def cvar_beta_variable(rv, beta):
"""
cvar of Beta variable on [0,1]
"""
qq = rv.ppf(beta)
scales, shapes = get_distribution_info(rv)[1:]
qq = (qq - scales["loc"]) / scales["scale"]
g1, g2 = shapes["a"], shapes["b"]
cvar01 = ((1 - betainc(1 + g1, g2, qq)) * gamma_fn(1 + g1) * gamma_fn(g2) /
gamma_fn(1 + g1 + g2)) / (beta_fn(g1, g2) * (1 - beta))
return cvar01 * scales["scale"] + scales["loc"]
def _gmw_k_constants(gamma, beta, k, norm='bandpass', dtype='float64'):
"""Laguerre polynomial constants & `coeff` term.
Higher-order GMWs are coded such that constants are pre-computed and reused
for any `w` input, since they remain fixed for said order.
"""
r = (2 * beta + 1) / gamma
c = r - 1
# compute `coeff`
if norm == 'bandpass':
coeff = np.sqrt(np.exp(gammaln_fn(r) + gammaln_fn(k + 1) -
gammaln_fn(k + r)))
elif norm == 'energy':
coeff = np.sqrt(2*pi * gamma * (2**r) *
np.exp(gammaln_fn(k + 1) - gammaln_fn(k + r)))
# compute Laguerre polynomial constants
L_consts = np.zeros(k + 1, dtype=dtype)
for m in range(k + 1):
fact = np.exp(gammaln_fn(k + c + 1) - gammaln_fn(c + m + 1) -
gammaln_fn(k - m + 1))
L_consts[m] = (-1)**m * fact / gamma_fn(m + 1)
k_consts = L_consts * coeff
if norm == 'bandpass':
k_consts *= 2
k_consts = k_consts.astype(dtype)
return k_consts
def morsefreq(gamma, beta, n_out=1):
"""Frequency measures for GMWs (with F. Rekibi).
`n_out` controls how many parameters are computed and returned, in the
following order: `wm, we, wi, cwi`, where:
wm: modal / peak frequency
we: energy frequency
wi: instantaneous frequency at time-domain wavelet's center
cwi: curvature of instantaneous frequency at time-domain wavelet's center
All frequency quantities are *radian*, opposed to linear cyclic (i.e. `w`
in `w = 2*pi*f`).
For BETA=0, the "wavelet" becomes an analytic lowpass filter, and `wm`
is not defined in the usual way. Instead, `wm` is defined as the point
at which the filter has decayed to one-half of its peak power.
# References
[1] Higher-Order Properties of Analytic Wavelets.
J. M. Lilly, S. C. Olhede. 2009.
https://sci-hub.st/10.1109/TSP.2008.2007607
[2] (c) Lilly, J. M. (2021), jLab: A data analysis package for Matlab,
v1.6.9, http://www.jmlilly.net/jmlsoft.html
https://github.com/jonathanlilly/jLab/blob/master/jWavelet/morsefreq.m
"""
wm = (beta / gamma)**(1 / gamma)
if n_out > 1:
we = (1 / 2**(1 / gamma)) * (gamma_fn((2*beta + 2) / gamma) /
gamma_fn((2*beta + 1) / gamma))
if n_out > 2:
wi = (gamma_fn((beta + 2) / gamma) /
gamma_fn((beta + 1) / gamma))
if n_out > 3:
k2 = _morsemom(2, gamma, beta, n_out=3)[-1]
k3 = _morsemom(3, gamma, beta, n_out=3)[-1]
cwi = -(k3 / k2**1.5)
if n_out == 1:
return wm
elif n_out == 2:
return wm, we
elif n_out == 3:
return wm, we, wi
return wm, we, wi, cwi
def find_radius(self):
n = len(self.states)
unit_ball_volume = np.sqrt(
np.pi)**self.ss.dim / gamma_fn(self.ss.dim / 2.0 + 1.0)
x_free_volume = self.ss.volume
gamma = (2*(1.0 + 1.0/self.ss.dim) * \
(x_free_volume/unit_ball_volume))**(1.0/self.ss.dim)
ball_radius = min(
gamma * ((np.log(n + 1) / (n + 1))**(1.0 / self.ss.dim)),
self.d_steer)
return ball_radius
def make_shower_ldf(energy, theta=0, phi=0, s=1):
