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All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

SU-E-T-478: Sliding Window Multi-Criteria IMRT Optimization

Purpose: To demonstrate a method for what-you-see-is-what-you-get multi-criteria Pareto surface navigation for step and shoot IMRT treatment planning. Methods: We show mathematically how multiple sliding window treatment plans can be averaged to yield a single plan whose dose distribution is the dosimetric average of the averaged plans. This is incorporated into the Pareto surface navigation based approach to treatment planning in such a way that as the user navigates the surface, the plans he/she is viewing are ready to be delivered (i.e. there is no extra ‘segment the plans’ step that often leads to unacceptable plan degradation in step and shoot Pareto surface navigation). We also describe how the technique can be applied to VMAT. Briefly, sliding window VMAT plans are created such that MLC leaves paint out fluence maps every 15 degrees or so. These fluence map leaf trajectories are averaged in the same way the static beam IMRT ones are. Results: We show mathematically that fluence maps are exactly averaged using our leaf sweep averaging algorithm. Leaf transmission and output factor corrections effects, which are ignored in this work, can lead to small errors in terms of the dose distributions not being exactly averaged even though the fluence maps are. However, our demonstrations show that the dose distributions are almost exactly averaged as well. We demonstrate the technique both for IMRT and VMAT. Conclusions: By turning to sliding window delivery, we show that the problem of losing plan fidelity during the conversion of an idealized fluence map plan into a deliverable plan is remedied. This will allow for multicriteria optimization that avoids the pitfall that the planning has to be redone after the conversion into MLC segments due to plan quality decline. David Craft partially funded by RaySearch Laboratories.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2015

Calibration of EBT2 film using a red-channel PDD method in combination with a modified three-channel technique

Purpose: Ashland Inc. EBT2 and EBT3 films are widely used in quality assurance for radiation therapy; however, there remains a relatively high degree of uncertainty [B. Hartmann, M. Martisikova, and O. Jakel, “Homogeneity of Gafchromic EBT2 film,” Med. Phys. 37, 1753–1756 (2010)]. Micke et al. (2011) recently improved the spatial homogeneity using all color channels of a flatbed scanner; however, van Hoof et al. (2012) pointed out that the corrected nonuniformity still requires further investigation for larger fields. To reduce the calibration errors and the uncertainty, the authors propose a new red-channel percentage-depth-dose method in combination with a modified three-channel technique. Methods: For the ease of comparison, the EBT2 film image used in the authors’ previous study (2012) was reanalyzed using different approaches. Photon beams of 6-MV were delivered to two different films at two different beam on times, resulting in the absorption doses of ranging from approximately 30 to 300 cGy at the vertical midline of the film, which was set to be coincident with the central axis of the beam. The film was tightly sandwiched in a 30{sup 3}-cm{sup 3} polystyrene phantom, and the pixel values for red, green, and blue channels were extracted from 234 points on the central axis of the beam and compared with the corresponding depth doses. The film was first calibrated using the multichannel method proposed by Micke et al. (2010), accounting for nonuniformities in the scanner. After eliminating the scanner and dose-independent nonuniformities, the film was recalibrated via the dose-dependent optical density of the red channel and fitted to a power function. This calibration was verified via comparisons of the dose profiles extracted from the films, where three were exposed to a 60° physical wedge field and three were exposed to composite fields, and all of which were measured in a water phantom. A correction for optical attenuation was implemented, and treatment plans of intensity modulated radiation therapy and volumetric modulated arc therapy were evaluated. Results: The method described here demonstrated improved accuracy with reduced uncertainty. The relative error compared with the measurements of a water phantom was less than 1%, and the overall calibration uncertainty was less than 2%. Verification tests revealed that the results were close to those of the authors’ previous study, and all differences were within 3%, except those with a high-dose gradient. The gamma pass rates (2%/2 mm) of the treatment plan evaluated using the method described here were greater than 99%, and no obvious stripe patterns were observed in the dose-difference maps. Conclusions: Spatial homogeneity was significantly improved via the calibration method described here. This technique is both convenient and time-efficient because it does not require cutting the film, and only two exposures are necessary.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2026

Assessing the Limitations of Self-Interaction-Corrected Functionals for Describing the Hydrated Electron

