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DOE OSTI2023

2D k -th nearest neighbour statistics: a highly informative probe of galaxy clustering

ABSTRACT Beyond standard summary statistics are necessary to summarize the rich information on non-linear scales in the era of precision galaxy clustering measurements. For the first time, we introduce the 2D k-th nearest neighbour (kNN) statistics as a summary statistic for discrete galaxy fields. This is a direct generalization of the standard 1D kNN by disentangling the projected galaxy distribution from the redshift-space distortion signature along the line-of-sight. We further introduce two different flavours of 2D kNNs that trace different aspects of the galaxy field: the standard flavour which tabulates the distances between galaxies and random query points, and a ‘DD’ flavour that tabulates the distances between galaxies and galaxies. We showcase the 2D kNNs’ strong constraining power both through theoretical arguments and by testing on realistic galaxy mocks. Theoretically, we show that 2D kNNs are computationally efficient and directly generate other statistics such as the popular two-point correlation function (2PCF), voids probability function, and counts-in-cell statistics. In a more practical test, we apply the 2D kNN statistics to simulated galaxy mocks that fold in a large range of observational realism and recover parameters of the underlying extended halo occupation distribution (HOD) model that includes velocity bias and galaxy assembly bias. We find unbiased and significantly tighter constraints on all aspects of the HOD model with the 2D kNNs, both compared to the standard 1D kNN, and the classical redshift-space 2PCF.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2014

Statistical Symbolic Execution with Informed Sampling

Symbolic execution techniques have been proposed recently for the probabilistic analysis of programs. These techniques seek to quantify the likelihood of reaching program events of interest, e.g., assert violations. They have many promising applications but have scalability issues due to high computational demand. To address this challenge, we propose a statistical symbolic execution technique that performs Monte Carlo sampling of the symbolic program paths and uses the obtained information for Bayesian estimation and hypothesis testing with respect to the probability of reaching the target events. To speed up the convergence of the statistical analysis, we propose Informed Sampling, an iterative symbolic execution that first explores the paths that have high statistical significance, prunes them from the state space and guides the execution towards less likely paths. The technique combines Bayesian estimation with a partial exact analysis for the pruned paths leading to provably improved convergence of the statistical analysis. We have implemented statistical symbolic execution with in- formed sampling in the Symbolic PathFinder tool. We show experimentally that the informed sampling obtains more precise results and converges faster than a purely statistical analysis and may also be more efficient than an exact symbolic analysis. When the latter does not terminate symbolic execution with informed sampling can give meaningful results under the same time and memory limits.

Reliability↗

DOE OSTI2022

Statistical Significance Testing for Mixed Priors: A Combined Bayesian and Frequentist Analysis

In many hypothesis testing applications, we have mixed priors, with well-motivated informative priors for some parameters but not for others. The Bayesian methodology uses the Bayes factor and is helpful for the informative priors, as it incorporates Occam’s razor via the multiplicity or trials factor in the look-elsewhere effect. However, if the prior is not known completely, the frequentist hypothesis test via the false-positive rate is a better approach, as it is less sensitive to the prior choice. We argue that when only partial prior information is available, it is best to combine the two methodologies by using the Bayes factor as a test statistic in the frequentist analysis. We show that the standard frequentist maximum likelihood-ratio test statistic corresponds to the Bayes factor with a non-informative Jeffrey’s prior. We also show that mixed priors increase the statistical power in frequentist analyses over the maximum likelihood test statistic. We develop an analytic formalism that does not require expensive simulations and generalize Wilks’ theorem beyond its usual regime of validity. In specific limits, the formalism reproduces existing expressions, such as the p-value of linear models and periodograms. We apply the formalism to an example of exoplanet transits, where multiplicity can be more than 10 7 . We show that our analytic expressions reproduce the $p$-values derived from numerical simulations. We offer an interpretation of our formalism based on the statistical mechanics. We introduce the counting of states in a continuous parameter space using the uncertainty volume as the quantum of the state. We show that both the $p$-value and Bayes factor can be expressed as an energy versus entropy competition.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Estimating Uncertainty in Simulated ENSO Statistics

