Source-linked AI summary
Multiple-output support vector regression with a firefly algorithm for interval-valued stock price index forecasting
Tao Xiong, Yukun Bao, Zhongyi Hu
TL;DR
The paper addresses the need for accurate interval-valued stock-index forecasts rather than point forecasts. It extends MSVR to forecast interval bounds jointly and uses FA to select its parameters, finding that FA-MSVR can outperform some established counterparts in forecast accuracy and competing-forecast accuracy.
Problem
Prior studies largely used point forecasts, while MSVR had rarely been evaluated for interval-valued time-series forecasting despite the importance of stock-price decision making.
Method
The study extends MSVR for interval-valued stock-index forecasting and uses a firefly algorithm to determine its parameters.
Results
FA-MSVR can statistically outperform some well-established counterparts in forecast accuracy and accuracy against competing forecasts.
Takeaways & Limitations
FA-MSVR is presented as a promising alternative for forecasting interval-valued financial time series.
Takeaways & Limitations
The study notes that applying the proposed model to other tough forecasting problems, such as exchange-rate forecasting, should be studied in the future.
Abstract
from arXiv · showhide
Highly accurate interval forecasting of a stock price index is fundamental to successfully making a profit when making investment decisions, by providing a range of values rather than a point estimate. In this study, we investigate the possibility of forecasting an interval-valued stock price index series over short and long horizons using multi-output support vector regression (MSVR). Furthermore, this study proposes a firefly algorithm (FA)-based approach, built on the established MSVR, for determining the parameters of MSVR (abbreviated as FA-MSVR). Three globally traded broad market indices are used to compare the performance of the proposed FA-MSVR method with selected counterparts. The quantitative and comprehensive assessments are performed on the basis of statistical criteria, economic criteria, and computational cost. In terms of statistical criteria, we compare the out-of-sample forecasting using goodness-of-forecast measures and testing approaches. In terms of economic criteria, we assess the relative forecast performance with a simple trading strategy. The results obtained in this study indicate that the proposed FA-MSVR method is a promising alternative for forecasting interval-valued financial time series.
Research Highlights
The study extends MSVR to interval-valued time-series forecasting, tunes its parameters with FA, and evaluates forecasting ability using statistical and economic criteria across one- and multi-step horizons.
- MSVR is extended to the scenario of interval-valued time series.
- The parameters of MSVR are tuned using the firefly algorithm.
- FA-MSVR forecasting ability is assessed using statistical and economic criteria.
- The experimental analysis uses one- and multi-step-ahead forecasts.
- FA-MSVR is identified as a promising method for interval forecasting of financial time series.
1. Introduction
Interval forecasts can represent stock-price variability and uncertainty, but prior work largely focused on point forecasts and standard SVR has limitations for jointly modeling interval bounds. The study addresses these gaps by extending MSVR and tuning it with FA for interval-valued stock-index forecasting.
- Research gap and motivation: Prior studies considered point forecasting rather than interval forecasting.
- Research gap and motivation: Interval forecasts account for variability and uncertainty relative to single-valued stock-price series.
- Research gap and motivation: Interval forecasts have been reported to achieve lower mean error and higher average accuracy ratio than traditional point forecasts.
- Proposed approach: The study extends MSVR to forecast lower and upper stock-index bounds simultaneously, using lagged intervals as inputs and two outputs for the bounds.
- Research gap and motivation: Standard SVR applications to interval-valued time series are limited because its single-output structure independently forecasts interval bounds without considering their interrelations.
- Proposed approach: FA-MSVR tunes MSVR parameters with a firefly algorithm and is compared with univariate, standard SVR, HoltI, VECM, and iMLP benchmarks.
- Contributions: The paper’s contributions include extending MSVR for interval-valued forecasting and assessing practicability with both statistical accuracy and economic criteria.
2. MSVR with an interval-valued stock index series
The paper formulates MSVR for interval-valued stock-index series by representing each interval through its bounds and jointly forecasting those bounds from lagged intervals. Its formulation uses a multi-output loss and an IRWLS solution, with FA selecting RBF-kernel parameters.
- 2.1 Construction of an interval-valued stock index series: Each interval is represented by lower and upper bounds, with the lower bound no greater than the upper bound.
- 2.1 Construction of an interval-valued stock index series: An interval can also be represented by its center and radius, where the radius is its half-range.
- 2.1 Construction of an interval-valued stock index series: An interval-valued time series is represented as a chronological sequence of interval-valued variables.
