def cdflomax(x, alpha, m0):
"""
Return CDF for local maxima for a zero-mean Gaussian process
Parameters
----------
x : array-like
evaluation points
alpha, m0 : real scalars
irregularity factor and zero-order spectral moment (variance of the
process), respectively.
Returns
-------
prb : ndarray
distribution function evaluated at x
Notes
-----
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.
Examples
--------
>>> import pylab
>>> import wafo.gaussian as wg
>>> import wafo.spectrum.models as wsm
>>> import wafo.objects as wo
>>> import wafo.stats as ws
>>> S = wsm.Jonswap(Hm0=10).tospecdata();
>>> xs = S.sim(10000)
>>> ts = wo.mat2timeseries(xs)
>>> tp = ts.turning_points()
>>> mM = tp.cycle_pairs()
>>> m0 = S.moment(1)[0]
>>> alpha = S.characteristic('alpha')[0]
>>> x = np.linspace(-10,10,200);
>>> mcdf = ws.edf(mM.data)
h = mcdf.plot(), pylab.plot(x,wg.cdflomax(x,alpha,m0))
See also
--------
spec2mom, spec2bw
"""
c1 = 1.0 / (sqrt(1 - alpha ** 2)) * x / sqrt(m0)
c2 = alpha * c1
return cdfnorm(c1) - alpha * exp(-x ** 2 / 2 / m0) * cdfnorm(c2)
def cdflomax(x, alpha, m0):
"""
Return CDF for local maxima for a zero-mean Gaussian process
Parameters
----------
x : array-like
evaluation points
alpha, m0 : real scalars
irregularity factor and zero-order spectral moment (variance of the
process), respectively.
Returns
-------
prb : ndarray
distribution function evaluated at x
Notes
-----
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
-------
>>> import pylab
>>> import wafo.gaussian as wg
>>> import wafo.spectrum.models as wsm
>>> import wafo.objects as wo
>>> import wafo.stats as ws
>>> S = wsm.Jonswap(Hm0=10).tospecdata();
>>> xs = S.sim(10000)
>>> ts = wo.mat2timeseries(xs)
>>> tp = ts.turning_points()
>>> mM = tp.cycle_pairs()
>>> m0 = S.moment(1)[0]
>>> alpha = S.characteristic('alpha')[0]
>>> x = np.linspace(-10,10,200);
>>> mcdf = ws.edf(mM.data)
h = mcdf.plot(), pylab.plot(x,wg.cdflomax(x,alpha,m0))
See also
--------
spec2mom, spec2bw
"""
c1 = 1.0 / (sqrt(1 - alpha ** 2)) * x / sqrt(m0)
c2 = alpha * c1
return cdfnorm(c1) - alpha * exp(-x ** 2 / 2 / m0) * cdfnorm(c2)