Abstract Large ensembles of model simulations are frequently used to reduce the impact of internal variability when evaluating climate models and assessing climate change induced trends. However, the optimal number of ensemble members required to distinguish model biases and climate change signals from internal variability varies across models and metrics. Here we analyze the mean, variance and skewness of precipitation and sea surface temperature in the eastern equatorial Pacific region often used to describe the El Niño–Southern Oscillation (ENSO), obtained from large ensembles of Coupled model intercomparison project phase 6 climate simulations. Leveraging established statistical theory, we develop and assess equations to estimate, a priori, the ensemble size or simulation length required to limit sampling‐based uncertainties in ENSO statistics to within a desired tolerance. Our results confirm that the uncertainty of these statistics decreases with the square root of the time series length and/or ensemble size. Moreover, we demonstrate that uncertainties of these statistics are generally comparable when computed using either pre‐industrial control or historical runs. This suggests that pre‐industrial runs can sometimes be used to estimate the expected uncertainty of statistics computed from an existing historical member or ensemble, and the number of simulation years (run duration and/or ensemble size) required to adequately characterize the statistic. This advance allows us to use existing simulations (e.g., control runs that are performed during model development) to design ensembles that can sufficiently limit diagnostic uncertainties arising from simulated internal variability. These results may well be applicable to variables and regions beyond ENSO.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Towards testing the theory of gravity with DESI: summary statistics, model predictions and future simulation requirements

Shortly after its discovery, General Relativity (GR) was applied to predict the behavior of our Universe on the largest scales, and later became the foundation of modern cosmology. Its validity has been verified on a range of scales and environments from the Solar system to merging black holes. However, experimental confirmations of GR on cosmological scales have so far lacked the accuracy one would hope for — its applications on those scales being largely based on extrapolation and its validity there sometimes questioned in the shadow of the discovery of the unexpected cosmic acceleration. Future astronomical instruments surveying the distribution and evolution of galaxies over substantial portions of the observable Universe, such as the Dark Energy Spectroscopic Instrument (DESI), will be able to measure the fingerprints of gravity and their statistical power will allow strong constraints on alternatives to GR. In this paper, based on a set of N-body simulations and mock galaxy catalogs, we study the predictions of a number of traditional and novel summary statistics beyond linear redshift distortions in two well-studied modified gravity models — chameleon f(R) gravity and a braneworld model — and the potential of testing these deviations from GR using DESI. These summary statistics employ a wide array of statistical properties of the galaxy and the underlying dark matter field, including two-point and higher-order statistics, environmental dependence, redshift space distortions and weak lensing. We find that they hold promising power for testing GR to unprecedented precision. The major future challenge is to make realistic, simulation-based mock galaxy catalogs for both GR and alternative models to fully exploit the statistic power of the DESI survey (by matching the volumes and galaxy number densities of the mocks to those in the real survey) and to better understand the impact of key systematic effects. Using these, we identify future simulation and analysis needs for gravity tests using DESI.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

A Parameter-masked Mock Data Challenge for Beyond-two-point Galaxy Clustering Statistics

The past few years have seen the emergence of a wide array of novel techniques for analyzing high-precision data from upcoming galaxy surveys, which aim to extend the statistical analysis of galaxy clustering data beyond the linear regime and the canonical two-point (2pt) statistics. We test and benchmark some of these new techniques in a community data challenge named “Beyond-2pt,” initiated during the Aspen 2022 Summer Program “Large-Scale Structure Cosmology beyond 2-Point Statistics,” whose first round of results we present here. The challenge data set consists of high-precision mock galaxy catalogs for clustering in real space, in redshift space, and on a light cone. Participants in the challenge have developed end-to-end pipelines to analyze mock catalogs and extract unknown (“masked”) cosmological parameters of the underlying ΛCDM models with their methods. The methods represented are density-split clustering, nearest neighbor statistics, BACCO power spectrum emulator, void statistics, LEFTfield field-level inference using effective field theory (EFT), and joint power spectrum and bispectrum analyses using both EFT and simulation-based inference. In this work, we review the results of the challenge, focusing on problems solved, lessons learned, and future research needed to perfect the emerging beyond-2pt approaches. The unbiased parameter recovery demonstrated in this challenge by multiple statistics and the associated modeling and inference frameworks supports the credibility of cosmology constraints from these methods. The challenge data set is publicly available, and we welcome future submissions from methods that are not yet represented.

