Mercurial > hg > octave-nkf
annotate scripts/polynomial/splinefit.m @ 15538:94d21131fefd
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| author | John W. Eaton <jwe@octave.org> |
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| date | Wed, 17 Oct 2012 11:51:35 -0400 |
| parents | 902a8e27b2a2 |
| children | 36dba9be680b |
| rev | line source |
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| 14509 | 1 ## Copyright (C) 2012 Ben Abbott, Jonas Lundgren |
| 2 ## | |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7 ## the Free Software Foundation; either version 3 of the License, or (at | |
| 8 ## your option) any later version. | |
| 9 ## | |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 16 ## along with Octave; see the file COPYING. If not, see | |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 18 | |
| 19 ## -*- texinfo -*- | |
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20 ## @deftypefn {Function File} {@var{pp} =} splinefit (@var{x}, @var{y}, @var{breaks}) |
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21 ## Fit a piecewise cubic spline with breaks (knots) @var{breaks} to the |
| 14509 | 22 ## noisy data, @var{x} and @var{y}. @var{x} is a vector, and @var{y} |
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23 ## a vector or N-D array. If @var{y} is an N-D array, then @var{x}(j) |
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24 ## is matched to @var{y}(:,@dots{},:,j). |
| 14509 | 25 ## |
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26 ## The fitted spline is returned as a piecewise polynomial, @var{pp}, and |
| 14509 | 27 ## may be evaluated using @code{ppval}. |
| 28 ## | |
| 29 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@var{x}, @var{y}, @var{p}) | |
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30 ## @var{p} is a positive integer defining the number of intervals along @var{x}, |
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31 ## and @var{p}+1 is the number of breaks. The number of points in each interval |
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32 ## differ by no more than 1. |
| 14509 | 33 ## |
| 34 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "periodic", @var{periodic}) | |
| 35 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "robust", @var{robust}) | |
| 36 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "beta", @var{beta}) | |
| 37 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "order", @var{order}) | |
| 38 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "constraints", @var{constraints}) | |
| 39 ## | |
| 40 ## The optional property @var{periodic} is a logical value which specifies | |
| 41 ## whether a periodic boundary condition is applied to the spline. The | |
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42 ## length of the period is @code{max (@var{breaks}) - min (@var{breaks})}. |
| 14509 | 43 ## The default value is @code{false}. |
| 44 ## | |
| 45 ## The optional property @var{robust} is a logical value which specifies | |
| 46 ## if robust fitting is to be applied to reduce the influence of outlying | |
| 47 ## data points. Three iterations of weighted least squares are performed. | |
| 48 ## Weights are computed from previous residuals. The sensitivity of outlier | |
| 49 ## identification is controlled by the property @var{beta}. The value of | |
| 50 ## @var{beta} is stricted to the range, 0 < @var{beta} < 1. The default | |
| 51 ## value is @var{beta} = 1/2. Values close to 0 give all data equal | |
| 52 ## weighting. Increasing values of @var{beta} reduce the influence of | |
| 53 ## outlying data. Values close to unity may cause instability or rank | |
| 54 ## deficiency. | |
| 55 ## | |
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56 ## The splines are constructed of polynomials with degree @var{order}. |
| 14509 | 57 ## The default is a cubic, @var{order}=3. A spline with P pieces has |
