Statistical methodology for high-dimensional and functional time series

1. Introduction and overview (1 hour)

• Motivate the study of functional and high-dimensional data with current data examples
• Summarize the available methodology
• Summarize the available technical contributions
• Provide outline of course

2. Review of scalar and multivariate methods (3 hours)

• Review of classical time series methods
• Decomposition into trend plus seasonality plus stationary errors
• The concept of stationarity and the autocorrelation function
• Predicting and estimating time series
• Data examples

3. Functional time series methods (10 hours)

• A primer on functional data and functional analysis
• Registration, representation and smoothing of functional data
• Functional time series models
• Prediction and estimation of functional time series

4. High-dimensional time series methods (9 hours)

•  A primer on random matrix theory
• High-dimensional phenomena
• Adding time series context
• Adding functional time series context
• Discuss possible new research projects

5. Conclusions and outlook (1 hour)

• Summarize contents of course

• Prepare for future research collaborations

References

[1] Aue, A., Dubart Nourinho, D., and Hormann, S. (2015). On the prediction of stationary functional time series, Journal of the American Statistical Association 110, 378–392.
[2] Bai, Z., and Silverstein J.W. (2010). Spectral Analysis of Large Dimensional Random Matrices (2nd et.). Springer-Verlag, New York.
[3] Bosq, D. (2000). Linear Processes in Function Spaces. Springer- Verlag, New York.
[4] Brillinger, D.R. (1975). Time Series: Data Analysis and Theory. Holt, Rinehart & Winston, New York.
[5] Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods (2nd ed.). Springer-Verlag, New York.
[6] Fan, J., and Yao, Q. (2003). Nonlinear Time Series. Springer-Verlag, New York.
[7] Hallin, M., Nisol, G., and Tavakoli, S. (2023). Factor models for high-dimensional functional time series I:Representation results. Journal of Time Series Analysis 44, 578–600.
[8] Hallin, M., Nisol, G., and Tavakoli, S. (2023). Factor models for high-dimensional functional time series II: Estimation and forecasting. Journal of Time Series Analysis 44, 601–621.
[9] Hsing, T., and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley, Chichester, UK.
[10] Liu, H., Aue, A., and Paul, D. (2015), On the Marcenko–Pastur law for linear time series, ˇ The Annals of Statistics 43, 675–712.
[11] Paul, D., and Aue, A. (2014). Random matrix theory in statistics: a review. Journal of Statistical Planning and Inference 150, 1–29.
[12] Silverman, B.W., and Ramsay, J.O. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer-Verlag, New York.
[13] Shumway, R.H., and Stoffer, D.A. (2006). Time Series Analysis and its Applications (2nd ed.). SpringerVerlag, New York.