Flow-based self-supervised density estimation for anomalous sound detection

As a number of companies worldwide are facing a shortage of skilled maintenance workers, the demand for an automatic sound-monitoring system has been increasing. Unsupervised anomaly detection methods are often adopted for this system since anomalous data can rarely be obtained [1, 2].

For unsupervised anomaly detection, generative models such as Variational Auto Encoder (VAE), which use approximate likelihood and reconstruction error to calculate anomaly scores, have been utilized. Normalizing Flows (NF) [3, 4] is another promising generative model thanks to its ability to perform exact likelihood estimation. However, this model fails at out-of-distribution detection since it assigns higher likelihood to smoother structured data [5, 6, 7].

Self-supervised classification-based approach is another way for detecting anomalies when sound data from multiple machines of the same machine type is available [8]. In self-supervised learning, a model is trained for a main task with another task called an auxiliary task to improve the performance on the main task. In self-supervised classification-based approach, the auxiliary task is to train a classifier that predicts a machine ID assigned to each machine. If the classifier misclassifies the machine ID of a sound data, the sound data is regarded as anomalous. Although this approach improves the detection performance on average, it shows significantly low scores on some machines.

In this paper, we propose a self-supervised density estimation method using NF. Our method uses sound data from one machine ID to detect anomalies (target data) and sound data from other machines of the same machine type (outlier data), and the model is trained to assign higher likelihood to the target data and lower likelihood to the outlier data. This method is a self-supervised approach because it improves the detection performance on one machine ID by introducing an auxiliary task in which the model discriminates the sound data of that machine ID (target data) from sound data of other machine IDs with the same machine type (outlier data). Also, since the method increases the likelihood of the target data, this method can also be similar to the unsupervised approach.

We evaluated the detection performance of our method using six types of machine sound data. Glow [9] and MAF [10] were utilized as NF models. Experimental results showed that our method outperformed unsupervised approaches while showing more stable detection performances compared to the self-supervised classification-based approach.

Anomalous sound detection is a task to identify whether a machine is normal or anomalous based on the anomaly score a trained model calculates from its sound. Each input sound data is determined as anomalous data if the anomaly score of the data exceeds a threshold value. We consider the unsupervised anomalous sound detection where only normal sound is available for training. We assume sound data of multiple machines with the same machine type is available. This is a realistic assumption since multiple machines of the same type are often installed in factories. This problem setting is the same as that in DCASE 2020 Challenge Task2 [11].

Normalizing Flows (NF) is a series of invertible transformations between an input data distribution $p({\mbox{\boldmath$x$}})$ and a known distribution $p({\mbox{\boldmath$z$}})$. Anomaly score can be calculated by the negative log likelihood (NLL) of the input data [16, 17, 18, 19]. However, this score is dependent on the smoothness of the input data and fails at out-of-distribution detection.

Kirichenko [14] proposed a loss function to distinguish the in-distribution data from the out-of-distribution data by using a supervised approach:

$$\begin{split}L=&\frac{1}{N_{D}}\sum_{{\mbox{\boldmath$x$}}\in D}\textrm{NLL}({\mbox{\boldmath$x$}})\\ &-\frac{1}{N_{OOD}}\sum_{{\mbox{\boldmath$x$}}\in OOD}\textrm{NLL}({\mbox{\boldmath$x$}})\cdot I[\textrm{NLL}({\mbox{\boldmath$x$}})<c],\end{split}$$

(1)

where $N_{D}$ is the number of the in-distribution data in each batch, $N_{ODD}$ is the number of the out-of-distribution data in each batch that satisfies the condition in the indicator function $I[\cdot]$, and $c$ is a threshold value.

To overcome the problems of NF models and the self-supervised classification-based approach, our method attempts to assign higher likelihood to the target data and lower likelihood to the outlier data using NF.

