🚀 go-pugleaf

In article <1jkb22INN8ob@pith.uoregon.edu> I wrote:

> I've got a researcher who is looking for a measure of 
> multi-modality (or "bi-polarity," if you will) such 
> that unimodal data would have a small score, and bimodal
> or trimodal or n-modal data would have a large score.
> 
> I'm at a loss for what to recommend -- any suggestions?

I am pleased to say that I got a number of very helpful
suggestions, and since several sci.math.stat readers requested
a copy of any replies I received, the following replies
are provided for your consideration. Since I've manually
prepared this compendium of replies, any errors in 
'digestification' are solely mine.

Thanks again to all who took the time to help!

Joe

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Date: Thu, 21 Jan 1993 08:03:00 -0700
From: mglacy@lamar.ColoState.EDU
To: joe@decoy.uoregon.edu

I've approached a similar problem by phrasing it in terms of amount
of variability. ( Could this sample have come from a population with
more than some specified amount of variability---measuring variability
with the Coefficient of Variation.)  I don't know if this translation
would work in in your situation.  If you do get some interesting ideas
about a direct test of modality, and they don't get posted to the net,
I'd appreciate a brief note about what you hear.

If the variability approach might work for you, I'd be glad to give
more details, a citation, etc.

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Date: Fri, 22 Jan 1993 15:28:11 -0500
From: fschwab@polaris.cv.nrao.edu (Fred Schwab)
To: joe@decoy.uoregon.edu (Joe St Sauver)

This might be of interest:

B. W. Silverman, "On a test for multimodality based on kernel density
estimates", MRC Tech. Summary Rep. 2181, U. of Wis. Math. Res. Ctr.,
Feb. 1981.  

Abstract: Kernel probability density estimates can be used to construct
a test of the hypothesis that the density underlying a given univariate
data set has at least k modes, for any given k>=1.  The test is based
on the critical value of the smoothing parameter for k modes to occur
in the estimate.  The theoretical properties of this test are  
investigated;
the asymptotic properties of the test statistic show that the test is
consistent.  Furthermore, ... 

Probably it was published, but I'm not sure where.  If you're interested
and have difficulty finding a copy of the report, or the published version
thereof, I could Xerox for you the copy that I have.

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Date: 23 Jan 1993 01:56:45 +0100 (CET)
From: Tomek Wyszomirski <TOMRYKI%PLEARN.BITNET@oregon.uoregon.edu>
Subject: Bimodality & bimodalizability

One of my colleagues has just passed me your question about bimodality
measure. I had the problem of quantifying bimodality when analysing a 
large body of results of plant competition simulations. My solution 
cWyszomirski T. 1992. Detecting and displaying size bimodality: kurtosis, 
skewness and bimodalizable distributions. Journal of Theoretical Biology, 
158(1): 109-128 (7 September 1992)! can be outlined as follows:
(i) we can accept the kurtosis coefficient as a bimodality index for
symmetric distributions. (ii) in skew distributions, skewness and kurtosis 
are not independent. So, we should modify the measure to separate the two 
characteristics. (iii) I propose the recipe: to perform skewness-removing 
(i.e.|g1|-minimizing) Box-Cox transformation and then to compute kurtosis
coefficient for BC-transformed data. Since power/log transformation may
sometimes create bimodality, I call this gBC (g2 for BC-transformed data)
coefficient an index of bimodalizability.

The above is a result of an entirely empirically-oriented approach.
Statistically rigorous attempts to separate skewness and kurtosis have 
been made e.g. by Balanda & MacGillivray c1990. Kurtosis and spread - 
Canadian Journal of Statistics, 18: 17-30 and their earlier papers!.

My gBC measure has some drawbacks, but I hope it will be useful for your
purpose. If I can help you any more, please feel free to ask me by e-mail.

If you (or others) try to apply my gBC coefficient, I would be very 
grateful for any comments on its performance.

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From: fschwab@daffy.CV.NRAO.EDU (Fred Schwab)
Subject: Re: Measures of Multi-Modality
Date: Sun, 24 Jan 93 16:44:46 EST

Joe,
    I was using one of the online data bases today and decided, for
fun, to see if there was anything recently published on multimodality
tests.  I came up with the following:

E. Mammen, J.S. Marron, and N.I. Fisher, "Some asymptotics for 
multimodality tests based on kernel density estimates", Probability 
Theory and Related Fields, v. 91 (Jan. 1992) 115-132.

Abstract: A test due to B.W. Silverman for modality of a probability 
density is based on counting modes of a kernel density estimator, and 
the idea of critical smoothing.  An asymptotic formula is given for the 
expected number of modes.  This, together with other methods, establishes 
the rate of convergence of the critically smoothed bandwidth.  These ideas 
are extended to provide insight concerning behaviour of the test based on 
bootstrap critical values.

     I had never heard of this journal before, but I see that the
university math library here (U.Va.) subscribes to it.  

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Date: Sun, 24 Jan 93 20:52 PST
From: jones@reed.edu (Albyn Jones)

you might look at the book by Titterington, Smith, and Makov
on finite mixture distributions.  i don't have the exact title
and so forth with me, if you need more info send me email.

this is a hard problem, in general, so there may not be a simple answer.

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