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Autorelevance function and other feature relevance measures for univariate time series

This paper proposes a model-agnostic framework utilizing Ghost variables and Shapley values to define autorelevance and partial autorelevance functions for measuring lag importance in univariate time series forecasting, demonstrating their effectiveness across seasonal ARMA and recurrent neural network models.

Original authors: Julian Cardenas, Jamie Arjona, Pedro Delicado

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Julian Cardenas, Jamie Arjona, Pedro Delicado

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for tomorrow. You have a super-smart computer program (a "black box" model) that looks at the temperature, wind, and rain from the last few days to make its guess. But here's the problem: the computer is so smart that it's also a bit of a mystery. You ask it, "Why did you predict rain?" and it just says, "Because of the data." It doesn't tell you which day's data mattered most. Was it yesterday's rain? Or the rain from last Tuesday?

This paper introduces a new set of tools to answer that question. The authors want to measure exactly how much each "lag" (each past day's data) contributes to the final prediction, regardless of whether the computer is using simple math or a complex neural network.

Here is a breakdown of their ideas using everyday analogies:

1. The Problem: The "Black Box" Forecast

In time series forecasting, the only inputs are usually past versions of the same thing (like past stock prices or past passenger numbers). Traditional tools (like the Autocorrelation Function) are like old-fashioned maps; they work great for simple, straight-line roads (linear models) but get confused when the road twists and turns into a complex, non-linear jungle (neural networks). We need a new way to see which past days are actually driving the prediction.

2. The Solution: Two New "Relevance" Rulers

The authors propose two main ways to measure importance, which they call Auto-Relevance Functions (ARF) and Partial Auto-Relevance Functions (PARF).

Method A: The "Ghost" Replacement (Ghost Variables)

Imagine you are baking a cake, and you want to know how important the eggs are.

The Standard Way: You bake the cake without eggs and see if it tastes bad.
The Problem with Time Series: You can't just "remove" yesterday's temperature. If you delete it, the timeline breaks, and the recipe makes no sense.
The Paper's Trick: Instead of deleting yesterday's temperature, you replace it with a "Ghost" temperature. This ghost is a smart guess of what the temperature would have been if you only looked at the days before and after it, ignoring yesterday entirely.
The Result: You bake the cake with the real temperature, then bake it again with the "Ghost" temperature. If the cake tastes totally different, then yesterday's temperature was crucial. If the cake tastes the same, yesterday didn't matter much.

The authors use this "Ghost" method to create a score (ARF) that tells you how much each past day matters.

Method B: The "Team Player" Score (Shapley Values)

Imagine a sports team where players (the past days) come in different combinations. Sometimes the team has just the top 3 players; sometimes it has the top 5.

The Concept: This method, based on a mathematical theory called Shapley values, asks: "How much does adding one specific player improve the team's score, no matter who else is on the team?"
The Application: They calculate how much the prediction error drops when you add "Day 1" to a group of other days, then do the same for "Day 2," and so on, across every possible combination of days.
The Result: This gives a "Partial Auto-Relevance Function" (PARF). It's like a fairness score that tells you exactly how much credit each past day deserves for the final prediction, even if it only matters when combined with other specific days.

3. The Secret Sauce: How to Fill in the Blanks

A major hurdle in this research is: What do you put in the "Ghost" slot or the "Missing Team Member" slot?

Old Way: Just put in a zero or the average temperature. This is like putting a dummy in the lineup; it breaks the flow of the game.
The Paper's Innovation: They use the forecasting model itself to guess the missing value. If you are missing "Yesterday's Temperature," the model looks at the other available days and predicts what yesterday should have been.
Why it works: This keeps the "story" of the time series intact. It's like asking the coach to simulate what would have happened if a player was missing, rather than just leaving an empty chair.

4. Did it Work? (The Results)

The authors tested these tools on two types of data:

Simulated Data: They created fake time series where they knew the answer (e.g., "We made a model where only Day 1 and Day 3 matter; Day 2 is useless").
- Result: Their tools correctly identified that Day 1 and 3 were stars and Day 2 was a dud, even when the prediction model was a complex neural network.
Real Data: They used the famous "Monthly Airline Passengers" dataset.
- Result: The tools correctly highlighted that the most important days to predict next month were 1 month ago, 12 months ago (last year), and 13 months ago. This matches human intuition about seasonal patterns.

5. The Bottom Line

The paper claims that these new methods (ARF and PARF) are:

Model-Agnostic: They work on simple math models and complex AI "black boxes" alike.
Accurate: They successfully find the important past days in both simple and tricky, non-linear situations.
Efficient: They are faster to calculate than previous methods that try to do the same thing.

In short, the authors built a "flashlight" that lets us see exactly which past moments are lighting up the future predictions, even when the computer doing the predicting is too complex to explain itself.

