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wlBlock.m
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wlBlock.m
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%**************************************************************************
% Hybrid denoising based on the HOS analysis, block thresholding and Wienar
% Filtering.
%**************************************************************************
%
% [USAGE]
% [dn org opt] = wlBlock(data,opt);
%
% [INPUTS]
% data: a structure including the information of input data
% opt: a structure including parameters needed for the CWT and
% denoising
%
% [OUTPUTS]
% dn: a structure including denoised signal and thresholded
% coefficients and block threshold_blocksize map.
% org: a structure including wavelet coefficients and scale info
% before thresholding.
% opt: parameter structure updated by a rough estimation of the
% arrival time.
%
% [References]
% [1] Mousavi S. M., and C. A. Langston (2016). Hybrid Seismic Denoising
% Using Wavelet Block Thresholding and Higher Order Statistics,
% Bulletin of Seismological Society of America, 106 (4), 1380-1393,
% DOI:10.1785/0120150345.
%
% [2] Mousavi S. M., C. A. Langston, and S. P. Horton (2016). Automatic
% microseismic denoising and onset detection using the synchrosqueezed
% continuous wavelet transform, Geophysics, 81 (4), V341-V355,
% DOI:10.1785/0120150345.
% [3] Cai, T. (1999). Adaptive wavelet estimation: A block thresholding and
% oracle inequality approach, Ann. Stat. 27, 898?924.
%
% [4] Cai, T., and H. Zhou (2009). A data-driven block thresholding approach
% to wave- let estimation, Ann. Stat. 37, no. 2, 569?595, doi: 10.1214/07-AOS538.
%
%--------------------------------------------------------------------------
% S. Mostafa Mousavi
% smousavi@memphis.edu
% Last time modified: June, 2, 2017
%---------------------------------------------------------------------------
%%% Copyright (c) 2015 S. Mostafa Mousavi
%%% All rights reserved.
%%% This software is provided for non-commercial research purposes only.
%%% No warranty is implied by this distribution. Permission is hereby granted
%%% without written agreement and without license or royalty fees, to use,
%%% copy, modify, and distribute this code and its documentation, provided
%%% that the copyright notice in its entirety appear in all copies of this
%%% code, and the original source of this code,
function [dn org opt] = wlBlock(data,opt);
%% wavelet Transforming
h = waitbar(0.1,'Wavelet Transforming...');
[wlCoef,as] = cwt_fw(data.noise,opt.type,opt.nv,data.dt);
waitbar(0.8,h,'Wavelet Transforming...'); close(h)
org.c = wlCoef; org.a = as;
[na n] = size(org.c);
if opt.at == 0
%% a rough arriaval time estimation
% this is based on a simplified form of the characteristic function introduced
% in [2].
ee = zeros(na,n);
for i = 1:na
v= real(wlCoef(i,:));
a = (v).^2 ;
b = (hilbert(v)).^2;
ee(i,:) = sqrt(a+b);
end
row = sum(abs(ee));
xr = smooth(row,0.01,'loess');
f = xr - 0.6*max(xr);
f(f < 0 )= 0;
n = length(f);
out = [];
for i = 2:n-1
if (f(i-1) == 0 & f(i+1)> 0 & f(i) == 0);
out = [out i+1];
end
end
opt.at=data.t(out(1))
end
%% HOS pre-processing
kk = zeros(na,1);
wk = org.c;
for i = 1:na
w = real(org.c(i,:));
Vk = 24/length(org.c(i,:)); % eq(12) in [1]
lk = (sqrt(Vk)/sqrt(1-0.9));
kr = (sum((w-mean(w)).^4)./length(w)) ./ (var(w,1).^2)-3; % calculating Kurtosis eq (11) in [1]
if abs(kr) <= lk*opt.gc;
wk(i,:) = 0;
end
end
wk(isnan(wk))=0;
%% Block Thresholding the Coefficients Scale-by-Scale
% Partitioning;
for i = 1:na
H{i} = wk(i,:);
end
% Thresholding
h = waitbar(0.30,'Block Denoising...')
for i = 1:na
waitbar(i./(na))
% level dependent noise level estimation from pr-signal noise
CH = H{i};
sigma = mad(abs(CH(:,1:round(opt.at))))./0.6745; % noise level
% estimation as discribed in page 1384 of [1].