Krause, Elisabeth [Univ. of Arizona, Tucson, AZ (U↗

NASA NTRS1976

Determination of statistics for any rotation of axes of a bivariate normal elliptical distribution

Transformation of statistics from a dimensional set to another dimensional set involves linear functions of the original set of statistics. Similarly, linear functions will transform statistics within a dimensional set such that the new statistics are relevant to a new set of coordinate axes. A restricted case of the latter is the rotation of axes in a coordinate system involving any two correlated random variables. A special case is the transformation for horizontal wind distributions. Wind statistics are usually provided in terms of wind speed and direction (measured clockwise from north) or in east-west and north-south components. A direct application of this technique allows the determination of appropriate wind statistics parallel and normal to any preselected flight path of a space vehicle. Among the constraints for launching space vehicles are critical values selected from the distribution of the expected winds parallel to and normal to the flight path. These procedures are applied to space vehicle launches at Cape Kennedy, Florida.

Falls, L. W.↗

NASA NTRS1986

Statistics of the geomagnetic secular variation for the past 5Ma

A new statistical model is proposed for the geomagnetic secular variation over the past 5Ma. Unlike previous models, the model makes use of statistical characteristics of the present day geomagnetic field. The spatial power spectrum of the non-dipole field is consistent with a white source near the core-mantle boundary with Gaussian distribution. After a suitable scaling, the spherical harmonic coefficients may be regarded as statistical samples from a single giant Gaussian process; this is the model of the non-dipole field. The model can be combined with an arbitrary statistical description of the dipole and probability density functions and cumulative distribution functions can be computed for declination and inclination that would be observed at any site on Earth's surface. Global paleomagnetic data spanning the past 5Ma are used to constrain the statistics of the dipole part of the field. A simple model is found to be consistent with the available data. An advantage of specifying the model in terms of the spherical harmonic coefficients is that it is a complete statistical description of the geomagnetic field, enabling us to test specific properties for a general description. Both intensity and directional data distributions may be tested to see if they satisfy the expected model distributions.

Constable, C. G.↗

NASA NTRS1988

Statistics of the geomagnetic secular variation for the past 5 m.y

Constable, C. G.↗

NASA NTRS1995

Modeling Ka-band low elevation angle propagation statistics

The statistical variability of the secondary atmospheric propagation effects on satellite communications cannot be ignored at frequencies of 20 GHz or higher, particularly if the propagation margin allocation is such that link availability falls below 99 percent. The secondary effects considered in this paper are gaseous absorption, cloud absorption, and tropospheric scintillation; rain attenuation is the primary effect. Techniques and example results are presented for estimation of the overall combined impact of the atmosphere on satellite communications reliability. Statistical methods are employed throughout and the most widely accepted models for the individual effects are used wherever possible. The degree of correlation between the effects is addressed and some bounds on the expected variability in the combined effects statistics are derived from the expected variability in correlation. Example estimates are presented of combined effects statistics in the Washington D.C. area of 20 GHz and 5 deg elevation angle. The statistics of water vapor are shown to be sufficient for estimation of the statistics of gaseous absorption at 20 GHz. A computer model based on monthly surface weather is described and tested. Significant improvement in prediction of absorption extremes is demonstrated with the use of path weather data instead of surface data.

Russell, Thomas A.↗

NASA NTRS1999

Parameter Estimation in Astronomy with Poisson-Distributed Data: The (CHI)2(gamma) Statistic - 1

