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CENDRe: Concept Extraction with Natural Domain Representations
Antonia Holzapfel, Andres Felipe Posada Moreno, Sebastian Trimpe
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Summary
The paper introduces CENDRe, a concept extraction method for Convolutional Neural Networks (CNNs) applied to time-series data. CENDRe addresses limitations of existing methods by automatically determining the number of concepts via silhouette-guided clustering and extracting concept localizations in both time and frequency domains using gradients of a presence score propagated through differentiable invertible mappings (e.g., Fourier transform). It evaluates representation and importance correctness on synthetic benchmarks and real bearing-fault datasets, demonstrating superior performance in identifying frequency-domain features compared to time-domain-only baselines.
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CENDRe → appliedto → Convolutional Neural Networks
confidence 95% · CENDRe, a concept extraction method for CNNs
CENDRe → extractsconceptsin → Frequency Domain
confidence 95% · CENDRe extracts concepts in both the time and frequency domains
CENDRe → extractsconceptsin → Time Domain
confidence 95% · CENDRe extracts concepts in both the time and frequency domains
CENDRe → evaluatedon → CWRU Bearing
confidence 90% · On real bearing-fault data... CWRU Bearing
ResNet1D-18 → istypeof → Convolutional Neural Networks
confidence 90% · ResNet1D-18
InceptionTime10 → istypeof → Convolutional Neural Networks
confidence 90% · We train three 1D CNN architectures: InceptionTime10
DenseNet1D-121 → istypeof → Convolutional Neural Networks
confidence 90% · DenseNet1D-121
CENDRe → uses → Fourier Transform
confidence 90% · propagated through a differentiable invertible mapping of the input such as a Fourier transform
CENDRe → uses →
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Abstract
Abstract:Convolutional neural networks (CNNs) are widely used for time-series classification, but their deployment in critical domains requires understanding the temporal and spectral patterns that drive their predictions. Concept extraction (CE) methods identify such patterns by analyzing representations within the models' latent space. However, existing time-series CE methods have three limitations: they operate only in the time domain and overlook frequency features, predefine the number of concepts, and produce localizations misaligned with the regions the model uses. We address these limitations by proposing CENDRe, a concept extraction method for CNNs. It first discovers concepts by clustering per-timestep latent representations in two stages, where silhouette-guided aggregation selects the number of concepts automatically. Then, it localizes each concept through gradients of a presence score that contrasts the latent representations with their prototypes, producing masks that concentrate on the regions driving the concept. These gradients, propagated through a differentiable invertible mapping of the input such as a Fourier transform, yield localizations for the same concepts in the frequency domain. Finally, each concept receives a relevance score that quantifies its contribution to each class. On synthetic benchmarks, CENDRe achieves representation correctness comparable to state-of-the-art CE methods and significantly higher importance correctness. On real bearing-fault data, CENDRe extracts the frequency bands driving the model's predictions, located in regions commonly inspected for fault diagnosis, producing evidence to assess the model that time-domain CE methods cannot.
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- Source: https://arxiv.org/abs/2607.29621v1
- Canonical: https://arxiv.org/abs/2607.29621v1
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CENDRe: Concept Extraction with Natural Domain Representations Antonia Holzapfel ∗ Andres Felipe Posada MorenoSebastian Trimpe Institute for Data Science in Mechanical Engineering (DSME) RWTH Aachen University Theaterstraße 35-39, 52062 Aachen, Germany Abstract Convolutional neural networks (CNNs) are widely used for time-series classifica- tion, but their deployment in critical domains requires understanding the temporal and spectral patterns that drive their predictions. Concept extraction (CE) methods identify such patterns by analyzing representations within the models’ latent space. However, existing time-series CE methods have three limitations: they operate only in the time domain and overlook frequency features, predefine the number of concepts, and produce localizations misaligned with the regions the model uses. We address these limitations by proposing CENDRe, a concept extraction method for CNNs. It first discovers concepts by clustering per-timestep latent representations in two stages, where silhouette-guided aggregation selects the number of concepts automatically. Then, it localizes each concept through gradients of a presence score that contrasts the latent representations with their prototypes, producing masks that concentrate on the regions driving the concept. These gradients, propagated through a differentiable invertible mapping of the input such as a Fourier transform, yield localizations for the same concepts in the frequency domain. Finally, each concept receives a relevance score that quantifies its contribution to each class. On synthetic benchmarks, CENDRe achieves representation correctness comparable to state-of-the-art CE methods and significantly higher importance correctness. On real bearing-fault data, CENDRe extracts the frequency bands driving the model’s predictions, located in regions commonly inspected for fault diagnosis, producing evidence to assess the model that time-domain CE methods cannot. 1 Introduction Deep learning, particularly convolutional neural networks (CNNs), has become a widespread choice for time-series classification and regression. Applications range from condition monitoring and predictive maintenance on industrial equipment [25,47] to early-warning and diagnostic systems in healthcare [26,40,14], settings where a wrong prediction can translate into costly misbehaviours. In such domains, stakeholders cannot simply take a prediction at face value: they need explanations to verify that it is based on plausible signal features and not on spurious correlations. For such inspection, explanations must be both faithful to the model’s behavior and expressed in an interpretable domain. Time-domain explanations cover at most half the story. Other domains, particularly the frequency domain, are often better suited to interpreting signals. Since models trained on time signals, especially CNNs, can learn features in either the time or frequency domain, explanation methods should also operate on both domains to make such features intelligible. Among XAI methods, concept methods [36] are post-hoc and grounded in the model’s own latent space, capturing its global behavior. Specifically, Concept Extraction (CE) discovers recurring latent ∗ Corresponding author: holzapfel@dsme.rwth-aachen.de Preprint. arXiv:2607.29621v1 [cs.LG] 31 Jul 2026 patterns that the model learned to distinguish across a dataset [24]. While original work in CE has focused on explaining images [15,22,37,38], recent works have transferred CE to time series [19,52]. However, these existing methods have three main limitations: First, their explanations live exclusively in the time domain, leaving any frequency-domain features the model uses unexplained. Second, the number of conceptsKmust be fixed before extraction, forcing a manual sweep that biases the resulting explanations. Yet K reflects how the model decides and should be determined by the data, not the user. Third, their localization methods produce masks that may either miss regions the model uses [38, 19] or cover regions it does not [52, 51]. These limitations restrict the practical usefulness of existing CE time-series methods, as highly relevant frequency-domain concepts are not covered, and strong priors on concept number and localization are included. We address these three limitations with CENDRe (Concept Extraction with Natural Domain Repre- sentations), a CE method for time-series CNNs that extracts concepts in any natural domain, that is, any invertible transform of the input in which signal content is directly interpretable, such as time or frequency. It does so by clustering timestep-wise activations in two stages, with silhouette-guided aggregation selecting the number of concepts automatically, rather than a user. Each concept is then localized through gradients of a contrastive presence score that compares the latent representations and concept prototypes, with respect to the input. These gradients produce masks that concentrate attribution on the regions influencing the concept. The same gradients, propagated through virtual inspection layers [48], yield concept localizations in other natural domains (for example, the fre- quency domain). We evaluate CENDRe on synthetic time-series with ground-truth concepts in both domains and on two natural bearing-fault datasets [28,45]. Across all settings, CENDRe extracts concepts in both the time and frequency domains, while the baselines are confined to the time domain. On time-domain concepts, CENDRe outperforms the baselines on importance correctness while recovering the same ground-truth concepts. On frequency-domain concepts, CENDRe recovers the ground-truth bands on synthetic data and extracts bands that include the characteristic fault frequencies experts use to identify each fault type on real bearing-fault data [28, 45]. To summarize, the main contributions in this work are: 1.CENDRe. We introduce a CE method for time-series CNNs that, to our knowledge, is the first to extract and visualize concepts in both the time and frequency domains. 2.Concept localization mechanism that transfers across domains. We backpropagate gradients of a contrastive presence score through differentiable invertible transforms of the input, yielding concept masks in any natural domain. 3.Automatic concept number selection. We develop a procedure that automatically identifies the number of relevant conceptsKfrom the latent structure of the data, providing insight into the model. This replaces the manual choice ofKwith the micro-cluster countJ, to which the extraction is substantially less sensitive. 4.Evaluation on synthetic and real datasets. On synthetic time-series, CENDRe substantially outperforms prior CE baselines on importance correctness and uniquely recovers ground- truth concepts in the frequency domain alongside the time domain. On real bearing-fault datasets, it identifies the characteristic fault frequencies experts use for diagnosis. 2 Related Work Concept-based explanations describe a model through human-interpretable patterns that correspond to directions or regions in its latent space [36]. Concept Extraction (CE) discovers such patterns automatically from data, without requiring a predefined example set. In the image domain, established CE methods include ACE [15], ECLAD [38], ConceptSHAP [51], and PACE [22]. In time-series, prior concept-based work has applied concept testing with TCAV [24] and manually curated concept sets [30,12]. Only two recent works perform concept extraction: ECLAD-ts [19] clusters per- timestep local aggregated descriptors at a probe layer, and MultiVISION [52] clusters segments defined by the effective receptive field (ERF) of highly activated neurons. CENDRe is a CE method for time-series CNNs that extracts concepts across natural domains, including the frequency domain. Despite the central role of frequency domain analysis in signal processing and related areas [35,3], no CE method operates in this domain. Spectral attribution methods, on the other hand, produce local explanations directly in the frequency domain either via perturbation or gradients. Perturbation 2 methods in the frequency or time-frequency domain [9,5] probe the model through targeted alterations of the spectrum, and virtual inspection layers [48] insert a differentiable, invertible transform into the forward graph so that attribution can be propagated to any chosen signal representation. These methods are local, explaining individual predictions rather than global model behavior. CENDRe adopts the virtual inspection layer mechanism to bring global concept explanations into the frequency domain, which, to our knowledge, has not been done before. In CE the number of conceptsKis typically fixed manually through sweeps or visual inspection [15, 22,52,38,19]. The same pattern holds in prototype-based XAI, where the per-class prototype count is fixed in advance [8,13], and silhouette scores appear only post hoc to evaluate prototype quality [10,31]. In general clustering, by contrast, silhouette analysis has long been proposed as a criterion for selectingK[41], alongside alternatives such as density-based methods [6], which we include as variants. To our knowledge, CENDRe is the first CE method to determine the number of concepts automatically, guided by silhouette scores, and thus avoiding the inductive bias introduced by manual concept number selection. To localize concepts in the input, existing time-series CE methods typically rely on two heuristics: either upsampling low-resolution activation maps to the input size [38,19], or taking the full ERF of the probe layer as a concept patch [52]. Upsampling attributes importance to the aggregation position even though its activation is driven by the full ERF, while the full-ERF patch can include regions the model did not meaningfully rely on. CENDRe avoids both caveats by taking the gradients with respect to the latent representation of the concept, propagated through virtual inspection layers, so that the attribution is concentrated on the regions that influence the concept, yielding compact localizations across domains. 