# Intelligent Aggregation of Single-Sensor Classifiers for Enhanced Structural Health Monitoring Networks

Mohamed Abdelillah Fidma, Jean-François Bercher, Franziska Schmidt

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## To cite this version:

Mohamed Abdelillah Fidma, Jean-François Bercher, Franziska Schmidt. Intelligent Aggregation of Single-Sensor Classifiers for Enhanced Structural Health Monitoring Networks. 8th International Conference on Machine Learning for Networking (MLN'2025), Selma Boumerdassi, Nour El Houda Yellas, Éric Renault, Dec 2025, Paris, France. hal-05451332

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https://hal.science/hal-05451332v1

Submitted on 9 Jan 2026

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# Intelligent Aggregation of Single-Sensor Classifiers for Enhanced Structural Health Monitoring Networks

Mohamed Abdelillah Fidma¹, Jean-François Bercher², and Franziska Schmidt¹

¹ EMGCU

Université Gustave Eiffel

Marne-la-Vallée, France

{mohamed.fidma,franziska.schmidt}@univ-eiffel.fr

² LIGM

Université Gustave Eiffel

Marne-la-Vallée, France

jf.bercher@esiee.fr

**Abstract.** Structural health monitoring (SHM) systems for large-scale infrastructures often rely on dense sensor networks, which are prone to faults, generate high-volume data, and require computationally efficient algorithms to ensure low-latency inference for real-time monitoring. To enhance overall network accuracy and robustness, aggregating the predictions of individual sensors provides a way to leverage complementary information across the network while mitigating sensor-level errors. In this study, we investigate intelligent aggregation strategies for single-sensor classifiers in SHM networks. We leverage acceleration time series data from the RT345 bridge dataset, collected from a real instrumented structure, to detect and classify structural damages. Individual sensor classifiers produce probabilistic predictions, which are then combined using different aggregation strategies. Soft averaging serves as a baseline, while stacking ensembles employs linear meta-classifiers (Logistic Regression) for interpretable per-sensor weighting and nonlinear meta-classifiers (Random Forest) to capture complex conditional dependencies across sensors, albeit at the cost of interpretability and stability. Experimental results demonstrate that meta-learning strategies significantly improve classification accuracy and robustness. We further evaluate prediction time, scalability, and model size, highlighting trade-offs between linear and nonlinear aggregation for real-time SHM applications. Finally, we extend the study by exploring alternative acceleration time series representations, showing that system efficiency can be improved without compromising damage detection performance.

**Keywords:** Structural Health Monitoring · Machine Learning · Meta-Learning · Stacking Classifier · Sensor Networks · Time Series Classification
# 1 Introduction

Monitoring large-scale civil structures equipped with dense sensor networks generates massive amounts of vibration data that must be transmitted, stored, and processed at a central node. This centralized paradigm imposes high communication, storage, and computational demands, which become particularly problematic in resource-constrained (edge) deployments with limited bandwidth and processing power. In addition, reliance on continuous high-throughput data transfer makes centralized systems vulnerable to communication bottlenecks and failures, which can compromise the timely and reliable assessment of structural health [4,12,18].

To address these limitations, our previous work [13] introduced a decentralized approach in which each sensor hosts a lightweight classifier operating directly on its local time series data. These per-sensor classifiers provide localized predictions of structural conditions, reducing dependence on centralized computation and improving resilience to communication failures. In that framework, the probabilistic outputs from individual sensors were aggregated using simple averaging at a gateway to produce a network-level prediction. However, the performance of single-sensor classifiers is inherently limited by the restricted information available at individual sensors and environmental variability. While this approach proved effective on the Z24 benchmark dataset, see [9,13], it faced challenges with the more complex RT345 bridge dataset [20], highlighting the need for enhancements toward a low-complexity, scalable SHM framework suitable for real-world bridge monitoring.

Potential enhancements can be summarized in three key directions:

1. **Time series representations:** exploring alternative encodings of vibration data to capture more discriminative structural patterns.

2. **Per-sensor base classifiers:** refining model selection and hyperparameters to improve robustness of single-sensor predictions.

3. **Sensor-level aggregation:** designing intelligent strategies to combine outputs from multiple sensors beyond naïve averaging.

This study focuses on the aggregation layer, investigating machine learning models to combine probabilistic outputs from single-sensor classifiers in SHM networks intelligently. Instead of assigning equal weights to all sensors, our framework leverages meta-learning to optimally aggregate predictions. Three aggregation strategies are evaluated: (i) soft averaging as a baseline, (ii) linear meta-classifiers (Logistic Regression) that provide interpretable per-sensor weighting, and (iii) nonlinear meta-classifiers (Random Forest) that capture complex conditional dependencies among sensors at the cost of interpretability.

