# Time Series Analysis

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Introduction A time series is the vector of realizations of a random variable ${\displaystyle X}$ over the time. Graphical representation Scatterplots show the development of the realizations of the underlying random variable over time. The horizontal axis represents the time ${\displaystyle t}$ (days, months, years) while the vertical axis shows the corresponding value ${\displaystyle x_{t}}$ of ${\displaystyle X}$. Example: Here are some examples from various fields of interest:

•  Price index for foreign services Berlin, 1st quarter 1977 to 4th quarter 1989 (1977:1-1989:4)

•  Number of phones in the US (measured in ${\displaystyle 1,000s}$) 1900 - 1970

•  Number of newly registered cars in Berlin 1977:1 - 1989:4

•  Number of cinema visitors 1977:1 - 1989:4

•  Fuel prices in Berlin 1977:1 - 1989:4

## The objectives of time series analysis

The above examples illustrate how different the behavior of a time dependent random variable can be. The understanding of these different temporal attributes in any application is the aim of time series analysis. Descriptive time series models are chosen so that they explain the characteristics of the series. A time series could be interpreted as the realization of a stochastic process hence one tries to find a stochastic model that could have generated the observed data. An important issue is the identification of influence factors, which may be time series them self. Stochastic time series modelling can help to understand such observed process. Also, assuming that the model remains valid in the future, it is possible forecast future observations (predictive times series models). In the following we consider descriptive time series models only.

## Components of time series

Time series are decomposed into its underlying driving components to show its characteristics:

• TrendGeneral long-run trend of the series.
• Periodic variationShort-run influences, which overlap the long-run development corresponding to a rigid model. If the period is one year, periodic variations are called seasonal variations.
• Irregular variation

Trend and periodic (seasonal) variation are the systematic components.