PrOuD: Probabilistic Outlier Detection Solution for TimeSeries Analysis of RealWorld Photovoltaic Inverters
Abstract
:1. Introduction
 Imbalanced anomaly data: Anomalies are rare events. Therefore, in most cases, only limited amounts of labeled anomalous data and patterns are available in realworld datasets. The imbalance problem leads to difficulties in optimizing models and evaluating their performance comprehensively.
 Unknown generative process of anomalies: Without prior knowledge of physical systems, information about the generative process of anomalous patterns, such as generative functions and hyperparameters, is mostly inaccessible. Detection models are thus optimized based solely on existing anomalies, leading to overfitting and difficulty in handling unseen anomaly types.
 Suspicious detection results: Traditional detection methods, such as hypothesis tests based on comparing calculated errors with threshold values, offer deterministic results, i.e., it is either abnormal or not. However, these methods often fall short in convincing experts or aiding in rootcause analysis.
 Expensive annotation: Annotating anomalies in realworld data is consistently a temporally and financially expensive task, particularly when dealing with highdimensional timeseries data from complex physical systems. Expert analysis is essential for extracting critical information about anomalous sources, including their duration, influenced channels, and root causes. The absence of accurate annotations can lead to misclassifying abnormal data as regular, resulting in erroneous assessments.
 We gave precise definitions of outlier, anomaly, novelty, and noise and mathematically analyzed their sources in multivariate time series from the perspective of Gaussiandistributed noise.
 We proposed generative methods for multivariate artificial time series and four anomalous patterns to address challenges to training and evaluating models when facing an insufficient number of samples.
 We proposed the PrOuD solution that describes a general workflow without being restricted to the specific application scenario or the neural network’s architecture. PrOuD is adaptive and can collaborate with various types of Bayesian neural networks. Compared with conventional probabilistic forecasting models and standalone detection models, PrOuD can provide domain experts with enhanced explanations and greater confidence in the detected results through an estimated outlier probability and the use of an explainable artificial intelligence technique. The experimental results on both artificial time series and realworld photovoltaic inverter data demonstrated high precision, fast detection, and interpretability of PrOuD.
 We published the code via (https://github.com/iesresearch/probabilisticoutlierdetectionfortimeseries (accessed on 5 November 2023)) for interested readers to generate artificial data and reimplement the experiments.
2. Related Work
3. Artificial Time Series with Synthetic Anomalies
3.1. Terms and Definitions
3.2. Artificial Time Series with Synthetic Anomalies
4. PrOuD: Probabilistic Outlier Detection Solution
4.1. Probabilistic Prediction Phase
4.2. Detection Phase
Algorithm 1 Isolation Forest Algorithm 

Algorithm 2 DBSCAN Algorithm 

4.3. Explainable Interactive Learning Phase
5. Experiments
5.1. Description of Datasets
5.2. Experimental Setups
5.3. Evaluation Metrics
5.4. Results
6. Conclusions
6.1. Interpretation of Findings
6.2. Limitations and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PrOuD  Probabilistic Outlier Detection 
CNN  Convolutional Neural Network 
LSTM  Long ShortTerm Memory 
RNN  Recurrent Neural Network 
GNN  Graphic Neural Network 
MSE  Mean Square Error 
HDNN  Heteroscedastic Deep Neural Network 
MC dropout  Monte Carlo dropout 
NLL  Negative Logarithm Likelihood 
iForest  Isolation Forest 
DBSCAN  Densitybased Spatial Clustering of Applications with Noise 
AUC  Area Under receiver operating characteristic Curve 
MTTD  Mean Time To Detect 
SHAP  SHapely Additive exPlanations 
MCDHDNN  MC Dropout Heteroscedastic Deep Neural Network 
VEHDNN  Voting Ensemble of Heteroscedastic Deep Neural Network 
VEHLSTM  Voting Ensemble of Heteroscedastic Long ShortTerm Memory 
Appendix A
Appendix A.1. MTTD Calculation
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NonStationary Noise $\tilde{\mathit{\u03f5}}\left(\mathit{t}\right)$  Sum of Noise ${\mathit{\u03f5}}_{0}\left(\mathit{t}\right)+\tilde{\mathit{\u03f5}}\left(\mathit{t}\right)$ 

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu},{\tilde{\sigma}}^{2}$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu},{\sigma}_{0}^{2}+{\tilde{\sigma}}^{2}$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu}\left(t\right),{\tilde{\sigma}}^{2}$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu}\left(t\right),{\sigma}_{0}^{2}+{\tilde{\sigma}}^{2}$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu},{\tilde{\sigma}}^{2}\left(t\right)$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu},{\sigma}_{0}^{2}+{\tilde{\sigma}}^{2}\left(t\right)$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu}\left(t\right),{\tilde{\sigma}}^{2}\left(t\right)$$

$$\mathcal{N}\left(\right)open="("\; close=")">\tilde{\mu}\left(t\right),{\sigma}_{0}^{2}+{\tilde{\sigma}}^{2}\left(t\right)$$

Phase  Inputs  Operations  Outputs 

1  Multivariate time series ${\mathbf{x}}_{i}$  Equations (20) and (21)  Predictive distribution $\mathcal{N}\left(\right)open="("\; close=")">{\widehat{\mu}}_{i},{\widehat{\sigma}}_{i}^{2}$ 
2  $\mathcal{N}\left(\right)open="("\; close=")">{\widehat{\mu}}_{i},{\widehat{\sigma}}_{i}^{2}$  Equations (22)–(24)  Outlier probability ${p}_{i}$ and anomaly segments 
3  ${p}_{i}$ and anomaly segments  Equation (25) or other explanation methods  Anomaly clusters and visualization results 
Pattern  Duration  Hyperparameters 

