R help - Mar 2012 - Idea/package to "linearize a curve" along the diagonal?

Hi,

I am trying to normalize some data. First I fitted a principal curve
(using the LCPM package), but now I would like to apply a
transformation so that the curve becomes a "straight diagonal line" on
the plot.  The data used to fit the curve would then be normalized by
applying the same transformation to it.

A simple solution could be to apply translations only (e.g., as done
after a fit using loess), but here rotations would have to be applied
as well. One could visualize this as the "stretching of a curve",
i.e., during such an "unfolding" process, both translations and
rotations would need to be applied.

Before I embark on this (which would require me remembering long
forgotten geometry principles) I was wondering if you can think of
packages or tricks that could make my life simpler?

Thanks for any input,

Emmanuel


Below I provide an example - the black curve is to be "brought" along
the diagonal, and all data points normal to a "small segment" (of the
black curve) would undergo the same transformation as it - what
"small" is remains to be defined though.

    tmp=rnorm(2000)
    X.1 = 5+tmp
    Y.1 = 5+ (5*tmp+rnorm(2000))
    tmp=rnorm(1000)
    X.2 = 9+tmp
    Y.2 = 40+ (1.5*tmp+rnorm(1000))
    X.3 = 7+ 0.5*runif(500)
    Y.3 = 15+20*runif(500)
    X = c(X.1,X.2,X.3)
    Y = c(Y.1,Y.2,Y.3)

    lpc1 = lpc(cbind(X,Y), scaled=FALSE, h=c(1,1) )
    plot(lpc1)

Aren't you just reinventing the inverse of a function?
---------------------------------------------------------------------------
Jeff Newmiller                        The     .....       .....  Go Live...
DCN:<jdnewmil at dcn.davis.ca.us>        Basics: ##.#.       ##.#.  Live
Go...
                                      Live:   OO#.. Dead: OO#..  Playing
Research Engineer (Solar/Batteries            O.O#.       #.O#.  with
/Software/Embedded Controllers)               .OO#.       .OO#.  rocks...1k
--------------------------------------------------------------------------- 
Sent from my phone. Please excuse my brevity.

Emmanuel Levy <emmanuel.levy at gmail.com> wrote:
>Hi,
>
>I am trying to normalize some data. First I fitted a principal curve
>(using the LCPM package), but now I would like to apply a
>transformation so that the curve becomes a "straight diagonal
line" on
>the plot.  The data used to fit the curve would then be normalized by
>applying the same transformation to it.
>
>A simple solution could be to apply translations only (e.g., as done
>after a fit using loess), but here rotations would have to be applied
>as well. One could visualize this as the "stretching of a curve",
>i.e., during such an "unfolding" process, both translations and
>rotations would need to be applied.
>
>Before I embark on this (which would require me remembering long
>forgotten geometry principles) I was wondering if you can think of
>packages or tricks that could make my life simpler?
>
>Thanks for any input,
>
>Emmanuel
>
>
>Below I provide an example - the black curve is to be "brought"
along
>the diagonal, and all data points normal to a "small segment" (of
the
>black curve) would undergo the same transformation as it - what
>"small" is remains to be defined though.
>
>    tmp=rnorm(2000)
>    X.1 = 5+tmp
>    Y.1 = 5+ (5*tmp+rnorm(2000))
>    tmp=rnorm(1000)
>    X.2 = 9+tmp
>    Y.2 = 40+ (1.5*tmp+rnorm(1000))
>    X.3 = 7+ 0.5*runif(500)
>    Y.3 = 15+20*runif(500)
>    X = c(X.1,X.2,X.3)
>    Y = c(Y.1,Y.2,Y.3)
>
>    lpc1 = lpc(cbind(X,Y), scaled=FALSE, h=c(1,1) )
>    plot(lpc1)
>
>______________________________________________
>R-help at r-project.org mailing list
>https://stat.ethz.ch/mailman/listinfo/r-help
>PLEASE do read the posting guide
>http://www.R-project.org/posting-guide.html
>and provide commented, minimal, self-contained, reproducible code.

It is possible that I do not see what you mean, but it seems like the following
code does what you want. The general version of this is likely to be rather more
difficult to do, but in theory the inverse function seems like what you are
trying to accomplish.

x <- 1:20
y <- x^2 + rnorm(20)

y.lm <- lm( y ~ I(x^2) + x )
plot( x, y )
lines( x, predict( y.lm ) )

qa <- coef(y.lm)["I(x^2)"]
qb <- coef(y.lm)["x"]
qc <- coef(y.lm)["(Intercept)"]

# define inverse of known model
f1 <- function( y ) { ( sqrt( 4*qa*( y -qc ) + qb^2 ) - qb ) / ( 2*qa ) }
f2 <- function( y ) { -( sqrt( 4*qa*( y -qc ) + qb^2 ) + qb ) / ( 2*qa ) }

plot( x, f1( y ) ) 


---------------------------------------------------------------------------
Jeff Newmiller                        The     .....       .....  Go Live...
DCN:<jdnewmil at dcn.davis.ca.us>        Basics: ##.#.       ##.#.  Live
Go...
                                      Live:   OO#.. Dead: OO#..  Playing
Research Engineer (Solar/Batteries            O.O#.       #.O#.  with
/Software/Embedded Controllers)               .OO#.       .OO#.  rocks...1k
--------------------------------------------------------------------------- 
Sent from my phone. Please excuse my brevity.



