Post-Hoc Uncertainty-Aware Explanations for Deployed Power Quality Disturbance Classifiers via Laplace Approximation

Yinsong Chen; Samson S. Yu; Kashem M. Muttaqi

cs.LGApr 15, 2026

Post-Hoc Uncertainty-Aware Explanations for Deployed Power Quality Disturbance Classifiers via Laplace Approximation

Authors: Yinsong Chen, Samson S. Yu, Kashem M. Muttaqi

Organizations: School of Engineering, Deakin University, Melbourne, 3216, VIC, Australia · ARC Training Centre in Energy Technologies for Future Grids, School of Engineering, University of Wollongong, Wollongong, 2522, NSW, Australia

Abstract

Deep learning classifiers achieve high accuracy in power quality disturbance (PQD) recognition, but existing explanation methods return a single deterministic attribution map and provide no measure of its reliability. This paper develops a post-hoc Bayesian explanation (B-explanation) method for trained PQD classifiers. A computationally efficient Laplace approximation converts the trained network into an approximate parameter posterior without retraining, and occlusion sensitivity is propagated through posterior samples to produce a distribution over disturbance-localization maps. Percentile summaries of this distribution yield explanations with distribution-free coverage bands: consensus summaries at low percentiles sharpen localization significantly for distinctive events such as sags, swells, and oscillatory transients, the band width indicates the reliability of each attribution, and the remaining disturbance types show class-dependent behavior. Explanation dispersion also increases under injected measurement noise and synthetic-to-field transfer, complementing predictive uncertainty. Experiments on a synthetic benchmark of 15 disturbance classes and on field-recorded sags compare the method with Monte Carlo dropout and deep ensembles under a common evaluation protocol, evaluate it against deterministic occlusion, LIME, and SHAP with localization and faithfulness metrics, and characterize the computational cost of explanation generation for grid monitoring applications.

Figures & tables

Study	Contribution	Limitation addressed in this work
Wang and Chen [ 37 ]	Deep CNN on raw waveforms; $>$ 99% accuracy on 15-class benchmark	No interpretability or uncertainty support
Machlev et al. [ 26 , 25 ]	XAI evaluation methodology for PQD classifiers; occlusion found most physically consistent	Deterministic explanations without a reliability measure
Liu et al. [ 19 , 21 , 22 , 20 ]	Attention, ensemble, and empirical-wavelet-transform-based recognition of complex and multiple PQDs	Accuracy-oriented black boxes; no explanation or uncertainty quantification
Baig et al. [ 4 , 3 ]	Image-transfer and multi-modal deep ensembles for PQD classification	Ensembles used for accuracy only; explanation reliability not considered
Bykov et al. [ 6 , 7 ]	B-LRP: posterior percentiles of LRP heatmaps for BNNs	Tied to LRP and Bayesian training; no ground-truth localization; image domain
Mulye and Valdenegro-Toro [ 28 ]	Variance of gradient explanations across uncertainty methods	No calibrated interpretation rule; no domain ground truth

Table 1: Closely related studies, their contributions, and the limitations addressed in this work.

Figure 1: Three-stage framework for PQD classification using conventional and deep learning methods.

Layer	Details	Activation
convolution_1	kernel: 1 $\times$ 3, channel: 32, stride: 1	ReLU
convolution_2	kernel: 1 $\times$ 3, channel: 32, stride: 1	ReLU
max_pooling_1	kernel: 1 $\times$ 3, stride: 1
batch_normalization_1	num_features: 32
convolution_3	kernel: 1 $\times$ 3, channel: 64, stride: 1	ReLU
convolution_4	kernel: 1 $\times$ 3, channel: 64, stride: 1	ReLU

Table 2: Architecture of the deep convolutional neural network (DCNN) of [ 37 ] used as the deployed classifier.

Panel of Figure 2 — Figure 2: DNNs with identical architectures and similar performance can yield markedly different explanations. Three models share the architecture in Table 2 but differ in parameters sampled from the Laplace posterior. All classify signal (a) correctly as “Flicker+Sag” with comparable accuracy and predictive entropy, yet generate distinct explanations (b–d).

