1. CARMA Models for Stochastic Variability (or how to read a PSD)
Jim Barrett School of Physics and Astronomy

2. Contents What is Stochastic Variabilty?
What is a PSD (Power Spectral Density)?
How do you read a PSD?
What is a CARMA Model?
What's the point?

3. Stochastic Variability
Random perturbations to a set of observations (noise)
Highly Complex or Poorly Understood systems can also appear stochastic.
Almost all modern statistical treatments rely on evenly sampled data

4. Stochastic Variability – White Noise

5. Stochastic Variability – AR1 Noise

6. Power Spectral Density (PSD)
A bit like a Fourier Transform...
Shows the 'power' in a signal as a function of frequency

7. How to Read a PSD - Knees

8. How to Read a PSD - Peaks

9. How to Read a PSD – Both

10. Noise Models
Model noise as a Multivariate Gaussian Process, and writing down the statistics is easy!
This is great but computationally expensive
We can transform the covariance matrix using a Continuous, Auto-Regressive, Moving Average (CARMA) model.

11. (Continuous Auto-Regressive Moving Average)
CARMA (in a nut shell)
(Continuous Auto-Regressive Moving Average)
Any observation can be described in terms of the most recent observations and some 'series wide' properties.
Parametrised in terms of the mean and variance of the series, and a number of correlation timescales.
The more timescales used, the more complex (and powerful) the model.

13. Example 1 - XB158

14. Example 1 - XB158
Barnard et al. (arXiv: )

15. Example 1 - XB158

16. Example 1 – XB158

17. Example 2 – A Companion for Aldebaran

18. Example 2 – A Companion for Aldebaran

19. Example 2 – A Companion for Aldebaran

20. Conclusions
It is important (and difficult) to understand and characterise stochastic variability
We can significantly speed up the characterisation process using CARMA models.
This method is very general!

21. Bonus – Variability of ζ Puppis

22. Bonus – Variability of ζ Puppis

23. Bonus – Variability of ζ Puppis

24. Bonus – Variability of ζ Puppis