3 Concept Extraction with Natural Domain Representations (CENDRe) CENDRe is a CE method for time-series CNNs that produces explanations across natural domains with a data-driven number of concepts. Realizing this raises three core technical challenges: (i) discovering the number of conceptsKfrom the data rather than imposing it, (i) localizing each concept on the regions the model actually uses, and (i) extending these localizations to natural domains beyond the input. The paragraphs below introduce our per-timestep latent representation, address each of these challenges, and conclude with a class-wise concept-importance score. Latent representation via LADs.For concept discovery, we need a post-hoc representation of the model’s latent space that captures patterns at multiple levels of abstraction and scales for clustering. We adapt LADs [38,19]: timestep-wise aggregations of activations across layers that leverage CNNs’ approximate translation equivariance, so similar local patterns yield similar LADs regardless of position. To our knowledge, no other CE representation combines these properties. As a first step, we build a per-timestep descriptor that summarizes the model’s local encoding at multiple levels of abstraction. For this, we consider a trained 1D CNNf : R T×D → R n out for supervised classification or regression on multivariate time-series of lengthTwithDchannels and n out targets. We select a subsetLof its internal layers as probe layers. Concept discovery operates on a datasetD ⊂ R T×D . For each input samplex∈D,a ℓ (x)∈ R T ℓ ×D ℓ denotes the activation map at probe layerℓ∈L. We linearly upsample each activation map to the longest probe lengthT ∗ via f U . For eachx∈Dand timestept∈0,...,T ∗ − 1, the LADz t (x)is the per-timestep slice of the concatenated upsampled maps, z t (x) = f U (a ℓ 1 (x))[t] ... f U (a ℓ |L| (x))[t] ∈ R D ∗ , D ∗ : = P ℓ∈L D ℓ .(1) Thus, similar LADs correspond to similar local evidence across the selected layers, and the collection of LADs over the dataset is the input to the first clustering stage. Concept discovery via clustering. CENDRe discovers latent concepts by clustering LADsz t (x) with agglomerative or density-based methods, which select the number of conceptsKautomatically and can capture non-spherical clusters expected in high-dimensional latent spaces. Applying such methods directly to all LADs is infeasible at scale, so we first compress LADs to micro-centroids, and then cluster those micro-centroids into concepts. 3 In the first stage, mini-batchk-means on LADs produces a set ofJ ≫ Kmicro-centroidsΓ = μ j J−1 j=0 with μ j ∈ R D ∗ , and we assign each LAD to its nearest micro-centroid, π(x,t) : =arg min j∈0,...,J−1 ∥z t (x)− μ j ∥ 2 2 .(2) In the second stage, we cluster theJmicro-centroids intoKconcepts via a mappingκ :0,...,J − 1 → 0,...,K − 1, so each conceptC k = μ j | κ(j) = kis a group of micro-centroids capturing similarly encoded patterns. By default, we use silhouette-guided hierarchical aggregation under cosine distance, takingKas the value that maximizes the average silhouette score across micro-centroids (CENDRe silhouette ) [33,41]. We additionally evaluate two variants in Sec. 5:k-means [43] directly on LADs with predefinedK (CENDRe kMeans ), and HDBSCAN [6] on the micro-centroids (CENDRe HDBSCAN ). Hyperparameters are listed in App. C.3. Concept masks. We localize each concept on each input through gradients with respect to an interpretable domain. Gradients concentrate the attribution on regions that relate to the concept, rather than covering the full receptive field or shrinking past the relevant evidence within it. Pulling the same gradient through any invertible differentiable transform allows the visualization of the concept in the corresponding domain (e.g. the input domain or the frequency domain). The presence score, defined below, captures what makes a LAD belong to conceptC k by contrasting its similarity to the assigned micro-centroid with the mean similarity to micro-centroids of the other concepts. To express this contrast, we partition the micro-centroid indices into the in-concept set J k : = κ −1 (k) ⊆ 0,...,J − 1and its complement ̄ J k : = 0,...,J − 1 k , and we collect the timesteps where the LAD ofxis inC k inT k (x) : = t ∈ 0,...,T ∗ − 1 | π(x,t) ∈ J k . To assess how similar a LAD is to a micro-centroid, we use the inverse-decay similarityd(z,μ) : = (1 +∥z − μ∥ 2 ) −1 ∈ (0, 1]between a LADzand a micro-centroidμ. For everyt ∈ T k (x), the presence score is then ρ k (x,t) : = d z t (x), μ π(x,t) − 1 | ̄ J k | X j∈ ̄ J k d z t (x), μ j .(3) A high presence score indicates strong concept membership: the LAD is close to its assigned micro- centroid and far from every micro-centroid of the other concepts. We aggregate it overT k (x)to obtain a sample-level concept presence ̄ρ k (x) : =|T k (x)| −1 P t∈T k (x) ρ k (x,t), with ̄ρ k (x) = 0when T k (x) =∅. To obtain a mask in an arbitrary domain we follow virtual inspection layers [48]: a differentiable invertible transformφis inserted into the computational graph asφ −1 ◦φfor attribution only, leaving the model’s forward mapping unchanged. The transform is fixed and has no learnable parameters, so no training is involved. We instantiateφin two ways: the input domain (φ = id) and the frequency domain (φ = F), whereF : R T×D → C F×D is the real-input fast Fourier transform (rFFT) withF : = ⌊T/2⌋ + 1andx f : = F (x). App. D.1.2 instantiates two further choices, the short-time Fourier and discrete wavelet transforms, which resolve concepts jointly in time and frequency. Through the inserted inverse transform, ̄ρ k becomes a function ofx f , and∇ x f ̄ρ k denotes its gradient atx f = φ(x). The corresponding masks are the sensitivities of ̄ρ k to the input and to the log-power-spectrum, respectively (derivation in App. A), m time k (x) : = ReLU ∇ x ̄ρ k (x) , m freq k (x) : = ReLU ℜ x f ⊙∇ x f ̄ρ k (x) .(4) The ReLU keeps the components where amplifying the signal, or its spectral power, increases the concept’s presence. Negative components mark evidence against the concept and are discarded for localization. Although LADs are indexed on the upsampled gridt ∈ 0,...,T ∗ − 1, masks are obtained by backpropagation to the original inputx ∈ R T×D or its rFFT imagex f ∈ C F×D . For visualization, a noise tunnel is applied on the masks [44] and they are smoothed by a moving average along the domain axis. Concept importance scores. For each classc, conceptC k , and input channelch, we ask whether the presence ofC k on channelchpushes the model toward or away from predictingc, and by how much. The importance scoreI c,k,ch ∈ [−1, 1]encodes this: magnitude reports the strength of the 4 effect. The sign on its own carries no meaning, but opposite signs across classes indicate that the concept pushes those classes apart. We computeI c,k,ch from the gradient of the per-class scorey c with respect to the input, weighted by the concept mask ofC k on channelch. For classification,y c (x) is the logit f (x) c . For regression, y c (x) is the c-th regression output. We measure the instance-level relevance as the overlap between the regions where the concept lives and the regions to whichy c is sensitive. Letm dom k,ch (x)ands dom c,ch (x)denote the channel slices of the mask and the sensitivity ofy c in domaindom∈time, freq. We writex ch ∈ R T andx f,ch ∈ C F for the time- and frequency-domain channels ofx. The relevance is then their inner product over the corresponding domain axis, r c,k,ch (x) : = m dom k,ch (x), s dom c,ch (x) . The sensitivitys dom c,ch (x)is obtained by backpropagation in the chosen domain:s time c,ch (x) = ∂y c (x)/∂x ch , ands freq c,ch (x) =ℜ x f,ch ⊙∇ x f,ch y c (x) (App. A). We smooths dom c,ch (x)with a noise tunnel to damp single-point gradient noise. Then, we averager c,k,ch (x)overD k,ch : = x ∈ D | m dom k,ch (x)̸= 0, the samples on whichC k leaves a non-zero mask on channelch, to obtain a dataset- level estimate of the concept’s effect. IfD k,ch =∅, we set the expectation to0. We divide the result by the largest such average across all (c ′ ,k ′ , ch ′ ) to make entries comparable on [−1, 1]: I c,k,ch = E x∼D k,ch [r c,k,ch (x) ] max c ′ ,k ′ ,ch ′ E x∼D k ′ ,ch ′ [r c ′ ,k ′ ,ch ′ (x) ] + ε , for a small ε > 0. To obtain a single, class-agnostic importance value per concept–channel pair (as required by down- stream metrics), we contrast the most-impacted class against the mean of the rest, I k,ch = 0.5 I c ⋆ (k,ch),k,ch − 1 n out − 1 X c̸=c ⋆ (k,ch) I c,k,ch ∈ [0, 1],(5) withc ⋆ (k, ch) : = arg max c |I c,k,ch |. Forn out = 1, we setI k,ch : = |I 0,k,ch | . Values close to1 identify concepts that selectively drive the prediction for at least one class, whereas values close to0 mark concepts whose contribution is either negligible or uniform across classes. 4 Experimental design A CE method should identify the recurring patterns a model learns, how they relate to the input, and how they influence the predictions. Yet, there is no direct ground truth on what a CNN relies on. Thus, we evaluate CE methods on synthetic datasets where each input is annotated with the masks of its class-discriminative primitives, assuming a model trained on these datasets relies on them. A faithful CE method should produce concepts tied to those primitives (representation) and assign higher importance to concepts on discriminative primitives than to unrelated ones (importance). We compare CE methods by running each through multiple trained models and measuring representa- tion and importance on the resulting concepts. In an experimental run, we train a CNN on a synthetic dataset, run the CE method on the same data to obtain concepts with time- and frequency-domain masks and per-class importance scores, and score these concepts on the validation data against the primitive masks. We repeat across datasets and seeds, then compare CE methods on representation and importance correctness on the synthetic data, and qualitatively on the real bearing-fault datasets CWRU and BearingPD by inspecting how concepts relate to characteristic fault frequencies. 4.1 Evaluation metrics To evaluate a CE method, we assess how much the resulting concepts align with the primitives through two correctness scores: representation, measuring how well concept masks align with primitive masks, and importance, measuring whether each concept’s importance score reflects its primitive’s function in the dataset. To this end, we propose two modifications over the evaluation procedure of [38,19]: We score concept alignment with the Relevance Mass Accuracy rather than the EDT score, and we compute the correctness metrics using soft variants (sRC, sIC) that avoid the sensitivity to an alignment threshold of the original Representation and Importance Correctness scores. 5 Alignment. For the alignment comparison between concepts and primitives to make sense, all masks must be on the same scale. CENDRe masksm dom k (x)can take any real value, so we normalize them per instance to ̃m dom k (x)∈ [0, 1]by clamping negatives to zero and dividing by the maximum, while baseline masks are already bounded or binary. On the ground-truth side,g dom p (x)∈0, 1is the binary mask of primitivep, withn p primitives andKextracted concepts in total. We score the alignment with the Relevance Mass Accuracy (RMA) [2], which for a continuous masku∈ R |dom|×D and binary referencev ∈0, 1 |dom|×D (with|time| = Tand|freq| = F) measures the fraction of u’s positive energy that falls inside v, RMA(u,v) = P t,ch ReLU(u t,ch )· 1[v t,ch > 0] P t,ch ReLU(u t,ch ) + ε ∈ [0, 1],(6) whereε > 0penalizes empty masks. We choose RMA over the EDT score because it accepts continuous masks directly and tolerates models that use only part of a primitive’s support. Representation and Importance metrics. With the concept and primitive alignments computed per instance, we aggregate them into one score per property. Following [19], we treat each pair(k, ch) of concept and channel as its own evaluation unit, and define its alignment to a pair(p, ch)of primitive and channel as the average RMA over the dataset,A p,k,ch : = E x∼D [RMA( ̃m dom k,ch (x), g dom p,ch (x))]∈ [0, 1] . For each primitive, Soft Representation Correctness (sRC) picks the concept whose alignment summed across channels is highest, and averages this over primitives and channels. Soft Importance Correctness (sIC) instead weights each concept’s class-agnostic importanceI k,ch from Eq.(5)by its best alignment to any primitive, sRC : = 1 n p D X p max k X ch A p,k,ch ,sIC : = P k,ch max p A p,k,ch I k,ch P k,ch |I k,ch | ∈ [0, 1].(7) 4.2 Experimental setup Models and training. We train three 1D CNN architectures: InceptionTime10 [21], ResNet1D- 18 [16], and DenseNet1D-121 [20]. Each model takes the standard hyperparameters of its reference implementation, with full configurations listed in App. C.1. We optimize the negative log-likelihood using AdamW [29] (learning rate10 −4 , weight decay10 −7 ) with early stopping on the validation loss (patience20, up to500epochs). Each dataset is split 80/20 with balanced sampling, and every training configuration is repeated over 11 seeds (0–10). Datasets. We use two synthetic binary-classification families with known ground-truth primitive masks: SYNTHETICLOCAL, which injects localized temporal shapes (square, circle, triangle) into a noisy sinusoidal baseline, and SYNTHETICFREQUENCY, which places class-discriminative peaks in specific frequency bands (mid, mid-high, high). For real-world signals we use CWRU Bearing [45], a rolling-element bearing fault dataset with four classes (healthy plus three fault locations) and two accelerometer channels (fan end and drive end). Both channels are sampled atf s = 12kHz and segmented into windows of T = 2048 samples with stride 1024 (50% overlap). The full dataset list is in Table 1 of App. B, with additional synthetic variants and a second bearing-fault benchmark, which are evaluated in App. D. Concept extraction hyperparameters. All CE methods are implemented in PyTorch [1] and require probe layers per model, listed in Table 4 of App. C. Each method runs on256samples with the same random seed as the corresponding model training, is fit on the model’s training partition and evaluated on its validation partition. The methods differ in how the number of conceptsKis set. For fixed-Kmethods (ECLAD-ts, MultiVISION, CENDRe kMeans ) we sweep overK ∈2, 4, 6, 8, 10 and report the configuration with the highest sRC. For methods that determineKautomatically (CENDRe silhouette , CENDRe HDBSCAN ) we take50micro-clusters, a choice whose effect the sensitivity analysis in App. E quantifies. CENDRe’s noise tunnel uses5samples with Gaussian perturbations of standard deviation0.01for instance masks,0.1for logit sensitivities, and a moving average of5for visualizations. 