[LB{i}, thres{i},T_H{i}] = BtThrsh(H{i},sigma); % block thresholding
end
% Assembeling
waitbar(0.30,h,'Preparing the output...')
N = size(wk,2);
t_decom = zeros(1,N);
tlMap = [0,0];
for i = 1:na
waitbar(i./(na))
t_decom = [t_decom;T_H{i}];
tlMap = [tlMap; LB{i}, thres{i}];
end
close(h)
clear h
[row col] = size(t_decom);
dnCoef = t_decom(2:row,:);
dnCoef(isnan(dnCoef)) = 0;
%% Recunstructing denoised signal
x = cwt_iw(dnCoef, opt.type,opt.nv, opt);
dn.c = dnCoef; dn.x = x; dn.m = tlMap;
function [L, lambda,tc] = BtThrsh(subC,sig)
% This function uses the optimal block size and threshold from Stein's
% unbiased risk estimate and threshols coefficients using the block
% thresholding improved by Wiener filtering.
%
% [INPUTS]
% subC; wavelet coefficients of the noisy observations
% sig; noise level estimation
%
% [Outputs]
% L: the optimal block length
% thr: the optimal threshold (Lamma)
% tc: the thresholded coefficients
%--------------------------------------------------------------------------
%% Estimating the optimal block size and threshold from SURE
[L,lambda] = optimalTL(subC,sig);
L = round(L);
%% partitioning coefficients ino blocks and thresholding coefficients
% in each block by James-Stein's shrinkage rule
nCol = length(subC);
tc = zeros(1,nCol);
m = 0; Td =0;
for i = 1:nCol
m = m + 1; S(m)=(abs(subC(i)).^2)-1;
end
s = sum(S(:));
Td = s/nCol;
Gammad = (log2(nCol^1.5))./sqrt(nCol);
if abs(Td) > Gammad
tc = zeros(1,nCol);
for j = 1:nCol/L
b2 = (j-1)*L;
bm = subC(1,b2+1:b2+L);
% eq (24)in [1] and page 904 of [3]
a = (1 - lambda*L*(sig)^2./sum(abs(bm(:)).^2));
f = a * (a > 0);
bt= f*bm; % Block Thresholding
% post-denoising
% Weiner filtering eq (25)in [1] and page 905 of [3]
dnWiener = bm.* (abs(bt).^2 ./ (abs(bt).^2 + L*(sig)^2));
tc(1,b2+1:b2+L) = dnWiener;
end
else
for i = 1:nCol
% eq (24)in [1] as on page 904 of [3]
a = (1 - (2*log10(nCol))./abs(subC(i))^2);
f = a * (a > 0);
bt= f*subC(i);% Block Thresholding
% post-denoising
% Weiner filtering eq (25)in [1] as on page 905 of [3]
dnWiener = subC(i).* (abs(bt).^2 ./ (abs(bt).^2 + (sig)^2));
tc(1,i) = dnWiener;
end
end
function [LB,lambda] = optimalTL(sc,sig)
% This function estimates the optimal threshold and block size using the
% Stein's unbiased risk estimation
%
% [INPUTS]
% subb: noisy subband
%
% [OUTPUTS]
% LB: Optimal block size
% lambda: Optimal threshold value
%--------------------------------------------------------------------------
% Computting different block sizes based on the restirictions on [4] page 10
Lmax = ceil(size(sc,2)^0.75);
p = floor(log2(Lmax));
for i = 1:p
L_set(i) = 2^i;
end
k = 0;
for L = L_set
k = k + 1;
[risk(k),thr(k)] = SURE(sc,L,sig);
end
% Obtain the optimal block size and the corresponding threshold
[guess,ibest] = min(risk);
lambda = thr(ibest); LB = L_set(ibest);
function [guess,lambda] = SURE(sc,L,sig)
% Finding an optimal threshold for a given block size.
