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Unsupervised Detection of Anomalous Sound based on Deep Learning and the Neyman-Pearson Lemma
Yuma Koizumi, Shoichiro Saito, Hisashi Uematsum Yuta Kawachi, Noboru Harada
TL;DR
Unsupervised-ADS must detect unknown anomalous sounds without anomalous training data, while conventional AE training focuses only on normal reconstruction error. The paper uses Neyman–Pearson optimization with rejection-sampled complementary anomalies to maximize TPR at low FPR, and reports improved low-FPR measures plus detection in three real environments.
Problem
Unknown anomalous sounds are rare, variable, and unavailable as exhaustive training data, while conventional AE objectives do not ensure increased anomaly scores for anomalies.
Method
The method treats ADS as a statistical hypothesis test, optimizing an AE objective for TPR under a low-FPR condition and simulating anomalies by rejection sampling.
Results
The proposed method achieved higher ρTPR and pAUC than AE across the reported conditions and accurately detected unknown anomalous sounds in three real environments.
Takeaways & Limitations
Neyman–Pearson-based AE training supports unsupervised detection of unknown anomalous sounds under low-FPR conditions.
Abstract
from arXiv · showhide
This paper proposes a novel optimization principle and its implementation for unsupervised anomaly detection in sound (ADS) using an autoencoder (AE). The goal of unsupervised-ADS is to detect unknown anomalous sound without training data of anomalous sound. Use of an AE as a normal model is a state-of-the-art technique for unsupervised-ADS. To decrease the false positive rate (FPR), the AE is trained to minimize the reconstruction error of normal sounds and the anomaly score is calculated as the reconstruction error of the observed sound. Unfortunately, since this training procedure does not take into account the anomaly score for anomalous sounds, the true positive rate (TPR) does not necessarily increase. In this study, we define an objective function based on the Neyman-Pearson lemma by considering ADS as a statistical hypothesis test. The proposed objective function trains the AE to maximize the TPR under an arbitrary low FPR condition. To calculate the TPR in the objective function, we consider that the set of anomalous sounds is the complementary set of normal sounds and simulate anomalous sounds by using a rejection sampling algorithm. Through experiments using synthetic data, we found that the proposed method improved the performance measures of ADS under low FPR conditions. In addition, we confirmed that the proposed method could detect anomalous sounds in real environments.
I. Introduction
Unsupervised-ADS targets unknown anomalous sounds without anomalous training data, but conventional objectives must balance TPR and FPR. The paper proposes Neyman–Pearson optimization with simulated anomalies to improve detection under low-FPR conditions.
- Motivation: Unsupervised-ADS detects unknown anomalous sounds that lack available training examples, unlike supervised detection of defined events.Industrial anomalies are rare, variable, and costly to collect exhaustively.
- Motivation: Anomalies are defined as patterns that do not conform to expected normal behavior, with the anomaly set complementing the normal set.Detection compares observed sounds with a normal model using an anomaly score and threshold.
- Motivation: TPR and FPR trade off when normal and anomalous anomaly-score distributions overlap, so effective training should reduce that overlap.Small thresholds increase both TPR and FPR, whereas large thresholds decrease both.
- Existing approach: Autoencoders minimize reconstruction error on normal sounds, but this objective does not ensure large anomaly scores for anomalous sounds.A generalized AE may reconstruct anomalous sounds well, limiting its ability to reduce false detections and improve ADS accuracy.
- Proposed approach: The proposed method treats ADS as a statistical hypothesis test and optimizes TPR under an arbitrary low-FPR condition.It defines anomalous sounds as the complement of normal sounds and simulates them with rejection sampling.
- Proposed approach: End-to-end AE training replaces the earlier feature-extractor and GMM arrangement, training both components to increase TPR and decrease FPR.The prior normal model did not guarantee these improvements, whereas the proposed AE supports joint optimization.
II. Conventional method
Conventional unsupervised-ADS uses a normal model and thresholded anomaly score, with an AE commonly scoring reconstruction error. Minimizing normal reconstruction error alone can fail to improve anomalous-sound separation.
- Outlier detection: Outlier-based ADS computes an anomaly score from the deviation between an observed sound and a normal model, then flags scores above threshold φ.The normal model is trained from normal sounds, and the decision rule assigns scores at or below φ to normal observations.
- Outlier detection: TPR measures correctly identified anomalies and FPR measures normal sounds incorrectly identified as anomalies; both depend on φ and trade off.Overlapping anomaly-score distributions make false detections unavoidable at every threshold.
- Autoencoder method: An AE encodes an input into a latent vector, decodes it, and uses reconstruction error as the observed sound’s anomaly score.The encoder and decoder are trained neural networks that learn an efficient input representation and reconstruction.
- Autoencoder method: The conventional AE objective minimizes average reconstruction error over normal training sounds, thereby reducing normal-sound anomaly scores.The training objective averages A(x_n, Θ) across N^(u) normal samples.
- Limitation: Because the conventional objective does not target anomalous sounds, a generalized AE may reconstruct them well and produce small anomaly scores.Consequently, the objective does not ensure reduced false detections or improved ADS accuracy.
