Alfred Inselberg, the inventor of parallel coordinates (pictured below) will be giving a talk at Columbia this Thursday at 11am. More information in the extended entry.
Columbia Vision and Graphics Center Lecture
Thursday, October 18, 2007, 11am
Schapiro Center, Interschool Lab
Columbia UniversityMultidimensional Visualization and its Applications
Alfred Inselberg
School of Mathematical Sciences, Tel Aviv University, Israel
Senior Fellow in Visualization -- San Diego Supercomputer Center, USAThe desire to understand the underlying geometry of multidimensional
problems motivated several visualization methodologies to augment our limited
3-dimensional perception. After a short overview, Parallel Coordinates are
rigorously developed, obtaining a 1-1 mapping between subsets of Euclidean
N-space and subsets of 2-space. It leads to representations of lines, flats,
curves, intersections, hypersurfaces, proximities and geometrical
construction algorithms. Convexity can be visualized in any dimension,
as well as non-orientability (Moebius strip) and other properties of
hypersurfaces. This is a visual multidimensional coordinate system with
applications to air traffic control, visual and automatic data mining,
and interactive models of complex systems.
This looks interesting! Too bad it's the same time as our seminar. I have two comments:
1. I don't know what x1,...,x10 are in the above graph, but I expect the graph would be much improved by ordering the variables and also by transforming using location-scale shifts. As it is, the most dramatic patterns in the graph are things like the flip in sign from x4 to x5. Some judicious recoding could allow this to be more of a true exploratory data analysis (EDA) tool.
As with many other statistical tools (regression, scatterplots, ...), a little preprocessing can make the tool much more powerful. There's no requirement that it be applied directly to raw data. (For that matter, some of my suggestions such as ordering and signing of variables can be automated. Perhaps we should write an R function that serves as a front end to these plots.)
2. For some more thoughts on parallel coordinate plots and their relation to EDA and Bayesian model checking, see Section 3.2 of this paper.
@Andrew: Regarding point 1, many people have studied this, see for example this. I've seen a poster at a conference and a talk on this, but only web pages appear when I search for this on Google. :)
The real need is to make the plots interactive. Parallel coordinates are more exploratory than presentational. Sorting, rescaling, inverting, querying, zooming add considerably to the power of the plots. Al will certainly show his interactive software in his talk and demonstrate how poweful such tools are (while being very entertaining at the same time). My group produced CASSATT by Sylvia Winkler, which shared the first Chambers Prize, but which has been superceded by the implementation of interactive parallel coordinates in Martin Theus's Mondrian.
@Andrew: You should have a look at our paper on parallel coordinates for model building:
Unwin AR, Volinsky, C., Winkler S. 2003. Parallel Coordinates for Exploratory Modelling Analysis. Computational Statistics & Data Analysis 43: 553-64.
There is also a recent conference paper of mine applying parallel coordinates to the top decathlon performances over the last 20 years, which shows some of what can be done with a range of parallel coordinate plots:
www.opus-bayern.de/uni-augsburg/volltexte/2007/659/
Anthony, is that conference paper really using parallel coordinates plots? Or (less provocatively) what is the difference between parallel coordinates and time series?
P.S. The more I look at the graph above, the more it seems inappropriate for parallel coordinates. The adjacent pairs of variables all seem to be exactly linearly related to each other. The resulting graph is dramatic but contains almost no information.
Aleks and Antony: I suspect that interactivity is great, but I'd still like good defaults, such as automated ordering, scaling, and signing of variables. Otherwise you're wasting your time doing these things interactively by hand that could've already been done automatically. Again, this is a general EDA principle, to look for patterns _after_ the basic operations have been done.
Hadley: isn't a time series plot a special case of parallel coordinates?
@Hadley: You're right. I'd forgotten that I hadn't included any parallel coordinate plots by event, just parallel boxplots.
The differences between parallel coordinate plots and time series plots are twofold; firstly, you can investigate distributions at particular time points (by switching to the boxplot view); secondly, parallel coordinate plots can't deal with time series collected at different sets of time points.
@Andrew: Good defaults are valuable, but should not be overrated. It is unrealistic to think that any default can be highly informative for the complex structures of multivariate continuous data. (After all, even the defaults for simple histograms are usually not very good.)
Leaving display definitions to software can be a copout. You have more knowledge of the data than the software and Incorporating your background knowledge of the dataset is important. For instance, with the decathlon data it makes sense to initially plot the events in order of competition. If I leave the ordering of ten variables to the software I have to spend time working out which variable is where and why.
Interaction does not mean clicking through routine commands, it means taking a series of context-dependent decisions to gain more information from the data. The user is in control, not the software. It also means looking at a large number of plots, not just one. Of course, my opinion is influenced by using fast and flexible interactive graphics software. If I had to write a line of code for each static display I'd have another opinion. Working with good interactive graphics software is different, you should try it.
An important principle of EDA is to look at the raw data (ideally be present when the data are collected). I would be wary of claiming that there are any "basic operations" that should be carried out without checking.
@antony: Inselberg has shown a tool ParallAX that does include very many interactive features. I have not, however, seen the alpha-blending that allows one to "see behind" when lots of lines overlap.
A remark on Hadley's comment. The picture is ripe with information. The points (one at infinity) represent the 9 linear relations between XI and X(I+1) which collectively specify a line in 10-D. This is an example of how in ||-coords relational information is concentrated into a patterns which here are the 9 points.
Parallel coordinates is a multidimensional coordinate system rather than a "plot".
See the most recent patterns representing Convex surfaces in ANY dimension, Moebius strip and much more
Enjoy
Al
Delighted to read the comments. The display above shows points on a line in 10-D hence the X1 ... X10. It is about 30 years old and much has happened since. Parallel coordinates is a multidimensional coordinate system and not a "plot" like pie-charts, bar-charts etc. The intent since its inception in 1959 is to concentrate relational information into patterns. Above for example there are 9 intersection points (one at infinity) which completely specify the line. Please see my website www.math.tau.ac.il/~aiisreal and the tutorials. Also there are several recent papers
(including one in LNCS 4647 Springer Sept. 2007) and my textbook to be released by Springer in the near future with the more recent results.
The remarks made are still very close to the
definition of ||-coords which is almost 50 years now. Still I will be happy to respond to any queries.
Al