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All sourcesNASA NTRSDOE OSTI
NASA NTRS2019

Approximate Entropy and Sample Entropy: A Comprehensive Tutorial

Approximate Entropy and Sample Entropy are two algorithms for determining the regularity of series of data based on the existence of patterns. Despite their similarities, the theoretical ideas behind those techniques are different but usually ignored. This paper aims to be a complete guideline of the theory and application of the algorithms, intended to explain their characteristics in detail to researchers from different fields. While initially developed for physiological applications, both algorithms have been used in other fields such as medicine, telecommunications, economics or Earth sciences. In this paper, we explain the theoretical aspects involving Information Theory and Chaos Theory, provide simple source codes for their computation, and illustrate the techniques with a step by step example of how to use the algorithms properly. This paper is not intended to be an exhaustive review of all previous applications of the algorithms but rather a comprehensive tutorial where no previous knowledge isrequired to understand the methodology.

Delgado-Bonal, Alfonso↗
DOE OSTI2014

Amputation effects on the underlying complexity within transtibial amputee ankle motion

The presence of chaos in walking is considered to provide a stable, yet adaptable means for locomotion. This study examined whether lower limb amputation and subsequent prosthetic rehabilitation resulted in a loss of complexity in amputee gait. Twenty-eight individuals with transtibial amputation participated in a 6 week, randomized cross-over design study in which they underwent a 3 week adaptation period to two separate prostheses. One prosthesis was deemed “more appropriate” and the other “less appropriate” based on matching/mismatching activity levels of the person and the prosthesis. Subjects performed a treadmill walking trial at self-selected walking speed at multiple points of the adaptation period, while kinematics of the ankle were recorded. Bilateral sagittal plane ankle motion was analyzed for underlying complexity through the pseudoperiodic surrogation analysis technique. Results revealed the presence of underlying deterministic structure in both prostheses and both the prosthetic and sound leg ankle (discriminant measure largest Lyapunov exponent). Results also revealed that the prosthetic ankle may be more likely to suffer loss of complexity than the sound ankle, and a “more appropriate” prosthesis may be better suited to help restore a healthy complexity of movement within the prosthetic ankle motion compared to a “less appropriate” prosthesis (discriminant measure sample entropy). Results from sample entropy results are less likely to be affected by the intracycle periodic dynamics as compared to the largest Lyapunov exponent. Adaptation does not seem to influence complexity in the system for experienced prosthesis users.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2020

Analyzing Changes in the Complexity of Climate in the Last Four Decades Using MERRA-2 Radiation Data

The energy balance of the Earth is controlled by the shortwave and longwave radiation emitted to space. Changes in the thermodynamic state of the system over time afect climate and are noticeable when viewing the system as a whole. In this paper, we study the changes in the complexity of climate in the last four decades using data from the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2). First, we study the complexity of the shortwave and longwave radiation felds independently using Approximate Entropy and Sample Entropy, observing that the rate of complexity change is faster for shortwave radiation. Then, we study the causality of those changes using Transfer Entropy to capture the non-linear dynamics of climate, showing that the changes are mainly driven by the variations in shortwave radiation. The observed behavior of climatic complexity could be explained by the changes in cloud amount, and we research that possibility by investigating its evolution from a complexity perspective using data from the International Satellite Cloud Climatology Project (ISCCP).

Delgado-Bonal, Alfonso↗
DOE OSTI2018

Towards a greater understanding of serrated flows in an Al-containing high-entropy-based alloy

A serrated flow, which occurs in a material undergoing mechanical deformation, is a complex process of great engineering significance. Here statistical, dynamical, and multifractal modeling and analyses were performed on the stress-time series to characterize and model the stress-drop behavior of an Al 0.5 CoCrCuFeNi high-entropy alloy (HEA). Results indicate that the spatiotemporal dynamics of the serrated flow is affected by changes in the strain rate and test temperature. The sample entropy, in general, was found to be the highest in the samples tested at 500°C. The higher complexity in the serrated flow at this temperature appeared to be associated with the stress-drop behavior that had intermediate values in terms of the maximum stress drop, the multifractality of the data set, and the histogram distributions. Moreover, the sample entropy was the lowest for the samples tested at 600 °C. The lower complexity values were associated with a wider multifractal spectrum and a less uniform and sparser distribution of the stress-drop magnitudes. In terms of the serration types, Type-C serrations were related to the lowest complexity values, widest multifractal spectra, and higher probability of exhibiting larger stress drops. Conversely, Type-A and B serrations were associated with the higher complexity, narrower spectra, and lower probability of higher stress drops. In conclusion, the body-centered-cubic (BCC) structure and the fully-ordered L1 2 nano-particles were found to emerge in the samples at 600°C and are thought to be linked to the decreased spatiotemporal correlations in the stressdrop behavior.