Simulating the hydrated electron using density functional theory is challenging due to the prevalence of self-interaction error in standard functionals. Hybrid functionals like PBEh(40) can reasonably describe the chemistry of an excess electron in water and partially mitigate self-interaction error by incorporating exact Hartree–Fock exchange, but they are computationally expensive making them impractical for large-scale and long-time ab initio molecular dynamics simulations. Explicit self-interaction correction schemes that are applied on an orbital-by-orbital basis offer a potential alternative when the correction is limited to the singly occupied molecular orbital obtained with a generalized gradient approximation functional. Here, we examine whether the Perdew–Zunger self-interaction correction scheme applied to the revPBE functional can provide a computationally efficient and physically sensible alternative to PBEh(40) for the hydrated electron. We find that functionals incorporating a self-interaction correction scheme should be viewed with caution when applied to the hydrated electron and its reactivity. Furthermore, we show that it is critical to consider extensive sampling and diverse chemical environments when validating their performance.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

A step in the direction of resolving the paradox of Perdew–Zunger self-interaction correction. II. Gauge consistency of the energy density at three levels of approximation

The Perdew-Zunger(PZ) self-interaction correction (SIC) was designed to correct the one-electron limit of any approximate density functional for the exchange-correlation (xc) energy, while yielding no correction to the exact functional. Unfortunately, it spoils the slowly-varying-in-space limits of the uncorrected approximate functionals, where those functionals are right by construction. The right limits can be restored by locally scaling down the energy density of the PZ SIC in many-electron regions, but then a spurious correction to the exact functional would be found unless the self-Hartree and exact self-xc terms of the PZ SIC energy density were expressed in the same gauge. Only the local density approximation satisfies the same-gauge condition for the energy density, which explains why the recent local-scaling SIC (LSIC) is found here to work excellently for atoms and molecules only with this basic approximation, and not with the more advanced generalized gradient approximations (GGAs) and meta-GGAs, which lose the Hartree gauge via simplifying integrations by parts. The transformation of energy density that achieves the Hartree gauge for the exact xc functional can also be applied to approximate functionals. Doing so leads to a simple scaled-down self-interaction (sdSIC) correction that is typically much more accurate than PZ SIC in tests for many molecular properties (including equilibrium bond lengths). The present work shows unambiguously that the largest errors of PZ SIC applied to standard functionals at three levels of approximation can be removed by restoring their correct slowlyvarying- density limits. Here, it also confirms the relevance of these limits to atoms and molecules.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Publisher Correction: Strongly-coupled quantum critical point in an all-in-all-out antiferromagnet

The original PDF version of the Article contained an error in the last sentence of the author affiliation information, which incorrectly read ‘Correspondence and requests for materials should be addressed to T.$.R. (e-mail: tfr@caltech.edu) or to Y.F. (e-mail: yejun@oist.jp)’. The correct version states ‘T.F.R.’ in place of “T.$.R.’. This has been corrected in the PDF version of the Article. The HTML version was correct from the time of publication.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Improving Planck calibration by including frequency-dependent relativistic corrections

The Planck satellite detectors are calibrated in the 2015 release using the 'orbital dipole', which is the time-dependent dipole generated by the Doppler effect due to the motion of the satellite around the Sun. Such an effect has also relativistic time-dependent corrections of relative magnitude 10{sup −3}, due to coupling with the 'solar dipole' (the motion of the Sun compared to the CMB rest frame), which are included in the data calibration by the Planck collaboration. We point out that such corrections are subject to a frequency-dependent multiplicative factor. This factor differs from unity especially at the highest frequencies, relevant for the HFI instrument. Since currently Planck calibration errors are dominated by systematics, to the point that polarization data is currently unreliable at large scales, such a correction can in principle be highly relevant for future data releases.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Exploring and enhancing the accuracy of interior-scaled Perdew–Zunger self-interaction correction

