Station arty in Time Series

B Head, D Pawar
Our aim here is to illustrate a few properties of stationary time series with supporting real life examples. Concepts of auto covariance and auto correlation are shown to be useful which can be easily introduced and conclusions have been drawn by testing methodology of hypothesis. In this article we have used production of wheat data of 1970 to 2002 at five locations in Marathwada region to illustrate most of properties theoretically established. Objectives of the Study 1. To develop theory of
more » ... develop theory of time series. 2. To develop algorithms for analyzing time series, which use the characterizing theorems. 3. To interpret the results of characterizations, in real economic and social terms. Basic Concepts Basic definitions and few properties of stationary time series are given in this section. Definition 2.1 A Time Series Let (Ω, С, Р) be a probability space let T be an index set. A real valued time series is a real valued function X(t ,ω) defined on T x Ω such that for each fixed t Є T, X(t, ω) is a random variable on (Ω, С, Р). The function X(t, ω) is written as X(ω) or X t and a time series considered as a collection {X t : t Є T}, of random variables [8]. Definition 2.2 Stationary Time Series A process whose probability structure does not change with time is called stationary. Broadly speaking a time series is said to be stationary, if there is no systematic change in mean i.e. no trend and there is no systematic change in variance. Definition 2.3 Strictly Stationary Time Series A time series is called strictly stationary, if their joint distribution function satisfy. F Xt1 X t2 ... X t n (X t1 X t2 ... X t n) = F X t1+ h X t2 + h ... X t n + h (X t1 X 2t ... X tn)... (1) Where, the equality must hold for all possible sets of indices ti and (ti + h) in the index set. Further the joint distribution depends only on the distance h between the elements in the index set and not on their actual values. Theorem 2.1 If {X t: t Є T}, is strictly stationary with E{ X t } < α and E{X t-μ } < β then , E(X t) = E(X t + h), for all t, h } ....... (2) E [(X t1-μ)(X t2-μ)] = E [(X t1 + h-μ)(X t2 + h-μ)] , for all t1 , t2, h Proof Proof follows from definition (2.3). In usual cases above equation (2) is used to determine that a time series is stationary i.e. there is no trend. Abstract In this paper, a time series {X(t, ), t Є T} on ( , C, P) is explained. Where X is a random variable (r. v.).The properties of stationary time series with supporting real life examples have been taken.Production of wheat series for 33 years from five districts of Marathwada region in Maharastra State were analyzed. A preliminary discussion of properties of time series precedes the actual application to production of wheat data.