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All sources NASA NTRS DOE OSTI

DOE OSTI2021

Machine Learning-Based Classification of Lignocellulosic Biomass from Pyrolysis-Molecular Beam Mass Spectrometry Data

High-throughput analysis of biomass is necessary to ensure consistent and uniform feedstocks for agricultural and bioenergy applications and is needed to inform genomics and systems biology models. Pyrolysis followed by mass spectrometry such as molecular beam mass spectrometry (py-MBMS) analyses are becoming increasingly popular for the rapid analysis of biomass cell wall composition and typically require the use of different data analysis tools depending on the need and application. Here, the authors report the py-MBMS analysis of several types of lignocellulosic biomass to gain an understanding of spectral patterns and variation with associated biomass composition and use machine learning approaches to classify, differentiate, and predict biomass types on the basis of py-MBMS spectra. Py-MBMS spectra were also corrected for instrumental variance using generalized linear modeling (GLM) based on the use of select ions relative abundances as spike-in controls. Machine learning classification algorithms e.g., random forest, k-nearest neighbor, decision tree, Gaussian Naïve Bayes, gradient boosting, and multilayer perceptron classifiers were used. The k-nearest neighbors (k-NN) classifier generally performed the best for classifications using raw spectral data, and the decision tree classifier performed the worst. After normalization of spectra to account for instrumental variance, all the classifiers had comparable and generally acceptable performance for predicting the biomass types, although the k-NN and decision tree classifiers were not as accurate for prediction of specific sample types. Gaussian Naïve Bayes (GNB) and extreme gradient boosting (XGB) classifiers performed better than the k-NN and the decision tree classifiers for the prediction of biomass mixtures. The data analysis workflow reported here could be applied and extended for comparison of biomass samples of varying types, species, phenotypes, and/or genotypes or subjected to different treatments, environments, etc. to further elucidate the sources of spectral variance, patterns, and to infer compositional information based on spectral analysis, particularly for analysis of data without a priori knowledge of the feedstock composition or identity.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Image-Based Methods to Score Fungal Pathogen Symptom Progression and Severity in Excised Arabidopsis Leaves

Image-based symptom scoring of plant diseases is a powerful tool for associating disease resistance with plant genotypes. Advancements in technology have enabled new imaging and image processing strategies for statistical analysis of time-course experiments. There are several tools available for analyzing symptoms on leaves and fruits of crop plants, but only a few are available for the model plant Arabidopsis thaliana (Arabidopsis). Arabidopsis and the model fungus Botrytis cinerea (Botrytis) comprise a potent model pathosystem for the identification of signaling pathways conferring immunity against this broad host-range necrotrophic fungus. Here, we present two strategies to assess severity and symptom progression of Botrytis infection over time in Arabidopsis leaves. Thus, a pixel classification strategy using color hue values from red-green-blue (RGB) images and a random forest algorithm was used to establish necrotic, chlorotic, and healthy leaf areas. Secondly, using chlorophyll fluorescence (ChlFl) imaging, the maximum quantum yield of photosystem II (Fv/Fm) was determined to define diseased areas and their proportion per total leaf area. Both RGB and ChlFl imaging strategies were employed to track disease progression over time. This has provided a robust and sensitive method for detecting sensitive or resistant genetic backgrounds. A full methodological workflow, from plant culture to data analysis, is described.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Target Selection and Validation of DESI Quasars

The Dark Energy Spectroscopic Instrument (DESI) survey will measure large-scale structures using quasars as direct tracers of dark matter in the redshift range 0.9 < z < 2.1 and using Lyα forests in quasar spectra at z > 2.1. We present several methods to select candidate quasars for DESI, using input photometric imaging in three optical bands (g, r, z) from the DESI Legacy Imaging Surveys and two infrared bands (W1, W2) from the Wide-field Infrared Survey Explorer. These methods were extensively tested during the Survey Validation of DESI. In this paper, we report on the results obtained with the different methods and present the selection we optimized for the DESI main survey. The final quasar target selection is based on a random forest algorithm and selects quasars in the magnitude range of 16.5 < r < 23. Visual selection of ultra-deep observations indicates that the main selection consists of 71% quasars, 16% galaxies, 6% stars, and 7% inconclusive spectra. Using the spectra based on this selection, we build an automated quasar catalog that achieves a fraction of true QSOs higher than 99% for a nominal effective exposure time of ~1000 s. With a 310 deg -2 target density, the main selection allows DESI to select more than 200 deg -2 quasars (including 60 deg -2 quasars with z > 2.1), exceeding the project requirements by 20%. The redshift distribution of the selected quasars is in excellent agreement with quasar luminosity function predictions.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