We train a model for each machine ID, where the data with that ID is the target data ${\mbox{\boldmath$x$}}_{t}$ and other machine sounds of the same machine type is the outlier data ${\mbox{\boldmath$x$}}_{e}$. Assume ${\mbox{\boldmath$x$}}_{t}$ and ${\mbox{\boldmath$x$}}_{e}$ consist of components specific to their machine IDs (${\mbox{\boldmath$x$}}_{t0}$, ${\mbox{\boldmath$x$}}_{e0}$) and components shared across the same machine type (${\mbox{\boldmath$x$}}_{c}$). The likelihood of ${\mbox{\boldmath$x$}}_{t}$ and ${\mbox{\boldmath$x$}}_{e}$ can be written as

$$p({\mbox{\boldmath$x$}}_{t})=p({\mbox{\boldmath$x$}}_{t0})p({\mbox{\boldmath$x$}}_{c}),$$

(2)

$$p({\mbox{\boldmath$x$}}_{e})=p({\mbox{\boldmath$x$}}_{e0})p({\mbox{\boldmath$x$}}_{c}).$$

(3)

If the model is trained to assign higher likelihood to ${\mbox{\boldmath$x$}}_{t}$ and lower likelihood to ${\mbox{\boldmath$x$}}_{e}$, only components specific to the machine ID (${\mbox{\boldmath$x$}}_{t0}$, ${\mbox{\boldmath$x$}}_{e0}$) will affect the likelihood. Therefore, we can improve the detection performance of a NF model by introducing an auxiliary task in which we train the NF model to discriminate the target data ${\mbox{\boldmath$x$}}_{t}$ from the outlier data ${\mbox{\boldmath$x$}}_{e}$ by the likelihood. When testing the model, the NLL is used as the anomaly score. The anomaly score will be higher when the data structure is different from the target data specific component ${\mbox{\boldmath$x$}}_{t0}$ or close to the outlier data ${\mbox{\boldmath$x$}}_{e0}$. The former case corresponds to anomaly detection by the unsupervised approach and the latter case to the self-supervised approach. Therefore, this idea can benefit from both the unsupervised and the self-supervised classification-based approach. As illustrated in Fig.1, the model is trained to decrease the NLL until the NLL of the outlier data reaches a threshold $c$. Then, the model is penalized so that the NLL of the outlier data can be higher than the target data.

This idea can be realized by using the loss function in (1), as

$$\begin{split}L=&\frac{1}{N_{target}}\sum_{{\mbox{\boldmath$x$}}\in target}\textrm{NLL}({\mbox{\boldmath$x$}})\\ &-\frac{1}{N_{outlier}}\sum_{{\mbox{\boldmath$x$}}\in outlier}\textrm{NLL}({\mbox{\boldmath$x$}})\cdot I[\textrm{NLL}({\mbox{\boldmath$x$}})<c],\end{split}$$

(4)

where $N_{target}$ is the number of the target data in each batch and $N_{outlier}$ is the number of the outlier data in each batch that satisfies the condition in $I[\cdot]$. Threshold $c$ is decided so that the NLL of the outlier data does not go to $+\infty$. The difference between (1) and (4) is that the loss function in (1) uses out-of-distribution data that has completely different data structures from the in-distribution data, while the loss function in (4) uses the outlier data of the same machine type as the target data. In (4), the NLL of the target data and the NLL of the outlier data that satisfies the condition in $I[\cdot]$ have the same impact on the loss. In unsupervised anomaly detection tasks, lower NLL of the target data can lead to better detection performances. Therefore, we modified (4) so that the model can prioritize decreasing the NLL of the target data, as

$$\begin{split}L=&\frac{1}{N_{target}}\sum_{{\mbox{\boldmath$x$}}\in target}\textrm{NLL}({\mbox{\boldmath$x$}})\\ &-k\cdot\frac{1}{N_{outlier}}\sum_{{\mbox{\boldmath$x$}}\in outlier}\textrm{NLL}({\mbox{\boldmath$x$}})\\ &\cdot I[\textrm{NLL}({\mbox{\boldmath$x$}})<c\land\textrm{NLL}({\mbox{\boldmath$x$}})<\textrm{max}(\textrm{NLL}({\mbox{\boldmath$x$}}_{t}))],\end{split}$$

(5)

where $0<k<1$ enhances decreasing the NLL of the target data. The second condition in $I[\cdot]$ is for removing the penalty if the NLL of the outlier data is larger than the maximum NLL of the target data ${\mbox{\boldmath$x$}}_{t}$ in each batch. In this case, the distinction between the target data and the outlier data can be completely made by the NLL, and the model should only focus on decreasing the NLL of the target data.

We proposed flow-based methods for anomalous sound detection that outperform unsupervised approaches while maintaining greater stability than the self-supervised classification-based approach. Experimental results demonstrated that our methods improve the detection performance by using the outlier data from the same machine type as the target data. Our future work will include the development of more efficient methods to decide the hyperparameters.