Re: DM: Parallel coordinates

[Alfred Inselberg]

Dear Warren,

Thanks for your thoughtful examples. However, "appears" is not what
distances are about just as what "appears" in projections is not what 
is
often the case. Specifically, in Parallel Coordinates (abbr. 
||-coords)
one can easily draw a hypercube of ANY dimension. Any polygonal line
within it is necessarily within the L1 (Manhattan metric) distance of 
the
side from any
other such polygonal line including those representing the vertices 
-- that
easily takes care of your examples and more. For Euclidean L2 
distance one 
constructs (i.e. draws) a sphere with any required radius -- that is
almost as easily done in ||-coords. There is a neat interior point
algorithm which I published and which easily and VISUALLY enables one 
to
decide if a 
point (or its polygonal line) is interior to the sphere. In this way 
one
can SEE (yes SEE) spherical neighborhoods in ANY dimension. Two 
papers in
the J. of Applied Math (April 1994) deal with all this and much more. 
It 
turns out that one can SEE (with some construction) the MINIMUM L1
Distance between two lines in ANY dimension. There is also a very 
simple 
algorithm there for constructing the L2 distance between any two 
points in 
||-coords. Some of these proximity results were used in Collision
Avoidance algorithms for Air Traffic Control (USA patents # 4,823,272,
# 5,058,024, # 5,173,861).

The easy definition of a point representation in ||-coords is NOT 
where
the strength of ||-coords lies. Rather it is in its ability to 
represent
RELATIONS provably without losing information. For example one of my
doctorate students recently found how to recognize if an object is 
CONVEX
in N-space. Of course, this is not "obvious" and many people do not
venture into these more challenging -- and most useful -- aspects   
of the Parallel Coordinates methodology.

Best regards

Alfred Inselberg






On Thu, 8 Jan 1998, Warren Sarle wrote:

> I wrote:
> > It is mathematically impossible except in degenerate cases to 
>display
> > 10-dimensional data in such a way that nearness of points in the 
>10D
> > space is visually preserved.
> 
> Alfred Inselberg <aiisreal@math.tau.ac.il> replied:
> > There is no problem doing this in Parallel Coordinates and for a 
>lot more
> > dimensions.
> 
> This raises some interesting questions. It is certainly true that a
> trained analyst can, with sufficient effort, assess distances in a
> parallel coordinates plot. But I don't think that the visual 
>impressions
> of distance provided by a parallel coordinates plot are a monotone
> function of actual Euclidean distance. Consider four cases in a 2D
> Euclidean space:
> 
>       X Y
>    A  0 0
>    B  1 1
>    C  0 1
>    D  1 0
> 
> The distance between A and B equals the distance between C and D.
> But a parallel coordinates plot of A and B looks like this:
> 
>       X  Y
>    0  ----
> 
> 
>    1  ----
> 
> while a parallel coordinates plot of C and D looks like this:
> 
>       X  Y
>    0  \  /
>        \/
>        /\
>    1  /  \
> 
> I think that C and D _appear_ closer than A and B. Whether this
> psychological impression is shared by other people is, of course,
> an empirical question. Does anyone know of experiments that have
> been done on the perception of distance in parallel coordinates 
>plots?
> 
> -- 
> 
> Warren S. Sarle       SAS Institute Inc.   The opinions expressed 
>here
> saswss@unx.sas.com    SAS Campus Drive     are mine and not 
>necessarily
> (919) 677-8000        Cary, NC 27513, USA  those of SAS Institute.
> * Do not send me unsolicited commercial, political, or religious 
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>