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