Is this normal? A new projection pursuit index to assess a sample against a multivariate null distribution

Dianne Cook
Econometrics and Business Statistics
Monash University
Joint with Ursula Laa, BOKU and Annalisa Calvi, Monash

Motivation

Image Nick Youngson CC BY-SA 3.0 Pix4free

Outline

• About projection pursuit
• The guided tour
• The PP index
• Examples

Projection pursuit and guided tour

Projection pursuit (PP) defines a quantitative criterion for the interestingness of a projection (a projection pursuit index), and searches the space of possible projections for the most interesting one to display. Principal component analysis can be considered to the a special case of PP.

Combining PP with a tour algorithm provides a way to watch the optimisation, and see more projections than just the best. This is called a projection pursuit guided tour.

Ways to make comparisons

Generate a reference sample (here it is from $4\text{-}D$ multivariate normal), and overlay with the new sample.

Or, generate a confidence region from the reference distribution (here a $4\text{-}D$ ellipse because reference distribution is multivariate normal).

BUT

How do you see that an observation is outside a confidence region, or the subspaces/directions it is outside?

How do you get the optimisation to ignore one of the point sets?

New PP Index

Let $P$ indicate a 2D projection basis ($p\times 2$). The reference sample is from $N_p(\mathbfit{\mu}, \Sigma)$. Mahalanobis distance measures the distance between an observation $\mathbfit{x}$ and the center $\mathbfit{\mu}$ using the variance-covariance matrix $\Sigma$.

$$d_M (\mathbfit{x}) = \sqrt{(\mathbfit{x}-\mathbfit{\mu}) \Sigma^{-1}(\mathbfit{x}-\mathbfit{\mu})^T}$$

We define the new PP index as

$$\sum_{\mathbfit{w} \in W} (\mathbfit{w} - \mathbfit{\mu}) P (P^T\Sigma P)^{-1}P^T(\mathbfit{w} - \mathbfit{\mu})^T.$$ where $W = \{\mathbfit{x}: (\mathbfit{x}-\mathbfit{\mu}) \Sigma^{-1}(\mathbfit{x}-\mathbfit{\mu})^T > c^2\}$ is the subset of points being compared with the normal.

Drawing ellipses in projections

Let $\mathbfit{x}$ be a $p\text{-}D$ vector. A $p\text{-}D$ ellipsoid corresponding to a given variance-covariance ($\Sigma$) and mean vector ($\mathbfit{\mu}$) is described by the equation

$$(\mathbfit{x}-\mathbfit{\mu}) \Sigma^{-1}(\mathbfit{x}-\mathbfit{\mu})^T = c^2$$

where $c$ is a constant that depends on a specific confidence level.

Let $\mathbfit{y}$ be a $2\text{-}D$ vector representing a point in $2\text{-}D$, and let $\mathbfit{\mu}_2$ denote the projected mean.

The projection of the $p\text{-}D$ ellipsoid onto the $2\text{-}D$ projection space described by $P$ has the equation

$$(\mathbfit{y} - \mathbfit{\mu}_2)(P^T \Sigma P)^{-1}(\mathbfit{y} - \mathbfit{\mu}_2)^T = c^2.$$

How it works

The null being from a normal population is represented in the projection using an ellipse, the projection of the $p$-D confidence ellipse.

Implementation

This method is implemented in the tourr package.

library(tourr)
animate_xy(samp, 
  guided_anomaly_tour(anomaly_index(),
  ellipse=vc_null), ellipse=vc_null, 
  axes = "bottomleft", half_range=5, 
  center=FALSE)

Application: Liver function tests

Similar to data on women’s liver function.

Can also work for longitudinal data - here being a sample from an aging male (simulated).

Application: weather extremes

• heatmap
• anomaly tour

Example from Marcus Mayrhofer and Peter Filzmoser (2023) using robust statistics to examine weather extremes in Vienna, Austria.

Extremes in different directions. Cluster the points that are considered extreme, and run projection pursuit on each group.

Summary and future directions

• This provides an alternative way to examine groups of observations relative to a reference distribtuion.
• It might be also be used for outlier detection.

• More approaches to grouping/clustering cases before projection pursuit could be explored.
• Incorporate other reference distributions, possibly with shapes represented by convex hulls.
• Some of the outlyingness methods in robust statistics could also be adapted into PP indexes specifically for outlier detection.

References and acknowledgements

• Calvi, Laa, Cook (2024) Is this normal? A new projection pursuit index to assess a sample against a multivariate null distribution
• Cook and Laa (2025) Interactively exploring high-dimensional data and models in R
• Wickham et al (2015) Visualizing statistical models: Removing the blindfold
• Flatland: A Romance of Many Dimensions (1884) Edwin Abbott
• R package: tourr.

Slides made in Quarto, with code on the website of the paper.

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.