6 5 Empirical results We report results on the SYNTHETICLOCAL and SYNTHETICFREQUENCY dataset families, and on the natural bearing-fault dataset CWRU. The experimental setup follows Section 4, with fur- ther evaluations in App. D. The main findings of our experiments are: (i) On SYNTHETICLOCAL, CENDRe extracts concept regions that match the ground truth qualitatively, with representation correctness similar to ECLAD-ts but higher importance correctness. (i) Frequency-domain concepts on SYNTHETICFREQUENCY localize in the ground-truth bands, with high representation and im- portance correctness. (i) Concepts extracted from CWRU separate cleanly by class and align with characteristic fault frequencies, evidencing that CENDRe transfers from controlled synthetic settings to real signals. CENDRe silhouette 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00 0.01 0 1 C 1 I 0 = 0 . 6 I 1 = − 0 . 6 0100200300400 0.000 0.005 01002003004000100200300400010020030040001002003004000100200300400 ECLAD-ts −0.5 0.0 0.5 1.0 1.5 C 0 I = − 0 . 6 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = − 1 . 0 02004000200400020040002004000200400 MultiVISION −0.5 0.0 0.5 1.0 1.5 C 0 I = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = 0 . 7 02004000200400020040002004000200400 Primitives 0.00.51.0 0 1 y: 0 0.00.51.0 y: 0 0.00.51.0 y: 0 0.00.51.0 y: 1 0.00.51.0 y: 1 0.00.51.0 y: 1 Time [s] squaretriangle Figure 1: Concepts extracted from a ResNet1D-18 trained on a SYNTHETICLOCAL square-triangle dataset. Within each grid, rows are concepts, columns are samples, and row labels show importance scores. The bottom row shows the ground-truth primitive locations for the same samples. CENDRe and ECLAD-ts extract the same input features, while MultiVISION shows no coherent pattern. Synthetic datasets.The SYNTHETICLOCAL family tests whether the compared CE methods extract the local cues time-series CNNs rely on for classification. Figure 1 shows CE on a representative square-triangle variant from this family. ECLAD-ts highlights the complete square defined by the ground-truth primitive. CENDRe picks up the same regions with importance scores of similar magnitude, but weights the corners more strongly, where the signal differs most recognizably from the baseline. MultiVISION, by contrast, returns no coherent pattern. Overall, CENDRe extracts the local cues without tuning the number of concepts, and additionally highlights the most important 7 structures within each concept. Figure 7 of App. D.1.1 repeats this comparison on two further architectures and the two remaining CENDRe variants, with the same outcome. The quantitative metrics in Figure 2 (top row) are consistent with this qualitative picture. For the Figure 2: Boxplots of sRC (left) and sIC (right) on SYNTHETICLOCAL (top) and SYNTHETICFRE- QUENCY (bottom); higher is better. On SYNTHETICLOCAL, CENDRe achieves sRC close to ECLAD-ts and higher sIC, while MultiVISION is weak on both metrics. On SYNTHETICFRE- QUENCY, CENDRe variants reach sRC and sIC values close to those on SYNTHETICLOCAL. CENDRe silhouette/HDBSCAN variants, sRC is slightly lower than ECLAD-ts’s on SYNTHETICLOCAL (k-means is essentially on par). Two factors plausibly contribute. First, the model exploits only a portion of the ground-truth primitive (a shortcut-learning pattern), and CENDRe silhouette/HDBSCAN stays faithful to that partial use, while ECLAD-ts’s broader masks cover the full primitive. Second, k-means and ECLAD-ts are reported at theKwith best sRC, which biases them toward human- recognizable concepts, whereas silhouette and HDBSCAN chooseKfrom the latent structure itself. The latter is also visible in sIC, which is higher for the CENDRe silhouette/HDBSCAN variants, meaning their masks better match the model’s per-timestep importance signal. MultiVISION underperforms on both metrics, in line with the lack of coherent concept structure observed qualitatively. The same ranking holds on the multichannel and confounded variants of the family (Figure 10 of App. D.1.5). Per-dataset values and pairwise significance tests are in App. D.1.3 and App. D.1.4. Across CENDRe variants the differences are small qualitatively (the three variants point to the same regions). Quantita- tively CENDRe silhouette and CENDRe HDBSCAN perform similarly, while CENDRe kMeans trades slightly higher sRC for slightly lower sIC, consistent with theK-selection effect noted above. We favor the silhouette variant because it reaches comparable quality in a single run without aKsweep, and picks a usableKacross datasets out of the box. It is also the more robust of the two to its remaining hyperparameter, the number of micro-clustersJ. Over three orders of magnitude,Jexplains0.02 of the sRC variance under CENDRe silhouette and0.14under CENDRe HDBSCAN , against0.70for the random seed (App. E). The SYNTHETICFREQUENCY family tests whether CE methods extract spectral cues exploited by the model. Among the compared methods, only CENDRe extends to the frequency domain, through its virtual inspection layers, so the baselines are omitted from this analysis. Figure 3 shows that the time-domain masks have no interpretable pattern, whereas the frequency-domain masks concentrate concept mass at specific bands, both per-sample and in the global concept correspondence. In the midhighFreq-highFreqvariant shown,C 0 aligns with a single high-frequency band, whileC 1 mixes mid-high and near-zero content, reflecting how the model composes information from several frequency components. When a model relies on frequency-related features, CENDRe captures them at both instance and global scale, turning uninterpretable time-domain masks into interpretable frequency explanations. Quantitatively, Figure 2 (bottom row) shows the same CENDRe behavior as the local case. CENDRe silhouette/HDBSCAN variants reach slightly lower sRC thank-means but higher 8 Time domain −0.05 0.00 0.05 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00 0.05 −0.05 0.00 0.05 C 1 I 0 = − 0 . 6 I 1 = 0 . 6 0.00.20.40.60.81.0 0.000 0.025 0.00.20.40.60.81.00.00.20.40.60.81.00.00.20.40.60.81.00.00.20.40.60.81.00.00.20.40.60.81.0 Time [s] Frequency domain 5 10 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.0000 0.0005 5 10 C 1 I 0 = − 0 . 6 I 1 = 0 . 6 050100150200 0.0000 0.0005 050100150200050100150200050100150200050100150200050100150200 Frequency [Hz] 0.00 0.05 0.10 0.15 C0 Channel 0 050100150200 0.00 0.05 0.10 0.15 C1 Frequency [Hz] Figure 3: Concepts extracted from a ResNet1D-18 trained on a SYNTHETICFREQUENCY midhighFreq-highFreqdataset. The two left grids show time-domain (top) and frequency-domain (bottom) masks; within each, rows are concepts, columns are samples, and row labels show im- portance scores. The right panel shows the global concept-frequency correspondence, obtained by aggregating the frequency masks across samples. Time-domain masks show no interpretable pattern, whereas frequency-domain masks localize in specific bands, as confirmed by the correspondence plot. sIC. Mean sRC is slightly below SYNTHETICLOCAL’s, while sIC is comparable. CENDRe thus extracts concepts in the frequency domain at quality close to the time domain. Natural datasets. On CWRU (Figure 4), concepts separate mostly by class.C 0 aligns with a sharp, dominant peak at∼2000Hz on the drive-end channel, and shows weaker activity at∼250and ∼1000Hz on the fan-end channel. ConceptsC 2 andC 3 correspond more weakly to bands around 800–1000Hz on the drive-end channel and on the fan-end channel, respectively. These specific bands give domain experts the information needed to assess whether the model is making predictions for the right frequency reasons. Analogous results on BearingPD (App. D.2.2) and on five datasets of the UCR archive [11] (App. D.2.1) support that CENDRe transfers from controlled synthetic settings to real signals across domains. 6 Conclusion We have presented CENDRe, a concept extraction method for time-series CNNs that addresses the three limitations of prior time-series CE methods. First, by propagating concept masks through virtual inspection layers, CENDRe extracts concepts in both time and frequency domains. The frequency masks recover the ground-truth bands on synthetic data and locate concept-specific bands on the natural bearing-fault datasets, where they support verification by domain experts. Second, silhouette-guided aggregation removes the need to fix the number of concepts by hand and delivers results comparable to the best configuration of a manualKsweep in a single run. This replaces the hyperparameterKwith the much less sensitive micro-cluster countJ. Third, the gradient-based localization concentrates on the regions that drive the concept, yielding representation correctness comparable to prior methods and higher importance correctness on the synthetic benchmarks. Being the first CE method to combine time and frequency domain concepts, CENDRe opens new avenues for future research. The same pipeline can be applied to other invertible representations such as time-frequency distributions. Architectures, by contrast, remain a current limitation: CENDRe’s per-timestep LADs rely on the approximate translation equivariance of 1D CNNs. Extending the latent representation to architectures without this property, such as transformers or state-space models, is a natural next step. Furthermore, the discovered concepts could feed back into the development cycle, for instance to guide model debugging or to flag suspicious reliance on shortcut cues. Domain- specific user studies, assessing whether CENDRe’s masks help experts within their own field, are also a natural direction for future work. 9 Time domain −5 0 5 C 0 — de I 0 = − 0 . 6 I 2 = 0 . 9 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1y: 2 ˆ y: 2y: 3 ˆ y: 3 0.000 0.025 −0.5 0.0 0.5 C 0 — fe I 2 = 1 . 0 I 3 = − 0 . 7 0.00 0.01 −0.5 0.0 0.5 C 1 — fe I 2 = 1 . 0 I 3 = − 0 . 8 0.00 0.01 −0.5 0.0 0.5 C 2 — fe I 2 = 0 . 2 0.000 0.005 −5 0 5 C 3 — de I 2 = 0 . 2 0.000.050.100.15 0.000 0.005 0.000.050.100.150.000.050.100.150.000.050.100.15 Time [s] Frequency domain 0 200000 400000 C 0 — de I 0 = − 0 . 6 I 2 = 0 . 9 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1y: 2 ˆ y: 2y: 3 ˆ y: 3 0.00 0.05 5000 10000 C 0 — fe I 2 = 1 . 0 I 3 = − 0 . 7 0.00 0.01 5000 10000 C 1 — fe I 2 = 1 . 0 I 3 = − 0 . 8 0.00 0.01 5000 10000 C 2 — fe I 2 = 0 . 2 0.00 0.01 0 200000 400000 C 3 — de I 2 = 0 . 2 0200040006000 0.00 0.01 020004000600002000400060000200040006000 Frequency [Hz] Concept correspondence 0.0 0.1 0.2 0.3 de C0C1C2C3 0200040006000 0.0 0.1 0.2 0.3 fe 020004000600002000400060000200040006000 Frequency [Hz] Figure 4: Concepts extracted from an InceptionTime10 trained on CWRU. The top grids show time-domain (left) and frequency-domain (right) masks; rows are concepts on channels with high importance and columns are samples. The bottom panel shows the concept-frequency correspondence per channel, with a fan end (fe) and drive end (de) channel. Each concept localizes almost exclusively on one class, with the frequency masks exposing characteristic fault frequencies. Acknowledgments and Disclosure of Funding We thank Franziska Hoffmann for her valuable help in designing the synthetic datasets. In addition, model training and experiments were partially performed with computing resources granted by RWTH Aachen University, under projects p0021919 and p0027430. This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 520460745. Funded by the European Union. This work has received funding from the European High Performance Computing Joint Undertaking (JU) and from the German Federal Ministry of Research, Technology and Space (BMFTR), the Ministry of Culture and Science of North Rhine-Westphalia (MKW NRW) and the Hessian Ministry of Science and Research, Arts and Culture (HMWK) under grant agreement No 101250682. References [1]Jason Ansel, Edward Yang, Horace He, Natalia Gimelshein, Animesh Jain, Michael Voznesen- sky, Bin Bao, Peter Bell, David Berard, Evgeni Burovski, Geeta Chauhan, Anjali Chourdia, Will Constable, Alban Desmaison, Zachary DeVito, Elias Ellison, Will Feng, Jiong Gong, Michael Gschwind, Brian Hirsh, Sherlock Huang, Kshiteej Kalambarkar, Laurent Kirsch, Michael Lazos, Mario Lezcano, Yanbo Liang, Jason Liang, Yinghai Lu, C. K. Luk, Bert Maher, Yunjie Pan, Christian Puhrsch, Matthias Reso, Mark Saroufim, Marcos Yukio Siraichi, Helen Suk, Shunting Zhang, Michael Suo, Phil Tillet, Xu Zhao, Eikan Wang, Keren Zhou, Richard Zou, Xiaodong Wang, Ajit Mathews, William Wen, Gregory Chanan, Peng Wu, and Soumith Chintala. Pytorch 2: Faster machine learning through dynamic python bytecode transformation and graph compi- 10 lation. In ACM International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS ’24, page 929–947. Association for Computing Machinery, 2024. doi: 10.1145/3620665.3640366. [2] Leila Arras, Ahmed Osman, and Wojciech Samek. Clevr-xai: A benchmark dataset for the ground truth evaluation of neural network explanations. Information Fusion, 81:14–40, 2022. [3]Karl Johan Åström and Richard Murray. Feedback systems: an introduction for scientists and engineers. Princeton university press, 2021. [4]James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. Journal of machine learning research, 13(2), 2012. [5]Thea Brüsch, Kristoffer K Wickstrøm, Mikkel N Schmidt, Tommy S Alstrøm, and Robert Jenssen. Explaining time series models using frequency masking. arXiv preprint arXiv:2406.13584, 2024. doi: 10.48550/arXiv.2406.13584. [6] Ricardo JGB Campello, Davoud Moulavi, and Jörg Sander. Density-based clustering based on hierarchical density estimates. In Pacific-Asia conference on knowledge discovery and data mining, pages 160–172. Springer, 2013. [7]Francesca Campolongo, Jessica Cariboni, and Andrea Saltelli. An effective screening design for sensitivity analysis of large models. Environmental modelling & software, 22(10):1509–1518, 2007. [8] Chaofan Chen, Oscar Li, Daniel Tao, Alina Barnett, Cynthia Rudin, and Jonathan K Su. This looks like that: deep learning for interpretable image recognition. Advances in neural information processing systems, 32, 2019. [9]Hyunseung Chung, Sumin Jo, Yeonsu Kwon, and Edward Choi. Time is not enough: Time- frequency based explanation for time-series black-box models. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, pages 394–403, 2024. doi: 10.1145/3627673.3679844. [10] Enyan Dai and Suhang Wang. Towards prototype-based self-explainable graph neural network. ACM Transactions on Knowledge Discovery from Data, 19(2):1–20, 2025. [11] Hoang Anh Dau, Eamonn Keogh, Kaveh Kamgar, Chin-Chia Michael Yeh, Yan Zhu, Shaghayegh Gharghabi, Chotirat Ann Ratanamahatana, Yanping, Bing Hu, Nurjahan Begum, Anthony Bagnall, Abdullah Mueen, and Hexagon-ML Batista, Gustavo. The ucr time series classification archive, October 2018. [12]Thomas Decker, Michael Lebacher, and Volker Tresp. Explaining Deep Neural Networks for Bearing Fault Detection with Vibration Concepts. In INDIN, pages 1–6, 2023. doi: 10.1109/INDIN51400.2023.10218170. [13]Jon Donnelly, Alina Jade Barnett, and Chaofan Chen. Deformable protopnet: An interpretable image classifier using deformable prototypes. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10265–10275, 2022. [14]Riccardo Fratti, Niccolò Marini, Manfredo Atzori, Henning Müller, Cesare Tiengo, and Franco Bassetto. A multi-scale cnn for transfer learning in semg-based hand gesture recognition for prosthetic devices. Sensors, 24(22):7147, 2024. [15] Amirata Ghorbani, James Wexler, James Y Zou, and Been Kim. Towards Automatic Concept- based Explanations. In NeurIPS, volume 32. Curran Associates, Inc., 2019. [16] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pages 770–778, 2016. [17] Jon Hills, Jason Lines, Edgaras Baranauskas, James Mapp, and Anthony Bagnall. Classification of time series by shapelet transformation. Data mining and knowledge discovery, 2014. 11 [18]Sture Holm. A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, pages 65–70, 1979. [19] Antonia Holzapfel, Andres Felipe Posada Moreno, and Sebastian Trimpe. Concept extraction for time series with eclad-ts. In World Conference on Explainable Artificial Intelligence, pages 90–112. Springer, 2025. [20]Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In CVPR, pages 4700–4708, 2017. [21]Hassan Ismail Fawaz, Benjamin Lucas, Germain Forestier, Charlotte Pelletier, Daniel F Schmidt, Jonathan Weber, Geoffrey I Webb, Lhassane Idoumghar, Pierre-Alain Muller, and François Petitjean. Inceptiontime: Finding alexnet for time series classification. Data Mining and Knowledge Discovery, 34(6):1936–1962, 2020. [22] Vidhya Kamakshi, Uday Gupta, and Narayanan C Krishnan. PACE: Posthoc Architecture- Agnostic Concept Extractor for Explaining CNNs. In IJCNN, pages 1–8, 2021. doi: 10.1109/ IJCNN52387.2021.9534369. [23]Dave S Kerby. The simple difference formula: An approach to teaching nonparametric correla- tion. Comprehensive Psychology, 3:11–IT, 2014. [24]Been Kim, Martin Wattenberg, Justin Gilmer, Carrie Cai, James Wexler, Fernanda Viegas, and Rory Sayres. Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV). In ICML, pages 2668–2677. PMLR, 2018. [25] Yejin Kim and Young-Keun Kim. Time-frequency multi-domain 1d convolutional neural network with channel-spatial attention for noise-robust bearing fault diagnosis. Sensors, 23(23): 9311, 2023. [26] Dimitri Kraft and Peter Rumm. Atrial fibrillation and atrial flutter detection using deep learning. Sensors, 25(13):4109, 2025. [27]Gregory Lee, Ralf Gommers, Filip Waselewski, Kai Wohlfahrt, and Aaron O’Leary. Pywavelets: A python package for wavelet analysis. Journal of Open Source Software, 4(36):1237, 2019. [28] Christian Lessmeier, James Kuria Kimotho, Detmar Zimmer, and Walter Sextro. Condition monitoring of bearing damage in electromechanical drive systems by using motor current signals of electric motors: A benchmark data set for data-driven classification. In PHM society European conference, volume 3, 2016. [29]Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017. [30]Anders Gjølbye Madsen, William Theodor Lehn-Schiøler, Áshildur Jónsdóttir, Bergdís Arnardóttir, and Lars Kai Hansen. Concept-based explainability for an eeg transformer model. In MLSP, pages 1–6. IEEE, 2023. [31] Helena Monke, Benjamin Sae-Chew, Benjamin Fresz, and Marco F Huber. From confusion to clarity: Protoscore-a framework for evaluating prototype-based xai. In Proceedings of the 2025 ACM Conference on Fairness, Accountability, and Transparency, pages 2215–2231, 2025. [32]Max D Morris. Factorial sampling plans for preliminary computational experiments. Quality control and applied statistics, 37(6):307–310, 1992. [33]Daniel Müllner. Modern hierarchical, agglomerative clustering algorithms. arXiv preprint arXiv:1109.2378, 2011. [34]Stephen Olejnik and James Algina. Generalized eta and omega squared statistics: measures of effect size for some common research designs. Psychological methods, 8(4):434, 2003. [35] Alan V Oppenheim. Discrete-time signal processing. Pearson Education India, 1999. [36] Eleonora Poeta, Gabriele Ciravegna, Eliana Pastor, Tania Cerquitelli, and Elena Baralis. Concept- based explainable artificial intelligence: A survey. ACM Computing Surveys, 2023. 12 [37]Andrés Felipe Posada-Moreno, Lukas Kreisköther, Tassilo Glander, and Sebastian Trimpe. Scale-preserving automatic concept extraction (SPACE). Machine Learning, 112(11):4495– 4525, 2023. doi: 10.1007/s10994-023-06373-2. [38] Andrés Felipe Posada-Moreno, Nikita Surya, and Sebastian Trimpe. ECLAD: Extracting Concepts with Local Aggregated Descriptors. Pattern Recognition, 147:110146, 2023. [39]Robert B Randall and Jerome Antoni. Rolling element bearing diagnostics—a tutorial. Me- chanical systems and signal processing, 25(2):485–520, 2011. [40]Swathy Ravi and Ashalatha Radhakrishnan. A hybrid 1d cnn-bilstm model for epileptic seizure detection using multichannel eeg feature fusion. Biomedical physics & engineering express, 10 (3):035040, 2024. [41]Peter J Rousseeuw. Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of computational and applied mathematics, 20:53–65, 1987. [42]Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global sensitivity analysis: the primer. John Wiley & Sons, 2008. [43] David Sculley. Web-scale k-means clustering. In Proceedings of the 19th international conference on World wide web, pages 1177–1178, 2010. [44] Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viégas, and Martin Wattenberg. Smooth- grad: removing noise by adding noise. arXiv preprint arXiv:1706.03825, 2017. [45]Wade A. Smith and Robert B. Randall. Rolling element bearing diagnostics using the Case West- ern Reserve University data: A benchmark study. Mechanical Systems and Signal Processing, 64–65:100–131, 2015. [46]Ilya M Sobol. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Mathematics and computers in simulation, 55(1-3):271–280, 2001. [47]Emmanuel Stathatos, Evangelos Tzimas, Panorios Benardos, and George-Christopher Vosniakos. Convolutional neural networks for raw signal classification in cnc turning process monitoring. Sensors, 24(5):1390, 2024. [48]Johanna Vielhaben, Sebastian Lapuschkin, Grégoire Montavon, and Wojciech Samek. Ex- plainable ai for time series via virtual inspection layers. Pattern Recognition, 150:110309, 2024. [49]Frank Wilcoxon. Individual comparisons by ranking methods. In Breakthroughs in statistics: Methodology and distribution, pages 196–202. Springer, 1992. [50] Moritz Wolter, Felix Blanke, Jochen Garcke, and Charles Tapley Hoyt. Ptwt-the pytorch wavelet toolbox. Journal of Machine Learning Research, 25(80):1–7, 2024. [51]Chih-Kuan Yeh, Been Kim, Sercan Arik, Chun-Liang Li, Tomas Pfister, and Pradeep Ravikumar. On completeness-aware concept-based explanations in deep neural networks. In NeurIPS, volume 33, pages 20554–20565, 2020. [52] Raneen Younis, Sergej Zerr, and Zahra Ahmadi. Multivariate time series analysis: An inter- pretable cnn-based model. In International Conference on Data Science and Advanced Analytics (DSAA), pages 1–10. IEEE, 2022. A Derivation of the spectral concept mask This appendix derives the frequency-domain CENDRe formulas of Sec. 3 as the sensitivity of a real scalar functional to the log-power spectrum. The same derivation applies both to the spectral concept mask (setting E = ̄ρ k ) and to the frequency-domain logit sensitivity (setting E = y c (·)). 13 Setup.Letx f =F (x)∈ C F×D be the rFFT of the time-domain signalx, and letE : C F×D → R be any real-valued, differentiable scalar functional. Writingx f = u + ivwithu = ℜ(x f )and v = ℑ(x f ), the power spectrum isP = |x f | 2 = u 2 + v 2 ∈ R F×D ≥0 . All operations below are element-wise over the F × D grid. Wirtinger preliminaries. Treatingx f andx f as formally independent, the Wirtinger derivatives are ∂ x f E : = 1 2 ∂ u − i∂ v E, ∂ x f E : = 1 2 ∂ u + i∂ v E. For real-valuedEwe have∂ x f E =∂ x f E. Following the convention adopted by standard automatic- differentiation libraries (e.g. PyTorch), we set∇ x f E : = ∂ x f E, so that the direction of real steepest ascent in the (u,v) parametrization is 2∇ x f E. Phase-fixed power perturbation.Writingx f = re iθ withr = √ Pand holding the phaseθfixed, dx f dP = e iθ 2 √ P = x f 2P , dx f dP = x f 2P . Chain rule. The complex chain rule gives dE dP = ∂ x f E⊙ dx f dP + ∂ x f E⊙ dx f dP = 1 2P ∂ x f E⊙ x f + ∂ x f E⊙x f , where the second equality uses∂ x f E = ∂ x f E. Since the two terms in the bracket are complex conjugates of each other, dE dP = 1 P ℜ x f ⊙∇ x f E . Log-power form. Multiplying through byPyields the sensitivity with respect tologP, which avoids the 1/P singularity at empty bins and gives a dimensionless quantity: dE d logP = ℜ x f ⊙∇ x f E .(8) CENDRe spectral mask and logit sensitivity.SubstitutingE = ̄ρ k (x f )into(8)and keeping only the positive contributions recovers the spectral concept mask of Sec. 3, m freq k (x) = ReLU ℜ(x f ⊙∇ x f ̄ρ k (x f )) . Similarly, settingE = y c (x)yields the frequency-domain logit sensitivitys freq c,ch (x) = ℜ( x f,ch ⊙ ∇ x f,ch y c (x)) used in the relevance score. B Synthetic Dataset Details We construct four synthetic binary-classification time-series datasets with known ground-truth concept masks (primitives), summarized in Table 1. SYNTHETICLOCAL and SYNTHETICFREQUENCY are reported in the main paper, while SYNTHETICLMC and SYNTHETICLCONF are included only here, as additional tests for per-channel attribution and confounder robustness. Table 1: The four synthetic datasets. Each is instantiated as several variants pairing primitives or sweeping a confounder ratio. NameDVariantsRole SYNTHETICLOCAL19Localized shapes SYNTHETICFREQUENCY16Frequency bands SYNTHETICLMC26Per-channel attribution SYNTHETICLCONF16Confounder robustness Each variant contains1,000samples of lengthT = 400. A base signal is generated by superimposing low-frequency sinusoidal components with additive Gaussian noise, then class-discriminative patterns 14 are injected at random, non-overlapping positions. Binary ground-truth masksg dom p ∈0, 1 |dom|×D (with|time| = Tfor time-domain patterns and|freq| = F = ⌊T/2⌋ + 1for frequency-domain patterns) record the exact locations of each injected pattern per channel, plus the complementary “absence” masks described below. Shared generation parameters are summarized in Table 2. Negatives as primitives. A model can learn “absence of square” or “absence of the frequency band” as a concept just as readily as their presence. We therefore treat the absence of every involved primitive as a primitive of its own, with a ground-truth mask covering the regions where the primitive is absent. This ensures that the correctness scores reward concepts that capture absence as well as presence. Table 2: Shared generation parameters for all synthetic datasets. ParameterValue Sample length T400 Samples per dataset1 000 Class balance50/50 Overlap allowedNo Shape datasets (SYNTHETICLOCAL / SYNTHETICLMC / SYNTHETICLCONF) Base signalNoisy sinusoidal (f s =1000 Hz) Base noise level0.05 Base components2 sinusoids, [0.5, 2] Hz Shape length40 time steps Shape amplitude1.0 Frequency datasets (SYNTHETICFREQUENCY) Base signalFrequency-domain generator (f s =400 Hz) Base noise level0.02 Base components8 sinusoids, [1, 5] Hz Peaks per band5 Peak amplitudeU[1.5, 2.0] Random phaseYes B.1SYNTHETICLOCAL — Localized Shape Patterns SYNTHETICLOCAL uses three primitive shapes of length 40: • Square: a constant-amplitude block. •Triangle: a triangular envelope that peaks at the center and decays linearly to zero at both ends. • Circle: a semi-elliptical (half-circle) envelope, producing a smooth bump. Each shape is injected at a uniformly random position, avoiding overlap with previously placed patterns. Nine variants (D=1) span three regimes: • Paired (3 variants). Each class is characterized by a different shape. Purpose: discriminate two class-specific shapes in a clean setting. Variants:square-circle,square-triangle, triangle-circle. •Single-shape vs nothing (3 variants). One class contains a single shape, the other contains only the noisy base signal. Purpose: probe presence detection of a single primitive against a background. Variants: square-nothing, triangle-nothing, circle-nothing. •Disjunctive vs nothing (3 variants).One class contains either of two shapes (cho- sen at random per sample), the other contains only the noisy base signal.Pur- pose:test whether the method recovers a concept defined as a disjunction of primitives.Variants:squareorcircle-nothing,squareortriangle-nothing, triangleorcircle-nothing. Figure 5 shows an example from the square-circle variant. 