Challenge Online Time Series Clustering For Demand Response A Theory to Break the ‘Curse of Dimensionality'

The advent of smart meters and advanced communication infrastructures catalyzes numerous smart grid applications such as dynamic demand response, and paves the way to solve challenging research problems in sustainable energy consumption. The space of solution possibilities are restricted primarily by the huge amount of generated data requiring considerable computational resources and efficient algorithms. To overcome this Big Data challenge, data clustering techniques have been proposed. Current approaches however do not scale in the face of the “increasing dimensionality” problem where a cluster point is represented by the entire customer consumption time series. To overcome this aspect we first rethink the way cluster points are created and designed, and then design an efficient online clustering technique for demand response (DR) in order to analyze high volume, high dimensional energy consumption time series data at scale, and on the fly. Our online algorithm is randomized in nature, and provides optimal performance guarantees in a computationally efficient manner. Unlike prior work we (i) study the consumption properties of the whole population simultaneously rather than developing individual models for each customer separately, claiming it to be a ‘killer’ approach that breaks the “curse of dimensionality” in online time series clustering, and (ii) provide tight performance guarantees in theory to validate our approach. Our insights are driven by the field of sociology, where collective behavior often emerges as the result of individual patterns and lifestyles.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2025

Integrating Crack Detection and Pipe Shape Optimization for Enhanced Sewage System Durability

Crack detection in underground reinforced concrete pipes has been essential in determining the state of stormwater infrastructure. Detection models have been implemented for detecting cracks and other defects in pipes using CCTV footage for stormwater drainage systems. In addition, Finite element models have been used to determine optimum shapes and pipe thickness for different boundary conditions such as header pipes in power plants. The concept of shape optimization emerges as a crucial factor in power plant design and operation, with the potential to maximize performance while minimizing the use of materials. Shape optimization not only enhances efficiency but also contributes to reducing the environmental footprint. This paper discusses the integration of both topics by using the cracks detected in underground pipes as boundary conditions for shape optimization of the pipes. A machine learning model has been developed which uses limited data for training and outlines the location of detected cracks. A shape optimization methodology is proposed in which ANSYS modules are used to analyze fluid flow and then optimize the shape of the pipe. The crack detection model developed has been applied to a crack detected in lab setting and machine learning model used has an accuracy of 98% using a random forest algorithm.

20 FOSSIL-FUELED POWER PLANTS↗

DOE OSTI2018

Evolutionary computation algorithms to coordinating order acceptance and batch delivery for an integrated supply chain scheduling

This paper proposes a novel approach of coordinating decisions in an integrated supply chain (ISC): coordinating order acceptance (OA) and batch delivery (BD) due to round trip transportation (RTT) and using third-party logistics (3PL) vehicles. The paper aims at trading off among accepted orders revenue, delivery costs as well as any penalties incurred in the ISC to maximize the total benefit. A novel mixed-integer programming is proposed for the problem. In addition, the paper provides a heuristic to form batches and develops a hybrid evolutionary computation algorithms based on particle swarm optimization (PSO) and genetic algorithm (GA) to solve the problem. An information sharing mechanism is improved and applied. To explore and to locate the proposed PSO in a better neighborhood, a local search is proposed. Taguchi experimental design is utilized to set the appropriate values of the algorithms’ parameters and random instances are generated to evaluate the performance of the algorithms. The paper investigates the profitability sensitivity of the problem to parameters and analyzes the effect of the changes in the parameters on the performance of our proposed algorithms. The attained results show the appropriate performance of our algorithms.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

Climate data for bioclimate research of Amazon Basin central region near the city of Manaus

