Component Guide#

A Component object models a simple distribution, typically a mutli-variate normal (i.e. Gaussian) distribution. A Component class encapsulates the parameters that define the distribution, the methods that determine the best parameters given input data and how the input data is interpreted.

For example, a SphereSpaceTimeComponent is defined by an age, birth mean and birth covariance matrix, parameterised by a single array parameters (as used in Paper 1). It also has methods for calculating the log probability of stars estimate_log_prob() and estimating its parameters maximize(). For more details, see the API. It also is able to reconstruct data covariance matrices from the input data rows.

Components can accept a list of membership probabilities, thereby the contribution of any datapoint is weighted accordingly during fits.

For example usages see Fitting a component.

Implemented Components#

SpaceComponent#

The space component fits a free 6D Gaussian to a set of 6D data points. This component cannot handle uncertainties and ignores any data beyond the first 6 columns.

This component class is useful for fitting to data where either uncertainties are not significant (e.g. when characterising the background) or for getting a first order characterisation of the environment of an association, the results of which may be used to initialise a more nuanced fit, thereby skipping a lot of computation.

This component is lightning quick to fit, because it doesn’t perform any parameter exploration. It calculates its mean and covariance using simple numpy functions.

The parameters array of a SpaceComponent is 42 elements long, the first 6 being the mean, the next 36 are the covariance matrix, flattened, i.e.:

parameters = np.hstack((mean, covariance.flatten()))

SphereSpaceTimeComponent#

The SphereSpaceTimeComponent has an age (hence the time in its name) and assumes the distribution of an association at birth to be spherical. The birth distribution is therefore a 6D Gaussian, spherical in position and spherical in velocity. The current day distribution (which determines the actual log probabilities of the data) is the birth distribution projected forward through the galactic potential by age Myr.

The parameters array parameterise the birth mean (first 6 elements), the birth covariance (the next 2) and the age. A standard deviation in position dxyz and in velocity duvw parameterise the birth covariance matrix. Therefore the parameters array looks like:

parameters = np.array([x, y, z, u, v, w, dxyz, duvw, age])

The component calculates its current day mean using the epicyclic approximation trace_epicyclic_orbit(), and transforms the birth covariance to the current day covariance using a jacobian matrix transform_covmatrix().

Implementing Custom Components#

All component classes derive from BaseComponent. Any custom derived component classes should also derive from this base class. The base class has some abstract methods and properties that your new class must implement:

  • maximize(): given input data, determine the best values for the component’s parameters

  • estimate_log_prob(): given your component’s current distribution, estimate the log probability of a data point. i.e. if the distribution is a normalised PDF, evaluate it at the location of the data point, and take the log of the result.

  • split(): split the current component into two similar, overlapping components that when combined more or less describe the original. This is the key mechanism used to by SimpleIntroducer to introduce new components.

  • set_parameters() - set the parameters

  • get_parameters() - get the parameters

  • n_params() - the number of parameters used. This is not necessarily the same as len(self.get_parameters()), for example SpaceComponent has duplicates of the covariance values in its parameter array. Calculations of BIC and AIC need to know this.

You may find it convenient to define further properties that convert the raw parameter array into the component aspects. For example, SphereSpaceTimeComponent has as properties mean, covariance and age. These properties should only depend on the component’s parameters. Defining them as properties (and not attributes) guarantees that the state of the component is defined in one place only, the parameters and removes the temptation of the user to attempt to modfiy the current day mean and covariance.

Note

The maximize() method should never make use of these properties, for two reasons. Firstly it doesn’t make algorithmic sense because it is trying to find the best values for parameters and update them, not evaluate how good the current values are. Secondly, there would be potential performance issues by redoing a potentially expensive operation during the many iterations.