Statistical Complexity Measure Research Articles

Extracting time-delays from time series via statistical complexity measures based on multi-scale transition networks

Extracting time-delays from time series via statistical complexity measures based on multi-scale transition networks

Evolution Equations Exhibiting H-Theorems related to the LMC Statistical Measures of Complexity

Abstract The LMC statistical measure of complexity (and other related measures that, in this work, are collectively referred to as “LMC measures”) have been applied, by researchers in physics and other areas, to the study of diverse problems. In spite of the intriguing results reported in those studies, relatively little attention has been devoted to characterize what types of dynamics lead to the optimization of the LMC measures. As a first, exploratory step in that direction, we consider an example of a family of evolution equations admitting an H-like theorem related to the LMC measures.

Complex Function-Based Fault Detection: A New Method to Measure the Complexity of Nonlinear Time Series

In this paper, we propose a new complexity computation based on the complex function. This measure exploits the dispersion Lempel–Ziv complexity (DLZC) and the dispersion statistical complexity measure based on Jensen–Shannon divergence (DCJS) of the analytic signal and constructs the binary complex function, called complex number complexity (CNC). The statistical measure depicts the complex system from different angle and mines more potential information. Through simulation experiments, we prove that the proposed method is able to detect data fluctuations more sensitively and accurately. For an application, the CNC is applied to fault detection which has different fault diameters and rotational speeds. The results deliver that the CNC measure can well represent the complexity of faulty bearings and has significant differences in any two fault types. The technique is effective to characterize different fault.

Statistical Complexity Analysis of Neurovascular Coupling with Cognitive Stimulation in Healthy Participants.

Neurovascular coupling (NVC) is the tight relationship between changes in cerebral blood flow and neural activation. NVC can be evaluated non-invasively using transcranial Doppler ultrasound (TCD)-measured changes in brain activation (cerebral blood velocity [CBv]) using different cognitive tasks and stimuli. This study used a novel approach to analyzing CBv changes occurring in response to 20 tasks from the Addenbrooke's Cognitive Examination III in 40 healthy individuals. The novel approach compared various information entropy families (permutation, Tsallis, and Rényi entropy) and statistical complexity measures based on disequilibrium. Using this approach, we found the majority of the attention, visuospatial, and memory tasks from the Addenbrooke's Cognitive Examination III that showed lower statistical complexity values when compared with the resting state. On the entropy-complexity (HC) plane, a receiver operating characteristic curve was used to distinguish between baseline and cognitive tasks using the area under the curve. Best area under the curve values were 0.91 ± 0.04, p = .001, to distinguish between resting and cognitively active states. Our findings show that brain hemodynamic signals captured with TCD can be used to distinguish between resting state (baseline) and cognitive effort (stimulation paradigms) using entropy and statistical complexity as an alternative method to traditional techniques such as coherent averaging of CBv signals. Further work should directly compare these analysis methods to identify the optimal method for analyzing TCD-measured changes in NVC.

Exploring the interplay of intrinsic fluctuation and complexity in intracellular calcium dynamics

The concentration of intracellular calcium ion (Ca2＋) exhibits complex oscillations, including bursting and chaos, as observed experimentally. These dynamics are influenced by inherent fluctuations within cells, which serve as crucial determinants in cellular decision-making processes and fate determination. In this study, we systematically explore the interplay between intrinsic fluctuation and the complexity of various dynamic states of intracellular cytosolic Ca2＋. To investigate this interplay, we employ complexity measures such as permutation entropy and statistical complexity. Using a stochastic chemical Langevin equation, we simulate the dynamics of cytosolic Ca2＋. Our findings reveal that permutation entropy and statistical complexity effectively characterize the diverse, dynamic states of cytosolic Ca2＋ and illustrate their interactions with intrinsic fluctuation. Permutation entropy analysis elucidates that the chaotic state is more sensitive to intrinsic fluctuation than the other periodic states. Furthermore, we identify distinct states of cytosolic Ca2＋ occupying specific locations within the theoretical bounds of the complexity–entropy causality plane. These locations indicate varying complexity and information content as intrinsic fluctuation varies. When adjusting the permutation order, the statistical complexity measure for the different periodic and chaotic states exhibits peaks in an intermediate range of intrinsic fluctuation values. Additionally, we identify scale-free or fractal patterns in this intermediate range, which are further corroborated by multifractal detrended fluctuation analysis. These high-complexity states likely correspond to optimal Ca2＋ dynamics with biological significance, revealing rich and complex dynamics shaped by the interplay of intrinsic fluctuation and complexity. Our investigation enhances our understanding of the intricate regulatory mechanisms governing intracellular Ca2＋ dynamics and how intrinsic fluctuation modulates the complexity of the various dynamics that play crucial roles in biological cells.