15 050100150200250300350400 Time step 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 Amplitude Class 0 square 050100150200250300350400 Time step Class 1 circle Figure 5: Example samples from the SYNTHETICLOCALsquare-circlevariant. Class 0 contains a square pattern (blue region) and class 1 a circle pattern (red region), each injected at a random position into a noisy sinusoidal base signal. B.2SYNTHETICFREQUENCY — Frequency Band Patterns SYNTHETICFREQUENCY uses frequency-domain patterns drawn from three bands: • midFreq: [70, 75] Hz • midhighFreq: [125, 130] Hz • highFreq: [170, 175] Hz Each band is realized by injecting5sinusoidal peaks with amplitudes drawn fromU [1.5, 2.0]and random phases into the discrete Fourier transform of the base signal, then inverting back to the time domain. Ground-truth masks are defined in the frequency domain, marking the bins within each band. Six variants (D=1) span two regimes: • Pure paired (3 variants). Each class is characterized by a different band, with no other peaks. Purpose: clean spectral discrimination, the frequency analogue of the paired regime in SYNTHETICLOCAL. Variants:midFreq-highFreq,midFreq-midhighFreq, midhighFreq-highFreq. •Paired + confounder (3 variants). Each pure-paired base is augmented with the third (unused) band injected at a50/50ratio, equally likely in both classes.Purpose: test whether the method ignores a spectrally prominent but uninformative band and focuses on the causal pair. Variants:midFreq-highFreq_conf-midhighFreq-50-50, midFreq-midhighFreq_conf-highFreq-50-50,midhighFreq-highFreq_conf-midFreq-50-50. Figure 6 shows an example from the midFreq-highFreq variant. We deliberately keep SYNTHETICFREQUENCY single-channel. With discriminative frequencies present in some channels and absent in others, the cross-channel structure is dominated by the low-frequency base signal, and a CNN trained on such a setup is too likely to learn local time- domain shapes induced by these residuals rather than the discriminative bands themselves. The synthetic benchmarks are only informative when model behavior is strongly predictable from the dataset construction, a property that is already strained on simpler datasets, which the multi-channel frequency setting would defeat. B.3SYNTHETICLMC — Multi-Channel concept localization SYNTHETICLMC reuses the three shape primitives of SYNTHETICLOCAL but signals are two-channel (D=2). Six variants in the following format exist: •Random-channel single-shape (3 variants). One class contains a single shape injected into a uniformly chosen channel, the other contains only the noisy base signal in both channels. Purpose: test whether the method recovers a channel-agnostic concept, de- fined by shape rather than channel index. Variants:square-nothing_ch2_random, triangle-nothing_ch2_random, circle-nothing_ch2_random. • Specific-channel paired (3 variants). Each of two class-discriminative shapes is as- signed to a designated channel, one shape per channel. Purpose: test per-channel 16 050100150200250300350400 Time step 0.03 0.02 0.01 0.00 0.01 0.02 Amplitude Class 0 time domain 050100150200250300350400 Time step 0.03 0.02 0.01 0.00 0.01 0.02 0.03 Amplitude Class 1 time domain 0255075100125150175200 Frequency (Hz) 0.0 0.5 1.0 1.5 |FFT| Class 0 frequency domain midFreq 0255075100125150175200 Frequency (Hz) 0.0 0.5 1.0 1.5 2.0 2.5 |FFT| Class 1 frequency domain highFreq Figure 6: Example samples from the SYNTHETICFREQUENCYmidFreq-highFreqvariant. Top: time-domain signals. Bottom: FFT magnitudes with ground-truth frequency bands highlighted (midFreq in blue, highFreq in red). Class 0 contains the midFreq pattern, class 1 the highFreq pattern. attribution, whether the method assigns each concept to the correct channel rather than mixing evidence across them. Variants:square-circle_ch2_specificChannels, square-triangle_ch2_specificChannels,triangle-circle_ch2_specificChannels. B.4SYNTHETICLCONF — Confounder Robustness SYNTHETICLCONF reuses the three shape primitives of SYNTHETICLOCAL (D=1) and adds a non-discriminative confounder shape on top of each paired base, with controlled class–confounder correlation. Six variants combine two paired bases with three confounder ratios: • Bases.square-circlewithtriangleas confounder, andtriangle-circlewith square as confounder. • Ratios.40/60,50/50, and60/40, where the ratio gives the proportion of confounder presence in class 0 versus class 1. Purpose: test robustness to spurious correlations. At50/50the confounder carries no label informa- tion, while at40/60and60/40it does, so a method that latches onto the prominent shape rather than the causal pair is penalized more strongly at the off-balance ratios. C Implementation details This appendix collects the implementation details needed to reproduce CENDRe and the baselines: backbone architectures, the probe layers used for concept extraction, and the CENDRe clustering hyperparameters. C.1 Model architectures Table 3 lists the exact hyperparameters of the three 1D CNN architectures used throughout the experi- ments (§4.2). All three are direct 1D adaptations of the corresponding standard image-classification architectures, with 2D convolutions replaced by 1D convolutions and 2D pooling by 1D pooling. C.2 Probe layers The choice of which layers to analyze constitutes the most critical hyperparameter for CE. Table 4 reports the specific layers selected for each network. In ECLAD-ts, probe layers are sampled at 17 Table 3: Architecture hyperparameters used for the three 1D CNN backbones. Defaults follow each architecture’s reference implementation. ModelConfiguration InceptionTime10 [21]10 inception modules, 32 filters per branch, base kernel size 40, with bottleneck layer and residual connections every three modules. ResNet1D-18 [16]Four stages with [2, 2, 2, 2] residual blocks, expansion factor 1. DenseNet1D-121 [20]Block configuration [6, 12, 24, 16], growth rate 32, compression factor 0.5, 64 initial features. uniform intervals throughout the architecture. MultiVISION instead relies on a single bottleneck layer, which prevents the analysis of blocks that residual connections might bypass. CENDRe adopts the same probe layers as ECLAD-ts. For MultiVISION, the neuron activation extraction threshold is set to the0.99quantile of the layer under examination, and activations are grouped viak-means clustering. Table 4: Layers used for CE with each method and model. ModelECLAD-ts and CENDRe InceptionTime10model.inception_block.inception_layers.n b .bottleneck, n b ∈6, 7, 8, 9 ResNet1D-18 model.layers.n b .1.relu n b ∈0, 1, 2, 3 DenseNet1D-121 model.features.transitionn b .conv, n b ∈1, 2, 3; model.features.denseblock4.block.15.conv2 ModelMultiVISION InceptionTime10model.inception_block. inception_layers.9.bottleneck ResNet1D-18 model.layers.2.1.relu DenseNet1D-121 model.features.transition3.conv C.3 CENDRe clustering hyperparameters The second stage of CENDRe silhouette and CENDRe HDBSCAN runs on theJ = 50micro-centroids returned by mini-batchk-means.CENDRe silhouette uses average linkage and selectsK ∈ 2,..., 10by the silhouette score, both evaluated under cosine distance. CENDRe HDBSCAN uses min_cluster_size = 5under cosine distance. Micro-centroids that HDBSCAN labels as noise are excluded from ̄ J k when computing ρ k . D Additional results This appendix extends Section 5 in two directions. §D.1 broadens the synthetic evaluation: it compares all three CENDRe variants across three architectures, visualizes concepts under two further invertible transforms, breaks the quantitative comparison down per dataset with significance tests, and extends it to two dataset families that test multichannel localization and confounder robustness. §D.2 extends the qualitative evaluation to five real-world datasets from the UCR archive and to a second bearing-fault dataset, BearingPD, on two architectures. D.1 Synthetic datasets We extend the synthetic evaluation of Section 5 along four lines. We first repeat the qualitative comparison of Figure 1 across all three CENDRe variants and three architectures. We then visualize concepts under two further invertible transforms, the short-time Fourier and the wavelet transform (§D.1.2). Next, we break the family-level quantitative comparison of Figure 2 down per dataset and test the resulting ordering for statistical significance (§D.1.3, §D.1.4). Finally, we extend the quantita- tive comparison to two appendix-only dataset families, SYNTHETICLMC and SYNTHETICLCONF, testing multichannel localization and confounder robustness. D.1.1 Qualitative comparison across CENDRe variants and architectures Section 5 reports the silhouette-guided aggregation variant (CENDRe silhouette ) as the default, ex- tracted from a single ResNet1D-18. Figure 7 extends that view in two directions, adding the two remaining CENDRe variants and repeating the extraction on two further architectures. It shows 18 ResNet1D-18 InceptionTime10 DenseNet1D-121 CENDRe silhouette 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.00 0.02 0 1 C 1 I 0 = 0 . 6 I 1 = − 0 . 6 0.00.20.40.60.81.0 0.00 0.01 0.00.20.40.60.81.0 Time [s] 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.000 0.025 0 1 C 1 I 0 = − 0 . 1 0.00.20.40.60.81.0 0.00 0.05 0.00.20.40.60.81.0 Time [s] 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.000 0.025 0 1 C 1 I 0 = − 0 . 6 I 1 = 0 . 6 0.00.20.40.60.81.0 0.00 0.01 0.00.20.40.60.81.0 Time [s] CENDRe HDBSCAN 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.00 0.02 0 1 C 1 I 0 = 0 . 6 I 1 = − 0 . 6 0.00.20.40.60.81.0 0.00 0.01 0.00.20.40.60.81.0 Time [s] 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.00 0.02 0 1 C 1 I 0 = − 0 . 2 0.00.20.40.60.81.0 0.00 0.05 0.00.20.40.60.81.0 Time [s] 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.00 0.02 0 1 C 1 I 0 = − 0 . 6 I 1 = 0 . 6 0.00.20.40.60.81.0 0.00 0.01 0.00.20.40.60.81.0 Time [s] CENDRe kMeans 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.00 0.02 0 1 C 1 I 0 = 1 . 0 I 1 = − 1 . 0 0.00.20.40.60.81.0 0.0000 0.0025 0.00.20.40.60.81.0 Time [s] 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.000 0.005 0 1 C 1 I 0 = 0 . 2 0.00.20.40.60.81.0 0.000 0.005 0.00.20.40.60.81.0 Time [s] 0 1 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0.000 0.005 0 1 C 1 I 0 = − 0 . 9 I 1 = 0 . 8 0.00.20.40.60.81.0 0.00 0.01 0.00.20.40.60.81.0 Time [s] ECLAD-ts −0.5 0.0 0.5 1.0 1.5 C 0 I = − 0 . 6 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = − 1 . 0 0200400 −0.5 0.0 0.5 1.0 1.5 C 0 I = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = − 0 . 2 0200400 −0.5 0.0 0.5 1.0 1.5 C 0 I = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = − 0 . 2 0200400 MultiVISION −0.5 0.0 0.5 1.0 1.5 C 0 I = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = 0 . 7 0200400 −0.5 0.0 0.5 1.0 1.5 C 0 I = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = 0 . 4 0200400 −0.5 0.0 0.5 1.0 1.5 C 0 I = 1 . 0 Importance Score y: 0 ˆ y: 0y: 1 ˆ y: 1 0200400 −0.5 0.0 0.5 1.0 1.5 C 1 I = 0 . 6 0200400 Figure 7: Concepts extracted by CENDRe silhouette , CENDRe HDBSCAN , CENDRe kMeans , ECLAD- ts, and MultiVISION (rows) from a ResNet1D-18, an InceptionTime10, and a DenseNet1D-121 (columns, in that order) trained on the same SYNTHETICLOCAL square-triangle dataset as Figure 1. For space, each panel shows the two most important concepts; within a panel, rows are concepts, columns are one class-0 and one class-1 sample, and row labels report the importance scores. Across architectures, ECLAD-ts isolates the square as a solid block, CENDRe recovers the same regions as ECLAD-ts but as graded masks whose largest values partially sit at the edges, and MultiVISION shows no correspondence to either primitive. On DenseNet1D-121 the CENDRe variants extract additional low-importance concepts (not shown) that further partition the background. the concepts extracted by the five CE methods (rows) from a ResNet1D-18, an InceptionTime10, and a DenseNet1D-121 (columns) trained on the same SYNTHETICLOCAL square-triangle dataset as Figure 1. CENDRe silhouette and CENDRe HDBSCAN set the number of concepts automatically; for CENDRe kMeans , ECLAD-ts, and MultiVISION we report the run whose number of concepts maximizes the RC score. For space, each panel shows the two most important concepts. The three CENDRe variants stay close to each other within every column, pointing to similar regions with comparable importance scores. Reporting CENDRe silhouette alone in the main text therefore loses 19 little. Against the baselines, the comparison reproduces the one in Section 5 on all three architectures. ECLAD-ts marks the square as a uniform block and its complement as a second concept, while CENDRe spreads graded mass over similar regions, with the largest values at the edges of the shapes. MultiVISION splits the signal into broad contiguous segments whose boundaries follow neither primitive. All three architectures yield concepts that map onto the primitives, with slight differences between them. On the ResNet1D-18 and the InceptionTime10, every method separates the discriminative shapes from the background. The DenseNet1D-121 follows the same pattern, with the CENDRe variants splitting the background further into additional low-importance concepts, left out of the figure for space. D.1.2 Visualizations in other natural domains In Section 3, we instantiate the transformφas a Fourier transform to visualize the concepts in the frequency domain, but mention how other invertible transforms could be used instead. While we don’t test them quantitatively, such transforms are straightforward to implement and can be used to visualize the same concepts in other domains, e.g., wavelet transforms or short-time Fourier transforms for time-frequency analysis. We have implemented such transforms by creating virtual inspection layers for short-time Fourier transform (STFT) and discrete wavelet transform (DWT). The STFT builds on the implementation of PyTorch [1], and the DWT on the differentiable filter banks of the PyTorch Wavelet Toolbox [50], which in turn builds on PyWavelets [27]. The STFT S : R T×D → C F×N×D applies the rFFT to overlapping, windowed segments of lengthWwith hop sizeH, producingF : = ⌊W/2⌋ + 1frequency bins acrossNtime frames. BecauseSis complex-valued likeF, the sensitivity in Eq. 4 carries over unchanged after substitutingx s : =S(x) forx f . Its derivation only requires a complex transform output, not a specific transform (App. A). We invertSexactly by choosing an analysis window that satisfies the nonzero-overlap-add condition. The DWTW : R T×D → R T×D decomposesxinto approximation and detail coefficients through an orthogonal wavelet filter bank. This filter bank admits an exact inverse by construction. Since Wis real-valued, unlikeFandS, its mask reduces to the plain ReLU gradient, following the same pattern asm time k in Eq. 4. We selected STFT and DWT because both are exactly invertible, matching the requirement virtual inspection layers place onφ. Other time-frequency transforms, such as the continuous wavelet transform, lack an exact inverse and are left for future work. In Figure 8 we show qualitative results of reports in the time-frequency domain using such transforms. Across the four views, both concepts land where the ground truth places them, but the choice ofφ decides how granular that visualization is. The FFT already isolatesC 0 in the midFreq band, while its C 1 mask spreads over a wide range with no dominant peak. The STFT resolvesC 1 into the highFreq band and shows it active at isolated time frames rather than across the whole record. ComparingC 0 in the FFT and STFT views shows that the lower activation magnitude of the concept corresponds with it being active in a shorter timespan. The DWT recovers the same time structure, but its highest band spans 100 to 200 Hz, so the highFreq band cannot be separated within it. D.1.3 Quantitative evaluation: summary per dataset In Figure 2 of Section 5, each box summarizes the sRC and sIC values across all datasets of a given family. Tables 5 and 6 report the same runs resolved per dataset, as mean±standard deviation with a 95%confidence interval over the33runs per cell (3architectures× 11seeds), with the best method per dataset in bold. The interval is the standard95%Student-tconfidence interval for the mean, ̄y± t 0.975,n−1 s/ √ n. The family-level ranking of Figure 2 holds dataset by dataset and is not an artifact of pooling. On SYNTHETICLOCAL, ECLAD-ts attains the highest sRC on all9datasets, with a family mean of0.741 [0.731, 0.751]separated from CENDRe kMeans ’s0.704 [0.695, 0.714], the next-best method. CENDRe silhouette and CENDRe HDBSCAN trail ECLAD-ts by0.09and0.11sRC respectively, on every dataset and with comparable spread, while MultiVISION is last on sRC everywhere. The ordering reverses in sIC, where CENDRe HDBSCAN is best on all nine datasets (family mean0.858), ahead of CENDRe silhouette (0.792) and MultiVISION (0.722), with CENDRe kMeans at0.658and ECLAD-ts at0.556. ECLAD-ts is also the least stable method under sIC, with per-dataset standard deviations up to0.305against0.140for the two automatic CENDRe variants. Its sRC lead therefore does not 20 0.00 0.05 C 0 I 0 = − 0 . 