;; Basis splines.
;; Liam Healy 2008-02-18 14:43:20EST basis-splines.lisp
;; Time-stamp: <2016-06-15 23:07:19EDT basis-splines.lisp>
;;
;; Copyright 2008, 2009, 2011, 2012, 2016 Liam M. Healy
;; Distributed under the terms of the GNU General Public License
;;
;; This program is free software: you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation, either version 3 of the License, or
;; (at your option) any later version.
;;
;; This program is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;; GNU General Public License for more details.
;;
;; You should have received a copy of the GNU General Public License
;; along with this program. If not, see .
(in-package :gsl)
;;; /usr/include/gsl/gsl_bspline.h
;;; Should be subclass of interpolation?
(defmobject basis-spline "gsl_bspline"
((order :sizet) (number-of-breakpoints :sizet))
"basis spline"
:gsl-version (1 9)
:documentation ; FDL
"Allocate a workspace for computing B-splines. The number of
breakpoints is given by number-of-breakpoints. This leads to n =
nbreak + k - 2 basis functions where k = order. Cubic B-splines are
specified by k = 4. The size of the workspace is O(5k + nbreak).")
;;; Constructing the knots vector
(defmfun knots (breakpoints workspace)
"gsl_bspline_knots"
(((mpointer breakpoints) :pointer) ((mpointer workspace) :pointer))
:gsl-version (1 9)
:documentation ; FDL
"Compute the knots associated with the given breakpoints and store
them in the workspace.")
(defmfun uniform-knots (a b workspace)
"gsl_bspline_knots_uniform"
((a :double) (b :double) ((mpointer workspace) :pointer))
:return (workspace)
:gsl-version (1 9)
:documentation ; FDL
"Compute knots uniformly on the interval [a, b] and store
them in the workspace.")
;;; Evaluation of B-splines
(defmfun evaluate
((workspace basis-spline) x
&key (B (grid:make-foreign-array 'double-float
:dimensions (number-of-coefficients workspace))))
"gsl_bspline_eval"
((x :double) ((mpointer B) :pointer) ((mpointer workspace) :pointer))
:definition :method
:outputs (B)
:gsl-version (1 9)
:documentation ; FDL
"Evaluate all B-spline basis functions at the position x and store
them in the GSL vector B, so that the ith element of B is B_i(x).
B must be of length n = nbreak + k - 2. This value is
also stored in the workspace. It is far more
efficient to compute all of the basis functions at once than to
compute them individually, due to the nature of the defining
recurrence relation.")
;;; Query settings of the B-spline
;;; These are not documented but are in
;;; /usr/include/gsl/gsl_bspline.h; are they officially supported?
(defmfun number-of-coefficients (bspline)
"gsl_bspline_ncoeffs"
(((mpointer bspline) :pointer))
:c-return :sizet
:gsl-version (1 9))
(defmfun order ((bspline basis-spline))
"gsl_bspline_order"
(((mpointer bspline) :pointer))
:definition :method
:c-return :sizet
:gsl-version (1 9))
(defmfun number-of-breakpoints (bspline)
"gsl_bspline_nbreak"
(((mpointer bspline) :pointer))
:c-return :sizet
:gsl-version (1 9)
:documentation "The number of breakpoints of the basis spline bspline.")
(defmfun breakpoint (i bspline)
"gsl_bspline_breakpoint"
((i :sizet) ((mpointer bspline) :pointer))
:c-return :double
:gsl-version (1 9)
:documentation "The ith breakpoint of the basis spline bspline.")
(defmfun greville-abscissa (i bspline)
"gsl_bspline_greville_abscissa"
((i :sizet) ((mpointer bspline) :pointer))
:c-return :double
:gsl-version (1 13)
:documentation "Returns the location of the @math{i}-th Greville abscissa for the given B-spline basis. For the ill-defined case when @math{k=1}, the implementation chooses to return breakpoint interval midpoints.")
;;; Examples and unit test
(defun bspline-example (&optional (ncoeffs 8))
(let* ((order 4) ; cubic
(ndata 200)
(nbreak (+ ncoeffs 2 (- order)))
(bw (make-basis-spline order nbreak))
(mw (make-fit-workspace ndata ncoeffs))
(rng (make-random-number-generator +mt19937+ 0))
(B (grid:make-foreign-array 'double-float :dimensions ncoeffs))
(c (grid:make-foreign-array 'double-float :dimensions ncoeffs))
(cov (grid:make-foreign-array 'double-float :dimensions (list ncoeffs ncoeffs)))
(w (grid:make-foreign-array 'double-float :dimensions ndata))
(x (grid:make-foreign-array 'double-float :dimensions ndata))
(y (grid:make-foreign-array 'double-float :dimensions ndata))
(Xmatrix (grid:make-foreign-array 'double-float :dimensions (list ndata ncoeffs)))
(sigma 0.1d0))
;; The data to be fitted.
(dotimes (i ndata)
(let* ((xi (coerce (* i (/ 15 (1- ndata))) 'double-float))
(yi (+ (* (cos xi) (exp (* -0.1d0 xi)))
(sample rng :gaussian :sigma sigma))))
(setf (grid:aref x i) xi
(grid:aref y i) yi
(grid:aref w i) (/ (expt sigma 2)))))
;; Uniform breakpoints [0, 15]
(uniform-knots 0.0d0 15.0d0 bw)
;; Fit matrix
(dotimes (i ndata)
;; Compute B_j
(evaluate bw (grid:aref x i) :b B)
;; Fill in row i of X
(dotimes (j ncoeffs)
(setf (grid:aref Xmatrix i j) (grid:aref B j))))
;; Do the fit
(linear-mfit Xmatrix y c w nil cov mw)
;; Return the smoothed curve
(loop for xi from 0.0d0 to 15.0d0 by 0.1d0
with yval and yerr
do
(evaluate bw xi :b B)
(multiple-value-setq (yval yerr)
(multi-linear-estimate B c cov))
collect yval into yvals
collect yerr into yerrs
finally (return (values yvals yerrs)))))
(save-test basis-spline (bspline-example))