Scheme	Post hoc	Extra cost	Posterior quality
MCMC	no	prohibitive at DCNN scale	asymptotically exact
VI [ 5 ]	no (retraining)	full retraining, $2\times$ parameters	good within family; mode-seeking
Deep ensemble [ 18 ]	no (multi-training)	one training run per posterior sample	multi-modal, well calibrated
MC-dropout [ 13 ]	partially (needs dropout layers)	negligible	tied to dropout architecture
SWAG [ 27 ]	no (needs training trajectory)	trajectory storage	local Gaussian, richer covariance
Laplace (diagonal) [ 33 , 10 ]	yes	one pass over $\mathcal{D}$	local Gaussian, curvature-limited

Table 3: Posterior-inference schemes against the constraints of a deployed classifier. “Post hoc” means applicable to an existing model without retraining.

Panel of Figure 3 — Figure 3: Uncertainty quantification of PQD classifier explanations with min–max normalized relevance scores. Rows show the MAP explanation followed by percentile B-explanations at $\alpha\in\{5,25,50,75,95\}$ ; the pair $\alpha=5$ and $\alpha=95$ bounds a $90\%$ explanation band, and $\alpha=25/50/75$ are the quartile and median summaries. Experimental setup as in Sections 3.1 and 4.1 .

Figure 4: Overview of the post-hoc B-explanation method for a deployed PQD classifier.

Panel of Figure 5 — Figure 5: Visualization of explanation multi-modality in PQD classifiers (disturbance: oscillatory transient). The first row presents the mean explanation for each cluster; the second row shows the t-SNE projection of the explanation distribution with highlighted cluster assignments. Experimental setup as in Sections 3.1 and 4.1 .

Symbol	Type	Meaning
$\theta_{\text{MAP}}$	input	checkpoint of the deployed classifier
$\mathcal{D}$	input	training set (for curvature accumulation only)
$x$	input	waveform to be explained, length $N$
$c$	input	class of interest
$w,v,b$	input	occlusion window, stride, baseline value
$S$	input	number of posterior samples

Table 4: Input and output variables of Algorithm 1 .

	DCNN(MAP)	Bayesian DCNN
Entropy	0.0045	0.0230 $\pm$ 0.0279
Accuracy	0.9820	0.9816 $\pm$ 0.0088

Table 5: Average entropy and accuracy of base models on the five test splits. For the Bayesian DCNN, mean and standard deviation are computed from 100 sampled models.

#	Disturbance Type	MAP	Bayesian Explanation
#	Disturbance Type	MAP	$\alpha{=}5$	$\alpha{=}25$	$\alpha{=}50$	$\alpha{=}75$	$\alpha{=}95$
1	Sag	$0.5799\pm 0.0140$	$0.6432\pm 0.0134$	$0.6148\pm 0.0108$	$0.6044\pm 0.0115$	$0.5963\pm 0.0126$	$0.5810\pm 0.0135$
2	Swell	$0.4489\pm 0.0131$	$0.6094\pm 0.0282$	$0.6116\pm 0.0224$	$0.5983\pm 0.0287$	$0.5673\pm 0.0254$	$0.4970\pm 0.0169$
3	Interruption	$0.3397\pm 0.0278$	$0.3243\pm 0.0236$	$0.2858\pm 0.0200$	$0.2608\pm 0.0153$	$0.2322\pm 0.0165$	$0.1707\pm 0.0159$
4	Harmonics	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$
5	Flicker	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$	$0.9984\pm 0.0000$

Table 6: Comparison of MAP and Bayesian explanations (at percentiles

\alpha=5,25,50,75,95

) based on relevance mass accuracy (mean

\pm

standard deviation).

#	Disturbance Type	MAP	Bayesian Explanation
#	Disturbance Type	MAP	$\alpha{=}5$	$\alpha{=}25$	$\alpha{=}50$	$\alpha{=}75$	$\alpha{=}95$
1	Sag	$0.6667\pm 0.0116$	$0.7014\pm 0.0180$	$0.6892\pm 0.0175$	$0.6852\pm 0.0169$	$0.6786\pm 0.0121$	$0.6658\pm 0.0115$
2	Swell	$0.6916\pm 0.0151$	$0.6981\pm 0.0171$	$0.7613\pm 0.0103$	$0.7731\pm 0.0177$	$0.7392\pm 0.0126$	$0.7052\pm 0.0108$
3	Interruption	$0.3779\pm 0.0320$	$0.3411\pm 0.0323$	$0.3263\pm 0.0378$	$0.2906\pm 0.0295$	$0.2460\pm 0.0285$	$0.1675\pm 0.0246$
4	Harmonics	$0.9973\pm 0.0001$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9970\pm 0.0001$
5	Flicker	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$	$0.9969\pm 0.0000$

Table 7: Comparison of MAP and Bayesian explanations (at percentiles

\alpha=5,25,50,75,95

) based on intersection over union score (mean

\pm

standard deviation).