Through extensive experiments on the RT345 bridge dataset, which includes both baseline and controlled damage scenarios, we assessed the impact of segment length and aggregation strategy on classification accuracy, computational cost, and scalability.

The main contributions of this work are as follows:
1. Development of an intelligent aggregation framework for single-sensor classifiers in SHM networks.

2. Systematic evaluation of aggregation strategies, highlighting trade-offs between interpretability, accuracy, and robustness.

3. Computational analysis of meta-classifiers, including prediction time, model size, and scalability for real-time SHM applications.

4. Exploration of alternative acceleration time series representations to improve system efficiency while preserving damage detection performance.

The proposed methodology provides a practical, interpretable, and computationally efficient framework for enhancing network-level SHM performance, demonstrating the potential of meta-learning and ensemble strategies for intelligent monitoring of large-scale civil infrastructures.

## 2 Related Work

Many work has explored decentralized and edge-based architectures in the context of SHM of civil structures. For example, Sharma et al. [14] proposed a decentralized wireless sensor network (WSN) that executes the Damage Localization Assurance Criterion (DLAC) algorithm locally on sensor nodes. By processing portions of the algorithm locally rather than transmitting all data to a central base station, their system reduced latency by up to 64.8% and energy consumption by up to 69.5% while accurately localizing structural damage. Similarly, Buckley et al. [6] introduced a low-power, IoT-driven EdgeSHM framework in which MEMS accelerometers and microcontrollers compute frequency- and time-domain features at the sensor nodes before sending processed information via long-range, low-power communication.

Although decentralized architectures reduce communication and computation burdens, they often do not aggregate information in an intelligent and efficient manner, which can lead to performance degradation. Moreover, in practical SHM systems, local sensor models may suffer from instability, noise sensitivity, and limited predictive power due to restricted local information. To mitigate these issues, ensemble learning has been applied in SHM. Madani et al. [17] proposed a stacking framework combining multiple classifiers—decision tree, support vector machine, and convolutional neural network—with a K-nearest neighbor meta-learner to improve vibration-based damage detection. Similarly, Asghari et al. [2] introduced a deep ensemble learning approach using multi-headed neural networks aggregated via a meta-learner. These methods successfully reduce uncertainty and improve classification accuracy. However, they focus on ensembling predictions from single sensors or single datasets, rather than aggregating outputs across multiple distributed sensors, leaving the challenge of robust network-level prediction largely unaddressed.

A recent trend in SHM system networks research is the adoption of federated learning (FL), which enables distributed training of ML models across sensor nodes without sharing raw data. Zhang et al. [1] proposed a personalized FL
framework where sensors act as clients training local models on-site while a cen-
tral server aggregates knowledge. This approach preserves data privacy, captures
correlations among sensor nodes, and achieves high damage detection accuracy
on bridge datasets (up to 97%). Yang et al. [21] extended FL to miter gate infras-
tructures, addressing communication, privacy, and computational constraints.
Wu et al. [19] developed a federated Convolutional neural network framework for
submerged structure-foundation systems, standardizing local vibration signals
and aggregating model knowledge without centralizing data. Cheema et al. [7]
further proposed a clustered federated learning (CFL) framework for population-
based SHM, where “population-based” refers to monitoring multiple bridges as
a collective system. Their approach clusters bridges according to data similarity,
trains cluster-specific models, and enables knowledge transfer to new bridges.

These FL-based approaches demonstrate the potential of network-level ML
aggregation for improving SHM performance under decentralized or resource-
constrained conditions. Nevertheless, FL primarily aggregates model weights
rather than predictions, which can be computationally intensive, less inter-
pretable, and sometimes unsuitable for edge deployment with limited resources
when powerful local classifiers are required. Moreover, in many SHM networks,
privacy concerns are not critical, making the overhead of FL less necessary.

Despite progress in decentralized SHM and federated frameworks, a criti-
cal gap remains in developing robust network-level aggregation strategies for
large-scale civil infrastructure SHM systems. Our work addresses this gap by
combining local classifiers through meta-learning-based aggregation. By aggre-
gating probabilistic predictions rather than model parameters, the proposed ap-
proach leverages decentralized processing and machine learning-driven aggrega-
tion, providing a lighter, simpler, and more interpretable edge deployment while
maintaining robustness, scalability, and accuracy.