Comparison of two statistical measures of complexity applied to ecological bipartite networks

Networks are a convenient way to represent many interactions among different entities as they provide an efficient and clear methodology to evaluate and organize relevant data. While there are many features for characterizing networks, a quantity seems rather elusive: Complexity. The quantification of the complexity of networks is nowadays a fundamental problem. Here, we present a novel tool for identifying the complexity of ecological networks. We compare the behavior of two relevant indices of complexity: K-complexity and Single Value Decomposition (SVD) entropy. For that, we use real data and two types of null models. Both null models consist of randomized networks built by swapping a controlled number of links of the original ones. We analyze 23 plant–pollinator and 19 host-parasite networks as case studies. Our results show that (a) it is necessary to calculate, not only the original network K-complexity and SVD entropy but also to calculate the corresponding indices of the randomized networks (b) the density and degree distribution are essential in the characterization of a network and the randomized networks are a suitable tool to detect the network complexity, and (c) plant–pollinator networks are more complex than host-parasite networks. We found that, for the first null model, K-complexity and SVD did not change with link swapping in both pollinator-plant and host-parasite networks. For the second null model, K-complexity for pollinator-plant networks generally decreased with an increasing number of links swapped (i.e. negative slope), showing that plant–pollinator networks lose complexity with increasing link swapping. In contrast, there was a positive slope between K-complexity and link swapping for host-parasite networks, showing that these networks are less complex than plant–pollinator networks. For both types of networks, in general, the slope between K-complexity and the number of links swapped became more positive with network density. Overall, SVD entropy was less responsive to link swapping. Our analyses show that although SVD entropy has been widely used to characterize network complexity, K-complexity is a more reliable tool. Additionally, they show that degree distribution and density are important drivers of network complexity and should be accounted for in future studies.

Detecting stochastic multiresonance in neural networks via statistical complexity measure

This paper employs statistical complexity measure (SCM) to investigate the occurrence of stochastic multiresonance (SMR) induced by noise and time delay in small-world neural networks coupled with FitzHugh-Nagumo (FHN) neurons. Our findings reveal that SCM exhibits four local maxima at four optimal noise levels, providing evidence for the occurrence of quadruple stochastic resonances. When time delay τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au$$\\end{document} is taken into account in the information transmission, under moderate noise levels, SCM shows several local maxima when τ=nTe\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au = nT_{e}$$\\end{document} with n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n$$\\end{document} being a positive integer and Te\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{e}$$\\end{document} being the period of subthreshold signal. This indicates the appearance of delay-induced SMR at the multiples of the period of subthreshold signal. Intriguingly, at low noise levels, a strong coherence between time delay and neuronal firing dynamics emerges at τ=nTe-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au = nT_{e} - 2$$\\end{document}, as confirmed by a series of SCM maxima at these time delays. Furthermore, the study demonstrates that by adjusting the degrees and sizes of small-world networks, as well as the coupling strength, it is possible to optimize the strength of delay-induced SMR, thus maximizing the detection capability of subthreshold signal. The research results may provide us with an effective approach for understanding the role of time delay in signal detection and information transmission.

Modification of Intertwining Logistic Map and a Novel Pseudo Random Number Generator

