# Re: Idempotence and "Replication Insensitivity" are equivalent ?

Date: 25 Sep 2006 08:45:44 -0700

Message-ID: <1159199143.967592.219690_at_m73g2000cwd.googlegroups.com>

I see what you mean.

Hmmm...

However, what one can observe is that for the same transformation this
invariance property does not hold for the AVG, as well, and this does
not seem to be a big problem ...

Both are invariant wrt linear transformations .... What do you think ?

-P

vc ha scritto:

*> pamelafluente_at_libero.it wrote:
**> > vc ha scritto:
*

> > > [...]

*> > > > In order to have a unique solution (and therefore equivalence), in the
**> > > > case we are dealing with finite set of numbers when the cardinality is
**> > > > not odd, the definition of MEDIAN is "completed" by *assuming* (it's
**> > > > an ASSUMPTION, a convention) as median the average of the 2 central
**> > > > terms of the ordered sequence of numbers:
**> > > >
**> > > > MEDIAN = ( x(n\2) + x(n\2 + 1) ) / 2
**> > >
**> > > The convention, while convenient in many cases, does not work if we
**> > > want to preserve invariance to monotonic transformations. Considering
**> > > the entire interval as the median is the only way to preserve the
**> > > invariance property.
**> >
**> > I have simply recalled the standard definition,
**> > but I have some "feeling" that this can be an interesting point.
**> >
**> > Can you make an example of what you have exactly in mind?
*

>

> Consider for example logarithmic(base 10) measurement scaling: (10,

*> 100, 1000, 10000) -> (1,2,3,4). Or income adjusted for inflation, or*

*> any kind of data transformation.*

>

> > Stated so generally it's not clear (to me) what you are referring to.

> > Invariance is sometimes used with different meanings,> > eg, in the case of the variance we say that it is

*> > translation-invariant.*

*> > What kind of "invariance" have you in mind ?*

>

> Let X is a randon variable, f is a transformation function and m is the

*> median. Then median invariance would mean that m(f(X)) = f(m(X)).*

Received on Mon Sep 25 2006 - 17:45:44 CEST