Stochastic Process Lecture 10

 Stochastic Process 

Lecture 10

Stochastic Realization

A stochastic realization is a model based on state space that employs randomization to represent a stochastic process. The stochastic realization involves identifying a system, usually a collection of equations, that can produce the same statistical behaviors as the specified stochastic process.

OR

Each member of the ensemble is a possible realization of the stochastic process.

For each fixed s ϵ S, X(t) corresponding to a function defined on T is called the sample path or stochastic realization of the process. 

Stochastic Process and its Probability Distribution

A stochastic process is a family of random variables X(t), t ϵ T, s ϵ S, where s belongs to sample space and t belongs to an index set, and it must be characterized by a cdf F(s, t) or pdf f(s, t) because it has a probability density function at every instant of time. The random process is unpredictable in nature, but there are some statistical characteristics that can be used to analyze the random process.

The CDF of a stochastic process is defined as

F(x, t) = P(Xt ≤ x)

F(x1, x2,⋯, xn; t1, t2, ⋯, tn) = P[Xt1≤x1, Xt2 ≤x2, ⋯, Xtn ≤xn]

    The pdf of (Xt) can be obtained by differentiating w.r.t. t:

$$

f(x1, x2,⋯, xn; t1, t2, ⋯, tn)  = ∂n /∂t1, ∂t2,⋯ ∂tn P(Xt1≤x1, Xt2≤x2, ..., Xtn≤xn)

The random process is unpredictable in nature, but there are some statistical characteristics that can be used to analyze the random process. 

These characteristics are:

1. The mean function of a stochastic process is the average value of the process (in our example, viewers) at time t. 

i. If t is constant and s varies

μx =E(Xt) = ∫x f(s,t) dx

μx  = ∫x f(x) dx

ii. If s is constant and t varies

The average value of the process (in our example, viewers) at different points of time (in our example, the average number of viewers on a particular day) is called the ensemble function.

2. The variance of the stochastic process is defined as:

Var(Xt) = σ²t

3. The auto-covariance of the stochastic function is defined as

γs,t = Cov(Xt, Xs)

4. It measures the joint variability of two random variables with itself at a pair of time points, s, t (s < t).

Where:

The autocorrelation of the function is defined as

The autocorrelation between Xt and Xt+τ is called ensemble.

Where:

τ is an increment/decrement in time.

Properties:

i. The autocorrelation is symmetrical

ρx.x+τ = ρx.x-τ

ii. The maximum value of  is zero

ρx.x + τ = E (Xt Xt+0)

ρx.x+τ =E(X^2t)

The autocorrelation between X1(t) and X2(t) is

• Read More: Classification of Stochastic Process