Emmanuel Levy <emmanuel.levy at gmail.com> wrote:
>Dear Jeff,
>
>I'm sorry but I do not see what you mean. It'd be great if you could
>let me know in more details what you mean whenever you can.
>
>Thanks,
>
>Emmanuel
>
>
>On 12 March 2012 00:07, Jeff Newmiller <jdnewmil at dcn.davis.ca.us>
>wrote:
>> Aren't you just reinventing the inverse of a function?
>>
>---------------------------------------------------------------------------
>> Jeff Newmiller ? ? ? ? ? ? ? ? ? ? ? ?The ? ? ..... ? ? ? ..... ?Go
>Live...
>> DCN:<jdnewmil at dcn.davis.ca.us> ? ? ? ?Basics: ##.#. ? ? ?
##.#. ?Live
>Go...
>> ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Live: ? OO#.. Dead: OO#..
>?Playing
>> Research Engineer (Solar/Batteries ? ? ? ? ? ?O.O#. ? ? ? #.O#. ?with
>> /Software/Embedded Controllers) ? ? ? ? ? ? ? .OO#. ? ? ? .OO#.
>?rocks...1k
>>
>---------------------------------------------------------------------------
>> Sent from my phone. Please excuse my brevity.
>>
>> Emmanuel Levy <emmanuel.levy at gmail.com> wrote:
>>
>>>Hi,
>>>
>>>I am trying to normalize some data. First I fitted a principal curve
>>>(using the LCPM package), but now I would like to apply a
>>>transformation so that the curve becomes a "straight diagonal
line"
>on
>>>the plot. ?The data used to fit the curve would then be normalized
by
>>>applying the same transformation to it.
>>>
>>>A simple solution could be to apply translations only (e.g., as done
>>>after a fit using loess), but here rotations would have to be
applied
>>>as well. One could visualize this as the "stretching of a
curve",
>>>i.e., during such an "unfolding" process, both
translations and
>>>rotations would need to be applied.
>>>
>>>Before I embark on this (which would require me remembering long
>>>forgotten geometry principles) I was wondering if you can think of
>>>packages or tricks that could make my life simpler?
>>>
>>>Thanks for any input,
>>>
>>>Emmanuel
>>>
>>>
>>>Below I provide an example - the black curve is to be
"brought" along
>>>the diagonal, and all data points normal to a "small
segment" (of the
>>>black curve) would undergo the same transformation as it - what
>>>"small" is remains to be defined though.
>>>
>>> ? ?tmp=rnorm(2000)
>>> ? ?X.1 = 5+tmp
>>> ? ?Y.1 = 5+ (5*tmp+rnorm(2000))
>>> ? ?tmp=rnorm(1000)
>>> ? ?X.2 = 9+tmp
>>> ? ?Y.2 = 40+ (1.5*tmp+rnorm(1000))
>>> ? ?X.3 = 7+ 0.5*runif(500)
>>> ? ?Y.3 = 15+20*runif(500)
>>> ? ?X = c(X.1,X.2,X.3)
>>> ? ?Y = c(Y.1,Y.2,Y.3)
>>>
>>> ? ?lpc1 = lpc(cbind(X,Y), scaled=FALSE, h=c(1,1) )
>>> ? ?plot(lpc1)
>>>
>>>______________________________________________
>>>R-help at r-project.org mailing list
>>>https://stat.ethz.ch/mailman/listinfo/r-help
>>>PLEASE do read the posting guide
>>>http://www.R-project.org/posting-guide.html
>>>and provide commented, minimal, self-contained, reproducible code.
>>

> I am trying to normalize some data. First I fitted a principal curve
> (using the LCPM package), but now I would like to apply a
> transformation so that the curve becomes a "straight diagonal
line" on
> the plot.  The data used to fit the curve would then be normalized by
> applying the same transformation to it.
It is unclear to me what you mean by "diagonal" but I suspect what 
you're looking for is to locate projected points onto the unfolded curve.

That is exactly what coordinates on the principal curve would give you.

Sorry if I misunderstood your point,

Yvonnick Noel
University of Brittany,
Rennes, France