The current research/data set deals with environmental data related to the region around the city of Manaus, Brazil, located in the Amazon Basin central region. The research problem encompasses the answers to the questions: 'What correlations can arise from the data of climate and fauna in the Amazon Basin?" "How useful is the data from instrumented stations related to that area, related to the fauna and climatic conditions?" To answer these questions, an initial data set was generated, taking into account the main pollutants with sources from NASA satellites, the Amazon Tall Tower Observatory (ATTO) experiment, and the GOAmazon project, from 2014 to 2015, along with other local data, such as the position of thermal electric power stations. It has been worth noting the data available was not coherent among the sources, due to different measuring time intervals, heights, and geographical locations. Equally important, for future bioclimatic research, a grid space of 5-km interval and a 30-minute time period was considered within the data set, covering a surface area within the boundaries of Latitude 2.00 S to 4.25 S and Longitude 59.00 to 61.25 W. The data from NASA satellites, ATTO, and GOAmazon do not have this kind of detailed grid and time mesh. To carry on, the machine learning (ML) Random Forest (RF) algorithm was selected and used, among other ML algorithms, to fill in the data into this new grid. The features considered were: data from NASA satellites and from the two projects above, covering the pollutants: ozone, NOx, particulates with diameter down to 10 nanometers, isoprene, CO, and acetonitrile. Moreover, distances and bearings to Manaus and 11 thermal power stations nearby, night and day time periods, dry and wet seasons, were also considered to compose the features. The interpolation produced by the Random Forest algorithm aimed at the pollutants: acetonitrile, ozone, NOx, Particulate N, isoprene, and CO, which are related to bioclimatic research commonly. In total, a data set was created with 51,148,800 rows, with 44 columns, 38 for the features and 6 for the targets.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Efficient Sampling of Complex Interdependent and Multiplex Networks

Efficient sampling of interdependent and multiplex infrastructure networks is critical for effectively applying failure and recovery algorithms in real-world settings, as well as to generate property-preserving reduced-order graph-based ensembles that address topological uncertainties. In this paper, we first explore the performance, i.e. the success in preserving graph properties, of graph sampling algorithms for interdependent and multiplex networks with synthetic and real-world graphs. We simulate sampling algorithms under different parameter settings. These settings include probabilistic graph generators, coupling patterns, and various performance metrics. Our results show that while Random Node and Random Walk sampling algorithms perform best for interdependent networks, Random Edge and Forest Fire sampling algorithms perform best for multiplex networks. Second, we propose and implement a novel similarity-based sampling algorithm for multiplex networks that samples only log(N) number of layers of an N-layer multiplex network while yielding computational savings with performance guarantees. Experimental results show that similarity sampling outperforms complete sampling of all layers while decreasing performance costs from a linear scale to a logarithmic one. Our results also indicate that similarity-based sampling outperforms complete sampling and random selection in nearly all scenarios when tested with real-world data.

Subasi, Omer↗

DOE OSTI2019

Vehicle energy consumption estimation using large scale simulations and machine learning methods

Vehicle energy consumption and powertrain operations for future vehicle powertrain technologies are predicted through full-vehicle simulations using Autonomie, a simulation tool developed at Argonne National Laboratory. Over 1 million vehicles, with different powertrain types (conventional and numerous electrified powertrains) and component technologies (advanced engines, batteries, electrical machines, light weighting) are simulated for outputs including vehicle fuel economy and cost. Despite the ability of Autonomic to predict fuel economy given various combination of vehicle inputs, the full-vehicle simulation is very time-consuming. It takes approximately 84 h to simulate for 33,060 vehicles even using distributed computing on a cluster of 128 worker nodes. Moreover, Autonomic is unable to populate a continuous output space through simulation since it only yields outputs with discrete values instead of an input-output relation. This paper proposes a novel large-scale learning and prediction process (LSLPP) via machine learning approaches to answer the need for a continuous space of vehicle fuel economy and a more efficient simulation process. By learning in a supervised fashion to discover structures in data, the machine learning approaches seek to train a model that can explain the data with which we can perform prediction over unseen data. As such, we are able to accomplish analytic continuation on the current discrete space of simulation results. In addition, we present a random sampling -based numerosity reduction algorithm incorporated in LSLPP that reduces simulation runs and improves efficiency of data procurement. We analytically derive a theoretical bound in order to guarantee the effectiveness of our algorithm. We also compare random sampling to stratified sampling and suggest sampling strategies. A user-friendly tool is developed to support the process. Our experimental results confirm that the proposed LSLPP is able to greatly accelerate prediction and analysis of vehicle fuel economy. The output of this study is used by the US government to evaluate the impact of its R&D funding on energy security and CO 2 emission as well as for the setting of Corporate Average Fuel Economy (CAFE) standards.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2020