36 MATERIALS SCIENCE↗
DOE OSTI2013

Optimizing human activity patterns using global sensitivity analysis

Implementing realistic activity patterns for a population is crucial for modeling, for example, disease spread, supply and demand, and disaster response. Using the dynamic activity simulation engine, DASim, we generate schedules for a population that capture regular (e.g., working, eating, and sleeping) and irregular activities (e.g., shopping or going to the doctor). We use the sample entropy (SampEn) statistic to quantify a schedule’s regularity for a population. We show how to tune an activity’s regularity by adjusting SampEn, thereby making it possible to realistically design activities when creating a schedule. The tuning process sets up a computationally intractable high-dimensional optimization problem. To reduce the computational demand, we use Bayesian Gaussian process regression to compute global sensitivity indices and identify the parameters that have the greatest effect on the variance of SampEn. Here we use the harmony search (HS) global optimization algorithm to locate global optima. Our results show that HS combined with global sensitivity analysis can efficiently tune the SampEn statistic with few search iterations. We demonstrate how global sensitivity analysis can guide statistical emulation and global optimization algorithms to efficiently tune activities and generate realistic activity patterns. Finally, though our tuning methods are applied to dynamic activity schedule generation, they are general and represent a significant step in the direction of automated tuning and optimization of high-dimensional computer simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Spoofing Cross-Entropy Measure in Boson Sampling

Cross-entropy (XE) measure is a widely used benchmark to demonstrate quantum computational advantage from sampling problems, such as random circuit sampling using superconducting qubits and boson sampling (BS). We present a heuristic classical algorithm that attains a better XE than the current BS experiments in a verifiable regime and is likely to attain a better XE score than the near-future BS experiments in a reasonable running time. The key idea behind the algorithm is that there exist distributions that correlate with the ideal BS probability distribution and that can be efficiently computed. The correlation and the computability of the distribution enable us to postselect heavy outcomes of the ideal probability distribution without computing the ideal probability, which essentially leads to a large XE. Our method scores a better XE than the recent Gaussian BS experiments when implemented at intermediate, verifiable system sizes. Much like current state-of-the-art experiments, we cannot verify that our spoofer works for quantum-advantage-size systems. However, we demonstrate that our approach works for much larger system sizes in fermion sampling, where we can efficiently compute output probabilities. Finally, we provide analytic evidence that the classical algorithm is likely to spoof noisy BS efficiently.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Bayesian optimized collection strategies for fatigue strength testing

Abstract A statistical framework is presented enabling optimal sampling and analysis of constant life fatigue data. Protocols using Bayesian maximum entropy sampling are built based on conventional staircase and stress step methods, reducing the requirement of prior knowledge for data collection. The Bayesian Staircase method shows improved parameter estimation efficiency, and the Bayesian Stress Step method shows equal accuracy to the standard method at larger step size allowing experimentalists to lessen concerns of loading history. Statistical methods for determining model suitability are shown, highlighting the influence of protocol. Experimental validation is performed, showing the applicability of the methods in laboratory testing.

36 MATERIALS SCIENCE↗
DOE OSTI2016

The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields

Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. There exist very few results for steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes due to the technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function. The proof is based on the Karhunen-Loève expansion of complex-valued Ornstein-Uhlenbeck process.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Relation Between the Defect Interactions and the Serration Dynamics in a Zr-Based Bulk Metallic Glass

For this study, the effects of thermal annealing and compressive strain rate on the complexity of the serration behavior in a Zr-based bulk metallic glass (BMG) was investigated. Here, as-cast and thermally-annealed (300 °C, 1 week) Zr 52.5 Cu 17.9 Ni 14.6 Al 10 Ti 5 BMG underwent room-temperature compression tests in the unconstrained condition at strain rates of 2 × 10 -5 s -1 and 2 × 10 -4 s -1 . The complexity of the serrated flow was determined, using the refined composite multiscale entropy technique. Nanoindentation testing and X-ray diffraction characterization were performed to assess the changes in the microstructure and mechanical properties of the BMG that occurred during annealing. The results indicated that the BMG did not crystallize during annealing in the prescribed heating condition. Nanoindentation tests revealed that annealing led to a significant increase in the depth-dependent nanoindentation hardness and Young’s modulus, which were attributed to the structural relaxation in the glass. Furthermore, both annealing and an increased strain rate resulted in a marked enhancement in the complexity of the serrated flow during compression. It was concluded that the increase in the sample entropy with increasing strain rate is related to an increase in the number of defect interactions during the serrated flow.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Characterizing the effect of incrementally increasing dry bulb temperature on linear and nonlinear measures of heart rate variability in nonpregnant, mid-gestation, and late-gestation sows