from scipy.special import gamma as gamma_fn
# critical energy in air, see EAS (4.17) and discussion
# on p.154.
E_critical = 84e6 # eV
# shower size is proportial to energy,
# see EAS (4.88)
N = energy/E_critical
# HACK:
# for muons, the normalization is about 3 orders lower
# than for photons. would be better to get a muon-specific
# LDF.
N *= 1e-3
eta = eta_fn(theta, N)
alpha = 2-s
# molliere radius, EAS p. 388
rM = 100 # m
# calculate the yucky gamma function term up front:
gamma_factor = gamma_fn(eta-alpha) / (gamma_fn(2-alpha)*gamma_fn(eta-2))
# unit vector pointing in the direction of the shower core:
n_hat = np.array([np.cos(phi)*np.sin(theta), np.sin(phi)*np.sin(theta), np.cos(theta)])
def ldf(points):
# promote the 2D vector points to 3d vector points
pts = np.hstack([points, np.zeros((len(points),1))])
# vector distance from point p to line
d = np.outer((n_hat.dot(pts.T)), n_hat) - pts
r = np.sqrt(np.sum(d**2, axis=1))
return N / (2*np.pi*rM**2) * (r/rM)**(-alpha) * (1 + r/rM)**(-(eta-alpha)) * gamma_factor
return ldf
def laguerre(x, k, c):
"""Generalized Laguerre polynomials. See `help(_gmw.morsewave)`.
LAGUERRE is used in the computation of the generalized Morse
wavelets and uses the expression given by Olhede and Walden (2002),
"Generalized Morse Wavelets", Section III D.
"""
x = np.atleast_1d(np.asarray(x).squeeze())
assert x.ndim == 1
y = np.zeros(x.shape)
for m in range(k + 1):
# Log of gamma function much better ... trick from Maltab's ``beta''
fact = np.exp(gammaln_fn(k + c + 1) - gammaln_fn(c + m + 1) -
gammaln_fn(k - m + 1))
y += (-1)**m * fact * x**m / gamma_fn(m + 1)
return y
def gmw_l2(gamma=3., beta=60., centered_scale=False, dtype='float64'):
"""Generalized Morse Wavelets, first order, L2(energy)-normalized.
See `help(_gmw.gmw)`.
"""
_check_args(gamma=gamma, beta=beta, allow_zerobeta=False)
wc = morsefreq(gamma, beta)
r = (2*beta + 1) / gamma
rgamma = gamma_fn(r)
(gamma, beta, wc, r, rgamma
) = _process_params_dtype(gamma, beta, wc, r, rgamma, dtype=dtype)
fn = _gmw_l2_gpu if USE_GPU() else (_gmw_l2_par if IS_PARALLEL() else _gmw_l2)
if centered_scale:
return lambda w: fn(S.atleast_1d(w * wc, dtype), gamma, beta, wc,
r, rgamma)
else:
return lambda w: fn(S.atleast_1d(w, dtype), gamma, beta, wc, r, rgamma)
def morsef(gamma, beta):
# normalized first frequency-domain moment "f_{beta, gamma}" of the
# first-order GMW
return (1 / (2*pi * gamma)) * gamma_fn((beta + 1) / gamma)
def log_likelihood(X, beta, gamma):
n = X.shape[0]
s = beta * n * np.log(gamma) + (beta - 1) * np.log(X).sum() - gamma * X.sum() - n * np.log(gamma_fn(beta))
return np.exp(s)
def likelihood(X, beta, gamma):
return ((gamma ** beta) ** (X.shape[0])) * np.exp(np.sum(np.log((X ** (beta - 1)) * np.exp(-gamma * X)))) / (gamma_fn(beta) ** (X.shape[0]))
def compute_A0(p0,q0):
inv_A0 = (2.**(0.5*(-1. - 2.*self.p0 + self.q0))*(gamma_fn(-self.p0 + self.q0/2.) + 2**self.p0*gamma_fn(self.q0/2.)))/np.sqrt(np.pi)
return 1./inv_A0
def _load_mass_function_parameters(self,mass_dict):
"""Load internal mass function parameters.
