Scaling Exponents of Time Series Data: A Machine Learning Approach
Abstract
:1. Introduction
- We present a novel modification to the R/S approach, highlighting the distinctions between fractional Lévy motions, fractional Brownian motions, and stock market data.
- We introduce a method for continuously estimating a scaling parameter via machine learning from time series data without employing sophisticated preprocessing methods.
- We propose a new technique for estimating the scaling exponent of fractional Lévy motion using machine learning models, demonstrating its effectiveness through extensive experiments.
- We show that traditional techniques like DFA and other traditional methods do not accurately depict the scaling parameter for time series that are close to fractional Lévy motion, emphasizing the potential for machine learning approaches in this realm.
2. Related Work
3. Methodology
3.1. Random Walks
3.2. Estimating the Hurst Exponent
3.2.1. R/S Analysis
3.2.2. Detrended Fluctuation Analysis (DFA)
- Integrate the time series: Calculate the cumulative sum of the deviations of the data points from their mean.
- Divide the integrated time series into non-overlapping segments of equal length n.
- Detrend the data: In each segment, fit a polynomial function (usually a linear function) and subtract it from the integrated time series.
- Calculate the root-mean-square fluctuations for each segment.
- Average the fluctuations over all segments and obtain the fluctuation function .
- Repeat steps 2–5 for various time scales (segment lengths) n.
- Analyze the scaling behavior of by plotting it against the time scale n on a log–log scale. A linear relationship indicates the presence of long-range correlations in the original time series.
- The Hurst exponent can be estimated from the slope of the log–log plot, providing information about the persistence or anti-persistence in the time series.
3.3. Machine Learning
3.3.1. Linear Models
3.3.2. Boost Regressors
- AdaBoost:AdaBoost, short for “Adaptive Boosting”, is a popular ensemble learning algorithm used in machine learning. It was developed to improve the performance of weak classifiers by combining them into a single, more accurate and robust classifier. The main idea behind AdaBoost is to iteratively train a series of weak classifiers on the data, assigning higher weights to misclassified instances at each iteration. This process encourages the subsequent classifiers to focus on the more challenging instances, ultimately leading to an ensemble model with an improved overall performance [13].
- CatBoost:CatBoost is a gradient boosting algorithm specifically designed to handle categorical features effectively. It was developed by Yandex researchers and engineers, and it is known for its high performance and accuracy in various machine learning tasks. CatBoost addresses the common challenges associated with handling categorical features, such as one-hot encoding, by employing an efficient, target-based encoding scheme called “ordered boosting”. This method reduces overfitting and improves generalization, leading to better results in many applications [29].
- LightGBM:LightGBM (Light Gradient Boosting Machine) is a gradient boosting framework developed by Microsoft that is designed to be more efficient and scalable than traditional gradient boosting methods. It is particularly well-suited for large-scale and high-dimensional data. LightGBM incorporates several key innovations, such as Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB), which significantly reduce memory usage and computational time while maintaining high accuracy [12].