Machine learning study of magnetism in uranium-based compounds

Actinide and lanthanide-based materials display exotic properties that originate from the presence of itinerant or localized f electrons and include unconventional superconductivity and magnetism, hidden order, and heavy-fermion behavior. Due to the strongly correlated nature of the 5f electrons, magnetic properties of these compounds depend sensitively on applied magnetic field and pressure, as well as on chemical doping. However, precise connection between the structure and magnetism in actinide-based materials is currently unclear. In this investigation, we established such structure-property links by assembling and mining two datasets that aggregate, respectively, the results of high-throughput density functional theory simulations and experimental measurements for the families of uranium- and neptunium-based binary compounds. Various regression algorithms were utilized to identify correlations among accessible attributes (features or descriptors) of the material systems and predict their cation magnetic moments and general forms of magnetic ordering. Descriptors representing compound structural parameters and cation f-subshell occupation numbers were identified as most important for accurate predictions. The best machine learning model developed employs the random forest regression algorithm. It can predict both spin and orbit moment size with root-mean-square error of 0.17 μ B and 0.19 μ B , respectively. Lastly, the random forest classification algorithm is used to predict the ordering (paramagnetic, ferromagnetic, and antiferromagnetic) of such systems with 76% accuracy.

36 MATERIALS SCIENCE↗

DOE OSTI2015

Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty

Reynolds Averaged Navier Stokes (RANS) models are widely used in industry to predict fluid flows, despite their acknowledged deficiencies. Not only do RANS models often produce inaccurate flow predictions, but there are very limited diagnostics available to assess RANS accuracy for a given flow configuration. If experimental or higher fidelity simulation results are not available for RANS validation, there is no reliable method to evaluate RANS accuracy. This paper explores the potential of utilizing machine learning algorithms to identify regions of high RANS uncertainty. Three different machine learning algorithms were evaluated: support vector machines, Adaboost decision trees, and random forests. The algorithms were trained on a database of canonical flow configurations for which validated direct numerical simulation or large eddy simulation results were available, and were used to classify RANS results on a point-by-point basis as having either high or low uncertainty, based on the breakdown of specific RANS modeling assumptions. Classifiers were developed for three different basic RANS eddy viscosity model assumptions: the isotropy of the eddy viscosity, the linearity of the Boussinesq hypothesis, and the non-negativity of the eddy viscosity. It is shown that these classifiers are able to generalize to flows substantially different from those on which they were trained. As a result, feature selection techniques, model evaluation, and extrapolation detection are discussed in the context of turbulence modeling applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

DeepHyper: Asynchronous Hyperparameter Search for Deep Neural Networks

Hyperparameters employed by deep learning (DL) methods play a substantial role in the performance and reliability of these methods in practice. Unfortunately, finding performance-optimizing hyperparameter settings is a notoriously difficult task. Hyperparameter search methods typically have limited production-strength implementations or do not target scalability within a highly parallel machine, portability across different machines, experimental comparison between different methods, and tighter integration with workflow systems. In this paper, we present DeepHyper, a Python package that provides a common interface for the implementation and study of scalable hyperparameter search methods. It adopts the Balsam workflow system to hide the complexities of running large numbers of hyperparameter configurations in parallel on high-performance computing (HPC) systems. We implement and study asynchronous model-based search methods that consist of sampling a small number of input hyperparameter configurations and progressively fitting surrogate models over the input-output space until exhausting a user-defined budget of evaluations. We evaluate the efficacy of these methods relative to approaches such as random search, genetic algorithms, Bayesian optimization, and hyperband on DL benchmarks on CPU- and GPU-based HPC systems.

