the last two decades ([25], [7], [42], [18], [38], [5], [15], [30], [41], [12]). However, in order to define a solution to a given problem, ANNs require the setting up of a series of system parameters, some of them are not always easy to determine. The ANN topology, the number of processing units, the algorithm for ANN training (and its corresponding variables) are just some of the parameters that require definition [9]. In this context, hybrid intelligent approaches have produced interesting results [19], [22], [1], [3], [9].
However, a dilemma arises from all these models regarding financial time series, known as random walk dilemma [38], where the predictions generated by such models show a characteristic one step delay regarding original time series data. This behavior has been seen as a dilemma regarding the financial time series representation, where it has been posed that the series follow a random walk like model and cannot, therefore, be predicted [20].
In this context, this work presents an Evolutionary Morphological-Rank-Linear Approach to overcome the random walk dilemma. The proposed Evolutionary Morphological-Rank-Linear Forecasting (EMRLF) method is inspired on Takens theorem [39] and consists of an intelligent hybrid model composed of a Morphological-Rank-Linear (MRL) [27] with a Modified Genetic Algorithm (MGA) [19], which searches for the particular time lags capable to optimally characterize the time series and estimates the initial (sub-optimal) parameters of the MRL filter (mixing parameter , rank , linear FIR filter and the MR filter coefficients). Then, each individual of the MGA population is improved by the Least Mean Squares (LMS) algorithm to further adjust the MRL filter parameters supplied by the MGA. After model training, the EMRLF method chooses the most fitted forecasting model, and performs a behavioral statistical test [9] in the attempt to adjust time phase distortions observed in financial time series.
An experimental analysis is conducted with the proposed method using four real world stock market time series (Directv Group Inc Stock Prices, Microsoft Corporation Stock Prices, Petrobras Company Stock Prices and Yahoo Inc Stock Prices), employing five well-known performance metrics to assess the performance of the method. The results achieved by the EMRLF method have shown a much better performance when compared to MultiLayer Perceptron (MLP) networks, and a better performance when compared to a previous hybrid model, named the Time-delay Added Evolutionary Forecasting (TAEF) method [9].
This paper is organized as follows. Section 2 presents the fundamentals of time series forecasting, the MGA description and the concepts of the MRL filter. Section 3 describes the proposed EMRLF method. Section 4 shows the performance measures used to compare assertiveness of all methods. In Section 5, simulations and experimental results are described with the EMRLF method, MLP networks and the TAEF algorithm [9] for two relevant stock market time series: the Coca-Cola Company and Microsoft Corporation Stock Prices. Section 6 presents, to conclude, the final remarks of this work.
Section snippets
The time series prediction problem
A time series is a sequence of observations about a given phenomenon, where it is observed in discrete or continuous space. In this work all time series will be considered time discrete and equidistant.
Usually, a time series can be defined bywhere t is the temporal index and N is the number of observations. The term will be seen as a set of temporal observations of a given phenomenon, orderly sequenced and equally spaced.
The aim of prediction techniques applied to a given
The proposed method
The proposed Evolutionary Morphological-Rank-Linear Forecasting (EMRLF) method consists of an intelligent hybrid model, which uses a Modified Genetic Algorithm (MGA) [19] to adjust the initial MRL filter parameters and then it uses the LMS algorithm to further improve the parameters supplied by the MGA. The advantage of those models is that not only they have linear and nonlinear components, but are quite attractive due to their simpler computational complexity when compared to other approaches
Performance evaluation
Many performance evaluation criteria are found in literature. However, most of the existing literature on time series prediction frequently employ only one performance criterion for prediction evaluation. The most widely used performance criterion is the Mean Squared Error , given bywhere N is the number of patterns, is the desired output for pattern j and is the predicted value for pattern j.
The measure may be used to drive the prediction
Simulations and experimental results
A set of four financial time series was used as a test bed for evaluation of the EMRLF method: Directv Group Inc Stock Prices, Microsoft Corporation Stock Prices, Petrobras Company Stock Prices and Yahoo Inc Stock Prices. All series investigated were normalized to lie within the range and divided into three sets according to Prechelt [29]: training set (first 50% of the points), validation set (second 25% of the points) and test set (third 25% of the points).
The MGA parameters used were
Conclusion
An evolutionary morphological-rank-linear approach was presented in order to overcome the random walk dilemma for financial time series forecasting. The experimental results used five different metrics for model evaluation, Mean Square Error (MSE), Mean Absolute Percentage Error (MAPE), Normalized Mean Square Error (NMSE), Prediction Of Change In Direction (POCID) and Average Relative Variance (ARV), demonstrating a consistent much better performance of the proposed model when compared to the
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