Source-linked AI summary
Exploring Interpretable LSTM Neural Networks over Multi-Variable Data
Tian Guo, Tao Lin, Nino Antulov-Fantulin
TL;DR
The paper addresses the difficulty of interpreting multi-variable RNN predictions when hidden states mix variables. It introduces variable-wise LSTM hidden states and mixture attention, with experiments reporting enhanced prediction performance and interpretation; high-order variable interactions remain future work.
Problem
Multi-variable RNNs have opaque mixed hidden states that make variable contributions and variable-wise temporal relevance difficult to distinguish, despite the need for interpretable forecasting.
Method
IMV-LSTM learns variable-wise hidden states and uses mixture attention with joint training to estimate network parameters, variable importance, and temporal importance.
Results
Experiments on real datasets report enhanced prediction performance and qualitative and quantitative evaluation of interpretation results.
Takeaways & Limitations
The framework supports end-to-end forecasting and knowledge extraction over multi-variable data, with extensions to other recurrent structures described.
Takeaways & Limitations
High-order effects such as variable interactions are left for future work through additional hidden-state rows and importance-vector elements.
Abstract
from arXiv · showhide
For recurrent neural networks trained on time series with target and exogenous variables, in addition to accurate prediction, it is also desired to provide interpretable insights into the data. In this paper, we explore the structure of LSTM recurrent neural networks to learn variable-wise hidden states, with the aim to capture different dynamics in multi-variable time series and distinguish the contribution of variables to the prediction. With these variable-wise hidden states, a mixture attention mechanism is proposed to model the generative process of the target. Then we develop associated training methods to jointly learn network parameters, variable and temporal importance w.r.t the prediction of the target variable. Extensive experiments on real datasets demonstrate enhanced prediction performance by capturing the dynamics of different variables. Meanwhile, we evaluate the interpretation results both qualitatively and quantitatively. It exhibits the prospect as an end-to-end framework for both forecasting and knowledge extraction over multi-variable data.
1. Introduction
The paper targets opaque hidden states in multi-variable RNNs, proposing an interpretable LSTM framework that jointly supports forecasting and variable- and time-specific importance interpretation.
- Motivation: Current RNN hidden states blend target and exogenous variables, making individual contributions to predictions difficult to distinguish.Mixed hidden states also neglect differences in variable dynamics, potentially hindering prediction performance.
- Objective: The paper seeks a unified framework for accurate forecasting and importance interpretation in multi-variable time series.The stated interpretations cover variable importance and variable-wise temporal importance.
- Method: IMV-LSTM uses variable-wise hidden states to capture individual variable dynamics and facilitate prediction and interpretation.The approach explores LSTM structure to encode separate hidden states for variables.
- Method: A mixture attention mechanism summarizes variable-wise hidden states and models the target’s generative process.The associated training method jointly learns network parameters, variable importance, and temporal importance.
- Evaluation: Extensive experiments against statistical, machine learning, and deep learning baselines report superior prediction performance and interpretability for IMV-LSTM.The framework’s idea is also described as applicable to GRU and stacked recurrent layers.
2. Related Work
Prior interpretable RNN research mainly uses attention or post-analysis, while related work also decomposes or partitions hidden states; this paper instead explores LSTM internals for joint forecasting and interpretation.
- Interpretability approaches: Existing interpretable RNN methods primarily use attention mechanisms or post-analysis of trained models.Attention commonly operates across time steps, while post-analysis includes perturbation- and gradient-based approaches.
- Interpretability approaches: Perturbation-based interpretation can produce samples different from the original data distribution, while gradient methods identify features to which outputs are sensitive.These methods analyze trained models rather than restructuring hidden states to encode variable-specific information.
- Positioning: This paper explores LSTM internal structure to learn accurate forecasts and importance measures simultaneously.The method is positioned as distinct from interpretation approaches that may compromise prediction performance.
- Hidden-state structure: Other work represents hidden states as matrices or tensors, increases network capacity, or partitions hidden layers into separately updated modules.These approaches motivate structural decompositions of recurrent hidden representations.
3. Interpretable Multi-Variable LSTM
IMV-LSTM restructures recurrent hidden states so each variable retains its own dynamics, then uses mixture attention to combine variable-wise histories for prediction and interpretation. The architecture also reduces parameter and computation complexity as the number of input variables grows.
- Network architecture: IMV-LSTM assigns each input variable a dedicated hidden-state vector within a hidden-state matrix, preserving variable-wise information for interpretation.The hidden state for the n-th variable has d dimensions, with total layer size D = N·d.
- Network architecture: IMV-Full and IMV-Tensor provide two realizations that retain variable-wise organization while differing in their gate and memory-cell update schemes.IMV-Full uses vectorization and can behave identically to standard LSTM, whereas IMV-Tensor uses tensor-dot operations and parallel variable-specific processing.