The Perdew–Zunger self-interaction correction (PZ-SIC) improves the performance of density functional approximations for the properties that involve significant self-interaction error (SIE), as in stretched bond situations, but overcorrects for equilibrium properties where SIE is insignificant. This overcorrection is often reduced by local scaling self-interaction correction (LSIC) of the PZ-SIC to the local spin density approximation (LSDA). Here, we propose a new scaling factor to use in an LSIC-like approach that satisfies an additional important constraint: the correct coefficient of the atomic number Z in the asymptotic expansion of the exchange–correlation (xc) energy for atoms. LSIC and LSIC+ are scaled by functions of the iso-orbital indicator z σ , which distinguishes one-electron regions from many-electron regions. LSIC+ applied to the LSDA works better for many equilibrium properties than LSDA-LSIC and the Perdew, Burke, and Ernzerhof generalized gradient approximation (GGA), and almost close to the strongly constrained and appropriately normed (SCAN) meta-GGA. LSDA-LSIC and LSDA-LSIC+, however, fail to predict interaction energies involving weaker bonds, in sharp contrast to their earlier successes. It is found that more than one set of localized SIC orbitals can yield a nearly degenerate energetic description of the same multiple covalent bond, suggesting that a consistent chemical interpretation of the localized orbitals requires a new way to choose their Fermi orbital descriptors. To make a locally scaled down SIC to functionals beyond the LSDA requires a gauge transformation of the functional’s energy density. Here, the resulting SCAN-sdSIC, evaluated on SCAN-SIC total and localized orbital densities, leads to an acceptable description of many equilibrium properties including the dissociation energies of weak bonds.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Optics measurement and correction during beam acceleration in the Relativistic Heavy Ion Collider

To minimize operational complexities, setup of collisions in high energy circular colliders typically involves acceleration with near constant β-functions followed by application of strong focusing quadrupoles at the interaction points (IPs) for the final beta-squeeze. At the Relativistic Heavy Ion Collider (RHIC) beam acceleration and optics squeeze are performed simultaneously. In the past, beam optics correction at RHIC has taken place at injection and at final energy with some interpolation of corrections into the acceleration cycle. Recent measurements of the beam optics during acceleration and squeeze have evidenced significant beta-beats which if corrected could minimize undesirable emittance dilutions and maximize the spin polarization of polarized proton beams by avoidance of higher-order multipole fields sampled by particles within the bunch. In this report the methodology now operational at RHIC for beam optics corrections during acceleration with simultaneous beta-squeeze will be presented together with measurements which conclusively demonstrate the superior beam control. As a valuable by-product, the corrections have minimized the beta-beat at the profile monitors so reducing the dominant error in and providing more precise measurements of the evolution of the beam emittances during acceleration.

43 PARTICLE ACCELERATORS↗
DOE OSTI2021

Elevating density functional theory to chemical accuracy for water simulations through a density-corrected many-body formalism

Abstract Density functional theory (DFT) has been extensively used to model the properties of water. Albeit maintaining a good balance between accuracy and efficiency, no density functional has so far achieved the degree of accuracy necessary to correctly predict the properties of water across the entire phase diagram. Here, we present density-corrected SCAN (DC-SCAN) calculations for water which, minimizing density-driven errors, elevate the accuracy of the SCAN functional to that of “gold standard” coupled-cluster theory. Building upon the accuracy of DC-SCAN within a many-body formalism, we introduce a data-driven many-body potential energy function, MB-SCAN(DC), that quantitatively reproduces coupled cluster reference values for interaction, binding, and individual many-body energies of water clusters. Importantly, molecular dynamics simulations carried out with MB-SCAN(DC) also reproduce the properties of liquid water, which thus demonstrates that MB-SCAN(DC) is effectively the first DFT-based model that correctly describes water from the gas to the liquid phase.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

SU-E-T-631: Preliminary Results for Analytical Investigation Into Effects of ArcCHECK Setup Errors

