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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↗
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↗
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↗
NASA NTRS1987

Efficiency of a model human image code

Hypothetical schemes for neural representation of visual information can be expressed as explicit image codes. Here, a code modeled on the simple cells of the primate striate cortex is explored. The Cortex transform maps a digital image into a set of subimages (layers) that are bandpass in spatial frequency and orientation. The layers are sampled so as to minimize the number of samples and still avoid aliasing. Samples are quantized in a manner that exploits the bandpass contrast-masking properties of human vision. The entropy of the samples is computed to provide a lower bound on the code size. Finally, the image is reconstructed from the code. Psychophysical methods are derived for comparing the original and reconstructed images to evaluate the sufficiency of the code. When each resolution is coded at the threshold for detection artifacts, the image-code size is about 1 bit/pixel.

Watson, Andrew B.↗
NASA NTRS1973

Magnetic properties of Dy2Ti2O7

Measurements were made of the magnetization, differential magnetic susceptibility, and magnetic entropy of powered samples of Dy2Ti2O7. The saturation magnetic moment is 4.7 + or - 0.2 Bohr magnetons per Dy ion, instead of 10 as predicted by Hund's rules. A temperature-independent magnetization is observed in the saturation region. Absolute values of magnetic entropy have been obtained for temperatures from 1.25 to 20 K, in applied fields up to 10.4 tesla. The magnetic entropy approaches a maximum value consistent with a ground-state multiplicity of 2. Low field magnetization and differential susceptibility data show a transition to antiferromagnetism near 1.35 K. A construction of the magnetic specific heat from the zero field entropy shows an anomaly near the same temperature.

Flood, D. J.↗
NASA NTRS1980

A hill-sliding strategy for initialization of Gaussian clusters in the multidimensional space

A hill sliding technique was devised to extract Gaussian clusters from the multivariate probability density estimate of sample data for the first step of iterative unsupervised classification. Each cluster was assumed to posses a unimodal normal distribution. A clustering function proposed distinguished elements of a cluster under formation from the rest in the feature space. Initial clusters were extracted one by one according to the hill sliding tactics. A dimensionless cluster compactness parameter was proposed as a universal measure of cluster goodness and used satisfactorily in test runs with LANDSAT multispectral scanner data. The normalized divergence, defined by the cluster divergence divided by the entropy of the entire sample data, was utilized as a general separability measure between clusters. An overall clustering objective function was set forth in terms of cluster covariance matrices, from which the cluster compactness measure could be deduced. Minimal improvement of initial data partitioning was evaluated by this objective function in eliminating scattered sparse data points. The hill sliding clustering technique developed herein has the potential applicability to decomposition any multivariate mixture distribution into a number of unimodal distributions when an appropriate distribution function to the data set is employed.

Park, J. K.↗
NASA NTRS1981

Pulsations in total columnar electron content

Radio signals from the ATS 6 beacon received at Boulder reveal small-amplitude, quasi-sinusoidal fluctuations with periods in the range of 10 to 50 s. Visual comparisons of these data (116 events for October 1974 to April 1975) shows a good correspondence with simultaneous geomagnetic pulsations at Boulder in two thirds of the cases for which Boulder magnetograms were available, but they do not necessarily correspond with magnetic pulsations on ATS 6. Spectral analyses, by the method of maximum entropy, were made on sample records. The principal results are the following: (1) The occurrence of the pulsations is higher on magnetically disturbed days. (2) The maximum likelihood of occurrence is around 2100 UT (1400 LT). (3) The dominant spectrum peaks of the radio fluctuations and geomagnetic field on the ground generally coincide. Cases are found also in which temporal characteristics of the spectra are similar. These results indicate a close association of the radio fluctuations with the Pc 3-4 type pulsations of the geomagnetic field on the ground. It is suggested that the radio fluctuations originate mainly in the F region of the ionosphere, while some of them could be due to plasmapause effects.

