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

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?

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

Stochastic Variability – White Noise

Stochastic Variability – AR1 Noise

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

How to Read a PSD - Knees

How to Read a PSD - Peaks

How to Read a PSD – Both

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.

(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.

Example 1 - XB158

Example 1 - XB158 Barnard et al. (arXiv:1501.01978)

Example 1 - XB158

Example 1 – XB158

Example 2 – A Companion for Aldebaran http://www.armaghplanet.com

Example 2 – A Companion for Aldebaran

Example 2 – A Companion for Aldebaran

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!

Bonus – Variability of ζ Puppis

Bonus – Variability of ζ Puppis

Bonus – Variability of ζ Puppis

Bonus – Variability of ζ Puppis