Information Theory and Predictability
Lecture 8: Statistical Predictability
1
CONCEPTUAL AND PRACTICAL BACKGROUND
In the following lectures we shall develop and explore a general theoretical framework derived from information theory to study practical dyn
Information Theory and Predictability.
Lecture 3: Stochastic Processes
1. Introduction
The major focus later in this course will be on statistical predictability of dynamical systems. In that context the primary issue we will consider is the evolution
of
Information Theory and Predictability
Lecture 6: Maximum Entropy Techniques
1
Philosophy
Often with random variables of high dimensional systems it is dicult to deduce the appropriate probability distribution from the given observations of
outcomes. We sh
Lecture 9: Theoretical approaches to
predictability
March 30, 2012
1
Small error case
The traditional theoretical approach to predictability pioneered in the late 1960s
and 1970s focusses on the growth of inntesimal errors within a dynamical system. As we
Information Theory and Predictability
Lecture 11: Information ow
1
Introduction
Within a general random dynamical system uncertainty can be thought of as
owing from one location, dimension or subspace to another. A good example of
such a ow is provided by
Information Theory and Predictability
Lecture 7: Gaussian Case
1
Gaussian distribution
The general multivariate form of this density for a random
x
n dimensional vector
can be written as
n
1/2
p(x) = [(2 ) det( )]
where
is the
nn
1
t
exp (x x) 1 (x x)
2
c
Information Theory and Predictability.
Lecture 2: Important functionals and their
properties
1. Elementary Properties of Entropy
We can write H (S ) as
H (S ) =
p(s) log
sA
1
p(s)
and since by the denition of a probability function 0 p(s) 1 this sum is
te
Information Theory and Predictability
Lecture 10: Predictability of some simple
models
1
Gaussian case
As was mentioned in Lecture 7 the case where probability distributions are
Gaussian is both practically relevant since many complex system distributions
Information Theory and Predictability
Lecture 5: Dierential entropy and continuous
outcome random variables
CONTINUOUS LIMIT
1.
Above we consider random variables with countable outcomes.
This may be
generalized to the case where outcomes are from a conti
Information Theory and Predictability.
Lecture 4: Optimal Codes
1. Introduction
We discussed codes briey in Lecture 1. We now turn to the subject in detail.
Denition 1. An encoding c(x) for a random variable X is a mapping from the
set of outcomes of the
Information Theory and Predictability.
Lecture 1: Introduction
1. Historical Background
The concept of entropy originated in thermodynamics in the 19th century where
it was intimately related to heat ow and central to the second law of the subject.
Later