Overview of ChronoEpilogi
This page summarizes some key elements of ChronoEpilogi. For an in-depth description of the algorithm, we refer to our paper [1].
Motivation: modeling large Multivariate Time Series
ChronoEpilogi is a Time Series Selection (TSS) algorithm : given a Multivariate Time Series (MTS), it selects appropriate dimensions to forecast a quantity of interest.
For instance, we have a MTS \(X\) of size \(T\times D\), with \(T\) timestamps and \(D\) Time Series recorded. We are interested in forecasting one of the dimensions that we will call the target TS \(X^1\), considering a lookback windows of \(L\) lags at horizon 1. Formally, we want to learn a model \(f\) to approximate \(\mathbb{E}[X^1_t | X^{1:D}_{t-L...t-1}]\). This problem is called one step ahead point forecasting. ChronoEpilogi extends to other similar tasks, see data and tasks.
Generally, using \(X^{1:D}_{t-L...t-1}\) is unadvisable when \(D\) is large: models trained on high dimensional data are less efficient and more likely to overfit. Only a portion of the MTS \(D\) dimensions might be useful for the forecasting task, and finding minimal, optimally predictive sets of TS is ChronoEpilogi's function. We distinguish Single TSS and Multiple TSS tasks.
Multiple TSS: Find all such sets.
Single TSS is useful to set of TS optimally suited for forecasting. Multiple TSS produces different sets of TS, which offer analysts a freedom of choice in which set is most suitable to use, given that they are predictively equivalent but may have different costs for the user. We also provide the different TS sets in a factorized form, called equivalence classes of TS. Each equivalence class contains TS that have equivalent information on the target TS, meaning that they can interchangeably be included in a TSS solution. Studying equivalences give knowledge on the structure of the MTS, hence providing interpretability.
ChronoEpilogi phases: single and multiple Time Series Selection
We provide utilities to solve both problems, through the keyword argument phases of the ChronoEpilogi class. The phases are consecutive: we first obtain a single solution (hence solve the Single TSS problem) then build equivalent solutions (TSS multiple).
The single TSS variant of ChronoEpilogi can be accessed with ChronoEpilogi(phases="FB"). It includes a Forward phase and a Backward phase. The forward phase iteratively builds a maximally predictive set of TS starting from an empty set. The backward phase eliminates potentially redundant TS from this set afterwards, ensuring that the returned set ChronoEpilogi.get_first_markov_boundary() is both minimal and maximally predictive.
The multiple TSS variant can be discovered with ChronoEpilogi(phases="FBEV"). After the first solution is found, the Equivalence phase looks for TS that can be swapped with TS of the first solution, while keeping predictive performance maximal. As a result, ChronoEpilogi builds an equivalence class of TS for each TS in the first solution. Other solutions can be obtained by picking one TS per equivalence class. After a Verification phase where false positive equivalences are removed, ChronoEpilogi.get_equivalence_classes() returns the equivalence classes.
A modular Time Series Selection algorithm
ChronoEpilogi relies on several modules that manipulate the data. Three modules are necessary for equivalence detection, the first two (Learning model and Association) are required for the Forward and Backward phases (single TSS problem).
While advanced use of ChronoEpilogi may require digging into each module, ChronoEpilogi has default configuration parameters which automatically choose appropriate ones depending on the type of the forecast quantity, the type of predictors and the data format.
Learning models
ChronoEpilogi relies on predictive models to assess whether a TS is beneficial to the forecasting task. The learning model has two functions:
1) To produce residuals (the errors of the model) when using a certain set of TS as input. Residuals contain information that "remains to be modeled", hence requires additional predictors. Both TSS-single and TSS-multiple variants depend on the residuals. 2) To enable statistical testing of model equivalence. This provides a method to end the Forward phase, remove conditionally independent (aka redundant) TS in the Backward phases, and possibly find equivalence classes during the Equivalence and Verification phases.
By default, ChronoEpilogi uses a linear model suited to the data at hand: a simple linear model for continuous targets, a Poisson GLM for count TS and a Logit GLM for binary TS.
In case that the provided models prove unsuitable for the data at hand, the LearningModel class is a wrapper that specifies what ChronoEpilogi main class expects from a learning model. Subclassing LearningModel requires filling the __init__, fit (model training/fitting), fittedvalues (returning predictions) and stopping_metric (testing for model equivalence) methods.
Associations
During the Forward phase of the algorithm, we build models by iteratively adding TS to our selected set. How to choose the next TS to include is the responsibility of the Association. Given a vector of model residuals (the information to model) and the remaining candidate TS, the Association produces a score for each TS, the higher the more informative the TS is to the residuals.
By default, ChronoEpilogi uses pearson correlation for continuous (numerical) TS, and one-way ANOVA for binary and count (categorical) TS as Association, depending on the provided TS types.
We note that the association is also used in the equivalence phase if ChronoEpilogi's equivalence_early_stopping is set to True, which it is by default.
Partial correlations
Searching for equivalent TS during the Equivalence phase is a lengthy process. We introduced heuristics to lessen computation time. The best empirical results were obtained using a partial correlation method, which requires its own module.
Similarily to the Association, the default parameter setting of ChronoEpilogi will choose the partial correlation appropriate for the data. For continuous values, pearson partial correlation is used. For mixed data, we use a mixed-type linear conditional independence test implemented in the tigramite library.
References
[1]. "ChronoEpilogi: scalable time-series variable selection with multiple solutions".