Disturbance family	Benefit of B-explanation	Best at
Sag, oscillatory transient	sharper localization via posterior consensus	$\alpha=5$
Swell	strongest localization gains, reached at moderately inclusive summaries	$\alpha=25$ – $50$
Interruption	adds a per-instance reliability measure to the MAP explanation	MAP or $\alpha=5$
Harmonics, flicker, composites	confirms explanation stability where localization is already near-saturated	$\alpha=50$
Impulsive transient, notch, spike	the $5$ – $95$ band flags low-confidence attributions for events below the occlusion resolution	$\alpha=50$ + $5$ – $95$ band

Table 8: Summary of B-explanation benefit by disturbance family and the percentile at which the strongest localization was observed. Higher percentiles are not recommended for localization; together with the lower percentiles they form the distribution-free

90\%

explanation band of Section 2.5 and serve as reliability diagnostics.

Scheme	Summary	RMA	IoU
Deterministic (MAP)		$0.6713\pm 0.0031$	$0.7175\pm 0.0024$
Laplace (proposed)	mean	$0.6684\pm 0.0018$	$0.7004\pm 0.0021$
	$\alpha=5$	$0.6911\pm 0.0032$ †	$0.7099\pm 0.0028$
	$\alpha=25$	$0.6850\pm 0.0029$ †	$0.7178\pm 0.0017$
	$\alpha=50$	$0.6806\pm 0.0029$ †	$0.7167\pm 0.0025$
	$\alpha=75$	$0.6751\pm 0.0026$ †	$0.7122\pm 0.0034$

Table 9: Total localization scores (mean

\pm

standard deviation over five test splits) for the three Bayesian schemes under the percentile protocol, against the deterministic MAP explanation. † marks a significant paired improvement over the deterministic baseline (paired over splits, Holm-corrected,

p<0.05

); test-set accuracy is within

0.2

percentage points of the MAP model for all schemes.

Method	RMA	IoU	Del. $\downarrow$	Ins. $\uparrow$	Mono.
Occlusion (MAP)	$0.669$	$0.715$	$0.294$	$0.450$	$\mathbf{0.220}$
B-explanation (mean)	$0.669$	$0.710$	$\mathbf{0.263}$	$\mathbf{0.461}$	$0.159$
B-explanation ( $\alpha=5$ )	$0.688$	$\mathbf{0.718}$	$0.294$	$0.453$	$0.167$
B-explanation ( $\alpha=95$ )	$0.663$	$0.712$	$0.270$	$0.459$	$0.175$
LIME	$\mathbf{0.754}$	$0.639$	$0.293$	$0.451$	$0.052$
KernelSHAP	$0.635$	$0.621$	$0.338$	$0.457$	$0.025$

Table 10: Localization and faithfulness comparison with deterministic attribution baselines (300 instances). Del. = deletion AUC (lower is better), Ins. = insertion AUC (higher is better), Mono. = window-level monotonicity. Bold marks the best value in each column.

Configuration	RMA ( $\alpha{=}5$ )	IoU ( $\alpha{=}25$ )	RMA (mean)	IoU (mean)
$S=10$ , $w=64$ , $v=8$	$0.6862\pm 0.0027$	$0.7171\pm 0.0087$	$0.6732\pm 0.0045$	$0.7094\pm 0.0047$
$S=25$ , $w=64$ , $v=8$	$0.6870\pm 0.0038$	$0.7154\pm 0.0077$	$0.6710\pm 0.0053$	$0.7079\pm 0.0073$
$S=50$ , $w=64$ , $v=8$	$0.6883\pm 0.0044$	$0.7169\pm 0.0076$	$0.6685\pm 0.0038$	$0.7027\pm 0.0075$
$S=100$ , $w=64$ , $v=8$	$0.6905\pm 0.0053$	$0.7163\pm 0.0086$	$0.6673\pm 0.0041$	$0.7032\pm 0.0038$
$S=200$ , $w=64$ , $v=8$	$0.6898\pm 0.0052$	$0.7200\pm 0.0072$	$0.6663\pm 0.0042$	$0.7020\pm 0.0053$
$w=16$ , $S=100$ , $v=8$	$0.6801\pm 0.0105$	$0.6323\pm 0.0071$	$0.6510\pm 0.0068$	$0.6244\pm 0.0079$

Table 11: Sensitivity of the localization metrics to the number of posterior samples and the occlusion window (mean

\pm

standard deviation over five test splits, 20 instances per class per split).