Applying the standard weighted mean formula, [Sigma (sub i)n(sub i)ssigma(sub i, sup -2)], to determine the weighted mean of data, n(sub i), drawn from a Poisson distribution, will, on average, underestimate the true mean by approx. 1 for all true mean values larger than approx.3 when the common assumption is made that the error of the i th observation is sigma(sub i) = max square root of n(sub i), 1).This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman'chi(sub 2) statistic, chi(sup 2, sub N) equivalent Sigma(sub i)((n(sub i) - y(sub i)(exp 2)) / max(n(sub i), 1), to compare Poisson - distributed data with model values, y(sub i), will typically predict a total number of counts that underestimates the true total by about 1 count per bin. Based on my finding that weighted mean of data drawn from a Poisson distribution can be determined using the formula [Sigma(sub i)[n(sub i) + min(n(sub i), 1)](n(sub i) + 1)(exp -1)] / [Sigma(sub i)(n(sub i) + 1)(exp -1))], I propose that a new chi(sub 2) statistic, chi(sup 2, sub gamma) equivalent, should always be used to analyze Poisson- distributed data in preference to the modified Neyman's chi(exp 2) statistic. I demonstrated the power and usefulness of,chi(sub gamma, sup 2) minimization by using two statistical fitting techniques and five chi(exp 2) statistics to analyze simulated X-ray power - low 15 - channel spectra with large and small counts per bin. I show that chi(sub gamma, sup 2) minimization with the Levenberg - Marquardt or Powell's method can produce excellent results (mean slope errors approx. less than 3%) with spectra having as few as 25 total counts.

Mighell, Kenneth J.↗

NASA NTRS2002

Scale Dependence of Statistics of Spatially Averaged Rain Rate Seen in TOGA COARE Comparison with Predictions from a Stochastic Model

A characteristic feature of rainfall statistics is that they in general depend on the space and time scales over which rain data are averaged. As a part of an earlier effort to determine the sampling error of satellite rain averages, a space-time model of rainfall statistics was developed to describe the statistics of gridded rain observed in GATE. The model allows one to compute the second moment statistics of space- and time-averaged rain rate which can be fitted to satellite or rain gauge data to determine the four model parameters appearing in the precipitation spectrum - an overall strength parameter, a characteristic length separating the long and short wavelength regimes and a characteristic relaxation time for decay of the autocorrelation of the instantaneous local rain rate and a certain 'fractal' power law exponent. For area-averaged instantaneous rain rate, this exponent governs the power law dependence of these statistics on the averaging length scale $L$ predicted by the model in the limit of small $L$. In particular, the variance of rain rate averaged over an $L \times L$ area exhibits a power law singularity as $L \rightarrow 0$. In the present work the model is used to investigate how the statistics of area-averaged rain rate over the tropical Western Pacific measured with ship borne radar during TOGA COARE (Tropical Ocean Global Atmosphere Coupled Ocean Atmospheric Response Experiment) and gridded on a 2 km grid depends on the size of the spatial averaging scale. Good agreement is found between the data and predictions from the model over a wide range of averaging length scales.

Kundu, Prasun K.↗

NASA NTRS1932

Statistics of the Bureau Veritas

Statistics are indispensable factors for the amelioration of safety. Through the reconciliation of accidents which may appear isolated to interested parties, they permit tracking of typical causes of accidents; conversely, they can prevent, after a serious accident due to some fortuitous cause, the taking of incautious measures under the pressure of public opinion, which always inclines to gauge the gravity of the causes by that of the results. Lastly, they permit appraisal of the efficacy of rules in force. We should add that statistics provide an agency of prevention for future accidents. A careful inspection of all signs of malfunction of material quite often prevents the occurrence of an accident. In this respect, many pilot's report, perfectly normal in every way as far as operation is concerned, can reveal much more interesting technical data than an accident, although it does not diminish the importance of statistics. Therefore, from the inception of its aeronautical service, at the end of 1922, the Bureau Veritas has kept annual statistics of all accidents which occurred in French civil aviation. In order to correctly perform their proper function, the statistics must be exact and sufficiently explicit and complete. To be exact, they must bear on all pertinent events, and on these alone. It is a matter then, first of all, defining the accident in such a way that no sinister detail bearing on the definition may escape control. The consideration of accidents to personnel only has appeared too limited. One of the essential qualities of statistics is to permit the taking of averages and in consequence to apply them ot a sufficiently large number; such is happily not the case in accidents to personnel.

