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Signature-Based Feature Learning for Human Activity Recognition: A Reproducible Machine Learning Study of Representation, Depth, and Model Choice

Kamal Jarrar, Jacky Cresson, Christian Paroissin

arXiv:2610.06553v1cs.AI

TL;DR

HAR representations need to capture temporal patterns that raw sensor values may not clearly express, while prior work leaves limitations in reproducibility and explicit treatment of cross-channel temporal interactions. This study compares signature-based representations under a reproducible, leakage-aware evaluation, finding that their performance benefit depends on representation and classifier choice.

  • Problem

    HAR windows can split movement cycles, and prior approaches provide limited reproducibility or do not explicitly capture higher-order temporal interactions between sensor channels.

    Source: §1.1 · p. 3, §1.2 · p. 4

  • Method

    The study systematically compares raw, time-augmented, and lead–lag representations across signature depths, model classes, and sensor configurations using a reproducible, leakage-aware evaluation.

    Source: §5.3 · p. 18, §6.1 · p. 19

  • Results

    Signature representations improve performance with Random Forest, but their effectiveness depends on representation design and classifier choice.

    Source: §6.1 · p. 19

  • Takeaways & Limitations

    Representation design is a central component of time-series modelling, and HAR improvements are better explained by temporal-structure encoding than model complexity alone.

    Source: §6.1 · p. 19

    The evaluation uses a single dataset, signature features can become high-dimensional at greater depths, and convolutional or recurrent models were not explored.

    Source: §5.3 · p. 18

Abstract

from arXiv · show

Human activity recognition (HAR) relies on transforming sensor signals into informative representations for classification. Although deep learning and handcrafted features are widely used, the role of representation itself is often not systematically isolated. Signature transforms provide a mathematically grounded way to encode temporal order and cross-channel interactions, but their value for HAR under a fully reproducible and leakage-aware framework remains unclear. To evaluate whether signature-based feature learning improves HAR performance compared with raw-signal baselines, and to assess the effects of embedding strategy, truncation depth, model choice, and sensor configuration. Experiments were conducted on the UCI HAR dataset using a fully reproducible pipeline with the original train--test split preserved and subject-disjoint validation to prevent leakage. Three representations were compared: raw flattened signals, time-augmented paths, and lead--lag transformed paths. Signature features were computed at multiple truncation depths and evaluated using multilayer perceptron (MLP) and Random Forest (RF) classifiers under identical preprocessing and validation procedures. A prior K-means-based feature reduction study was also reproduced for comparison. Signature-based representations improved performance when paired with RF models, the best configuration was time-augmented six-channel signatures at depth 6 using entropy-based RF achieving 0.858 accuracy and 0.859 macro F1, outperforming the strongest raw baseline (0.816 accuracy). Lead--lag representations were competitive at moderate depths but did not surpass the best time-augmented models. MLP models did not exceed raw baselines. Signature-based feature learning can improve HAR, but its benefit depends on alignment between representation design and classifier choice.

1 Introduction

HAR must turn sensor windows into representations that capture temporal change and interactions between channels. This study evaluates signature features against existing approaches under controlled, reproducible conditions, motivated by gaps in prior comparisons.

1.1 Human Activity Recognition (HAR)

  • Short windows can split a movement cycle, making it harder to distinguish similar activities and increasing the importance of how each window is represented.Raw values preserve measurements but do not clearly expose movement order or relationships between channels.Source: §1.1 · p. 3

1.2 Existing Approaches and Previous Work

  • Prior HAR approaches include manually designed statistical features and deep models that learn from raw inputs, but flattened inputs do not directly convey temporal structure.The paper also discusses Ismi et al.’s K-means feature reduction and Naive Bayes pipeline on UCI HAR’s 561 handcrafted features.Source: §1.2 · p. 3
  • Prior K-means-based work provides limited implementation detail and no public code, constraining reproducibility and rigorous comparison.The authors also note that the method does not explicitly capture higher-order temporal interactions between sensor channels.Source: §1.2 · p. 4