Args:
mass_function (str): Mass function name (see class description for options).
mass_dict (dict): Dictionary of additional parameters.
"""
if self.mass_function_name=='Sheth-Tormen':
# Load mass function parameters and normalization
self.p_ST = 0.3
self.a_ST = 0.707
self.A_ST = (1.+2.**(-self.p_ST)*gamma_fn(0.5-self.p_ST)/np.sqrt(np.pi))**(-1.)*np.sqrt(2.*self.a_ST/np.pi)
# Compute the spherical collapse threshold of Nakamura-Suto, 1997.
Om_mz = self.cosmology._Omega_m()
dc0 = (3./20.)*pow(12.*np.pi,2./3.);
self.delta_c = dc0*(1.+0.012299*np.log10(Om_mz));
elif self.mass_function_name=='Tinker':
self.delta_c = 1.686 # critical density for collapse
# MASS FUNCTION PARAMETERS
## Compute model parameters from interpolation given odelta value
odelta = mass_dict['tinker_overdensity']
alpha=np.asarray([0.368,0.363,0.385])
beta0=np.asarray([0.589,0.585,0.544])
gamma0=np.asarray([0.864,0.922,0.987])
phi0=np.asarray([-0.729,-0.789,-0.910])
eta0=np.asarray([-0.243,-0.261,-0.261])
odeltas=np.asarray([200,300,400])
self.alpha=interp1d(odeltas,alpha)(odelta)
beta0=interp1d(odeltas,beta0)(odelta)
gamma0=interp1d(odeltas,gamma0)(odelta)
phi0=interp1d(odeltas,phi0)(odelta)
eta0=interp1d(odeltas,eta0)(odelta)
self.beta = beta0*self.a**-0.2
self.phi = phi0*self.a**0.08
self.eta = eta0*self.a**-0.27
self.gamma = gamma0*self.a**0.01
# BIAS FUNCTION PARAMETERS
y = np.log10(odelta);
self.fit_A = 1.0 + 0.24*y*np.exp(-np.power(4./y,4.));
self.fit_a = 0.44*y-0.88;
self.fit_B = 0.183;
self.fit_b = 1.5;
self.fit_C = 0.019+0.107*y+0.19*np.exp(-np.power(4./y,4.));
self.fit_c = 2.4;
elif self.mass_function_name=='Crocce':
self.delta_c = 1.686 # critical density for collapse
if self.cosmology.name=='Quijote':
if self.verb: print('Using fitted parameters for the Crocce mass function from Quijote simulations ')
# Optimal values for the Quijote simulations
self.pA = 0.729
self.pa = 2.355
self.pb = 0.423
self.pc = 1.318
else:
# Optimal values from original simulations
self.pA = 0.58*self.a**0.13
self.pa = 1.37*self.a**0.15
self.pb = 0.3*self.a**0.084
self.pc = 1.036*self.a**0.024
elif self.mass_function_name=='Bhattacharya':
self.delta_c = 1.686 # critical density for collapse
def compute_A0(p0,q0):
inv_A0 = (2.**(0.5*(-1. - 2.*self.p0 + self.q0))*(gamma_fn(-self.p0 + self.q0/2.) + 2**self.p0*gamma_fn(self.q0/2.)))/np.sqrt(np.pi)
return 1./inv_A0
if self.cosmology.name=='Quijote':
if self.verb: print('Using fitted parameters for the Bhattacharya mass function from Quijote simulations ')
# Optimal values for the Quijote simulations
self.a0 = 0.77403116
self.p0 = 0.63685683
self.q0 = 1.66263337
self.A0 = compute_A0(self.p0,self.q0)
elif self.cosmology.name=='Abacus':
if self.verb: print('Using fitted parameters for the Bhattacharya mass function from Abacus simulations ')
# Optimal values for the Quijote simulations
self.a0 = 0.86648878
self.p0 = 1.30206972
self.q0 = 1.97133804
self.A0 = 0.35087244
else:
# Optimal values from original paper
self.a0 = 0.788
self.p0 = 0.807
self.q0 = 1.795
self.A0 = compute_A0(self.p0,self.q0)
else:
raise NameError('Mass function %s not currently implemented!'%self.mass_function_name)