Balaprakash, Prasanna↗

DOE OSTI2024

Improved Subseasonal Forecasting of Extreme Polar Vortices Using Machine Learning

Our research was focused on forecasting the position and shape of the winter stratospheric polar vortex at a subseasonal timescale of 15 days in advance. To achieve this, we employed both statistical and neural network machine learning techniques. The analysis was performed on 42 winter seasons of reanalysis data provided by NASA giving us a total of 6,342 days of data. The state of the polar vortex for determined by using geometric moments to calculate the centroid latitude and the aspect ratio of an ellipse fit onto the vortex. Timeseries for thirty additional precursors were calculated to help improve the predictive capabilities of the algorithm. Feature importance of these precursors was performed using random forest to measure the predictive importance and the ideal number of precursors. Then, using the precursors identified as important, various statistical methods were tested for predictive accuracy with random forest and nearest neighbor performing the best. An echo state network, a type of recurrent neural network that features sparsely connected hidden layer and a reduced number of trainable parameters that allows for rapid training and testing, was also implemented for the forecasting problem. Hyperparameter tuning was performed for each methods using a subset of the training data. The algorithms were trained and tuned on the first 41 years of data, then tested for accuracy on the final year. In general, the centroid latitude of the polar vortex proved easier to predict than the aspect ratio across all algorithms. Random forest outperformed other statistical forecasting algorithms overall but struggled to predict extreme values. Forecasting from echo state network suggested a strong predictive capability past 15 days, but further work is required to fully realize the potential of recurrent neural network approaches.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2015

Fast generation of sparse random kernel graphs

The development of kernel-based inhomogeneous random graphs has provided models that are flexible enough to capture many observed characteristics of real networks, and that are also mathematically tractable. We specify a class of inhomogeneous random graph models, called random kernel graphs, that produces sparse graphs with tunable graph properties, and we develop an efficient generation algorithm to sample random instances from this model. As real-world networks are usually large, it is essential that the run-time of generation algorithms scales better than quadratically in the number of vertices n. We show that for many practical kernels our algorithm runs in time at most ο(n(logn)²). As an example, we show how to generate samples of power-law degree distribution graphs with tunable assortativity.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

COE Task Order 4. Final Report: Automated Threat Recognition

Lawrence Livermore National Laboratory (LLNL) was tasked by the Department of Homeland Security (DHS) Science and Technology Directorate (S&T) with developing an automated target recognition algorithm for finding objects of interest in checked baggage at airports. The project used x-ray computed tomography (CT) data collected on a medical scanner of various simulated checked bags. Target objects of interest include clay, rubber, and saline in various concentrations. The final results are 93.9% probability of detection (P D ) with 11.9% probability of false alarm (PFA) on all targets and 100% detection for pseudo target sheets. This result is achieved by using a three-stage algorithm made up of segmentation, post-processing, and feature extraction and object classification. The segmentation proved to be the most important stage and it involves merging two distinct segmentation results followed by a set of algorithms for cleaning up the merged segmentations. The first segmentation algorithm operates on planar 10x10 voxel slabs (two- dimensional segments) in the three spatial planes. The segmenter calculates a probability that each slab in the image is a part of a target, determines if the probability was above a threshold, and then merges all connected slabs together. The second segmentation algorithm is an ensemble segmenter, which generates a set of possible segmentations through random permutations and computes an average behavior. The distinct properties of these segmenters results in a merged segmentation with superior results than either segmenter alone. The merged segmentation is “cleaned up” using post processing techniques to achieve the final labeled images. The final classification stage made use of a random forest classifier algorithm to discriminate targets from non-targets based on a set of features of the segmented objects partially from the voxel slabs and partially from the individual voxels. Features include standard statistics such as mean, median, and standard deviations of the density, features capturing some aspects of texture, and some features about the surrounding bag structure. In addition to the detection and false alarm results above, we show how the algorithms can be trained to “overfit” to the limited data set provided and achieve near perfect results (P D =100% and P FA =0%). This over-training is not recommended for long-term operation, but instead is meant to demonstrate how test design is an important factor in aviation security. Possible improvements to the algorithm and the testing methodology (including blind testing) are also presented.

