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This paper analyzes and examines the general ability of Support Vector Machine (SVM) models to correctly predict and trade daily EUR exchange rate directions. Seven models with varying kernel functions are considered. Each SVM model is benchmarked against traditional forecasting techniques in order to ascertain its potential value as out-of-sample forecasting and quantitative trading tool. It is found that hyperbolic SVMs perform well in terms of forecasting accuracy and trading results via a simulated strategy. This supports the idea that SVMs are promising learning systems for coping with nonlinear classification tasks in the field of financial time series applications.


Introduction
Support Vector Machines (SVMs) have proven to be a principled and very powerful supervised learning system that since its introduction (Cortes and Vapnik (1995)) has outperformed many systems in a variety of applications, such as text categorization (Joachims (1998)), image processing (Quinlan et al. (2004)), and bioinformatic problems (Brown et al. (1999)). Subsequent applications in time series prediction (M¨uller et al. (1999)) indicate the potential that SVMs have with respect to economics and finance. In predicting Australian foreign exchange rates, Kamruzzaman and Sarker (2003b) showed that a moving average-trained SVM has advantages over an Artificial Neural Network (ANN) based model, which was shown to have advantages over ARIMA models (2003a). Furthermore, Kamruzzaman et al. (2003) had a closer look at SVM regression and investigated how it performs with different standard kernel functions. It was found that Gaussian Radial Basis Function (RBF) and
polynomial kernels appear to be a better choice in forecasting the Australian foreign exchange market than linear or spline kernels. Although Gaussian kernels
are adequate measures of similarity when the representation dimension of the space remains small, they fail to reach their goal in high dimensional spaces (Francois et al. (2005)).We will examine the general ability of SVMs to correctly classify daily EUR/GBP, EUR/JPY and EUR/USD exchange rate directions. It is more useful for traders and risk managers to predict exchange rate fluctuations than their levels. To predict that the level of the EUR/USD, for instance, is close to the level today is trivial. On the contrary, to determine if the market will rise or fall is much more complex and interesting. Since SVM performance mostly depends on choosing the right kernel, we empirically verify the use of customized p-Gaussians by comparing them with a
range of standard kernels. The remainder is organized as follows: Section 2 outlines the procedure for obtaining an explanatory input dataset. Section 3 formulates the SVM as applied to exchange rate forecasting and presents the kernels used. Section 4 describes the benchmarks and trading metrics used for model evaluation. Section 5 gives the empirical results. The conclusion, as well as brief directions for future research, are given in Section 6.

This paper presents the analytic recursive formulas of Crame´r-Rao lower bound (CRLB) for the switching models system, in which the target moves either with a constant velocity or with a constant speed and a constant turn rate. For the case of two-observer bearings-only maneuvering target tracking, a reliable maneuver detection method is investigated and then utilized to approximate the theoretic CRLB. Finally, to demonstrate the agreement between the approximated CRLB using the proposed maneuver detection method and the theoretic one, a large number of Monte-Carlo runs under different maneuvering scenarios are conducted. Correctness of the analytic recursive formulas of CRLB and effectiveness of the proposed maneuver detection method are verified from these simulations.


1. Introduction
For a general nonlinear filtering problem, the optimal recursive state estimator in the Bayesian sense needs the complete posterior density of the state. As is well known, such a problem has no analytic closed-form solution. As a consequence, the solution of the nonlinear filtering is usually approximated by means of extended
Kalman filter, unscented Kalman filter [9], particle filter [16] and so on. In practical applications, many researchers usually attempt to derive the theoretically best achievable second-order error performance for nonlinear filtering to evaluate these filters, and we call it Crame´r-Rao lower bound (CRLB).

The Crame´r-Rao lower bound, defined as the inverse of the Fisher information matrix (FIM), represents an objective lower limit of cognizability of parameters in constant or random parameter estimation. It has been widely used in many cases, such as bearings-only tracking [13,20,21], ballistic target tracking [8] and so on. Due to its ability of predicting the best achievable performance, it is usually utilized as a benchmark to evaluate the performance of an estimation algorithm and can provide guidance to improve the experimental design as well. Therefore, the discussion on CRLB is always a hot topic in the field of target motion analysis (TMA).

A complete history of the developments of the CRLB for target tracking would involve more than sixty publications (e.g., [3–7,10,12–15,17–19]), and its attention varies from single target to multiple targets, from constant parameter estimation to random parameter estimation, from clutter-free to clutter, etc. In many practical tracking scenarios, due to high frequency of maneuver occurrence, the corresponding discussion
on the CRLB seems necessary and imminent. However, few reports on the CRLB of maneuvering target tracking can be found so far, even so, most of them are based on such an assumption that the model history is already known [13]. Therefore, the aim of this paper is to investigate the performance bounds of the switching-models-based bearings-only maneuvering target tracking system, in which the target moves either with a constant velocity or with a constant speed and a constant turn rate. It is noted that, in bearing-only tracking, if a single observer is collecting angular measurements, the target state becomes observable only if the observer ‘‘outmaneuvers” the target, i.e., observer motion is one derivative higher than that of target and a component of this motion is perpendicular to the line of sight. However, for a two-observer bearings-only tracking system to be discussed in this paper, it is not the case, and the target is always observable only if the target does not move on the line connecting the two observers, thus the observability will not be addressed later.

This work takes inspiration mainly from Farina et al. [7], in which the detection probability of target was supposed to be less than unity. Thus we suppose in this paper that there is the possibility of target’s maneuver or non-maneuver at each sampling period. On the basis of such assumption, the recursive theoretic formulas
of CRLB for two-observer bearings-only maneuvering target are derived. Since the calculation of the theoretic CRLB relies on the exponentially growing number of maneuver/non-maneuver sequences, a reliable maneuver detector is developed to reduce the possible sequences to further approximate the theoretic CRLB.

The rest of this paper is arranged as follows. The target dynamics and measurement models for bearingsonly target tracking are formulated in Section 2. In Section 3, the recursive theoretic formulas of CRLB for two-observer bearings-only maneuvering target are derived, and moreover, a reliable maneuver detector for bearings-only tracking is developed and utilized to approximate the theoretic CRLB. Finally, performance evaluation and conclusions are given in Sections 4 and 5, respectively.