SC740 SEMINAR REPORT 14 for Frederick L. Jones

PRESENTERS: Dr. Jiu Ding and Dr. Lawrence Mead

TOPIC: Maximum Entropy Method: Ideas and Applications

 

OVERVIEW

Noting the importance of Maximum Entropy Method to scientific research in such fields as Thermal Physics and Statistical Physics, Dr. Lawrence Mead provided a general mathematical introduction to the Maximum Entropy function, before Dr. Jiu Ding provided an overview of the Maximum Entropy Method.

The Maximum Entropy Method is a useful numerical scheme for constructing the unknown density function, and it can also be used to compute invariant measures of chaotic dynamics.

 

AN INTRODUCTION TO MAXIMUM ENTROPY

Dr. Lawrence Mead provided a general mathematical introduction to the Maximum Entropy function. Dr. Mead noted that in order to construct a maximum entropy function, one must reconstruct a density function p(x) from limited information. The density function p(x) defines power moments as follows :

ó 1

m n = ô x n p(x) dx

õ

The least biased or most likely function p(x) is the one that maximizes the following function:

ó 1 N ó 1

S = - ô p(x) [ ln p(x) - 1] dx + å l n [ m n - ô x M p(x) dx ]

õ 0 M=1 õ 0

 

 

THE MAXIMUM ENTROPY METHOD

Dr. Jiu Ding provided an overview of the Maximum Entropy Method. The Maximum Entropy Method is a useful numerical scheme for constructing the unknown density function, and it can also be used to compute invariant measures of chaotic dynamics.

Dr. Ding covered entropy transforms such as the Birkhoff (1933) I.E.T., i.e,

 

1 N-1 ó _______

lim --- å xI (Sn(x)) = ô ( 1 / p Ö p (1 - x) ) dx

N® ¥ N n=0 õ I

and entropy operators such as the Probenius-Penon operator, i.e,

 

ó ó

ô P f(x) dx = ô f(x) dx

õ A õ S-1 (A)

 

SUMMARY AND CONCLUSIONS

Noting the importance of Maximum Entropy Method to scientific research in such fields as Thermal Physics and Statistical Physics, Dr. Lawrence Mead provided a general mathematical introduction to the Maximum Entropy function, before Dr. Jiu Ding provided an overview of the Maximum Entropy Method.

The Maximum Entropy Method is a useful numerical scheme for constructing the unknown density function, and it can also be used to compute invariant measures of chaotic dynamics.