Okuzawa, T.↗
NASA NTRS1973

Suitability of the rare-earth compounds Dy2Ti2O7 and Gd3Al5O12 for low temperature (4K-20K) magnetic refrigeration cycle

Measurements were made of the magnetic entropy and magnetization of powered samples of the compounds Dy2Ti2O7 and Gd3Al5O12. The magnetization was measured for temperatures at and below 4.2 K, in applied fields ranging to 7.0 tesla. Isothermal changes in magnetic entropy were measured for temperatures from 1.2 to 20 K, in applied fields up to 10 tesla. The results of the measurements are consistent with a doublet ground state for Dy2Ti2O7, and an eight-fold degenerate ground state for Gd3Al5O12. Absolute values of magnetic entropy have been obtained at the lower temperatures, permitting the isotherms to be properly located in the S-H plane with the use of adiabatic magnetization data. The iso-field lines in the S-T plane were determined. The results indicate that Dy2Ti2O7 can absorb a maximum of 71 + or - 4 joules/kg of heat at 4.2 K, while Gd3Al5O12 can absorb 233 + or - joules/kg at the same temperature. The large difference between the two is most likely a result of crystal field interactions in the dysoprosium compound. Both materials can be cycled adiabatically between 4.2 and 20 K.

Flood, D. J.↗
NASA NTRS1989

Unstable vortices in the near region of an internal flow cavity

This paper presents experimental data taken in the forward region of a separated internal free-shear layer produced in an internal cavity flowfield. It has been found that in the region very near the forward restrictor, experimental velocity profiles agree closely with the exact Stuart instability velocity profile with various values of a steepness parameter. Reynolds shear-stress profiles suggest the presence of counter-rotating longitudinal vortices. Spectral analysis by the maximum entropy method of the time samples within the vortices indicates subharmonic and harmonic components of the fundamental frequency, with a weak indication of the fundamental frequency itself.

Isaacson, L. K.↗
NASA NTRS1991

Some practical universal noiseless coding techniques, part 3, module PSl14,K+

The algorithmic definitions, performance characterizations, and application notes for a high-performance adaptive noiseless coding module are provided. Subsets of these algorithms are currently under development in custom very large scale integration (VLSI) at three NASA centers. The generality of coding algorithms recently reported is extended. The module incorporates a powerful adaptive noiseless coder for Standard Data Sources (i.e., sources whose symbols can be represented by uncorrelated non-negative integers, where smaller integers are more likely than the larger ones). Coders can be specified to provide performance close to the data entropy over any desired dynamic range (of entropy) above 0.75 bit/sample. This is accomplished by adaptively choosing the best of many efficient variable-length coding options to use on each short block of data (e.g., 16 samples) All code options used for entropies above 1.5 bits/sample are 'Huffman Equivalent', but they require no table lookups to implement. The coding can be performed directly on data that have been preprocessed to exhibit the characteristics of a standard source. Alternatively, a built-in predictive preprocessor can be used where applicable. This built-in preprocessor includes the familiar 1-D predictor followed by a function that maps the prediction error sequences into the desired standard form. Additionally, an external prediction can be substituted if desired. A broad range of issues dealing with the interface between the coding module and the data systems it might serve are further addressed. These issues include: multidimensional prediction, archival access, sensor noise, rate control, code rate improvements outside the module, and the optimality of certain internal code options.

Rice, Robert F.↗
NASA NTRS2018

Uncertainty in Heart Rate Complexity Metrics Caused by R-Peak Perturbations

Heart rate complexity (HRC) is a proven metric for gaining insight into human stress and physiological deterioration. To calculate HRC, the detection of the exact instance of when the heart beats, the R-peak, is necessary. Electrocardiogram (ECG) signals can often be corrupted by environmental noise (e.g., from electromagnetic interference, movement artifacts), which can potentially alter the HRC measurement, producing erroneous inputs which feed into decision support models. Current literature has only investigated how HRC is affected by noise when R-peak detection errors occur (false positives and false negatives). However, the numerical methods used to calculate HRC are also sensitive to the specific location of the fiducial point of the R-peak. This raises many questions regarding how this fiducial point is altered by noise, the resulting impact on the measured HRC, and how we can account for noisy HRC measures as inputs into our decision models. This work uses Monte Carlo simulations to systematically add white and pink noise at different permutations of signal-to-noise ratios (SNRs), time segments, sampling rates, and HRC measurements to characterize the influence of noise on the HRC measure by altering the fiducial point of the R-peak. Using the generated information from these simulations provides improved decision processes for system design which address key concerns such as permutation entropy being a more precise, reliable, less biased, and more sensitive measurement for HRC than sample and approximate entropy.

