is there a reference to a journal paper for the selection of the smoothing parameter p = 1(1 + h3/60) in Matlab command csaps

% Fit cubic spline
%h=average distance between CP points;
p = 1/(1 + h^3/0.6);   % Satisfactory smoothing for evenly spaced data
CPvalues=[];
CPvalues=csaps(CPowe,CPOWE,p);
% Is there a journal reference for the choice of smoothing parameter p = 1/(1 + h^3/0.6)

[Rik]

References are generally found at the bottom of the documentation page.

Where did you get this code?

[Branko Celler]

There are no references at the dottom of the documentation page. There are no references anywhere that I can see.

The code is a snippet from my own program where I use the recommended value of the smoothing parameter p as recommended in the documentation. I am trying to find a journal reference for this formula for p.

Regards

Branko Celler

[Rik]

Then I'm afraid I can't help you.

[Walter Roberson]

@John D'Errico -- would you happen to have encountered information about this choice of smoothing parameter when you were doing your spline work?

Answer 1

[Drew]

The algorithms section of the csaps doc page https://www.mathworks.com/help/curvefit/csaps.html begins with this

• csaps is an implementation of the Fortran routine SMOOTH from PGS.

The reference to PGS in the first point is resolved by another doc page https://www.mathworks.com/help/curvefit/curve-fitting-toolbox-splines-and-matlab-splines.html which says

• Curve Fitting Toolbox™ spline functions contain versions of the essential MATLAB® programs of the B-spline package (extended to handle also vector-valued splines) as described in A Practical Guide to Splines, (Applied Math. Sciences Vol. 27, Springer Verlag, New York (1978), xxiv + 392p; revised edition (2001), xviii+346p), hereafter referred to as PGS.

So, PGS refers to the book A Practical Guide to Splines by Carl de Boor. The acknowledgements section of the https://www.mathworks.com/help/curvefit/curve-fitting-toolbox-splines-and-matlab-splines.html page goes on to say:

• MathWorks® would like to acknowledge the contributions of Carl de Boor to the Curve Fitting Toolbox spline functions. Professor de Boor authored the Spline Toolbox™ from its first release until Version 3.3.4 (2008).
• Professor de Boor received the John von Neumann Prize in 1996 and the National Medal of Science in 2003. He is a member of both the American Academy of Arts and Sciences and the National Academy of Sciences. He is the author of A Practical Guide to Splines (Springer, 2001).

Regarding your choice of p taken from the suggestions in the documentation, the "p - Smoothing Parameter" section of the csaps doc page https://www.mathworks.com/help/curvefit/csaps.html thoroughly describes the tradeoffs involved in choosing p. The explanation in the doc includes this, which is where you obtained the suggestion:

• "The favorable range for p is often near 1/(1 + h^3/6), where h is the average spacing of the data sites. The function chooses a default value for p within this range. For uniformly spaced data, you can expect a close fit with p = 1(1 + h^3/60) and some satisfactory smoothing with p = 1/(1 + h^3/0.6). "

At the high level, the reasoning behind the formula lies in the desire to achieve a smooth curve while maintaining a balance between fitting the data closely and avoiding overfitting. By incorporating the average spacing, h, into the formula, this equation takes into account the density of data points.

The m-file help for csaps has more info which includes this section

help csaps

....

% When P is 0, the smoothing spline is the least-squares straight line fit
%     to the data, while, at the other extreme, i.e., when P is 1, it is the
%     `natural' or variational cubic spline interpolant. The transition region
%     between these two extremes is usually only a rather small range of values
%     for P and its location strongly depends on the data sites. It is in this
%     small range that P is chosen when it is not supplied, or when an empty
%     P or a negative P is input.

The algorithms section of the csaps doc page https://www.mathworks.com/help/curvefit/csaps.html ends with this statement:

• The calculation of the smoothing spline requires solving a linear system whose coefficient matrix has the form p*A + (1-p)*B, with the matrices A and B depending on the data sites x. The default value of p makes p*trace(A) equal (1-p)*trace(B).

The above sheds some light on the choice of the default p.

In conclusion, more info can be obtained from:

• csaps doc page https://www.mathworks.com/help/curvefit/csaps.html
• The book A Practical Guide to Splines (Springer 2001) by Carl de Boor. The SMOOTH function, which is the basis for csaps, appears in chapter 14.
• The m-file help for the csaps function, using "help csaps"
• Reading the csaps code, using "open csaps"

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