State Space Models and the Kalman Filter

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Takeaways:

• Bayesian statistics offers a robust framework for econometric modeling.
• State space models provide a comprehensive way to understand time series data.
• Gaussian random walks serve as a foundational model in time series analysis.
• Innovations represent external shocks that can significantly impact forecasts.
• Understanding the assumptions behind models is key to effective forecasting.
• Complex models are not always better; simplicity can be powerful.
• Forecasting requires careful consideration of potential disruptions. Understanding observed and hidden states is crucial in modeling.
• Latent abilities can be modeled as Gaussian random walks.
• State space models can be highly flexible and diverse.
• Composability allows for the integration of different model components.
• Trends in time series should reflect real-world dynamics.
• Seasonality can be captured through Fourier bases.
• AR components help model residuals in time series data.
• Exogenous regression components can enhance state space models.
• Causal analysis in time series often involves interventions and counterfactuals.
• Time-varying regression allows for dynamic relationships between variables.
• Kalman filters were originally developed for tracking rockets in space.
• The Kalman filter iteratively updates beliefs based on new data.
• Missing data can be treated as hidden states in the Kalman filter framework.
• The Kalman filter is a practical application of Bayes' theorem in a sequential context.
• Understanding the dynamics of systems is crucial for effective modeling.
• The state space module in PyMC simplifies complex time series modeling tasks.

Chapters:

00:00 Introduction to Jesse Krabowski and Time Series Analysis

04:33 Jesse's Journey into Bayesian Statistics

10:51 Exploring State Space Models

18:28 Understanding State Space Models and Their Components

40:39 Composability of State Space Models

48:36 Understanding Trends and Derivatives

52:35 The Importance of Seasonality in Time Series

56:41 Components of Time Series Analysis

01:00:46 Exogenous Regression in State Space Models

01:06:41 Impulse Response Functions and Causality

01:11:30 Why Kalman Filter Is So Powerful

01:24:28 Future Directions and Applications

Thank you to my Patrons for making this episode possible!

Yusuke Saito, Avi Bryant, Ero Carrera, Giuliano Cruz, Tim Gasser, James Wade, Tradd Salvo, William Benton, James Ahloy, Robin Taylor,, Chad Scherrer, Zwelithini Tunyiswa, Bertrand Wilden, James Thompson, Stephen Oates, Gian Luca Di Tanna, Jack Wells, Matthew Maldonado, Ian Costley, Ally Salim, Larry Gill, Ian Moran, Paul Oreto, Colin Caprani, Colin Carroll, Nathaniel Burbank, Michael Osthege, Rémi Louf, Clive Edelsten, Henri Wallen, Hugo Botha, Vinh Nguyen, Marcin Elantkowski, Adam C. Smith, Will Kurt, Andrew Moskowitz, Hector Munoz, Marco Gorelli, Simon Kessell, Bradley Rode, Patrick Kelley, Rick Anderson, Casper de Bruin, Philippe Labonde, Michael Hankin, Cameron Smith, Tomáš Frýda, Ryan Wesslen, Andreas Netti, Riley King, Yoshiyuki Hamajima, Sven De Maeyer, Michael DeCrescenzo, Fergal M, Mason Yahr, Naoya Kanai, Steven Rowland, Aubrey Clayton, Jeannine Sue, Omri Har Shemesh, Scott Anthony Robson, Robert Yolken, Or Duek, Pavel Dusek, Paul Cox, Andreas Kröpelin, Raphaël R, Nicolas Rode, Gabriel Stechschulte, Arkady, Kurt TeKolste, Gergely Juhasz, Marcus Nölke, Maggi Mackintosh, Grant Pezzolesi, Avram Aelony, Joshua Meehl, Javier Sabio, Kristian Higgins, Alex Jones, Gregorio Aguilar, Matt Rosinski, Bart Trudeau, Luis Fonseca, Dante Gates, Matt Niccolls, Maksim Kuznecov, Michael Thomas, Luke Gorrie, Cory Kiser, Julio, Edvin Saveljev, Frederick Ayala, Jeffrey Powell, Gal Kampel, Adan Romero, Will Geary, Blake Walters, Jonathan Morgan, Francesco Madrisotti, Ivy Huang, Gary Clarke, Robert Flannery, Rasmus Hindström, Stefan, Corey Abshire and Mike Loncaric.

Links from the show:

• Jesse on GitHub: https://github.com/jessegrabowski
• Jesse on LinkedIn: www.linkedin.com/in/jessegrabowski
• Jesse on Google Scholar: https://scholar.google.com/citations?user=vOCjGPwAAAAJ&hl=en
• State space presentation repo: https://github.com/jessegrabowski/statespace-presentation/tree/main
• Try the statespace module on pymc-experimental: https://github.com/pymc-devs/pymc-experimental
• Durbin, James, and Siem Jan Koopman. Time series analysis by state space methods, Oxford, 2012: https://academic.oup.com/book/16563?login=false
• Hyndman, Rob and George Athanasopoulos, Forecasting: Principals and Practice, 3rd Edition. Otexts, 2018: https://otexts.com/fpp3/
• Roger Labbe, Kalman and Bayesian Filters in Python: https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
• Quantecon.org: https://quantecon.org/
• Lecture on Kalman Filtering: https://python.quantecon.org/kalman.html
• Mamba – Linear-Time Sequence Modeling with State Spaces (state spaces in machine learning): https://arxiv.org/abs/2312.00752
• Paper explanation: https://www.youtube.com/watch?v=9dSkvxS2EB0
• Good lecture on the statistics of the Kalman filter: https://www.youtube.com/watch?v=8lPBkkbtNW8
• And on structural state space models: https://www.youtube.com/watch?v=2vf-d_fRCXs

Transcript

This is an automatic transcript and may therefore contain errors. Please get in touch if you're willing to correct them.