orthogonal natural cubic spline Files

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 Copyright (c) 2016, Tzu-Yu Liu, Yun S. Song


 Orthogonal Natural Cubic Spline:
 Matrix factorization is applied to the B-spline basis to 
 construct an orthonormal basis. Data can be approximated by a 
 linear combination of the orthonormal basis functions. Constraints 
 are imposed such that the fitted function is linear at the boundary 
 points, as in natural cubic spline.

 1) To apply orthogonal natural cubic spline to your data, prepare your
   data in the following format.
   INPUT
	x: n by 1 time points, where n is the number of time points
	y: n by 1 responses
       knots: num_knots by 1, position of knots
       num_knots: number of knots
       degree: polynomial degree
       smspline: impose penalty on the squared second derivative
  
   Apply the function OsplineFit_LinearBoundary to the data. 
   OUTPUT
   Ospoinemodel: a data structure that contains the the fitted model

 2) The Orthogonal Natural Cubic Spline utilizes generalized 
   cross-validation to decide the regularization parameter, proposed by 
   David Ruppert.
   Ruppert, David. "Selecting the number of knots for penalized splines." 
   Journal of computational and graphical statistics (2012).
   https://people.orie.cornell.edu/davidr/matlab/

 3) An example can be found in the file test.m

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