Technical Summary: Autorelevance Function and Other Feature Relevance Measures for Univariate Time Series

Problem Statement
The increasing adoption of machine learning models for time series forecasting has created a critical need for interpretability, particularly regarding how lagged observations influence predictions. While classical statistical tools like the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) effectively describe linear temporal structures, they are ill-suited for explaining the behavior of modern "black-box" forecasting models (e.g., neural networks). Existing feature importance methods often struggle in time series contexts due to the high correlation between predictors (lags) and the lack of model-agnostic strategies for handling "absent" features in coalition-based explanations. Standard imputation techniques (e.g., zero-filling or unconditional means) often fail to preserve the temporal dynamics essential to forecasting, leading to potentially misleading relevance estimates.

Methodology
The authors propose a model-agnostic framework to quantify the global relevance of lagged inputs in univariate time series forecasting. The methodology integrates two primary approaches: Ghost Variables and Shapley Values, adapted specifically for forecasting tasks where inputs are exclusively lagged values of the series.

Ghost Variables Approach (Autorelevance Function - ARF):
- Adapted from the Ghost Variables methodology, this approach evaluates the relevance of a specific lag $h$ by replacing the actual lagged value $y_{t-h}$ with an imputed "ghost" value $\hat{y}^*_{t-h}$ .
- The ghost value is generated by an auxiliary imputation model (specifically, a multiple linear regression) that predicts $y_{t-h}$ using the remaining $p-1$ lags.
- Relevance is quantified by comparing the prediction error of the original model against the model using the ghost value. The authors define two metrics:
  - ARF $_{MSPD}$ : Based on the Mean Squared Prediction Difference.
  - ARF $_{MSE}$ : Based on the change in Mean Squared Error (MSE) contributions.
Shapley Value Approach (Partial Autorelevance Function - PARF):
- This approach treats lags as players in a cooperative game, calculating the average marginal contribution of each lag across all possible coalitions (subsets of lags).
- Imputation Strategy: A key innovation is the treatment of absent features in a coalition. Instead of generic imputation, the authors propose replacing absent lags with a one-step-ahead forecast generated by the same forecasting model being analyzed. This preserves the temporal coherence of the series.
- Value Function: The value function is defined based on MSE improvement relative to a baseline (sample mean).
- Metrics:
  - PARF $_{MSE}$ : Standardized Shapley values based on MSE contributions.
  - PARF $_{MSPD}$ : Standardized Shapley values based on squared prediction differences.
- Computational Efficiency: Unlike local SHAP methods that require permutation analysis for every observation, this framework computes relevance globally by averaging over the test sample, significantly reducing computational burden.

Key Contributions

Novel Imputation for Time Series: The paper introduces a model-driven one-step-ahead imputation scheme for handling absent features in coalition-based methods. This strategy aligns the relevance analysis with the dynamics of the forecasting model itself, offering a more natural definition of coalitions in sequential settings compared to static imputations.
Model-Agnostic Lag Relevance: The proposed ARF and PARF measures are applicable to both classical statistical models (e.g., ARMA) and complex black-box models (e.g., Recurrent Neural Networks), providing a unified framework for interpretability.
Integration of Additive Importance Measures: The work connects the practical needs of time series explainability with the theoretical foundations of additive importance measures (AIM), specifically utilizing MSE as the loss function, which is standard in forecasting.

Results
The authors evaluated their proposals using simulated data (linear ARMA, nonlinear regime-switching models) and a real-world dataset (Monthly Airline Passengers).

Linear Simulations (AR(3) and ARMA(3,3)): In linear settings, both ARF and PARF successfully identified the true non-zero lags (lags 1 and 3) while assigning negligible relevance to irrelevant lags (lag 2), even when the fitted model was a neural network.
Nonlinear Simulations: In a nonlinear regime-switching model, the relevance measures (particularly PARF with MSE contributions) correctly identified the dominant lags (1, 12, and 13) that drove the data-generating process, demonstrating robustness to nonlinearity.
Real Data (Airline Passengers): Applied to the classic airline dataset, the methods consistently highlighted lags 1, 12, and 13 as the most relevant, aligning with the known seasonal structure. While the neural network model showed slightly different patterns than the SARIMA model, the relevance measures still captured the fundamental lag structure.
Imputation Comparison: The model-based one-step-ahead imputation generally outperformed linear imputation in nonlinear and complex scenarios, providing clearer identification of dominant lag structures.

Significance and Claims
The paper claims that the proposed framework offers a practical and effective solution for interpreting time series forecasting models. The authors emphasize that:

The methods successfully recover expected lag structures in almost all tested cases, including black-box neural networks.
The one-step-ahead imputation strategy provides a natural way to define coalitions in sequential settings, avoiding the heuristic nature of generic imputations.
The approach is computationally more efficient than standard local SHAP methods while maintaining interpretability.
The framework is particularly suitable for time series because it respects the temporal order and dynamics of the data.

The authors remain modest regarding limitations, noting that the strategy relies on the forecasting model having sufficiently good predictive performance; poor models may introduce noise through the imputation process, leading to misleading relevance estimates. Future work is suggested to explore alternative relevance formulations, the inclusion of exogenous predictors, and extension to multivariate time series.