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A Time-Aware Bag-of-Receptive-Fields for Interpretable Irregular Time Series Classification
Francesco Spinnato
TL;DR
Irregular time series pose classification challenges because sampling intervals, observation coverage, and lengths vary, while existing methods can be limited in interpretability. i-borf extends borf with time-weighted aggregation and normalization, and benchmark results show improvement over the unweighted baseline while retaining borf’s explanation framework.
Problem
Irregular time series have non-uniform sampling, missing observations, and variable lengths, while effective interpretable classifiers for this setting are limited.
Source: §1 · p. 1Method
i-borf extends borf by weighting local aggregation and normalization by time deltas and using a sliding-window weighted-variance recurrence.
Source: §1 · p. 2, §4.2 · p. 6, §4.4 · p. 7Results
On the PYRREGULAR benchmark, time-weighting improves over the unweighted baseline, and i-borf retains borf’s local-saliency and global-pattern-importance explanations.
Source: §6 · p. 12Takeaways & Limitations
i-borf combines time-aware symbolic pattern extraction with deterministic, interpretable features and explanation analyses for irregular time series.
Source: §1 · p. 2, §6 · p. 12Time awareness is limited to segment and window statistics, and delta weighting assumes each observation represents the preceding interval, which may not suit point events or clinician-triggered measurements.
Source: §6 · p. 12
Abstract
from arXiv · showhide
Irregular time series, characterized by non-uniform sampling intervals, missing observations, and variable lengths, are ubiquitous in healthcare, mobility, and environmental monitoring, yet effective and interpretable classifiers for this setting are limited. Existing approaches often rely on imputation, which can obscure the temporal structure of the data, or require complex neural architectures that are opaque and difficult to explain. In this work, we extend the Bag-Of-Receptive-Fields (BORF), a fast, deterministic, and interpretable transform for time series, to the irregular setting. Our key contribution is a time-weighted normalization scheme in which each observation is weighted proportionally to its associated time delta, making pattern extraction sensitive to the actual temporal distribution of samples rather than only their index position. This requires deriving an efficient sliding-window recurrence for the time-weighted standard deviation, preserving the linear time complexity of BORF. We benchmark the resulting method against state-of-the-art irregular time series classifiers on datasets from the PYRREGULAR repository, demonstrating competitive classification performance with the added benefit of human-interpretable explanations.
1 Introduction
Irregular time series are common but create classification challenges because sampling intervals, observation coverage, and series lengths vary. The paper introduces i-borf, extending borf with time-weighted statistics while retaining its symbolic, deterministic, and interpretable structure.
1 Introduction
- Irregular sampling, partial observation, and variable lengths affect local-pattern meaning and the validity of classifier explanations.Such data arise in healthcare, mobility, environmental monitoring, and industrial sensing.Source: §1 · p. 1
- Existing irregular-TSC studies often use narrow benchmarks or artificially remove observations, while a unified benchmark found regular-data methods can remain competitive without explicit timestamp use.The benchmark evaluates methods under a unified representation and experimental protocol.Source: §1 · pp. 1–2
- Index-based borf statistics assume equal temporal support, so dense bursts can dominate while sparse intervals spanning longer durations are underrepresented.The resulting counts of symbolic local patterns are inspectable, but uniform averaging summarizes observations rather than elapsed time.Source: §1 · p. 2
- i-borf replaces index-based local statistics with time-weighted aggregation and normalization, deriving a moving-window weighted-variance recurrence for receptive fields.It preserves borf’s symbolic, deterministic, interpretable structure without imputing missing values or resampling onto a regular grid.Source: §1 · p. 2
2 Related Work
Prior irregular classifiers span continuous-time, recurrent, graph, and attention-based paradigms, while generalist methods can perform competitively but remain difficult to interpret. This work extends symbolic dictionary methods and borf’s explanation framework to irregular time series.
2 Related Work
- Irregular classifiers include neural ODE/CDE, time-decay or missingness-aware recurrent, graph-based, and continuous-time attention or imputation-oriented models, which can require substantial model selection and remain difficult to interpret,,,,.Generalist methods such as rocket and borf have also performed competitively on irregular benchmarks,,.Source: §2 · p. 3
- Dictionary methods build bags of symbolic words, but existing SAX-based approaches assume regular, fixed-length inputs; i-borf replaces uniform aggregation with time-weighted aggregation for irregular series,,,,.The paper identifies this as the first extension of the SAX-based dictionary paradigm to irregular time series.Source: §2 · p. 3
- borf combines pattern counts with a linear classifier for local saliency maps and global pattern importance, and i-borf extends its SHAP-based explanation framework to irregular time series,.The passage notes that comparable explanation mechanisms for irregular series remain largely unexplored.Source: §2 · p. 3
3 Background
The background defines irregular signals and classification, then describes borf’s symbolic receptive-field representation. borf converts receptive fields into counted SAX words, while its regular-sampling normalization motivates the paper’s time-aware replacement.