Purpose: As three-dimensional diode arrays increase in popularity for patient-specific quality assurance for intensity-modulated radiation therapy (IMRT), it is important to evaluate an array’s susceptibility to setup errors. The ArcCHECK phantom is set up by manually aligning its outside marks with the linear accelerator’s lasers and light-field. If done correctly, this aligns the ArcCHECK cylinder’s central axis (CAX) with the linear accelerator’s axis of rotation. However, this process is prone to error. This project has developed an analytical expression including a perturbation factor to quantify the effect of shifts. Methods: The ArcCHECK is set up by aligning its machine marks with either the sagittal room lasers or the light-field of the linear accelerator at gantry zero (IEC). ArcCHECK has sixty-six evenly-spaced SunPoint diodes aligned radially in a ring 14.4 cm from CAX. The detector response function (DRF) was measured and combined with inverse-square correction to develop an analytical expression for output. The output was calculated using shifts of 0 (perfect alignment), +/−1, +/−2 and +/−5 mm. The effect on a series of simple inputs was determined: unity, 1-D ramp, steps, and hat-function to represent uniform field, wedge, evenly-spaced modulation, and single sharp modulation, respectively. Results: Geometric expressions were developed with perturbation factor included to represent shifts. DRF was modeled using sixth-degree polynomials with correlation coefficient 0.9997. The output was calculated using simple inputs such as unity, 1-D ramp, steps, and hat-function, with perturbation factors of: 0, +/−1, +/−2 and +/−5 mm. Discrepancies have been observed, but large fluctuations have been somewhat mitigated by aliasing arising from discrete diode placement. Conclusion: An analytical expression with perturbation factors was developed to estimate the impact of setup errors on an ArcCHECK phantom. Presently, this has been applied to simple functions, but is being developed to handle more realistic clinical cases.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2024

The impact of low-mode symmetry on inertial fusion energy output in the burning plasma state

Indirect Drive Inertial Confinement Fusion Experiments on the National Ignition Facility (NIF) have achieved a burning plasma state with neutron yields exceeding 170 kJ, roughly 3 times the prior record and a necessary stage for igniting plasmas. The results are achieved despite multiple sources of degradations that lead to high variability in performance. Results shown here, for the first time, include an empirical correction factor for mode-2 asymmetry in the burning plasma regime in addition to previously determined corrections for radiative mix and mode-1. Analysis shows that including these three corrections alone accounts for the measured fusion performance variability in the two highest performing experimental campaigns on the NIF to within error. Here we quantify the performance sensitivity to mode-2 symmetry in the burning plasma regime and apply the results, in the form of an empirical correction to a 1D performance model. Furthermore, we find the sensitivity to mode-2 determined through a series of integrated 2D radiation hydrodynamic simulations to be consistent with the experimentally determined sensitivity only when including alpha-heating.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Energy Frontier Research Centers: Center for the Computational Design of Functional Layered Materials (CCDM) August 1, 2014 - July 31, 2018; Center for Complex Materials from First Principles (CCM) August 1, 2018 - July 31, 2021 (Final Report)

The mission of the DOE Energy Frontier Research Centers CCDM (2014-2018) and CCM (2018-2021) was to theoretically develop, computationally apply, and experimentally validate electronic structure methods for all materials, with a focus on the complex materials, especially layered and two-dimensional materials, strongly-correlated materials, and liquid water. This was achieved by over 200 published journal articles authored by about 17 senior investigators from physics and chemistry and from theory, computation, and experiment, plus their collaborators. In particular, the Centers confirmed the predictive power of the SCAN (strongly constrained and appropriately normed) density functional, which was constructed to satisfy 17 known exact constraints and several appropriate norms. Without being fitted to real bonded systems, and at a modest computational cost, SCAN correctly predicted covalent, ionic, metallic, hydrogen, and van der Waals bonds in many challenging materials. SCAN gave an improved description of defects in semiconductors, surface properties of metals, seven phases of ice, liquid water, liquid and supercooled silicon, subtle structural distortions in ferroelectrics, formation energies and structural predictions for solids, and critical pressures for structural phase transitions. Perhaps most remarkably, SCAN correctly described some strongly-correlated materials that were previously believed to be beyond the reach of density-functional approximations. SCAN is the only density functional that correctly predicts the band gap closing under chemical doping of the cuprate high-temperature superconducting materials. SCAN also predicts a landscape of competing stripe and magnetic phases in the cuprates. For some materials with some codes, SCAN has convergence problems that are greatly reduced by the CCM-developed r 2 SCAN, without loss of accuracy or rigor. SCAN and r 2 SCAN still make some self-interaction error, which is greatly reduced by the CCDM/ CCM-developed local orbital scaling correction (LOSC). These Centers further proved that the fundamental energy gaps of a solid from an orbital energy difference and from total energy differences are the same for a large class of generalized Kohn-Sham (GKS) functionals, including SCAN and standard hybrid functionals, and that symmetry breaking arises when a dynamic density fluctuation drops to zero frequency. The Centers identified new mechanisms for catalysis in layered materials with ions intercalated between the layers, investigated charge density waves both in model systems and in real layered materials, studied changes of band gap with the number of layers, and explored topological ultrathin films, bent nanoribbons, and defects.

