Mercurial > hg > octave-nkf
annotate scripts/polynomial/splinefit.m @ 19775:c8cb111e727e
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Wed, 14 Jan 2015 13:54:28 -0500 |
| parents | d63878346099 |
| children | 4197fc428c7d |
| rev | line source |
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1 ## Copyright (C) 2012-2013 Ben Abbott, Jonas Lundgren |
| 14509 | 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 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@var{x}, @var{y}, @var{p}) |
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22 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "periodic", @var{periodic}) |
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23 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "robust", @var{robust}) |
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24 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "beta", @var{beta}) |
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25 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "order", @var{order}) |
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26 ## @deftypefnx {Function File} {@var{pp} =} splinefit (@dots{}, "constraints", @var{constraints}) |
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27 ## |
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28 ## Fit a piecewise cubic spline with breaks (knots) @var{breaks} to the |
| 14509 | 29 ## noisy data, @var{x} and @var{y}. @var{x} is a vector, and @var{y} |
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30 ## is a vector or N-D array. If @var{y} is an N-D array, then @var{x}(j) |
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31 ## is matched to @var{y}(:,@dots{},:,j). |
| 14509 | 32 ## |
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33 ## The fitted spline is returned as a piecewise polynomial, @var{pp}, and |
| 14509 | 34 ## may be evaluated using @code{ppval}. |
| 35 ## | |
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36 ## @var{p} is a positive integer defining the number of intervals along @var{x}, |
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37 ## and @var{p}+1 is the number of breaks. The number of points in each interval |
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38 ## differ by no more than 1. |
| 14509 | 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 |
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62 ## linear constraints on the fit. The structure has three fields, @qcode{"xc"}, |
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63 ## @qcode{"yc"}, and @qcode{"cc"}. |
| 14509 | 64 ## |
| 65 ## @table @asis | |
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66 ## @item @qcode{"xc"} |
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67 ## Vector of the x-locations of the constraints. |
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68 ## |
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69 ## @item @qcode{"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 ## |
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73 ## @item @qcode{"cc"} |
| 14509 | 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 | |
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95 function pp = splinefit (x, y, breaks, varargin) |
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96 if (nargin > 3) |
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97 n = cellfun ("isclass", varargin, "char"); |
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98 varargin(n) = lower (varargin(n)); |
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99 try |
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100 props = struct (varargin{:}); |
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101 catch |
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102 print_usage (); |
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103 end_try_catch |
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104 else |
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105 props = struct (); |
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106 endif |
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107 fields = fieldnames (props); |
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108 for f = 1:numel (fields) |
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109 if (! any (strcmp (fields{f}, |
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110 {"periodic", "robust", "beta", "order", "constraints"}))) |
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111 error ("splinefit:invalidproperty", |
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112 "unrecognized property '%s'", fields{f}); |
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113 endif |
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114 endfor |
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115 args = {}; |
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116 if (isfield (props, "periodic") && props.periodic) |
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117 args{end+1} = "p"; |
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118 endif |
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119 if (isfield (props, "robust") && props.robust) |
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120 args{end+1} = "r"; |
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121 endif |