% [INPUTS]
% subb: inputted noisy subband
% L: inputted block size
% sig: noise level estimation
%
% [OUTPUTS]
% guess: outputted minimum risk
% th: outputted threshold value
%---------------------------------------------------------------------------
sc = real(sc);
% Compute the square sum of all coefficients in a block and generate a threshold series
if L == 1
sb = sc(:).^2;
if min(sb) > 0
Thres = [sb; 0];
end
else
% the square sum of all atoms in each block
nCol = length(sc);
p = mod(nCol, L);
padSubb = padarray(sc,[0 L-p], 'post');
blc_n = (length(padSubb)./L);
sb = zeros(blc_n,1);
m = 0;
% the square sum of the atoms in each block
for i = 1:blc_n
bm = padSubb(1,((i-1)*L)+1:i*L);
m = m + 1;
sb(m) = sum(bm(:).^2);
end
end
%% Generating the threshold vector
LammaF = floor(2*L*log10(nCol)); % [4], page 10
upth = max(sb):LammaF;
upth = upth';
Thres = [0; abs(sb); upth];
Thres = Thres(find(Thres >= 1)); % Yu et.al (2008)
%% Computing the threshold corresponding to the minimum risk
risk = zeros(length(Thres),1); m = 0;
blc_num = length(sb); Thres = Thres';
for th = Thres
risk_temp = 0;
for k = 1:blc_num
% eq(20) in [1]
R = (sig^2)*(L + ((((th^2)*(L.^2))-2*th*L*(L-2))/(sb(k)/(sig^2)))*((sb(k)/(sig^2))>th*L) + ((sb(k)/(sig^2))-2*L)*((sb(k)/(sig^2))<=th*L));
risk_temp = risk_temp+ R;
end
m = m + 1; risk(m) = risk_temp;
end
[guess,ibest] = min(risk);
lambda = Thres(ibest);
function [Wx,as] = cwt_fw(x, type, nv, dt, opt)
% Forward continuous wavelet transform, discretized, as described
% in Mallat, S., Wavelet Tour of Signal Processing 3rd ed.Sec. 4.3.3.
%
% [INPUTS]
% x: input signal vector.
% type: wavelet type, string
% nv: number of voices
% dt: sampling period
% opt: options structure
%
% [OUTPUTS]
% Wx: [na x n] size matrix (rows = scales, cols = times)
% as: na length vector containing the associated scales
%
%---------------------------------------------------------------------------------
% Modified after a wavelet transform by Eugene Brevdo
%---------------------------------------------------------------------------------
opt = struct();
opt.rpadded = 0;
x = x(:); % Turn into column vector
n = length(x);
% Padding the signal
N = 2^(1+round(log2(length(x)+eps)));
n1 = floor((N-n)/2);
n2 = n1;if (mod(2*n1+n,2)==1), n2 = n1 + 1; end
xl = padarray(x(:), n1, 'pre');
xr = padarray(x(:), n2, 'post');
x = [xl(1:n1); x(:); xr(end-n2+1:end)];
% Choosing more than this means the wavelet window becomes too short
noct = log2(N)-1;
assert(noct > 0 && mod(noct,1) == 0);
assert(nv>0 && mod(nv,1)==0);
assert(dt>0);
assert(~any(isnan(x)));
na = noct*nv;
as = 2^(1/nv) .^ (1:1:na);
Wx = zeros(na, N);
x = x(:).';
xh = fft(x);
% for each octave
for ai = 1:na
a = as(ai);
psih = wfilth(type, N, a, opt);
xcpsi = ifftshift(ifft(psih .* xh));
Wx(ai, :) = xcpsi;
end
% Shorten W to proper size (remove padding)
if (~opt.rpadded)
Wx = Wx(:, n1+1:n1+n);
end
% Output a for graphing purposes, scale by dt
as = as * dt;
function x = cwt_iw(Wx, type, nv, opt)
% The inverse wavelet transform
%
% Implements Eq. (4.67) of Mallat, S., Wavelet Tour of Signal Processing 3rd ed.
%
% Inputs:
% Wx: wavelet transform of a signal, see help cwt_fw
% type: wavelet used to take the wavelet transform,
% see help cwt_fw and help wfiltfn
% opt: options structure used for forward wavelet transform.
