What mathematical methods are there to find anomalies in time series data? Which relevant scientific publications can you recommend?
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.
how do i pick the right one for my data?
Choosing the right method for detecting anomalies in your time series data depends on various factors, such as the nature of the data, the type of anomalies you are looking for, and the accuracy requirements.

Here are some guidelines to help you pick the right method:

1    Understand the nature of your data:
    You should understand the characteristics of your time series data, such as its seasonality, trend, and noise. For example, if your data has a strong seasonal component, you may want to consider methods that can handle seasonality.

2    Determine the type of anomalies you are looking for:
    Different methods may be suitable for different types of anomalies, such as sudden changes, outliers, or recurring patterns. For example, if you are looking for outliers, you may want to consider statistical methods such as Z-score or Grubbs' test.

3    Evaluate the performance of different methods:
    You should compare the performance of different methods on your data to choose the one that works best. You can use metrics such as precision, recall, or F1-score to evaluate the accuracy of the methods.

4    Consider the computational requirements:
    Some methods may be computationally expensive, especially for large datasets. You should consider the computational requirements of the methods and choose the one that is scalable and efficient.

 5   Seek expert advice:
    If you are unsure which method to use or how to apply it to your data, you can seek advice from experts in the field or consult relevant scientific publications.

Overall, selecting the right method for detecting anomalies in time series data requires a combination of domain knowledge, data exploration, and experimentation.