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122 if (isfield (props, "beta")) |
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123 if (0 < props.beta && props.beta < 1) |
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124 args{end+1} = props.beta; |
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125 else |
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126 error ("splinefit:invalidbeta", "invalid beta parameter (0 < beta < 1)"); |
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127 endif |
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128 endif |
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129 if (isfield (props, "order")) |
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130 if (props.order >= 0) |
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131 args{end+1} = props.order + 1; |
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132 else |
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133 error ("splinefit:invalidorder", "invalid order"); |
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134 endif |
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135 endif |
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136 if (isfield (props, "constraints")) |
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137 args{end+1} = props.constraints; |
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138 endif |
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139 if (nargin < 3) |
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140 print_usage (); |
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141 elseif (! isnumeric (breaks) || ! isvector (breaks)) |
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142 print_usage (); |
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143 endif |
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144 pp = __splinefit__ (x, y, breaks, args{:}); |
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145 endfunction |
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146 |
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147 |
| 14509 | 148 %!demo |
| 149 %! % Noisy data | |
| 150 %! x = linspace (0, 2*pi, 100); | |
| 151 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 152 %! % Breaks | |
| 153 %! breaks = [0:5, 2*pi]; | |
| 154 %! % Fit a spline of order 5 | |
| 155 %! pp = splinefit (x, y, breaks, "order", 4); | |
| 156 %! clf () | |
| 157 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 158 %! xlabel ("Independent Variable") | |
| 159 %! ylabel ("Dependent Variable") | |
| 160 %! title ("Fit a piece-wise polynomial of order 4"); | |
| 161 %! legend ({"data", "fit", "breaks"}) | |
| 162 %! axis tight | |
| 163 %! ylim auto | |
| 164 | |
| 165 %!demo | |
| 166 %! % Noisy data | |
| 167 %! x = linspace (0,2*pi, 100); | |
| 168 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 169 %! % Breaks | |
| 170 %! breaks = [0:5, 2*pi]; | |
| 171 %! % Fit a spline of order 3 with periodic boundary conditions | |
| 172 %! pp = splinefit (x, y, breaks, "order", 2, "periodic", true); | |
| 173 %! clf () | |
| 174 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 175 %! xlabel ("Independent Variable") | |
| 176 %! ylabel ("Dependent Variable") | |
| 177 %! title ("Fit a periodic piece-wise polynomial of order 2"); | |
| 178 %! legend ({"data", "fit", "breaks"}) | |
| 179 %! axis tight | |
| 180 %! ylim auto | |
| 181 | |
| 182 %!demo | |
| 183 %! % Noisy data | |
| 184 %! x = linspace (0, 2*pi, 100); | |
| 185 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 186 %! % Breaks | |
| 187 %! breaks = [0:5, 2*pi]; | |
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188 %! % Constraints: y(0) = 0, y'(0) = 1 and y(3) + y"(3) = 0 |
| 14509 | 189 %! xc = [0 0 3]; |
| 190 %! yc = [0 1 0]; | |
| 191 %! cc = [1 0 1; 0 1 0; 0 0 1]; | |
| 192 %! con = struct ("xc", xc, "yc", yc, "cc", cc); | |
| 193 %! % Fit a cubic spline with 8 pieces and constraints | |
| 194 %! pp = splinefit (x, y, 8, "constraints", con); | |
| 195 %! clf () | |
| 196 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 197 %! xlabel ("Independent Variable") | |
| 198 %! ylabel ("Dependent Variable") | |
| 199 %! title ("Fit a cubic spline with constraints") | |
| 200 %! legend ({"data", "fit", "breaks"}) | |
| 201 %! axis tight | |
| 202 %! ylim auto | |
| 203 | |
| 204 %!demo | |
| 205 %! % Noisy data | |
| 206 %! x = linspace (0, 2*pi, 100); | |
| 207 %! y = sin (x) + 0.1 * randn (size (x)); | |
| 208 %! % Breaks | |
| 209 %! breaks = [0:5, 2*pi]; | |
| 210 %! xc = [0 0 3]; | |
| 211 %! yc = [0 1 0]; | |
| 212 %! cc = [1 0 1; 0 1 0; 0 0 1]; | |
| 213 %! con = struct ("xc", xc, "yc", yc, "cc", cc); | |
| 214 %! % Fit a spline of order 6 with constraints and periodicity | |
| 215 %! pp = splinefit (x, y, breaks, "constraints", con, "order", 5, "periodic", true); | |
| 216 %! clf () | |
| 217 %! plot (x, y, "s", x, ppval (pp, x), "r", breaks, ppval (pp, breaks), "+r") | |
| 218 %! xlabel ("Independent Variable") | |
| 219 %! ylabel ("Dependent Variable") | |
| 220 %! title ("Fit a 5th order piece-wise periodic polynomial with constraints") | |
| 221 %! legend ({"data", "fit", "breaks"}) | |
| 222 %! axis tight | |
| 223 %! ylim auto | |
| 224 | |
| 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 |