III. Proposed method
The proposed method treats unsupervised ADS as a statistical hypothesis test and trains an autoencoder-based model to improve detection under controlled false-positive rates. It simulates anomalous sounds as the complement of normal sounds using rejection sampling, enabling Neyman–Pearson or AUC-based optimization.
- A. Objective function for anomaly detection based on the Neyman-Pearson lemma: The method defines ADS optimality with a Neyman–Pearson objective that maximizes TPR while controlling or reducing FPR.The threshold is selected for a target FPR, while the model parameters are optimized using the resulting detection objective.
- B. Anomalous sound simulation using an autoencoder: Anomalous sounds are defined as the complement of normal sounds and simulated with rejection sampling to estimate TPR without anomalous training data.Candidate sounds are sampled from the distribution of various machine sounds and accepted when their anomaly score is sufficiently high.
- B. Anomalous sound simulation using an autoencoder: The encoder maps various machine sounds into a normalized Gaussian latent distribution, while a generator reconstructs them for anomalous-sound simulation.The normal latent distribution is modeled with a GMM, and the generator reconstructs sampled latent vectors into input-space sounds.
- C. Detailed description of training procedure: The training procedure uses normal and various-machine-sound data to learn encoder and generator parameters for the proposed anomaly-detection model.The procedure is organized as a three-step training process and outputs the encoder and decoder parameters.
- AUC maximization: Rejection sampling supplies anomalous data needed to extend the method from Neyman–Pearson optimization to AUC maximization.The AUC-based variant replaces the Neyman–Pearson objective with an objective computed from simulated anomalous sounds.
IV. Experiments
The experiments compare NP-PROP and AUC-PROP with established anomaly-detection methods across synthetic and real-environment sound settings. NP-PROP performs especially strongly under low-FPR conditions, while both proposed methods clearly detect anomalous sounds in real environments.
- Experimental setup: The evaluation compares NP-PROP and AUC-PROP with AE, VAE, VAEGAN, and CONV-PROP using FNN and 1D-CRNN architectures.The experiments use synthetic anomalous sounds and three real-environment verification experiments.
- Experimental setup: The study evaluates AUC, ρTPR, and partial AUC, with ρTPR and partial AUC measuring performance under low-FPR conditions.The evaluation includes Collision, Sustain, and Mix sound sets.
- Synthetic-data results: NP-PROP achieves higher ρTPR and partial AUC than AE across all reported conditions, while AUC-PROP achieves high AUC and partial AUC.ROC curves on the Mix dataset show substantially higher low-FPR TPRs for NP-PROP than for the other methods.
- Synthetic-data results: AUC-PROP achieves higher TPR than the other methods at moderate and high FPR conditions, whereas NP-PROP is strongest at low FPR.The reported pattern is attributed to the different objective functions used by the proposed methods.
- Synthetic-data results: In FNN experiments, VAE scores lower than AE and VAEGAN scores lower than all other methods, while NP-CONV scores lower than DNN-based methods.These results favor simple reconstruction error over the more complex VAE and VAEGAN procedures for FNN scoring, and favor DNN normal models over GMM-based ones in this comparison.
- Real-environment results: In real-environment experiments, NP-PROP and AUC-PROP detect anomalous sounds above the threshold while normal sounds remain below it.AE and VAE show false positives or small threshold margins in some equipment recordings, while VAEGAN misses the air-blower-pump anomaly.
V. Conclusions
The paper reports an AE-based unsupervised-ADS method that detects unknown anomalous sounds and identifies remaining challenges for real-environment deployment.
- The proposed method accurately detects unknown anomalous sounds in three real environments.
- Its objective function increases TPR under a low FPR condition by formulating ADS as a statistical hypothesis test using the Neyman-Pearson lemma.
- Rejection sampling simulates anomalous sounds by treating them as the complement of the normal-sound set.
- Future work includes extending the approach to detect both known and unknown anomalous sounds when only partial anomalous samples are available.Training on only part of the anomalous sounds does not guarantee performance for unknown anomalous sounds.
- The experiments used a simple detection rule, motivating machine- or context-specific rules and trainable post-processing for anomaly scores.
A. List of Symbols
The symbol list defines notation for the paper's objectives, models, variables, distributions, indices, and matrix operations.
- Functions and operators include the objective function J, anomaly score A, binary decision H, encoder E, decoder D, generator G, and Gaussian distribution N.
- Model parameters are denoted by Θ, with ΘE, ΘD, and ΘG for encoder, decoder, and generator parameters, and Υ for Gaussian-mixture parameters.
- Core variables include x for the input vector, y for the state variable, z for the latent vector, φ for the anomaly-score threshold, and ρ for the desired FPR.
- Additional notation covers Gaussian-mixture means, covariances, weights, mixture counts, time frames, training samples, minibatches, dimensions, step size, context windows, and anomaly decisions.
- Indices and superscripts distinguish observation time τ, training sample n, Gaussian component k, and normal, anomalous, or various sounds.