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Hyperspectral images encompass abundant information and provide unique characteristics for material classification. However, the labeling of training samples can be challenging in hyperspectral image classification. To address this problem, this study proposes a framework named flexible Gabor-based superpixel-level unsupervised linear discriminant analysis (FG-SuULDA) to extract the most informative and discriminating features for classification. First, a number of 3-D flexible Gabor filters are rigorously designed using an asymmetric sinusoidal wave to sufficiently characterize the spatial-spectral structure in hyperspectral images. Then, an unsupervised linear discriminant analysis strategy guided by the entropy rate superpixel (ERS) segmentation algorithm, called SuULDA, is skillfully introduced to reduce the extracted large amount of FG features. The SuULDA method not only boosts the classification capability but also increases the peculiarity of features, with the aid of superpixel information. Finally, the achieved features are imported to the popular support vector machine classifier. The proposed FG-SuULDA framework is applied to four real hyperspectral image data sets, and the experiments constantly prove that our FG-SuULDA is superior to several state-of-the-art methods in both classification performance and computational efficiency, especially with scarce training samples. The codes of this work are available at http://jiasen.tech/papers/ for the sake of reproducibility.

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e., on the standard sphere and Gromoll-Meyer exotic sphere . In this case, and are homeomorphic but not diffeomorphic. The standard sphere can be constructed as the quotient manifold with the so-called -action of S3, whereas the Gromoll-Meyer exotic sphere as the quotient manifold with respect to the so-called -action of S3. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism . The stochastic flow can be regarded as the same stochastic flow on S7, but viewed in Gromoll-Meyer differential structure. The flow on and the corresponding flow on constructed in this paper have the same regularities. There is no difference between the stochastic flow's appearance on S7 viewed in standard differential structure and the appearance of the same stochastic flow viewed in the Gromoll-Meyer differential structure. Furthermore, since the inverse mapping h-1 is differentiable on , the Riemannian metric tensor on , i.e., the pull-back of the Riemannian metric tensor G on the standard sphere , is also differentiable. This fact implies, for instance, the fact that the Fokker-Planck equation associated with the stochastic flow and the Fokker-Planck equation associated with the stochastic differential equation have the same regularities provided that the function β is C1-differentiable. Therefore both differential structures on S7 give the same description of the dynamics of the distribution function of the stochastic process understudy on seven spheres.

It is common to assess the "memory strength" of a stationary process looking at how fast the normalized log-determinant of its covariance submatrices (i.e., entropy rate) decreases. In this work, we propose an alternative characterization in terms of the normalized trace-inverse of the covariance submatrices. We show that this sequence is monotonically non-decreasing and is constant if and only if the process is white. Furthermore, while the entropy rate is associated with one-sided prediction errors (present from past), the new measure is associated with two-sided prediction errors (present from past and future). This measure can be used as an alternative to Burg's maximum-entropy principle for spectral estimation. We also propose a counterpart for non-stationary processes, by looking at the average trace-inverse of subsets.

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map another infinite transitive graph onto $G_n$, then we can improve the strategy by using a comparison with the random walk on this transitive graph (instead of that of the regular tree), and we obtain a lower bound of the form $1/h \log n$, where $h$ is the entropy rate associated with the walk on the transitive graph. We call this the entropic lower bound. It was recently proved that in the case of the tree, this entropic lower bound is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibit cutoff at the entropic time. In this paper, we provide a generalization of the result by providing a sufficient condition on the spectra the random walks on $G_n$ under which the random walk exhibit cutoff at the entropic time. It applies notably to anisotropic random walks on random $d$-regular graphs and to random walks on random $n$-lifts of a base graph (including non-reversible walks)., Comment: 45 pages, non-reversible random walks added, accepted for publication in Journal of the Institute of Mathematics of Jussieu

Chaotic-based S-box image encryption schemes promise to be a practical solution for securing digital images. However, the high-dimensional continuous chaotic has increased the algorithm’s complexity. Recent alternatives that focused on double or multiple S-box es approaches, on the other hand, have been proven vulnerable to differential attacks. This paper presents an efficient and secure chaotic-based S-box image encryption scheme. Firstly, a single S-box with a size of $10\times 26$ was constructed by using a low-dimensional chaotic system. Without a complex mathematical operation, the constructed single S-box has obvious efficiency advantages and achieved a higher image entropy rate than recent double or multiple S-box es. Secondly, a new dynamic encryption step method is proposed to solve the high correlation and deterministic problems in multiple S-box encryptions. Under the control of the dynamic encryption step algorithm, it effectively destroys the correlation between the source image’s pixels. The experimental results and security analysis show that the proposed scheme enjoys higher security and is more efficient to secure digital images in real-world applications.

In this article, we integrate the spatial-spectral information of hyperspectral image (HSI) samples into nonnegative matrix factorization (NMF) for affinity matrix learning to address the issue of HSI clustering. This technique consists of three main components: 1) oversegmentation for computing the spectral-spatial affinity matrix; 2) NMF with the guidance of the obtained affinity matrix; and 3) density-based spectral clustering on the final affinity matrix. First, the HSI is oversegmented into superpixels via the entropy rate superpixel algorithm. The spectral-spatial affinity matrix is defined based on the class-consistency assumption of all the HSI samples in each superpixel and the similar HSI samples between adjacent superpixels. Second, to integrate the spectral-spatial information into NMF, the obtained affinity matrix is used to guide the iterative process of NMF. The spectral-spatial affinity matrix is then weighted by the affinity matrix in the obtained low-dimensional subspace to form the final affinity matrix. Third, density-based spectral clustering is applied to the final affinity matrix to obtain clustering maps. Experimental results on three public benchmark HSIs demonstrate that the proposed method is superior to the considered state-of-the-art baseline methods on both the computational cost and clustering accuracy.

