John D'Errico

No. SLM does NOT use truncated power functions! These are generally a numerically poor way to implement a spline, even only a cubic spline.

SLM uses a Hermite formulation, which personally, I've always preferred as they are easy to look at and understand the shape just by looking at the parameters since they are parameterized by function values and derivatives at the knots. Its just my personal preference though, since a spline by any other basis is still a spline.

As far as monotonicity or concavity applying only over a restricted range, you can do so using SLM directly. Read the help for slmset. It says (in part):

'increasing' - controls monotonicity of the function
= 'off' --> No part of the spline is constrained to be an
increasing function.
= 'on' --> the function will be increasing over its entire domain.
= vector of length 2 --> denotes the start and end points of a
region of the curve over which it is monotone increasing.
= array of size nx2 --> each row of which denotes the start
and end points of a region of the curve over which it is
monotone increasing.

So if you wish a constraint to apply only over a portion of the curve, you can specify the interval. This same approach applies to the concavity constraints. If that constraint would indicate the curve be monotonic over only part of a knot interval, I do stretch it to require monotonicity over a complete knot interval. So you cannot stop midway between a pair of knots. In that case, simply add a knot.

Erick - sorry, but I'm not doing active research in this area, and I don't have access to any new papers. I recall a paper by Wright and Wegman about constrained regression splines, but that dates back over 30 years. The work I did in developing SLM was mainly my own, based on my consulting activities, though I never published any papers on the topic.

Thank you, John. According to your suggestion, I have fitted my data again, but unfortunately, the results is worse than seperately fitting the two parts of the curve. I made a simple simulation, but I still got the similar result. Would you please take a few minutes to have a look at my simulation data, I will send it to you by email. Thank you.

I found the code written by Meyer on this web site "http://www.stat.colostate.edu/~meyer/srrs.htm", but it was written in R code, not matlab. I think I should learn R code these days. Meyer also did not give out how to fit the data with half convex and half concave, it seems still a long way for me to get out. Would you please give me some suggestion? Thank you.

I am looking forward to your new reply, Thank you again.

Yeah, it works =) Thank u so much!!

Congrats again for the useful function John. I just have a qestion: i'm trying to use fminsearchcon in a 5 parameter function i've created:

function error=fit(parameters)
x1=parameters(1);
x2=parameters(2);
x3=parameters(3);
x4=parameters(4);
x5=parameters(5);
(...) Here there is a long script but, at the end, the important thing are the input and outputs (...)

What I wonder is to find the parameters that optimize the error function. These parameters are interelated by the non-linear constrain:

-2.3*(x4^2+x3)^0.5+x5+x2>0

The problem is that I don't know how to implement this constrain, the examples I have seen are just with one parameter. I have tried this:

x0=[0.5,0.05,0.05,1,0.5];
lb=[0,0,0,0,0];
ub=[2,1,1,10,5];
[x,fval]=fminsearchcon(@fit,x0,lb,ub,[],[],@(x2,x3,x4,x5) 2.3*(x4^2+x3)^0.5-x5-x2)

But, of course, it doesn't recognise what is x2, x3, x4, or x5 because it is defined nowhere.

Any clue?? Thanks in advance! :)