Napoli, Nicholas J.↗
NASA NTRSDate not supplied

Adaptive Stress Testing of Trajectory Predictions in Flight Management Systems

To find failure events and their likelihoods in flight-critical systems, we investigate the use of an advanced black-box stress testing approach called adaptive stress testing. We analyze a trajectory predictor from a developmental commercial flight management system which takes as input a collection of lateral waypoints and en-route environmental conditions. Our aim is to search for failure events relating to inconsistencies in the predicted lateral trajectories. The intention of this work is to find likely failures and report them back to the developers so they can address and potentially resolve shortcomings of the system before deployment. To improve search performance, this work extends the adaptive stress testing formulation to be applied more generally to sequential decision-making problems with episodic reward by collecting the state transitions during the search and evaluating at the end of the simulated rollout. We use a modified Monte Carlo tree search algorithm with progressive widening as our adversarial reinforcement learner. The performance is compared to direct Monte Carlo simulations and to the cross-entropy method as an alternative importance sampling baseline. The goal is to find potential problems otherwise not found by traditional requirements-based testing. Results indicate that our adaptive stress testing approach finds more failures and finds failures with higher likelihood relative to the baseline approaches.

adaptive stress testing↗
NASA NTRSDate not supplied

Does Collection Time Bias the Ecology of Cleanroom Air Samples?

