Definition and examples of time series

Transcription

1 Definiion and examples of ime series A ime series is a sequence of daa poins being recorded a specific imes. Formally, le,,p be a probabiliy space, and T an index se. A real valued sochasic process is a real valued funcion X ( ) such ha for each fixed T, X ( ) is a random variable on,,p. When he index se T corresponds o ime indices (discree or coninuous), we will call X ( ) a ime series.

2 For fixed, X ( ) is a random variable. For fixed, X ( ) is a real valued funcion of ha is called a realizaion of he ime series. When we look a he plo of a recorded ime series, we are looking a one realizaion ou of he collecion (ensemble) of all possible realizaions. We ypically suppress he and wrie X. For a discree ime series, he se of imes T is a discree se, and he measuremens are ypically a successive imes spaced a uniform inervals. Coninuous ime series are obained when observaions are recorded over some ime inerval, e.g. 0,1.

3 Time series daa have a naural emporal ordering. This makes i disinc from common daa problems, where here is no naural ordering of he observaions, and from spaial daa analysis, where he observaions ypically relae o geographic locaions. A ime series model will generally reflec he fac ha observaions close ogeher in ime will be more closely relaed han observaions furher apar. In addiion, ime series models will ofen make use of he naural one-way ordering of ime so ha values for a given period will be expressed as being derived in some way from pas values, raher han fuure values.

4 Examples of ime series he daily closing value of he Dow Jones index, he annual flow volume of he River Nile a Aswan, daily air emperaure or monhly precipiaion in a specific locaion, he annual yield of corn in Iowa, he size of an organism, measured daily, annual U.S. populaion daa, daily closing sock prices, weekly ineres raes, naional income, sales figures, innumerable oher sequences based on indusrial, economic, and social phenomena, and sudies in medicine, geophysics, and engineering.

5 This series appears o be saionary in he mean as i varies abou a fixed level.

6 This series does no vary abou a fixed level bu insead exhibis an overall upward rend. Moreover, he variance of he series increases as he level of he series increases. The series is nonsaionary.

7 This quarerly series is repeiive in naure due o seasonal variaions. This is an example of a seasonal ime series. There also appears o be an upward rend.

8 This plo displays anoher phenomenon of nonsaionariy due o a change in srucure in he series from some exernal disurbance. Such exernal disurbances are ermed inervenions or ouliers.

9 Objecives of ime series analysis Time series analysis seeks o draw inferences from he daa. To do so, one ses up a probabiliy model o represen he daa. Afer he model parameers are esimaed and he fi of he model is esed, he model may be used in a variey of ways, depending on he paricular field of applicaion.

10 One use of a ime series model is o describe and explain general characerisics of he series. As an example, he series may be represened as a sum of componens represening rend (long-erm movemens), seasonaliy (periodic movemens due o seasonal variaion), and random flucuaions. An applicaion is an economic ime series such as unemploymen raes, where i is imporan o separae seasonal flucuaions from he long-erm rend. This process is known as seasonal adjusmen. Oher objecives of ime series analysis are o use he model o forecas fuure values, and o conrol he series by adjusing parameers.

11 Mehods of ime series analysis may be divided ino wo main classes: (a) frequency domain mehods (specral analysis o examine cyclic behavior which need no be relaed o seasonaliy) (b) ime domain mehods (auocorrelaion and cross-correlaion analysis o examine dependence over ime) Alhough he wo mehods are mahemaically equivalen in he sense ha he auocorrelaion funcion and he specrum funcion form a Fourier ransform pair, here are occasions when one approach is more advanageous han he oher. We consider ime domain mehods.