Towards Accurate and Interpretable Time-series Forecasting: A Polynomial Learning Approach

Bo Liu; Shao-Bo Lin; Changmiao Wang; Xiaotong Liu

cs.LGMar 3, 2026

Towards Accurate and Interpretable Time-series Forecasting: A Polynomial Learning Approach

Authors: Bo Liu, Shao-Bo Lin, Changmiao Wang, Xiaotong Liu

Organizations: Center for Intelligent Decision-Making and Machine Learning, School of Management, Xi’an Jiaotong University, Xi’an, China · The 39th Research Institute of China Electronics Technology Group Corporation, Xi’an, China · Shenzhen Research Institute of Big Data, ShenZhen, China

Abstract

Time series forecasting enables early warning and has driven asset performance management from traditional planned maintenance to predictive maintenance. However, the lack of interpretability in forecasting methods undermines users' trust and complicates debugging for developers. Consequently, interpretable time-series forecasting has attracted increasing research attention. Nevertheless, existing methods suffer from several limitations, including insufficient modeling of temporal dependencies, lack of feature-level interpretability to support early warning, and difficulty in simultaneously achieving the accuracy and interpretability. This paper proposes the interpretable polynomial learning (IPL) method, which integrates interpretability into the model structure by explicitly modeling original features and their interactions of arbitrary order through polynomial representations. This design preserves temporal dependencies, provides feature-level interpretability, and offers a flexible trade-off between prediction accuracy and interpretability by adjusting the polynomial degree. We evaluate IPL on simulated and Bitcoin price data, showing that it achieves high prediction accuracy with superior interpretability compared with widely used explainability methods. Experiments on field-collected antenna data further demonstrate that IPL yields simpler and more efficient early warning mechanisms.

Figures & tables

Figure 1: A comparison of different interpretable time-series forecasting methods

Component	Meaning
$s$	Total degree of the polynomial.
$m$	Degree of the current monomial ( $0\leq m\leq s$ ).
$T$	Total length of the time series.
$d$	Feature dimension at each time point.
$t_{k}$	Time index of the $k$ -th variable ( $1\leq t_{k}\leq T$ ).
$i_{k}$	Feature index of the $k$ -th variable ( $1\leq i_{k}\leq d$ ).

Table 1: Explanation of the polynomial formula components

\min_{u\in\mathbb{R}^{n},v\in\mathbb{R}^{T}}f(v)\hskip 9.24994pt\text{s.t.}\hskip 9.24994ptAu-v=0,

(4)

Algorithm 1 Interpretable Polynomial Learning for Time Series

Figure 2: Importance of time-related target variables (“-x6” denotes a 6-dimensional input formed by combining the original features with one lagged target variable, “-x7” with two lagged target variables, and so on).

Panel of Figure 3 — Figure 3: Top important features identified by different interpretability methods (note: features shown on the horizontal axis that are present in the objective function are highlighted in red; red and blue bars in the chart represent negative and positive values, respectively).

Metrics	Original input with three added input variables $x_{6},x_{7},x_{8}$ (Dimension = 8)				Original input only (Dimension = 5)
Metrics	IPL	SHAP	LIME	ARIMAX	IPL	SHAP	LIME	ARIMAX
Feature interpretability
Feature overlap ratio	0.7143	0.4286	0.1429	0.7143	1.0000	0.4286	0.7143	0.8571
Ranking similarity	1.0000	0.6699	0.7881	0.8929	1.0000	0.8571	0.6847	0.4324
Value similarity	0.9979	0.1734	0.4723	0.9835	0.9992	0.2759	0.2831	0.8487
Prediction accuracy (MSE) on top 10 features

Table 2: Performance comparison of interpretability methods under different evaluation metrics

Figure 4: Radar charts comparing the performance of interpretability methods (closer proximity to the outer boundary indicates stronger capability in that dimension).

Figure 5: Perturbation analysis (the most important feature selected by each method is indicated in parentheses).

Figure 6: Analysis of Bitcoin price dynamics

Panel of Figure 7 — Figure 7: Importance of the lagged time-related target variables

Features	Methods	Class 1 Performance			Accuracy
Features	Methods	Precision	Recall	F1-score	Accuracy
Top 3	LIME	0.73	0.51	0.60	0.64
	SHAP	0.76	0.56	0.65	0.69
	ARIMAX (2,0,2)	0.58	0.53	0.55	0.53
	IPL	0.75	0.74	0.74	0.73
Top 1	LIME	0.72	0.71	0.71	0.70

Table 3: Classification performance comparison for Target_

3

period

Variable	Description	Range / Note
$x_{1}$	Position angle value	$0^{\circ}\sim 360^{\circ}$ , obtained by combining $x_{4}$ and $x_{5}$
$x_{2}$	Speed	$0\sim 2000$ r/min, collected by tachometers
$x_{3}$	Current ratio	$0\sim 100\%$ , ratio of collected current to rated current
$x_{4}$	Coarse code value	$0\sim 65535$ , 16-bit binary, resolver feedback
$x_{5}$	Fine code value	$0\sim 65535$ , 16-bit binary, resolver feedback
$x_{6}$	Label	$-1$ = normal, $1$ = abnormal

Table 4: Description of the field-collected antenna data

Figure 8: Distribution of target labels across different feature values

Panel of Figure 9 — Figure 9: Relationship between IPL based feature sparsity and prediction performance

Figure 10: Early-warning mechanisms designed using feature importance rankings derived from different methods

Figure 11: Early-warning results based on IPL with varying tree depths ( $k$ )

Sample index	Early warning features			Labels at the current time $t_{1}$			Labels at the next time $t_{2}$			$T_{n}$
	$x_{2}x_{3}$	$x_{6}$	$x_{2}$	Predicted	Predicted	Ground	Predicted	Predicted	Ground
	$x_{2}x_{3}$	$x_{6}$	$x_{2}$	$(k=2)$	$(k=3)$	truth	$(k=2)$	$(k=3)$	truth
5034, 5048	91.0	-1	13.0	-1	1	1	-1	1	1	2
5065, 5082	96.0	-1	12.0	-1	1	1	-1	1	1	10

Table 5: Justification for setting

k=3

: triggering alarms for anomalies during sustained normal operation

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As forecasts increasingly drive decisions in fields such as energy, transportation, and healthcare, understanding the historical data behind these predictions has become as crucial as the predictions themselves. Although existing interpretable-by-design forecasters reveal their internal structures, they offer no guarantee that these structures faithfully reflect the underlying evidence driving the predictions. In contrast, while faithfulness-oriented methods explicitly verify model behavior, they are almost exclusively designed for post-hoc classification tasks. To bridge this gap, we propose IB-Forecast, an inherently interpretable multivariate time-series forecasting framework. It decomposes forecasting into a learned periodic component and a residual component computed with explainable masks over input tokens. With a budget-constrained information bottleneck, end-to-end optimization enables users to directly control explanation sparsity. With a rigorous faithfulness evaluation protocol, extensive experiments demonstrate that IB-Forecast matches the forecasting error of leading black-box models while providing faithful explanations at no additional inference cost. Furthermore, under a matched sparsity budget, these native explanations consistently surpass gradient-based, occlusion-based, and optimization-based baselines across all evaluated datasets. Ultimately, whereas the native explanations of existing interpretable forecasters exhibit poor faithfulness, IB-Forecast guarantees high explanation fidelity, requiring only 14-20% of the observations to deliver low-error predictions.

Abstract

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