BUS606: Operations and Supply Chain Management

A time series consists of a set of observations ordered in time, on a given phenomenon (target variable). Usually the measurements are equally spaced, e.g. by year, quarter, month, week, day. The most important property of a time series is that the ordered observations are dependent through time, and the nature of this dependence is of interest in itself. Examples of time series are the gross national product, the unemployment rate, or the daily closing value of the Dow Jones index. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data.

Time series data have a natural temporal ordering. This makes time series analysis distinct from other common data analysis problems, in which there is no natural ordering of the observations (e.g. explaining people's income relative to their education level, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. house prices). A time series model will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values.

Formally, a time series is defined as a set of random variables indexed in time, $\left\{X_{t}, \ldots, X_{T}\right\}$. In this regard, an observed time series is denoted by $\left\{X_{t}, \ldots, X_{T}\right\}$, where the sub-index indicates the time to which the observation $x_{t}$ pertains. The first observed value $x_{t}$ can be interpreted as the realization of the random variable $x_{t}$, which can also be written as $X(t=1, \omega)$ where $\omega$ denotes the event belonging to the sample space. Similarly, $x_{2}$ is the realization of $x_{2}$, and so on. The T-dimensional vector of random variable can be characterized by different probability distribution.

For socio-economic time series the probability space is continuous, and the time measurements are discrete. The frequency of measurements is said to be high when it is daily, weekly or monthly and to be low when the observations are quarterly or yearly.