In this paper, we propose a novel approach, termed as regularized least squares fuzzy support vector regression, to handle financial time series forecasting. Two key problems in financial time series forecasting are noise and non-stationarity. Here, we assign a higher membership value to data samples that contain more relevant information, where relevance is related to recency in time. The approach requires only a single matrix inversion. For the linear case, the matrix order depends only on the dimension in which the data samples lie, and is independent of the number of samples. The efficacy of the proposed algorithm is demonstrated on financial datasets available in the public domain.
Over the last decade, support vector machines (SVMs) have emerged as the paradigm of choice for pattern classification and regression (Burges, 1998, Cristianini and Shawe-Taylor, 2000, Smola, 1996 and Vapnik, 1998). SVMs emerged from research in statistical learning theory on how to trade off structural complexity against empirical risk. The “maximum margin SVM” is amongst the most popular of SVM classifiers. It aims to minimize an upper bound on the generalization error through maximizing the margin between two disjoint half planes (Burges, 1998 and Vapnik, 1998).
Support vector regression (SVR) aims to fit a linear regressor through a given set of data points, where the points may be in the pattern space or in a higher dimensional feature space. Determining such a regressor requires solving a quadratic minimization problem subject to linear inequality constraints, which is a convex programming task (Burges, 1998).
Typical technical data involved in financial time series prediction are close value (price of the last performed trade during the day), highest traded price during the day, lowest traded price during the day, and volume (total number of traded stock during the day) (Pissarenko, 2002). Financial time series illustrate regime shifting, i.e. their statistical properties vary with time (the process is time-varying (Hellström and Holmström, 1998 and Pissarenko, 2002)).
A common property of financial time series is volatility clustering, i.e. large changes tend to succeed large ones, while small changes are followed by small ones. This, combined with a lack of long-term stationarity, suggests that a conventional SVM approach that lays equal emphasis on samples in a sequence would find it does hard to capture any input–output relationship inherent in the data. It also indicates that any attempt at predicting a future sample from past ones might benefit by giving more emphasis to recent samples, as against older ones. Some fuzzy SVM techniques have attempted to overcome this problem (e.g. Jayadeva et al., 2004, Lin and Wang, 2002 and Lin and Wang, 2005).
Taking motivation from Jayadeva et al., 2004, Lin and Wang, 2002, Lin and Wang, 2005 and Tay and Cao, 2002, we propose a novel approach to support vector regression for financial time series forecasting, whose preliminary ideas have been reported in Jayadeva, Khemchandani, and Chandra (2006). This approach is motivated by the observation that in a non-stationary time series, the dependency between the input variables and the output does not remain constant over time. Specifically, recent data samples provide more relevant information than more distant ones. The proposed approach assigns fuzzy membership values to data samples. This not only helps capture the input–output relationship in a better way, but also reduces overfitting. The proposed approach requires only one matrix inversion to determine the regressor. Experimental results on selected financial datasets available in the public domain show that the proposed approach not only yields comparable results, but is also faster than other reported approaches.
The paper is organized as follows: Section 2 briefly dwells on support vector regression and also introduces the notation that is employed in the rest of the paper. Section 3 discusses least squares support vector machines. Section 4 proposes regularized least squares fuzzy support vector regression, and contains some theoretical results. Section 5 is devoted to experimental results. Section 6 is devoted to concluding remarks.
In this paper, we have proposed a novel approach to support vector regression for financial forecasting, termed as regularized least squares fuzzy SVR (RLFSVR). RLFSVR uses knowledge of the noise and non-stationarity associated with the financial time series data samples, to improve generalization. RLFSVR requires only a single matrix inversion for finding the regressor, regardless of the kernel used. When a linear kernel is used, an added benefit is that the order of the matrix to be inverted is the input dimension, and does not depend on the number of data samples. Rectangular kernels can be used to reduce the computational burden in the case of nonlinear kernels. Experimental results demonstrate the efficacy of the regressor.