3.3.3. Multi Layer Perceptron
3.3.4. Error Analysis
4. Machine Learning Training/Validation
4.1. Training Data
4.2. Training the Machine Learning Models
4.3. Validating the Trained Models
4.3.1. Fractional Brownian Motion
Window Length | Alg. Nolds Hurst | Alg. Nolds DFA | Alg. Hurst Hurst | Alg. Hurst Hurst Simplified |
---|---|---|---|---|
10 | 0.2916 ± 0.01531 | 0.0751 ± 0.24194 | - | - |
25 | 0.3302 ± 0.01983 | 0.09547 ± 0.04252 | - | - |
50 | 0.36642 ± 0.01355 | 0.10151 ± 0.02645 | - | - |
100 | 0.40063 ± 0.01175 | 0.10392 ± 0.019 | 0.21317 ± 0.01494 | 0.114 ± 0.01291 |
200 | 0.41373 ± 0.00816 | 0.06301 ± 0.01224 | 0.19125 ± 0.01136 | 0.10718 ± 0.00966 |
350 | 0.42249 ± 0.00697 | 0.04495 ± 0.0069 | 0.18045 ± 0.01012 | 0.10363 ± 0.00875 |
Window Length | Ridge fBm | Ridge fLm | Ridge Both |
---|---|---|---|
10 | 0.28137 ± 0.00024 | 0.28134 ± 0.00071 | 0.28136 ± 0.00045 |
25 | 0.28157 ± 0.00033 | 0.28144 ± 0.0004 | 0.28161 ± 0.00022 |
50 | 0.28125 ± 0.00051 | 0.28137 ± 0.00056 | 0.28124 ± 0.00043 |
100 | 0.28144 ± 0.00088 | 0.28123 ± 0.00122 | 0.28138 ± 0.00089 |
200 | 0.28126 ± 0.00011 | 0.28131 ± 0.00071 | 0.28122 ± 0.00053 |
350 | 0.2814 ± 0.00012 | 0.28132 ± 0.0005 | 0.28137 ± 0.00038 |
Window Length | Lasso fBm | Lasso fLm | Lasso both |
10 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
25 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
50 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
100 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
200 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
350 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
Window Length | AdaBoost fBm | AdaBoost fLm | AdaBoost Both |
10 | 0.22244 ± 0.01316 | 0.2262 ± 0.013 | 0.21941 ± 0.01388 |
25 | 0.16691 ± 0.01471 | 0.19494 ± 0.00965 | 0.17723 ± 0.01186 |
50 | 0.14028 ± 0.01172 | 0.17266 ± 0.00652 | 0.14778 ± 0.00961 |
100 | 0.11962 ± 0.01022 | 0.16265 ± 0.00337 | 0.13449 ± 0.00574 |
200 | 0.12036 ± 0.00652 | 0.16359 ± 0.00161 | 0.1346 ± 0.00317 |
350 | 0.12079 ± 0.00461 | 0.16256 ± 0.00117 | 0.13533 ± 0.00224 |
Window Length | CatBoost fBm | CatBoost fLm | CatBoost Both |
10 | 0.16014 ± 0.02044 | 0.21121 ± 0.01504 | 0.1782 ± 0.01751 |
25 | 0.08767 ± 0.01857 | 0.1851 ± 0.01472 | 0.11832 ± 0.01546 |
50 | 0.05087 ± 0.01546 | 0.16244 ± 0.01481 | 0.07659 ± 0.0142 |
100 | 0.02484 ± 0.01112 | 0.13197 ± 0.01461 | 0.04776 ± 0.01179 |
200 | 0.026 ± 0.00839 | 0.13222 ± 0.0105 | 0.04831 ± 0.00879 |
350 | 0.0257 ± 0.00625 | 0.13197 ± 0.00765 | 0.0479 ± 0.00651 |
Window Length | LightGBM fBm | LightGBM fLm | LightGBM Both |
10 | 0.16011 ± 0.02052 | 0.21073 ± 0.01527 | 0.17831 ± 0.01753 |
25 | 0.09248 ± 0.01834 | 0.18331 ± 0.01444 | 0.12301 ± 0.01514 |