- 2.2 MSVR for interval-valued stock index forecasting: The MSVR model uses 2d lagged-interval inputs and two outputs corresponding to forecasts of the lower and upper bounds.
- 2.2 MSVR for interval-valued stock index forecasting: MSVR uses a nonlinear feature transformation and a penalty coefficient C that trades off regularization against error reduction.
- 2.2 MSVR for interval-valued stock index forecasting: Its quadratic epsilon-insensitive cost function accounts for all outputs when constructing each regressor and yields a single support-vector set across dimensions.
- 2.2 MSVR for interval-valued stock index forecasting: The MSVR solution is obtained through an iterative reweighted least squares procedure with quasi-Newton optimization and backtracking until convergence.
- 2.2 MSVR for interval-valued stock index forecasting: Using an RBF kernel, FA-MSVR determines the MSVR parameters C, ε, and σ.
3. The proposed FA-MSVR method
The proposed FA-MSVR combines multi-output support vector regression with a firefly algorithm to select MSVR parameters for interval-valued stock-index forecasting.
- Firefly algorithm: In the firefly algorithm, fireflies are attracted according to brightness, which is determined by a fitness function.Movement combines attraction toward a more attractive firefly with randomized movement, with attractiveness controlled by distance and absorption.
- Firefly algorithm: The absorption coefficient controls the search regime: γ → 0 yields broadly shared attractiveness, whereas γ →∞ approaches random search.Intermediate parameter settings can allow the firefly algorithm to outperform particle swarm optimization and random search.
- FA-MSVR formulation: FA-MSVR applies a firefly algorithm to determine the parameters of the established MSVR model.The paper presents the formulation and procedure for FA-MSVR and provides pseudocode for its implementation.
- Parameter selection: Using an RBF kernel, FA-MSVR searches exponentially scaled spaces for C, σ, and ε.The three decision variables are searched over log-scaled ranges defined in the methodology.
- Fitness evaluation: Each firefly’s fitness is based on MSVR prediction performance measured by the interval average relative variance.Lower ARVI values indicate better forecasts; the metric compares fitted and true intervals using their lower and upper bounds.
4. Research design
The research design evaluates interval forecasts for three global stock indices using hold-out testing, multiple horizons, cross-validation, statistical tests, and trading-based criteria.
- Data description: The study uses daily interval-valued data for the S&P 500, FTSE 100, and Nikkei 225 indices.Each daily interval is defined by the lowest and highest index values, and the data are expressed on a log scale.
- Data description: For each index, the first two-thirds of observations form the estimation sample and the remainder form the hold-out sample.For the S&P 500 example, the estimation period runs from July 19, 2010 to December 1, 2011, followed by 174 hold-out observations.
- Model evaluation: Five-fold cross-validation is used during training to avoid over-fitting, followed by out-of-sample evaluation on hold-out samples.Forecasts are produced across the hold-out sample and compared with observed values to assess predictive performance.
- Forecasting strategy: The design evaluates one-step and iterated multi-step forecasts at horizons h = 1, 3, and 5.The iterated strategy reuses each predicted value as input for forecasting the subsequent point until the target horizon.
- Evaluation criteria: Statistical assessment combines ARVI with ANOVA and Tukey’s HSD tests, while economic assessment uses interval forecasts in a simple trading strategy.The trading evaluation compares average annualized returns and the percentage of trades producing positive returns.
5. Experimental results
Across three indices and multiple horizons, FA-MSVR generally provides the strongest statistical and economic forecasting performance, while statistical models require less computation.
- Statistical accuracy: FA-MSVR ranks as the best-performing method across the three indices and three prediction horizons.The comparison includes FA-MSVR, VECM, iMLP, HoltI, and SVR using interval-valued forecasting experiments.
- Statistical accuracy: FA-MSVR outperforms SVR consistently, supporting simultaneous fitting of both lower and upper interval bounds.The study attributes the contrast to the MSVR’s joint treatment of the interval components, whereas SVR models them independently.
- Statistical significance: All ANOVA results are significant at the 0.05 level, indicating differences among the five forecasting methods.Tukey’s HSD comparisons further identify pairwise differences at the 0.05 level.
- Statistical significance: FA-MSVR performs best at 95% confidence, except for the S&P 500 at h = 3 and the Nikkei 225 at h = 1.VECM and iMLP outperform HoltI in three statistically significant scenarios, while their differences are usually nonsignificant.
- Computational cost: Statistical models are less computationally expensive than artificial-intelligence models, with HoltI requiring negligible computational effort.Among the artificial-intelligence methods, FA-MSVR is less expensive than iMLP and is the computational winner over SVR.
