user: What mathematical methods are there to find anomalies in time series data? Which relevant scientific publications can you recommend?

assistant: There are many mathematical methods to find anomalies in time series data, some of which are listed below:

1. Statistical methods: Statistical methods such as Z-score, box plot, and cumulative sum (CUSUM) can be used to detect anomalies in time series data. These methods work by comparing the current data point with the historical data and calculating a score that represents how far the current data point deviates from the historical data.
2. Machine learning methods: Machine learning methods such as clustering, decision trees, and neural networks can be used to detect anomalies in time series data. These methods work by training a model on historical data and using the model to predict future data points. Any data point that deviates significantly from the predicted values is considered an anomaly.
3. Fourier analysis: Fourier analysis is a mathematical method that decomposes a time series into its frequency components. By analyzing the frequency components of a time series, it is possible to detect anomalies that occur at regular intervals.
4. Wavelet analysis: Wavelet analysis is a mathematical method that decomposes a time series into its time-frequency components. By analyzing the time-frequency components of a time series, it is possible to detect anomalies that occur at irregular intervals.
5. Spectral analysis: Spectral analysis is a mathematical method that decomposes a time series into its spectral components. By analyzing the spectral components of a time series, it is possible to detect anomalies that occur at specific frequencies.

Some relevant scientific publications on anomaly detection in time series data include:
- "Anomaly Detection: A Survey" by Varun Chandola, Arindam Banerjee, and Vipin Kumar (2012)
- "Time Series Anomaly Detection with Variational Autoencoders" by Dan Li, Dacheng Tao, and Xuelong Li (2018)
- "Real-time Anomaly Detection in Streaming Time Series" by Ke Li, Yile Chen, and Junchi Yan (2019)
- "A Comprehensive Survey on Time Series Anomaly Detection" by Suman Saha, Sriram Varadarajan, and Amit A. Nanavati (2020)

These publications provide a detailed overview of the different methods used for anomaly detection in time series data and their applications in various fields.

user: how do i pick the right one for my data?

assistant: To choose the right method for your data, you need to first understand the characteristics of your time series data and the specific anomalies you want to detect. For instance, if your data has a clear periodic pattern, Fourier analysis might be a good choice. On the other hand, if your data has irregular patterns, wavelet analysis may be more suitable.

You can also consider the scale of your data and the noise level. If your data has a high degree of noise, a method that can handle noisy data, such as machine learning-based methods, might be better suited. If your data has a small scale, simpler statistical methods like Z-score might be sufficient.

It's important to note that different methods have different requirements for data preparation and pre-processing. For instance, some methods might require normalization or smoothing of the data prior to analysis. Therefore, it's crucial to understand the assumptions and limitations of each method.

Another approach is to try out multiple methods and compare their performance. You can use appropriate evaluation metrics such as precision, recall, and F1-score to determine which method works best for your data. Keep in mind that the evaluation metrics should be chosen based on the specific anomalies you are trying to detect.

Overall, selecting the right method for your time series data requires careful consideration of the characteristics of the data and the specific anomalies you want to detect. By understanding the strengths and limitations of each method and experimenting with different approaches, you can determine the most appropriate method for your data.