LOMAXCDF CDF for local maxima for a zero-mean Gaussian process CALL: F = lomaxcdf(x,alpha,m0) F = distribution function evaluated at x alpha = irregularity factor m0 = zero-order spectral moment (variance of the process) The cdf is calculated from an explicit expression involving the standard-normal cdf. This relation is sometimes written as a convolution M = sqrt(m0)*( sqrt(1-a^2)*Z + a*R ) where M denotes local maximum, Z is a standard normal r.v., R is a standard Rayleigh r.v., and "=" means equality in distribution. Note that all local maxima of the process are considered, not only crests of waves. Example: S = jonswap(10); xs = spec2sdat(S,10000); mM = tp2mm(dat2tp(xs)); m0 = spec2mom(S,1); alpha = spec2char(S,'alpha'); x = linspace(-10,10,200).'; empdistr(mM(:,2),[x,lomaxcdf(x,alpha,m0)]) See also spec2mom, spec2bw
001 function F = lomaxcdf(x,alpha,m0) 002 %LOMAXCDF CDF for local maxima for a zero-mean Gaussian process 003 % 004 % CALL: F = lomaxcdf(x,alpha,m0) 005 % 006 % F = distribution function evaluated at x 007 % alpha = irregularity factor 008 % m0 = zero-order spectral moment (variance of the process) 009 % 010 % 011 % The cdf is calculated from an explicit expression involving the 012 % standard-normal cdf. This relation is sometimes written as a convolution 013 % 014 % M = sqrt(m0)*( sqrt(1-a^2)*Z + a*R ) 015 % 016 % where M denotes local maximum, Z is a standard normal r.v., 017 % R is a standard Rayleigh r.v., and "=" means equality in distribution. 018 % 019 % Note that all local maxima of the process are considered, not 020 % only crests of waves. 021 % 022 % Example: 023 % S = jonswap(10); 024 % xs = spec2sdat(S,10000); 025 % mM = tp2mm(dat2tp(xs)); 026 % m0 = spec2mom(S,1); 027 % alpha = spec2char(S,'alpha'); 028 % x = linspace(-10,10,200).'; 029 % empdistr(mM(:,2),[x,lomaxcdf(x,alpha,m0)]) 030 % 031 % See also spec2mom, spec2bw 032 033 % Tested on Matlab 6.0 034 % History: 035 % Revised pab Feb2004 036 % -extended example 037 % By jr 31.03.2001 038 039 error(nargchk(3,3,nargin)) 040 c1 = 1/(sqrt(1-alpha^2))*x./sqrt(m0); 041 c2 = alpha*c1; 042 043 F = wnormcdf(c1,0,1)-alpha*exp(-x.^2/2/m0).*wnormcdf(c2,0,1); 044
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