@erick: A good entry to the literature on regression splines might be J.O. Ramsay, "Monotone regression splines in action", which is heavily cited by later works. A copy can be obtained for free from http://tinyurl.com/cqekokb

A superb package. Well done! It is perfect for fitting a curve to in-orbit test (IOT) antenna pattern measured data points, used in validating the performance of our satellite antennas. Thanks very much.

Peter Simon
Space Systems/Loral
Antenna Subsystems Operations

Eshwar - I considered offering a 5th order or higher fit when I wrote the code. I did not do so however, for valid reasons, at least what I considered valid ones.
- I've only rarely ever needed more than 3rd order. In one such case we were modeling paper paths through a copier, and the path needed to be smoother than a cubic spline could offer.
- Higher orders than a cubic are a serious problem for many of the most useful constraints one may want to apply. Monotonicity for example, is done using a set of necessary constraints based on an inequality from a Fritsch and Carlson paper. Higher orders than cubic however will not allow such a nice solution, so we would have problems ensuring true monotonicity. The curvature constraints would also be more difficult to satisfy.
So in the end, I chose not to implement higher orders than cubic. Sorry.

Hello John,
Thanks for this great tool.
I do have a question, I am not sure if it has been already answered. I would like pre-define the knots to be the points where the derivative of the curve changes sign. Is there an option for that? I am using the piecewise linear SLM fit, I can send you an example of my data if needed.
Thanks,
Judith

Mustafa - I'm glad you like it.
The initial knot placement is as you supply it, or if you supplied nothing, then the default is used. By default, the knots chosen are equally spaced from min(x) to max(x), with 6 knots in total, so there would be 4 interior knots to vary. The optimizer then varies only those interior knots.
There is no overt randomness in the optimization or the initial values though. For example:
x = rand(100);
y = exp(x);
clear functions
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 1.369497 seconds.
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 0.176209 seconds.
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 0.127645 seconds.
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 0.123504 seconds.
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 0.121428 seconds.
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 0.123522 seconds.
tic,slm = slmengine(x,y,'knots',6,'interiorknots','free');toc
Elapsed time is 0.126947 seconds.
The difference in time for the subsequent calls is small, but non-zero. That variability is purely due to your CPU as it is constantly running various other things in the background.
ALWAYS do such time tests multiple times, and ignore the first couple of calls from your sample, as the first time a function is called, MATLAB takes extra time to cache the JIT parsed code. Above, see that I did a clear functions first to clear the function cache. The first call was quite slow, then after the second call, the time needed was far less, and quite consistent.

I'm confused. If you found this page, you should be able to click on the "Download Submission" button. Once you have done that, unzip the file, and add the resulting directory to your search path using either addpath & savepath or pathtools.

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