Condition	Accuracy	RMA ( $\alpha{=}5$ )	IoU ( $\alpha{=}25$ )	Band	Std
Clean	$0.983$	$0.665$	$0.708$	$0.117$	$0.046$
30 dB	$0.503$	$0.603$	$0.611$	$0.138$	$0.048$
30 dB + harm.	$0.460$	$0.602$	$0.602$	$0.125$	$0.043$
20 dB	$0.300$	$0.602$	$0.610$	$0.134$	$0.046$
20 dB + harm.	$0.279$	$0.603$	$0.607$	$0.129$	$0.044$
10 dB	$0.124$	$0.600$	$0.584$	$0.155$	$0.053$

Table 12: Robustness under injected noise and background harmonics (

7{,}500

instances per condition). Band = mean width of the

5

–

95

explanation band; Std = mean explanation standard deviation.

Panel of Figure 6 — Figure 6: MAP and Bayesian explanations for real-world Sag signal prediction (signal 1). Both MAP models, trained on the synthetic dataset with different random seeds, classify the signal correctly as “Sag”.

Panel of Figure 7 — Figure 7: MAP and Bayesian explanations for real-world Sag signal prediction (signal 2). Both MAP models, trained on the synthetic dataset with different random seeds, classify the signal correctly as “Sag”.

OOD group	$n$	Pred. entropy	Mutual info.	Band width	Expl. std
47 Hz	150	$\mathbf{0.730}$	$0.649$	$0.536$	$0.524$
52 Hz	150	$\mathbf{0.795}$	$0.695$	$0.483$	$0.459$
Real-world	104	$0.614$	$0.550$	$\mathbf{0.734}$	$0.708$

Table 13: Out-of-distribution separation (AUROC against the in-distribution reference of 150 instances) for predictive-uncertainty and explanation-dispersion statistics. Bold marks the best-separating statistic for each group.

Device	Forward pass	MAP explanation	B-expl. $S{=}10$	B-expl. $S{=}100$
CPU	$1.3$ ms	$126$ ms	$1.29$ s	$12.8$ s
GPU	$0.9$ ms	$93$ ms	$0.97$ s	$9.7$ s

Table 14: Wall-clock cost of explaining one waveform (

N=640

w=64

v=8

Explainable Artificial Intelligence

Explainable artificial intelligence develops ways to make AI decisions understandable, testable, and useful to people.

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A Unified Framework for Uncertainty-Aware Explainable Artificial Intelligence: A Case Study in Power Quality Disturbance Classification

Yinsong Chen, Samson S. Yu, Zhong Li, Chee Peng Lim

School of Engineering, Deakin University, Melbourne, 3216, VIC, Australia · Faculty of Mathematics and Computer Science, FernUniversität in Hagen, 58084, Germany · Department of Computing Technologies, Swinburne University of Technology, Hawthorn, 3122, VIC, Australia

Bayesian Neural Networks Feature Attribution Explainable Artificial Intelligence Uncertainty Quantification

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Post-hoc explainable AI (XAI) methods typically produce deterministic attribution maps, whereas Bayesian neural networks (BNNs) induce a distribution over explanations. Capturing the variability of this distribution is important for uncertainty-aware decision-making. This paper formalises the explanation distribution as the push-forward measure of the BNN posterior through any Lipschitz-continuous attribution operator. It further proposes the uncertainty-aware relevance attribution operator (UA-RAO), a general family of operators that summarises the explanation distribution using the mean, variance, coefficient of variation, quantiles, and set-theoretic aggregation measures. Theoretical support is provided through Monte Carlo accessibility and Wasserstein approximation bounds. The framework is evaluated on a 15-class power quality disturbance (PQD) classification benchmark, comparing three BNN approximations paired with three attribution operators using relevance mass accuracy and intersection-over-union as localisation metrics. Results show that deep ensembles with the mean UA-RAO improve localisation over the deterministic baseline, while other UA-RAO summaries reveal uncertainty patterns absent from point-estimate attributions. Qualitative results on measured signals further suggest that these patterns generalise beyond the synthetic training distribution. The framework is domain-agnostic and can be applied to any BNN paired with a Lipschitz-continuous attribution operator.

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