Time series analysis

Time series analysis is process of decomposition of time series data into several components which we conduct analysis and conclude based on. There are two broad objectives for analyses:

• understanding and modeling stochastic process for the data.
• forecasting.

Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test theories that the current values of one or more independent time series affect the current value of another time series, this type of analysis of time series is not called “time series analysis”, which focuses on comparing values of a single time series or multiple dependent time series at different points in time. [^Imdadullah. “Time Series Analysis”.Basic Statistics and Data Analysis. Retrieve. 2 January 2014]

There are also several typical components in time series: trend, season, and cycle. Those components are easily modeled but we have to ensure that our data has a sufficiently good quality without autocorrelation between the realizations of the time series. Unfortunately, autocorrelation is so common in financial world.

The decomposition can fundamentally be conducted by using additive decomposition as follows:

\[\text{Time series} = \text{Trend factor} + \text{Season factor} + \text{Error term}\]

Adjustment for the model can be necessarily performed, depending on how the data are shown in plots. For example, if we observe that the season factor increases with the trend factor, we can adjust our framework into \(\text{trend factor} \times \text{season factor}\). This is so-called multiplicative decomposition.

Methods for time series analysis

Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include autocorrelation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.

Additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary process has a certain structure which can be described using a small number of parameters (for example, using an \(AR\) or \(MA\) model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.

Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate.