Mixture Guide

A Mixture object is a linear combination of Component objects that describes a complex distribution. The Mixture object state describes the distribution in terms of the Component objects (which have their own paramters) and a list of weights that describe the relative importance of each Component. Using the Expectation-Maximization algorithm, a Mixture object can fit itself to an input data array of shape (n_samples, n_features) where a “sample” is a single star (or can be treated as such, e.g. a combined binary) and a feature is any data we’re trying to fit to (position, velocity, magnitude, apparent-magnitude, abundances, inferred-age, etc.). The Mixture object will remain completely agnostic of the features being fit. The implementation of how features are fit to is left to the BaseComponent class.

See Fitting a Mixture for an example script and Fitting a mixture for instructions on how to use the command line tool fit-mixture.

Implemented Mixtures

ComponentMixture

Chronostar’s sole implemented Mixture Model matches the interface of scikit-learn’s mixture models as closely as possible, inheriting from sklearn.mixture.BaseMixture. For reference of a class that inherits from BaseMixture, see sklearn.mixture.GaussianMixture.

The key method of the ComponentMixture is its fit() method. In this instance, it calls fit() which is a scikit-learn method. This performs the Expectation-Maximization algorithm which we explain briefly here.

The Expectation-Maximization (EM) algorithm consists of two steps which are executed iteratively until convergence is reached. Underlying this algorithm is the assumption that each data point as a true membership to one of the components, or equivalently, for each data point, there is one component that is responsible for it. The E-step’s purpose is to estimate these membership probabilities as a value between 0 and 1. This is done by, for each data point, calculating the probability that it was generated by each component’s distribution (i.e. evaluate each Component’s PDF at the position of the data point) and then normalizing these probabilities such that they sum to 1. The E-step is implemented by scikit-learn’s base Class which calls the abstract method _estimate_log_prob which is defined in SKLComponentMixture.

Eqiupped with membership probabilites, the M-step can maximize the parameters of each Component in isolation. Since each Component has its own defined maximize() method, the M-step implementation loops over the mixture’s components, maximizing them one by one.