Linear transformations of variance/covariance matrices

Parois, Pascal; Lutz, Martin

doi:10.1107/s0108767311018216

Cited by 4 publications

(7 citation statements)

References 25 publications

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“…(16), (19) and (27) in vector form. Assuming the state vectors X, A and B are multi-normally distributed with distributions described by covariance matrices X , A and B then, the covariance matrix for the mixing state X is given as follows (Parois and Lutz, 2011):…”

Section: Multivariate Normal Distributions For Mixture Typesmentioning

confidence: 99%

Separating mixtures of aerosol types in airborne High Spectral Resolution Lidar data

et al. 2014

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Abstract. Knowledge of aerosol type is important for determining the magnitude and assessing the consequences of aerosol radiative forcing, and can provide useful information for source attribution studies. However, atmospheric aerosol is frequently not a single pure type, but instead occurs as a mixture of types, and this mixing affects the optical and radiative properties of the aerosol. This paper extends the work of earlier researchers by using the aerosol intensive parameters measured by the NASA Langley Research Center airborne High Spectral Resolution Lidar (HSRL-1) to develop a comprehensive and unified set of rules for characterizing the external mixing of several key aerosol intensive parameters: extinction-to-backscatter ratio (i.e., lidar ratio), backscatter color ratio, and depolarization ratio. We present the mixing rules in a particularly simple form that leads easily to mixing rules for the covariance matrices that describe aerosol distributions, rather than just single values of measured parameters. These rules can be applied to infer mixing ratios from the lidar-observed aerosol parameters, even for cases without significant depolarization. We demonstrate our technique with measurement curtains from three HSRL-1 flights which exhibit mixing between two aerosol types, urban pollution plus dust, marine plus dust, and smoke plus marine. For these cases, we infer a time-height cross-section of extinction mixing ratio along the flight track, and partition aerosol extinction into portions attributed to the two pure types.

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Section: Multivariate Normal Distributions For Mixture Typesmentioning

confidence: 99%

Separating mixtures of aerosol types in airborne High Spectral Resolution Lidar data

et al. 2014

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“…(ii) Restraints on the direction and/or magnitude of the principal axes to be similar to, or the average of, other atoms, optionally in a local coordinate system: ULIJ, URIGU, UALIGN, UTLS, UVOL, UEQIV. These new restraints are implemented in CRYSTALS (Parois et al, 2015; Version 14.6720 and above) and full details are provided here so that they may be re-implemented in other tools as required.…”

Section: Implementation Of Displacement Parameter Restraintsmentioning

confidence: 99%

“…In CRYSTALS, for numerical convenience and speed, the six derived ADPs @U=@U ij (3 Â 3 matrix) are vectorized as nine element vectors (Parois & Lutz, 2011). The resulting six vectors (one for each partial derivative) are stacked to form a 9 Â 6 matrix @U:…”

Section: Derivation Of the Restraintsmentioning

confidence: 99%

“…There are some limitations to this approach: firstly, a TLS model must replace the displacement parameters of at least four atoms in order to provide a simpler description of the displacements using fewer parameters; and secondly, the TLS fit calculation will fail if all the atoms lie in a circle or on any conic section (Cruickshank, 1956;Schomaker & Trueblood, 1968), which rules out simple groups such as CF 3 (one can add the next-neighbour atom to make the TLS fit possible; see x4.1) and larger symmetric groups such as phenyl. However, the program CRYSTALS (Parois et al, 2015) has a mechanism to handle the instability in some cases using eigendecomposition, and in practice a TLS fit on a phenyl ring is often possible.…”

Section: Introductionmentioning

confidence: 99%

“…absolute or relative distance, angle, planarity, chiral volume, and geometric similarity). This article presents a series of new ADP restraints implemented in CRYSTALS [Parois, Cooper & Thompson (2015), Chem. Cent.…”

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An enhanced set of displacement parameter restraints in CRYSTALS

2018

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Crystallographic restraints are widely used during refinement of small-molecule and macromolecular crystal structures. They can be especially useful for introducing additional observations and information into structure refinements against low-quality or low-resolution data (e.g. data obtained at high pressure) or to retain physically meaningful parameter values in disordered or unstable refinements. However, despite the fact that the anisotropic displacement parameters (ADPs) often constitute more than half of the total model parameters determined in a structure analysis, there are relatively few useful restraints for them, examples being Hirshfeld rigid-bond restraints, direct equivalence of parameters and SHELXL RIGU-type restraints. Conversely, geometric parameters can be subject to a multitude of restraints (e.g. absolute or relative distance, angle, planarity, chiral volume, and geometric similarity). This article presents a series of new ADP restraints implemented in CRYSTALS [Parois, Cooper & Thompson (2015), Chem. Cent. J. 9, 30] to give more control over ADPs by restraining, in a variety of ways, the directions and magnitudes of the principal axes of the ellipsoids in locally defined coordinate systems. The use of these new ADPs results in more realistic models, as well as a better user experience, through restraints that are more efficient and faster to set up. The use of these restraints is recommended to preserve physically meaningful relationships between displacement parameters in a structural model for rigid bodies, rotationally disordered groups and low-completeness data.

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Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement

Midgley

Bourhis

Dolomanov

et al. 2021

Acta Crystallogr A Found Adv

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When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-spherical? The assumption that this particular term is very close to zero is generally made in non-spherical refinements (for example, implementations of Hirshfeld atom refinement or transferable aspherical atom models), unless the form factors are refinable parameters (for example multipole modelling). To evaluate this general approximation for one specific method, a numerical differentiation was implemented within the NoSpherA2 framework to calculate the derivatives of the structure factors in a Hirshfeld atom refinement directly as accurately as possible, thus bypassing the approximation altogether. Comparing wR 2 factors and atomic parameters, along with their uncertainties from the approximate and numerically differentiating refinements, it turns out that the impact of this approximation on the final crystallographic model is indeed negligible.

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Linear transformations of variance/covariance matrices

Cited by 4 publications

References 25 publications

Separating mixtures of aerosol types in airborne High Spectral Resolution Lidar data

Separating mixtures of aerosol types in airborne High Spectral Resolution Lidar data

An enhanced set of displacement parameter restraints in CRYSTALS

Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement

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