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fitting - Implementation of smoothing splines function



I have some problems in writing a module for spline smoothing. Actually, I have been trying for about two weeks. My listing is here:


SplSmooth[data_, knots_, lambda_, degree_] := 
  Module[{M, Knots, NKnots, NBasis, X, Dsq, a},
   M = Length@data;
   Knots = Flatten@{Table[1, {i, 1, degree}], knots,Table[M, {i,1,degree}]};
   NKnots = Length@Knots;
   NBasis = NKnots - degree - 1;
   X = Table[
     Evaluate @ BSplineBasis[{degree, Knots}, n, t] // N, {t, 1, M}, 
       {n, 0, NBasis - 1}];

   Dsq = Differences[X, 2];
   a=Inverse[Transpose[X].X + lambda*Transpose[Dsq].Dsq // N].Transpose[X].data // N;
   Return[X.a]
   ];

When I try to place a knot in every point in my data, numerical errors arise, such as:



Inverse::luc: Result for Inverse of badly conditioned matrix {{1.251,-0.1255,-0.251,0.0836667,0.0418333,0.,0.,0.,0.,0.,<<72>>},<<9>>,<<72>>} may contain significant numerical errors. >>



Obviously, the corresponding result is wrong (I can see it from the plot). It seems that the matrix to be inverted is ill-conditioned:



a = Inverse[Transpose[X].X + lambda*Transpose[Dsq].Dsq // N].Transpose[X].data // N;

but now comes the other problem. I use equidistant knots (let's say with 7 points distance) to overcome this problem. But then sometimes the algorithm works with:


Knots = Flatten @ {Table[1, {i, 1, degree}], knots, Table[M, {i, 1, degree}]};

and some other times works with


Knots = Flatten @ {Table[1, {i, 0, degree}], knots, Table[M, {i ,0, degree}]};

Now, I think that there is some kind of problem in BSplineBasis function.


Q: Can you spot the problem please? Or has anyone of you implemented a simillar function in the past with BSplineBasis function?





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