[R] OT -- isotonic regression subject to bound constraints.
Rolf Turner
r.turner at auckland.ac.nz
Mon Dec 11 19:15:48 CET 2017
Well, I could argue that it's not *completely* OT since my question is
motivated by an enquiry that I received in respect of a CRAN package
"Iso" that I wrote and maintain.
The question is this: Given observations y_1, ..., y_n, what is the
solution to the problem:
minimise \sum_{i=1}^n (y_i - y_i^*)^2
with respect to y_1^*, ..., y_n^* subject to the "isotonic" constraint
y_1^* <= y_2^* <= ... <= y_n^* and the *additional8 bound constraint
a <= y_1^* and y_n^* <= b, where a and b are given constants?
I have googled around a bit (unsuccessfully) and have asked this
question on crossvalidated a couple of days ago, with no response whatever.
So I thought that I might try the super-knowledgeable R community, in
the hope that someone out there might be able to tell me something useful.
Note that the question can be expressed as finding the projection of the
point (y_1, ..., y_n) onto the intersection of the isotonic cone and
the hypercube [a,b]^n.
At first I thought that protecting onto the isotonic cone and then
projection that result onto the hypercube might work, but I am now
pretty sure that is hopelessly naive.
Any hints? Ta.
cheers,
Rolf Turner
--
Technical Editor ANZJS
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276
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