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241
lib/smooth1q.m
普通文件
241
lib/smooth1q.m
普通文件
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function [z,s] = smooth1q(y,s,varargin)
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%SMOOTH1Q Quick & easy smoothing.
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% Z = SMOOTH1Q(Y,S) smoothes the data Y using a DCT- or FFT-based spline
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% smoothing method. Non finite data (NaN or Inf) are treated as missing
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% values.
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%
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% S is the smoothing parameter. It must be a real positive scalar. The
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% larger S is, the smoother the output will be. If S is empty (i.e. S =
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% []), it is automatically determined by minimizing the generalized
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% cross-validation (GCV) score.
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%
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% Z = SMOOTH1Q(...,'robust') carries out a robust smoothing that
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% minimizes the influence of outlying data.
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%
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% Z = SMOOTH1Q(...,'periodic') assumes that the data to be smoothed must
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% be periodic.
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%
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% [Z,S] = SMOOTH1Q(...) also returns the calculated value for the
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% smoothness parameter S so that you can fine-tune the smoothing
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% subsequently if required.
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%
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% SMOOTH1Q is a simplified and quick version of SMOOTHN for 1-D data. If
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% you want to smooth N-D arrays use <a
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% href="matlab:web('http://www.mathworks.com/matlabcentral/fileexchange/25634')">SMOOTHN</a>.
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%
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% Notes
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% -----
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% 1) SMOOTH1Q works with regularly spaced data only. Use SMOOTH1 for non
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% regularly spaced data.
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% 2) The smoothness parameter used in this algorithm is determined
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% automatically by minimizing the generalized cross-validation score.
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% See the references for more details.
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%
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% References
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% ----------
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% 1) Garcia D, Robust smoothing of gridded data in one and higher
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% dimensions with missing values. Computational Statistics & Data
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% Analysis, 2010.
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% <a
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% href="matlab:web('http://www.biomecardio.com/pageshtm/publi/csda10.pdf')">PDF download</a>
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% 2) Buckley MJ, Fast computation of a discretized thin-plate smoothing
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% spline for image data. Biometrika, 1994.
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% <a
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% href="matlab:web('http://biomet.oxfordjournals.org/content/81/2/247')">Link</a>
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%
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% Examples:
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% --------
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% % Simple curve
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% x = linspace(0,100,200);
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% y = cos(x/10)+(x/50).^2 + randn(size(x))/10;
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% z = smooth1q(y,[]);
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% plot(x,y,'r.',x,z,'k','LineWidth',2)
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% axis tight square
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%
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% % Periodic curve with ouliers and missing data
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% x = linspace(0,2*pi,300);
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% y = cos(x)+ sin(2*x+1).^2 + randn(size(x))/5;
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% y(150:155) = rand(1,6)*5;
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% y(10:40) = NaN;
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% subplot(121)
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% z = smooth1q(y,1e3,'periodic');
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% plot(x,y,'r.',x,z,'k','LineWidth',2)
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% axis tight square
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% title('Non robust')
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% subplot(122)
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% z = smooth1q(y,1e3,'periodic','robust');
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% plot(x,y,'r.',x,z,'k','LineWidth',2)
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% axis tight square
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% title('Robust')
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%
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% % Lima<EFBFBD>on
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% t = linspace(0,2*pi,300);
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% x = cos(t).*(.5+cos(t)) + randn(size(t))*0.05;
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% y = sin(t).*(.5+cos(t)) + randn(size(t))*0.05;
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% z = smooth1q(complex(x,y),[],'periodic');
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% plot(x,y,'r.',real(z),imag(z),'k','linewidth',2)
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% axis equal tight
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%
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% See also SMOOTHN, SMOOTH1.
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%
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% -- Damien Garcia -- 2012/08, revised 2014/02/26
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% website: <a
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% href="matlab:web('http://www.biomecardio.com')">www.BiomeCardio.com</a>
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%-- Check input arguments
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error(nargchk(2,4,nargin));
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assert(isvector(squeeze(y)),...
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['Y must be a 1-D array. Use <a href="matlab:web(''',...
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'http://www.mathworks.com/matlabcentral/fileexchange/25634'')">SMOOTHN</a> for non vector arrays.'])
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if isempty(s)
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isauto = 1;
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else
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assert(isnumeric(s),'S must be a numeric scalar')
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assert(isscalar(s) && s>0,...