1.3 Proposed Strategy

  • The study systematically tests raw, time-augmented, and lead–lag representations across signature depths, classifiers, and sensor configurations using leakage-aware evaluation.Its framework includes subject-disjoint evaluation and a pre-registered protocol, public code, model documentation, and a Docker environment.Source: §1.3 · p. 4

1.4 Plan of the Paper

  • The paper moves from its mathematical framework to HAR experiments, a reproduced K-means comparison, discussion of findings and limitations, and conclusions.Appendices provide dataset and representation details, implementation and evaluation protocols, and reproducibility information.Source: §1.4 · pp. 4–5

2 Mathematical Framework: Temporal Data Streams and Signatures

The framework represents discrete sensor streams as continuous paths, applies embeddings and time augmentation, and computes path signatures. It also describes signature computation and properties before defining signatures for embedded discrete streams.

2 Mathematical Framework: Temporal Data Streams and Signatures

  • The framework introduces temporal streams, embeddings, augmentations, continuous-path signatures, their properties, and signatures extended to discrete streams.Source: §2 · p. 5

2.1 Temporal Data Streams

  • A d-dimensional temporal data stream is a map from the discrete time scale T to R^d.Source: §2.1 · p. 5

2.2 Embedding of Temporal Data Streams

  • A discrete stream can be embedded as a continuous stream; the standard embedding uses linear interpolation, while lead–lag is another embedding choice.Source: §2.2 · pp. 5–6

2.3 Augmentation of Temporal Data Streams

  • Time augmentation appends the time coordinate to each signal value, and it can be applied at either the discrete or continuous level.Source: §2.3 · p. 6

2.4 Signature of a Continuous Path

  • The signature of a continuous path is a tensor series of iterated integrals, which is truncated to a chosen depth for computation.The truncated signature can be identified with a finite-dimensional vector.Source: §2.4 · p. 7

2.5 Explicit Computation of the Signature for Piecewise Linear Functions

  • For piecewise linear paths, the paper describes computing signature components on intervals and combining them across concatenated segments with Chen’s formula.Source: §2.5 · pp. 7–8

2.6 Properties of the Signature

  • Under mild conditions satisfied by augmented continuous data streams, the signature characterizes the path; stated properties also simplify computations for periodic or scaling signals.Source: §2.6 · p. 8

2.7 Signature of a d-Temporal Data Stream

  • A discrete stream’s signature is defined with respect to a one-to-one embedding, making the embedding choice part of the signature construction.Source: §2.7 · p. 8

3 Signature-Based Features for Human Activity Recognition

The section evaluates signature-based representations of UCI HAR sensor windows across embeddings, truncation depths, classifiers, and sensor configurations. The strongest results come from time-augmented signatures with Random Forest, while performance varies by classifier and configuration.

3.1 Structure of the HAR Dataset

  • The dataset contains six activity classes from 30 participants, with 127-step windows and six inertial channels; an accelerometer-only three-channel version supports sensor comparisons.The six channels are three-axis body acceleration and three-axis gyroscope measurements.Source: §3.1 · p. 9

3.2 Main Strategy

  • The strategy embeds each temporal stream as a continuous path, computes its signature, truncates at depth k, and classifies the resulting features.The study compares time augmentation and lead–lag embeddings with MLP and RF classifiers.Source: §3.2 · pp. 9–10, Figure 1 · p. 9

3.3.2 Random Forest Models with Signature-Based Representations

  • The best RF results used time augmentation and improved monotonically with depth under entropy; lead–lag remained competitive at moderate depths but did not surpass them.Entropy-based time-augmented RF reached 0.858 test accuracy and 0.859 macro F1 at depth 6; lead–lag entropy RF reached 0.839 accuracy at depth 4.Source: §3.3.2 · pp. 11–12, Table 2 · p. 12

