A time series, also denominated historical series, is a sequence of data obtained in regular intervals of time during a specific period. In the analysis of a time series, one first wants to model the study phenomenon and, from this, to describe the behaviour of the series, to make estimates, and, in the end, to evaluate the factors that may have influenced the behaviour of the series, with the objective of defining cause-effect relationships between two or more series. For this, there is a set of available statistical techniques which depend upon the defined model (or that estimated for the series), the type of the study series, and of the objective of the work. To analyse trends, it is possible to adjust polynomial regression models based on the whole series or on the neighbourhood of a specific point. This can also be done with mathematical functions. A seasonal phenomenon is defined as the one that occurs regularly in fixed periods of time and, if there is seasonality considered as deterministic in the series, one can use regression models which include functions like seno or cosseno to the variable time. In the analysis of the behaviour of a time series without trend and seasonality, the auto-regressive models (AR) or models which incorporate moving averages (ARMA) can be used. When trend is present, one can use auto-regressive models integrated with moving averages (ARIMA) and to incorporate the seasonality component the SARIMA models are used. The generalized linear models constitute another class of models. In this group of statistical models, the response variable is a counting process and the independent variables are those which are candidates to explain the behaviour of the series throughout the time. This class of models is indicated when the study variables do not follow the Normal distribution, mainly because they are counting processes. These models represent a group of probability distributions known as exponential family of distributions that incorporates many additive functions like the linear regression, Poisson, logistic, log-linear, etc. The generalized additive models are an extension of this class of models, in which each independent variable analysed does not enter in the model with its own value, but adopting a non parametric function in a non specific manner, which is estimated from smoothing curves.
Time series; Models, statistical; Trends; Seasonality