3 Single-Sensor Classification

**3.1 Methodology**

In the proposed SHM framework, each sensor operates independently, with a
dedicated classifier trained on its segmented time series data. The primary ob-
jective is to assess the structural condition based solely on local measurements,
thereby reducing reliance on centralized computation and enabling potential
edge deployment.

In our previous work [13], we investigated the impact of segment length and
downsampling on the performance of these per-sensor classifiers. This analysis
helps to understand how temporal resolution and data reduction affect classifi-
cation accuracy, as well as other practical considerations for edge deployment.

To capture temporal patterns efficiently, we adopt the state-of-the-art ROCKET
(RandOm Convolutional KErnel Transform) model [10]. ROCKET applies a
large number of random convolutional kernels to the input time series, trans-
forming it into a feature space where simple linear classifiers can achieve high ac-
curacy. Formally, for a time series segment $x \in \mathbb{R}^T$, ROCKET generates a transformed feature vector $f \in \mathbb{R}^{2K}$ using K random convolutional kernels $\{k_i\}_{i=1}^K$. Each kernel produces two features via global max pooling and the proportion of positive values (PPV):

$$f_{2i-1} = \max(x * k_i), \quad f_{2i} = \text{PPV}(x * k_i), \quad i = 1, \dots, K \quad (1)$$

where * denotes convolution, and

$$\text{PPV}(\mathbf{y}) = \frac{|\{j : y_j > 0\}|}{|\mathbf{y}|} \quad (2)$$

computes the proportion of positive values in $\mathbf{y}$. These features are then fed into a logistic regression classifier, which outputs probabilistic predictions for each damage class.

We selected ROCKET for its combination of computational efficiency, scalability, and state-of-the-art performance on time series classification tasks. Its lightweight feature extraction and linear classifier make it suitable for edge deployment, where low-latency inference is critical. However, for the RT345 dataset (see section 5) and across multiple segment lengths, the average per-sensor prediction accuracy did not exceed 50% (see Figure 3 below), highlighting the need for intelligent aggregation across sensors.

## 3.2 Base Classifier Outputs

For each segment, the trained ROCKET classifier produces a vector of class probabilities:

$$\mathbf{p}_s = [p_{s,1}, p_{s,2}, \dots, p_{s,C}]^\top, \quad \sum_{c=1}^{C} p_{s,c} = 1 \quad (3)$$

where $s$ indexes the sensor and $C$ is the number of damage classes. For a network of $S$ sensors, these probability vectors are concatenated to form the meta-feature vector for aggregation:

$$\mathbf{P} = [\mathbf{p}_1^\top, \mathbf{p}_2^\top, \dots, \mathbf{p}_S^\top]^\top \in \mathbb{R}^{S \cdot C} \quad (4)$$

These probabilistic outputs serve as input for subsequent aggregation strategies, enabling the combination of predictions across the sensor network to enhance overall damage detection performance.

# 4 Aggregation Strategies

## 4.1 Soft Averaging

Soft averaging is a simple and interpretable method for aggregating probabilistic predictions from multiple sensors at the network gateway. In this approach, all sensors are assigned equal weight and their class probability estimates are
averaged to produce the final prediction. Formally, for S sensors and C damage classes, the aggregated probability for class c is given by:

$$ \hat{p}_c = \frac{1}{S} \sum_{s=1}^{S} p_{s,c}, \quad c = 1, \dots, C, \qquad (5) $$

where $p_{s,c}$ denotes the probability assigned to class c by sensor s. The final predicted class is then obtained as:

$$ \hat{y} = \arg \max_{c} \hat{p}_{c}. \qquad (6) $$

The theoretical foundation for soft averaging lies in ensemble learning theory, where combining multiple diverse classifiers can reduce variance and improve generalization [11]. According to Condorcet's Jury Theorem [16], if individual classifiers are better than random and make independent errors, the ensemble accuracy approaches 100% as the number of classifiers increases. While perfect independence is rarely achieved in practice, even moderate diversity can yield substantial improvements.

Soft averaging serves as a baseline for evaluating more advanced aggregation strategies.

**Segment Length and Ensemble Performance** The effectiveness of soft averaging depends critically on segment length and the correlation structure of sensor errors:

- **Long segments:** Extended temporal windows provide richer contextual information, resulting in higher individual sensor accuracy. However, sensors tend to make correlated errors (e.g., all sensors misclassify similar structural events), limiting the benefit of ensemble averaging.