Microbial monitoring of astromaterials collections has taken on increased importance with the return of biologically sensitive samples from the asteroids Ryugu and Bennu and the initiation of the Mars Sample Return Program. Terrestrial bacteria and fungi can alter the mineralogy and organic composition of our collections causing irreversible contamination of pristine samples and increasing the risk of false positives for life detection measurements. NASA has conducted routine microbial monitoring of its existing collections since 20181. Initial monitoring focused on surface samples collected with foam swabs. Although, airborne microbiology is often decoupled from surface microbiology in the built environment2 culture-based air sampling techniques like impactors were not compliant with existing contamination control requirements. Bringing organic rich media, gelatin or liquids into curation cleanrooms presents an unacceptable risk to pristine samples. In 2022 NASA purchased a materials complaint air sampler and began collecting air samples from the cleanrooms in addition to surface samples3. The new instrument uses an electret filter to collect samples that are suitable for cultivating organisms or for direct DNA sequencing. Preliminary DNA sequencing results appeared to indicate that longer sampling times biased the microbial community in favor of hearty, spore-forming bacteria3. We present the results of a study comparing overnight sampling (17 hours) to short (1 hour) sampling of unoccupied curation cleanrooms. The results will help us optimize our monitoring protocols and develop a more detailed inventory of the ecology of astromaterials curation cleanrooms. Methods: We analyzed 72 paired air samples from six different cleanrooms including the meteorite processing lab (ISO 7 equivalent, 16 samples), the lunar lab (ISO 6 equivalent, 10 samples), the stardust lab (ISO 5 equivalent 14 samples), the OSIRIS-REx lab (ISO 5 equivalent, 12 samples), the Hayabusa2 lab (ISO 5 equivalent, 14 samples), and the Genesis lab (ISO 4 equivalent, 6 samples). All the samples were collected with an InnovaPrep Bobcat air sampler operating at a sampling rate of 200 L/min. The sampler operates for 5 minutes out of every 20 minute period. Half of the samples were collected by filtering 3,000L (15 min. of active sampling) of air across an electret filter for one hour. The rest of the samples were collected by filtering approximately 51,000 L air across the filter overnight (~17 hours, 255 min. of active sampling). Cells were eluted from the filter using 6-7 ml of pressurized 0.15% tween 20 in PBS (phosphate buffered saline). This liquid was used to cultivate bacteria according to previously published methods1,4,5 and for DNA extraction and next generation sequencing. DNA was extracted with a Qiagen MagAttract PowerMicrobiome kit6. To identify bacteria and archaea, the 16S rRNA gene was amplified using Earth Microbiome primers for the V4 region 7. The amplified DNA was sequenced on an Illumina MiSeq using a V3 reagent kit. The resulting sequences were processed using DADA2 and QIIME2 as implemented on the EDGE bioinformatics platform8–10. Results: Only two of the 72 samples had no amplifiable DNA. Amplified DNA concentrations ranged from 2.67 – 0.272 ng/µl. The median concentration of amplified DNA for the 1 hour samples was 0.770 ± 0.368 ng/µl. The median concentration of amplified DNA for the overnight samples was 0.877 ± 0.434 ng/µl. On average the overnight samples had slightly more sequences (58,960 vs. 59,456) and ASV’s (amplicon sequence variants) (60 vs 64.5) than the one hour samples, but these differences are not statistically significant. The most abundant ASV in every sample mapped to the genus Cupravidus. ASV’s mapping to the genuses Bacillus, Schlegelella, Thermus, and Staphylococcus were also common. Discussion and Future Work: Alpha diversity statistics like Shannon Entropy and Faith Phylogenetic Diversity are used to describe the diversity of organisms in a single sample. If a longer sampling time was biasing the data, we would expect to see a change in these diversity statistics vs. sample time. However, we did not observe this in our data. The median Shannon entropy was slightly higher for the overnight samples (3.773 vs 3.611) as was the Faith Phylogenetic Diversity (4.042 vs 3.596), but both values were within a standard deviation of each other for the two sampling times (Fig. 1). It is unlikely, that the longer sampling time is introducing bias into our data. We do observe a significant decrease in diversity when comparing the air samples by lab. The Genesis lab (ISO 4 equivalent) has a lower median number of ASV’s (45.5) than the other labs (62). Median values for Shannon Entropy (3.717 vs. 3.430) and Faith Phylogenetic Diversity (3.796 vs. 3.548) are also lower for Genesis, but those values are with one standard deviation of each other for the different sampling times. This is consistent with previous culture-based results suggesting that the environment in cleanrooms tends to select for a core group of organisms capable of surviving under dry, low nutrient, conditions. The presence of the ASV’s mapping to Cupravidus and Thermus in our sequencing blanks and controls suggests that several of the most common organisms in our samples represent contaminants from the reagents used to perform the DNA extractions and sequencing. Further work is needed to identify these contaminants, remove them from our data and recalculate the diversity statistics. This is a systematic error. Therefore, we do not expect removing the sequencing contaminants to change our conclusions. Longer air sample collection times appear to result in slightly higher diversity and do not bias the results towards “hardy” bacteria like spore-formers. Based on these preliminary results we conclude that sampling at least 3,000 liters of air is sufficient to capture the microbial diversity of cleanrooms, and that air samples can also be collected overnight without negatively impacting diversity. These results allow us to be flexible when designing microbial monitoring plans so that they do not interfere with routine lab activity. References: 1. Regberg, A. B. et al. 49th Lunar and Planetary Science Conference (2018). 2. The United States Pharmacopeial Convention. USP General Chapter <1116> (2013). 3. Regberg, A. B., et al. 54th Lunar and Planetary Science Conference (2023). 4. Regberg, A. B. et al. 53rd Lunar and Planetary Science Conference ( 2022). 5. Davis, R. E.,et al. 50th Lunar and Planetary Science Conference (2019). 6. Qiagen. MagAttract® PowerMicrobiome® DNA/RNA EP Kit Handbook. (2018). 7. Walters, W. et al. mSystems 1, (2015). 8. Callahan, B. J. et al. Nat. Methods 13, 581–583 (2016). 9. Hall, M. & Beiko, R. G. Microbiome Analysis: Methods and Protocols113–129 (Springer, 2018). 10. Philipson, C. et al. Bio-Protoc. 7, e2622 (2017).

A. B. Regberg↗
NASA NTRS1976

Coding isotropic images

Rate distortion functions for two-dimensional homogeneous isotropic images are compared with the performance of 5 source encoders designed for such images. Both unweighted and frequency weighted mean square error distortion measures are considered. The coders considered are differential PCM (DPCM) using six previous samples in the prediction, herein called 6 pel (picutre element) DPCM; simple DPCM using single sample prediction; 6 pel DPCM followed by entropy coding; 8 x 8 discrete cosine transform coder, and 4 x 4 Hadamard transform coder. Other transform coders were studied and found to have about the same performance as the two transform coders above. With the mean square error distortion measure DPCM with entropy coding performed best. The relative performance of the coders changes slightly when the distortion measure is frequency weighted mean square error. The performance of all the coders was separated by only about 4 dB.

Oneal, J. B., Jr.↗
NASA NTRS1981

A modified adaptive delta modulator

This paper presents the results of a study in data compression of adaptive delta modulated video signals. The Song mode ADM is first investigated at a sampling rate of 16 Mbits/s and shown not to produce enough redundancy to warrant entropy encoding. A modified adaptive delta modulator algorithm is presented that operates at a sampling rate of 16 Mbits/s and does produce sufficient redundancy to yield a 40-50 percent data compression by using a simple code on 4 bit data blocks. Other techniques such as field interpolation and direct substitution are shown to increase the possible data compression further without noticeable degradation in the two input images used in this investigation. The effects of channel errors in the transmission of packet video over a computer network are considered. A leaky integrator is used to reduce the effects of channel errors in the data bits. It is shown that the effects of channel errors can be reduced by field interpolating those packets that can be shown to contain errors.

Barba, J.↗
NASA NTRS1996

Monitoring of Time-Dependent System Profiles by Multiplex Gas Chromatography with Maximum Entropy Demodulation

The maximum entropy technique was successfully applied to the deconvolution of overlapped chromatographic peaks. An algorithm was written in which the chromatogram was represented as a vector of sample concentrations multiplied by a peak shape matrix. Simulation results demonstrated that there is a trade off between the detector noise and peak resolution in the sense that an increase of the noise level reduced the peak separation that could be recovered by the maximum entropy method. Real data originated from a sample storage column was also deconvoluted using maximum entropy. Deconvolution is useful in this type of system because the conservation of time dependent profiles depends on the band spreading processes in the chromatographic column, which might smooth out the finer details in the concentration profile. The method was also applied to the deconvolution of previously interpretted Pioneer Venus chromatograms. It was found in this case that the correct choice of peak shape function was critical to the sensitivity of maximum entropy in the reconstruction of these chromatograms.

Becker, Joseph F.↗
NASA NTRS1991

An information theory approach for evaluating earth radiation budget (ERB) measurements - Nonuniform sampling of diurnal longwave flux variations

Satellite measurements are subject to a wide range of uncertainties due to their temporal, spatial, and directional sampling characteristics. An information-theory approach is suggested to examine the nonuniform temporal sampling of ERB measurements. The information (i.e., its entropy or uncertainty) before and after the measurements is determined, and information gain (IG) is defined as a reduction in the uncertainties involved. A stochastic model for the diurnal outgoing flux variations that affect the ERB is developed. Using Gaussian distributions for the a priori and measured radiant exitance fields, the IG is obtained by computing the a posteriori covariance. The IG for the monthly outgoing flux measurements is examined for different orbital parameters and orbital tracks, using the Earth Observing System orbital parameters as specific examples. Variations in IG due to changes in the orbit's inclination angle and the initial ascending node local time are investigated.

Halyo, Nesim↗
NASA NTRS2007

Space Invariant Independent Component Analysis and ENose for Detection of Selective Chemicals in an Unknown Environment

In this paper, we present a space invariant architecture to enable the Independent Component Analysis (ICA) to solve chemical detection from two unknown mixing chemical sources. The two sets of unknown paired mixture sources are collected via JPL 16-ENose sensor array in the unknown environment with, at most, 12 samples data collected. Our space invariant architecture along with the maximum entropy information technique by Bell and Sejnowski and natural gradient descent by Amari has demonstrated that it is effective to separate the two mixing unknown chemical sources with unknown mixing levels to the array of two original sources under insufficient sampled data. From separated sources, they can be identified by projecting them on the 11 known chemical sources to find the best match for detection. We also present the results of our simulations. These simulations have shown that 100% correct detection could be achieved under the two cases: a) under-completed case where the number of input (mixtures) is larger than number of original chemical sources; and b) regular case where the number of input is as the same as the number of sources while the time invariant architecture approach may face the obstacles: overcomplete case, insufficient data and cumbersome architecture.

ENose↗