In this paper, based upon the appearance of patterns derived from a time series, we have investigated the suitability of multiscale entropy (MSE) technique for complexity quantification of cardiac rhythms in chronic pathological conditions. MSE analysis was developed to quantify the complexity of a wide variety of biomedical signals. Here, sample entropy (SampEn) technique was evaluated across multiple spatio-temporal scales. In SampEn, to find the appearance of repetitive patterns in multi-dimensional phase space, the threshold value ‘s’ is pre-fixed as 0.2. However, the cardiac rhythms of some pathologies are characterized with considerable erratic beat-to-beat fluctuations, and hence, in accordance with that, the patterns concealed in the pathologic cardiac rhythms spread across a wider region of multidimensional phase space. But, fixed threshold value ‘s’ assigns a fewer similarity pattern inside a circle of fixed dimensions, and hence, the higher entropy rate is associated with the chronic pathologic cardiac rhythms when compared to healthy cardiac rhythms. This flaw of SampEn is present in MSE, which leads to the wrong estimation of complexity associated with a time series. The outcome of this issue is clearly visible at low time scales, where period-to-period fluctuations in chronic pathologic cardiac rhythms and in randomized time series are significantly increased. In this present study, MSE analysis was performed over synthetic simulated database comprising of (white noise) WN and (power noise) PN signals. Further, MSE analysis was performed on the RR-interval series collected from (normal sinus rhythm) NSR group, and patients affected by (Atrial Fibrillation) AF. A fixed number of data samples ‘M’ of 10,000 were considered for each type of time series. Here, it is being observed that at some time scales, MSE assigns higher entropy to the WN and AF group, rather than PN and NSR group respectively, which is a wrong estimation of complexity. However, both the groups are discriminated efficiently by this algorithm. Further, it is concluded that MSE measure both the entropy and short term variations associated with a time series, but unable to investigate the real complexity (meaning full structural organization) present in a signal.

A long-standing question in applications of dynamical systems theory is how to distinguish noisy signals from chaotic steady states. Information-theoretic measures hold promise to resolve this problem. We apply two such measures to numerically computed phase-space trajectories of continuous-state nonlinear oscillators: forecasting or statistical complexity, which quantifies the minimum memory required for the optimal prediction of discrete observables, and the entropy rate, which quantifies their intrinsic unpredictability. We estimate empirical generating partitions to obtain discrete observables faithfully representing continuous-state chaotic time series. We focus on the problem of distinguishing stochastically perturbed periodic orbits from chaotic attractors that exist at nearby parameter values, in a region of the parameter space where a strange invariant set exists. We find that a stochastically perturbed, stable, high-period ( $$p=15$$ ) orbit of a periodically driven Duffing oscillator admits high values of both information measures, making it difficult to distinguish it from chaotic states at adjacent parameters, even with small noise. However, for a low-period ( $$p=3$$ ) orbit, such a distinction becomes easier, as both measures admit considerably lower values compared to a chaotic attractor at a nearby parameter. Furthermore, the forecasting complexity of the selected periodic orbits increases with noise as they undergo a transition to “noise-induced chaos.” For sufficiently high noise levels, our ability to distinguish chaos from noise depends on model-order selection when estimating forecasting complexity and also on the exact choice of discrete observables used to encode phase-space trajectories.

In this present communication, we proposed the mathematical model for non-Newtonian liquid (Ree-Eyring model) towards a stretched surface. The Buongiorno model is used in the modelling of flow problem subject to magnetohydrodynamics, entropy generation, nonlinear thermal radiation, homogeneous-heterogeneous reactions, Brownian motion, Joule heating and thermophoresis diffusion. Convective boundary conditions are implemented at the stretched surface. With the assistance of rheological expression of Ree-Eyring model, we construct the governing flow expressions. The energy equation is developed via Brownian diffusion, nonlinear thermal radiation, Joule heating and thermophoresis. Two different types of species (homogeneous-heterogeneous reactions) are considered for the analysis of mass transport. The flow is solely induced due to a nonlinear stretched surface. The electrical conducting and chemically reactive fluid is considered for the analysis towards a stretched surface. The appropriate transformations are implemented to transform governing PDEs into a set of coupled ODEs. Shooting method has been applied to get the solutions of obtained highly non-linear ODEs. The thermodynamics second law is employed to model and calculate total entropy rate. Impact of influential variables on the flow field (velocity), gradient of velocity, temperature, gradient of temperature, concentration and entropy generation rate are studied. The engineering quantities like velocity, temperature gradients are numerical discussed in tabular form. Furthermore, the entropy rate and Bejan number show contrast impact versus larger Brinkman number.

From the perspective of an experimentalist, we discuss the comparison framework that ensures the same information rate in probabilistic shaping (PS) signals. Theoretically, the information rate can be tuned by adjusting the target entropy, scaling the signal bandwidth and adjusting FEC code rate. In practice, the information rate tuning should be based on the hardware feasibility (e.g., DAC resolution, available analog bandwidth, cost of rate adaptation). Adjusting a constellation entropy rate without considering FEC redundancy will cause an over-estimation of the PS system performance gain. To avoid the over-estimation issue, the modified pre-FEC threshold for PS signal is also proposed to estimate the system performance accurately. The modified pre-FEC threshold is obtained from the FEC code rate of PS signal. Moreover, we experimentally investigated the adaptive entropy allocation scheme in a millimeter-wave analog radio over fiber fronthaul using the modified pre-FEC threshold. By grouping the subcarriers into PS units and using the same QAM order, the proposed scheme can reduce the required data frame length as well as the processing complexity. Notably, this method requires neither dynamic FEC coding rate nor bandwidth adjusting method. Up to 2.5-dB power margins and a smoother pre-FEC performance among PS units are achieved in a frequency selective fading channel. If the modified FEC threshold were not used, it would induce up to 0.7-dB over-estimation in the received sensitivity. This indicates the importance of employing the proposed methods for comparison when FEC is used in combination with PS.

