The geometry of random drift IV. Random time substitutions and stationary densities

1978 ◽  
Vol 10 (3) ◽  
pp. 563-569 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Distance Measure ◽  
Spherical Geometry ◽  
Edge Length ◽  
Random Time ◽  
Random Drift ◽  
Multiple Allele ◽  
Original Process ◽  
Frequency Space ◽  
Gradient Type ◽  
Time Substitution

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.

The geometry of random drift IV. Random time substitutions and stationary densities

1978 ◽  
Vol 10 (03) ◽  
pp. 563-569 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Distance Measure ◽  
Spherical Geometry ◽  
Edge Length ◽  
Random Time ◽  
Random Drift ◽  
Multiple Allele ◽  
Original Process ◽  
Frequency Space ◽  
Gradient Type ◽  
Time Substitution

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.

The geometry of random drift I. Stochastic distance and diffusion

1977 ◽  
Vol 9 (02) ◽  
pp. 238-249 ◽  
Author(s):  
Peter L. Antonelli ◽  
Curtis Strobeck
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Diffusion Process ◽  
Covariance Matrix ◽  
Distance Measure ◽  
Multinomial Distribution ◽  
Random Drift ◽  
Frequency Space ◽  
Simple Part ◽  
The Matrix

A stochastic distance measure is defined for a general diffusion process on a parameter space X. This distance is defined by where (gij ) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij ) is the fundamental tensor of the Riemannian space (X, gij ), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift with n alleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher to n alleles. The geometry associated with the diffusion equation for random drift with n alleles is that of a part of an (n − 1)-sphere of radius two. We also show that the diffusion equation for random drift is not spherical Brownian motion, although it is approximated by it near the centroid of frequency space.

The geometry of random drift I. Stochastic distance and diffusion

1977 ◽  
Vol 9 (2) ◽  
pp. 238-249 ◽  
Author(s):  
Peter L. Antonelli ◽  
Curtis Strobeck
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Diffusion Process ◽  
Covariance Matrix ◽  
Distance Measure ◽  
Multinomial Distribution ◽  
Random Drift ◽  
Frequency Space ◽  
The Matrix ◽  
Angular Transformation

A stochastic distance measure is defined for a general diffusion process on a parameter space X. This distance is defined by where (gij) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij) is the fundamental tensor of the Riemannian space (X, gij), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift with n alleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher to n alleles. The geometry associated with the diffusion equation for random drift with n alleles is that of a part of an (n − 1)-sphere of radius two. We also show that the diffusion equation for random drift is not spherical Brownian motion, although it is approximated by it near the centroid of frequency space.

The geometry of random drift II. The symmetry of random genetic drift

1977 ◽  
Vol 9 (2) ◽  
pp. 250-259 ◽  
Author(s):  
Peter L. Antonelli ◽  
Jared Chapin ◽  
G. Mark Lathrop ◽  
Kenneth Morgan
Keyword(s):  
Velocity Field ◽  
Genetic Drift ◽  
Gene Frequency ◽  
Radial Symmetry ◽  
Divergence Times ◽  
Random Genetic Drift ◽  
Initial State ◽  
Random Drift ◽  
Arithmetic Average ◽  
Frequency Space

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher–Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

The geometry of random drift II. The symmetry of random genetic drift

1977 ◽  
Vol 9 (02) ◽  
pp. 250-259 ◽  
Author(s):  
Peter L. Antonelli ◽  
Jared Chapin ◽  
G. Mark Lathrop ◽  
Kenneth Morgan
Keyword(s):  
Velocity Field ◽  
Genetic Drift ◽  
Gene Frequency ◽  
Radial Symmetry ◽  
Divergence Times ◽  
Random Genetic Drift ◽  
Initial State ◽  
Random Drift ◽  
Arithmetic Average ◽  
Frequency Space

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher–Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

Determination of the Best Emulsion for the Microscopy of Crystals

1981 ◽  
Vol 39 ◽  
pp. 32-33
Author(s):  
David A. Grano ◽  
Kenneth H. Downing
Keyword(s):  
Spatial Frequency ◽  
Information Transfer ◽  
Signal To Noise Ratio ◽  
Experimental Situation ◽  
Photographic Emulsion ◽  
Modulation Transfer ◽  
Signal To Noise ◽  
Frequency Space ◽  
Noise Ratio

The retrieval of high-resolution information from images of biological crystals depends, in part, on the use of the correct photographic emulsion. We have been investigating the information transfer properties of twelve emulsions with a view toward 1) characterizing the emulsions by a few, measurable quantities, and 2) identifying the “best” emulsion of those we have studied for use in any given experimental situation. Because our interests lie in the examination of crystalline specimens, we've chosen to evaluate an emulsion's signal-to-noise ratio (SNR) as a function of spatial frequency and use this as our critereon for determining the best emulsion.The signal-to-noise ratio in frequency space depends on several factors. First, the signal depends on the speed of the emulsion and its modulation transfer function (MTF). By procedures outlined in, MTF's have been found for all the emulsions tested and can be fit by an analytic expression 1/(1+(S/S0)2). Figure 1 shows the experimental data and fitted curve for an emulsion with a better than average MTF. A single parameter, the spatial frequency at which the transfer falls to 50% (S0), characterizes this curve.

Feature selection of the armature winding broken coils in synchronous motor using genetic algorithm and mahalanobis distance

2012 ◽  
Vol 57 (3) ◽  
pp. 829-835 ◽  
Author(s):  
Z. Głowacz ◽  
J. Kozik
Keyword(s):  
Genetic Algorithm ◽  
Feature Selection ◽  
Mahalanobis Distance ◽  
Distance Measure ◽  
Synchronous Motor ◽  
Medical Diagnostics ◽  
Motor Current ◽  
Feature Spaces ◽  
Multidimensional Feature Spaces ◽  
Selection Of

The paper describes a procedure for automatic selection of symptoms accompanying the break in the synchronous motor armature winding coils. This procedure, called the feature selection, leads to choosing from a full set of features describing the problem, such a subset that would allow the best distinguishing between healthy and damaged states. As the features the spectra components amplitudes of the motor current signals were used. The full spectra of current signals are considered as the multidimensional feature spaces and their subspaces are tested. Particular subspaces are chosen with the aid of genetic algorithm and their goodness is tested using Mahalanobis distance measure. The algorithm searches for such a subspaces for which this distance is the greatest. The algorithm is very efficient and, as it was confirmed by research, leads to good results. The proposed technique is successfully applied in many other fields of science and technology, including medical diagnostics.

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