Volmerange, Andre↗

NASA NTRS2006

Jet Noise Diagnostics Supporting Statistical Noise Prediction Methods

The primary focus of my presentation is the development of the jet noise prediction code JeNo with most examples coming from the experimental work that drove the theoretical development and validation. JeNo is a statistical jet noise prediction code, based upon the Lilley acoustic analogy. Our approach uses time-average 2-D or 3-D mean and turbulent statistics of the flow as input. The output is source distributions and spectral directivity. NASA has been investing in development of statistical jet noise prediction tools because these seem to fit the middle ground that allows enough flexibility and fidelity for jet noise source diagnostics while having reasonable computational requirements. These tools rely on Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solutions as input for computing far-field spectral directivity using an acoustic analogy. There are many ways acoustic analogies can be created, each with a series of assumptions and models, many often taken unknowingly. And the resulting prediction can be easily reverse-engineered by altering the models contained within. However, only an approach which is mathematically sound, with assumptions validated and modeled quantities checked against direct measurement will give consistently correct answers. Many quantities are modeled in acoustic analogies precisely because they have been impossible to measure or calculate, making this requirement a difficult task. The NASA team has spent considerable effort identifying all the assumptions and models used to take the Navier-Stokes equations to the point of a statistical calculation via an acoustic analogy very similar to that proposed by Lilley. Assumptions have been identified and experiments have been developed to test these assumptions. In some cases this has resulted in assumptions being changed. Beginning with the CFD used as input to the acoustic analogy, models for turbulence closure used in RANS CFD codes have been explored and compared against measurements of mean and rms velocity statistics over a range of jet speeds and temperatures. Models for flow parameters used in the acoustic analogy, most notably the space-time correlations of velocity, have been compared against direct measurements, and modified to better fit the observed data. These measurements have been extremely challenging for hot, high speed jets, and represent a sizeable investment in instrumentation development. As an intermediate check that the analysis is predicting the physics intended, phased arrays have been employed to measure source distributions for a wide range of jet cases. And finally, careful far-field spectral directivity measurements have been taken for final validation of the prediction code. Examples of each of these experimental efforts will be presented. The main result of these efforts is a noise prediction code, named JeNo, which is in middevelopment. JeNo is able to consistently predict spectral directivity, including aft angle directivity, for subsonic cold jets of most geometries. Current development on JeNo is focused on extending its capability to hot jets, requiring inclusion of a previously neglected second source associated with thermal fluctuations. A secondary result of the intensive experimentation is the archiving of various flow statistics applicable to other acoustic analogies and to development of time-resolved prediction methods. These will be of lasting value as we look ahead at future challenges to the aeroacoustic experimentalist.

Bridges, James E.↗

NASA NTRS2010

Establishing Consensus Turbulence Statistics for Hot Subsonic Jets

Many tasks in fluids engineering require knowledge of the turbulence in jets. There is a strong, although fragmented, literature base for low order statistics, such as jet spread and other meanvelocity field characteristics. Some sources, particularly for low speed cold jets, also provide turbulence intensities that are required for validating Reynolds-averaged Navier-Stokes (RANS) Computational Fluid Dynamics (CFD) codes. There are far fewer sources for jet spectra and for space-time correlations of turbulent velocity required for aeroacoustics applications, although there have been many singular publications with various unique statistics, such as Proper Orthogonal Decomposition, designed to uncover an underlying low-order dynamical description of turbulent jet flow. As the complexity of the statistic increases, the number of flows for which the data has been categorized and assembled decreases, making it difficult to systematically validate prediction codes that require high-level statistics over a broad range of jet flow conditions. For several years, researchers at NASA have worked on developing and validating jet noise prediction codes. One such class of codes, loosely called CFD-based or statistical methods, uses RANS CFD to predict jet mean and turbulent intensities in velocity and temperature. These flow quantities serve as the input to the acoustic source models and flow-sound interaction calculations that yield predictions of far-field jet noise. To develop this capability, a catalog of turbulent jet flows has been created with statistics ranging from mean velocity to space-time correlations of Reynolds stresses. The present document aims to document this catalog and to assess the accuracies of the data, e.g. establish uncertainties for the data. This paper covers the following five tasks: Document acquisition and processing procedures used to create the particle image velocimetry (PIV) datasets. Compare PIV data with hotwire and laser Doppler velocimetry (LDV) data published in the open literature. Compare different datasets acquired at roughly the same flow conditions to establish uncertainties. Create a consensus dataset for a range of hot jet flows, including uncertainty bands. Analyze this consensus dataset for self-consistency and compare jet characteristics to those of the open literature. One final objective fulfilled by this work was the demonstration of a universal scaling for the jet flow fields, at least within the region of interest to aeroacoustics. The potential core length and the spread rate of the half-velocity radius were used to collapse of the mean and turbulent velocity fields over the first 20 jet diameters in a highly satisfying manner.