50 | 0.05323 ± 0.01538 | 0.15624 ± 0.01414 | 0.08313 ± 0.01379 |
100 | 0.02802 ± 0.01073 | 0.12563 ± 0.01423 | 0.05224 ± 0.01161 |
200 | 0.02904 ± 0.00821 | 0.12482 ± 0.01025 | 0.05178 ± 0.00821 |
350 | 0.02863 ± 0.00605 | 0.12465 ± 0.0075 | 0.05151 ± 0.00608 |
Window Length | MLP fBm | MLP fLm | MLP Both |
10 | 0.1656 ± 0.02182 | 0.21432 ± 0.01481 | 0.17702 ± 0.01761 |
25 | 0.09959 ± 0.01788 | 0.17948 ± 0.01469 | 0.1326 ± 0.01475 |
50 | 0.06085 ± 0.01463 | 0.14635 ± 0.01535 | 0.08806 ± 0.01467 |
100 | 0.03473 ± 0.00988 | 0.12899 ± 0.01635 | 0.05588 ± 0.01101 |
200 | 0.03578 ± 0.00755 | 0.12915 ± 0.01183 | 0.05784 ± 0.00798 |
350 | 0.03586 ± 0.00541 | 0.12858 ± 0.00873 | 0.05516 ± 0.00615 |
4.3.2. Fractional Lévy Motion, = 0.5
Window Length | Alg. Nolds Hurst | Alg. Nolds DFA | Alg. Hurst Hurst | Alg. Hurst Hurst Simplified |
---|---|---|---|---|
10 | 0.28517 ± 0.01622 | 0.42035 ± 0.2706 | - | - |
25 | 0.26316 ± 0.02612 | 0.47115 ± 0.0471 | - | - |
50 | 0.26357 ± 0.01986 | 0.45586 ± 0.03011 | - | - |
100 | 0.28779 ± 0.01684 | 0.45104 ± 0.02172 | 0.21913 ± 0.01621 | 0.24127 ± 0.01786 |
200 | 0.31121 ± 0.01129 | 0.46071 ± 0.01456 | 0.22713 ± 0.01267 | 0.24231 ± 0.01428 |
350 | 0.33902 ± 0.00942 | 0.44371 ± 0.01141 | 0.23226 ± 0.01066 | 0.24178 ± 0.01233 |
Window Length | Ridge fBm | Ridge fLm | Ridge both |
---|---|---|---|
10 | 0.28139 ± 0.0003 | 0.28116 ± 0.00077 | 0.28123 ± 0.00053 |
25 | 0.28133 ± 0.00053 | 0.28131 ± 0.00052 | 0.28135 ± 0.00032 |
50 | 0.28134 ± 0.00085 | 0.28135 ± 0.00079 | 0.28131 ± 0.00057 |
100 | 0.28141 ± 0.0015 | 0.28124 ± 0.00148 | 0.28122 ± 0.00111 |
200 | 0.28136 ± 8 | 0.28122 ± 0.00079 | 0.28121 ± 0.00058 |
350 | 0.28133 ± 6 | 0.28109 ± 0.00055 | 0.28109 ± 0.00041 |
Window Length | Lasso fBm | Lasso fLm | Lasso Both |
10 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
25 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
50 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
100 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
200 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
350 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
Window Length | AdaBoost fBm | AdaBoost fLm | AdaBoost Both |
10 | 0.28501 ± 0.00965 | 0.27944 ± 0.00736 | 0.28093 ± 0.0092 |
25 | 0.2994 ± 0.0117 | 0.27436 ± 0.00838 | 0.2817 ± 0.01012 |
50 | 0.30762 ± 0.01202 | 0.26953 ± 0.00996 | 0.28504 ± 0.01142 |
100 | 0.32061 ± 0.01262 | 0.26418 ± 0.01149 | 0.28198 ± 0.01247 |
200 | 0.3201 ± 0.00709 | 0.26357 ± 0.00652 | 0.28126 ± 0.00638 |
350 | 0.31737 ± 0.00536 | 0.26173 ± 0.0047 | 0.27918 ± 0.00468 |
Window Length | CatBoost fBm | CatBoost fLm | CatBoost Both |
10 | 0.29105 ± 0.01685 | 0.26013 ± 0.01451 | 0.27373 ± 0.01506 |
25 | 0.31954 ± 0.02005 | 0.22098 ± 0.01589 | 0.25413 ± 0.01666 |
