1. CARMA Models for Stochastic Variability (or how to read a PSD) Jim Barrett University of Birmingham

2. Contents What is Stochastic Variabilty? What is a PSD? 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

7. Power Spectral Density (PSD) A bit like a Fourier Transform... Shows the 'power' in a signal as a function of frequency Serves as an excellent visual tool when investigating stochastic variability

8. How to Read a PSD - Knees

9. How to Read a PSD - Peaks

10. How to Read a PSD – Both

11. 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.

12. CARMA (in a nut shell) 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.

14. The Pipeline Use the banded, CARMA representation of the covariance matrix to write down a Likelihood function Baye's Theorem! Use posterior distributions on the various noise characteristics to perform (and inform) whatever science you want to do

15. Example 1 - XB158

16. Barnard et al. (arXiv:1501.01978)

17. Example 1 - XB158

18. Example 1 – XB158

19. Example 2 – Variability of ζ Puppis

23. Conclusions It can be important (and difficult) to understand stochastic variability Modelling noise as a multivariate Gaussian is an excellent way to go, but its expensive Using CARMA, we can render these problems computationally tractable