- Economic performance: The highest profitable-trade rate is 84.44% for FA-MSVR with k = 2 and h = 3.Reported annualized returns range from 22.87% to 64.33% across methods and choices of k and h.
- Economic performance: FA-MSVR outperforms competitors for nearly all trading-strategy choices of h and k, with iMLP the exception at k = 3 and h = 1.Across the three datasets, profitable trades usually outnumber losses and expected annualized returns exceed 21%.
6. Conclusions
The study introduces FA-MSVR for interval-valued forecasting of three broad market indices and evaluates it across statistical, economic, and forecasting-horizon criteria. Results support FA-MSVR as a promising alternative, while future work extends evaluation to other financial series and econometric models.
- Contribution: FA-MSVR combines multi-output support vector regression with firefly-algorithm parameter optimization for interval forecasting.
- Experimental design: The experiments cover the S&P 500, FTSE 100, and Nikkei 225 using statistical and economic criteria across one- and multi-step-ahead horizons.
- Findings: FA-MSVR can statistically outperform some established counterparts in forecast accuracy and competing-forecast accuracy measures.
- Findings: FA-MSVR also successfully generates profits under a simple trading strategy, supporting its promise for interval-valued financial time-series forecasting.
- Future work: Future research should examine other difficult financial interval-valued series, including exchange rates, and incorporate econometric models relating stock prices to technical indicators.
Appendix A: FA-MSVR vs. PSO-MSVR vs. GA-MSVR
Appendix A explains the rationale for selecting the firefly algorithm for MSVR parameter selection and frames comparisons with PSO and GA within the study’s forecasting scope.
- Comparison design: The appendix compares FA-MSVR with PSO-MSVR and GA-MSVR using prediction accuracy, economic criteria, and computational time.
- Study scope: The study focuses on interval-valued stock-index forecasting with MSVR over short and long horizons, compared against iMLP, HoltI, and VECM.
- Selection rationale: Existing evidence on FA’s superiority over GA and PSO motivates its use for MSVR parameter selection.
- Presentation choice: The authors note that a fuller FA, PSO, and GA comparison could make the main text redundant, so detailed results are placed in the appendix.
Appendix A.
The appendix finds broadly similar economic performance among FA-MSVR, PSO-MSVR, and GA-MSVR, while FA-MSVR is the least computationally expensive in the reported case.
- Computational cost: FA-MSVR is the least expensive method in the reported computational comparison, while GA-MSVR is much more expensive than FA-MSVR and PSO-MSVR.
- Statistical criteria: FA-MSVR and PSO-MSVR produce marginally more accurate forecasts than GA-MSVR, but their mutual accuracy difference is unclear.
- Economic criteria: FA-MSVR, PSO-MSVR, and GA-MSVR are almost tied under the economic criteria.
- Interpretation: Because prediction-accuracy and economic results cannot be conclusively combined, the authors draw only the computational-cost conclusion for choosing FA.
Caption page
The caption material identifies a trading strategy for the hold-out sample and indicates that the relevant sample discussion appears later.
- Evaluation: The evaluation uses a trading strategy for the hold-out sample.
- Scope: No outcome or comparison is stated in the supplied caption material.
- Cross-reference: The text indicates that the hold-out-sample discussion appears in a later section.
Figures
The figures and tables document interval-valued stock-index data, the FA-MSVR procedure, sample construction, forecasting comparisons, trading-strategy evaluation, and computational-time comparisons.
- Data and interval construction: Daily interval-valued S&P 500 observations are constructed from higher-frequency index data, with intervals illustrated across the December 2012 sample.The figure distinguishes 10-minute S&P 500 index values from corresponding daily intervals.
- FA-MSVR procedure: FA-MSVR initializes fireflies representing MSVR parameter settings, evaluates their fitness, moves them according to attractiveness, and outputs the best-ranked solution.The parameter vector includes C, σ, and ε, while light intensity determines comparisons between fireflies.
- Experimental design: The S&P 500 interval series is divided into an earlier estimation sample and a later hold-out sample for model assessment.The displayed interval series marks the estimation and hold-out portions separately.
- Forecasting comparisons: ARVI figures compare different forecasting methods for interval-valued S&P 500, FTSE 100, and Nikkei 225 index series.The comparisons are presented separately for each broad market index.
- Evaluation: The evaluation includes ANOVA and multiple-comparison results, trading-strategy performance, and required elapsed time across prediction horizons.The trading analysis reports average annualized returns and the percentage of trades with positive annualized returns.