8 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1 0.0 0.1 0.00 0.05 C 1 I 1 = 0 . 4 0.00.20.40.60.81.0 0.00 0.05 0.00.20.40.60.81.00.00.20.40.60.81.00.00.20.40.60.81.0 Time [s] (a) Time domain 0 10 20 C 0 I 0 = − 0 . 8 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00 0.05 0 10 20 C 1 I 1 = 0 . 4 050100150200 0.00 0.01 050100150200050100150200050100150200 Frequency [Hz] (b) Frequency domain (rFFT) 0 50 100 150 200 Freq [Hz] C 0 I 0 = − 0 . 8 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00.20.40.60.81.0 0 50 100 150 200 Freq [Hz] C 1 I 1 = 0 . 4 0.00.20.40.60.81.00.00.20.40.60.81.00.00.20.40.60.81.0 Time [s] (c) Time-frequency domain (STFT) 0 25 50 100 200 Freq [Hz] C 0 I 0 = − 0 . 8 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00.20.40.60.8 0 25 50 100 200 Freq [Hz] C 1 I 1 = 0 . 4 0.00.20.40.60.80.00.20.40.60.80.00.20.40.60.8 Time [s] (d) Time-frequency domain (wavelet) Figure 8: The same two concepts, extracted with CENDRe silhouette from an InceptionTime10 trained on the SYNTHETICFREQUENCYmidFreq-highFreqvariant, visualized under four choices of the transformφ. The time-domain masks (a) carry no interpretable pattern, while all three spectral views placeC 0 in the ground-truth midFreq band at[70, 75]Hz. The rFFT (b) resolves that band as a sharp peak, but leavesC 1 ambiguous, spreading low mask values over a wide frequency range. The STFT (c) removes the ambiguity:C 1 concentrates in the highFreq band at[170, 175]Hz and is active only at isolated time frames, whereasC 0 persists across the full record. The DWT (d) reproduces the same time structure at coarser frequency resolution, since its highest dyadic band spans100to200Hz and cannot separate the highFreq band inside it. Table 5: Distributional summary (mean± std, 95% CI) of sRC across CE methods and datasets. CENDRe silhouette CENDRe HDBSCAN CENDRe kMeans ECLAD-tsMultiVISION mean± std95% CImean± std95% CImean± std95% CImean± std95% CImean± std95% CI SYNTHETICLOCAL circle-nothing0.731± 0.035[0.718, 0.743]0.716± 0.030[0.705, 0.726]0.774± 0.049[0.756, 0.791]0.824± 0.060[0.803, 0.846]0.584± 0.070[0.559, 0.609] square-circle0.599± 0.053[0.581, 0.618]0.561± 0.035[0.549, 0.574]0.662± 0.066[0.639, 0.686]0.723± 0.094[0.690, 0.756]0.450± 0.052[0.431, 0.468] triangle-circle0.622± 0.073[0.596, 0.648]0.575± 0.061[0.553, 0.597]0.672± 0.087[0.642, 0.703]0.689± 0.099[0.654, 0.724]0.443± 0.057[0.423, 0.463] square-nothing0.710± 0.028[0.701, 0.720]0.711± 0.034[0.699, 0.724]0.757± 0.046[0.741, 0.774]0.813± 0.057[0.792, 0.833]0.600± 0.045[0.584, 0.616] square-triangle0.597± 0.066[0.573, 0.620]0.563± 0.049[0.545, 0.580]0.639± 0.062[0.617, 0.661]0.674± 0.087[0.643, 0.704]0.453± 0.039[0.439, 0.467] triangle-nothing0.742± 0.054[0.723, 0.762]0.729± 0.053[0.710, 0.747]0.804± 0.059[0.783, 0.825]0.838± 0.061[0.817, 0.860]0.609± 0.031[0.598, 0.620] squareorcircle-nothing0.617± 0.054[0.598, 0.637]0.620± 0.035[0.607, 0.633]0.669± 0.064[0.646, 0.691]0.705± 0.046[0.688, 0.721]0.526± 0.042[0.511, 0.541] triangleorcircle-nothing0.623± 0.063[0.601, 0.646]0.618± 0.069[0.592, 0.643]0.677± 0.067[0.653, 0.700]0.701± 0.040[0.687, 0.715]0.525± 0.040[0.511, 0.540] squareortriangle-nothing0.618± 0.060[0.597, 0.640]0.600± 0.073[0.573, 0.627]0.684± 0.073[0.658, 0.710]0.703± 0.040[0.689, 0.717]0.533± 0.035[0.521, 0.546] Mean0.651± 0.078[0.642, 0.660]0.633± 0.082[0.623, 0.642]0.704± 0.084[0.695, 0.714]0.741± 0.091[0.731, 0.751]0.525± 0.077[0.516, 0.534] SYNTHETICFREQUENCY midFreq-highFreq0.503± 0.042[0.488, 0.518]0.504± 0.042[0.489, 0.519]0.593± 0.058[0.573, 0.614]– midFreq-highFreq _conf-midhighFreq-50-50 0.490± 0.050[0.472, 0.508]0.490± 0.049[0.473, 0.508]0.554± 0.054[0.535, 0.573]– midFreq-midhighFreq0.532± 0.034[0.520, 0.544]0.532± 0.034[0.520, 0.544]0.585± 0.052[0.567, 0.603]– midFreq-midhighFreq _conf-highFreq-50-50 0.503± 0.036[0.490, 0.516]0.502± 0.038[0.489, 0.516]0.570± 0.047[0.553, 0.586]– midhighFreq-highFreq0.546± 0.037[0.533, 0.559]0.546± 0.036[0.533, 0.559]0.583± 0.047[0.566, 0.599]– midhighFreq-highFreq _conf-midFreq-50-50 0.488± 0.031[0.477, 0.499]0.489± 0.030[0.478, 0.500]0.540± 0.049[0.523, 0.558]– Mean0.510± 0.044[0.504, 0.517]0.511± 0.044[0.504, 0.517]0.571± 0.054[0.563, 0.578]– Table 6: Distributional summary (mean± std, 95% CI) of sIC across CE methods and datasets. CENDRe silhouette CENDRe HDBSCAN CENDRe kMeans ECLAD-tsMultiVISION mean± std95% CImean± std95% CImean± std95% CImean± std95% CImean± std95% CI SYNTHETICLOCAL circle-nothing0.806± 0.091[0.774, 0.839]0.850± 0.100[0.814, 0.886]0.666± 0.140[0.616, 0.716]0.684± 0.283[0.584, 0.784]0.686± 0.137[0.637, 0.734] square-circle0.721± 0.127[0.676, 0.766]0.816± 0.140[0.767, 0.866]0.605± 0.098[0.570, 0.639]0.550± 0.250[0.461, 0.638]0.735± 0.120[0.692, 0.777] triangle-circle0.794± 0.123[0.751, 0.838]0.849± 0.121[0.806, 0.893]0.653± 0.145[0.602, 0.705]0.474± 0.298[0.369, 0.580]0.743± 0.135[0.695, 0.791] square-nothing0.799± 0.103[0.763, 0.836]0.873± 0.090[0.840, 0.905]0.668± 0.135[0.620, 0.716]0.466± 0.234[0.383, 0.549]0.711± 0.078[0.683, 0.739] square-triangle0.766± 0.097[0.731, 0.800]0.842± 0.111[0.803, 0.881]0.626± 0.110[0.587, 0.665]0.596± 0.228[0.515, 0.677]0.731± 0.104[0.694, 0.768] triangle-nothing0.784± 0.125[0.740, 0.828]0.873± 0.115[0.832, 0.914]0.661± 0.174[0.599, 0.723]0.599± 0.287[0.497, 0.701]0.741± 0.083[0.712, 0.771] squareorcircle-nothing0.819± 0.098[0.784, 0.854]0.870± 0.094[0.835, 0.905]0.712± 0.128[0.667, 0.758]0.551± 0.193[0.483, 0.619]0.707± 0.100[0.672, 0.743] triangleorcircle-nothing0.812± 0.098[0.778, 0.847]0.884± 0.084[0.853, 0.915]0.676± 0.123[0.633, 0.720]0.558± 0.305[0.450, 0.666]0.739± 0.065[0.717, 0.762] squareortriangle-nothing0.827± 0.097[0.792, 0.861]0.869± 0.095[0.834, 0.904]0.651± 0.128[0.606, 0.697]0.523± 0.217[0.446, 0.600]0.706± 0.092[0.673, 0.738] Mean0.792± 0.110[0.779, 0.805]0.858± 0.107[0.846, 0.871]0.658± 0.134[0.642, 0.673]0.556± 0.262[0.526, 0.586]0.722± 0.105[0.710, 0.734] SYNTHETICFREQUENCY midFreq-highFreq0.802± 0.096[0.768, 0.836]0.804± 0.096[0.770, 0.837]0.694± 0.127[0.649, 0.739]– midFreq-highFreq _conf-midhighFreq-50-50 0.776± 0.121[0.733, 0.819]0.775± 0.123[0.731, 0.818]0.690± 0.114[0.650, 0.731]– midFreq-midhighFreq0.820± 0.111[0.781, 0.860]0.819± 0.111[0.780, 0.859]0.744± 0.161[0.687, 0.801]– midFreq-midhighFreq _conf-highFreq-50-50 0.792± 0.109[0.754, 0.831]0.790± 0.111[0.751, 0.830]0.690± 0.108[0.651, 0.728]– midhighFreq-highFreq0.791± 0.103[0.754, 0.828]0.789± 0.103[0.752, 0.825]0.709± 0.144[0.658, 0.761]– midhighFreq-highFreq _conf-midFreq-50-50 0.770± 0.128[0.724, 0.815]0.772± 0.127[0.726, 0.817]0.679± 0.116[0.638, 0.720]– Mean0.792± 0.112[0.776, 0.807]0.791± 0.112[0.776, 0.807]0.701± 0.130[0.683, 0.719]– transfer to sIC: broader masks that cover the ground-truth primitive, selected at theKwith the best sRC, do not track the model’s per-timestep importance signal. 21 On SYNTHETICFREQUENCY, only the CENDRe variants are applicable, since the baselines have no frequency-domain counterpart, which the tables mark as “–”. Both automatic variants lose about 0.06sRC to CENDRe kMeans and gain about0.09sIC, on every dataset of the family, the same trade observed on SYNTHETICLOCAL. CENDRe silhouette and CENDRe HDBSCAN behave similarly on SYNTHETICFREQUENCY (family means within0.001sRC and sIC of one another) though with11 seeds their per-run scores no longer coincide everywhere, matching exactly in42%of the (dataset, architecture, seed) combinations. Compared with SYNTHETICLOCAL, mean sRC is lower (0.510 against0.651for CENDRe silhouette ) while mean sIC is unchanged (0.792in both cases). Frequency- domain concepts thus track the model’s importance signal as well as time-domain ones, while matching the ground-truth band less tightly. D.1.4 Quantitative evaluation: statistical significance Section 5 compares the methods by reading the boxplots of Figure 2, where an ordering is visible but not tested. We test that ordering here, with no additional experiments. Each run is one combination of dataset, architecture, and seed, and every applicable method is evaluated on all of them, so the scores form matched pairs. We test each architecture separately over11seeds, giving99matched pairs per method pair on SYNTHETICLOCAL (9datasets× 11seeds) and66on SYNTHETICFREQUENCY (6datasets× 11seeds). Runs in which CENDRe HDBSCAN returned no concepts are dropped from the pairs involving it. We compare each pair with a two-sided Wilcoxon signed-rank test [49], and Holm-correct thep-values over all pairs within a family, metric, and architecture [18]. We accompany each test with the matched-pairs rank-biserial correlationr[23], the share of pairs favoring the row method minus the share favoring the column method. Figure 9 reports both, with cell color givingr(red for row above column, blue for row below) and cell text the Holm-adjustedp-value. On SYNTHETICFREQUENCY, only the three CENDRe variants enter the comparison, since the baselines have no frequency-domain counterpart. Of the78tests in Figure 9 (10method pairs on SYNTHETICLOCAL and3on SYNTHETICFREQUENCY, per architecture and metric),65reach p < 0.01, and the sign ofrfollows the ordering of §D.1.3 throughout. The tests therefore confirm the reading of Figure 2. The13pairs that do not separate concentrate in three places, with two isolated exceptions. The first is CENDRe silhouette against CENDRe HDBSCAN on SYNTHETICFREQUENCY, which separates on neither metric nor any architecture (p ≥ 0.014across all six tests). This matches their family means in §D.1.3, which differ by less than0.001in both metrics. On SYNTHETICLOCAL the two criteria separate everywhere except in sRC on the InceptionTime10 (p = 0.25). Both methods select the number of centroids automatically and visualize concepts by the same means, so similar behaviour on some datasets is expected. The second is CENDRe kMeans against ECLAD-ts in sRC on SYNTHETICLOCAL, indistinguishable on the DenseNet1D-121 (p = 0.71) and the ResNet1D-18 (p = 0.10) but separated on the Incep- tionTime10. The gap of0.035sRC between their family means is therefore carried by a single architecture. In sIC the same pair separates on all three, with ECLAD-ts behind. The two methods share the same clustering step, so their scores differ mainly through the mode of concept visualization. The visualizations converge when the CENDRe masks are dense, which can occur on individual architecture-dataset combinations. The third is the InceptionTime10 in sIC on SYNTHETICFREQUENCY, where no pair of CENDRe variants reaches the1%level (pbetween0.011and0.015). The other two architectures separate CENDRe kMeans from both automatic variants there, so this reflects the backbone rather than the metric. The two remaining cells both involve MultiVISION in sIC on SYNTHETICLOCAL: against CENDRe silhouette on the DenseNet1D-121 (p = 0.055) and against CENDRe kMeans on the Incep- tionTime10 (p = 0.18). Both are consistent with MultiVISION ranking mid-field on sIC rather than last, and neither affects the ordering among the CENDRe variants or against ECLAD-ts. D.1.5 Multichannel localization and confounder robustness We extend the quantitative evaluation of Figure 2 to two synthetic datasets: SYNTHETICLMC (D=2, per-channel attribution) and SYNTHETICLCONF (D=1, paired with a third shape as confounder at 40/60,50/50, and60/40ratios). Figure 10 reports sRC and sIC across all variants and the three 22 (A)(B)(C)(D)(E) CENDRE ACsilhouette (A) CENDRE HDBSCAN (B) CENDRE kMeans (C) ECLAD-ts (D) MultiVISION (E) 3.82e-11*8.35e-10*1.83e-04*5.70e-17* 7.27e-15*8.89e-13*5.70e-17* 7.11e-015.70e-17* 5.70e-17* syntheticlocal DenseNet1D121 (A)(B)(C)(D)(E) 2.49e-012.09e-14*5.70e-17*6.11e-17* 1.61e-12*3.47e-15*2.11e-14* 3.88e-14*5.70e-17* 5.70e-17* InceptionTime10 (A)(B)(C)(D)(E) 9.67e-12*2.95e-16*5.70e-17*5.70e-17* 8.62e-17*5.70e-17*5.70e-17* 1.01e-015.70e-17* 5.70e-17* ResNet1D18 (A)(B)(C) CENDRE ACsilhouette (A) CENDRE HDBSCAN (B) CENDRE kMeans (C) 5.87e-014.92e-12* 4.92e-12* syntheticfrequency DenseNet1D121 (A)(B)(C) 3.25e-012.36e-05* 2.36e-05* InceptionTime10 (A)(B)(C) 3.82e-015.15e-12* 5.15e-12* ResNet1D18 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 rank-biserial r (row > col) (a) sRC (A)(B)(C)(D)(E) CENDRE ACsilhouette (A) CENDRE HDBSCAN (B) CENDRE kMeans (C) ECLAD-ts (D) MultiVISION (E) 1.08e-11*5.52e-07*4.17e-07*5.53e-02 6.66e-15*8.76e-12*1.08e-11* 9.65e-04*1.37e-04* 9.41e-07* syntheticlocal DenseNet1D121 (A)(B)(C)(D)(E) 7.50e-05*6.73e-06*7.54e-09*2.20e-04* 2.19e-10*3.63e-11*1.42e-09* 6.59e-03*1.84e-01 6.95e-05* InceptionTime10 (A)(B)(C)(D)(E) 1.06e-10*1.02e-15*1.27e-13*1.04e-11* 2.29e-16*5.19e-14*5.19e-14* 3.61e-04*9.09e-09* 2.35e-08* ResNet1D18 (A)(B)(C) CENDRE ACsilhouette (A) CENDRE HDBSCAN (B) CENDRE kMeans (C) 9.62e-013.58e-11* 3.58e-11* syntheticfrequency DenseNet1D121 (A)(B)(C) 1.45e-021.11e-02 1.45e-02 InceptionTime10 (A)(B)(C) 7.47e-011.09e-09* 8.12e-10* ResNet1D18 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 rank-biserial r (row > col) (b) sIC Figure 9: Pairwise Holm-corrected Wilcoxon signed-rank tests between CE methods, per archi- tecture, on the runs of Figure 2, for (a) sRC and (b) sIC. Within each