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# lomaxcdf

## PURPOSE

CDF for local maxima for a zero-mean Gaussian process

## SYNOPSIS

F = lomaxcdf(x,alpha,m0)

## DESCRIPTION

``` 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)])

## CROSS-REFERENCE INFORMATION

This function calls:
 wnormcdf Normal cumulative distribution function error Display message and abort function.
This function is called by:

## SOURCE CODE

```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 %
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```

Mathematical Statistics
Centre for Mathematical Sciences
Lund University with Lund Institute of Technology

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