[Rd] Canberra distance

[Prof Brian Ripley]

This is cetainly ancient R history.  The essence of the formula was 
last changed
-	    dist += fabs(x[i1] - x[i2])/(x[i1] + x[i2]);
+	    dist += fabs(x[i1] - x[i2])/fabs(x[i1] + x[i2]);

in October 1998.  The help page description came later.

The
            dist += fabs(x[i1] - x[i2])/(x[i1] + x[i2]);
form was there as 'canberra' in the first CVS archive in September 
1997 (as src/library/mva/src/dist.c) so it looks like one of R&R was 
the original author and this could be called pre-history.

On Sun, 7 Feb 2010, Bill.Venables at csiro.au wrote:

> That is interesting.  The first of these, namely
>
> sum(|x_i - y_i|) / sum(x_i + y_i)
>
> is now better known in ecology as the Bray-Curtis distance.  Even more interesting is the typo in Henry & Stevens "A Primer of Ecology in R" where the Bray Curtis distance formula is actually the Canberra distance  (Eq. 10.2 p. 289).  There seems to be a certain slipperiness of definition in this field.
>
> What surprises me most is why ecologists still cling to this way of doing things,  It is one of the few places I know of where the analysis is justified purely heuristically and not from any kind of explicit model for the ecological processes under study.
>
> Bill Venables.
>
>
>
> ________________________________________
> From: r-devel-bounces at r-project.org [r-devel-bounces at r-project.org] On Behalf Of Duncan Murdoch [murdoch at stats.uwo.ca]
> Sent: 07 February 2010 03:00
> To: genolini at u-paris10.fr
> Cc: r-devel at r-project.org
> Subject: Re: [Rd] Canberra distance
>
> On 06/02/2010 11:31 AM, Christophe Genolini wrote:
>> The definition I use is the on find in the book "Cluster analysis" by
>> Brian Everitt, Sabine Landau and Morven Leese.
>> They cite, as definition paper for Canberra distance, an article of
>> Lance and Williams "Computer programs for hierarchical polythetic
>> classification" Computer Journal 1966.
>> I do not have access, but here is the link :
>> http://comjnl.oxfordjournals.org/cgi/content/abstract/9/1/60
>> Hope this helps.
>>
>
> I do have access to that journal, and that paper gives the definition
>
> sum(|x_i - y_i|) / sum(x_i + y_i)
>
> and suggests the variation
>
> sum( [|x_i - y_i|) / (x_i + y_i) ] )
>
> It doesn't call either one the Canberra distance; it calls the first one
> the "non-metric coefficient" and doesn't name the second.  (I imagine
> the Canberra name came from the fact that the authors were at CSIRO in
> Canberra.)
>
> So I'd agree your definition is better, but I don't know if it can
> really be called the "Canberra distance".
>
> Duncan Murdoch
>
>> Christophe
>>> On 06/02/2010 10:39 AM, Christophe Genolini wrote:
>>>> Hi the list,
>>>>
>>>> According to what I know, the Canberra distance between X et Y is :
>>>> sum[ (|x_i - y_i|) / (|x_i|+|y_i|) ] (with | | denoting the function
>>>> 'absolute value')
>>>> In the source code of the canberra distance in the file distance.c,
>>>> we find :
>>>>
>>>>     sum = fabs(x[i1] + x[i2]);
>>>>     diff = fabs(x[i1] - x[i2]);
>>>>     dev = diff/sum;
>>>>
>>>> which correspond to the formula : sum[ (|x_i - y_i|) / (|x_i+y_i|) ]
>>>> (note that this does not define a distance... This is correct when
>>>> x_i and y_i are positive, but not when a value is negative.)
>>>>
>>>> Is it on purpose or is it a bug?
>>> It matches the documentation in ?dist, so it's not just a coding
>>> error.  It will give the same value as your definition if the two
>>> items have the same sign (not only both positive), but different
>>> values if the signs differ.
>>>
>>> The first three links I found searching Google Scholar for "Canberra
>>> distance" all define it only for non-negative data.  One of them gave
>>> exactly the R formula (even though the absolute value in the
>>> denominator is redundant), the others just put x_i + y_i in the
>>> denominator.
>>>
>>> None of the 3 papers cited the origin of the definition, so I can't
>>> tell you who is wrong.
>>>
>>> Duncan Murdoch
>>>
>>>
>
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595