Time Series Models

Overview


Time series modeling relies on the techniques of statistical analysis, however, the points in a time series are generally not independent of each other. This puts certain limitations on the analysis, and leads to methods used to identify trends and seasonality.

The Standard Model


The standard approach to time series modeling is to find a function that transforms a time series into a white noise series.
{% X_1,X_2,...,X_n \rightarrow W_1, W_2,...,W_n %}
A white noise series is one in which

  • {% \mathbb{E}(W_i) = 0 %}
  • {% \mathbb{E}(W_i * W_i) = constant %}
  • {% \mathbb{E}(W_i * W_j) = 0 %}
(see Ljung-Box test for a test of white noise)


The transformation that accomplishes this goal contains all the predictable structure of the time series. In the context of information theory, a white noise series is such that there is no information relevant to the realization of the nth in the first (n-1) items.

Examples

  • Random Walk

Time Series Frameworks


  • Classical Decomposition : models a time series as the sum of a trend, seasonal factors, and a de-trended random component.
  • ARIMA :
  • Co-integration : modeling of multiple time series that share a common trend.

Additional Topics


  • Multivariable extends the basic ideas of time series analysis to cases where there are multiple variables present. Whereas a single variable time series can only use past values of the time series in analysis, a multi-variable time series can explore relationships among the variables.
  • Diagnostics - is the process of analyzing the results of the modeling process to determine the fit of the model to the available data. Is used to help pick the best model among a set of candidates, as well as to assess the quality of the selected model.
  • Forecasting
  • Simulating Time Series - uses Monte Carlo style techniques to generate random simulations a given time series, which can be used to simplify model analysis.
  • Spectral analysis of a time series is the method of applying Fourier series analysis to a time series. It is complicated by the notion of randomness in the signal, or likewise and signal and noise considerations.
  • Machine Learning