## Time series analysis

## What is a time series analysis?

A time series analysis deals with the inferential statistical analysis of corresponding time series and with the prediction of trends on their future development. Time series analysis is a special form of regression analysis. The time series is a chronologically ordered sequence of numbers or of observations in which the arrangement of characteristic values results compellingly from the time sequence under consideration (thus share prices and stock exchange prices can be considered in general, as well as population development, the election intention surveys, weather data, interest and price index). Individual points in time are combined into a set of observation points in time T and exactly one observation is used for each point in time t out of T. The observation points in time T are then combined into a set of observation points in time.

Time series occur in all fields of science. In time series, the data are not continuous but discrete and exist at finite time intervals. A time series can be obtained by sampling. Time points can be equidistantly spaced at constant intervals (such as every 5 seconds) or at other regularities (such as weekdays) or irregularly spaced. A single data point can consist of a single number (scalar value) or a multitude of numerical values (tuple). However, all data points must be composed of certain individual values in the same way.

Typically, time series arise from the interaction of regular and random causes. Regular causes can vary periodically and, on the other hand, contain long-term trends. The random influences can often be described as "noise".

## In which areas are there time series analyses

In financial mathematics and in financial economics there are time series, such as for stock market prices and liquidity developments. Such series also exist in econometrics (for gross national product and the unemployment rate). Such series developments also occur in biometrics (for the EEG). There are also such time series analyses in meteorology, in remote sensing (vegetation development and aspect sequence) and in polemology (quantitative peace research).

## What are the components of time series analysis

In time series, there are often typical patterns in the series values. These are due to certain influencing factors that have the same effect. The patterns can be divided into certain components and the time series can be broken down into such components. There are trend components, seasonal components, cyclical components and residual components. Which component is to be considered depends directly on the corresponding type of time series.

## Time series analysis models

With the modern methods of time series analysis, even the future can be predicted with the help of observations from the past. This is possible with consumption figures, share prices or temperatures. There are different models that can be applied:

### AR models:

In general, autoregressive models are models that take into account serial correlation of corresponding observations. Future observations can be predicted and estimated. AR models are the simplest form of time series analyses.

### ARIMA models:

The ARIMA models are extensions of AR models, the so-called autoregressive-integrated-moving-average models. Not only the past observations of the time series are taken into account, but also the unobserved errors of this time series. Temporal dependencies are described that only briefly influence a time series.

### GARCH models:

GARCH models are general-autoregressive-conditional-heteroscedasticity models. Variations are estimated over time in this model. The variance of the observations is in the foreground. For example, the volatility of share prices can be predicted.