08 HYDROGEN↗
DOE OSTI2020

Development and Implementation of a CTF Code Verification Suite.

CTF is a thermal hydraulic subchannel code developed to predict light water reactor (LWR) core behavior. It is a version of Coolant Boiling in Rod Arrays (COBRA) developed by Oak Ridge National Laboratory (ORNL) and North Carolina State University (NCSU) and used in the Consortium for the Advanced Simulation of LWRs (CASL). Here, the existing CTF code verification matrix is expanded, which ensures that the code is a faithful representation of the underlying mathematical model. The suite of code verification tests are mapped to the underlying conservation equations of CTF and significant gaps are addressed. As such, five new problems are incorporated: isokinetic advection, conduction, pressure drop, convection, and pipe boiling. Convergence behavior and numerical errors are quantified for each of the tests and all tests converge at the correct rate to their corresponding analytic solution. A new verification utility that generalizes the code verification process is used to incorporate these problems into the CTF automated test suite.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

Author Correction: Proton irradiation-decelerated intergranular corrosion of Ni-Cr alloys in molten salt

In the original version of this article, numbers for the beam current densities were incorrectly given as 1.5, 2.0, and 2.5 mA cm -2 in various locations, instead of the correct values 0.3, 0.4, and 0.5 mA cm -2 . This was owing to a measurement error coming from the indirect correspondence between the Faraday cup and the beam profile monitor (BPM) on the accelerator as the proton beam traveling along the beamline resulted in mismatching between calculated values and real beam currents, which were obtained by ex-situ calibration. The following changes have been made in the correct version. The eighth sentence of the Results, and the figure legend of Fig. 2j state '0.3, 0.4, 0.5 mA cm -2 ' in place of '1.5, 2.0, 2.5 mA cm -2 '. The figure legend of Fig. 1j–l states '0.5, 0.4, 0.3 mA cm -2 ' in place of '2.5, 2.0, 1.5 mA cm -2 '. Figure 1e states '0.5 mA cm -2 ' in place of '2.5 mA cm -2 ', Fig. 1h states '0.4 mA cm -2 ' in place of '2.0 mA cm -2 ', and Fig. 1k states ‘0.3 mA cm -2 ' in place of ‘1.5 mA cm -2 '. Figure 2c states '0.5 mA cm -2 ' in place of '2.5 mA cm -2 ', Fig. 2d states '0.4 mA cm -2 ' in place of '2.0 mA cm -2 ', and Fig. 2e states '0.3 mA cm -2 ' in place of '1.5 mA cm -2 '. Figure 2j and k states ‘0.3 mA cm -2 ', '0.4 mA cm -2 ', '0.5 mA cm -2 ' in place of '1.5 mA cm -2 ', '2.0 mA cm -2 ', and '2.5 mA cm -2 ', respectively. The figure legend of Fig. 3b states '0.4 mA cm -2 ' in place of '2.0 mA cm -2 '. The figure legend of Supplementary Fig. 1 states '0.5 mA cm -2 ' in place of '2.5 mA cm -2 '. This has been corrected in the PDF and HTML versions of the Article.

36 MATERIALS SCIENCE↗
DOE OSTI2014

SU-E-T-396: Dosimetric Accuracy of Proton Therapy for Patients with Metal Implants in CT Scans Using Metal Deletion Technique (MDT) Artifacts Reduction