This paper presents a systematic approach to identify and quantify the types of structures featured by packettraces in communication networks. Our approach leverages an information-theoretic methodology, based oniterative randomization and compression of the packet trace, which allows us to systematically remove andmeasure dimensions of structure in the trace. In particular, we introduce the notion oftrace complexitywhichapproximates the entropy rate of a packet trace. Considering several real-world traces, we show that tracecomplexity can provide unique insights into the characteristics of various applications. Based on our approach,we also propose a traffic generator model able to produce a synthetic trace that matches the complexity levelsof its corresponding real-world trace. Using a case study in the context of datacenters, we show that insightsinto the structure of packet traces can lead to improved demand-aware network designs: datacenter topologiesthat are optimized for specific traffic patterns., European Union. Horizon 2020 Research and Innovation Programme (Agreement 864228 AdjustNet: Self-Adjusting Networks)

Feature-based image watermarking is a well-developed area of digital image watermarking. The key advantage of such an approach is the ability to guard against the local desynchronization attacks. However, research in this area usually suffers from the following limitations: (1) The feature points focus too much on the high contrast region and are distributed unevenly. (2) The feature point detector is sensitive to the texture region, and some noise feature points are always detected in the texture region. Based on superpixel image segmentation and quaternion wavelet transform, we propose a digital image watermarking approach which is highly robust against common image processing operations and local geometric transformations. First, the entropy rate superpixel segmentation is used to segment the host image into several homogeneous regions. Then, for each segment region, image feature points are extracted using the SIFER detector, and the affine invariant local regions are constructed adaptively. Finally, the digital watermark is embedded into the local regions by modulating the invariant modulus coefficients. Experimental results are provided to illustrate the efficiency of the proposed image watermarking, especially for noise attacks and local desynchronization attacks.

We study event-triggered control for stabilization of unstable linear plants over rate-limited communication channels subject to unknown, bounded delay. On one hand, the timing of event triggering carries implicit information about the state of the plant. On the other hand, the delay in the communication channel causes information loss, as it makes the state information available at the controller out of date. Combining these two effects, we show a phase transition behavior in the transmission rate required for stabilization using a given event-triggering strategy. For small values of the delay, the timing information carried by the triggering events is substantial, and the system can be stabilized with any positive rate. When the delay exceeds a critical threshold, the timing information alone is not enough to achieve stabilization and the required rate grows. When the loss of information due to the communication delay perfectly compensates the implicit information carried by the triggering events, the delay equals the inverse of the entropy rate of the plant, and we obtain the same rate requirement prescribed by the data-rate theorem. When the delay is larger than this threshold, the required rate becomes larger than that required by the data-rate theorem. We also provide an explicit construction yielding a sufficient rate for stabilization, and generalize our results to vector systems. The results do not rely on any a priori probabilistic model of the delay or the initial conditions.

True random number generators (TRNGs) play a fundamental role in cryptographic systems. This brief presents a new and efficient method to generate true random numbers on field programmable gate array (FPGA) by utilizing the random jitter of free-running oscillators as a source of randomness. The free-running oscillator rings incorporate programmable delay lines (PDLs) to generate large variation of the oscillations and to introduce jitter in the generated ring oscillators clocks. The main advantage of the proposed TRNG utilizing PDLs is to reduce correlation between several equal length oscillator rings, and thus improve the randomness qualities. In addition, a Von Neumann corrector as post-processor is employed to remove any bias in the output bit sequence. The validation of the proposed approach is demonstrated on Xilinx Spartan-3A FPGAs. The proposed TRNG occupies 528 slices, achieves 6 Mb/s throughput with 0.999 per bit entropy rate, and passes all the national institute of standards and technology (NIST) statistical tests.

Here, we communicate the peristalsis of single- and multi-walled CNTs through nonlinear porous, non-uniform propagating channel boundaries. Non-uniform channel boundaries are of asymmetric characteristics. Flow equations are modeled by taking variable viscosity, linear and nonlinear porous medium (i.e., Darcy and Darcy–Forchheimer), nonlinear thermal radiation, mixed convection, heat generation and absorption aspects. Convective heat transfer aspects are at the convectively heated surface. The entropy generation rate is modeled via the second law of thermodynamics. Modeled equations are simplified with the help of long wavelength assumption. Dimensionless system of equations with respective boundary conditions is solved numerically via built-in shooting algorithm in Mathematica 8 software. Further, these numerical results are directly received in the form of curves. Such curves are made for velocity, temperature, pressure gradient, trapping, entropy rate and Bejan number. Heat transfer rate at lower and upper walls is achieved through bar charts.

Work is devoted to studying of a linear mode thermodynamic – a mode which is actively investigated now. One of the main concepts of a linear mode – local entropy rate of production. The purpose of given article consists in expansion of a circle of problems for which it is possible to calculate a local entropy rate of production, namely its definition, using the experimental “time-temperature” curves of heating/cooling. “Time-temperature” curves heating or cooling are widely used in non-stationary thermophysical experiments at studying properties of substances and materials: phase transitions of the first and second sort, a thermal capacity, thermal diffusivity. The quantitative substantiation of the formula for calculation of the local entropy rate of production in which it is used thermogram (change of temperature from time) which is received by a method of pulse electric heating is resulted. Initial time dependences of electric capacity and temperature are measured on the sample of niobium in a microsecond range simultaneously. Conformity of two dependences of the local entropy rate of production from time is shown: one is calculated under the known formula in which the brought electric capacity is used; another is calculated, using the thermogram.

Clustering is one of fundamental tasks of data mining and machine learning. Due to the limitation of cluster assumption, lots of clustering algorithms perform poorly on some datasets against their assumptions, especially high-dimensional big data. This paper presents a maximum average entropy-rate based correlation clustering algorithm which is a kind of a graph-based correlation clustering. The objective function of original correlation clustering is decomposed into several single cluster optimizations and the limitation of big data in correlation clustering is removed by the neighboring connected graph. In algorithm implementation, the optimization of proper neighbor searching and correlation clustering are performed by heuristic neighbor searching and cluster generating respectively, and there is also an efficient graph-iterated implementation on distributed computation platform. Compared with other clustering algorithms, the proposed clustering algorithm is more flexible in cluster assumption, when accelerating the clustering process. In an experimental study we demonstrate the performance of the proposed algorithms on several datasets. The proposed clustering algorithm performed better than the other six clustering algorithms on the highest f1-measure and purity values, while its running time on high-dimensional big data is much lower than other clustering algorithms.

We study random string-duplication systems, which we call Polya string models. These are motivated by a class of mutations that are common in most organisms and lead to an abundance of repeated sequences in their genomes. Unlike previous works that study the combinatorial capacity of string-duplication systems, or in a probabilistic setting, various string statistics, this work provides the exact entropy rate or bounds on it, for several probabilistic models. The entropy rate determines the compressibility of the resulting sequences, as well as quantifying the amount of sequence diversity that these mutations can create. In particular, we study the entropy rate of noisy string-duplication systems, including the tandem-duplication, end-duplication, and interspersed-duplication systems, where in all cases we study duplication of length 1 only. Interesting connections are drawn between some systems and the signature of random permutations, as well as to the beta distribution common in population genetics.

