[R] "FANNY" function in R package "cluster"

Aamir M intuitionist at gmail.com
Tue May 31 22:07:11 CEST 2005

```Martin> there is no 'm' in the book there, but they talk about the
Martin> exponent "^ 2" used in some places {but "^ 1" in other places},
Martin> notably in   5.2 "Why did we choose FANNY?"

Martin> There is no "fuzziness" parameter defined there, so can you be
Martin> more specific?

Martin> Is it the exponent  2 in  u_{jv}^2 ?
Martin> That one is currently fixed at 2, and yes, that could be made a
Martin> parameter though K & R warn against going all the way to "1"
Martin> where their algorithm can happend to converge very slowly.

Yes, that is what I am referring to. If you refer to equation (1) in
section 4.1 of K&R (1990), where the FANNY objective function is
defined, you can see that the membership values are all raised to the
power two. In fact, the choice of raising them to the power 2 is
arbitrary. Rather, the value of this exponent should be a user
specified parameter. This  is called the "m parameter" or the
"fuzziness parameter" in Fuzzy k-Means.

Now that you mentioned it, I see that K&R did in fact comment on this
in section 5.2. K&R say that setting m=1 will cause slower
convergence; in Fuzzy k-Means, setting m=1 will cause a hard
clustering (minumum fuzziness), and setting m=infinity will cause
maximum fuzziness (i.e. all cluster membership values will be equal to
1/k). They go on to say that "exponents equal to 2 seem to be a
reasonable choice, as is confirmed by actual clustering analyses." I
do not know about FANNY, but in Fuzzy k-Means, studies have shown that
values of the exponents between 1 and 2 can lead to better results
than the rather arbitrary choice of m=2.

Aamir> Is there, then, any way to compute the FANNY
Aamir> clustering membership values of a test data point
Aamir> without affecting the clustering membership values of
Aamir> the training data? Perhaps there are enough
Aamir> conditions to use the objective function as a way of
Aamir> computing the membership values of the test data?

Martin> That's an interesting proposal, at least the way I choose to
understand you :-)

Martin> Yes, why not look at the objective function C {eq.(1), p.182}

Martin> One could think of optimizing it with respect to new data only,
Martin> by keeping all "old data" memberships.
Martin> For that to work, one would need the n dissimilarites
Martin>     d[i', j]   where  i'  : `index for' new data
Martin> 	       j = 1,..,n : indices for training data.
Martin> Is this feasible in your situation?

Yes, this would be feasible, I think. If I understand it correctly,
this would just involve recomputing the DAISY dissimilarity matrix on
the combined set of both training data and test data. It seems that
the resulting optimization problem would also be uniquely solvable.

Martin> Alternatively, when we *did* assume ``all continuous'' data
Martin> *and* the use of simple Euclidean distances,
Martin> we could easily compute the cluster centers, determine (by
Martin> minimization!) memberships for new observations.

The problem of "predicting" fuzzy cluster memberships for new data
appears to be much simpler in Euclidean space; one could just compare
the new data to the cluster centers computed in Fuzzy k-Means.
Unfortunately, the data I'm working with is not all continuous.

Martin> In any case that needs some assumptions (and code!) currently
Martin> not part of fanny().

I'll have to work on this. Thought I'm guessing fanny() is written in
FORTRAN, which I cannot (yet) program in.

- Aamir

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