50 | 0.32883 ± 0.01971 | 0.19311 ± 0.01505 | 0.22873 ± 0.01656 |
100 | 0.32947 ± 0.01705 | 0.18257 ± 0.01354 | 0.21623 ± 0.01513 |
200 | 0.3283 ± 0.01265 | 0.18206 ± 0.00866 | 0.21506 ± 0.01052 |
350 | 0.3265 ± 0.00934 | 0.18128 ± 0.00607 | 0.21404 ± 0.00751 |
Window Length | LightGBM fBm | LightGBM fLm | LightGBM Both |
10 | 0.28999 ± 0.01713 | 0.25899 ± 0.01505 | 0.2733 ± 0.01536 |
25 | 0.31583 ± 0.02002 | 0.22418 ± 0.016 | 0.25836 ± 0.01676 |
50 | 0.32294 ± 0.02003 | 0.1996 ± 0.01483 | 0.23572 ± 0.01697 |
100 | 0.30969 ± 0.01737 | 0.18867 ± 0.01346 | 0.2209 ± 0.01607 |
200 | 0.30965 ± 0.0128 | 0.1884 ± 0.00876 | 0.22123 ± 0.01096 |
350 | 0.30776 ± 0.00946 | 0.18759 ± 0.00615 | 0.22019 ± 0.00783 |
Window Length | MLP fBm | MLP fLm | MLP Both |
10 | 0.30555 ± 0.01746 | 0.26038 ± 0.01426 | 0.27412 ± 0.0149 |
25 | 0.33234 ± 0.01994 | 0.23591 ± 0.01552 | 0.25557 ± 0.01708 |
50 | 0.33114 ± 0.02127 | 0.20713 ± 0.01606 | 0.23219 ± 0.01847 |
100 | 0.29554 ± 0.01763 | 0.18192 ± 0.01455 | 0.23114 ± 0.01596 |
200 | 0.29431 ± 0.01297 | 0.18123 ± 0.00948 | 0.2301 ± 0.01107 |
350 | 0.29222 ± 0.00954 | 0.18008 ± 0.00668 | 0.22919 ± 0.00788 |
4.3.3. Fractional Lévy Motion, = 1.0
Window Length | Alg. Nolds Hurst | Alg. Nolds DFA | Alg. Hurst Hurst | Alg. Hurst Hurst Simplified |
---|---|---|---|---|
10 | 0.28169 ± 0.01582 | 0.14867 ± 0.24577 | - | - |
25 | 0.26134 ± 0.02507 | 0.16672 ± 0.04256 | - | - |
50 | 0.29846 ± 0.01826 | 0.15482 ± 0.02765 | - | - |
100 | 0.34383 ± 0.01472 | 0.15946 ± 0.01826 | 0.11358 ± 0.01295 | 0.09771 ± 0.0146 |
200 | 0.37387 ± 0.00979 | 0.15343 ± 0.01281 | 0.09689 ± 0.00983 | 0.09633 ± 0.01395 |
350 | 0.40277 ± 0.00817 | 0.14777 ± 0.00979 | 0.08666 ± 0.0082 | 0.08423 ± 0.01146 |
Window Length | Ridge fBm | Ridge fLm | Ridge Both |
---|---|---|---|
10 | 0.28117 ± 0.00027 | 0.28126 ± 0.00072 | 0.28145 ± 0.00048 |
25 | 0.28131 ± 0.00043 | 0.28134 ± 0.00046 | 0.2813 ± 0.00027 |
50 | 0.2812 ± 0.00069 | 0.28136 ± 0.00067 | 0.28124 ± 0.00051 |
100 | 0.28138 ± 0.00121 | 0.28139 ± 0.00133 | 0.28141 ± 0.00101 |
200 | 0.28127 ± 0.00013 | 0.28133 ± 0.00081 | 0.28124 ± 0.0006 |
350 | 0.28129 ± 0.00011 | 0.28135 ± 0.00052 | 0.28129 ± 0.00039 |
Window Length | Lasso fBm | Lasso fLm | Lasso Both |
10 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
25 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
50 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
100 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
200 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
350 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
Window Length | AdaBoost fBm | AdaBoost fLm | AdaBoost Both |
10 | 0.22433 ± 0.0129 | 0.22269 ± 0.01196 | 0.21992 ± 0.01335 |