panel, the upper block is SYNTHETICLOCAL and the lower block SYNTHETICFREQUENCY, and the three column groups are the DenseNet1D-121, the InceptionTime10, and the ResNet1D-18. Only the upper triangle is shown, since the test is symmetric. Cell color gives the matched-pairs rank-biserial correlationr(red for row above column, blue for row below), and cell text the Holm-adjustedp-value, with an asterisk marking p < 0.01. On SYNTHETICFREQUENCY, only the three CENDRe variants enter the comparison, since the baselines have no frequency-domain counterpart. architectures. These two families run over3seeds (0–2), rather than the11of the main families. The ranking is similar to the main paper. CENDRe silhouette and CENDRe HDBSCAN trade slightly lower sRC for noticeably higher sIC than ECLAD-ts, CENDRe kMeans is similar to ECLAD-ts on both metrics, and MultiVISION underperforms on sRC. Across these settings, CENDRe HDBSCAN reaches slightly higher sIC than CENDRe silhouette , consis- tent with density-based clustering being better suited to the non-spherical structures one expects in latent spaces. We still report silhouette-guided aggregation as the default in the main text be- cause HDBSCAN occasionally requires per-dataset tuning of its neighborhood parameters when its 23 Figure 10: Quantitative evaluation on SYNTHETICLMC (top, per-channel attribution) and SYNTHET- ICLCONF (bottom, paired with confounders). The ranking mirrors Figure 2: CENDRe silhouette and CENDRe HDBSCAN achieve sRC close to ECLAD-ts and higher sIC, CENDRe kMeans tracks ECLAD-ts on both metrics, and MultiVISION is weak on sRC. single-pass run fails to find any dense region (see failure rates below); on more complex problems where this tuning effort is acceptable, HDBSCAN may be the stronger choice. CENDRe HDBSCAN occasionally fails to converge, with HDBSCAN labeling all50micro-centroids as noise, on11/297 SYNTHETICLOCAL and5/54SYNTHETICLCONF runs (all on InceptionTime10 or ResNet1D-18, concentrated ontriangle-circlebases). No failures occur on SYNTHETICFREQUENCY (0/198) or SYNTHETICLMC (0/54). Failed runs are excluded from the boxplot statistics, so the boxplots are conditional on convergence. D.2 Natural datasets We complement the CWRU results of Section 5 with two further natural-data evaluations. We extend the qualitative evaluation to five univariate datasets from the UCR archive, covering cardiac, automotive, motion, robotic, and semiconductor signals (§D.2.1). We then extract concepts from a second bearing-fault dataset, Paderborn Bearing (BearingPD), on two architectures, to test whether CENDRe recovers the same fault-relevant frequency bands under a different rig and sensor setup. D.2.1 UCR We extend the qualitative evaluation to five univariate datasets of the UCR archive [11], covering cardiac, automotive, motion, robotic, and semiconductor signals. For each dataset we train the three architectures of Section 4.2 and extract concepts from one of them, using CENDRe silhouette with the hyperparameters of Section 4.2. Each report shows the time-domain masks, the frequency-domain masks, and the global concept-frequency correspondence. The archive distributes most of its datasets without the sampling rate of the original recording. Where the rate is undocumented we plot the frequency axis in cycles per sample, underf s = 1, rather than assume a rate. Where it is documented we state it in the corresponding paragraph and label the time and frequency axes in seconds and Hz. The choice affects the labeling of the axes alone, since the masks are computed on the sampled series either way. ECG200[11] traces the electrical activity recorded during a single heartbeat, on one electrocar- diogram channel. The two classes are a normal heartbeat and a myocardial infarction. Series have T = 96samples. CENDRe silhouette extractsK = 4concepts, which partition the beat into consecutive segments.C 0 andC 1 carry the largest and opposite-signed importance, withC 0 activating on the plateau that follows the main negative deflection, mostly in they = 1series, andC 1 on the deflection itself.C 2 andC 3 cover the late and the early part of the beat at lower importance. In frequency, all 24 four concepts keep their mass below0.25cycles per sample and peak near0.02, so the model reads the morphology of the beat rather than high-frequency content. Time domain −2.5 0.0 2.5 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.000 0.005 −2.5 0.0 2.5 C 1 I 0 = − 0 . 9 I 1 = 0 . 9 0.000 0.005 −2.5 0.0 2.5 C 2 I 1 = − 0 . 5 0.000 0.005 −2.5 0.0 2.5 C 3 I 0 = 0 . 0 020406080 0.0000 0.0025 020406080020406080020406080020406080020406080 Frequency domain 1000 2000 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00 0.01 1000 2000 C 1 I 0 = − 0 . 9 I 1 = 0 . 9 0.00 0.01 1000 2000 C 2 I 1 = − 0 . 5 0.000 0.005 1000 2000 C 3 I 0 = 0 . 0 0.00.10.20.30.40.5 0.000 0.002 0.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.5 0.00 0.05 0.10 0.15 0.20 C0 Channel 0 0.00 0.05 0.10 0.15 0.20 C1 0.00 0.05 0.10 0.15 0.20 C2 0.00.10.20.30.40.5 0.00 0.05 0.10 0.15 0.20 C3 Figure 11: Concept extraction from a DenseNet1D-121 trained on ECG200 (two classes: normal heartbeat, myocardial infarction), reaching91.67%test accuracy. Top left: time-domain concept masks. Bottom left: frequency-domain concept masks. Right: global concept-frequency correspon- dence. The concepts split the beat into consecutive segments, and none of them reads content above 0.25 cycles per sample. FordA[11] contains engine noise measurements from an automotive subsystem, with the task of deciding whether a given symptom is present. Series haveT = 500samples, recorded under typical operating conditions and with minimal contamination. CENDRe silhouette extractsK = 2 concepts with opposite class importance.C 0 concentrates on short high-amplitude bursts andC 1 on the remaining oscillation, the two alternating along the window instead of splitting it into contiguous segments. Both confine their frequency mass to the band below0.15cycles per sample, which carries the dominant spectral peak of the signal, and both peak near 0.05. Within that band the two profiles differ in shape:C 0 concentrates its mass around the0.05peak and decays quickly above it, whileC 1 spreads a broader shoulder that persists toward 0.15. GunPoint[11] tracks the horizontal position of the centroid of an actor’s right hand, extracted from video of one of two motions. In the gun class the actor draws a replica gun from a hip holster, points it at a target for about one second, and returns it. In the point class the actor performs the same motion with an extended index finger and no prop. Series haveT = 150samples from two actors. The motion was recorded at30Hz, so the axes are labeled in seconds and in Hz over[0, 15]. CENDRe silhouette extractsK = 4concepts, all of which localize on the transitions of the hand trajectory and none on the plateau where the hand is held on target.C 0 andC 1 cover the rising edge around1–2s andC 2 the falling edge around3–4s in they = 0series, whileC 3 covers both edges in they = 1series. Every concept peaks between0.3and0.6Hz and carries no mass above2Hz, on the order of the one-second gesture. The extraction recovers the draw and the return as separate concepts, which is the distinction the two motions are defined by, as has been observed in literature [17]. SonyAIBORobotSurface1[11] contains accelerometer readings from a Sony AIBO robot, taken while the robot walks on one of two surfaces. The two classes are cement and carpet. The robot 25 Time domain −2 0 2 C 0 I 0 = − 0 . 9 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.000 0.001 −2 0 2 C 1 I 0 = 0 . 6 I 1 = − 0 . 6 0100200300400500 0.0000 0.0005 01002003004005000100200300400500010020030040050001002003004005000100200300400500 Frequency domain 20000 40000 C 0 I 0 = − 0 . 9 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.000 0.005 20000 40000 C 1 I 0 = 0 . 6 I 1 = − 0 . 6 0.00.10.20.30.40.5 0.0000 0.0025 0.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 C0 Channel 0 0.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 C1 Figure 12: Concept extraction from an InceptionTime10 trained on FordA (two classes: symptom present, symptom absent), reaching92.48%test accuracy. Top left: time-domain concept masks. Bottom left: frequency-domain concept masks. Right: global concept-frequency correspondence. Both concepts peak near0.05cycles per sample;C 0 concentrates its mass around that peak, whileC 1 spreads a broader shoulder across the band below 0.15. Time domain 0 2 C 0 I 0 = 1 . 0 I 1 = − 0 . 8 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00 0.05 0 2 C 1 I 0 = 0 . 6 0.000 0.025 0 2 C 2 I 0 = 0 . 4 0.00 0.02 0 2 C 3 I 0 = 0 . 1 012345 0.000 0.005 012345012345012345012345012345 Time [s] Frequency domain 5000 10000 C 0 I 0 = 1 . 0 I 1 = − 0 . 8 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.0 0.5 5000 10000 C 1 I 0 = 0 . 6 0.0 0.2 5000 10000 C 2 I 0 = 0 . 4 0.0 0.1 5000 10000 C 3 I 0 = 0 . 1 051015 0.00 0.05 051015051015051015051015051015 Frequency [Hz] 0.0 0.2 0.4 0.6 0.8 C0 Channel 0 0.0 0.2 0.4 0.6 0.8 C1 0.0 0.2 0.4 0.6 0.8 C2 051015 0.0 0.2 0.4 0.6 0.8 C3 Frequency [Hz] Figure 13: Concept extraction from an InceptionTime10 trained on GunPoint (two classes: gun draw, point), reaching100%test accuracy. Top left: time-domain concept masks. Bottom left: frequency- domain concept masks. Right: global concept-frequency correspondence. All four concepts localize on the draw and the return of the hand, and none on the plateau in between. carries roll, pitch, and yaw accelerometers, of which the archive distributes thexaxis alone. Series have T = 70 samples. The accelerometer samples at 125 Hz, so the axes are labeled in seconds and in Hz over[0, 62.5]. CENDRe silhouette extractsK = 2concepts that separate by class rather than by band.C 0 activates on they = 1series andC 1 on they = 0series, each spread over the whole0.55s window instead of a contiguous segment. Both concentrate their mass between5and25Hz, peak near8Hz, and decay above30Hz, in the range of the step cadence of the walk. Within that band the two profiles differ in shape:C 0 splits its mass into two sharp peaks near8and14Hz, whileC 1 pairs the 8 Hz peak with a lower but longer ridge that extends to about 30 Hz. 26 Time domain −2 0 2 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.0000 0.0025 −2 0 2 C 1 I 1 = − 0 . 3 0.00.20.4 0.000 0.001 0.00.20.40.00.20.40.00.20.40.00.20.40.00.20.4 Time [s] Frequency domain 500 1000 1500 C 0 I 0 = − 1 . 0 I 1 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.0000 0.0025 500 1000 1500 C 1 I 1 = − 0 . 3 0204060 0.000 0.002 02040600204060020406002040600204060 Frequency [Hz] 0.00 0.02 0.04 0.06 0.08 0.10 0.12 C0 Channel 0 0204060 0.00 0.02 0.04 0.06 0.08 0.10 0.12 C1 Frequency [Hz] Figure 14: Concept extraction from a DenseNet1D-121 trained on SonyAIBORobotSurface1 (two classes: cement, carpet), reaching97.50%test accuracy. Top left: time-domain concept masks. Bottom left: frequency-domain concept masks. Right: global concept-frequency correspondence. Both concepts peak near8Hz within the gait band;C 0 adds a second sharp peak near14Hz, while C 1 spreads a lower ridge up to about 30 Hz. Wafer[11] contains inline process control measurements from semiconductor fabrication, each series recorded by one sensor while one tool processes one silicon wafer. The two classes are normal and abnormal, with abnormal covering10.7%of the training and12.1%of the test series. Series haveT = 152samples. CENDRe silhouette extractsK = 5concepts. Four of them cover consecutive stages of the process trace, withC 1 andC 4 on the initial transient,C 3 on the plateau, andC 2 on the final ramp. The fifth,C 0 , carries the largest importance magnitude with opposite sign across the two classes and spans the whole window of they = 1series. The frequency masks are broader than on the other four datasets:C 2 ,C 3 , andC 4 peak below0.03cycles per sample, whileC 0 andC 1 cover the full band,C 1 as a near-uniform comb. That comb is the signature of the step-like transients of the trace rather than of a narrowband feature, which is what a purely spectral summary of this model would misread. D.2.2 BearingPD We complement the CWRU example in the main text with concept extraction on a second natural bearing-fault dataset. Paderborn Bearing (BearingPD) [28] contains accelerometer recordings from a motor test rig at Paderborn University, covering healthy bearings and bearings with artificially induced or naturally degraded inner and outer race damage; we use a three-class setup (healthy, inner-race fault, outer-race fault). Recordings are sampled atf s = 64kHz on a single vibration channel and segmented into non-overlapping windows ofT = 5000samples. Training follows Section 4.2, with two additions: Inputs are standardized by the global mean and standard deviation of the training set, and Gaussian noise of standard deviation0.01is added to the standardized signal during training as augmentation. On BearingPD on the InceptionTime10 model (Figure 16a), the extracted concepts partly separate by class, though some concepts activate in more than one class with differing activation strengths. The frequency-domain masks and the global concept-correspondence concentrate concept mass on four distinguishable bands, with markedly different distributions across concepts:C 0 spans all four, with its largest mass at0.6kHz and comparable mass in the2and3.4kHz bands;C 1 andC 2 concentrate on the low band, peaking at1.3kHz, and retain only weak activity at2and3.4kHz; all three carry a narrow peak near10kHz, and none shows a distinct peak above it. All four bands fall inside the structural-resonance region typically used for envelope analysis on this kind of bearing rig [39], and the per-concept differences within that region reflect that the model has partitioned the resonance content into class-specific sub-bands rather than treating it as a single feature. These specific bands give domain experts the information needed to assess whether the model is reasoning over fault-relevant spectral content, and the agreement with the literature regions supports that CENDRe transfers from controlled synthetic settings to real signals. 27 Time domain 0 5 C 0 I 0 = 1 . 0 I 1 = − 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.00 0.01 0 5 C 1 I 0 = 0 . 2 0.000 0.025 0 5 C 2 I 0 = − 0 . 1 0.00 0.01 0 5 C 3 I 0 = − 0 . 1 0.00 0.01 0 5 C 4 I 0 = − 0 . 