Time Series Data and Statistical Forecasting Mothods

Time Series Data

Time series data comes from a variety of data generating processes. There are also different formulations and views of time series data.

Time series data can be formulated as a sequence of vector functions of time ¹. There are many different types of tasks on time series data, for example,

• classification,
• anomaly detection, and
• forecasting.

In this chapter, we focus on the forecasting problem.

The Forecasting Problem

To make it easier to formulate the forecasting problem, we group the time series features based on the role they play in a forecasting problem. Given a dataset \(\mathcal D\), with

1. \(y^{(i)}_t\), the sequential variable to be forecasted,
2. \(x^{(i)}_t\), exogenous data for the time series data,
3. \(u^{(i)}_t\), some features that can be obtained or planned in advance,

where \({}^{(i)}\) indicates the \(i\)th variable, \({}_ t\) denotes time. In a forecasting task, we use \(y^{(i)} _ {t-K:t}\), \(x^{(i) _ {t-K:t}}\), and \(u^{(i)} _ {t-K:t+H}\), to forecast the future \(y^{(i)} _ {t+1:t+H}\). In these notations, \(K\) is the input sequence length and \(H\) is the forecast horizon.

A forecasting model \(f\) will use \(x^{(i)} _ {t-K:t}\) and \(u^{(i)} _ {t-K:t+H}\) to forecast \(y^{(i)} _ {t+1:t+H}\).

In the section Time Series Forecasting Tasks, we will discuss more details of the forecasting problem.

Methods of Forecasting Methods

Januschowsk et al proposed a framework to classify the different forecasting methods². We illustrate the different methods in the following charts.

flowchart TB

subgraph Objective

params_shared["Parameter Shared Accross Series"]

params_shared --"True"-->Global
params_shared --"False"-->Local

uncertainty["Uncertainty in Forecasts"]
uncertainty --"True"--> Probabilistic["Probabilistic Forecasts:\n forecasts with predictive uncertainty"]
uncertainty --"False"--> Point["Point Forecasts"]

computational_complexity["Computational Complexity"]

end



subgraph Subjective

structural_assumptions["Strong Structural Assumption"] --"Yes"--> model_driven["Model-Driven"]
structural_assumptions --"No"--> data_driven["Data-Driven"]

model_comb["Model Combinations"]

discriminative_generative["Discriminative or Generative"]

theoretical_guarantees["Theoretical Guarantees"]

predictability_interpretability["Predictability and Interpretibility"]

end

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1. Dorffner G. Neural networks for time series processing. Neural Network World 1996; 6: 447–468. ↩

2. Januschowski T, Gasthaus J, Wang Y, Salinas D, Flunkert V, Bohlke-Schneider M et al. Criteria for classifying forecasting methods. International journal of forecasting 2020; 36: 167–177. ↩

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Contributors: LM