- **Short segments:** Reduced temporal context decreases individual sensor accuracy, but different sensors capture complementary local dynamics and respond differently to noise. This leads to more independent prediction errors, which soft averaging can effectively cancel out, yielding improved ensemble accuracy.

**Sensor Disagreement and the “Wisdom of the Crowd” Effect** Sensor disagreement quantifies the diversity of predictions across the sensor network. It is computed as the average pairwise disagreement rate:

$$ \text{Disagreement} = \frac{2}{S(S-1)} \sum_{i<j} \frac{1}{N} \sum_{n=1}^{N} \mathbf{1}\{\hat{y}_{i}^{(n)} \neq \hat{y}_{j}^{(n)}\} \qquad (7) $$

where N is the number of test samples, $\hat{y}_{i}^{(n)}$ denotes the predicted class by sensor i for sample n, and $\mathbf{1}\{\cdot\}$ is the indicator function. High disagreement indicates that sensors provide complementary information, enabling the ensemble to leverage diverse perspectives—a phenomenon known as the “Wisdom of the Crowd” effect. This diversity is particularly valuable when individual classifiers have comparable (but imperfect) accuracy.
## 4.2 Stacking

To further exploit the complementary information across sensors, we propose a stacking-based aggregation strategy. Unlike soft averaging, where all sensors contribute equally, stacking learns to assign different weights to sensors depending on their reliability and relevance in each damage classification scenario. In this way, noisy or less informative sensors can be automatically down-weighted, improving robustness.

In ensemble learning, *stacking* (or stacked generalization) refers to a two-level framework where the outputs of several base learners are combined by a higher-level model, called the *meta-classifier* [15]. The base classifiers provide predictions (in our case, per-sensor class probabilities), which are used as input features for the meta-classifier on the network gateway. This meta-classifier then learns how to best integrate the base outputs, capturing sensor-specific relevance and interdependencies (Figure 1).

<figure><img src="image_3.png" /><figcaption>Fig. 1: Stacking the class probabilities from all sensors at the gateway to train the meta-classifier.</figcaption></figure>

**Linear Meta-Classifier (Logistic Regression)** In the linear setting, we use Multinomial Logistic Regression (LR) as the meta-classifier. Each sample is represented by the concatenation of all sensor probability vectors. For *S* sensors and *C* classes, this yields an input of dimension *S* × *C*. For instance, with *S* = 30 sensors and *C* = 7 classes, the meta-classifier receives 210 features per sample.

Formally, the logistic regression model learns a coefficient matrix of size (*C*, *S* × *C*), where each coefficient can be interpreted as the influence of a specific sensor-class probability on the final prediction. This makes LR particularly attractive in SHM contexts, since the learned weights provide interpretability by highlighting which sensors (and which class assignments) are most influential.
The predicted probability for class c is computed using the softmax function:

$$ \hat{p}_c = \frac{\exp(\mathbf{w}_c^\top \mathbf{x} + b_c)}{\sum_{j=1}^{C} \exp(\mathbf{w}_j^\top \mathbf{x} + b_j)}, \quad c = 1, \dots, C \qquad (8) $$

where $\mathbf{x} \in \mathbb{R}^{S \cdot C}$ is the input vector of concatenated sensor probabilities, $\mathbf{w}_c$ is the weight vector for class c, and $b_c$ is the bias term. The predicted class is then:

$$ \hat{y} = \arg \max_{c} \hat{p}_c \qquad (9) $$

This softmax-based multinomial logistic regression allows the meta-classifier to assign different weights to each sensor's contribution for each class, capturing cross-sensor interactions while maintaining interpretability.

Two variants are considered:

- **Full probability input:** The LR is trained on the complete probability distribution across all classes and sensors. This enables the model to capture cross-sensor interactions, e.g., "If Sensor 5 assigns low probability to *baseline* but Sensor 7 assigns high probability to *damage_2*, then predict *damage_2*."

- **True-class probability input:** The LR is trained only on the probabilities assigned to the true class for each sensor. In this case, LR coefficients directly represent per-sensor weights for each class.

**Nonlinear Meta-Classifier (Random Forest)** While logistic regression assumes linear relationships, it cannot capture more complex interactions among sensor outputs. To address this limitation, we employ a Random Forest (RF) meta-classifier, which can model nonlinear dependencies and conditional rules between sensors.

RF is particularly well-suited to this task because it can naturally encode logical interactions (e.g., AND/OR conditions) between sensor predictions. For example:

IF (Sensor1 predicts Damage_2 > 0.5) AND (Sensor2 predicts Damage_3 > 0.6) THEN predict Damage_2.