Abstract Characterizing the sow physiological response to an increased heat load is essential for effective heat stress mitigation. The study objective was to characterize the effects of a 400-min heating episode on sow heart rate variability (HRV) at different reproductive stages. HRV is a commonly used noninvasive proxy measure of autonomic function. Twenty-seven sows were enrolled in the study according to their gestation stage at time of selection: 1) nonpregnant (NP; n = 7), 2) mid-gestation (MID; 57.3 ± 11.8 d gestation; n = 11), and 3) late-gestation (LATE; 98.8 ± 4.9 d gestation; n = 8). The HRV data utilized in the study were collected from each pig as the dry bulb temperature in the room increased incrementally from 19.84 ± 2.15 °C to 35.54 ± 0.43 °C (range: 17.1–37.5 °C) over a 400-min period. After data collection, one 5-min set of continuous heart rate data were identified per pig for each of nine temperature intervals (19–20.99, 21–22.99, 23–24.99, 25–26.99, 27–28.99, 29–30.99, 31–32.99, 33–34.99, and 35–36.99 °C). Mean inter-beat interval length (RR), standard deviation of r-r intervals (SDNN), root mean square of successive differences (RMSSD), high frequency spectral power (HF), sample entropy (SampEn), short-term detrended fluctuation analysis (DFAα1), and three measures (%REC, DET, LMEAN) derived from recurrence quantification analysis were calculated for each data set. All data were analyzed using the PROC GLIMMIX procedure in SAS 9.4. Overall, LATE sows exhibited lower RR than NP sows (P < 0.01). The standard deviation of r–r intervals and RMSSD differed between each group (P < 0.01), with LATE sows exhibiting the lowest SDNN and RMSSD and NP sows exhibiting the greatest SDNN and RMSSD. Late-gestation sows exhibited lower HF than both MID and NP sows (P < 0.0001), greater DFA values than NP sows (P = 0.05), and greater DET compared to MID sows (P = 0.001). Late-gestation also sows exhibited greater %REC and LMEAN compared to MID (P < 0.01) and NP sows (all P < 0.01). In conclusion, LATE sows exhibited indicators of greater autonomic stress throughout the heating period compared to MID and NP sows. However, temperature by treatment interactions were not detected as dry bulb increased. Future studies are needed to fully elucidate the effect of gestational stage and increasing dry bulb temperature on sow HRV.

Agriculture↗
DOE OSTI2014

SU-E-J-261: Statistical Analysis and Chaotic Dynamics of Respiratory Signal of Patients in BodyFix

Purpose: To quantify respiratory signal of patients in BodyFix undergoing 4DCT scan with and without immobilization cover. Methods: 20 pairs of respiratory tracks recorded with RPM system during 4DCT scan were analyzed. Descriptive statistic was applied to selected parameters of exhale-inhale decomposition. Standardized signals were used with the delay method to build orbits in embedded space. Nonlinear behavior was tested with surrogate data. Sample entropy SE, Lempel-Ziv complexity LZC and the largest Lyapunov exponents LLE were compared. Results: Statistical tests show difference between scans for inspiration time and its variability, which is bigger for scans without cover. The same is for variability of the end of exhalation and inhalation. Other parameters fail to show the difference. For both scans respiratory signals show determinism and nonlinear stationarity. Statistical test on surrogate data reveals their nonlinearity. LLEs show signals chaotic nature and its correlation with breathing period and its embedding delay time. SE, LZC and LLE measure respiratory signal complexity. Nonlinear characteristics do not differ between scans. Conclusion: Contrary to expectation cover applied to patients in BodyFix appears to have limited effect on signal parameters. Analysis based on trajectories of delay vectors shows respiratory system nonlinear character and its sensitive dependence on initial conditions. Reproducibility of respiratory signal can be evaluated with measures of signal complexity and its predictability window. Longer respiratory period is conducive for signal reproducibility as shown by these gauges. Statistical independence of the exhale and inhale times is also supported by the magnitude of LLE. The nonlinear parameters seem more appropriate to gauge respiratory signal complexity since its deterministic chaotic nature. It contrasts with measures based on harmonic analysis that are blind for nonlinear features. Dynamics of breathing, so crucial for 4D-based clinical technologies, can be better controlled if nonlinear-based methodology, which reflects respiration characteristic, is applied. Funding provided by Varian Medical Systems via Investigator Initiated Research Project.

