MCQMC Methods for Multivariate Statistical Distributions

This paper deals with the application of certain classes of fractional calculus operators in statistical distributions. The images of product combinations of special functions under the calculus of operators are applied to certain general ized forms of univariate and multivariate statistical distributions. Further results giving the expectations, cummulative functions and characteristic functions of such special function distributions are also obtained.

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Long-term variations in individual particle types in the aerosol of Phoenix, Arizona, as determined using automated electron microprobe analysis and multivariate statistical techniques

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149544 ◽

1993 ◽

Vol 51 ◽

pp. 742-743

Author(s):

Karen A. Katrinak ◽

James R. Anderson ◽

Peter R. Buseck

Keyword(s):

Electron Microprobe ◽

Fine Fraction ◽

Coarse Fraction ◽

Motor Vehicles ◽

Multivariate Statistical Methods ◽

Multivariate Statistical ◽

X Ray ◽

Mineral Particles ◽

Anthropogenic Air Pollutants

Aerosol samples were collected in Phoenix, Arizona on eleven dates between July 1989 and April 1990. Elemental compositions were determined for approximately 1000 particles per sample using an electron microprobe with an energy-dispersive x-ray spectrometer. Fine-fraction samples (particle cut size of 1 to 2 μm) were analyzed for each date; coarse-fraction samples were also analyzed for four of the dates.The data were reduced using multivariate statistical methods. Cluster analysis was first used to define 35 particle types. 81% of all fine-fraction particles and 84% of the coarse-fraction particles were assigned to these types, which include mineral, metal-rich, sulfur-rich, and salt categories. "Zero-count" particles, consisting entirely of elements lighter than Na, constitute an additional category and dominate the fine fraction, reflecting the importance of anthropogenic air pollutants such as those emitted by motor vehicles. Si- and Ca-rich mineral particles dominate the coarse fraction and are also numerous in the fine fraction.

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Angular Calibration of Ribosomal Subuntts in Factor Space

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100181063 ◽

1990 ◽

Vol 48 (1) ◽

pp. 462-463

Author(s):

Minakhi Pujari ◽

Joachim Frank

Keyword(s):

Three Dimensional ◽

Carbon Layer ◽

Uranyl Acetate ◽

Particle Analysis ◽

Factor Space ◽

Multivariate Statistical ◽

Ribosomal Subunits ◽

Layer Method ◽

Dimensional Reconstruction ◽

Number Of Particles

In single-particle analysis of macromolecule images with the electron microscope, variations of projections are often observed that can be attributed to the changes of the particle’s orientation on the specimen grid (“rocking”). In the multivariate statistical analysis (MSA) of such projections, a single factor is often found that expresses a large portion of these variations. Successful angle calibration of this “rocking factor” would mean that correct angles can be assigned to a large number of particles, thus facilitating three-dimensional reconstruction.In a study to explore angle calibration in factor space, we used 40S ribosomal subunits, which are known to rock around an axis approximately coincident with their long axis. We analyzed micrographs of a field of these particles, taken with 20° tilt and without tilt, using the standard methods of alignment and MSA. The specimen was prepared with the double carbon-layer method, using uranyl acetate for negative staining. In the MSA analysis, the untilted-particle projections were used as active, the tilted-particle projections as inactive objects. Upon tilting, those particles whose rocking axes are parallel to the tilt axis will change their appearance in the same way as under the influence of rocking. Therefore, each vector, in factor space, joining a tilted and untilted projection of the same particle can be regarded as a local 20-degree calibration bar.

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Molecular Views of Ice-Embedded Lumbricus Terrestris Erythrocruorin Obtained By Invariant Classification

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100181002 ◽

1990 ◽

Vol 48 (1) ◽

pp. 450-451

Author(s):

Michael schatz ◽

Joachim Jäger ◽

Marin van Heel

Keyword(s):

Image Interpretation ◽

Lumbricus Terrestris ◽

Reference Image ◽

Large Set ◽

Multivariate Statistical ◽

Data Set ◽

Low Contrast ◽

Negative Stain ◽

Reference Images ◽

Statistical Resolution

Lumbricus terrestris erythrocruorin is a giant oxygen-transporting macromolecule in the blood of the common earth worm (worm "hemoglobin"). In our current study, we use specimens (kindly provided by Drs W.E. Royer and W.A. Hendrickson) embedded in vitreous ice (1) to avoid artefacts encountered with the negative stain preparation technigue used in previous studies (2-4).Although the molecular structure is well preserved in vitreous ice, the low contrast and high noise level in the micrographs represent a serious problem in image interpretation. Moreover, the molecules can exhibit many different orientations relative to the object plane of the microscope in this type of preparation. Existing techniques of analysis requiring alignment of the molecular views relative to one or more reference images often thus yield unsatisfactory results.We use a new method in which first rotation-, translation- and mirror invariant functions (5) are derived from the large set of input images, which functions are subsequently classified automatically using multivariate statistical techniques (6). The different molecular views in the data set can therewith be found unbiasedly (5). Within each class, all images are aligned relative to that member of the class which contributes least to the classes′ internal variance (6). This reference image is thus the most typical member of the class. Finally the aligned images from each class are averaged resulting in molecular views with enhanced statistical resolution.

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