[R] Need to fit a regression line using orthogonal residuals

[Bill.Venables at csiro.au]

Jonathon Kopecky asks:

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Jonathon Kopecky
Sent: Tuesday, 30 January 2007 5:52 AM
To: r-help at stat.math.ethz.ch
Subject: [R] Need to fit a regression line using orthogonal residuals

I'm trying to fit a simple linear regression of just Y ~ X, but both X
and Y are noisy.  Thus instead of fitting a standard linear model
minimizing vertical residuals, I would like to minimize
orthogonal/perpendicular residuals.  I have tried searching the
R-packages, but have not found anything that seems suitable.  I'm not
sure what these types of residuals are typically called (they seem to
have many different names), so that may be my trouble.  I do not want to
use Principal Components Analysis (as was answered to a previous
questioner a few years ago), I just want to minimize the combined noise
of my two variables.  Is there a way for me to do this in R?
[WNV] There's always a way if you are prepared to program it.  Your
question is a bit like asking "Is there a way to do this in Fortran?"
The most direct way to do it is to define a function that gives you the
sum of the perpendicular distances and minimise it using, say, optim().
E.g.
	ppdis <- function(b, x, y) sum((y - b[1] - b[2]*x)^2/(1+b[2]^2))
	b0 <- lsfit(x, y)$coef  # initial value
	op <- optim(b0, ppdis, method = "BFGS", x=x, y=y)
	op  # now to check the results
	plot(x, y, asp = 1)  # why 'asp = 1'?? exercise
	abline(b0, col = "red")
	abline(op$par, col = "blue")
There are a couple of things about this you should be aware of, though
First, this is just a fiddly way of finding the first principal
component, so your desire not to use Principal Component Analysis is
somewhat thwarted, as it must be.
Second, the result is sensitive to scale - if you change the scales of
either x or y, e.g. from lbs to kilograms, the answer is different.
This also means that unless your measurement units for x and y are
comparable it's hard to see how the result can make much sense.  A
related issue is that you have to take some care when plotting the
result or orthogonal distances will not appear to be orthogonal.
Third, the resulting line is not optimal for either predicting y for a
new x or x from a new y.  It's hard to see why it is ever of much
interest.
Bill Venables.


Jonathon Kopecky
University of Michigan

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