By combining multiple decision trees, RF creates flexible, nonlinear decision boundaries and improves robustness against overfitting. However, this gain in flexibility comes at the cost of reduced interpretability compared to LR.

# 5 Dataset and Preprocessing

## 5.1 Dataset Description

For this study, we use the RT345 dataset [20], collected from the Route 345 Bridge over Big Sucker Brook in Waddington, NY. The bridge was instrumented prior to demolition, providing an opportunity to record structural responses under both baseline (healthy) and controlled damage conditions.
The north span was instrumented with thirty dual-axis accelerometers and twenty strain transducers, with the accelerometers distributed uniformly along the five stringers. For our analysis, only the acceleration data are used to detect and classify structural damages, and the strain measurements are excluded. Figure 2 illustrates the spatial layout of the accelerometers along the stringers.

Several damage scenarios were imposed, including forced settlement at a rocker bearing and removal of bolts from diaphragm connections as illustrated in Table 1. For each scenario, a truck traversed the bridge three times at typical local road speeds (see Figure 2), and at least three vibration time histories were recorded to capture variability in ambient excitations while maintaining consistent structural conditions. This ensures generalizability for per-sensor classifier training and evaluation.

<table><caption>Table 1: Overview of imposed damage scenarios on RT345 bridge.</caption><thead><tr><td>Scenario</td><td>Description</td></tr></thead><tbody><tr><td>Initial baseline</td><td>structural response prior to imposed damage</td></tr><tr><td>Damage Scenario 1</td><td>imposed displacement at bearing of ~3 mm</td></tr><tr><td>Damage Scenario 2</td><td>imposed displacement at bearing of ~5 mm</td></tr><tr><td>Second baseline</td><td>hydraulic jack removed from bearing</td></tr><tr><td>Damage Scenario 3</td><td>removal of four bolts at intermediate diaphragm connection</td></tr><tr><td>Damage Scenario 4</td><td>removal of the remaining two bolts at the diaphragm connection</td></tr><tr><td>Damage Scenario 5</td><td>removal of all six bolts at intermediate diaphragm connection</td></tr><tr><td>Damage Scenario 6</td><td>removal of all six bolts at the end diaphragm connection</td></tr></tbody></table>

## 5.2 Preprocessing and Segmentation

Before analysis, we preprocessed the dataset to ensure data quality and consistency:

- Missing data were handled, and we excluded the last damage scenario due to excessive missing values. Damage Scenario 6 contains numerous missing values across several time series due to sensor failures. To maintain the robustness of the proposed approach, this scenario has been excluded from the study.

- Outliers were clipped using a 4-standard deviation (z-score) threshold.

- Signals were segmented into fixed-length windows to capture sufficient temporal context.

- Each axis of each sensor was normalized to ensure fair per-sensor classifier training and consistent meta-learning aggregation.

These preprocessing steps enable robust training of per-sensor classifiers and effective aggregation at the network level, while maintaining generalizability across varying ambient excitations.
<figure><img src="image_4.png" /><figcaption>Fig. 2: Location of instrumentation and truck wheel positions in static load tests: a) cross section; b) framing plan [20]</figcaption></figure>

# 6 Experimental Results and Evaluation

## 6.1 Results

**Segment Length and Ensemble Performance** The effectiveness of ensemble strategies strongly depends on the choice of segment length and the correlation of errors across sensors. Shorter segments tend to yield lower per-sensor accuracy but higher sensor disagreement, whereas longer segments improve single-sensor accuracy but reduce the diversity that ensembles can exploit.
To account for the limited dataset size and the inherent variability in train/test splits, we conducted 5-fold stratified cross-validation with a fixed random seed (42). We report mean accuracy across splits for both individual sensors and aggregated ensemble predictions. Table 2 summarizes the per-sensor accuracies, network-aggregated accuracies, and average disagreement across different segment lengths for the two base classifiers.

Figure 3 illustrates the trends for both ROCKET and STSF models.

- **Short segments:** Although individual sensor accuracy decreases, disagreement across sensors increases. This diversity enables ensembles to achieve substantial performance gains, sometimes approaching the accuracy of longer segments, but with reduced computational cost.

- **Long segments:** While per-sensor accuracy is higher, correlated errors limit the improvement obtainable through ensembling.

For the remainder of this study, we use ROCKET with the current configuration as the base classifier.