3.4 Best Model Performance and Class-Level Evaluation

  • 0.858 test accuracy and 0.859 macro F1 came from six-channel time-augmented signatures at depth 6 with entropy-based RF, exceeding the 0.816 raw RF baseline.The best signature model correctly classified 2529 instances; the raw RF baseline’s 0.816 accuracy used the Gini criterion.Source: Table 2 · p. 12, §3.3.4 · p. 13, §3.3.3 · p. 13

3.3.3 Baseline Models with Raw Flattened Signals

  • Raw flattened inputs omit explicit temporal encoding; six-channel RF achieved the strongest baseline, with 0.816 test accuracy and 0.817 macro F1 under Gini.Six-channel raw MLP with ReLU achieved 0.794 accuracy and 0.799 macro F1.Source: §3.3.3 · pp. 12–13, Table 3 · p. 13

3.3.4 Comparative Analysis Across Configurations

  • Random Forest consistently outperformed MLP across all representation types and configurations, and time-augmented Random Forests achieved strongest overall performance, reaching accuracy 0.858 with entropy-criterion truncation depth increasing monotonically.Six-channel inputs generally yielded higher performance for baseline models and Random Forests, but three-channel MLP signature representations could achieve higher peak performance, particularly with time augmentation.Source: §3.3.4 · p. 13

3.3.1 MLP Models with Signature-Based Representations

  • MLP results depended on representation and activation: ReLU outperformed Tanh, and three-channel time augmentation reached 0.734 test accuracy at depth 4.For six-channel ReLU MLP, lead–lag at depth 3 reached 0.652 accuracy and 0.637 macro F1; the three-channel ReLU time-augmented result exceeded all six-channel signature MLP settings.Source: §3.3.1 · p. 11, §3.3.4 · p. 13

3.4 Best Model Performance and Class-Level Evaluation

  • For the best RF, macro precision, recall, and F1 were 0.861, 0.858, and 0.859; SITTING had the lowest class F1 at 0.773.WALKING had F1 of 0.929, while WALKING UPSTAIRS and WALKING DOWNSTAIRS had F1 scores of 0.902 and 0.854.Source: Table 4 · p. 14, §3.4 · p. 14

4 Comparison with Ismi et al. K-means-based Features

The study reproduces the K-means-based feature reduction results of Ismi et al. (2016), whose method was not sufficiently specified for checking. The reproduced results were weaker than those reported in the original study, while the proposed model exceeded both across the evaluated configurations.

4 Comparison with Ismi et al. K-means-based Features

  • The original results lacked enough implementation detail to check the method, and the reproduction did not obtain the performance reported by its authors.Across most k values, reproduced results were lower than the original study’s reported results.Source: §4 · pp. 14–15
  • The reproduced all-handcrafted-features baseline achieved 69.79%, rising to 80.10% after K-means-based reduction at k = 70.The reproduced result was weaker than the original study’s reported baseline of 76.77% and maximum of 83.35% at k = 70.Source: §4 · pp. 14–15
  • 85.816% correctly classified by the proposed model exceeded both the reproduced and original K-means-based results across evaluated configurations.The reproduced feature-reduction result reached 80.10% at k = 70; the original study reported 83.35% at k = 70.Source: §4 · p. 15

5 Discussion

Signature features improve HAR conditionally: outcomes depend on representation, depth, classifier, and sensor configuration. The study contrasts these results with prior feature reduction and reports a reproducible, controlled evaluation alongside limits in dataset scope, feature dimensionality, and model coverage.