The challenges from Distributed Denial-of-Service (DDoS) attacks are severe and still increasing significantly. We observe that the existing entropy-based methods only consider the probability distribution of traffic flows that have high false negative rates. On the other hand, sophisticated attack strategies, increasing attack strength and dynamic nature of network traffic patterns make it more difficult to detect the DDoS attacks with high accuracy. In this letter, we present an accurate approach, entropy rate measurement (ERM), to detect DDoS attacks. The proposed approach is based on the differences between the probability distributions and the number of flows. Both theoretical proofs and the results of experiments using real datasets demonstrate that our method has high detection accuracy rate compared to the existing measurements.

Modeling for boundary layer stagnation point flow of Williamson fluid is developed. Electrically conducting liquid in presence of constant magnetic field is considered. Fluid is conducting. Induced magnetic field is accounted. Energy equation is modeled subject to radiative heat flux, heat source/sink and dissipation. Concentration equation for binary chemical reaction with activation energy is examined. Volumetric entropy rate is computed employing second law of thermodynamics. Nonlinear system numerically solved. Outcomes of velocity, temperature, entropy generation and concentration are carefully examined. Nusselt number and skin friction coefficient are numerically discussed. The obtained results are matched in an excellent manner.

The overarching goal of the present work is to contribute to the understanding of the relations between fetal heart rate (FHR) temporal dynamics and the well-being of the fetus, notably in terms of predicting \M{the evolution of lactate, pH and} cardiovascular decompensation (CVD). It makes uses of an established animal model of human labor, where fourteen near-term ovine fetuses subjected to umbilical cord occlusions (UCO) were instrumented to permit regular intermittent measurements of metabolites (lactate and base excess), pH, and continuous recording of electrocardiogram (ECG) and systemic arterial blood pressure to identify CVD during UCO. ECG-derived FHR was digitized at the sampling rate of 1000 Hz and resampled to 4 Hz, as used in clinical routine. We focused on four FHR variability features which are tunable to temporal scales of FHR dynamics, robustly computable from FHR sampled at 4Hz and within short-time sliding windows, hence permitting a time-dependent, or local, analysis of FHR which helps to deal with signal noise. Results show the sensitivity of the proposed features for early detection of CVD, correlation to metabolites and pH, useful for early acidosis detection, and the importance of coarse time scales (2.5 to 8 seconds) which are not disturbed by the low FHR sampling rate. Further, we introduce the performance of an individualized self-referencing metric of the distance to a healthy state, based on a combination of the four features. We demonstrate that this novel metric, applied to clinically available FHR temporal dynamics alone, accurately predicts the time occurrence of CVD which heralds a clinically significant degradation of the fetal health reserve to tolerate the trial of labor.

Ocular activity is known to be sensitive to variations in mental workload, and recent studies have successfully related the distribution of eye fixations to the mental load. This study aimed to verify the effectiveness of the spatial distribution of fixations as a measure of mental workload and its sensitivity to different types of demands imposed by the task: mental, temporal, and physical. To test the research hypothesis, two experimental studies were run: Experiment 1 evaluated the sensitivity of an index of spatial distribution (Nearest Neighbor Index; NNI) to changes in workload. A sample of 30 participants participated in a within-subject design with different types of task demands (mental, temporal, physical) applied to Tetris game; Experiment 2 investigated the accuracy of the index through the analysis of 1-min epochs during the execution of a visual-spatial task (the “spot the differences” puzzle game). Additionally, NNI was compared to a better-known ocular mental workload index, the entropy rate. The data analysis showed a relation between the NNI and the different workload levels imposed by the tasks. In particular: Experiment 1 demonstrated that increased difficulty, due to higher temporal demand, led to a more dispersed pattern with respect to the baseline, whereas the mental demand led to a more grouped pattern of fixations with respect to the baseline; Experiment 2 indicated that the entropy rate and the NNI show a similar pattern over time, indicating high mental workload after the first minute of activity. That suggests that NNI highlights the greater presence of fixation groups and, accordingly, the entropy indicates a more regular and orderly scanpath. Both indices are sensitive to changes in workload and they seem to anticipate the drop in performance. However, the entropy rate is limited by the use of the areas of interest, making it impossible to apply it in dynamic contexts. Conversely, NNI works with the entire scanpath and it shows sensitivity to different types of task demands. These results confirm the NNI as a measure applicable to different contexts and its potential use as a trigger in adaptive systems implemented in high-risk settings, such as control rooms and transportation systems.

We leverage the interplay between microscopic variability and macroscopic order to connect physical descriptions across scales directly from data, without underlying equations. We reconstruct a state space by concatenating measurements in time, building a maximum entropy partition of the resulting sequences, and choosing the sequence length to maximize predictive information. Trading non-linear trajectories for linear, ensemble evolution, we analyze reconstructed dynamics through transfer operators. The evolution is parameterized by a transition timeτ: capturing the source entropy rate at smallτand revealing timescale separation with collective, coherent states through the operator spectrum at largerτ. Applicable to both deterministic and stochastic systems, we illustrate our approach through the Langevin dynamics of a particle in a double-well potential and the Lorenz system. Applied to the behavior of the nematode wormC. elegans, we derive a “run-and-pirouette” navigation strategy directly from posture dynamics. We demonstrate how sequences simulated from the ensemble evolution capture both fine scale posture dynamics and large scale effective diffusion in the worm’s centroid trajectories and introduce a top-down, operator-based clustering which reveals subtle subdivisions of the “run” behavior.POPULAR SUMMARYComplex structure is often composed from a limited set of relatively simple building blocks; such as novels from letters or proteins from amino acids. In musical composition, e.g., sounds and silences combine to form longer time scale structures; motifs form passages which in turn form movements. The challenge we address is how to identify collective variables which distinguish structures across such disparate time scales. We introduce a principled framework for learning effective descriptions directly from observations. Just as a musical piece transitions from one movement to the next, the collective dynamics we infer consists of transitions between macroscopic states, like jumps between metastable states in an effective potential landscape.The statistics of these transitions are captured compactly by transfer operators. These operators play a central role, guiding the construction of maximally-predictive short-time states from incomplete measurements and identifying collective modes via eigenvalue decomposition. We demonstrate our analysis in both stochastic and deterministic systems, and with an application to the movement dynamics of an entire organism, unravelling new insight in long time scale behavioral states directly from measurements of posture dynamics. We can, in principle, also make connections to both longer or shorter timescales. Microscopically, postural dynamics result from the fine scale interactions of actin and myosin in the muscles, and from electrical impulses in the brain and nervous system. Macroscopically, behavioral dynamics may be extended to longer time scales, to moods or dispositions, including changes during aging, or over generations due to ecological or evolutionary adaptation. The generality of our approach provides opportunity for insights on long term dynamics within a wide variety of complex systems.