3 Background
- An irregular signal pairs observations with increasing timestamps whose successive intervals need not be constant; irregular datasets collect such signals for classification into categorical labels.Irregularity can also include partial observation and mismatches in sampling, length, or alignment.Source: §3 · p. 4
3.1 Bag-Of-Receptive-Fields
- borf is a deterministic, interpretable transform that represents each signal by counts of symbolic patterns extracted from receptive fields rather than by the raw subsequences.This extends Bag-of-Patterns with receptive fields and SAX,,.Source: §3.1 · p. 4
- Receptive fields select ordered observations using window size, dilation, and stride; dilation varies within-field spacing, while stride sets the spacing between fields.Each field is segmented, summarized by means, normalized using its enclosing field’s statistics, and discretized into a SAX word that is counted.Source: §3.1 · pp. 4–5
- Counted symbolic features remain interpretable because their contributions can be mapped to patterns and temporal locations, and moving-average lookups reduce repeated local-statistic computation.Regular SAX normalization ignores timestamps and uses unweighted segment means and window statistics, motivating the targeted time-aware replacement.Source: §3.1 · p. 5
4 Irregular Bag-Of-Receptive-Fields
i-borf extends BORF to irregular time series by replacing uniform aggregation and normalization with time-weighted counterparts, while retaining the BORF pipeline’s later extraction and counting steps. A sliding-window recurrence makes the weighted statistics efficient.
4.1 Time-Delta Weights
- Each observation receives a weight equal to the time elapsed since its predecessor, so temporal aggregates account for unequal sampling intervals.The implementation sets the first observation’s weight equal to the second’s; alternative boundary conventions are left for future work.Source: §4.1 · p. 6
4.2 Time-Weighted Piecewise Aggregate Approximation
- i-borf computes each PAA segment mean as a time-weighted average, reducing to BORF’s uniform mean when all time deltas are equal.Source: §4.2 · p. 6
4.3 Time-Weighted Standardization
- Time-weighted standardization reduces densely sampled observations’ influence, while receptive-field boundaries, dilation, stride, and counts remain index-based.Source: §4.3 · p. 7
4.4 Sliding-Window Weighted Standard Deviation
- The weighted-variance recurrence updates overlapping windows in constant time per slide, preserving O(m) computation for fixed dilation.For dilation greater than one, the recurrences run independently on interleaved subsequences.Source: §4.4 · pp. 7–8
5 Experiments
The experiments compare i-borf with irregular time-series classifiers and BORF variants across irregular datasets, then demonstrate local and global explanations on ABF.
5 Experiments
- The evaluation uses 15 PYRREGULAR datasets with missing, uneven, or ragged sampling, and compares i-borf with reference classifiers and borf+rc using a Ridge head.Datasets with only unequal length or temporal shift are excluded; the Ridge-head borf variant helps isolate the weighting scheme.Source: §5 · p. 8
5.1 Results
- i-borf leads macro-F1 with average rank 3.20, ahead of borf+rc at 4.20, and is not statistically tied to any competing approach.For accuracy, i-borf also has the best average rank, 3.80, though its separation from competitors is less pronounced.Source: §5.1 · pp. 8–9, Figure 3 · p. 9
- On 15 datasets, i-borf achieves best or tied-best F1 on 6 and improves over borf+rc on 8, ties on 6, and loses on one.It also improves over the original borf on 12 datasets.Source: Table 1 · p. 9, §5.1 · p. 9
5.2 Explainability
- Local explanations color observations by SHAP contribution and show a key not-contained pattern; the highlighted region is earlier for Alembic and later for Flask.The Alembic pattern decreases slowly, while the Flask pattern increases slowly; each resembles the opposite class's skew.Source: Figure 5 · p. 10, §5.2 · pp. 10–11
- On ABF, Alembics, Bowls, and Flasks differ by positive, zero, and negative temporal-axis skew, respectively.The paper computes SHAP values for a Ridge classifier on i-borf features and projects pattern importances back to timestamps.Source: §5.2 · p. 10, Figure 4 · p. 10
- Global explanations associate slowly decreasing patterns with Alembics, opposite patterns with Flasks, and less-skewed intermediate patterns with Bowls.The display pairs temporal alignments of important patterns with their class frequencies and SHAP values.Source: Figure 6 · p. 11, §5.2 · p. 11
6 Conclusions
i-borf extends borf with time-weighted statistics while retaining its symbolic, interpretable structure. Experiments show improvement over the unweighted baseline, while the method’s time awareness and weighting assumptions remain limited.
6 Conclusions
- i-borf’s time-weighted aggregation and normalization improve over the unweighted baseline on the PYRREGULAR benchmark.The transform remains linear-time, deterministic, and interpretable, using a sliding-window weighted standard deviation recurrence.Source: §6 · p. 12
- The retained borf explanation framework supports local saliency maps and global pattern-importance analyses, absent from the competing irregular time-series classifiers considered.The paper describes tracing influential symbolic features to timestamped temporal regions.Source: §6 · p. 12
- Time awareness applies only to segment and window statistics; receptive-field construction and word counts remain index-based.Source: §6 · p. 12
- Time-delta weighting assumes each observation represents the preceding interval, which may not suit point events or clinician-triggered measurements and does not explicitly model partial observation.Source: §6 · p. 12
A Proof of Proposition 1
The proof tracks a sliding window’s total weight and weighted mean, then uses successive substitutions and algebraic identities to derive the variance update.
A Proof of Proposition 1
- The window’s total weight is the sum of its time deltas, and its weighted mean is the time-delta-weighted sum of observations divided by that weight.Source: A Proof of Proposition 1 · p. 14
- Advancing the window updates its total weight by removing the outgoing observation’s delta and adding the incoming observation’s delta.Source: A Proof of Proposition 1 · p. 14
- The derivation computes the change in the window statistic, substitutes the weighted-mean relation, and groups outgoing and incoming observation terms.It then substitutes the resulting expression into earlier equations and applies algebraic identities.Source: A Proof of Proposition 1 · p. 14