TRANSMAXDIST is the jWavelet module of jLab.

 TRANSMAXDIST  Distributions of wavelet transform maxima in noise.
 
    This function is part of 'element analysis' described in Lilly (2017), 
    "Element analysis: a wavelet-based method for analyzing time-localized
    events in noisy time series", submitted.  Available at www.jmlilly.net.
   
    [COUNT,BINS]=TRANSMAXDIST(GAMMA,BETA,ALPHA,FS,R,N,M) returns the 
    histogram of wavelet transform maxima magnitudes, for a length N time 
    series having spectral slope -2*ALPHA transformed at frequencies FS 
    using a (GAMMA,BETA) wavelet, based on a simulation having N*M points.
 
    Here GAMMA, BETA, ALPHA, and R are all scalars, or are all arrays of 
    the same length as FS.  FS is a frequency array computed by MORSESPACE.
 
    R is the ratio between each frequency FS and the next.  This will be 
    constant and greater than one when FS is computed by MORSESPACE.  As 
    as described in Appendix C of Lilly (2017), R=FS(n)./FS(n+1) for all n. 
 
    COUNT is the number of transform maxima observed at each frequency in
    the magnitude bins BINS.  COUNT is a LENGTH(BINS) x LENGTH(FS) matrix.
 
    Transform maxima values, as output in BINS, are normalized such that
    the expected squared magnitude of the wavelet transform of noise occurs
    at unity.  BINS thus corresponds to the normalized event magnitude.  
 
    TRANSMAXDIST works by simulating a vector whose statistical properties 
    mimic those of the wavelet transform and the four adjacted points, thus 
    avoiding the need to explicitly compute the transform.  The choice of
    e.g. M=1000 simulates a transform 1000 times as long as time series of 
    interest, which itself is of length N. 
 
    [COUNT,BINS,RATE]=TRANSMAXDIST(...) also returns the RATE, the
    normalized reversed cumulative density function.  RATE gives the 
    expected number of transform maxima occuring in a time series of length
    N having a magnitude greater than the corresponding bin value.
 
    [COUNT,BINS,RATE,SIGMA]=TRANSMAXDIST(...) also returns the theoretical 
    covariance matrix SIGMA from which the Monte Carlo simulations are 
    constructed.  SIGMA is an array of length 5 x 5 x LENGTH(FS). 
 
    Note that if the covariance matrix is not positive definite, as can 
    happen due to numerical complications for extreme BETA and GAMMA 
    choices, then COUNT and RATE will both consist entirely of NaNs.
    _______________________________________________________________________
 
    Additional options
 
    TRANSMAXDIST(...,BINS) alternately uses BINS for the bin centers 
    instead of the default choice, which is set to LINSPACE(0,6,400)'.
 
    By default, TRANSMAXDIST performs a simulation for each of the scale
    frequencies in FS.  TRANSMAXDIST(...,'extrapolate') instead computes
    the distribution only for the highest scale frequency, then 
    extrapolates these values to all other scale frequencies with a scaling
    law.  GAMMA, BETA, ALPHA, and R must all be scalars in this case.
   
    For details, see Lilly (2017).
 
    See also MAXPROPS, TRANSMAX, ISOMAX, MAX2EDDY.
 
    'transmaxdist --t' runs some tests.
 
    Usage: [count,bins]=transmaxdist(ga,be,al,fs,r,N,M); 
           [count,bins,rate,sigma]=transmaxdist(ga,be,al,fs,r,N,M);
           [count,bins,rate,sigma]=transmaxdist(ga,be,al,fs,r,N,M,'extrap');
    __________________________________________________________________
    This is part of JLAB --- type 'help jlab' for more information
    (C) 2017 J.M. Lilly --- type 'help jlab_license' for details

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