The stability for magnetic field to the solution of the Riemann problem for the polytropic fluid in a variable cross-section duct is discussed. By the vanishing magnetic field method, the stable solutions are determined by comparing the limit solutions with the solutions of the Riemann problem for the polytropic fluid in a duct obtained by the entropy rate admissibility criterion.

Oversegmentation is a preprocessing step for many computer vision and remote sensing tasks, such as object recognition and image understanding. In this paper, a method called agglomerative oversegmentation (AOS) is proposed. AOS first designs a dual similarity of the connected pixels by considering both the pixel and region levels. The entropy rate of random walks is then employed on the image plane, and the gain of each pair of connected pixels is estimated. Finally, a fast agglomerative algorithm is developed to oversegment the image using an entropy rate gain function. Two challenging datasets are utilized in the experiments which show the promising performance of AOS. In contrast to the state-of-the-art algorithms, the oversegmentation of AOS is more consistent with the geometric structures of the target objects, especially for the objects with complex texture on remote sensing images.

Background and objective Improvement of high performance thermal systems for heat transport augmentation has become quite prevalent nowadays. Various works have been performed to pick up a comprehension of the heat transport execution for their practical utilization to heat transport augmentation. Therefore, the nanomaterial has been used in flow of Ree-Eyring fluid between two rotating disks for thermal conductivity enhancement of base fluid. Heat transfer characteristics are discussed through viscous dissipation and heat source/sink. Behaviors of Brownian motion and thermophoresis are also examinted. Physical behaviors of irreversibility in nanofluid with Arrhenius activation energy are also accounted. Methods The nonlinear systems lead to ordinary differential problems through implementation of appropriate transformations. The relevant problems are tackled by (OHAM) Optimal homotopic method for series solutions. Results Effects of various physical parameters on the velocity, entropy rate, Bejan number, concentration and temperature are discussed graphically. Skin friction coefficient and gradient of temperature are numerically examined and discussed with various parameters. Conclusions Entropy generation rate is control by minimizing the values of Brinkman number and stretching parameter. Entropy rate and Bejan number show the dual behaviors against Eckert number. Both decay near the lower disk while reverse holds near the upper disk. Entropy rate and Bejan number show similar behaviors for Weissenberg number.

Here, cubic autocatalysis in radiative flow of power law fluid is addressed. Flow is examined by a stretched surface. Heat generation/absorption, nonlinear radiation and dissipation effect are utilized in modeling of energy equation. Through implementation of second thermodynamics law, the entropy rate is calculated. Further entropy generation is explored with respect to homogeneous and heterogeneous mass concentrations. Nonlinear system is tackled via NDSolve of MATHEMATICA. Impacts of pertinent variables associated with flow are physically discussed for the velocity, entropy generation, temperature and concentration fields. Clearly, entropy augmentation rate is controlled through Brinkman number. Main key points are presented.

Nanomaterials belong to a new class of materials with increased thermo-physical characteristics and heat transport performance. A comprehensive spectrum of applications in science and mechanical engineering could possibly help from them. Therefore keeping such effectiveness in mind we pointed to explore the characteristics of nanomaterial in non-Darcy mixed convective flow of viscous nanofluid through effective and without effective Prandtl number towards a stretched surface. Through implementation of second thermodynamics law total entropy rate is obtained. Stagnation point flow is pointed out. Shooting technique is implemented to tackle the nonlinear equations for numerical results. The variations in entropy generation, velocity, Bejan number and temperature fields for different estimations of flow parameters like mixed convection parameter (0.0–0.4), inverse Darcy number (0.0–0.4), Inertia parameter (1.0–5.8), velocity ratio parameter (0.0–0.4) and nanoparticle volume fraction (0.00–0.8) are discussed and the results are graphically presented. Skin friction and Nusselt number of considered flow are discussed via graphically sketch. Moreover, it is examined that velocity of fluid particles increases for both effective and without effective Prandtl number with rising estimations of nanoparticles volume fractions. It is also examined that entropy rate shows dual behavior verses nanoparticles volume fraction. Entropy rate enhances in case of effective Prandtl number, while decays in case without effective Prandtl number.

Process mining algorithms fall in two classes: imperative miners output flow diagrams, showing all possible paths, whereas declarative miners output constraints, showing the rules governing a process. But given a log, how do we know which of the two to apply? Assuming that logs exhibiting a large degree of variability are more suited for declarative miners, we can attempt to answer this question by defining a suitable measure of the variability of the log. This paper reports on an exploratory study into the use of entropy measures as metrics of variability. We survey notions of entropy used, e.g. in physics; we propose variant notions likely more suitable for the field of process mining; we provide an implementation of every entropy notion discussed; and we report entropy measures for a collection of both synthetic and real-life logs. Finally, based on anecdotal indications of which logs are better suited for declarative/imperative mining, we identify the most promising measures for future studies. For estimating overall entropy, global block and k-nearest neighbour estimators of entropy appear most promising and excel at identifying noise in logs. For estimating entropy rate we identify Lempel–Ziv and certain variants of k-block estimators performing well, and note that the former is more stable, but sensitive to noise, while the latter is less stable, being sensitive to cut-off constraints determining block size.

