Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion
The fractal dimension (<i>D</i>) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional B...
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MDPI AG
2024-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/1/50 |
| _version_ | 1797343989223915520 |
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| author | Yen-Ching Chang |
| author_facet | Yen-Ching Chang |
| author_sort | Yen-Ching Chang |
| collection | DOAJ |
| description | The fractal dimension (<i>D</i>) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian motion (2D FBM) is commonly used because it has specific physical meaning and only contains the finite-valued parameter (a real value from 0 to 1) of the Hurst exponent (<i>H</i>). More usefully, <i>H</i> and <i>D</i> possess the relation of <i>D</i> = 3 − <i>H</i>. The accuracy of the maximum likelihood estimator (MLE) is the best among estimators, but its efficiency is appreciably low. Lately, an efficient MLE for the Hurst exponent was produced to greatly improve its efficiency, but it still incurs much higher computational costs. Therefore, in the paper, we put forward a deep-learning estimator through classification models. The trained deep-learning models for images of 2D FBM not only incur smaller computational costs but also provide smaller mean-squared errors than the efficient MLE, except for size 32 × 32 × 1. In particular, the computational times of the efficient MLE are up to 129, 3090, and 156248 times those of our proposed simple model for sizes 32 × 32 × 1, 64 × 64 × 1, and 128 × 128 × 1. |
| first_indexed | 2024-03-08T10:55:48Z |
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| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-08T10:55:48Z |
| publishDate | 2024-01-01 |
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| series | Fractal and Fractional |
| spelling | doaj.art-f833ea315da34032a1874d63f7f4c9522024-01-26T16:36:10ZengMDPI AGFractal and Fractional2504-31102024-01-01815010.3390/fractalfract8010050Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian MotionYen-Ching Chang0Department of Medical Informatics, Chung Shan Medical University, Taichung 40201, TaiwanThe fractal dimension (<i>D</i>) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian motion (2D FBM) is commonly used because it has specific physical meaning and only contains the finite-valued parameter (a real value from 0 to 1) of the Hurst exponent (<i>H</i>). More usefully, <i>H</i> and <i>D</i> possess the relation of <i>D</i> = 3 − <i>H</i>. The accuracy of the maximum likelihood estimator (MLE) is the best among estimators, but its efficiency is appreciably low. Lately, an efficient MLE for the Hurst exponent was produced to greatly improve its efficiency, but it still incurs much higher computational costs. Therefore, in the paper, we put forward a deep-learning estimator through classification models. The trained deep-learning models for images of 2D FBM not only incur smaller computational costs but also provide smaller mean-squared errors than the efficient MLE, except for size 32 × 32 × 1. In particular, the computational times of the efficient MLE are up to 129, 3090, and 156248 times those of our proposed simple model for sizes 32 × 32 × 1, 64 × 64 × 1, and 128 × 128 × 1.https://www.mdpi.com/2504-3110/8/1/50deep learningdeep-learning estimatortwo-dimensional fractional Brownian motionfractal dimensionmaximum likelihood estimatorHurst exponent |
| spellingShingle | Yen-Ching Chang Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion Fractal and Fractional deep learning deep-learning estimator two-dimensional fractional Brownian motion fractal dimension maximum likelihood estimator Hurst exponent |
| title | Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion |
| title_full | Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion |
| title_fullStr | Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion |
| title_full_unstemmed | Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion |
| title_short | Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion |
| title_sort | deep learning estimators for the hurst exponent of two dimensional fractional brownian motion |
| topic | deep learning deep-learning estimator two-dimensional fractional Brownian motion fractal dimension maximum likelihood estimator Hurst exponent |
| url | https://www.mdpi.com/2504-3110/8/1/50 |
| work_keys_str_mv | AT yenchingchang deeplearningestimatorsforthehurstexponentoftwodimensionalfractionalbrownianmotion |