What mathematical methods are there to find anomalies in time series data? Which relevant scientific publications can you recommend?
There are several techniques used for detecting anomalies in time series data, including:

1. Segmentation: This involves dividing the time series into different segments and analyzing each segment separately. Anomalies may be detected by comparing the mean or median values of the segments with their respective historical averages or medians.
2. Time-series analysis: This involves using statistical tools such as autocorrelation, cross correlation, and spectral analysis to analyze the temporal patterns in the time series data. These tools can help identify any unusual trends or outliers that could indicate an anomaly.
3. Pattern recognition: This involves identifying specific patterns in the time series data that might suggest an anomaly. For example, if a stock price increases rapidly over a short period of time, it may be indicative of a market manipulation attempt.
4. Clustering: This involves applying clustering algorithms to the time series data to group similar points together. Anomalies can then be identified by looking at the distribution of these clusters.
5. Bayesian statistics: This involves using Bayesian inference to estimate the probability of an event occurring based on the observed data. Anomaly detection can be performed by calculating the posterior probabilities of different events under different models.
6. Neural networks: This is a type of machine learning algorithm that can be trained to detect anomalies in time series data. The neural network learns from the training data to recognize patterns and make predictions.
7. Support vector machines (SVMs): This is another machine learning algorithm that can be used for anomaly detection in time series data. SVMs use kernel functions to map the time series data into a high-dimensional space, where they can be classified using linear classifiers.
8. Random forests: This is a ensemble learning method that combines multiple decision trees to improve the accuracy of anomaly detection. It can be useful when dealing with complex datasets with many variables.
9. K-nearest neighbors (KNN): This is a simple yet effective technique for detecting anomalies in time series data. It works by finding the k-nearest neighbors of a given point in the time series data and classifying it according to its distance to the nearest neighbor.
10. Association rules: This is a graphical tool that allows users to explore relationships between different elements in a dataset. Association rules can be used to discover interesting patterns and potential anomalies.
11. Chi-square test: This is a statistical test that can be used to determine whether two samples come from the same population. It can also be used to check for significant differences between the means of two groups.
12. T-test: This is a statistical test that can be used to compare the means of two populations. It can also be used to check for significant differences between the standard deviations of two groups.
13. Mann-Whitney U test: This is a nonparametric test that can be used to compare the distributions of two independent samples. It can be particularly useful when dealing with small sample sizes.
14. Logistic regression: This is a statistical model that can be used to predict binary outcomes based on continuous variables. It can be used for predicting the likelihood of an event happening based on past observations.
15. Naïve Bayes classification: This is a simple but effective classification algorithm that uses Bayes' theorem to calculate the probability of an event occurring based on the available evidence. It can be used for anomaly detection in time series data.
16. Decision tree: This is a tree-based visualization tool that can be used for exploring complex data sets. It can be helpful for identifying potential anomalies and making decisions.
17. Regression tree: This is a tree-based visualization tool that can be used for exploring complex data sets. It can be helpful for identifying potential anomalies and making decisions.
18. Neural network: This is a type of machine learning algorithm that can be used for anomaly detection in time series data. The neural network learns from the training data to recognize patterns and make predictions.
19. Support vector machines (SVMs): This is another machine learning algorithm that can be used for anomaly detection in time series data. SVMs use kernel functions to map the time series data into a high-dimensional space, where they can be classified using linear classifiers.
20. Random forests: This is a ensemble learning method that combines multiple decision trees to improve the accuracy of anomaly detection. It can be useful when dealing with complex datasets with many variables.