To get a deep understanding of a diffusion process and realizing the most efficient methods for investigating a real network, has always been of great interest and utility to us. In this work, we aim to study and compare our mobility in a network using a normal random walk and a long-range navigation strategy such as Levy Walk. We study the Global Mean First Traverse Distance (GMFTD) and the entropy rate for this long-range navigation process and later, compare with a normal walk strategy. For this study, GMFTD is utilized instead of Global Mean First Passage Time (GMFPT). The reason for such a choice is the fact that for more accurate and precise results, we need to also take into account the cost of a large step in case of a long-range navigation strategy. Next, we continue by calculating the entropy rate in both procedures and we derive the corresponding expressions of these quantities on a Treelike fractal. Eventually, we show that while GMFTD decreases in a long-range navigation system, entropy rate will increase. We perform an analytical comparison between the two cases and clearly demonstrate the superiority of a Levy Walk strategy. Eventually, our simulation results, based on the derived expressions, give us a very reliable estimation for the optimum value of exponent parameter, α, which are in strong agreement with each other for both cases of a minimum GMFTD and a maximum entropy rate.

This paper provides a comprehensive analysis of the following optimization problem: Maximize the entropy rate generated by a Markov chain over a connected graph of order $n$ and subject to a prescribed stationary distribution. First, we show that this problem is strictly convex with global optimum lying in the interior of the feasible space. Second, using Lagrange multipliers, we provide a closed-form expression for the maxentropic Markov chain as a function of an $n$ -dimensional vector, referred to as the maxentropic vector; we provide a provably converging iteration to compute this vector. Third, we show that the maxentropic Markov chain is reversible, compute its entropy rate and describe special cases, among other results. Fourth, through analysis and simulations, we show that our proposed procedure is more computationally efficient than semidefinite programming methods. Finally, we apply these results to robotic surveillance problems. We show realizations of the maxentropic Markov chains over prototypical robotic roadmaps and find that maxentropic Markov chains outperform minimum mean hitting time Markov chains for the so-called “intelligent intruders” with short attack durations. A comprehensive analysis of the following optimization problem: maximize the entropy rate generated by a Markov chain over a connected graph of order $n$ and subject to a prescribed stationary distribution.

Here we address entropy generation in mixed convective magnetohydrodynamics chemically reactive flow of second-grade nanomaterial over a heated stretched sheet. Second-grade nanomaterial is considered electrically conducting. Characteristics of heat transfer are discussed through nonlinear heat flux, viscous dissipation and heat source/sink. Novel behaviors of thermophoresis and Brownian diffusion are also accounted. Nonlinear flow expressions are first converted into ordinary differential one through appropriate transformations. Optimal homotopy method is used for solution development. Impacts of different flow variables on the velocity, entropy generation rate, Bejan number, temperature and concentration are discussed through graphs. Surface drag force and gradient of temperature are graphically discussed. Present results are also compared with published literature. Velocity and temperature fields monotonically boost up via higher second-grade parameter. Entropy rate increases via larger Brinkmann number and magnetic parameter. Bejan number is maximum for (Br = 0) and then it gradually decays for increasing the value of Brinkmann number.

In the present paper, formulation and computations are made to investigate the flow of nanofluid past a heated stretching cylinder. Flow in porous space is specified by Darcy–Forchheimer model. Water with silver (Ag), copper (Cu), copper oxide (CuO), titanium oxide (TiO2) and aluminum oxide (Al2O3) are considered as nanofluids. Heat transport mechanism is expressed through nonlinear radiation and non-uniform heat generation/absorption. Convective condition is also utilized for heat transport phenomenon. Through implementation of second law of thermodynamics the total entropy generation is calculated. Moreover, entropy generation for fluid friction and heat transfer is discussed. This study is specially investigated for the impact of Darcy–Forchheimer relation with entropy generation subject to distinct flow parameters. Optimal homotopy algorithm computes the nonlinear strong differential systems. Plots reflecting outcomes of influential variables are examined. The obtained results indicate that entropy rate directly depends on Brinkman number, temperature difference parameter and Forchheimer number. Estimations of skin friction coefficient and heat transfer rate are also carried out. It is found that magnitude of rate of heat transfer and skin friction coefficient enhances via volume fraction of nanoparticles.

Predictability of behavior is an important characteristic in many fields including biology, medicine, marketing, and education. When a sequence of actions performed by an individual can be modeled as a stationary time-homogeneous Markov chain the predictability of the individual’s behavior can be quantified by the entropy rate of the process. This article compares three estimators of the entropy rate of finite Markov processes. The first two methods directly estimate the entropy rate through estimates of the transition matrix and stationary distribution of the process. The third method is related to the sliding-window Lempel–Ziv compression algorithm. The methods are compared via a simulation study and in the context of a study of interactions between mothers and their children.

Compared with more and more steganography techniques motivated by abundant compressed speech, steganalysis is still a challenging task. Many existing studies are based on a single dimensional feature model and it is difficult to have a wide range of applicability. In this paper, a hybrid Markov model is proposed, which is based on the correlation of fixed codebook parameters in speech codec between pulses in a given track. And then, two detecting methods based on entropy are given. One is designed as a single-pulse position based entropy detection method (SPBE). The other is pulse-position pairs based entropy detection method (PPBE). Simultaneously, to solve the problem of inaccurate calculation of the entropy rate of finite length samples, corrected conditional entropy (CCE) is used as an estimate of the Markov chain entropy rate. Experiments show that CCE and entropy are highly complementary, and both can be employed as classification features to achieve better steganalysis results. Finally, the performance of the proposed detection methods is evaluated and compared with the existing detection methods. Results prove that the two methods proposed in this paper are suitable for online and real-time steganographic detection, especially for small-size samples.

In this note, we use the Perron–Frobenius theorem to obtain the Rényi's entropy rate for a time-inhomogeneous Markov chain whose transition matrices converge to a primitive matrix. As direct corollaries, we also obtain the Rényi's entropy rate for asymptotic circular Markov chain and the Rényi's divergence rate between two time-inhomogeneous Markov chains.