25 | 0.18802 ± 0.01813 | 0.20001 ± 0.01125 | 0.19137 ± 0.01486 |
50 | 0.18038 ± 0.0187 | 0.19031 ± 0.0111 | 0.17882 ± 0.01649 |
100 | 0.17227 ± 0.01964 | 0.1828 ± 0.00976 | 0.17062 ± 0.01527 |
200 | 0.19107 ± 0.01682 | 0.19405 ± 0.00888 | 0.18627 ± 0.01331 |
350 | 0.1832 ± 0.01297 | 0.18972 ± 0.00662 | 0.17945 ± 0.01015 |
Window Length | CatBoost fBm | CatBoost fLm | CatBoost Both |
10 | 0.16251 ± 0.0228 | 0.1735 ± 0.0155 | 0.16663 ± 0.01876 |
25 | 0.10053 ± 0.02435 | 0.12608 ± 0.01514 | 0.11452 ± 0.01884 |
50 | 0.08678 ± 0.0223 | 0.10853 ± 0.0148 | 0.09634 ± 0.01796 |
100 | 0.0805 ± 0.01776 | 0.09093 ± 0.0143 | 0.08429 ± 0.01576 |
200 | 0.09697 ± 0.01578 | 0.11339 ± 0.01109 | 0.10593 ± 0.01341 |
350 | 0.08847 ± 0.01262 | 0.10591 ± 0.009 | 0.09767 ± 0.01095 |
Window Length | LightGBM fBm | LightGBM fLm | LightGBM Both |
10 | 0.16298 ± 0.02281 | 0.17226 ± 0.0156 | 0.1669 ± 0.01887 |
25 | 0.10358 ± 0.02465 | 0.12709 ± 0.01515 | 0.11801 ± 0.01879 |
50 | 0.08901 ± 0.02287 | 0.11128 ± 0.01457 | 0.0991 ± 0.01792 |
100 | 0.08219 ± 0.01884 | 0.09293 ± 0.01406 | 0.08662 ± 0.01601 |
200 | 0.10263 ± 0.01659 | 0.11551 ± 0.01118 | 0.10916 ± 0.01362 |
350 | 0.09302 ± 0.01335 | 0.10787 ± 0.00902 | 0.10049 ± 0.01107 |
Window Length | MLP fBm | MLP fLm | MLP Both |
10 | 0.16992 ± 0.02453 | 0.17416 ± 0.01483 | 0.1626 ± 0.01924 |
25 | 0.0989 ± 0.02508 | 0.12536 ± 0.01521 | 0.11192 ± 0.01802 |
50 | 0.087 ± 0.02189 | 0.10809 ± 0.01586 | 0.09365 ± 0.01821 |
100 | 0.08741 ± 0.01875 | 0.09523 ± 0.01581 | 0.09705 ± 0.01728 |
200 | 0.10958 ± 0.01624 | 0.11189 ± 0.01236 | 0.12114 ± 0.01475 |
350 | 0.10021 ± 0.01313 | 0.10521 ± 0.00936 | 0.11197 ± 0.01189 |
4.3.4. Fractional Lévy Motion, = 1.5
Window Length | Alg. Nolds Hurst | Alg. Nolds DFA | Alg. Hurst Hurst | Alg. Hurst Hurst Simplified |
---|---|---|---|---|
10 | 0.27084 ± 0.01484 | 0.23363 ± 0.21841 | - | - |
25 | 0.28817 ± 0.02327 | 0.23679 ± 0.0342 | - | - |
50 | 0.33359 ± 0.01618 | 0.25416 ± 0.01981 | - | - |
100 | 0.3782 ± 0.01311 | 0.25933 ± 0.01619 | 0.21506 ± 0.01824 | 0.21175 ± 0.02937 |
200 | 0.40683 ± 0.00869 | 0.23166 ± 0.01267 | 0.20716 ± 0.01488 | 0.20741 ± 0.02327 |
350 | 0.43147 ± 0.00722 | 0.21754 ± 0.01074 | 0.20099 ± 0.01288 | 0.20322 ± 0.02032 |
Window Length | Ridge fBm | Ridge fLm | Ridge Both |
---|---|---|---|
10 | 0.28124 ± 0.00023 | 0.2811 ± 0.00061 | 0.28131 ± 0.00042 |
25 | 0.28142 ± 0.0004 | 0.2813 ± 0.0004 | 0.28145 ± 0.00026 |
50 | 0.28148 ± 0.00062 | 0.28155 ± 0.00058 | 0.28168 ± 0.00045 |
100 | 0.28118 ± 0.00108 | 0.28203 ± 0.00117 | 0.28168 ± 0.00088 |
200 | 0.28143 ± 0.00013 | 0.28202 ± 0.00069 | 0.28194 ± 0.00052 |
350 | 0.28158 ± 9 | 0.28202 ± 0.00046 | 0.28208 ± 0.00034 |
Window Length | Lasso fBm | Lasso fLm | Lasso Both |