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'The smoothing parameter S must be a scalar >0')
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isauto = 0;
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end
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%-- Order (use m>=2, m = 2 is recommended)
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m = 2; % Note: order of the smoothing process, can be modified
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%-- Options ('robust' and/or 'periodic')
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isrobust = 0; method = 'dct'; % default options
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%--
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if nargin>2
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assert(all(cellfun(@ischar,varargin)),...
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'The options must be ''robust'' and/or ''periodic''.')
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varargin = lower(varargin);
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if nargin==3
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idx = ismember({'robust','periodic'},varargin);
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assert(any(idx),...
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'The options must be ''robust'' and/or ''periodic''.')
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if idx(1), isrobust = 1; else method = 'fft'; end
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else % nargin = 4
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assert(all(ismember(varargin,{'robust','periodic'})),...
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'The options must be ''robust'' and/or ''periodic''.')
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isrobust = 1;
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method = 'fft';
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end
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end
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n = length(y);
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siz0 = size(y);
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y = y(:).';
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%-- Weights
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W0 = ones(siz0);
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I = isfinite(y); % missing data (NaN or Inf values)
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if any(~I) % replace the missing data (for faster convergence)
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X = 1:n;
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x = X(I); xi = X(~I);
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y(~I) = interp1(x,y(I),xi,'linear','extrap');
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end
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W0(~I) = 0; % weights for missing data are 0
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W = W0;
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%-- Eigenvalues
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switch method
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case 'dct'
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Lambda = 2-2*cos((0:n-1)*pi/n);
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case 'fft'
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Lambda = 2-2*cos(2*(0:n-1)*pi/n);
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end
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%-- Smoothing process
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nr = 3; % Number of robustness iterations
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for k = 0:nr*isrobust
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if isrobust && k>0
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tmp = sqrt(1+16*s);
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h = sqrt(1+tmp)/sqrt(2)/tmp;
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W = W0.*bisquare(y,z,I,h);
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end
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if ~all(W==1) % then use an iterative method
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tol = Inf;
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zz = y;
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while tol>1e-3
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switch method
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case 'dct'
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Y = dct(W.*(y-zz)+zz);
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case 'fft'
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Y = fft(W.*(y-zz)+zz);
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end
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if isauto
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fminbnd(@GCVscore,-10,30,optimset('TolX',.1));
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else
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Gamma = 1./(1+s*Lambda.^m);
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switch method
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case 'dct'
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z = idct(Gamma.*Y);
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case 'fft'
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if isreal(y)
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z = ifft(Gamma.*Y,'symmetric');
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else
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z = ifft(Gamma.*Y);
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end
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end
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end
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tol = norm(zz-z)/norm(z);
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zz = z;
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end
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else %---
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% No missing values, non robust method => Direct fast method
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%---
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switch method
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case 'dct'
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Y = dct(y);
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case 'fft'
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Y = fft(y);
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end
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if isauto
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fminbnd(@GCVscore,-10,30,optimset('TolX',.1));
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else
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Gamma = 1./(1+s*Lambda.^m);
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end
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switch method
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case 'dct'
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z = idct(Gamma.*Y);
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case 'fft'
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if isreal(y)
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z = ifft(Gamma.*Y,'symmetric');
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else
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z = ifft(Gamma.*Y);
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end
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end
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end
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end
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z = reshape(z,siz0);
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function GCVs = GCVscore(p)
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s = 10^p;
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Gamma = 1./(1+s*Lambda.^m);
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if any(W)
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switch method
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case 'dct'
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z = idct(Gamma.*Y);
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case 'fft'
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if isreal(y)
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z = ifft(Gamma.*Y,'symmetric');
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else
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z = ifft(Gamma.*Y);
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end
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end
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RSS = norm(sqrt(W).*(y-z))^2;
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else % No missing values, non robust method => Direct fast method
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RSS = norm(Y.*(Gamma-1))^2;
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end
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TrH = sum(Gamma);
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GCVs = RSS/(1-TrH/n)^2;
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end
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end
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function W = bisquare(y,z,I,h)
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r = y-z; % residuals
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MAD = median(abs(r(I)-median(r(I)))); % median absolute deviation
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u = abs(r/(1.4826*MAD)/sqrt(1-h)); % studentized residuals
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W = (1-(u/4.685).^2).^2.*((u/4.685)<1); % bisquare weights
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end
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