%
% Output:
% x: the signal, as reconstructed from Wx
%
%---------------------------------------------------------------------------------
% Modified after a wavelet transform written by Eugene Brevdo
%---------------------------------------------------------------------------------
opt = struct()
[na, n] = size(Wx);
% Padding the signal
N = 2^(1+round(log2(n+eps)));
n1 = floor((N-n)/2);
n2 = n1;if (mod(2*n1+n,2)==1), n2 = n1 + 1; end
Wxp = zeros(na, N);
Wxp(:, n1+1:n1+n) = Wx;
Wx = Wxp; clear Wxp;
noct = log2(N)-1;
as = 2^(1/nv) .^ (1:1:na);
assert(mod(noct,1) == 0);
assert(nv>0 && mod(nv,1)==0);
% the admissibility coefficient Cpsi
switch type
case 'shannon',
Cpsi = log(2);
otherwise
psihfn = wfiltfn(type, opt);
Cpsi = quadgk(@(x) (conj(psihfn(x)).*psihfn(x))./x, 0, Inf);
end
% Normalize
Cpsi = Cpsi / (4*pi);
x = zeros(1, N);
for ai=1:na
a = as(ai);
Wxa = Wx(ai, :);
psih = wfilth(type, N, a, opt);
% Convolution theorem
Wxah = fft(Wxa);
xah = Wxah .* psih;
xa = ifftshift(ifft(xah));
x = x + xa/a;
end
% Take real part and normalize by log_e(a)/Cpsi
x = log(2^(1/nv))/Cpsi * real(x);
% Keep the unpadded part
x = x(n1+1: n1+n);
function [psih] = wfilth(type, N, a, opt)
% Outputs the FFT of the wavelet of family 'type' with parameters
% in 'opt', of length N at scale a: (psi(-t/a))^.
%
% [Inputs]
% type: wavelet type
% N: number of samples to calculate
% a: wavelet scale parameter
% opt: wavelet options
% opt.dt: delta t
%
% [Outputs]
% psih: wavelet sampling in frequency domain
%---------------------------------------------------------------------------------
opt = struct();
k = 0:(N-1);
xi = zeros(1, N);
xi(1:N/2+1) = 2*pi/N*[0:N/2];
xi(N/2+2:end) = 2*pi/N*[-N/2+1:-1];
psihfn = wfiltfn(type, opt);
psih = psihfn(a*xi);
% Normalizing
psih = psih * sqrt(a) / sqrt(2*pi);
% Center around zero in the time domain
psih = psih .* (-1).^k;
function [psihfn] = wfiltfn(type, opt)
% Wavelet transform function of the wavelet filter in question,
% fourier domain.
%
% [Input]
% type: string (see below)
% opt: options structure, e.g. struct('s',1/6,'mu',2)
%
% [Output]
% psihfn: mother wavelet function ( mexican hat, morlet, shannon, or hermitian)
% Example:
% psihfn = wfiltfn('bump', struct('mu',1,'s',.5));
% plot(psihfn(-5:.01:5));
%---------------------------------------------------------------------------------
switch type
case 'mhat', % mexican hat
if ~isfield(opt,'s'), s = 1; else s = opt.s; end
psihfn = @(w) -sqrt(8)*s^(5/2)*pi^(1/4)/sqrt(3)*w.^2.*exp(-s^2*w.^2/2);
case 'morlet',
% can be used with synsq for large enough s (e.g. >5)
if ~isfield(opt,'mu'), mu = 2*pi; else mu = opt.mu; end
cs = (1+exp(-mu^2)-2*exp(-3/4*mu^2)).^(-1/2);
ks = exp(-1/2*mu^2);
psihfn = @(w)cs*pi^(-1/4)*(exp(-1/2*(mu-w).^2)-ks*exp(-1/2*w.^2));
case 'shannon',
psihfn = @(w)exp(-i*w/2).*(abs(w)>=pi & abs(w)<=2*pi);
case 'hhat', % hermitian hat
psihfn = @(w)2/sqrt(5)*pi^(-1/4)*w.*(1+w).*exp(-1/2*w.^2);
% case 'mostafa',
% load ss
% if ~isfield(opt,'mu'), mu = 5; else mu = opt.mu; end
% if ~isfield(opt,'s'), s = 1; else s = opt.s; end
% psihfnorig = @(w)(0.0720*w.^8)+(0.2746*w.^7)+(0.2225*w.^6)+(-0.2781*w.^5)+(-0.3884*w.^4)+(0.0735*w.^3)+(-0.3354*w.^2)+(-0.0043*w)+(0.3675);
% psihfn = @(w) psihfnorig((w-mu)/s);
otherwise
error('Unknown wavelet type: %s', type);
end
function y = mad(x,flag)
%MAD Mean/median absolute deviation.