10 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
25 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
50 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
100 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
200 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
350 | 0.28137 ± 0.0 | 0.28137 ± 0.0 | 0.28137 ± 0.0 |
Window Length | AdaBoost fBm | AdaBoost fLm | AdaBoost Both |
10 | 0.17954 ± 0.00739 | 0.17841 ± 0.00603 | 0.17501 ± 0.0072 |
25 | 0.13308 ± 0.00729 | 0.16829 ± 0.00594 | 0.15288 ± 0.00664 |
50 | 0.11901 ± 0.00597 | 0.16298 ± 0.00575 | 0.13967 ± 0.00629 |
100 | 0.10492 ± 0.00603 | 0.16836 ± 0.0062 | 0.13647 ± 0.00699 |
200 | 0.10557 ± 0.00475 | 0.1677 ± 0.00428 | 0.13728 ± 0.00471 |
350 | 0.10388 ± 0.00248 | 0.16646 ± 0.00301 | 0.13614 ± 0.00312 |
Window Length | CatBoost fBm | CatBoost fLm | CatBoost Both |
10 | 0.12585 ± 0.01533 | 0.13493 ± 0.01381 | 0.12543 ± 0.01354 |
25 | 0.09966 ± 0.02066 | 0.11572 ± 0.01648 | 0.10284 ± 0.01716 |
50 | 0.09861 ± 0.01676 | 0.10646 ± 0.0161 | 0.09885 ± 0.0167 |
100 | 0.11132 ± 0.01521 | 0.10493 ± 0.01517 | 0.10261 ± 0.01542 |
200 | 0.10422 ± 0.01406 | 0.10233 ± 0.01111 | 0.09901 ± 0.01235 |
350 | 0.10423 ± 0.01236 | 0.10089 ± 0.00847 | 0.09761 ± 0.00979 |
Window Length | LightGBM fBm | LightGBM fLm | LightGBM Both |
10 | 0.12474 ± 0.01531 | 0.13449 ± 0.01421 | 0.12653 ± 0.01365 |
25 | 0.09164 ± 0.02074 | 0.11827 ± 0.01639 | 0.10419 ± 0.01662 |
50 | 0.09421 ± 0.01709 | 0.1128 ± 0.01605 | 0.1014 ± 0.01649 |
100 | 0.10036 ± 0.01603 | 0.10926 ± 0.0156 | 0.10413 ± 0.01581 |
200 | 0.09374 ± 0.0143 | 0.10574 ± 0.01141 | 0.09917 ± 0.01238 |
350 | 0.0929 ± 0.01228 | 0.10395 ± 0.00866 | 0.09765 ± 0.00978 |
Window Length | MLP fBm | MLP fLm | MLP Both |
10 | 0.10349 ± 0.01432 | 0.13397 ± 0.01323 | 0.12935 ± 0.01348 |
25 | 0.09597 ± 0.02127 | 0.11178 ± 0.01572 | 0.11672 ± 0.01739 |
50 | 0.08883 ± 0.01777 | 0.11371 ± 0.0167 | 0.12015 ± 0.018 |
100 | 0.09425 ± 0.01593 | 0.13946 ± 0.01676 | 0.09106 ± 0.01621 |
200 | 0.08884 ± 0.01365 | 0.13815 ± 0.01241 | 0.0876 ± 0.01283 |
350 | 0.08686 ± 0.01138 | 0.1367 ± 0.00935 | 0.08641 ± 0.01017 |
5. Finance Experiments
5.1. The Scaling Exponent of Financial Data
5.2. Results
5.3. Summary & Discussion
6. Conclusions
- We trained a range of machine learning models on both fractional Brownian and fractional Lévy motions with different Hurst/scaling exponents and different Lévy indices. We used the known scaling exponent as the ground truth for the value to be predicted by the machine learning algorithms, i.e., the output of the models. The features, or the input, are time series data from the discussed stochastic processes scaled to the unit interval .
