FRACTAL DIMENSION ESTIMATION WITH PERSISTENT HOMOLOGY: A COMPARATIVE STUDY

Estimating fractal dimensions: A comparative review and open source implementations

The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper, we present a self-contained and comprehensive introduction to the fractal dimension. We collect and present various numerical estimators and focus on the three most promising ones: generalized entropy, correlation sum, and extreme value theory. We then perform an extensive quantitative evaluation of these estimators, comparing their performance and precision using different datasets and comparing the impact of features like length, noise, embedding dimension, and falsify-ability, among many others. Our analysis shows that for synthetic noiseless data, the correlation sum is the best estimator with extreme value theory following closely. For real experimental data, we found the correlation sum to be more strongly affected by noise vs the entropy and extreme value theory. The recent extreme value theory estimator seems powerful as it has some of the advantages of both alternative methods. However, using four different ways for checking for significance, we found that the method yielded "significant" low-dimensional results for inappropriate data like stock market timeseries. This fact, combined with some ambiguities we found in the literature of the method applications, has implications for both previous and future real-world applications using the extreme value theory approach, as, for example, the argument for small effective dimensionality in the data cannot come from the method itself. All algorithms discussed are implemented as performant and easy to use open source code via the DynamicalSystems.jl library.

Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology

Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, n-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.

Grading your models: Assessing dynamics learning of models using persistent homology

Assessing model accuracy for complex and chaotic systems is a non-trivial task that often relies on the calculation of dynamical invariants, such as Lyapunov exponents and correlation dimensions. Well-performing models are able to replicate the long-term dynamics and ergodic properties of the desired system. We term this phenomenon "dynamics learning." However, existing estimates based on dynamical invariants, such as Lyapunov exponents and correlation dimensions, are not unique to each system, not necessarily robust to noise, and struggle with detecting pathological errors, such as errors in the manifold density distribution. This can make meaningful and accurate model assessment difficult. We explore the use of a topological data analysis technique, persistent homology, applied to uniformly sampled trajectories from constructed reservoir models of the Lorenz system to assess the learning quality of a model. A proposed persistent homology point summary, conformance, was able to identify models with successful dynamics learning and detect discrepancies in the manifold density distribution.

Persistent homology of fractional Gaussian noise

In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent (H). The coefficients of the birth and death curves of the k-dimensional topological holes (k-holes) at a given threshold depend on H which is almost not affected by finite sample size. We show that the distribution function of a lifetime for k-holes decays exponentially and the corresponding slope is an increasing function versus H and, more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost H-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution (M_{n}) for n≥1 reveal a dependency on H, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.

An exploration of fractal-based prognostic model and comparative analysis for second wave of COVID-19 diffusion

The coronavirus disease 2019 (COVID-19) pandemic has fatalized 216 countries across the world and has claimed the lives of millions of people globally. Researches are being carried out worldwide by scientists to understand the nature of this catastrophic virus and find a potential vaccine for it. The most possible efforts have been taken to present this paper as a form of contribution to the understanding of this lethal virus in the first and second wave. This paper presents a unique technique for the methodical comparison of disastrous virus dissemination in two waves amid five most infested countries and the death rate of the virus in order to attain a clear view on the behaviour of the spread of the disease. For this study, the data set of the number of deaths per day and the number of infected cases per day of the most affected countries, the USA, Brazil, Russia, India, and the UK, have been considered in the first and second waves. The correlation fractal dimension has been estimated for the prescribed data sets of COVID-19, and the rate of death has been compared based on the correlation fractal dimension estimate curve. The statistical tool, analysis of variance, has also been used to support the performance of the proposed method. Further, the prediction of the daily death rate has been demonstrated through the autoregressive moving average model. In addition, this study also emphasis a feasible reconstruction of the death rate based on the fractal interpolation function. Subsequently, the normal probability plot is portrayed for the original data and the predicted data, derived through the fractal interpolation function to estimate the accuracy of the prediction. Finally, this paper neatly summarized with the comparison and prediction of epidemic curve of the first and second waves of COVID-19 pandemic to visualize the transmission rate in the both times.

Autoreservoir computing for multistep ahead prediction based on the spatiotemporal information transformation

We develop an auto-reservoir computing framework, Auto-Reservoir Neural Network (ARNN), to efficiently and accurately make multi-step-ahead predictions based on a short-term high-dimensional time series. Different from traditional reservoir computing whose reservoir is an external dynamical system irrelevant to the target system, ARNN directly transforms the observed high-dimensional dynamics as its reservoir, which maps the high-dimensional/spatial data to the future temporal values of a target variable based on our spatiotemporal information (STI) transformation. Thus, the multi-step prediction of the target variable is achieved in an accurate and computationally efficient manner. ARNN is successfully applied to both representative models and real-world datasets, all of which show satisfactory performance in the multi-step-ahead prediction, even when the data are perturbed by noise and when the system is time-varying. Actually, such ARNN transformation equivalently expands the sample size and thus has great potential in practical applications in artificial intelligence and machine learning.