5.1 Effect of Representation, Truncation Depth, and Model Choice

  • 0.858 accuracy and 0.859 macro F1 for the best RF signature model exceeded the strongest raw baseline (0.816 accuracy), while MLP signatures did not beat raw inputs.Across configurations, RF outperformed MLP under subject-disjoint evaluation.Source: §5.1 · pp. 16–17
  • Time augmentation achieved the highest overall performance, especially with RF, while lead–lag remained competitive at moderate depths, particularly for MLP.The discussion attributes this balance to temporal-order preservation with limited dimensional expansion.Source: §5.1 · p. 17
  • Increasing signature depth consistently improved RF results, whereas MLP gains plateaued or declined as deeper, higher-dimensional features could hinder optimization.The authors treat truncation depth as a key modeling hyperparameter.Source: §5.1 · p. 17
  • Six-channel inputs produced the best overall results, although accelerometer-only data approximated full inertial performance considerably, particularly for MLP.The discussion identifies gyroscope signals as complementary classification information.Source: §5.1 · p. 17

5.2 Comparison with Prior Feature Selection Approaches

  • The reproduced K-means pipeline reached 80.10% at k = 70, below the original study’s reported result, amid incomplete reporting that limits exact reproducibility.The comparison frames the signature framework as addressing temporal representation rather than only redundancy reduction.Source: §5.2 · pp. 17–18

5.3 Strengths and Limitations

  • The study reports a preregistered, publicly documented pipeline with subject-disjoint validation and controlled comparisons across representation, depth, model, and sensor configuration.The pipeline also includes public code, a model card, and a Docker image for replication.Source: §5.3 · p. 18
  • The evaluation uses one dataset, signature features can become high-dimensional, and convolutional or recurrent architectures were not tested.These boundaries constrain generalizability and leave architecture-dependent representation comparisons unexplored.Source: §5.3 · p. 18

6 Conclusion and Perspectives

The study concludes that signature-based HAR performance depends on representation design and classifier choice, with Random Forest and time-augmented signatures performing best overall. It proposes broader evaluations, hybrid deep-learning approaches, dimensionality-reduction strategies, and adaptive embeddings as future directions.

6.1 Conclusion

  • Signature representations outperform raw baselines with suitable models, and Random Forest achieves the best overall performance; time augmentation and deeper signatures are especially effective for nonparametric models.MLP performance is more sensitive to high-dimensional representations.Source: §6.1 · p. 19
  • The conclusion treats representation design as central to time-series modeling, explaining HAR improvements through temporal-structure encoding rather than model complexity alone.A fully specified, reproducible framework is presented as a foundation for future research.Source: §6.1 · p. 19

6.2 Perspectives

  • Future evaluation should cover more datasets, including real-world and clinical HAR and variable-length or less-structured sequences, to test generalizability beyond UCI HAR.Source: §6.2 · p. 19
  • The authors propose combining explicit signature features with convolutional or recurrent networks, including deep-signature methods that learn representations and classifications end to end.Source: §6.2 · p. 19
  • Log-signatures, feature selection, or regularization are proposed to reduce the dimensionality and computational challenges of high truncation depths while preserving informative structure.Source: §6.2 · p. 19
  • Adaptive or learned embeddings are proposed as more flexible alternatives to fixed time-augmentation and lead–lag designs.Source: §6.2 · pp. 19–20

Declaration of Competing Interests

The authors declare no financial or personal affiliations that may have biased the research.

Declaration of Competing Interests

Data Availability

The study uses a publicly available UCI HAR dataset and reports that no additional data were generated or analyzed. The appendix describes the dataset and representations used in the experiments.

Data Availability

  • The dataset is publicly available through the UCI Machine Learning Repository, and no additional data were generated or analyzed.Source: Data Availability · p. 20

A Dataset and data representation

A.1 Dataset

The study uses the UCI HAR dataset of smartphone inertial recordings, with six activity classes and a preserved train–test split. It compares six-channel and acceleration-only inputs, while reserving the provided handcrafted features for baseline reproduction.

