Fractional calculus has gained a lot of attention in the last couple of years. Researchers have discovered that processes in various fields follow fractional dynamics rather than ordinary integer-ordered dynamics, meaning that the corresponding differential equations feature non-integer valued derivatives. There are several arguments for why this is the case, one of which is that fractional derivatives inherit spatiotemporal memory and/or the ability to express complex naturally occurring phenomena. Another popular topic nowadays is machine learning, i.e., learning behavior and patterns from historical data. In our ever-changing world with ever-increasing amounts of data, machine learning is a powerful tool for data analysis, problem-solving, modeling, and prediction. It has provided many further insights and discoveries in various scientific disciplines. As these two modern-day topics hold a lot of potential for combined approaches in terms of describing complex dynamics, this article review combines approaches from fractional derivatives and machine learning from the past, puts them into context, and thus provides a list of possible combined approaches and the corresponding techniques. Note, however, that this article does not deal with neural networks, as there is already extensive literature on neural networks and fractional calculus. We sorted past combined approaches from the literature into three categories, i.e., preprocessing, machine learning and fractional dynamics, and optimization. The contributions of fractional derivatives to machine learning are manifold as they provide powerful preprocessing and feature augmentation techniques, can improve physically informed machine learning, and are capable of improving hyperparameter optimization. Thus, this article serves to motivate researchers dealing with data-based problems, to be specific machine learning practitioners, to adopt new tools, and enhance their existing approaches.

In this article, we investigate the applicability of quantum machine learning for classification tasks using two quantum classifiers from the Qiskit Python environment: the variational quantum circuit and the quantum kernel estimator (QKE). We provide a first evaluation on the performance of these classifiers when using a hyperparameter search on six widely known and publicly available benchmark datasets and analyze how their performance varies with the number of samples on two artificially generated test classification datasets. As quantum machine learning is based on unitary transformations, this paper explores data structures and application fields that could be particularly suitable for quantum advantages. Hereby, this paper introduces a novel dataset based on concepts from quantum mechanics using the exponential map of a Lie algebra. This dataset will be made publicly available and contributes a novel contribution to the empirical evaluation of quantum supremacy. We further compared the performance of VQC and QKE on six widely applicable datasets to contextualize our results. Our results demonstrate that the VQC and QKE perform better than basic machine learning algorithms, such as advanced linear regression models (Ridge and Lasso). They do not match the accuracy and runtime performance of sophisticated modern boosting classifiers such as XGBoost, LightGBM, or CatBoost. Therefore, we conclude that while quantum machine learning algorithms have the potential to surpass classical machine learning methods in the future, especially when physical quantum infrastructure becomes widely available, they currently lag behind classical approaches. Our investigations also show that classical machine learning approaches have superior performance classifying datasets based on group structures, compared to quantum approaches that particularly use unitary processes. Furthermore, our findings highlight the significant impact of different quantum simulators, feature maps, and quantum circuits on the performance of the employed quantum estimators. This observation emphasizes the need for researchers to provide detailed explanations of their hyperparameter choices for quantum machine learning algorithms, as this aspect is currently overlooked in many studies within the field. To facilitate further research in this area and ensure the transparency of our study, we have made the complete code available in a linked GitHub repository.

Measures of signal complexity, such as the Hurst exponent, the fractal dimension, and the Spectrum of Lyapunov exponents, are used in time series analysis to give estimates on persistency, anti-persistency, fluctuations and predictability of the data under study. They have proven beneficial when doing time series prediction using machine and deep learning and tell what features may be relevant for predicting time-series and establishing complexity features. Further, the performance of machine learning approaches can be improved, taking into account the complexity of the data under study, e.g., adapting the employed algorithm to the inherent long-term memory of the data. In this article, we provide a review of complexity and entropy measures in combination with machine learning approaches. We give a comprehensive review of relevant publications, suggesting the use of fractal or complexity-measure concepts to improve existing machine or deep learning approaches. Additionally, we evaluate applications of these concepts and examine if they can be helpful in predicting and analyzing time series using machine and deep learning. Finally, we give a list of a total of six ways to combine machine learning and measures of signal complexity as found in the literature.