% Y = MAD(X) returns the mean absolute deviation of the values in X. For
% vector input, Y is MEAN(ABS(X-MEAN(X)). For a matrix input, Y is a row
% vector containing the mean absolute deviation of each column of X. For
% N-D arrays, MAD operates along the first non-singleton dimension.
%
% MAD(X,1) computes Y based on medians, i.e. MEDIAN(ABS(X-MEDIAN(X)).
% MAD(X,0) is the same as MAD(X), and uses means.
%
% MAD treats NaNs as missing values, and removes them.
%
% See also VAR, STD, IQR.
% References:
% [1] L. Sachs, "Applied Statistics: A Handbook of Techniques",
% Springer-Verlag, 1984, page 253.
% Copyright 1993-2004 The MathWorks, Inc.
% $Revision: 2.10.2.2 $ $Date: 2004/01/24 09:34:28 $
% The output size for [] is a special case, handle it here.
if isequal(x,[])
y = NaN;
return;
end;
if nargin < 2
flag = 0;
end
% Figure out which dimension nanmean will work along.
sz = size(x);
dim = find(sz ~= 1, 1);
if isempty(dim)
dim = 1;
end
% Need to tile the output of nanmean to center X.
tile = ones(1,ndims(x));
tile(dim) = sz(dim);
if flag
% Compute the median of the absolute deviations from the median.
y = nanmedian(abs(x - repmat(nanmedian(x), tile)));
else
% Compute the mean of the absolute deviations from the mean.
y = nanmean(abs(x - repmat(nanmean(x), tile)));
end
function y = nanmedian(x,dim)
%NANMEDIAN Median value, ignoring NaNs.
% M = NANMEDIAN(X) returns the sample median of X, treating NaNs as
% missing values. For vector input, M is the median value of the non-NaN
% elements in X. For matrix input, M is a row vector containing the
% median value of non-NaN elements in each column. For N-D arrays,
% NANMEDIAN operates along the first non-singleton dimension.
%
% NANMEDIAN(X,DIM) takes the median along the dimension DIM of X.
%
% See also MEDIAN, NANMEAN, NANSTD, NANVAR, NANMIN, NANMAX, NANSUM.
% Copyright 1993-2004 The MathWorks, Inc.
% $Revision: 2.12.2.2 $ $Date: 2004/01/24 09:34:33 $
if nargin == 1
y = prctile(x, 50);
else
y = prctile(x, 50,dim);
end
function m = nanmean(x,dim)
%NANMEAN Mean value, ignoring NaNs.
% M = NANMEAN(X) returns the sample mean of X, treating NaNs as missing
% values. For vector input, M is the mean value of the non-NaN elements
% in X. For matrix input, M is a row vector containing the mean value of
% non-NaN elements in each column. For N-D arrays, NANMEAN operates
% along the first non-singleton dimension.
%
% NANMEAN(X,DIM) takes the mean along the dimension DIM of X.
%
% See also MEAN, NANMEDIAN, NANSTD, NANVAR, NANMIN, NANMAX, NANSUM.
% Copyright 1993-2004 The MathWorks, Inc.
% $Revision: 2.13.4.2 $ $Date: 2004/01/24 09:34:32 $
% Find NaNs and set them to zero
nans = isnan(x);
x(nans) = 0;
if nargin == 1 % let sum deal with figuring out which dimension to use
% Count up non-NaNs.
n = sum(~nans);
n(n==0) = NaN; % prevent divideByZero warnings
% Sum up non-NaNs, and divide by the number of non-NaNs.
m = sum(x) ./ n;
else
% Count up non-NaNs.
n = sum(~nans,dim);
n(n==0) = NaN; % prevent divideByZero warnings
% Sum up non-NaNs, and divide by the number of non-NaNs.
m = sum(x,dim) ./ n;
end
function y = prctile(x,p,dim)
%PRCTILE Percentiles of a sample.
% Y = PRCTILE(X,P) returns percentiles of the values in X. P is a scalar
% or a vector of percent values. When X is a vector, Y is the same size
% as P, and Y(i) contains the P(i)-th percentile. When X is a matrix,
% the i-th row of Y contains the P(i)-th percentiles of each column of X.
