What is a 'bagplot', or 'bivariate boxplot'? I've found a paper which introduces the multidimensional (bivariate here) version of the boxplot - a bagplot. What is that bagplot exactly? I can see the series of nested polygons based on vertices, one of those polygons being declared as a bagplot. What is the idea of nested polygon building? Which of the polygons is the bagplot (central or holding the average number of points)? Do the edges of a bagplot possess some useful properties (like specifically dividing the point set)?
 A: Here's an example with notes:
Here is the article The Bagplot: A Bivariate Boxplot by Peter J. Rousseeuw , Ida Ruts & John W. Tukey from The American Statistician: http://venus.unive.it/romanaz/ada2/bagplot.pdf
From the abstract of that article:

The “depth median” is the deepest location, and it is  surrounded by a
  “bag” containing the n/2 observations with largest depth. Magnifying
  the bag by a factor 3 yields the “fence” (which is not plotted).
  Observations between the bag and the fence are marked by a light gray
  loop, whereas observations outside the fence are flagged as outliers.
  The bagplot visualizes the location, spread, correlation, skewness,
  and tails of the data.

Here's an illustration of the key parts:

Additional discussion can be found in the following:


*

*https://fenix.tecnico.ulisboa.pt/downloadFile/395137801290/paper.pdf

*http://www.thl.fi/publications/morgam/theses/kontto_2007.pdf

*http://www.sciencedirect.com/science/article/pii/S1470160X14004695

*http://search.r-project.org/library/aplpack/doc/bagplot.pdf

*ftp://ftp.univates.br/ensino/Softwares/R%20matem%C3%A1tica/R/R-2.14.2/library/aplpack/pdf/bagplot.pdf

*http://eprints.maynoothuniversity.ie/6085/1/State%20of%20the%20Art%20in%20Patterns%20for%20Point%20Cluster%20Analysis_EtienneDevogeleMcArdleICCSA2014.pdf

*https://en.wikipedia.org/wiki/Bagplot

*https://books.google.com.au/books?id=66gQCi5JOKYC&pg=PA59&redir_esc=y#v=onepage&q&f=false

*https://wis.kuleuven.be/stat/robust/papers/1999/struyfrousseeuw-depthcharacterizesdistribution-jmv.pdf
From the help docs of the aplpack package (for R users): 

A bagplot is a bivariate generalization of the well known boxplot. It
  has been proposed by Rousseeuw, Ruts, and Tukey. In the bivariate case
  the box of the boxplot changes to a convex hull, the bag of bagplot.
  In the bag are 50 percent of all points. The fence separates points in
  the fence from points outside. It is computed by increasing the the
  bag. The loop is defined as the convex polygon containing all points
  inside the fence. If all points are on a straight line you get a
  classical boxplot. bagplot() plots bagplots that are very similar to
  the one described in Rousseeuw et al. Remarks: The two dimensional
  median is approximated. There are known difficulties with small data
  sets (But I think it is not wise to make a (graphical) summary of e.g.
  10 points.)
In case people want to plot multiple (overlappIng) bagplots, it is
  convenient if the plots are semi-transparent. For this reason the
  transparency flag has been added to the bagplot command. If
  transparency==TRUE the alpha layer is set to '99' (hex). This causes
  the bagplots to appear semi-transparent, but ONLY if the output device
  is PDF and opened using: pdf(file="filename.pdf", version="1.4"). For
  this reason, the default is transparency==FALSE. This feature as well
  as the arguments to specify different colors has been proposed by
  Wouter Meuleman.

And an example:
library(aplpack)
attach(mtcars)
bagplot(wt, mpg, xlab="Car Weight", 
        ylab="Miles Per Gallon",
        main="Bagplot Example", 
        transparency = TRUE, 
        show.whiskers = FALSE, 
        # note that data a 'fence' separates inliers from outliers, 
        # and a 'loop' indicates the points outside the bag but 
        # inside the fence. In the 'bag' are 50 percent of all 
        # points
        show.loophull = TRUE, # draw 'loop'?
        show.baghull = TRUE)  # draw 'bag'?

 
related questions:


*

*https://stackoverflow.com/questions/29501282/plot-multiple-series-of-data-into-a-single-bagplot-with-r    

*https://stackoverflow.com/questions/31893559/r-adding-alpha-bags-to-a-2d-or-3d-scatterplot

*https://stackoverflow.com/questions/30301108/plot-a-circle-convex-hull-arround-a-given-percentage-of-points/
