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Free-knot spline approximation

version (100 KB) by Bruno Luong
Least squares approximation of 1D data using free-knots spline


Updated 30 Nov 2016

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The purpose of this function is to provide a flexible and robust fit to one-dimensional data using free-knot splines. The knots are free and able to cope with rapid change in the underlying model. Knot removal strategy is used to fit with only a small number of knots.
Optional L2-regularization on the derivative of the spline function can be used to enforce the smoothness.
Shape preserving approximation can be enforced by specifying the lower and upper bounds of the derivative(s) of the spline function on sub-intervals. Furthermore specific values of the spline function and its derivative can be specified on a set of discrete data points.
I did not test QUADPROG engine, but I have implemented it. Any feedback is welcome.

Cite As

Bruno Luong (2020). Free-knot spline approximation (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (26)

Bruno Luong

Michael, have you look at the code testBSFK with caseid == 2, there is a constraint of zeros slope at x==0 (line 91).
Just replace x by respectively the left and right points, and you might even add the same constraints for the second derivative

Bruno, could you kindly add any simple example how to apply specific constraints: zero 1st derivative before and after data points? I need to hold 1st derivative equal zero outside of data interval.

Long Cheng


Dear Bruno,

I want to force the spline to be straight line between the first fixed knot and the second one.
Is this possible?

Gaute Hope

How do you prefer this work to be cited?

Gaute Hope


Thank you, any possible modification to make it work for k=1? if not, do you know any other method for constant piece wise approximation?

Bruno Luong

To Fasta:

Yes for k=1; the gradient wrt knots are dirac like, and the gradient method used by here cannot handle it correctly.

Yes, it's a major work to extend to 2D.


I think there is an error when you use piecewise constant (k=1). The results are not the expected. Is it ok?

Also I would like to know if it would be very difficult to modify it and use it for surfaces instead of lines (1D -> 2D).




I´m always getting the following warning message:

Warning: Options LargeScale = 'off' and Algorithm = 'trust-region-reflective'
conflict. Ignoring Algorithm and running active-set algorithm. To run
trust-region-reflective, set LargeScale = 'on'. To run active-set without this
warning, set Algorithm = 'active-set'.

What am I doing wrong?

Bruno Luong


not exactly like you want but you can enforce the y-value between two knots to be zero:

y = sin(x);
y = y + 0.1*randn(size(y));

nknots = 5;
lo = -inf(1,nknots);
up = +inf(1,nknots);
lo(3) = 0;
up(3) = 0;
shape = struct('p',0,'lo',lo,'up',up);
options = struct('shape', shape,'animation', 1, 'knotremoval','none');


Brilliant package Bruno. Quick question.

I know you can fix any given knot, but is it possible to fix a given not to a y-value but let the least squares find the best x-value for it?

I'm using 4 knots/3 lines to represent my data but I would always like the 3rd knot to have y=0. While this condition is met some of the time (by chance) ideally I would like to enforce it.

Bruno Luong

Nima, it is not rotational invariant. Because the fit is carried out using the least-squares to the ordinate data (y) only.

Nima Zarif

Hey Bruno thanks for the awesome package, it works great!
I just have one problem, is this approximation rotation invariant? Since when I intentionally rotate my data points the knots are completely different with increased fit error compared to the baseline profile! please let me know if I am doing sth wrong

Hi Bruno,

Nice package, many great features!

I am having a problem with the 'startingknots' parameter.

When I type this:


I get this:

BSFK starts
??? Error using ==> sparse
Sparse matrix sizes must be non-negative integers less than MAXSIZE as defined by
COMPUTER. Use HELP COMPUTER for more details.

Error in ==> BSFK>BuildDineqMat at 1623
D = sparse(row,col,val,m,n);

Error in ==> BSFK>UpdateConstraints at 2039
[D LU X] = BuildDineqMat(t, knotidx, k, shape);

Error in ==> BSFK>InitPenalization at 2077
smoothing = UpdateConstraints(smoothing, t, shape, pntcon, periodic);

Error in ==> BSFK at 393
smoothing = InitPenalization(y, t, k, d, lambda, p, regmethod, ...

However, if I use 'chebyschev' instead of a vector of starting knots, it works. But it misses some important knots.

Any ideas?



Bruno Luong

Michael, I'll reply in the appropriate place (minmaxfilt)

Michael Teo

Does it handle NaN data?

ePeriod = 3;
aData = [ 5;1;3;NaN;8;2;3;NaN;1;9 ];
minmaxfilt(eData, ePeriod, 'max', 'valid')];

Actual output:

If we take NaN as a empty data, the expected output is:

Bruno Luong

Just discover an issue with continuous regularization. In the mean time, please use the discrete regularization

Bruno Luong

Periodic spline is now available

x yuij


Add regularization to make quadprog more robust. This is important change for those who use QUADPROG as engine.

quadprog is functioning (blind codding before)

Fix a bug when checking for knot collision

Fix the bug when starting knots are provided

Fix the bug for continuous regularization

Fix a small bug (eigs with 'sa' option requires true symmetric matrix, which is now always the case by symmetrizing)

A more robust conversion in pp form is implemented

Remove some redundant code, modify test program

New feature: Periodic spline

fix a bug with parsing k and nknots
Spline order can be as low as k=1 (piecewise constant fit)

fixed small bug when calling QP engine minqdef

Correct a bug in UpdateConstraints that did not update the knot positions. Precasting data to double. Update more frequently the scaling matrix. Reduce the Lagrange's tolerance to detect active set of QPC solver

Singular constraints will issue a warning (instead of an error). Refine the Gauss-Newton direction. Fix few minor bugs.

Change the description.

Correct another bug in the Jacobian calculation (constrained case)

Point-wise constraints. Discover an error of the Jacobian formula in [Schutze/Schwetlick 97] paper, modify the calculation accordingly. This concern only the constrained fitting.

A major enhancement with shape preserving splines

Change title and description

Remove NaN data before fitting, change TRY/CATCH ME syntax for better compatibility (tested under 2006B), estimate automatic of the noise standard deviation

Update description, more options added to control the fit, discrete regularization

MATLAB Release Compatibility
Created with R2009b
Compatible with any release
Platform Compatibility
Windows macOS Linux