% For N-D arrays, PRCTILE operates along the first non-singleton
% dimension.
%
% Y = PRCTILE(X,P,DIM) calculates percentiles along dimension DIM. The
% DIM'th dimension of Y has length LENGTH(P).
%
% Percentiles are specified using percentages, from 0 to 100. For an N
% element vector X, PRCTILE computes percentiles as follows:
% 1) The sorted values in X are taken as the 100*(0.5/N), 100*(1.5/N),
% ..., 100*((N-0.5)/N) percentiles.
% 2) Linear interpolation is used to compute percentiles for percent
% values between 100*(0.5/N) and 100*((N-0.5)/N)
% 3) The minimum or maximum values in X are assigned to percentiles
% for percent values outside that range.
%
% PRCTILE treats NaNs as missing values, and removes them.
%
% Examples:
% y = prctile(x,50); % the median of x
% y = prctile(x,[2.5 25 50 75 97.5]); % a useful summary of x
%
% See also IQR, MEDIAN, NANMEDIAN, QUANTILE.
% Copyright 1993-2004 The MathWorks, Inc.
% $Revision: 2.12.4.4 $ $Date: 2004/01/24 09:34:55 $
if ~isvector(p) || numel(p) == 0
error('stats:prctile:BadPercents', ...
'P must be a scalar or a non-empty vector.');
elseif any(p < 0 | p > 100)
error('stats:prctile:BadPercents', ...
'P must take values between 0 and 100');
end
% Figure out which dimension prctile will work along.
sz = size(x);
if nargin < 3
dim = find(sz ~= 1,1);
if isempty(dim)
dim = 1;
end
dimArgGiven = false;
else
% Permute the array so that the requested dimension is the first dim.
nDimsX = ndims(x);
perm = [dim:max(nDimsX,dim) 1:dim-1];
x = permute(x,perm);
% Pad with ones if dim > ndims.
if dim > nDimsX
sz = [sz ones(1,dim-nDimsX)];
end
sz = sz(perm);
dim = 1;
dimArgGiven = true;
end
% If X is empty, return all NaNs.
if isempty(x)
if isequal(x,[]) && ~dimArgGiven
y = nan(size(p),class(x));
else
szout = sz; szout(dim) = numel(p);
y = nan(szout,class(x));
end
else
% Drop X's leading singleton dims, and combine its trailing dims. This
% leaves a matrix, and we can work along columns.
nrows = sz(dim);
ncols = prod(sz) ./ nrows;
x = reshape(x, nrows, ncols);
x = sort(x,1);
nonnans = ~isnan(x);
% If there are no NaNs, do all cols at once.
if all(nonnans(:))
n = sz(dim);
if isequal(p,50) % make the median fast
if rem(n,2) % n is odd
y = x((n+1)/2,:);
else % n is even
y = (x(n/2,:) + x(n/2+1,:))/2;
end
else
q = [0 100*(0.5:(n-0.5))./n 100]';
xx = [x(1,:); x(1:n,:); x(n,:)];
y = zeros(numel(p), ncols, class(x));
y(:,:) = interp1q(q,xx,p(:));
end
% If there are NaNs, work on each column separately.
else
% Get percentiles of the non-NaN values in each column.
y = nan(numel(p), ncols, class(x));
for j = 1:ncols
nj = find(nonnans(:,j),1,'last');
if nj > 0
if isequal(p,50) % make the median fast
if rem(nj,2) % nj is odd
y(:,j) = x((nj+1)/2,j);
else % nj is even
y(:,j) = (x(nj/2,j) + x(nj/2+1,j))/2;
end
else
q = [0 100*(0.5:(nj-0.5))./nj 100]';
xx = [x(1,j); x(1:nj,j); x(nj,j)];
y(:,j) = interp1q(q,xx,p(:));
end
end
end
end
% Reshape Y to conform to X's original shape and size.
szout = sz; szout(dim) = numel(p);
y = reshape(y,szout);
end
% undo the DIM permutation
if dimArgGiven
y = ipermute(y,perm);
end
% If X is a vector, the shape of Y should follow that of P, unless an
% explicit DIM arg was given.
if ~dimArgGiven && isvector(x)
y = reshape(y,size(p));
end
function [c,ww] = smooth(varargin)
%SMOOTH Smooth data.
