wave_matlab.zip_matlab inv.m_wave

%WAVE_SIGNIF Significance testing for the 1D Wavelet transform WAVELET % % [SIGNIF,FFT_THEOR] = ... % wave_signif(Y,DT,SCALE,SIGTEST,LAG1,SIGLVL,DOF,MOTHER,PARAM) % % INPUTS: % % Y = the time series, or, the VARIANCE of the time series. % (If this is a single number, it is assumed to be the variance...) % DT = amount of time between each Y value, i.e. the sampling time. % SCALE = the vector of scale indices, from previous call to WAVELET. % % % OUTPUTS: % % SIGNIF = significance levels as a function of SCALE % FFT_THEOR = output theoretical red-noise spectrum as fn of PERIOD % % % OPTIONAL INPUTS: % *** Note *** setting any of the following to -1 will cause the default % value to be used. % % SIGTEST = 0, 1, or 2. If omitted, then assume 0. % % If 0 (the default), then just do a regular chi-square test, % i.e. Eqn (18) from Torrence & Compo. % If 1, then do a "time-average" test, i.e. Eqn (23). % In this case, DOF should be set to NA, the number % of local wavelet spectra that were averaged together. % For the Global Wavelet Spectrum, this would be NA=N, % where N is the number of points in your time series. % If 2, then do a "scale-average" test, i.e. Eqns (25)-(28). % In this case, DOF should be set to a % two-element vector [S1,S2], which gives the scale % range that was averaged together. % e.g. if one scale-averaged scales between 2 and 8, % then DOF=[2,8]. % % LAG1 = LAG 1 Autocorrelation, used for SIGNIF levels. Default is 0.0 % % SIGLVL = significance level to use. Default is 0.95 % % DOF = degrees-of-freedom for signif test. % IF SIGTEST=0, then (automatically) DOF = 2 (or 1 for MOTHER='DOG') % IF SIGTEST=1, then DOF = NA, the number of times averaged together. % IF SIGTEST=2, then DOF = [S1,S2], the range of scales averaged. % % Note: IF SIGTEST=1, then DOF can be a vector (same length as SCALEs), % in which case NA is assumed to vary with SCALE. % This allows one to average different numbers of times % together at different scales, or to take into account % things like the Cone of Influence. % See discussion following Eqn (23) in Torrence & Compo. % % %---------------------------------------------------------------------------- % Copyright (C) 1995-1998, Christopher Torrence and Gilbert P. Compo % University of Colorado, Program in Atmospheric and Oceanic Sciences. % This software may be used, copied, or redistributed as long as it is not % sold and this copyright notice is reproduced on each copy made. This % routine is provided as is without any express or implied warranties % whatsoever. %---------------------------------------------------------------------------- function [signif,fft_theor] = ... wave_signif(Y,dt,scale1,sigtest,lag1,siglvl,dof,mother,param); if (nargin < 9), param = -1;, end if (nargin < 8), mother = -1;, end if (nargin < 7), dof = -1;, end if (nargin < 6), siglvl = -1;, end if (nargin < 5), lag1 = -1;, end if (nargin < 4), sigtest = -1;, end if (nargin < 3) error('Must input a vector Y, sampling time DT, and SCALE vector') end n1 = length(Y); J1 = length(scale1) - 1; scale(1:J1+1) = scale1; s0 = min(scale); dj = log(scale(2)/scale(1))/log(2.); if (n1 == 1) variance = Y; else variance = std(Y)^2; end if (sigtest == -1), sigtest = 0;, end if (lag1 == -1), lag1 = 0.0;, end if (siglvl == -1), siglvl = 0.95;, end if (mother == -1), mother = 'MORLET';, end mother = upper(mother); % get the appropriate parameters [see Table(2)] if (strcmp(mother,'MORLET')) %---------------------------------- Morlet if (param == -1), param = 6.;, end k0 = param; fourier_factor = (4*pi)/(k0 + sqrt(2 + k0^2)); % Scale-->Fourier [Sec.3h] empir = [2.,-1,-1,-1]; if (k0 == 6), empir(2:4)=[0.776,2.32,0.60];, end elseif (strcmp(mother,'PAUL')) %-------------------------------- Paul if (param == -1), param = 4.;, end m = param; fourier_factor = 4*pi/(2*m+1); empir = [2.,-1,-1,-1]; if (m == 4), empir(2:4)=[1.132,1.17,1.5];, end elseif (strcmp(mother,'DOG')) %--------------------------------- DOG if (param == -1), param = 2.;, end m = param; fourier_factor = 2*pi*sqrt(2./(2*m+1)); empir = [1.,-1,-1,-1]; if (m == 2), empir(2:4) = [3.541,1.43,1.4];, end if (m == 6), empir(2:4) = [1.966,1.37,0.97];, end else error('Mother must be one of MORLET,PAUL,DOG') end period = scale.*fourier_factor; dofmin = empir(1); % Degrees of freedom with no smoothing Cdelta = empir(2); % reconstruction factor gamma_fac = empir(3); % time-decorrelation factor dj0 = empir(4); % scale-decorrelation factor freq = dt ./ period; % normalized frequency fft_theor = (1-lag1^2) ./ (1-2*lag1*cos(freq*2*pi)+lag1^2); % [Eqn(16)] fft_theor = variance*fft_theor; % include time-series variance signif = fft_theor; if (dof == -1), dof = dofmin;, end if (sigtest == 0) % no smoothing, DOF=dofmin [Sec.4] dof = dofmin; chisquare = chisquare_inv(siglvl,dof)/dof; signif = fft_theor*chisquare ; % [Eqn(18)] elseif (sigtest == 1) % time-averaged significance if (length(dof) == 1), dof=zeros(1,J1+1)+dof;, end truncate = find(dof < 1); dof(truncate) = ones(size(truncate)); dof = dofmin*sqrt(1 + (dof*dt/gamma_fac ./ scale).^2 ); % [Eqn(23)] truncate = find(dof < dofmin); dof(truncate) = dofmin*ones(size(truncate)); % minimum DOF is dofmin for a1 = 1:J1+1 chisquare = chisquare_inv(siglvl,dof(a1))/dof(a1); signif(a1) = fft_theor(a1)*chisquare; end elseif (sigtest == 2) % time-averaged significance if (length(dof) ~= 2) error('DOF must be set to [S1,S2], the range of scale-averages') end if (Cdelta == -1) error(['Cdelta & dj0 not defined for ',mother, ... ' with param = ',num2str(param)]) end s1 = dof(1); s2 = dof(2); avg = find((scale >= s1) & (scale <= s2)); % scales between S1 & S2 navg = length(avg); if (navg == 0) error(['No valid scales between ',num2str(s1),' and ',num2str(s2)]) end Savg = 1./sum(1 ./ scale(avg)); % [Eqn(25)] Smid = exp((log(s1)+log(s2))/2.); % power-of-two midpoint dof = (dofmin*navg*Savg/Smid)*sqrt(1 + (navg*dj/dj0)^2); % [Eqn(28)] fft_theor = Savg*sum(fft_theor(avg) ./ scale(avg)); % [Eqn(27)] chisquare = chisquare_inv(siglvl,dof)/dof; signif = (dj*dt/Cdelta/Savg)*fft_theor*chisquare; % [Eqn(26)] else error('sigtest must be either 0, 1, or 2') end return % end of code