In this study, we present a novel approach to estimating the Hurst exponent of time series data using a variety of machine learning algorithms. The Hurst exponent is a crucial parameter in characterizing long-range dependence in time series, and traditional methods such as Rescaled Range (R/S) analysis and Detrended Fluctuation Analysis (DFA) have been widely used for its estimation. However, these methods have certain limitations, which we sought to address by modifying the R/S approach to distinguish between fractional Lévy and fractional Brownian motion, and by demonstrating the inadequacy of DFA and similar methods for data that resembles fractional Lévy motion. This inspired us to utilize machine learning techniques to improve the estimation process. In an unprecedented step, we train various machine learning models, including LightGBM, MLP, and AdaBoost, on synthetic data generated from random walks, namely fractional Brownian motion and fractional Lévy motion, where the ground truth Hurst exponent is known. This means that we can initialize and create these stochastic processes with a scaling Hurst/scaling exponent, which is then used as the ground truth for training. Furthermore, we perform the continuous estimation of the scaling exponent directly from the time series, without resorting to the calculation of the power spectrum or other sophisticated preprocessing steps, as done in past approaches. Our experiments reveal that the machine learning-based estimators outperform traditional R/S analysis and DFA methods in estimating the Hurst exponent, particularly for data akin to fractional Lévy motion. Validating our approach on real-world financial data, we observe a divergence between the estimated Hurst/scaling exponents and results reported in the literature. Nevertheless, the confirmation provided by known ground truths reinforces the superiority of our approach in terms of accuracy. This work highlights the potential of machine learning algorithms for accurately estimating the Hurst exponent, paving new paths for time series analysis. By marrying traditional finance methods with the capabilities of machine learning, our study provides a novel contribution towards the future of time series data analysis.

This paper shows if and how the predictability and complexity of stock market data changed over the last half-century and what influence the M1 money supply has. We use three different machine learning algorithms, i.e., a stochastic gradient descent linear regression, a lasso regression, and an XGBoost tree regression, to test the predictability of two stock market indices, the Dow Jones Industrial Average and the NASDAQ (National Association of Securities Dealers Automated Quotations) Composite. In addition, all data under study are discussed in the context of a variety of measures of signal complexity. The results of this complexity analysis are then linked with the machine learning results to discover trends and correlations between predictability and complexity. Our results show a decrease in predictability and an increase in complexity for more recent years. We find a correlation between approximate entropy, sample entropy, and the predictability of the employed machine learning algorithms on the data under study. This link between the predictability of machine learning algorithms and the mentioned entropy measures has not been shown before. It should be considered when analyzing and predicting complex time series data, e.g., stock market data, to e.g., identify regions of increased predictability.

Machine learning methods, such as Long Short-Term Memory (LSTM) neural networks can predict real-life time series data. Here, we present a new approach to predict time series data combining interpolation techniques, randomly parameterized LSTM neural networks and measures of signal complexity, which we will refer to as complexity measures throughout this research. First, we interpolate the time series data under study. Next, we predict the time series data using an ensemble of randomly parameterized LSTM neural networks. Finally, we filter the ensemble prediction based on the original data complexity to improve the predictability, i.e., we keep only predictions with a complexity close to that of the training data. We test the proposed approach on five different univariate time series data. We use linear and fractal interpolation to increase the amount of data. We tested five different complexity measures for the ensemble filters for time series data, i.e., the Hurst exponent, Shannon’s entropy, Fisher’s information, SVD entropy, and the spectrum of Lyapunov exponents. Our results show that the interpolated predictions consistently outperformed the non-interpolated ones. The best ensemble predictions always beat a baseline prediction based on a neural network with only a single hidden LSTM, gated recurrent unit (GRU) or simple recurrent neural network (RNN) layer. The complexity filters can reduce the error of a random ensemble prediction by a factor of 10. Further, because we use randomly parameterized neural networks, no hyperparameter tuning is required. We prove this method useful for real-time time series prediction because the optimization of hyperparameters, which is usually very costly and time-intensive, can be circumvented with the presented approach.