# Anomaly Detection in High-Dimensional Time Series Data with Scaled Bregman Divergence
## 2 Materials and Methods
### Algorithm and Computational Complexity Analysis
#### 2.3.1 Anomaly Detection Algorithm
As shown in Figure 8, with squared error loss, PLsBD detected both anomaly intervals but also produced some false positives, while RuLSIF misclassified all points as anomalies, and the mean divergence score was flat, making it ineffective in anomaly detection. These results underscore the importance of selecting appropriate loss functions and the mixing parameter \(\alpha\) for effective anomaly detection. Overall, PLsBD outperformed RuLSIF, demonstrating greater robustness and minimal sensitivity to \(\alpha\) and the choice of loss functions in this context. \[\hat{\boldsymbol{\theta}}=(\widehat{\mathbf{H}}+\lambda\mathbf{I}_{n})^{-1} \widehat{\mathbf{h}}_{tr},\]4 Anomaly detection results in Yahoo Dataset using the uLSIF, RuLSIF, and PLsBD methods with the \(L_{2}\) loss function; the mixing parameter \(\alpha\) is set to be 0.1 for RuLSIF and PLsBD. The black bars indicate the frequencies that an observation is flagged as an anomaly in 100 runs. Red dashed vertical lines mark the true anomaly indices. \[\hat{\boldsymbol{\theta}}=(\widehat{\mathbf{H}}+\lambda\mathbf{I}_{n})^{-1} \widehat{\mathbf{h}}_{tr},\]5 _Cont_. \[\hat{\boldsymbol{\theta}}=(\widehat{\mathbf{H}}+\lambda\mathbf{I}_{n})^{-1} \widehat{\mathbf{h}}_{tr},\]6 Anomaly detection results in Yahoo Synthetic 11 using the RuLSIF and PLsBD methods with the \(L_{1}\) loss function; the mixing parameter \(\alpha\) is set to be 0.5. The black bars indicate the frequencies that an observation is flagged as an anomaly in 50 runs. Red dashed vertical lines mark the true anomaly indices. **High-dimensional Biomedical Benchmark Data. ** We consider four biomedical-related datasets, _Diabetes_, _Heart_, _BreastCancer_, and _Thyroid_, available from [PERSON]'s benchmark Repository [43]. This benchmark dataset varies in size and complexity and has been widely used to test the performance of different anomaly detection algorithms. Each dataset consists of a feature matrix, where rows represent data points and columns represent features, and a label vector, where (1) indicates normal samples and (\(-1\)) denotes anomalies. Table 1 summarizes the number of features and observations for each data point. In our setup, the training set contains only normal samples (label 1), and the test set includes all normal samples along with a randomly sampled fraction (20%) of anomalies. To ensure a robust evaluation of the anomaly detection algorithms, we performed 20 trials using fixed training and testing splits. For each trial, the window size was set to 10, the step size to 5, and the threshold to 0.9. Other parameters, including the number of kernel functions and the use of 5-fold cross-validation, were kept consistent with the previous data analysis. Algorithm 1 was then applied to identify all potential anomaly candidates using different values of \(\alpha\) (0.01, 0.1, 0.5, and 0.9), while also measuring the computation time for each configuration. To evaluate the performance of anomaly detection algorithms, we use the area under the curve (AUC) of the receiver operating characteristic (ROC) curve, a widely adopted metric in anomaly detection studies. The AUC summarizes a model's ability to produce relative scores that distinguish between positive and negative instances across all classification thresholds. By capturing the trade-off between the True Positive Rate (TPR) and False Positive Rate (FPR) across thresholds, AUC provides a consistent, threshold-independent measure, making it especially suitable for comparing models in scenarios with imbalanced data. Table 2 summarizes the mean AUC values over 20 trials for the diabetes and heart datasets (Table 2), and the ROC curve for the diabetes data is shown in \[\hat{\boldsymbol{\theta}}=(\widehat{\mathbf{H}}+\lambda\mathbf{I}_{n})^{-1} \widehat{\mathbf{h}}_{tr},\]7 These \begin{table}
\begin{tabular}{l c c c c} \hline
**Dataset** & **Diabetes** & **Heart** & **Thyroid** & **Breast Cancer** \\ \hline
**Number of Features** & 8 & 13 & 5 & 9 \\
**Number of Observations** & 768 & 270 & 215 & 263 \\ \hline \end{tabular}
\end{table}