<figure><img src="image_5.png" /><figcaption>Fig. 3: Effects of segment length on individual sensor accuracy, sensor disagreement, and ensemble performance using soft averaging across multiple train/test splits: ROCKET (left) vs. STSF (right).</figcaption></figure>

**Meta-Classifier Weights and Interpretability** To investigate how the linear meta-classifier combines sensor predictions, Figures 4 and 5 visualize the learned weights. Figure 4 shows the logistic regression coefficients assigned to each sensor's probability of the actual class. The heatmap reveals that certain sensors consistently provide more discriminative information for specific damage classes. Noisy or ambiguous sensors are assigned lower weights, confirming that the linear meta-classifier effectively down-weights unreliable sources. For example, the meta-classifier heavily relies on sensor 11's probability estimate when predicting damage 2.

Figure 5 shows, interestingly, that contributions are not limited to the actual class probabilities: cross-class contributions are also observed. For instance,
<table><caption>Table 2: Per-sensor, aggregated performance accuracies and average disagreement across segment lengths for ROCKET and STSF classifiers.</caption><thead><tr><th>Model</th><th>Configuration</th><th>Segment Length</th><th>Avg. Per-Sensor Accuracy.</th><th>Aggregated Accuracy</th><th>Avg. Disagreement</th></tr></thead><tbody><tr><td rowspan="8">ROCKET</td><td rowspan="8">50 kernels LR max_iter=50, solver=lbfgs</td><td>3500</td><td>0.496</td><td>0.730</td><td>0.594</td></tr><tr><td>2500</td><td>0.408</td><td>0.641</td><td>0.656</td></tr><tr><td>2048</td><td>0.404</td><td>0.667</td><td>0.660</td></tr><tr><td>1500</td><td>0.388</td><td>0.660</td><td>0.686</td></tr><tr><td>800</td><td>0.338</td><td>0.603</td><td>0.735</td></tr><tr><td>600</td><td>0.344</td><td>0.694</td><td>0.757</td></tr><tr><td>400</td><td>0.323</td><td>0.660</td><td>0.771</td></tr><tr><td rowspan="8">STSF</td><td rowspan="8">50 estimators</td><td>3500</td><td>0.570</td><td>0.842</td><td>0.507</td></tr><tr><td>2500</td><td>0.473</td><td>0.615</td><td>0.569</td></tr><tr><td>2048</td><td>0.427</td><td>0.625</td><td>0.556</td></tr><tr><td>1500</td><td>0.424</td><td>0.533</td><td>0.456</td></tr><tr><td>800</td><td>0.369</td><td>0.573</td><td>0.657</td></tr><tr><td>600</td><td>0.373</td><td>0.658</td><td>0.717</td></tr><tr><td>400</td><td>0.356</td><td>0.627</td><td>0.716</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 4: Per-sensor weights learned by the logistic regression meta-classifier, highlighting the relative contribution of each sensor to ensemble decisions.</figcaption></figure>

when detecting damage 3, the classifier leverages information not only from the probability of class 3 but also from probabilities associated with classes 5 and 6. This reflects inter-sensor and inter-class relationships that naïve soft averaging cannot capture.
<figure><img src="image_7.png" /><figcaption>Fig. 5: Heatmap of all-class weights per sensor. Rows correspond to sensors, columns to class probabilities, showing how each contributes to predicting each damage class.</figcaption></figure>

## 6.2 Comparison of Aggregation Strategies

Figures 6 and 8 illustrate the performance of the aggregation methods across different segment lengths: soft averaging, linear meta-classification (multinomial logistic regression with `max_iter=100` and `lbfgs` solver), and nonlinear meta-classification (random forest with 50 decision trees).

For this evaluation, we conducted 5 random stratified train/test splits using fixed seeds (42 + *i*, with *i* = 0, ..., 4) to ensure reproducibility. For each split, per-sensor ROCKET classifiers were trained independently, and their predictions were aggregated according to the selected method. The resulting accuracies, along with the average disagreements, are reported as mean ± standard deviation across splits in Table 3.

<figure><img src="image_8.png" /><figcaption>Fig. 6: Comparison of Soft, Linear, and Nonlinear Meta-Classifier accuracy versus segment length.</figcaption></figure>
Figure 7 shows an example of the confusion matrix obtained after aggregation using the proposed Linear Meta-Classifier.

- **Linear meta-classifiers:** By applying logistic regression weights to the full class probability vectors, they consistently outperform soft averaging — particularly for short segments—by learning sensor-specific contributions that reflect both reliability and class relevance.