Chaotic maps have been widely studied in the field of cryptography for their complex dynamics. However, chaos-based cryptosystems have not been widely used in practice. One important reason is that the following requirements of practical engineering applications are not taken into account: computational complexity and difficulty of hardware implementation. In this paper, based on the demand for information security applications, we modify the local structure of the three-dimensional Intertwining Logistic chaotic map to improve the efficiency of software calculation and reduce the cost of hardware implementation while maintaining the complex dynamic behavior of the original map. To achieve the goal by reducing the number of floating point operations, we design a mechanism that can be decomposed into two processes. One process is that the input parameters value of the original system is fixed to 2k by Scale index analysis. The other process is that the transcendental function of the original system is replaced by a nonlinear polynomial. We named the new map as “Simple intertwining logistic”. The basic chaotic dynamic behavior of the new system for controlling parameter is qualitatively analyzed by bifurcation diagram and Lyapunov exponent; the non-periodicity of the sequence generated by the new system is quantitatively evaluated by using Scale index technique based on continuous wavelet change. Fuzzy entropy (FuzzyEn) is used to evaluate the randomness of the new system in different finite precision digital systems. The analysis and evaluation results show that the optimized map could achieve the designed target. Then, a novel scheme for generating pseudo-random numbers is proposed based on new map. To ensure its usability in cryptographic applications, a series of analysis are carried out. They mainly include key space analysis, recurrence plots analysis, correlation analysis, information entropy, statistical complexity measure, and performance speed. The statistical properties of the proposed pseudo random number generator (PRNG) are tested with NIST SP800-22 and DIEHARD. The obtained results of analyzing and statistical software testing shows that, the proposed PRNG passed all these tests and have good randomness. In particular, the speed of generating random numbers is extremely rapid compared with existing chaotic PRNGs. Compared to the original chaotic map (using the same scheme of random number generation), the speed is increased by 1.5 times. Thus, the proposed PRNG can be used in the information security.

Time-delay identification from chaotic time series via statistical complexity measures based on ordinal pattern transition networks

Time-delay identification from chaotic time series via statistical complexity measures based on ordinal pattern transition networks

Refined composite multivariate multiscale complexity-entropy causality plane analysis for gas-liquid two-phase flow

Abstract This paper presents a refined composite multivariate multiscale complexity-entropy causality plane (RCMMCECP) to explore the dynamics features of gas–liquid two-phase flow. Firstly, we employ a series of typical nonlinear time series to confirm the effectiveness of the RCMMCECP, including seven chaotic systems, two random processes, and one periodic process. The comparison results of the proposed method and conventional multivariate multiscale complexity-entropy causality plane (MMCECP) confirm the stability performance of the proposed RCMMCECP. Above all, the RCMMCECP enhances the reliability of the statistical complexity measure over large time scales and exhibits good continuity and noise-resistant ability in multiscale analysis. Then, we employ the RCMMCECP to analyze the upstream and downstream conductance signals. The experimental results demonstrate that the RCMMCECP can characterize the change of complexity and structural stability in the gas-liquid two-phase flow evolution process, effectively revealing its dynamics features.

Detecting the subthreshold signal in a neural network via statistical complexity measure

This paper proposes an information theory approach for detecting the subthreshold signal in a small-world network composed of Fitz Hugh-Nagumo (FHN) neurons. Statistical complexity measure (SCM) and normalized Shannon-entropy (NSE) have been defined based on the specific and nonconsecutive firing time intervals series, and employed to quantify the stochastic multiresonance (SMR) phenomena in this small-world neural network. The results show that there are several maxima of SCM and several minima of NSE at various optimal noise levels, which is regarded as the signature of the occurrence of SMR. This also implies that the subthreshold signal can be accurately detected across multiple levels of noise. More intriguingly, we have discovered that the degree of this neural network can induce the generation of multiple resonance-like behaviors. In addition, our findings demonstrate that the SCM outperforms the traditional signal-to-noise ratio (SNR) in accurately identifying firing dynamical regularity under certain noisy conditions. Furthermore, the SCM can capture subtle resonance-like behavior induced by the degree, which the SNR is unable to detect. Thus, the SCM could have potential power on detecting weak signals in neural networks.

A novel hybrid feature extraction approach of marine vessel signal via improved empirical mode decomposition and measuring complexity

The feature extraction of marine vessel-radiated noise (MVRN) under the complex ocean background is explored. To this end, a hybrid approach is presented based on the analysis of MVRN in subspaces of intrinsic mode functions (IMF) extracted using the improved empirical mode decomposition (IEMD) and measuring complexity. The restriction of the end-effect is an important problem when employing the EMD algorithm. In this study, first, to reduce the end-effects, an IEMD algorithm based on the correlation expansion model is proposed. Then, a comparative study of IEMD, classic EMD (CEMD), and EMD by other expansion methods are conducted on several signals. Next, IEMD, CEMD, and variational mode decomposition (VMD) algorithms are utilized to extract a group of IMFs for three types of MVRN. Later, one obtained IMF from each method that contains the most dominant information is selected. Lastly, two statistical complexity measures (i.e., permutation entropy (PE) and slope entropy (SlopEn)) are used as the features of the chosen IMF to improve the underwater signal separability and stability. Experimental results indicate that the suggested approach (IEMD-PE/SlopEn) can effectively extract the feature information of underwater signals. Additionally, it has a better ability to discriminate between various kinds of MVRN.

