Question

What is stochastic process in time series analysis?

Answer 1

Stochastic process is a random process. Stochastic processes are often used in modelling in time series data. A time series is a discreet path of  a discrete time stochastic process.

A (discrete) time series [a (discrete) stochastic process] is a sequence of random numbers (or vectors) indexed by the integers:y₀,y₁,…

  y₁,y₂,…

  …,y_-1,y₀,y₁,…

The objective of time series analysis is to infer the characteristics of the stochastic process from a data sample and any additional information we have about the process.

In order to be able to use data to draw inferences about the stochastic process that generated these data, there has to be some characteristics of the stochastic process that remain stable over time: E.g., the mean, the variance …

A covariance stationary stochastic process is a stochastic process with the following properties –

      E(y_t) = ? for all t

      Var(y_t) = ?² for all t

      Cov(y_t,y_t-s) = ?_s for all t,s

A strictly stationary stochastic process is a stochastic process whose fidi’s are time invariant –

for all integers m,n,t₁,…,t_n and real numbers ?₁,…,?_n.

EXAMPLE:

A poisson process is the cannonical example of discreet state space stochastic process.

N(t),t [0,) ith rate if it is a counting process and N(0)=0.