- We validated the trained models for different lengths of input windows using, again, fractional Brownian and fractional Lévy motions. The results show that in most cases the trained machine learning models outperform classical algorithms (such as R/S analysis) to estimate the scaling exponent of both fractional Brownian and fractional Lévy motions.
- We then took three asset time series, i.e., Dow Jones, S&P500, and NASDAQ, and applied a slightly modified version of R/S analysis to these datasets to show that these data signals are more akin to fractional Lévy motions than fractional Brownian motions in nature. The reason for doing this was to argue that certain classical algorithms cannot correctly estimate the scaling exponents of these datasets because, as shown in the previous step, compared to the trained models, they suffer from large errors in estimating the scaling exponent of fractional Lévy motions.
- In a final step, we analyzed the scaling exponent of the previously named three assets in a sliding window manner, to show and discuss the applicability of the trained models and classical algorithms to estimate the scaling behavior of time series data. Our research shows that results from the literature might be wrong in estimating the scaling exponent using detrended fluctuation analysis (DFA) and drawing conclusions from it. To do this, we first reconstructed the scaling behavior using DFA, which coincides with the results from the literature. We then found that the trained machine learning algorithms do not reproduce the scaling behavior from the literature, even though we showed that the assets under study are closer to a fractional Lévy motion, and that our trained models can better estimate the scaling exponent of stochastic processes like these.
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Fractional Lévy Motion and Its Scaling Behavior
Appendix B. Additional Plots, Finance Experiments
Appendix B.1. Additional Plots Dow Jones
Appendix B.2. Additional Plots S&P500
Appendix B.3. Additional Plots NASDAQ
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Window Length and Type of Training Data | CV-Score Ridge Regressor | CV-Score Lasso Regressor | CV-Score AdaBoost Regressor | CV-Score CatBoost Regressor | CV-Score LightGBM Regressor | CV-Score MLP Regressor |
---|---|---|---|---|---|---|
10, fBm | <0.0001 | <0.0001 | 0.29876 | 0.47271 | 0.47396 | 0.45155 |
25, fBm | <0.0001 | <0.0001 | 0.53221 | 0.73295 | 0.72234 | 0.69416 |
50, fBm | <0.0001 | <0.0001 | 0.66082 | 0.84480 | 0.84369 | 0.82256 |
100, fBm | <0.0001 | <0.0001 | 0.73564 | 0.91811 | 0.91329 | 0.90260 |
10, fLm | <0.0001 | <0.0001 | 0.29913 | 0.41482 | 0.42049 | 0.41858 |
25, fLm | <0.0001 | <0.0001 | 0.36256 | 0.53270 | 0.52886 | 0.52027 |
50, fLm | <0.0001 | <0.0001 | 0.40205 | 0.60694 | 0.59713 | 0.56785 |
100, fLm | <0.0001 | <0.0001 | 0.42368 | 0.65468 | 0.64698 | 0.60597 |
10, both | <0.0001 | <0.0001 | 0.29751 | 0.41519 | 0.41749 | 0.41090 |
25, both | <0.0001 | <0.0001 | 0.43414 | 0.59296 | 0.58660 | 0.55943 |
50, both | <0.0001 | <0.0001 | 0.51247 | 0.69218 | 0.68393 | 0.64834 |
100, both | <0.0001 | <0.0001 | 0.56500 | 0.76081 | 0.75103 | 0.73408 |
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Raubitzek, S.; Corpaci, L.; Hofer, R.; Mallinger, K. Scaling Exponents of Time Series Data: A Machine Learning Approach. Entropy 2023, 25, 1671. https://doi.org/10.3390/e25121671
Raubitzek S, Corpaci L, Hofer R, Mallinger K. Scaling Exponents of Time Series Data: A Machine Learning Approach. Entropy. 2023; 25(12):1671. https://doi.org/10.3390/e25121671
Chicago/Turabian StyleRaubitzek, Sebastian, Luiza Corpaci, Rebecca Hofer, and Kevin Mallinger. 2023. "Scaling Exponents of Time Series Data: A Machine Learning Approach" Entropy 25, no. 12: 1671. https://doi.org/10.3390/e25121671