This research article deals with entropy generation and heat generation/absorption analysis for MHD stagnation point flow of Sisko-nanomaterial towards stretchable sheet. Homogeneous-heterogeneous reactions are considered. Nanomaterial comprises thermophoresis and Brownian diffusion effect. Through implementation of second law of thermodynamics the total entropy generation is calculated. In addition, entropy generation for fluid friction, mass transfer and heat transfer is discussed. This study is specially investigated for the impact of homogeneous-heterogeneous reactions with entropy generation subject to distinct flow parameters. The nonlinear systems are computed. Influence of pertinent flow variables on the velocity, entropy, concentration and temperature are discussed. Coefficient of skin friction (velocity gradient) and Nusselt number (temperature gradient) are calculated and examined graphically. The obtained results indicate that entropy rate directly depends upon Brinkman number, diffusion parameter, temperature and concentration difference parameters, magnetic parameter and volumetric entropy number.

The storage and the transmission of messages in big data are discussed in this paper, where message importance is taken into account. To this end, we propose to use non-parametric message importance measure (NMIM) as a measure of message importance, which can characterize the uncertainty of random events like Shannon entropy and Renyi entropy. We prove that NMIM sufficiently describes the two key characters of big data, i.e., the rare events finding and the large diversities of events. Based on NMIM, we then propose an effective compressed encoding mode for data storage, and discuss the transmission of messages over some typical channel models with limited message importance loss. Our numerical results show that the proposed strategy occupies less storage space without losing too much important information, and the maximum received entropy rate increases with the increasing of message importance loss until it reaches saturation, which contributes to designing of better practical communication system.

Face detection plays an important role in many computer vision applications. In recent years, much research has focused on extending the well-established Adaboost face detector algorithm for multi-view face detection. However, detecting in-plane and out-of-plane rotated faces simultaneously is still a challenging task today. In this paper, a very robust multi-view face detection algorithm, with its core functionality based on frontal face detection, is proposed to simultaneously detect in-plane and out-of-plane rotated faces. Moreover, only the training data in the frontal face is needed and we do not require the training data from different in-plane or out-of-plane rotation angles. In the proposed algorithm, first, techniques such as the skin filter and entropy rate superpixels (ERSs) are applied to obtain face candidates. Then, angle compensation and refinement are applied to improve the accuracy of face detection in the in-plane case. Moreover, the symmetry extension technique, i.e., extending the face candidate with its flipped version to create a face similar to the frontal one, is applied to detect out-of-plane faces without the need of training data. Simulations on the FEI, Pointing'04, BaoFace, Group, Utrecht, and WWW datasets demonstrate the proposed algorithm's effectiveness and superior performance as compared to state-of-the-art face detection methods.

We evaluate the (ɛ, τ) entropy of chaotic laser outputs generated by an optically injected semiconductor laser for physical random number generation. The vertical resolution ɛ and sampling time τ are numerically optimized by comparing the (ɛ, τ) entropy with the Kolmogorov–Sinai entropy, which is estimated from the Lyapunov exponents using linearized model equations. We then investigate the dependence of the (ɛ, τ) entropy on the optical injection strength of the laser system. In addition, we evaluate the (ɛ, τ) entropy from the experimentally obtained chaotic temporal waveforms in an optically injected semiconductor laser. Random bits with an entropy close to one bit per sampling point are extracted to satisfy the conditions of physical random number generation. We find that the extraction of the third-most significant bit from eight-bit experimental chaotic data results in an entropy of one bit per sample for certified physical random number generation.

Entropy optimization in convective viscous fluids flow due to a rotating cone is explored. Heat expression with heat source/sink and dissipation is considered. Irreversibility with binary chemical reaction is also deliberated. Nonlinear system is reduced to ODEs by suitable variables. Newton built in shooting procedure is adopted for numerical solution. Salient features velocity filed, Bejan number, entropy rate, concentration and temperature are deliberated. Numerical outcomes for velocity gradient and mass and heat transfer rates are displayed through tables. Assessments between the current and previous published outcomes are in an excellent agreement. It is noted that velocity and temperature show contrasting behavior for larger variable viscosity parameter. Entropy rate and Bejan number have reverse effect against viscosity variable. For rising values of thermal conductivity variable both Bejan number and entropy optimization have similar effect.

The consensusability and global optimality problems are solved for the discrete-time linear multiagent system (MAS) with marginally stable and strictly unstable dynamics. A unified framework is proposed by capturing the maximal disc-guaranteed gain margin (GGM) of the discrete-time linear quadratic regulator (LQR). Sufficient and necessary conditions on consensusability are established. Two bounds of the consensus region are derived only in terms of the unstable eigenvalues of the agent' dynamics. For the single-input MAS, by proving that the radius of the consensus region exactly equals the reciprocal of the Mahler measure of the agent' dynamics, we incidentally reveal the relation between the maximal GGM and the intrinsic entropy rate of the system dynamics for single-input discrete-time linear systems. By employing the inverse optimal control approach, it is proved that the globally optimal consensus is achieved, if and only if the associated Laplacian matrix is a simple matrix and all its nonzero eigenvalues can be radially projected into a specific subset of the consensus region. Moreover, the limitation on the eigenvalues vanishes for the marginally stable MAS. As an application of the global optimality, the minimum-energy-distributed consensus control problem is solved for the marginally stable MAS. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.

We introduce an index based on information theory to quantify the stationarity of a stochastic process.The index compares on the one hand the information contained in the increment at the time scale $\tau$ of the process at time $t$ with, on the other hand, the extra information in the variable at time $t$ that is not present at time $t-\tau$. By varying the scale $\tau$, the index can explore a full range of scales. We thus obtain a multi-scale quantity that is not restricted to the first two moments of the density distribution, nor to the covariance, but that probes the complete dependences in the process. This index indeed provides a measure of the regularity of the process at a given scale.Not only is this index able to indicate whether a realization of the process is stationary, but its evolution across scales also indicates how rough and non-stationary it is.We show how the index behaves for various synthetic processes proposed to model fluid turbulence, as well as on experimental fluid turbulence measurements., Comment: Entropy, MDPI, 2021

Choosing the stochastic process with the entropy is attributable to increase the least amount of information to the problem under consideration. As a result, the entropy rate for stochastic processes must be determined. In the current correspondence, we will describe the rate of ‘useful’ conditional Renyi information measure, as well as, show that rate for ‘useful’ Renyi Information, the chain rule holds. Therefore, for the rate of ‘useful’ conditional Renyi information and also for an ergodic Markov chain, we will present a relationship and use it to extract the rate of ‘useful’ Renyi information. At last, we will show that Shannon's information rate is the rate for ‘useful’ Renyi information.

Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider here this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the applicability of the large matrix limit: as the dynamic range increases, a phase transition occurs whereby the random matrix theory result is avoided, and long relaxation times ensue, in the entire `ordered' phase. We furthermore show that this phase transition is accompanied by a drop in the entropy rate and a peak in complexity, as measured by predictive information (Bialek, Nemenman, Tishby Neural Computation 13(21) 2001). Extending the Markov model to a Hidden Markov model (HMM), we show that observable sequences inherit properties of the hidden sequences, allowing HMMs to be understood in terms of more accessible Markov models. We then apply our findings to fMRI data from 820 human subjects scanned at wakeful rest. We show that the data can be quantitatively understood in terms of the random model, and that brain activity lies close to the phase transition when engaged in unconstrained, task-free cognition -- supporting the brain criticality hypothesis in this context., Comment: 12 pages; v2: expanded introduction and added references, addition of eigenvalue scatter plots

We present a generically applicable four-step process for quantifying information movement in complex networks. (1) Construction of local entropy rate and specific entropy rate. Local entropy rate is a continuous, time-dependent measure that quantifies the information gained at time t on observing x(t) given the recent past. There is a statistically responsible procedure for specifying “recent”. Specific entropy rate is a related time-dependent locally determined measure that gives an estimate of uncertainty at time t. (2) Construct specific transfer entropy (i.e., a time-dependent generalization of epoch-determined transfer entropy) that gives a state- and time-resolved quantification of the predictive input of a candidate input system on a candidate output system. (3) Construct a time-dependent network adjacency matrix. Specific transfer entropy can be used to populate the adjacency matrix characterizing a network. In the case of multichannel EEG/MEG recordings, the nodes are electrodes, and specific transfer entropy quantifies information movement between electrodes. In this analysis, the adjacency matrix is real, time-dependent and asymmetric. Any of a large number of measures commonly used to characterize an adjacency matrix can be used. The result \(\Lambda (t)\) is a scalar function of time. (4) Identify hierarchical transition chronometries in \(\Lambda (t)\). The simple directive “find transitions in \(\Lambda (t)\)” is unacceptably naive. Dynamically meaningful transitions are timescale-dependent. In this analysis, \(\Lambda (t)\) is embedded and the structure of this embedded object is examined by quadrant scans of the corresponding recurrence diagram. A hierarchy of transitions can be identified by manipulating the embedding dimension. We note that \(\Lambda (t)\) can serve as the order parameter in phase transition experiments in which time is the tuning parameter.

Human induced pluripotent stem cell-derived cardiac spheroids (iPSC-CSs) in 3D possess tremendous potential for treating heart diseases and screening drugs for their cardiac effect. The beating pattern (including beating frequency and amplitude) of iPSC-CSs is a direct indicator of their health and function. However, detecting the beating pattern of 3D cardiac spheroid is not well studied and the probes commonly used for labeling cardiomyocytes for their beating pattern detection is toxic during long-term culture. Here, we reveal that the beating pattern of 3D iPSC-CSs can be conveniently detected/quantified by calculating the relative change of entropy in all the frames/images of non-fluorescent optical signal without labeling any cells. The entropy rate superpixel segmentation method is used for image segmentation in frames containing multiple or aggregated iPSC-CSs to identify individual iPSC-CSs, enabling rapid detection/quantification of the beating pattern of each iPSC-CS. Moreover, the responses of iPSC-CSs to both anticancer and cardiac drugs can be reliably detected with the image entropy-based label-free method in terms of their beating patterns. This novel label-free approach may be valuable for convenient and efficient functional evaluation of 3D and 2D cardiac constructs, which is important not only for drug screening but also the advancement of manufacturing functional cardiac constructs to treat heart diseases.

Here we analyze the Marangoni convective magnetohydrodynamic flow of second grade liquid. Heat transportation is discussed through Joule heating and viscous dissipation. Characteristics of thermo-diffusion and diffusion-thermo are also considered. Gibbs–Marangoni effect is the solutal transfer along the boundary between liquids as a result of gradient of surface tension. Furthermore Benard–Marangoni convection is the temperature dependence phenomenon. Irreversibility communication is developed through thermodynamic second law. Characteristics of entropy optimization with chemical reaction are discussed. Nonlinear system is converted to ordinary system. Optimal Homotopy analysis method (OHAM) is employed to get convergent solutions. Variation of different sundry variables on velocity field, concentration, entropy rate, Bejan number, and temperature are scrutinized. Larger Marangoni ratio variable boosted the velocity field. Velocity field is reduced for higher magnetic variable. An augmentation occurs in temperature versus Dufour number. Temperature distribution boosted against magnetic and fluid variables. Concentration gets reduced versus larger Soret number. Higher Marangoni ratio variable decays the concentration. Larger approximation of magnetic variable enhances entropy rate. Bejan number and entropy rate have opposite trend for fluid variable. Entropy rate boosted via higher Brinkman number.

Szilard's now-famous single-molecule engine was only the first of three constructions he introduced in 1929 to resolve several challenges arising from Maxwell's demon paradox. Given that it has been thoroughly analyzed, we analyze Szilard's remaining two demon models. We show that the second one, though a markedly different implementation employing a population of distinct molecular species and semipermeable membranes, is informationally and thermodynamically equivalent to an ideal gas of the single-molecule engines. One concludes that (i) it reduces to a chaotic dynamical system-called the Szilard Map, a composite of three piecewise linear maps and associated thermodynamic transformations that implement measurement, control, and erasure; (ii) its transitory functioning as an engine that converts disorganized heat energy to work is governed by the Kolmogorov-Sinai entropy rate; (iii) the demon's minimum necessary "intelligence" for optimal functioning is given by the engine's statistical complexity; and (iv) its functioning saturates thermodynamic bounds and so it is a minimal, optimal implementation. We show that Szilard's third construction is rather different and addresses the fundamental issue raised by the first two: the link between entropy production and the measurement task required to implement either of his engines. The analysis gives insight into designing and implementing novel nanoscale information engines by investigating the relationships between the demon's memory, the nature of the "working fluid," and the thermodynamic costs of erasure and measurement.