Depth and outlyingness measures

Depth and outlyingness functions are related concepts proposed to order the observations of a dataset, extend the concept of median and trimmed statistics to multivariate and functional data and to detect outliers.

Depth

Depth measures are functions that assign, to each possible observation, a value measuring how deep is that observation inside a given distribution (usually the distribution is approximated by a dataset). This function has it maximum value towards a “center” of the distribution, called the median of the depth. This allows a extension of the concept of median to multivariate or functional data. These functions also provide a natural order of the data, which is required to apply methods such as the boxplot or the trimmed mean.

The interface of a depth function is given by the following class:

skfda.exploratory.depth.Depth()

Abstract class representing a depth function.

The following classes implement depth functions for functional data:

skfda.exploratory.depth.IntegratedDepth(*[, ...])	Functional depth as the integral of a multivariate depth.
skfda.exploratory.depth.BandDepth()	Implementation of Band Depth for functional data.
skfda.exploratory.depth.ModifiedBandDepth()	Implementation of Modified Band Depth for functional data.
skfda.exploratory.depth.DistanceBasedDepth([...])	Functional depth based on a metric.

Most of them support functional data with more than one dimension on the domain and on the codomain.

Multivariate depths

Some utilities, such as the MagnitudeShapePlot require computing a non-functional (multivariate) depth pointwise. Moreover, some functional depths, such as the integrated depth are defined using multivariate depths. Thus we also provide some multivariate depth functions:

skfda.exploratory.depth.multivariate.ProjectionDepth()	Computes Projection depth.
skfda.exploratory.depth.multivariate.SimplicialDepth()	Simplicial depth.

Outlyingness

The concepts of depth and outlyingness are (inversely) related. A deeper datum is less likely an outlier. Conversely, a datum with very low depth is possibly an outlier. The following interface (which is very similar to the one used for depths) is used to define an outlyingness measure:

skfda.exploratory.depth.Outlyingness()

Abstract class representing an outlyingness function.

Multivariate outlyingness

We provide the classical Stahel-Donoho outlyingness measure for the univariate data case:

skfda.exploratory.depth.multivariate.StahelDonohoOutlyingness()

Computes Stahel-Donoho outlyingness.

Conversion

As depth and outlyingness are closely related, there are ways to convert one into the other. The following class define a depth based on an outlyingness measure.

skfda.exploratory.depth.OutlyingnessBasedDepth(...)

Computes depth based on an outlyingness measure.