A.1 Dataset

  • UCI HAR contains smartphone accelerometer and gyroscope recordings from 30 participants performing six activities, segmented into 128-step windows with 50% overlap.Signals were acquired at 50 Hz, so each window spans 2.56 seconds.Source: §A.1 · p. 23
  • Inputs comprise six body-acceleration and gyroscope channels, with additional experiments using only the three acceleration channels.Source: §A.1 · p. 23
  • The original split contains 7,352 training and 2,947 test windows, with the test set excluded from model development and validation.Source: §A.1 · p. 23
  • Primary analyses compare raw and signature-based representations, while the dataset’s 561 handcrafted features are used only to reproduce a baseline.Source: §A.1 · p. 23

A.2 Data Representation

The study compares flattened signals with time-augmented and lead–lag path representations under identical experimental conditions. It evaluates these representations using MLP and Random Forest classifiers.

A.2 Data Representation

  • Flattened six-channel signals form 768-dimensional vectors, while the acceleration-only setting yields 384 dimensions.Source: §A.2 · p. 23
  • Appending normalized time creates seven-channel or four-channel paths for the six- and three-sensor-channel settings, respectively.Source: §A.2 · pp. 23–24
  • The lead–lag transform expands time-augmented paths to twice the dimension and length to encode temporal dependencies and cross-channel interactions.For example, a (128, 7) path becomes (256, 14).Source: §A.2 · p. 24

B Model Implementation Details

  • The MLP experiments compare signature features with raw flattened signals across six-channel ReLU and three-channel ReLU or Tanh configurations.The network uses fully connected layers with batch normalization and dropout.Source: B Model Implementation Details · p. 24
  • Random Forest experiments compare Gini and entropy splits on six-channel signatures and raw inputs, and test entropy-based time-augmented signatures at depths 2–6.Source: B Model Implementation Details · p. 24

B.1 Training Protocol

Model development uses a fixed, subject-disjoint validation split while preserving the original UCI HAR test set for final evaluation. The section specifies complementary evaluation metrics and reproducibility measures.

B.1 Training Protocol

  • The original 7,352/2,947 train–test split is preserved, and subject-disjoint validation uses 15% of training data to prevent cross-subject leakage.The split uses seed 2025 and is reused across experiments; the test set is reserved for final evaluation.Source: §B.1 · p. 24
  • Features are standardized using only the training subset, and model selection uses validation macro F1 with early stopping for both classifier types.MLP stopping uses an eight-epoch patience; Random Forest trees stop after eight non-improving iterations.Source: §B.1 · p. 24

B.2 Evaluation Metrics

  • Macro F1 is the primary model-selection metric because it gives equal weight to each of the six classes, while accuracy measures the proportion correctly classified.The evaluation also reports class-wise precision, recall, and F1, confusion matrices, and 95% bootstrap confidence intervals for accuracy and macro F1.Source: §B.2 · p. 25
  • The baseline reproduction uses mean accuracy from 10-fold stratified cross-validation, consistent with the original study.Source: §B.2 · p. 25

B.3 Reproducibility Framework

  • A public protocol, code repository, model card, and Docker container support transparent and reproducible experiments.Fixed random seeds and systematically stored configurations and outputs support deterministic execution and traceability.Source: §B.3 · p. 26

C About Ismi et al. approach using K-means-Based Feature Reduction

The study reproduces an earlier K-means feature-reduction and Naive Bayes pipeline using UCI HAR’s handcrafted features. It documents implementation assumptions because the original specification is incomplete.

C About Ismi et al. approach using K-means-Based Feature Reduction

  • The reproduced pipeline clusters the transposed 561-feature matrix, treating each feature as an observation, then averages features within clusters before Naive Bayes classification.The reduced representation has k dimensions for each of k values from 10 to 70.Source: C About Ismi et al. approach using… · p. 26
  • The prior study reports 76.77% accuracy with all 561 features and a maximum 83.35% after reduction at k = 70.Source: C About Ismi et al. approach using… · p. 26
  • Because the original implementation omits details such as preprocessing and parameter settings, the reproduction follows its published description and documents assumptions.The reproduced K-means uses random initialization, one initialization, five maximum iterations, and random state 42.Source: C About Ismi et al. approach using… · p. 26

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