of a Higherorder Markov Chain By Zhang Shixiao Supervisors Prof ChiKwong Li and Dr JorTing Chan Content 1 Introduction Background 2 Higherorder Markov Chain 3 Conclusion 1 Introduction ID: 675133
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Slide1
Stationary Probability Vectorof a Higher-order Markov Chain
By Zhang Shixiao
Supervisors: Prof. Chi-Kwong Li and Dr. Jor-Ting ChanSlide2
Content1. Introduction: Background
2.
Higher-order Markov
Chain
3. ConclusionSlide3
1. Introduction: BackgroundMatrices are widely used in both science and engineering.
In statistics
Stochastic process: flow direction of a particular system or process.
Stationary distribution: limiting
behavior of a stochastic
process.Slide4
Discrete Time-HomogeneousMarkov Chains
A
stochastic process with a discrete finite state space
S
A
unit sum vector
X
is said to be a
stationary probability distribution
of a finite Markov Chain if
P
X
=
X
where Slide5
Discrete Time-HomogeneousMarkov Chains
In other words
a
coutinuous
function
f
:
which preserves at least one fixed point. Slide6
2. Higher-order Markov Chain
a stochastic process with a sequence of random variables,
, which takes on a finite set
called the state set of the
process
Definition 2.1
Suppose the probability independent of time satisfying
Slide7
2. Higher-order Markov Chain
Definition 2.2
Write
to be a
three-order
n
-
dimensional
tensor where and define an n-dimensional column vector
Slide8
2. Higher-order Markov Chain
Example:
is
a
three
-order 2-dimensional
tensor where
and
Slide9
Conditions forInfinitely Many Solutions over the Simplex
Theorem 2.1
Now
we are
considering
where
all
Slide10
Conditions forInfinitely Many Solutions over the Simplex
Then one of the following holds
If
,
then we must have two solutions
or
to the above equation.
If , then we must have infinitely many solutions, namely, every
with
is a solution to the above equation.
Otherwise, we must have a unique solution.Slide11
Conditions forInfinitely Many Solutions over the Simplex
Then we want to extend the condition for infinitely many solutions for
case
Slide12
Main Theorem 2.2
would have infinitely many solutions over the whole
set
if and only if
Slide13
Main Theorem 2.2Slide14
Main Theorem 2.2Slide15
Main Theorem 2.2Slide16
Main Theorem 2.2Proof:Sufficiency:
For
, infinitely
many solutions
Slide17
Main Theorem 2.2
Slide18
Main Theorem 2.2Slide19
Main Theorem 2.2Slide20
OtherGiven any two solutions lying on the interior of1-dimensional face of the boundary of the simplex, then the whole 1-dimensional face must be a set of collection of solutions to the above
equation.
Conjecture:
given any
k+1
solutions lying in the interior of the
k
-dimensional face of the simplex, then any point lying in the whole
k-dimensional face, including the vertexes and boundaries, will be a solution to the equation.Slide21
3. ConclusionSlide22
Thank you!