ASSESSMENT OF SECTOR BOND, EQUITY INDICES AND GREEN BOND INDEX USING INFORMATION THEORY QUANTIFIERS AND CLUSTERS TECHNIQUES

Green bonds are financial assets similar to classic debt securities used to finance sustainable investments. Given this, they are a long-term investment alternative that effectively contributes to the planet’s future by preserving the environment and encouraging sustainable development. This research encompasses a rich dataset of equity and bond sectors, general indices, and the S&P Green Bond Index. We estimate the permutation entropy [Formula: see text], an appropriate statistical complexity measure [Formula: see text], and Fisher Information measure [Formula: see text]. Therefore, we employ these complexity measures to construct two 2D maps, the complexity-entropy causality plane ([Formula: see text] ×[Formula: see text]) and the Shannon–Fisher causality plane ([Formula: see text] ×[Formula: see text]). Also, we use the information theory quantifiers to rank these indices’ efficiency analogous to the complexity hierarchy. From a mathematical point of view, the complexity-entropy causality plane (CECP) is a map that considers the global analysis, while the SFCP is a map that simultaneously feels the global and local analysis. Our findings reveal that both 2D maps indicated the most efficient (b_info_tech) and least efficient (b_energy) assets. There are peculiarities in the ranking performed considering the information theory quantifiers used to build each map due to the mathematical distinction that underlies the construction of each map. Moreover, we applied two clustering approaches ([Formula: see text]-means and Hierarchical cluster) that categorically converged in the indication of four distinct groups, which allowed us to verify that, in an overview, equities present a unique dynamic when compared to bonds and the Green bond index.

ASSESSMENT THE PREDICTABILITY IN THE PRICE DYNAMICS FOR THE TOP 10 CRYPTOCURRENCIES: THE IMPACTS OF RUSSIA–UKRAINE WAR

This paper has investigated the predictability of the top 10 cryptocurrencies’ price dynamics, ranked by their daily market capitalization and trade volume, via the information theory quantifiers. Our analysis considers the Complexity-entropy causality plane to study the temporal evolution of the price of these cryptocurrencies and their respective locations along this 2D map, bearing in mind after and during the Russia–Ukraine war. Moreover, we apply the permutation entropy and the Jensen–Shannon statistical complexity measure to rank these cryptocurrencies similarly to a complexity hierarchy. Our findings reflect that the Russian–Ukraine war affects the informational efficiency of cryptocurrency dynamics. Specifically, the cryptocurrencies notably showed a decrease in informational inefficiency (USD-coin, Binance-USD, BNB, Dogecoin, and XRP). At the same time, the cryptocurrencies with more expressiveness for the financial market, considering the volume traded and the capitalized market, were strongly impacted, presenting an increase in informational inefficiency (Tether, Cardano, Ethereum, and Bitcoin). It clarifies the potential of cryptocurrencies to mitigate exogenous shocks and their capability to use with portfolio selection, risk diversification and herding behavior.

Analysis of the Dispersion Havrda–Charvat Entropy Plane in Financial Time Series

This paper introduces a new statistical tool: dispersion Havrda–Charvat entropy plane, which is used to analyze the complexity characteristics of time series. The Havrda–Charvat entropy with one parameter can provide flexibility in applications and provide more information about time series. The dispersion entropy algorithm is a fast and powerful algorithm for evaluating time series, which has been proposed in recent years. The statistical complexity measure defined by Jensen–Shannon divergence reflects the chaotic degrees of complex systems. The dispersion Havrda–Charvat entropy plane is constructed using the above conceptions. The performance of the dispersion entropy plane is evaluated by simulated chaotic processes and fractional Brownian motions, and then we apply the method to stock data. This demonstrates that dispersion Havrda–Charvat entropy plane can distinguish the intensive properties of time series well and is a powerful method to classify stock markets. In addition, the multiscale measure is experimented, the results show that it can eliminate the noise contained in the data and effectively extract the information contained in time series with different time scales.

Detection of cardiac arrhythmia patterns in ECG through H × C plane.