- Complexity: IMV-Full reduces parameters by (N − 1)D+(1−1/N)D·D, while IMV-Tensor reduces them by 4(N −1)D+4(1−1/N)D·D relative to standard LSTM.These reductions are stated for layers of size D processing N-variable time series.
- Complexity: IMV-Full has update complexity O(D2 + N · D), whereas IMV-Tensor has O(D2/N + D), making the latter 1/N of standard-LSTM complexity.The paper states that increasing the number of input variables leads to substantial parameter and computation reductions in the IMV-LSTM family.
- Mixture attention: Mixture attention first summarizes each variable’s hidden-state sequence temporally, then merges the resulting variable-wise states through variable attention.The two attention stages are assembled into a probabilistic mixture model for learning, prediction, and interpretation.
- Learning to interpret and predict: Joint training learns network parameters and importance vectors without post-processing, while attention histograms alone remain difficult to interpret reliably.The learned importance vectors represent global variable and variable-wise temporal relations, unlike instance-specific attention values.
4. Experiments
Experiments evaluate IMV-LSTM on PM2.5, PLANT, and SML forecasting tasks against statistical, machine-learning, and deep-learning baselines. The results assess prediction accuracy, variable and temporal importance, domain-knowledge agreement, and performance after selecting important variables.
- Datasets: The experiments use PM2.5, PLANT, and SML datasets, covering meteorological, photovoltaic power-production, and indoor-temperature forecasting.PM2.5 contains 41,700 sequences; PLANT contains 20,842 sequences; SML uses 16 exogenous series sampled every minute.
- Baselines and evaluation: IMV-LSTM is compared with statistical, machine-learning, and deep-learning baselines using RMSE and MAE with standard errors.The baselines include STRX, ARIMAX, RF, XGT, ENET, DUAL, and RETAIN.
- Prediction performance: IMV-LSTM outperforms baselines by around 80% at most, while XGT has comparable performance with deep-learning baselines on PLANT and SML.The reported errors are average RMSE and MAE with standard errors.
- Prediction performance: IMV-Full and IMV-Tensor have comparable performance, while mixture over independent variable-wise hidden states in IMV-Tensor achieves the best performance.Both models use a single network structure, unlike the composite architectures used by some baselines.
- Interpretation: Variable-importance values converge during training, identify an importance ranking, and align with domain knowledge across the PM2.5, PLANT, and indoor-environment datasets.Examples include wind speed, pressure, snow, and rain for PM2.5; irradiance, cloud cover, wind speed, humidity, and temperature for PLANT; and humidity, CO2, and lighting for indoor data.
- Interpretation: Variable-wise temporal importance reveals distinct dynamics, including short-history effects and long-term correlations such as approximately 5 hours for PM2.5 and 13 hours for cloud cover.Lighter colors indicate data contributing more to prediction, and different variables exhibit different decay patterns.
5. Conclusion and Discussion
The paper presents IMV-Full and IMV-Tensor for interpretable prediction on multivariable time series, combining variable-wise dynamics with variable and temporal importance. It identifies extending the framework to model variable interactions as future work.
- IMV-Full and IMV-Tensor use hidden-state matrices to quantify variable and variable-wise temporal importance relative to the target.
- The framework is designed to support both interpretable prediction and improved forecasting by representing individual variable dynamics.
- Modeling higher-order effects such as variable interactions is identified as future work, although the paper states these effects could be represented with additional hidden-state rows and importance-vector elements.
6. Appendix
The appendix provides proof details, dataset and evaluation information, variable-ranking descriptions, and discussion of how IMV-LSTM represents multivariable dynamics and importance. It reports improved performance over traditional hidden-state baselines and discusses learned variable selection.
- Proof: The EM derivation states that, after the E-step, the entropy term is constant, so the M-step minimizes the corresponding loss function.
- Experiments: NASDAQ contains 81 exogenous corporate series, uses the NASDAQ 100 index as target, and splits 35,100, 2,730, and 2,730 minute-level observations into training, validation, and test sets.
- Experiments: Variable rankings are reported in decreasing order of learned importance or attention values across NASDAQ, PLANT, and SML datasets.
- Discussion: IMV-Full and IMV-Tensor outperform baselines with traditional hidden states, which mix all input variables and do not explicitly capture individual dynamics.
- Discussion: IMV-Tensor models variables individually and uses mixture attention to capture interactions, enabling interpretation at both variable and temporal levels.
- Discussion: Selecting variables using learned importance values mostly preserves prediction performance and yields lower errors on two datasets, whereas Pearson-correlation selection increases errors.