| 58 ## P+@var{order} degrees of freedom. With periodic boundary conditions | |
| 59 ## the degrees of freedom are reduced to P. | |
| 60 ## | |
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61 ## The optional property, @var{constaints}, is a structure specifying |
| 14509 | 62 ## linear constraints on the fit. The structure has three fields, "xc", |
| 63 ## "yc", and "cc". | |
| 64 ## | |
| 65 ## @table @asis | |
| 66 ## @item "xc" | |
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67 ## Vector of the x-locations of the constraints. |
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68 ## |
| 14509 | 69 ## @item "yc" |
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70 ## Constraining values at the locations, @var{xc}. |
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71 ## The default is an array of zeros. |
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72 ## |
| 14509 | 73 ## @item "cc" |
| 74 ## Coefficients (matrix). The default is an array of ones. The number of | |
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75 ## rows is limited to the order of the piecewise polynomials, @var{order}. |
| 14509 | 76 ## @end table |
| 77 ## | |
| 78 ## Constraints are linear combinations of derivatives of order 0 to | |
| 79 ## @var{order}-1 according to | |
| 80 ## | |
| 81 ## @example | |
| 82 ## @group | |
| 83 ## @tex | |
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84 ## $cc(1,j) \cdot y(xc(j)) + cc(2,j) \cdot y\prime(xc(j)) + ... = yc(:,\dots,:,j)$. |
| 14509 | 85 ## @end tex |
| 86 ## @ifnottex | |
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87 ## cc(1,j) * y(xc(j)) + cc(2,j) * y'(xc(j)) + ... = yc(:,...,:,j). |
| 14509 | 88 ## @end ifnottex |
| 89 ## @end group | |
| 90 ## @end example | |
| 91 ## | |
| 92 ## @seealso{interp1, unmkpp, ppval, spline, pchip, ppder, ppint, ppjumps} | |
| 93 ## @end deftypefn | |
| 94 | |
| 95 %!demo | |
| 96 %! % Noisy data | |
| 97 %! x = linspace (0, 2*pi, 100); | |
| 98 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 99 %! % Breaks | |
| 100 %! breaks = [0:5, 2*pi]; | |
| 101 %! % Fit a spline of order 5 | |
| 102 %! pp = splinefit (x, y, breaks, "order", 4); | |
| 103 %! clf () | |
| 104 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 105 %! xlabel ("Independent Variable") | |
| 106 %! ylabel ("Dependent Variable") | |
| 107 %! title ("Fit a piece-wise polynomial of order 4"); | |
| 108 %! legend ({"data", "fit", "breaks"}) | |
| 109 %! axis tight | |
| 110 %! ylim auto | |
| 111 | |
| 112 %!demo | |
| 113 %! % Noisy data | |
| 114 %! x = linspace (0,2*pi, 100); | |
| 115 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 116 %! % Breaks | |
| 117 %! breaks = [0:5, 2*pi]; | |
| 118 %! % Fit a spline of order 3 with periodic boundary conditions | |
| 119 %! pp = splinefit (x, y, breaks, "order", 2, "periodic", true); | |
| 120 %! clf () | |
| 121 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 122 %! xlabel ("Independent Variable") | |
| 123 %! ylabel ("Dependent Variable") | |
| 124 %! title ("Fit a periodic piece-wise polynomial of order 2"); | |
| 125 %! legend ({"data", "fit", "breaks"}) | |
| 126 %! axis tight | |
| 127 %! ylim auto | |
| 128 | |
| 129 %!demo | |
| 130 %! % Noisy data | |
| 131 %! x = linspace (0, 2*pi, 100); | |
| 132 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 133 %! % Breaks | |
| 134 %! breaks = [0:5, 2*pi]; | |
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135 %! % Constraints: y(0) = 0, y'(0) = 1 and y(3) + y"(3) = 0 |
| 14509 | 136 %! xc = [0 0 3]; |
| 137 %! yc = [0 1 0]; | |
| 138 %! cc = [1 0 1; 0 1 0; 0 0 1]; | |
| 139 %! con = struct ("xc", xc, "yc", yc, "cc", cc); | |
| 140 %! % Fit a cubic spline with 8 pieces and constraints | |
| 141 %! pp = splinefit (x, y, 8, "constraints", con); | |
| 142 %! clf () | |
| 143 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 144 %! xlabel ("Independent Variable") | |
| 145 %! ylabel ("Dependent Variable") | |