% Z = SMOOTH(Y) smooths data Y using a 5-point moving average.
%
% Z = SMOOTH(Y,SPAN) smooths data Y using SPAN as the number of points used
% to compute each element of Z.
%
% Z = SMOOTH(Y,SPAN,METHOD) smooths data Y with specified METHOD. The
% available methods are:
%
% 'moving' - Moving average (default)
% 'lowess' - Lowess (linear fit)
% 'loess' - Loess (quadratic fit)
% 'sgolay' - Savitzky-Golay
% 'rlowess' - Robust Lowess (linear fit)
% 'rloess' - Robust Loess (quadratic fit)
%
% Z = SMOOTH(Y,METHOD) uses the default SPAN 5.
%
% Z = SMOOTH(Y,SPAN,'sgolay',DEGREE) and Z = SMOOTH(Y,'sgolay',DEGREE)
% additionally specify the degree of the polynomial to be used in the
% Savitzky-Golay method. The default DEGREE is 2. DEGREE must be smaller
% than SPAN.
%
% Z = SMOOTH(X,Y,...) additionally specifies the X coordinates. If X is
% not provided, methods that require X coordinates assume X = 1:N, where
% N is the length of Y.
%
% Notes:
% 1. When X is given and X is not uniformly distributed, the default method
% is 'lowess'. The 'moving' method is not recommended.
%
% 2. For the 'moving' and 'sgolay' methods, SPAN must be odd.
% If an even SPAN is specified, it is reduced by 1.
%
% 3. If SPAN is greater than the length of Y, it is reduced to the
% length of Y.
%
% 4. In the case of (robust) lowess and (robust) loess, it is also
% possible to specify the SPAN as a percentage of the total number
% of data points. When SPAN is less than or equal to 1, it is
% treated as a percentage.
%
% For example:
%
% Z = SMOOTH(Y) uses the moving average method with span 5 and
% X=1:length(Y).
%
% Z = SMOOTH(Y,7) uses the moving average method with span 7 and
% X=1:length(Y).
%
% Z = SMOOTH(Y,'sgolay') uses the Savitzky-Golay method with DEGREE=2,
% SPAN = 5, X = 1:length(Y).
%
% Z = SMOOTH(X,Y,'lowess') uses the lowess method with SPAN=5.
%
% Z = SMOOTH(X,Y,SPAN,'rloess') uses the robust loess method.
%
% Z = SMOOTH(X,Y) where X is unevenly distributed uses the
% 'lowess' method with span 5.
%
% Z = SMOOTH(X,Y,8,'sgolay') uses the Savitzky-Golay method with
% span 7 (8 is reduced by 1 to make it odd).
%
% Z = SMOOTH(X,Y,0.3,'loess') uses the loess method where span is
% 30% of the data, i.e. span = ceil(0.3*length(Y)).
%
% See also SPLINE.
% Copyright 2001-2012 The MathWorks, Inc.
if nargin < 1
error(message('curvefit:smooth:needMoreArgs'));
end
if nargout > 1 % Called from the GUI cftool
ws = warning('off', 'all'); % turn warning off and record the previous warning state.
[lw,lwid] = lastwarn;
lastwarn('');
else
ws = warning('query','all'); % Leave warning state alone but save it so resets are no-ops.
end
% is x given as the first argument?
if nargin==1 || ( nargin > 1 && (length(varargin{2})==1 || ischar(varargin{2})) )
% smooth(Y) | smooth(Y,span,...) | smooth(Y,method,...)
is_x = 0; % x is not given
y = varargin{1};
y = y(:);
x = (1:length(y))';
else % smooth(X,Y,...)
is_x = 1;
y = varargin{2};
x = varargin{1};
y = y(:);
x = x(:);
end
% is span given?
span = [];
if nargin == 1+is_x || ischar(varargin{2+is_x})
% smooth(Y), smooth(X,Y) || smooth(X,Y,method,..), smooth(Y,method)
is_span = 0;
else
% smooth(...,SPAN,...)
is_span = 1;
span = varargin{2+is_x};
end
% is method given?
