WikiMath

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Analysis or prediction

The study of time series consists in the study of time-stamped series of observations of quantities of interest:

• t-n: "prior" or "lag time",
• t: "current time" or "point of reference".
• t+n: "future" or "forecast time".

There are two main types of time series studies:

• prediction (time series forecasting): predicting the future evolution of the time series from the knowledge of the past.
• analysis (time series analysis): we want to understand the data set: classification or clustering, anomaly detection...

This understanding can eventually help in preconditioning, but not necessarily. It is often technically complex, requiring assumptions and a modeling effort, and frequently involves a decomposition of the original signal into its various components, such as

• the height, the level: the base value for the series if it were a horizontal line (the mean, the constant in a constant regression...)
• the trend: the increase or decrease, often linear, of the series over time, i.e. the slope in a linear regression,
• seasonality: cyclical or repeated behavior that occurs over time,
• noise: variability in the observations that cannot be explained by the model,

the last three components being optional.

What to predict?

We can try to predict the value of the observation during the next time interval (t+1, "one-step forecasting"), or for a larger time interval ("multi-step forecasting").

One can also try to predict only one observation, such as temperature ("univariate time series"), or be interested in the prediction of several quantities in parallel ("multivariate time series"). Univariate time series are generally the only ones studied, the multivariate approach being generally much more delicate to deal with.

The sliding window, prediction by machine learning

The sliding window (or lag method) allows to reformulate the prediction for time series into a machine learning problem.

To do so, it is sufficient to use one (or more) previous time step(s) as input variables, and the next time step as output variable. The number of previous time steps is called the window width or the lag size.