| 146 %! title ("Fit a cubic spline with constraints") | |
| 147 %! legend ({"data", "fit", "breaks"}) | |
| 148 %! axis tight | |
| 149 %! ylim auto | |
| 150 | |
| 151 %!demo | |
| 152 %! % Noisy data | |
| 153 %! x = linspace (0, 2*pi, 100); | |
| 154 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 155 %! % Breaks | |
| 156 %! breaks = [0:5, 2*pi]; | |
| 157 %! xc = [0 0 3]; | |
| 158 %! yc = [0 1 0]; | |
| 159 %! cc = [1 0 1; 0 1 0; 0 0 1]; | |
| 160 %! con = struct ("xc", xc, "yc", yc, "cc", cc); | |
| 161 %! % Fit a spline of order 6 with constraints and periodicity | |
| 162 %! pp = splinefit (x, y, breaks, "constraints", con, "order", 5, "periodic", true); | |
| 163 %! clf () | |
| 164 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 165 %! xlabel ("Independent Variable") | |
| 166 %! ylabel ("Dependent Variable") | |
| 167 %! title ("Fit a 5th order piece-wise periodic polynomial with constraints") | |
| 168 %! legend ({"data", "fit", "breaks"}) | |
| 169 %! axis tight | |
| 170 %! ylim auto | |
| 171 | |
| 172 function pp = splinefit (x, y, breaks, varargin) | |
| 173 if (nargin > 3) | |
| 174 n = cellfun (@ischar, varargin, "uniformoutput", true); | |
| 175 varargin(n) = lower (varargin(n)); | |
| 176 try | |
| 177 props = struct (varargin{:}); | |
| 178 catch | |
| 179 print_usage (); | |
| 180 end_try_catch | |
| 181 else | |
| 182 props = struct (); | |
| 183 endif | |
| 184 fields = fieldnames (props); | |
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185 for f = 1:numel (fields) |
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186 if (! any (strcmp (fields{f}, |
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187 {"periodic", "robust", "beta", "order", "constraints"}))) |
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188 error ("splinefit:invalidproperty", |
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189 "unrecognized property '%s'", fields{f}); |
| 14509 | 190 endif |
| 191 endfor | |
| 192 args = {}; | |
| 193 if (isfield (props, "periodic") && props.periodic) | |
| 194 args{end+1} = "p"; | |
| 195 endif | |
| 196 if (isfield (props, "robust") && props.robust) | |
| 197 args{end+1} = "r"; | |
| 198 endif | |
| 199 if (isfield (props, "beta")) | |
| 200 if (0 < props.beta && props.beta < 1) | |
| 201 args{end+1} = props.beta; | |
| 202 else | |
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203 error ("splinefit:invalidbeta", "invalid beta parameter (0 < beta < 1)"); |
| 14509 | 204 endif |
| 205 endif | |
| 206 if (isfield (props, "order")) | |
| 207 if (props.order >= 0) | |
| 208 args{end+1} = props.order + 1; | |
| 209 else | |
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210 error ("splinefit:invalidorder", "invalid order"); |
| 14509 | 211 endif |
| 212 endif | |
| 213 if (isfield (props, "constraints")) | |
| 214 args{end+1} = props.constraints; | |
| 215 endif | |
| 216 if (nargin < 3) | |
| 217 print_usage (); | |
| 218 elseif (! isnumeric (breaks) || ! isvector (breaks)) | |
| 219 print_usage (); | |
| 220 endif | |
| 221 pp = __splinefit__ (x, y, breaks, args{:}); | |
| 222 endfunction | |
| 223 | |
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224 |
| 14509 | 225 %!shared xb, yb, x |
| 226 %! xb = 0:2:10; | |
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227 %! yb = 2*rand (size (xb)) - 1; |
| 14509 | 228 %! x = 0:0.1:10; |
| 229 | |
| 230 %!test | |
| 231 %! y = interp1 (xb, yb, x, "linear"); | |
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232 %! assert (ppval (splinefit (x, y, xb, "order", 1), x), y, 15 * eps ()); |
| 14509 | 233 %!test |
| 234 %! y = interp1 (xb, yb, x, "spline"); | |
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235 %! assert (ppval (splinefit (x, y, xb, "order", 3), x), y, 15 * eps ()); |
| 14509 | 236 %!test |
| 237 %! y = interp1 (xb, yb, x, "spline"); | |
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238 %! assert (ppval (splinefit (x, y, xb), x), y, 15 * eps ()); |
| 14509 | 239 |