method = [];
if nargin >= 2+is_x+is_span
% smooth(...,Y,method,...) | smooth(...,Y,span,method,...)
method = varargin{2+is_x+is_span};
end
t = length(y);
if t == 0
c = y;
ww = '';
if nargout > 1
ww = lastwarn;
lastwarn(lw,lwid);
warning(ws); % turn warning back to the previous state.
end
return
elseif length(x) ~= t
warning(ws); % reset warn state before erroring
error(message('curvefit:smooth:XYmustBeSameLength'));
end
if isempty(method)
diffx = diff(x);
if uniformx(diffx,x,y)
method = 'moving'; % uniformly distributed X.
else
method = 'lowess';
end
end
% realize span
if span <= 0
warning(ws); % reset warn state before erroring
error(message('curvefit:smooth:spanMustBePositive'));
end
if span < 1, span = ceil(span*t); end % percent convention
if isempty(span), span = 5; end % smooth(Y,[],method)
idx = 1:t;
sortx = any(diff(isnan(x))<0); % if NaNs not all at end
if sortx || any(diff(x)<0) % sort x
[x,idx] = sort(x);
y = y(idx);
end
c = NaN(size(y));
ok = ~isnan(x);
switch method
case 'moving'
c(ok) = moving(x(ok),y(ok),span);
case {'lowess','loess','rlowess','rloess'}
robust = 0;
iter = 5;
if method(1)=='r'
robust = 1;
method = method(2:end);
end
c(ok) = lowess(x(ok),y(ok),span, method,robust,iter);
case 'sgolay'
if nargin >= 3+is_x+is_span
degree = varargin{3+is_x+is_span};
else
degree = 2;
end
if degree < 0 || degree ~= floor(degree) || degree >= span
warning(ws); % reset warn state before erroring
error(message('curvefit:smooth:invalidDegree'));
end
c(ok) = sgolay(x(ok),y(ok),span,degree);
otherwise
warning(ws); % reset warn state before erroring
error(message('curvefit:smooth:unrecognizedMethod'));
end
c(idx) = c;
if nargout > 1
ww = lastwarn;
lastwarn(lw,lwid);
warning(ws); % turn warning back to the previous state.
end
%--------------------------------------------------------------------
function c = moving(x,y, span)
% moving average of the data.
ynan = isnan(y);
span = floor(span);
n = length(y);
span = min(span,n);
width = span-1+mod(span,2); % force it to be odd
xreps = any(diff(x)==0);
if width==1 && ~xreps && ~any(ynan), c = y; return; end
if ~xreps && ~any(ynan)
% simplest method for most common case
c = filter(ones(width,1)/width,1,y);
cbegin = cumsum(y(1:width-2));
cbegin = cbegin(1:2:end)./(1:2:(width-2))';
cend = cumsum(y(n:-1:n-width+3));
cend = cend(end:-2:1)./(width-2:-2:1)';
c = [cbegin;c(width:end);cend];
elseif ~xreps
% with no x repeats, can take ratio of two smoothed sequences
yy = y;
yy(ynan) = 0;
nn = double(~ynan);
ynum = moving(x,yy,span);
yden = moving(x,nn,span);
c = ynum ./ yden;
else
% with some x repeats, loop
notnan = ~ynan;
yy = y;
yy(ynan) = 0;
c = zeros(n,1);
for i=1:n
if i>1 && x(i)==x(i-1)
c(i) = c(i-1);
continue;
end
R = i; % find rightmost value with same x
while(R<n && x(R+1)==x(R))
R = R+1;
end
hf = ceil(max(0,(span - (R-i+1))/2)); % need this many more on each side
hf = min(min(hf,(i-1)), (n-R));
L = i-hf; % find leftmost point needed
while(L>1 && x(L)==x(L-1))
L = L-1;
end
R = R+hf; % find rightmost point needed
while(R<n && x(R)==x(R+1))
R = R+1;
end
c(i) = sum(yy(L:R)) / sum(notnan(L:R));
end
end
%--------------------------------------------------------------------
function c = lowess(x,y, span, method, robust, iter)
% LOWESS Smooth data using Lowess or Loess method.
%
% The difference between LOWESS and LOESS is that LOWESS uses a
% linear model to do the local fitting whereas LOESS uses a