Bridges, James↗

NASA NTRS2015

Statistical Engineering in Air Traffic Management Research

NASA is working to develop an integrated set of advanced technologies to enable efficient arrival operations in high-density terminal airspace for the Next Generation Air Transportation System. This integrated arrival solution is being validated and verified in laboratories and transitioned to a field prototype for an operational demonstration at a major U.S. airport. Within NASA, this is a collaborative effort between Ames and Langley Research Centers involving a multi-year iterative experimentation process. Designing and analyzing a series of sequential batch computer simulations and human-in-the-loop experiments across multiple facilities and simulation environments involves a number of statistical challenges. Experiments conducted in separate laboratories typically have different limitations and constraints, and can take different approaches with respect to the fundamental principles of statistical design of experiments. This often makes it difficult to compare results from multiple experiments and incorporate findings into the next experiment in the series. A statistical engineering approach is being employed within this project to support risk-informed decision making and maximize the knowledge gained within the available resources. This presentation describes a statistical engineering case study from NASA, highlights statistical challenges, and discusses areas where existing statistical methodology is adapted and extended.

Wilson, Sara R.↗

NASA NTRS2018

A Statistical Study of Solar Particle Events in Flux and Dose

The high-energy protons from solar energetic particle (SEP) events present a hazard to space systems: damage to science instruments/electronics/materials or to astronauts. A reliable estimate of the high-energy proton environment is critical to assure mission success. Important characteristics of an SEP event are fluence, peak flux, energy spectrum, time to reach the peak flux, time to reach peak dose, and properties of the cumulative dose profile after an event starts. All of these characteristics are important to understand in order to design space missions properly for both robotic and human missions. Because of the unpredictable and sporadic nature of SEP events, statistical models are often used to represent the SEP parameters described above. In a study by Jun et al. (2007), the statistics of event fluences, durations, and time intervals between events were investigated using the then available historical SEP dataset obtained from the instruments onboard the IMP-8 spacecraft. Since then, a more comprehensive SEP dataset based off of IMP-8 and GOES called Reference Data Set Version 2.0 (RDSv2.0) has become available covering the SEP events up to Year 2015 under a framework of the European Space Agency's (ESA's) Solar Energetic Particle Environment Modelling (SEPEM) project (Jiggens et al., 2018). The main objectives of this statistical study of SEP events are two-fold: First, the statistics of peak fluxes, event fluences, durations, and time intervals will be re-visited by using RDSv2.0; Second, the statistical analyses of flux and dose timing will be performed using the same dataset RDSv2.0. The results of this study will address the statistical properties of all key parameters for designing a spacecraft or a human mission where the SEP environment is an important consideration.

Whitman, K.↗

NASA NTRS2019

Poisson vs. Gaussian statistics for sparse X-ray data: Application to the soft X-ray spectrometer

Reliable results when fitting X-ray data require proper consideration of the statistics involved. We probe the impact of Gaussian versus Poisson statistics at low count levels using both the standard χ^(2) method and maximum likelihood based on Poisson (C) statistics. The difference is studied and quantified through simulated spectra with known properties. We then test the results through analysis of Mn Kα calibration data taken with the flight spare microcalorimeter for the Hitomi soft X-ray spectrometer. Through comparison with simulations, our results show that the χ^(2) method tends to give overly optimistic estimates of the detector energy resolution, in particular when there are few counts. Given an energy resolution of ∼5 eV and a line with about 100 photons, the line width becomes ∼10% lower in the χ^(2) method than in Poisson statistics. This is a consequence of the uncertainties being dominated by counting statistics, and therefore highlights the need to choose the appropriate fit statistic.

Shinya Yamada↗