42 ENGINEERING↗

DOE OSTI2016

Random functions via Dyson Brownian Motion: progress and problems

We develope a computationally efficient extension of the Dyson Brownian Motion (DBM) algorithm to generate random function in C{sup 2} locally. We further explain that random functions generated via DBM show an unstable growth as the traversed distance increases. This feature restricts the use of such functions considerably if they are to be used to model globally defined ones. The latter is the case if one uses random functions to model landscapes in string theory. We provide a concrete example, based on a simple axionic potential often used in cosmology, to highlight this problem and also offer an ad hoc modification of DBM that suppresses this growth to some degree.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

A chemistry-informed hybrid machine learning approach to predict metal adsorption onto mineral surfaces

Historically, surface complexation model (SCM) constants and distribution coefficients (K d ) have been employed to quantify mineral-based retardation effects controlling the fate of metals in subsurface geologic systems. Our recent SCM development workflow, based on the Lawrence Livermore National Laboratory Surface Complexation/Ion Exchange (L-SCIE) database, illustrated a community FAIR data approach to SCM development by predicting uranium(VI)-quartz adsorption for a large number of literature-mined data. Here, we present an alternative hybrid machine learning (ML) approach that shows promise in achieving equivalent high-quality predictions compared to traditional surface complexation models. At its core, the hybrid random forest (RF) ML approach is motivated by the proliferation of incongruent SCMs in the literature that limit their applicability in reactive transport models. Our hybrid ML approach implements PHREEQC-based aqueous speciation calculations; values from these simulations are automatically used as input features for a random forest (RF) algorithm to quantify adsorption and avoid SCM modeling constraints entirely. Named the LLNL Speciation Updated Random Forest (L-SURF) model, this hybrid approach is shown to have applicability to U(VI) sorption cases driven by both ion-exchange and surface complexation, as is shown for quartz and montmorillonite cases. The approach can be applied to reactive transport modeling and may provide an alternative to the costly development of self-consistent SCM reaction databases.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

DOE OSTI2012

A BAYESIAN APPROACH TO LOCATING THE RED GIANT BRANCH TIP MAGNITUDE. II. DISTANCES TO THE SATELLITES OF M31

In 'A Bayesian Approach to Locating the Red Giant Branch Tip Magnitude (Part I)', a new technique was introduced for obtaining distances using the tip of the red giant branch (TRGB) standard candle. Here we describe a useful complement to the technique with the potential to further reduce the uncertainty in our distance measurements by incorporating a matched-filter weighting scheme into the model likelihood calculations. In this scheme, stars are weighted according to their probability of being true object members. We then re-test our modified algorithm using random-realization artificial data to verify the validity of the generated posterior probability distributions (PPDs) and proceed to apply the algorithm to the satellite system of M31, culminating in a three-dimensional view of the system. Further to the distributions thus obtained, we apply a satellite-specific prior on the satellite distances to weight the resulting distance posterior distributions, based on the halo density profile. Thus in a single publication, using a single method, a comprehensive coverage of the distances to the companion galaxies of M31 is presented, encompassing the dwarf spheroidals Andromedas I-III, V, IX-XXVII, and XXX along with NGC 147, NGC 185, M33, and M31 itself. Of these, the distances to Andromedas XXIV-XXVII and Andromeda XXX have never before been derived using the TRGB. Object distances are determined from high-resolution tip magnitude posterior distributions generated using the Markov Chain Monte Carlo technique and associated sampling of these distributions to take into account uncertainties in foreground extinction and the absolute magnitude of the TRGB as well as photometric errors. The distance PPDs obtained for each object both with and without the aforementioned prior are made available to the reader in tabular form. The large object coverage takes advantage of the unprecedented size and photometric depth of the Pan-Andromeda Archaeological Survey. Finally, a preliminary investigation into the satellite density distribution within the halo is made using the obtained distance distributions. For simplicity, this investigation assumes a single power law for the density as a function of radius, with the slope of this power law examined for several subsets of the entire satellite sample.

79 ASTRONOMY AND ASTROPHYSICS↗