WEBVTT
Kind: captions
Language: en
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Now we are moving into the fourth definition
that is auto correlation. Auto correlation
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function. The way we have defined the covariance
function. Now we are defining the auto correlation
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function. It is defined with the notation
R of s,t that is nothing but or we can write
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it in the terms of expectation. Expectation
of X of t, X of s minus expectation of X of
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t into expectation of X of s divided by the
square root of variance of X of t and the
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square root of variance of X of s. So the
numerator can be written covariance of X of
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a t,s divided by square root of variance of
X of t square root of variance of X of s.
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So this is going to be used with the notation
R of s,t and this is going to be auto correlation
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function for the random variable X of t and
X of s.
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So it's basically describes the correlation
between values of process at the different
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time points s and t. Sometimes we assume the
- we assume R of s,t depends only on absolute
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of t minus s. In the later case when you are
discussing the stationary process it is going
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to be depends only on the interval length
not the actual time. Therefore the R of s,t
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is going to be depend only on the length that
t minus s in absolute not the actual s and
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t. Therefore by assuming R of s,t is going
to be a only depends on t minus s we can have
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R of instead of for two variables I can use
the one variable as the R of tau that is nothing
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but the expectation of X of t minus mu multiplied
by X of t plus tau minus mu the expectation
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of that product divided by Sigma square.
So here I have made the one more assumption
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the m of t that is nothing but the expectation
of X of t that is going to be mu and the variance
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of X of t is going to be Sigma square. With
that assumption only the R of tau is going
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to be expectation of this product divided
by Sigma square where the variance of X of
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t is going to be not a function of t it is
a constant that is Sigma square and similarly
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the mean function expectation of X of t is
going to be mu that is also independent of
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t therefore I can simplify this R of s,t the
product expectation minus individual expectation
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that can be simplified as expectation of this
product.
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So basically this is evaluated at X of t and
X of t plus tau and that difference is going
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to be tau and this is also going to be even
function that means it has R of tau is same
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as R of minus tau and this auto correlation
function is used in time series analysis as
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well as a signal processing. In the signal
processing we assume that the signal the corresponding
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time series satisfying the stationary property
therefore the stationarity property implies
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the auto correlation function is going to
be depends only on the length of the interval
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not the actual time. Therefore, this R of
tau will be used in the signal processing
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as well as in general time series analysis
also.
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The fifth definition so we are covering the
different definitions which we want the fifth
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definition first we started with the mean
function. Second we started with the second
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order stochastic process. Then third we start
- third we have given the covariance function
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and the fourth we have given the auto correlation
function. Now we are giving the definition
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that is independent increments.
If for every t1 less than t2 less than tn
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the random variables X of t2 minus X of t1,
X of t3 minus X of t2 so on till X of tn minus
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X of tn minus 1 are mutually independent random
variables for all n then we say the corresponding
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stochastic process
is having independent increment property.
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So whenever you take a few t's t1,t2, tn and
the increments that is X of t2 minus X of
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t1 like that till X of tn minus Xtn minus
1 so these are all going to be the increment
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and each one is random variable therefore
the increment is also going to be a random
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variable and you have n such random variables
and suppose these n random variables are mutually
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independent random variable for all n. So
this is mixed for one n like that if you go
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for all n if this property is satisfied then
we can conclude the corresponding stochastic
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process having the property of independent
increments.
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So the independent increment that does not
imply some other properties but here what
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we are saying is the increment satisfies the
mutually independent property. That means
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if you find out the CDF of the joint CDF of
this random variable that is same as the product
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of the individual CDF that is property satisfied
by all the n then you can conclude that stochastic
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process has the independent increment.
The next property or the next definition is
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Ergodic property. What is the meaning of Ergodic
property? It says the time average of a function
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along a realization or sample exists almost
everywhere and is related to the space average.
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What it means? Whenever the system or the
stochastic process is Ergodic the time average
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is the same for all almost initial points
that is the process evolved for a longer time
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forgets its initial state. So statistical
sampling can be performed at one instant across
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a group of identical processes or sampled
over time on a single process with the no
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change in the measured result. We will discuss
the Ergodic property for the Markov process
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in detail later but this Ergodic property
is going to be very important when you study
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the Markov property or when you study the
stationarity property. Therefore these Ergodic
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properties always goes along with the stationarity
property or goes along with the Markov property
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therefore the stochastic property is going
to behave in a different way and that we are
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going to discuss later. The most important
stationary process that is a strict sense
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stationary process. First let me start with
the strict sense stationary process of order
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n. Then I will define the strict sense stationary
process for all order n or there is a strict
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sense stationary process itself. If for arbitrary
t1, t2 and so on tn the joint distribution
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of the random vector that is X of t1, X of
t2 and so on X of tn and the another random
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vector that is X of t1 plus h comma X of t2
plus h and so on X of tn plus h are the same
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for all h which is greater than 0 then we
say the stochastic process
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is a strict sense stationary of order n because
here we restricted with the n random variable.
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So we take a n random variable taken at the
points t1, t2, and tn and to find out the
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joint distribution of X of t1, X of t2 and
X of tn so you can find out what is a joint
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distribution of this n random variable. Also
you find the joint distribution of n random
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variable shifted by h that means earlier the
random variable X of t1 now you have a random
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variable X of t1 plus h with the same shift
h you do it with the t2. Therefore the random
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variable X of t2 plus h. Similarly the nth
random variable is X of tn earlier. Now you
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have a random variable X of tn plus h. So
you have another random vector with n random
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variables and find out the joint distribution
of that. If the joint distribution of this
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first n random variable as well as the joint
distribution of the shifted by h that random
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variable if both the distributions are same
that means they are identically distributed.
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The joint distributions are going to be identical
then you can conclude this stochastic process
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is a strict sense stochastic process of order
n because you use the n random variable.
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If this is going to be satisfied the above
property is going to be satisfied for all
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n then you can conclude the stochastic process
is going to be a strict sense stationary process
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for any integer n.
This is going to be a strict sense stationary
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process for any integer n. So we start to
cross checking the joint distribution of n
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random variable so if it is satisfying by
only with the maximum sum integer then it
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is going to be a strict sense stationary process
of order that n. If it is going to be satisfied
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for all n then for any integer n then it is
going to be call it as just strict sense stationary
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process.