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derekbeaton / OuRS

Outliers and Robust Structures
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OuRS: Outliers and Robust Structures

OuRS

OuRS is short for Outliers and Robust Structures. The goal of this package is to provide a variety of multivariate outlier detection and robust subspace techniques. While there are several other packages to do so in R, this package was developed to meet specific needs not currently met by those packages. OuRS includes, for examples, outlier detection in multivariate categorical or mixed data, and simplified approaches to high dimensional (i.e., (n << P)) data, as well as approaches to identify sources (variables) of outlierness (of observations).

As of now, the OuRS package is under active development. However, several approaches we make available in OuRS are stable and available for use, namely:

Installation

Because OuRS is under development it is not yet available as a complete package, though it can still be installed. OuRS depends on the GSVD package (GSVD). For now, the simplest approach to installation is through the devtools or remote packages. NOTE both packages are more aggressive about warnings and treat them as errors, and thus prevents installation (see here, for example). So to install OuRS, you will need to set an environment variable for the remotes package (there may be equivalents in devtools if that is preferred):

remotes::install_github("derekbeaton/GSVD")
Sys.setenv(R_REMOTES_NO_ERRORS_FROM_WARNINGS = TRUE)
remotes::install_github("derekbeaton/OuRS/OuRS")

Set up

Prior to running the examples, we require several packages to be loaded. The analyses are all performed in OuRS, which depends on GSVD. We also require cellWise to access data for this README.

Examples

Here we provide examples with brief explanations of the three previously mentioned approaches: MCD and a GenMCD.

MCD

The MCD algorithm (Hubert & Debruyne, 2009) is an outlier detection and robust covariance technique. The MCD’s goal is to find a covariance matrix with a robust location (mean) and minimum scatter (covariance). The objective of the MCD is to find a minimum determinant (i.e., product of the eigenvalues), which reflects a minimum scatter. The MCD primarily relies on subsampling to find some subsample (H) that provides a minimum determinant. Generally the MCD works as follows. Given some matrix (\mathbf{X}) with (I) rows and (J) columns, we randomly subsample (\mathbf{X}) to have (H) rows, referred to as (\mathbf{X}_{H}). The size of (H) is controlled by the user through the (\alpha) parameter which must exist between .5 and 1:

  1. For (\mathbf{X}{H}), compute the column-wise mean as (\boldsymbol{\mu}{H}) and the covariance as (\mathbf{S}_{H}).

  2. Compute the squared Mahalanobis distances for each observation in (\mathbf{X}) as (m{i} = (x{i} - \boldsymbol{\mu}{H}) \mathbf{S}{H}^{-1} (x{i} - \boldsymbol{\mu}{H})^{T}).

  3. Set (\mathbf{X}_{H}) as the subset of (H) observations with the smallest Mahalanobis distances computed from Step 2.

  4. Compute and store the determinant of (\mathbf{S}{H}) which is the covariance matrix of (\mathbf{X}{H}).

The steps above are repeated for either a pre-specified number of iterations or until convergence (i.e., the determinant in Step 4 has little to no change). Once completed, the (H) set with a minimum determinant reflects the sub-sample with the robust location (robust mean), minimum scatter (robust covariance) from the data in (\mathbf{X}). Furthermore, a set of robust Mahalanobis distances are computed from the robust covariance matrix. The robust Mahalanobis distances can then be used to identify outliers.

Below we provide an example of the MCD for a routinely used data set for the MCD: the philips dataset from the cellWise package. While there are numerous outputs from the algorithm, we only show a graph of the standard vs. robust Mahalanobis distances for the philips data. The philips data are used as a typical example because of the “masking” effect, wherein there are outliers that exist but are not detectable by Mahalanobis distance. This is because they outlyingness of these observations exist, typically, in a lower variance subspace. Thus the MCD and robust Mahalanobis distance “unmasks” these observations.

data("philips")
# mostly default parameters for now, but with the lowest size
# of H, i.e., alpha=.5, and many iterations
mcd.results <- continuous_mcd(philips, alpha = 0.5)
dd_plot(mcd.results, dist_transform = "sqrt")

As can be seen in the above figure, there are observations with a large robust Mahalanobis distance, but relatively small Mahalanobis distance; these are the observations that were “masked”.

GenMCD

One of the primary motivations behind the creation of the OuRS package was the development of an extension to the MCD. We initially extended the MCD so that it could be applied to categorical (and similar data). However, our extension of the MCD can incorporate data of almost any data type (categorical, ordinal, continuous, counts), either as a homogeneous set or heterogeneous variables (i.e., mixed data). We have made a preprint of this available on bioRxiv.

The example we provide here is strictly categorical data. The data here are a toy data set of genetic variables (single nucleotide polymorphisms, a.k.a. SNPs).

load("OuRS/data/SNPS.rda")
summary(SNPS[, 1:2])
##     SNP.1              SNP.2          
##  Length:60          Length:60         
##  Class :character   Class :character  
##  Mode  :character   Mode  :character
# mostly default parameters for now, but with the lowest size
# of H, i.e., alpha=.5, and many iterations
catmcd.results <- categorical_mcd(SNPS, alpha = 0.5)
dd_plot(catmcd.results, dist_transform = "sqrt")

Additional materials

Please see the (Publications)[./Publications] and (Presentations)[./Presentations] directories for more examples of some of our outlier and robust structures work.