- **Nonlinear meta-classifiers:** Random forests capture higher-order dependencies between sensors and provide additional accuracy gains in certain complex scenarios. However, these gains come at the expense of interpretability and stability, as shown by larger performance variability.

To simulate bandwidth-constrained scenarios, we also evaluated the models using downsampled time series (factor 2). Results in Figure 8 show that down-sampling reduces computational and communication costs while having minimal impact on ensemble performance, especially when short segments are used.

<figure><img src="image_9.png" /><figcaption>Fig. 7: Confusion matrix of the aggregated probabilities with the linear meta classifier</figcaption></figure>

## 6.3 Computational Considerations

Beyond prediction accuracy, computational efficiency is crucial for real-time SHM deployment, where multiple sensors require rapid inference under limited hardware and bandwidth constraints [3].

- **Prediction time:** LR prediction consists of a single matrix-vector multiplication followed by a softmax operation, whereas RF requires traversing multiple decision paths across all trees, aggregating predictions, and performing majority voting or probability averaging [5].
<figure><img src="image_10.png" /><figcaption>Fig. 8: Accuracy comparison of Soft, Linear, and Nonlinear Meta-Classifiers across segment lengths after downsampling time series by a factor of 2.</figcaption></figure>

<table><caption>Table 3: Comparison of aggregation strategies across segment lengths. Values are mean ± standard deviation over 5 random splits.</caption><thead><tr><th rowspan="2">Segment</th><th rowspan="2">Soft Averaging</th><th colspan="2">Linear</th><th rowspan="2">Avg. Disagreement</th></tr><tr><th>Meta-Classifier</th><th>Meta-Classifier</th></tr></thead><tbody><tr><td>3500</td><td>0.705 ± 0.054</td><td>0.705 ± 0.054</td><td>0.726 ± 0.107</td><td>0.606 ± 0.029</td></tr><tr><td>2500</td><td>0.554 ± 0.052</td><td>0.577 ± 0.049</td><td>0.546 ± 0.062</td><td>0.655 ± 0.028</td></tr><tr><td>2048</td><td>0.631 ± 0.054</td><td>0.637 ± 0.042</td><td>0.562 ± 0.110</td><td>0.639 ± 0.009</td></tr><tr><td>1024</td><td>0.586 ± 0.048</td><td>0.626 ± 0.049</td><td>0.603 ± 0.033</td><td>0.725 ± 0.010</td></tr><tr><td>800</td><td>0.555 ± 0.021</td><td>0.616 ± 0.031</td><td>0.647 ± 0.049</td><td>0.739 ± 0.009</td></tr><tr><td>600</td><td>0.682 ± 0.043</td><td>0.713 ± 0.034</td><td>0.695 ± 0.055</td><td>0.756 ± 0.008</td></tr><tr><td>400</td><td>0.666 ± 0.024</td><td>0.694 ± 0.025</td><td>0.687 ± 0.015</td><td>0.776 ± 0.003</td></tr></tbody></table>

The computational complexity of LR per sample is $O(n_{features} \times n_{classes}) = O(S \cdot C^2)$, dominated by dense linear algebra operations. For the RF, each prediction requires $O(n_{trees} \times d_{avg})$ operations, where $d_{avg}$ denotes the average tree depth. Despite a similar order of magnitude in arithmetic operations, RF inference incurs substantially higher constant factors due to sequential branching, irregular memory access, and poor cache locality—making LR significantly faster and more suitable for real-time edge execution [5].

- **Model size and memory footprint:** LR maintains a compact weight matrix of size $n_{classes} \times n_{features}$, with $n_{features} = S \times C$. For $S = 30$ and $C = 7$, this results in $7 \times 210 = 1,470$ parameters. In contrast, RF stores complete tree structures, including split thresholds, feature indices, and leaf values across all trees [5]. As a result, the serialized model size of LR remains nearly constant across segment lengths, while RF grows substantially for shorter segments because each tree stores more detailed splits to capture higher-resolution patterns.

Figure 9 illustrates the model size on disk for LR and RF meta-classifiers. LR consistently maintains a small footprint, whereas RF's size increases with
shorter segment lengths, reaching up to ~560 KB, yielding ratios of over 40× compared to LR.

<figure><img src="image_11.png" /><figcaption>Fig. 9: Serialized model size on disk for LR and RF meta-classifiers across different segment lengths.</figcaption></figure>

# 7 Time Features representation for Damage Detection

Building on the demonstrated utility of aggregation strategies, we explored the use of time-domain features extracted from acceleration time series for damage detection in the SHM network. Unlike raw time series approaches, time-domain features provide a compact yet informative representation of the signal, preserving key discriminative information for classification while reducing computational complexity.