Purpose: To evaluate the dosimetric accuracy for proton therapy patients with metal implants in CT using metal deletion technique (MDT) artifacts reduction. Methods: Proton dose accuracies under CT metal artifacts were first evaluated using a water phantom with cylindrical inserts of different materials (titanium and steel). Ranges and dose profiles along different beam angles were calculated using treatment planning system (Eclipse version 8.9) on uncorrected CT, MDT CT, and manually-corrected CT, where true Hounsfield units (water) were assigned to the streak artifacts. In patient studies, the treatment plans were developed on manually-corrected CTs, then recalculated on MDT and uncorrected CTs. DVH indices were compared between the dose distributions on all the CTs. Results: For water phantom study with 1/2 inch titanium insert, the proton range differences estimated by MDT CT were with 1% for all beam angles, while the range error can be up to 2.6% for uncorrected CT. For the study with 1 inch stainless steel insert, the maximum range error calculated by MDT CT was 1.09% among all the beam angles compared with maximum range error with 4.7% for uncorrected CT. The dose profiles calculated on MDT CTs for both titanium and steel inserts showed very good agreements with the ones calculated on manually-corrected CTs, while large dose discrepancies calculated using uncorrected CTs were observed in the distal end region of the proton beam. The patient study showed similar dose distribution and DVHs for organs near the metal artifacts recalculated on MDT CT compared with the ones calculated on manually-corrected CT, while the differences between uncorrected and corrected CTs were much pronounced. Conclusion: In proton therapy, large dose error could occur due to metal artifact. The MDT CT can be used for proton dose calculation to achieve similar dose accuracy as the current clinical practice using manual correction.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

Bone-marrow densitometry: Assessment of marrow space of human vertebrae by single energy high resolution-quantitative computed tomography

Purpose: Accurate noninvasive assessment of vertebral bone marrow fat fraction is important for diagnostic assessment of a variety of disorders and therapies known to affect marrow composition. Moreover, it provides a means to correct fat-induced bias of single energy quantitative computed tomography (QCT) based bone mineral density (BMD) measurements. The authors developed new segmentation and calibration methods to obtain quantitative surrogate measures of marrow-fat density in the axial skeleton. Methods: The authors developed and tested two high resolution-QCT (HR-QCT) based methods which permit segmentation of bone voids in between trabeculae hypothesizing that they are representative of bone marrow space. The methods permit calculation of marrow content in units of mineral equivalent marrow density (MeMD). The first method is based on global thresholding and peeling (GTP) to define a volume of interest away from the transition between trabecular bone and marrow. The second method, morphological filtering (MF), uses spherical elements of different radii (0.1–1.2 mm) and automatically places them in between trabeculae to identify regions with large trabecular interspace, the bone-void space. To determine their performance, data were compared ex vivo to high-resolution peripheral CT (HR-pQCT) images as the gold-standard. The performance of the methods was tested on a set of excised human vertebrae with intact bone marrow tissue representative of an elderly population with low BMD. Results: 86% (GTP) and 87% (MF) of the voxels identified as true marrow space on HR-pQCT images were correctly identified on HR-QCT images and thus these volumes of interest can be considered to be representative of true marrow space. Within this volume, MeMD was estimated with residual errors of 4.8 mg/cm{sup 3} corresponding to accuracy errors in fat fraction on the order of 5% both for GTP and MF methods. Conclusions: The GTP and MF methods on HR-QCT images permit noninvasive localization and densitometric assessment of marrow fat with residual accuracy errors sufficient to study disorders and therapies known to affect bone marrow composition. Additionally, the methods can be used to correct BMD for fat induced bias. Application and testing in vivo and in longitudinal studies are warranted to determine the clinical performance and value of these methods.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2016

Numerically stable, scalable formulas for parallel and online computation of higher-order multivariate central moments with arbitrary weights

Formulas for incremental or parallel computation of second order central moments have long been known, and recent extensions of these formulas to univariate and multivariate moments of arbitrary order have been developed. Such formulas are of key importance in scenarios where incremental results are required and in parallel and distributed systems where communication costs are high. We survey these recent results, and improve them with arbitrary-order, numerically stable one-pass formulas which we further extend with weighted and compound variants. We also develop a generalized correction factor for standard two-pass algorithms that enables the maintenance of accuracy over nearly the full representable range of the input, avoiding the need for extended-precision arithmetic. We then empirically examine algorithm correctness for pairwise update formulas up to order four as well as condition number and relative error bounds for eight different central moment formulas, each up to degree six, to address the trade-offs between numerical accuracy and speed of the various algorithms. Finally, we demonstrate the use of the most elaborate among the above mentioned formulas, with the utilization of the compound moments for a practical large-scale scientific application.

97 MATHEMATICS AND COMPUTING↗