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基于李雅普诺夫的睡眠脑电动力学分析 核心算法程序。

GENERAL DESCRIPTION

This directory contains a MATLAB program to solve a linear inverse
problem using Minimum Relative Entropy Inversion. The code is described in
Neupauer R.M. and B. Borchers, A MATLAB Implementation of the Minimum
Relative Entropy Method for Linear Inverse Problems, submitted to
Computers & Geosciences.


LIST OF MATLAB FUNCTIONS

The following files are needed to use the MRE program:

 mre.m - main function

 bisection.m - performs the bisection method to calculate the values
		of the Lagrange multipliers, beta
 domre.m - mre subroutine that controls the main calculations
 donewton.m - performs newton's method to calculate the values of the
		Lagrange multipliers, lambda
 gensol.m - generates a random solution according to the posterior pdf
		of the model parameters
 getfk.m - calculates the value of F (Eq. 14)
 getjac.m - calculate the value of the Jacobian (Eq. 16)
 golden.m - performs golden section search in the damped newton method
 qinv.m - computes the inverse of the posterior cdf of the model parameters
 rscale.m - performs row scaling
 snfcn.m - expected value equation used to find values of the Lagrange
		multipliers, beta (Eq. 3)


HOW TO USE THE MRE PROGRAM

How to run mre.m for a general linear inverse problem

To solve a linear inverse problem using the MRE method, run the function
mre.m.  The syntax for running the function is:

[x,lambda,beta,a,p5,p95]=mre(G,d,upper,lower,expvalue,noise)

The INPUTS are:
G - G matrix of size N by M (G)
d - data vector of length N (d) 
upper - vector of upper bounds on the model parameters, length M (u)
lower - vector of lower bounds on the model parameters, length M (l)
expvalue - vector of expected values of the model parameters, length M (s)
noise - 2-element row vector of standard deviations of measurement error
	noise(1) = standard deviation of the additive error (epsilon_a)
	noise(2) = standard deviation of the multiplicative (epsilon_m)

The OUTPUTS are:
x = model solution vector of length M (mhat)
labmda = vector of Lagrange multpliers, lambda, length N
beta = vector of Lagrange mulipliers, beta, length M
a = vector a a's, length M (see Section 2)
p5 = vector containing the 5th percentile probability level for each
	element of the model solution vector, length M
p95 = vector containing the 95th percentile probability level for each
	element of the model solution vector, length M


The user may wish to adjust some parameters in the function 'mre.m'.
These parameters are defined on lines 51-59 of mre.m. The parameters are:
leftbegin = lower limit of the search region for the bisection method
rightbegin = upper limit of the search region for the bisection method
tolbeta = tolerance used in determining the Lagrange multipliers, beta
maxiter = maximum number of iterations used to determine the Lagrange
		multipliers beta
tollam = tolerance used in calculating the Lagrange multipliers lambda
lamiter = maximum number of iterations used to determine the Lagrange
		multipliers lambda
tolls = tolerance for golden section search
nearzero = value below which an asymptotic approximation is used for
	a parameter whose absolute value is much smaller than 1
large = value above which an asymptotic approximation is used for a 
	parameter whose value is much greater than 1; and also
	the negative of the value below which an asymptotic approximation 
	is used for a parameter whose value is much smaller than -1




HOW TO GENERATE A SINGLE REALIZATION FROM THE MRE OUTPUT

To generate a realization of the model solution based on the output from
'mre.m', use the function 'gensol.m'. The syntax for running the function is

soln=gensol(G,lower,upper,a,beta)

where G, lower, upper, a, and beta are output from 'mre.m' and are
defined above; and soln is the vector containing the realization of
the model solution.