60 APPLIED LIFE SCIENCES↗
NASA NTRS2020

On the Use of Complexity Algorithms: a Cautionary Lesson from Climate Research

Complexity algorithms provide information about datasets which is radically different from classical moment statistics. Instead of focusing on the divergences from central values, they quantify other characteristics such as order, pattern repetitions, or the existence of attractors. However, those analyses must be done with the proper statistical treatment, which is, unfortunately, not always the case. In this contribution, I provide an example of the hazards of applying complexity measures without sufficient care by correcting a previously published analysis that aimed to quantify the complexity of climate. I clarify some misconceptions about the use of Sample Entropy and revise the incorrect assessments and conclusions drawn from the previous misapplication of the methods.

Delgado-Bonal, Alfonso↗
DOE OSTI2025

𝑁-dimensional maximum-entropy tomography via particle sampling

We propose a modified maximum-entropy (MENT) algorithm for six-dimensional phase space tomography. The algorithm uses particle sampling and low-dimensional density estimation to approximate large sets of high-dimensional integrals in the original MENT formulation. We implement this approach using Markov Chain Monte Carlo (MCMC) sampling techniques and demonstrate convergence of six-dimensional MENT on both synthetic and measured data.

Hoover, Austin [Oak Ridge National Laboratory (ORN↗
DOE OSTI2020

Thermodynamic stability and contributions to the Gibbs free energy of nanocrystalline Ni 3 Fe

In this work, the heat capacities of nanocrystalline Ni 3 Fe and control materials with larger crystallites were measured from 0.4–300 K. The heat capacities were integrated to obtain the enthalpy, entropy, and Gibbs free energy and to quantify how these thermodynamic functions are altered by nanocrystallinity. From the phonon density of states (DOS) measured by inelastic neutron scattering, we find that the Gibbs free energy is dominated by phonons and that the larger heat capacity of the nanomaterial below 100 K is attributable to its enhanced phonon DOS at low energies. Besides electronic and magnetic contributions, the nanocrystalline material has an additional contribution at higher temperatures, consistent with phonon anharmonicity. The nanocrystalline material shows a stronger increase with temperature of both the enthalpy and entropy compared to the bulk sample. Its entropy exceeds that of the bulk material by 0.4 $k_B$/atom at 300 K. This is insufficient to overcome the enthalpy of grain boundaries and defects in the nanocrystalline material, making it thermodynamically unstable with respect to the bulk control material.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Entropy-driven Optimal Sub-sampling of Fluid Dynamics for Developing Machine-learned Surrogates

Optimal sub-sampling of large datasets from fluid dynamics simulations is essential for training reduced-order machine learned models. A method using Shannon entropy was developed to weight flow features according to their level of information content, such that the most informative features can be extracted and used for training a surrogate model. The method is demonstrated in the canonical flow over a cylinder problem simulated with OpenFOAM. Both time-independent predictions and temporal forecasting were investigated as well as two types of prediction targets: local per-grid-point predictions and global per-time-step predictions. When tested on training a surrogate model, results indicate that our entropy-based sampling method typically outperforms random sampling and yields more reproducible results in less iterations. Finally, the method was used to train a surrogate model for modeling turbulence in magnetohydrodynamic flows, which revealed various challenges and opportunities for future research.

Brewer, Wes↗
DOE OSTI2025

Average Rényi entanglement entropy in Gaussian boson sampling

Recently, many experiments have been conducted with the goal of demonstrating a quantum advantage over classical computation. One popular framework for these experiments is Gaussian boson sampling, where quadratic photonic input states are interfered via a linear optical unitary and subsequently measured in the Fock basis. In this paper, we study the modal entanglement of the output states in this framework just before the measurement stage. Specifically, we compute Page curves as measured by various Rényi- α entropies, where the Page curve describes the entanglement between two partitioned groups of output modes averaged over all linear optical unitaries. We derive these formulas for α = 1 (i.e., the von Neumann entropy) and, more generally, for all positive integer α , in the asymptotic limit of infinite number of modes and for input states that are composed of single-mode-squeezed-vacuum state with equal squeezing strength. We then analyze the limiting behaviors when the squeezing is small and large. Having determined the averages, we then explicitly calculate the Rényi- α variance for integers α > 1 and are able to show that these entropies are weakly typical. Published by the American Physical Society 2025

Youm, Jason (ORCID:0009000057597782)↗