The aim of this study is to formulate a new methodology based upon informational tools to detect patients with cardiac arrhythmias. As it is known, sudden death is the consequence of a final arrhythmia, and here lies the relevance of the efforts aimed at the early detection of arrhythmias. The information content in the time series from an electrocardiogram (ECG) signal is conveyed in the form of a probability distribution function, to compute the permutation entropy proposed by Bandt and Pompe. This selection was made seeking its remarkable conceptual simplicity, computational speed, and robustness to noise. In this work, two well-known databases were used, one containing normal sinus rhythms and another one containing arrhythmias, both from the MIT medical databank. For different values of embedding time delay τ, normalized permutation entropy and statistical complexity measure are computed to finally represent them on the horizontal and vertical axes, respectively, which define the causal plane H×C. To improve the results obtained in previous works, a feature set composed by these two magnitudes is built to train the following supervised machine learning algorithms: random forest (RF), support vector machine (SVM), and k nearest neighbors (kNN). To evaluate the performance of each classification technique, a 10-fold cross-validation scheme repeated 10 times was implemented. Finally, to select the best model, three quality parameters were computed, namely, accuracy, the area under the receiver operative characteristic (ROC) curve (AUC), and the F1-score. The results obtained show that the best classification model to detect the ECG coming from arrhythmic patients is RF. The values of the quality parameters were at the same levels reported in the available literature using a larger data set, thus supporting this proposal that uses a very small-sized feature space to train the model later used to classify. Summarizing, the attained results show the possibility to discriminate both groups of patients, with normal sinus rhythm or arrhythmic ECG, showing a promising efficiency in the definition of new markers for the detection of cardiovascular pathologies.

Quantum Statistical Complexity Measure as a Signaling of Correlation Transitions.

We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signaling function of quantum order–disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in correlation distributions. We apply our measure on two exactly solvable Hamiltonian models: the -Quantum Ising Model (in the single-particle reduced state), and on Heisenberg XXZ spin- chain (in the two-particle reduced state). We analyze its behavior across quantum phase transitions for finite system sizes, as well as in the thermodynamic limit by using Bethe Ansatz technique.

Optimal robust mean and location estimation via convex programs with respect to any pseudo-norms

We consider the problem of robust mean and location estimation with respect to any pseudo-norm of the form\(x\in {{\mathbb {R}}}^d\mapsto \left\| x\right\| _S = \sup _{v\in S}\bigl < v,x \bigr>\) where S is any symmetric subset of \({{\mathbb {R}}}^d\). We show that the deviation-optimal minimax sub-Gaussian rate for confidence \(1-\delta \) is $$\begin{aligned} \max \left( \frac{\ell ^*(\Sigma ^{1/2}S)}{\sqrt{N}}, \sup _{v\in S}\left\| \Sigma ^{1/2}v\right\| _2\sqrt{\frac{\log (1/\delta )}{N}}\right) \end{aligned}$$where \(\ell ^*(\Sigma ^{1/2}S)\) is the Gaussian mean width of \(\Sigma ^{1/2}S\) and \(\Sigma \) the covariance of the data. This improves the entropic minimax lower bound from Lugosi and Mendelson (Probab Theory Relat Fields 175(3–4):957–973, 2019) and closes the gap characterized by Sudakov’s inequality between the entropy and the Gaussian mean width for this problem. This shows that the right statistical complexity measure for the mean estimation problem is the Gaussian mean width. We also show that this rate can be achieved by a solution to a convex optimization problem in the adversarial and \(L_2\) heavy-tailed setup by considering minimum of some Fenchel–Legendre transforms constructed using the median-of-means principle. We finally show that this rate may also be achieved in situations where there is not even a first moment but a location parameter exists.

Statistical complexity of quantum circuits

In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely, the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resources in the circuit, we introduce a magic resource measure based on the $(p,q)$ group norm, which quantifies the amount of magic resource in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.

Temporal Vector Visibility Graph: A Tool for Complexity Analysis of Multivariate Time Series

The family of visibility algorithms provides a new point of view to describe time series by transforming them into networks. In this paper, we propound a new visibility algorithm named Temporal Vector Visibility graph [Formula: see text], which maps the multivariate time series to a directed network. Computed by the [Formula: see text] and [Formula: see text] degree distributions obtained from the [Formula: see text], these statistics such as the Kullback–Leibler divergence (KLD), the normalized Shannon entropy and the statistical complexity measure, are introduced to assess the complexity of time series. Furthermore, we also apply the Multivariate Multiscale Entropy Plane (MMEP) to evaluate the complexity of multivariate time series. The experimental results of eight different types of time series verify the effectiveness of our method. Subsequently, this method is employed to explore the complexity characteristics of financial time series and classify different stock markets. Our research reveals that this method is capable of investigating the physical structures of financial time series.