1 050100150 0.00 0.05 050100150050100150050100150050100150050100150 Frequency domain 5000 10000 C 0 I 0 = 1 . 0 I 1 = − 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 1 ˆ y: 1 0.000 0.005 5000 10000 C 1 I 0 = 0 . 2 0.000 0.025 5000 10000 C 2 I 0 = − 0 . 1 0.00 0.05 5000 10000 C 3 I 0 = − 0 . 1 0.000 0.025 5000 10000 C 4 I 0 = − 0 . 1 0.00.10.20.30.40.5 0.000 0.005 0.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.50.00.10.20.30.40.5 0.00 0.05 0.10 0.15 C0 Channel 0 0.00 0.05 0.10 0.15 C1 0.00 0.05 0.10 0.15 C2 0.00 0.05 0.10 0.15 C3 0.00.10.20.30.40.5 0.00 0.05 0.10 0.15 C4 Figure 15: Concept extraction from a ResNet1D-18 trained on Wafer (two classes: normal, abnormal), reaching100%test accuracy. Top left: time-domain concept masks. Bottom left: frequency-domain concept masks. Right: global concept-frequency correspondence. Four concepts follow the stages of the process trace, whileC 0 spans the whole window and carries the strongest class-discriminative importance. On BearingPD on the DenseNet1D-121 model (Figure 16b), CENDRe recovers the same four bands, distributed differently across the concepts. The concepts again separate only partly by class, with some activating in more than one class at differing strengths.C 0 places its largest mass at3.4kHz, with the low pair close behind and a clear peak near10kHz;C 1 is dominated by a sharp peak at 1.3kHz, roughly twice the height of any peak in the other two concepts;C 2 carries the0.6and 1.3kHz peaks at comparable height, with weaker mass at3.4and10kHz. Both architectures resolve the low band into the same0.6and1.3kHz pair, and in both, one concept is dominated by a single member of the pair while the others carry both at comparable height, though the concept indices playing each role differ. Just as for InceptionTime10, these bands give domain experts the information needed to assess whether the model is reasoning over fault-relevant spectral content. Recovering them under a second architecture supports that CENDRe transfers from controlled synthetic settings to real signals independently of the backbone. E Hyperparameter sensitivity The CENDRe silhouette/HDBSCAN variants replace the number of conceptsKwith the number of micro-clustersJ. We investigate the sensitivity of CENDRe toJby sweepingJwith every other hyperparameter fixed at its main-experiment value. The grid is bounded by the method itself: the upper bound is the number of LADs in one mini-batch (J max = 3200in our configuration), since k-means cannot form more clusters than it sees in one partial fit, and the lower boundJ min = 11 follows from the agglomerative candidate search over up to10concepts. Within[J min ,J max ] we sample10log-spaced values, snapped to include the main-experiment valueJ = 50, giving 28 Time domain 0 5 C 0 I 0 = − 0 . 8 I 1 = − 0 . 5 I 2 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 2 ˆ y: 2y: 2 ˆ y: 2 0.0000 0.0025 0 5 C 1 I 0 = 0 . 6 0.000 0.002 0 5 C 2 I 1 = 0 . 4 0.000.020.040.060.08 0.000 0.001 0.000.020.040.060.080.000.020.040.060.080.000.020.040.060.080.000.020.040.060.080.000.020.040.060.08 Time [s] Frequency domain 0 200000 C 0 I 0 = − 0 . 8 I 1 = − 0 . 5 I 2 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 2 ˆ y: 2y: 2 ˆ y: 2 0.000 0.005 0 200000 C 1 I 0 = 0 . 6 0.00 0.01 0 200000 C 2 I 1 = 0 . 4 0100002000030000 0.00 0.01 01000020000300000100002000030000010000200003000001000020000300000100002000030000 Frequency [Hz] 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 C0 Channel 0 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 C1 0100002000030000 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 C2 Frequency [Hz] (a) InceptionTime10, whereC 0 carries the2and3.4kHz bands alongside the low pair, whileC 1 andC 2 concentrate on the low pair. Time domain 0 5 C 0 I 0 = − 0 . 7 I 2 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 2 ˆ y: 2y: 2 ˆ y: 2 0.0000 0.0005 0 5 C 1 I 0 = − 0 . 6 I 1 = 0 . 8 0.0000 0.0005 0 5 C 2 I 0 = 0 . 6 0.000.020.040.060.08 0.0000 0.0005 0.000.020.040.060.080.000.020.040.060.080.000.020.040.060.080.000.020.040.060.080.000.020.040.060.08 Time [s] Frequency domain 0 200000 C 0 I 0 = − 0 . 7 I 2 = 1 . 0 Importance Score y: 0 ˆ y: 0y: 0 ˆ y: 0y: 1 ˆ y: 1y: 1 ˆ y: 1y: 2 ˆ y: 2y: 2 ˆ y: 2 0.0000 0.0025 0 200000 C 1 I 0 = − 0 . 6 I 1 = 0 . 8 0.000 0.001 0 200000 C 2 I 0 = 0 . 6 0100002000030000 0.0000 0.0025 01000020000300000100002000030000010000200003000001000020000300000100002000030000 Frequency [Hz] 0.000 0.002 0.004 0.006 0.008 0.010 C0 Channel 0 0.000 0.002 0.004 0.006 0.008 0.010 C1 0100002000030000 0.000 0.002 0.004 0.006 0.008 0.010 C2 Frequency [Hz] (b) DenseNet1D-121, whereC 0 peaks at3.4kHz,C 1 is dominated by a sharp1.3kHz peak, andC 2 carries the 0.6–1.3 kHz pair at comparable height. Figure 16: Concept extraction on BearingPD (three classes: healthy, inner-race fault, outer-race fault) for two architectures, each trained with seed0to100%validation accuracy. Top left of each panel: time-domain concept masks, which localize between high-amplitude transients and pick up patterns that align with the periodic impact structure of the signal. Bottom left: frequency-domain concept masks, whose mass concentrates on four bands at roughly0.6–1.3,2,3.4, and10kHz. Right: global concept-frequency correspondence per channel, where both architectures show a peak near10kHz. The two architectures recover the same four bands, all inside the structural-resonance region commonly used for envelope-based bearing diagnostics [39], and differ only in how the bands distribute across concepts. 29 10 1 10 2 10 3 0.5 0.6 0.7 sRC circle-nothing ch1 specificChannels 10 1 10 2 10 3 0.5 0.6 0.7 squareorcircle-nothing ch1 specificChannels 10 1 10 2 10 3 0.5 0.6 0.7 midFreq-highFreq ch1 specificChannels 10 1 10 2 10 3 0.5 0.6 0.7 midFreq-highFreq ch1 specificChannels conf-midhighFreq-50-50 10 1 10 2 10 3 0.2 0.4 0.6 0.8 1.0 sIC 10 1 10 2 10 3 0.2 0.4 0.6 0.8 1.0 10 1 10 2 10 3 0.2 0.4 0.6 0.8 1.0 10 1 10 2 10 3 0.2 0.4 0.6 0.8 1.0 10 1 10 2 10 3 n clusters 0 10 20 30 40 K 10 1 10 2 10 3 n clusters 0 10 20 30 40 10 1 10 2 10 3 n clusters 0 10 20 30 40 10 1 10 2 10 3 n clusters 0 10 20 30 40 CE Method ACsilhouette HDBSCAN Figure 17: Response of sRC (top row) and sIC (bottom row) to the number of micro-clustersJ, per dataset, for CENDRe silhouette and CENDRe HDBSCAN . Markers are the mean over11seeds and shading is one standard deviation, with all panels on a commony-range. The dashed vertical line is the value J = 50 used in the main experiments. J ∈ 11, 21, 50, 73, 137, 257, 483, 907, 1704, 3200. The sweep covers two datasets from each of the main synthetic families and the11seeds of the main experiments, reused at every level ofJso that seed acts as a block rather than an independent replicate. Runs in which the extractor returned no concepts are excluded, which affects CENDRe HDBSCAN at small J only. Figure 17 shows the seed mean and standard deviation of sRC and sIC per dataset. CENDRe silhouette is flat over three orders of magnitude ofJon all four datasets, except at the two smallest values, where too few micro-clusters remain for the candidate search. CENDRe HDBSCAN matches it on SYNTHETICFREQUENCY, but on SYNTHETICLOCAL its sIC collapses above a few hundred micro- clusters, with a widening seed spread. We quantify the sensitivity with two standard measures, both taken under a log-uniform distribution ofJon[J min ,J max ], which the log-spaced grid samples uniformly [4]. Writing the score asy(J,ε), where ε collects everything the seed controls, the first-order Sobol index [46, 42] S J 1 = Var J (E ε [y | J ]) Var(y) (9) is the share of the total variance explained by the choice ofJ, andS seed 1 exchanges the roles ofJ andε. We estimate Equation 9 from the paired design as a two-way variance decomposition over levels ofJand seeds, in the bias-correctedω 2 form [34], weighting the levels by their log spacing. A variance share carries no direction, so we add the elementary effects of Morris [32]. Taking the factor aslog 2 J, which the geometric grid samples at near-uniform steps, the effect of the step fromJ j to J j+1 under seed s is E s,j = y s,j+1 − y s,j log 2 J j+1 − log 2 J j .(10) Their meanμ E reads as the average change in the score per doubling ofJ, in the units of that score. We report the signed mean rather than the absolute-value variantμ ∗ [7], since with a single swept factor the sign carries the direction of the drift. Tables 7a, 7b, and 7c report the three measures. For CENDRe silhouette ,Jexplains on average0.022 of the variance in sRC,0.072in sIC, and0.010in the selectedKitself, against0.702,0.577, and 0.159for the seed. The mean elementary effect onKis a negligible+0.02concepts per doubling of J, consistent with the near-zero effect already observed on sRC and sIC. ChoosingJis therefore a smaller decision than the run-to-run variation the method already tolerates, for the score and for Kalike. CENDRe HDBSCAN is more sensitive on average in sRC and sIC (S J 1 = 0.144and0.320). 30 Its selectedKtracksJfar more directly, withS J 1 = 0.416on average and a mean elementary effect reaching+3.67concepts per doubling ofJon CIRCLE-NOTHING. This uncontrolled growth inKplausibly explains the sIC collapse noted above. Sensitivity toJ, in the score and in the selectedK, is therefore a property of the density-based cluster mode rather than of CENDRe itself. The one exception is sIC onsquareorcircle-nothing, where the silhouette variant also reaches S J 1 = 0.23. Table 7: Sensitivity to the number of micro-clustersJ, for sRC, sIC, and K. The columnsS J 1 and S seed 1 are the first-order Sobol indices of Equation 9 forJand for the random seed, andμ E is the mean elementary effect of Equation 10, in units of the reported score per doubling ofJ. The final row of each panel averages each index over datasets. (a) Sensitivity of sRC to J . CENDRe silhouette CENDRe HDBSCAN DatasetS J 1 S seed 1 μ E S J 1 S seed 1 μ E circle-nothing0.0090.666-0.00210.0810.383+0.0078 squareorcircle-nothing0.0080.696+0.00050.4480.045+0.0174 midFreq-highFreq 0.0690.591+0.00090.0360.746+0.0032 midFreq-highFreq _conf-midhighFreq-50-50 0.0000.854+0.00030.0120.910+0.0022 Mean0.0220.702-0.00010.1440.521+0.0076 (b) Sensitivity of sIC to J . CENDRe silhouette CENDRe HDBSCAN DatasetS J 1 S seed 1 μ E S J 1 S seed 1 μ E circle-nothing0.0000.633-0.00290.5990.047-0.1073 squareorcircle-nothing0.2330.479+0.01180.6400.094-0.0932 midFreq-highFreq 0.0560.290+0.00030.0410.207+0.0066 midFreq-highFreq _conf-midhighFreq-50-50 0.0000.907-0.00160.0000.729+0.0029 Mean0.0720.577+0.00190.3200.269-0.0478 (c) Sensitivity of K to J . CENDRe silhouette CENDRe HDBSCAN DatasetS J 1 S seed 1 μ E S J 1 S seed 1 μ E circle-nothing0.0410.635+0.08800.5890.041+3.6664 squareorcircle-nothing0.0000.000-0.01080.4700.001+2.7356 midFreq-highFreq 0.0000.000+0.00000.3560.026+0.1462 midFreq-highFreq _conf-midhighFreq-50-50 0.0000.000+0.00000.2480.010+0.1250 Mean0.0100.159+0.01930.4160.020+1.6683 F Computational resources The experiments span two systems, an institutional HPC cluster and a local workstation. Each HPC GPU node provides two Intel Xeon CPUs (24physical cores each,192GB RAM), four NVIDIA A100 (40 GB) GPUs, and an InfiniBand interconnect. The workstation has dual Intel Xeon Gold 6330 CPUs (2×28 cores,112logical threads at2.00GHz,1TiB RAM) and two NVIDIA A100-PCIE- 40GB GPUs.Model training ran entirely on the cluster. The CE and synthetic-metric experiments ran on both systems, so their runtimes are reported as wall-clock measurements taken on the workstation under controlled conditions and aggregated over the full sweep. F.1 Model training Model training was performed on3architectures over27dataset configurations, with11seeds on the 15SYNTHETICLOCAL and SYNTHETICFREQUENCY datasets and3seeds on the12SYNTHETICLMC and SYNTHETICLCONF datasets, totaling603trained models. Training consumed a total of40.64 GPU hours. 31 F.2 Concept extraction and synthetic metrics Runtimes below are wall-clock measurements collected on the workstation from controlled batches on a single partition with25allocated CPUs and no concurrent GPU users. Both CE and synthetic-metric runtimes are reported per (architecture, method) as the mean over3seeds on the SYNTHETICLOCAL square-triangle variant, summed over theK-sweep used in the comparison. For the AC-clustered CENDRe variants (silhouette, HDBSCAN),Kis determined in a single pass, so a single run replaces the sweep, and the tables reflect this (they are6–9×faster than the fixed-Kbaselines). The experiments on ECLAD-ts and MultiVISION were done on the SYNTHETICLOCAL family (9datasets,11seeds) and the SYNTHETICLMC and SYNTHETICLCONF families (12datasets,3 seeds), which correspond to9× 11 + 12× 3 = 135runs per architecture. The CENDRe variants additionally run on the SYNTHETICFREQUENCY family (6datasets,11seeds), making for a total of 135 + 66 = 201 runs per architecture. Table 8: Mean CE runtime per (architecture, method), in seconds, over3seeds on a representative SYNTHETICLOCAL variant. Fixed-Kmethods (ECLAD-ts, MultiVISION, CENDRe kMeans ) sum over K ∈2, 4, 6, 8, 10; AC variants run once. ArchitectureECLAD-tsMultiVISIONCENDRe kMeans CENDRe silhouette CENDRe HDBSCAN InceptionTime101021201181815 ResNet1D-1888103911111 DenseNet1D-1212513212274035 Concept extraction. Aggregated over the full sweep, CE consumed68.54GPU hours. A further 29.97GPU hours were spent on exploratory and repeated runs during development, an amount that does not grow with the seed count. Synthetic metrics. The representation-correctness (RC) and importance-correctness (IC) metrics are computed jointly in a single pass, so the table reports one runtime per (architecture, method) covering both metrics together. Aggregated over the full sweep, synthetic-metric computation con- Table 9: Mean synthetic-metric runtime (RC and IC computed jointly) per (architecture, method), in seconds, over3seeds on a representative SYNTHETICLOCAL variant. Fixed-Kmethods sum over K ∈2, 4, 6, 8, 10; AC variants run once. ArchitectureECLAD-tsMultiVISIONCENDRe kMeans CENDRe silhouette CENDRe HDBSCAN InceptionTime10108743334229 ResNet1D-18107553082323 DenseNet1D-121111724647750 sumed95.08GPU hours. As for CE, an additional39.58GPU hours were spent during development, independent of the seed count. In total, across model training, CE and synthetic-metric experiments, the computational resources used to produce this paper amount to approximately 274 GPU hours. 32