Maximum Entropy Fitting package
ilavsky
This is Maximum Entropy Package for solving problems which can be written as linear equation: I = G f, where I is measured signal, G is response matrix and f is a model distribution. It has been used for interpretation of a size distribution from small-angle scattering data, which involves the inversion of an integral equation for which there is no exact solution. It can likely be used for number of similar problems.
Expressing the size distribution as a histogram, f, it is possible to rewrite the scattering equation as a linear equation.
I = G f
The matrix component, G, describes the assumed morphology of the scatterers underlying the measured data, I. Solution of the linear scattering equation by a direct matrix inversion is not unique due to the high condition number. In short, their MaxEnt method maximizes the configurational entropy of the histogram subject to the scattering calculated from that histogram fitting the measured data to within the experimental errors. These two constraints are imposed simultaneously through the use of a Lagrange multiplier.
The maximum entropy method was developed by Jennifer Potton, et al. (Potton, J. A., G. J. Daniell, et al. (1988). "Particle-Size Distributions from Sans Data Using the Maximum-Entropy Method." Journal of Applied Crystallography 21: 663-668.; Potton, J. A., G. J. Daniell, et al. (1988). "A New Method for the Determination of Particle-Size Distributions from Small-Angle Neutron-Scattering Measurements." Journal of Applied Crystallography 21: 891-897.), and further advanced by one of the authors (Jemian, P. R., J. R. Weertman, et al. (1991). "Characterization of 9cr-1movnb Steel by Anomalous Small-Angle X-Ray-Scattering." Acta Metallurgica Et Materialia 39(11): 2477-2487.). It relies on a maximum entropy engine of Skilling and Bryan (Bryan, R. K. and J. Skilling (1980). "Deconvolution by Maximum-Entropy, as Illustrated by Application to the Jet of M87." Monthly Notices of the Royal Astronomical Society 191(1): 69; Bryan, R. K. and J. Skilling (1986). "Maximum-Entropy Image-Reconstruction from Phaseless Fourier Data." Optica Acta 33(3): 287-299.; Skilling, J. and R. K. Bryan (1984). "Maximum-Entropy Image-Reconstruction - General Algorithm." Monthly Notices of the Royal Astronomical Society 211(1): 111).
Note: this code has been converted and optimized by Pete Jemian from original Basic code by J. Potton to Fortran and then C. Later I have converted this code to Igor code. As such it is very clumsy and difficult to read. It is my hope that this code will be soon replaced by much cleaner code.
Instructions to use are in comments at the top of the procedure file.
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Expressing the size distribution as a histogram, f, it is possible to rewrite the scattering equation as a linear equation.
I = G f
The matrix component, G, describes the assumed morphology of the scatterers underlying the measured data, I. Solution of the linear scattering equation by a direct matrix inversion is not unique due to the high condition number. In short, their MaxEnt method maximizes the configurational entropy of the histogram subject to the scattering calculated from that histogram fitting the measured data to within the experimental errors. These two constraints are imposed simultaneously through the use of a Lagrange multiplier.
The maximum entropy method was developed by Jennifer Potton, et al. (Potton, J. A., G. J. Daniell, et al. (1988). "Particle-Size Distributions from Sans Data Using the Maximum-Entropy Method." Journal of Applied Crystallography 21: 663-668.; Potton, J. A., G. J. Daniell, et al. (1988). "A New Method for the Determination of Particle-Size Distributions from Small-Angle Neutron-Scattering Measurements." Journal of Applied Crystallography 21: 891-897.), and further advanced by one of the authors (Jemian, P. R., J. R. Weertman, et al. (1991). "Characterization of 9cr-1movnb Steel by Anomalous Small-Angle X-Ray-Scattering." Acta Metallurgica Et Materialia 39(11): 2477-2487.). It relies on a maximum entropy engine of Skilling and Bryan (Bryan, R. K. and J. Skilling (1980). "Deconvolution by Maximum-Entropy, as Illustrated by Application to the Jet of M87." Monthly Notices of the Royal Astronomical Society 191(1): 69; Bryan, R. K. and J. Skilling (1986). "Maximum-Entropy Image-Reconstruction from Phaseless Fourier Data." Optica Acta 33(3): 287-299.; Skilling, J. and R. K. Bryan (1984). "Maximum-Entropy Image-Reconstruction - General Algorithm." Monthly Notices of the Royal Astronomical Society 211(1): 111).
Note: this code has been converted and optimized by Pete Jemian from original Basic code by J. Potton to Fortran and then C. Later I have converted this code to Igor code. As such it is very clumsy and difficult to read. It is my hope that this code will be soon replaced by much cleaner code.
Instructions to use are in comments at the top of the procedure file.
Project Details
Current Project Release
Maximum Entropy Fitting package IGOR.6.02.x-1.0
| Release File: | Maximum Entropy v1.00.zip (14.15 KB) |
| Version: | IGOR.6.02.x-1.0 |
| Version Date: | |
| Version Major: | 1 |
| Version Patch Level: | 0 |
| OS Compatibility: | Mac-Intel Windows |
| Release Notes: | Original release of Maximum Entropy package |
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Hello sir,
Can we use the MEM to do the integral inversion of the following function:
F(k)=∫p(x)g(k,x)dx, limit(0,∞)
Suppose g(k,x)=2*pi^2*(c)^4*exp(-2*x^2*pi^2*(c)^2)*(1+Besselj(0,2*pi*x*d))
. We need to find p(x). How can we do it?
Thanks: Rabindra
December 30, 2018 at 02:45 pm - Permalink