I  12–48 h  $\alpha =0.2$, $\beta =0.05$, $b=0$ 
II  12–48 h  $\alpha =0.2$, $\beta =0.005$, $b=0.005$ 
III  12–48 h  $\alpha =0.1$, $\beta =0.01$, $b=0$ 
IV  10–15 days  $\alpha =0.05$, $\beta =0.001$, $b=0$ 
ID  MCDHDNN  VEHDNN  VEHLSTM  

F1 Score  Precision  Recall  F1 Score  Precision  Recall  F1 Score  Precision  Recall  
I1  0.933  0.914  0.953  0.932  0.910  0.954  $\mathbf{0}.\mathbf{936}$  $\mathbf{0.930}$  0.943 
I2  $\mathbf{0.943}$  $\mathbf{0.942}$  0.944  0.927  0.910  $\mathbf{0.946}$  0.925  0.939  0.912 
I3  0.968  $\mathbf{0.978}$  0.958  $\mathbf{0.969}$  0.977  $\mathbf{0.961}$  0.953  0.966  0.941 
I4  0.967  $\mathbf{0.973}$  0.962  $\mathbf{0.974}$  0.967  $\mathbf{0}.\mathbf{982}$  0.891  0.950  0.840 
I5  $\mathbf{0.960}$  0.959  0.962  0.958  0.953  $\mathbf{0.964}$  0.905  $\mathbf{0.991}$  0.832 
AUC  MTTD  ${\mathrm{MSE}}_{\mathrm{test}}$  AUC  MTTD  ${\mathrm{MSE}}_{\mathrm{test}}$  AUC  MTTD  ${\mathrm{MSE}}_{\mathrm{test}}$  
I1  $0.989$  13.5  $1.4\times {10}^{2}$  $\mathbf{0.990}$  $\mathbf{10.6}$  $1.4\times {10}^{2}$  0.971  14.2  $\mathbf{1.3}\times {\mathbf{10}}^{\mathbf{2}}$ 
I2  $\mathbf{0.989}$  6.1  $1.8\times {10}^{2}$  $0.985$  $\mathbf{5.0}$  $2.1\times {10}^{2}$  0.976  7.1  $\mathbf{1}.\mathbf{5}\times {\mathbf{10}}^{\mathbf{2}}$ 
I3  $\mathbf{0.998}$  27.3  $8.3\times {10}^{4}$  $0.994$  $\mathbf{24.2}$  $8.3\times {10}^{4}$  0.980  41.9  $\mathbf{8.1}\times {\mathbf{10}}^{\mathbf{4}}$ 
I4  $0.988$  109.0  $1.9\times {10}^{3}$  $\mathbf{0.996}$  $\mathbf{90.0}$  $1.8\times {10}^{3}$  0.932  290.0  $\mathbf{4.3}\times {\mathbf{10}}^{\mathbf{4}}$ 
I5  $0.990$  26.3  $1.1\times {10}^{3}$  $\mathbf{0.995}$  $\mathbf{24.4}$  $1.1\times {10}^{3}$  0.972  51.8  $\mathbf{9.2}\times {\mathbf{10}}^{\mathbf{4}}$ 
Model  F1 Score  Precision  Recall  AUC  MTTD  ${\mathbf{MSE}}_{\mathbf{train}}$  ${\mathbf{MSE}}_{\mathbf{test}}$ 

MCDHDNN  0.700 ($0.168$)  $\mathbf{0.610}$ ($0.197$)  0.902 ($0.042$)  0.974 ($0.024$)  75.1 ($98.0$)  0.002 ($0.0008$)  35.1 ($82.2$) 
VEHDNN  $\mathbf{0.705}$ ($0.160$)  0.601 ($0.189$)  $\mathbf{0.903}$ ($0.051$)  0.972 ($0.020$)  $\mathbf{65.6}$ ($79.4$)  $\mathbf{0.001}$ ($0.0005$)  41.1 ($80.7$) 
VEHLSTM  0.547 ($0.223$)  0.545 ($0.239$)  0.632 ($0.255$)  0.867 ($0.137$)  183.2 ($281.5$)  0.004 ($0.0018$)  $\mathbf{0.033}$ ($0.016$) 
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He, Y.; Huang, Z.; Vogt, S.; Sick, B. PrOuD: Probabilistic Outlier Detection Solution for TimeSeries Analysis of RealWorld Photovoltaic Inverters. Energies 2024, 17, 64. https://doi.org/10.3390/en17010064
He Y, Huang Z, Vogt S, Sick B. PrOuD: Probabilistic Outlier Detection Solution for TimeSeries Analysis of RealWorld Photovoltaic Inverters. Energies. 2024; 17(1):64. https://doi.org/10.3390/en17010064
Chicago/Turabian StyleHe, Yujiang, Zhixin Huang, Stephan Vogt, and Bernhard Sick. 2024. "PrOuD: Probabilistic Outlier Detection Solution for TimeSeries Analysis of RealWorld Photovoltaic Inverters" Energies 17, no. 1: 64. https://doi.org/10.3390/en17010064