2. Re distances on grids "really" being in multidimensional space, and not
just being the distances projected onto the flat plane that it's convenient
to represent on a page:
I've just made a little model with cocktail sticks that assures me that,
just as the distance between two points in 2-dimensional space is, by
Pythagoras:
2 2
square root of { (x2-X1) + (y2 - y1) }
so, when you go to 2 points anywhere in 3 dimensions, Pythagoras's Theorem
gives the distance between the points as:
2 2 2
square root of { (x2-X1) + (y2 - y1) + (z2 - z1) }
and, pretty obviously, if you have more dimensions, you just add another term
2
of the form: (x2-x1)
to the sum that you accumulate before you take the square root.
I expect you really knew that!
So, if we are talking about distances between points in the factor space of
an orthogonal PCA, the terms such as x1, x2, x3, etc are the set of factor
loadings for Element 1, and the terms such as y1, y2, y3, etc are the set
of factor loadings for Element 2, and you can find the distance between the
positions in which they would plot, by squaring the differences in their
scores on each factor, and then adding these terms up, and finally taking
the square root.
Now J. Maxwell Legg has written that Patrick Slater developed a routine
along these lines for Ingrid, but he refers to it as probabilistic. I
don't see what can be probabilistic about it, once you've got the factor
scores (which, e.g., Minitab would give you). (This no doubt shows my
ignorance, but it seems to me that it's as deterministic as you'd expect
from Pythagoras!)
But of course the whole enterprise is probabilistic, in that if you did the
grid all over again you'd probably get different factors and factor
scores.
But at this stage of deriving a distance, aren't we just wanting to know
how far it is between those points, and the fuzziness of the entire
enterprise has receded into the background ofour minds. (?)
Am I missing something?
Tony Downing,
Dept. of Psychology, University of Newcastle upon Tyne, England.
A.C.Downing@ncl.ac.uk
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