## 7.1 Feature Extraction

From each sensor segment, we extracted a comprehensive set of standard time-domain features, including:

- **Autocorrelation (ACF)** up to lag *N*, capturing repeating temporal patterns in the vibration signals.

- **Partial autocorrelation (PACF)** to quantify direct dependencies between time points.

- **Autoregressive (AR) coefficients** obtained from fitting an AR model of order *p* to each segment.

- **Scalar statistical features** such as mean, standard deviation, absolute sum of changes, change quantiles, and percentage of recurring values.
Feature extraction was performed in parallel using the `tsfresh` library [8] to enhance computational efficiency. This process generated a fixed-length feature vector for each segment and sensor, enabling the effective use of traditional classifiers, such as logistic regression and random forests.

## 7.2 Results

Using these time-domain features, we trained a baseline classifier (Multinomial Logistic Regression) for each sensor and evaluated the proposed ensemble strategies, including soft averaging, linear meta-classifiers, and nonlinear meta-classifiers.

Figure 10 summarizes the classification performance across different segment lengths.

<figure><img src="image_12.png" /><figcaption>Fig. 10: Accuracy comparison of ensemble strategies using time-domain features across different segment lengths. Boxplots represent results over multiple random train/test splits.</figcaption></figure>

Interestingly, carefully engineered time-domain features combined with a simple baseline classifier and intelligent aggregation strategies achieved significant improvements in aggregated accuracy across all segment lengths. This demonstrates that, even without complex deep learning models, the combination of meaningful features and sensor-level aggregation can yield robust and efficient damage detection in SHM networks.

## 8 Discussion and conclusions

We considered a sensor network for Structural health monitoring and a decentralized approach (edge computing) in which each sensor hosts a lightweight classifier operating directly on its local time series. The aggregation of individual sensors' predictions enables improving accuracy and robustness [13]. In the present paper, we first focus on intelligent aggregation strategies for single-sensor classifiers in SHM networks, and secondly, consider alternative features for time
series representations. We leverage acceleration time series data from a real in-
strumented structure, the RT345 bridge dataset, to detect and classify structural
damages.

The results from both raw time series aggregation and time-domain feature-
based approaches highlight the effectiveness of learned aggregation strategies in
SHM networks. Key observations include:

- **Linear meta-classifiers (Logistic Regression):** Provide a good trade-off between predictive performance, interpretability, and computational efficiency. The learned weights reveal the relative contribution of each sensor and class probability to the final ensemble prediction, enabling insights into the most informative sensors and damage-specific patterns. Linear aggregation consistently improves over naïve soft averaging, particularly for short segment lengths where sensor-level predictions are more variable.

- **Nonlinear meta-classifiers (Random Forests):** Capture complex inter-sensor dependencies and nonlinear feature interactions, providing additional accuracy gains in challenging scenarios. However, this comes at the cost of interpretability, higher memory usage, and longer prediction times, which may limit deployment in resource-constrained SHM systems.

- **Time-domain feature representation:** Handcrafted features extracted from each sensor segment reduce the input dimensionality while preserving discriminative information. When combined with baseline classifiers and intelligent aggregation, time-domain features achieve competitive or superior aggregated accuracy compared to raw time series models, especially for shorter segments. This demonstrates that feature engineering remains a viable and computationally efficient alternative to deep learning for SHM tasks.

- **Edge deployment considerations:** Linear models trained on time-domain features are lightweight, enable real-time inference with sub-millisecond latency, and can run on memory- and power-constrained devices such as IoT gateways (e.g. Raspberry Pi systems). This makes them highly suitable for wireless sensor networks where energy efficiency and low-latency decisions are critical.

- **Sensor disagreement and reliability:** Across all approaches, sensors producing noisy or ambiguous predictions are automatically down-weighted by the aggregation strategies, highlighting the benefit of learned combination over naïve averaging. This enhances robustness to sensor variability and failure.

Overall, this paper demonstrates that combining single-sensor classifiers with learned aggregation strategies —linear or nonlinear— significantly enhances network-level SHM performance. Importantly, these results confirm that careful feature engineering combined with intelligent sensor fusion can match or even surpass the performance of more complex deep learning approaches, providing both practical and theoretical insights for SHM system design. Futur work should focus on exploring alternative base classifiers, compact yet richer time series representation from many domains to further enhance robustness and scalability.
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