def _odes(self, x,y,args): derivs = [y[1]/x**2] if (x>0) else [0] derivs.append(-9.0*x**2*np.exp(y[0])) u = y[1] phi=y[0] phi0=args[0] eta=args[1] B = args[2] C = 1 - ((1-B)/eta**2) if phi>0: inf = gu(1.5,phi/eta**2)*np.exp(phi/eta**2)*g(1.5) if inf!=inf or inf==np.inf: inf = (phi/eta**2)**0.5 inf2 = gu(1.5,phi0/eta**2)*np.exp(phi0/eta**2)*g(1.5) if inf2!=inf2 or inf2==np.inf: inf2 = (phi0/eta**2)**0.5 dydt =[u/(x**2), -9*(x**2)*(self.mmb*(np.exp(phi)*g(1.5)*gl(1.5,phi)-(2.0/3.0)*phi**(1.5)*B- (4.0/15.0)*C*phi**(2.5)) + self.mmpe* ((1-B)*eta**3 *inf))/ (self.mmb*(np.exp(phi0)*g(1.5)*gl(1.5,phi0)-(2.0/3.0)*phi0**(1.5)*B- (4.0/15.0)*C*phi0**(2.5))+self.mmpe* ((1-B)*eta**3 *inf2))] if phi<0: inf_neg = np.exp(phi/eta**2)*g(1.5) inf2 = gu(1.5,phi0/eta**2)*np.exp(phi0/eta**2)*g(1.5) if inf2!=inf2 or inf2==np.inf: inf2 = (phi0/eta**2)**0.5 dydt =[u/(x**2), -9*(x**2)*(self.mmpe*((1-B)*eta**3 *inf_neg))/(self.mmb*(np.exp(phi0)*g(1.5)*gl(1.5,phi0)-(2.0/3.0)*phi0**(1.5)*B- (4.0/15.0)*C*phi0**(2.5))+self.mmpe* ((1-B)*eta**3 *inf2))] return dydt
def solveNeuralNetworks():
print(" X shape = ", X.shape, "\n Theta1 shape =", Theta1.shape)
z2 = np.matmul(X, Theta1.T)
a2 = g(z2)
print(" a2 shape = ", a2.shape, "\n Theta2 shape =", Theta2.shape)
a2 = np.hstack([np.zeros([5000, 1]), a2])
z3 = np.matmul(a2, Theta2.T)
h = g(z3)
predict = np.argmax(h.T, axis=0)
def anderson_darlin(seq, F, alpha=.05, n=None, verbose=True):
"""
Statistical test of whether a given sample of data is drawn from
a given probability distribution.
Inputs:
- seq: Array of integers
- F: Cumulative distribution function
- alpha: Float desirible level of significance
- n: Integer number of regions (default is None)
- verbose: Boolean; If set to true then print logs
Outputs:
- Boolean; If hypothesis is ejected
"""
seq = np.array(seq)
if n is None:
n = len(seq)
else:
seq = np.random.choice(seq, n)
seq_sorted = sorted(seq)
s = 0
for i in range(1, n + 1):
f = F(seq_sorted[i - 1])
s += (2*i - 1) * np.log(f) / (2*n) + \
(1 - (2*i - 1) / (2*n)) * np.log(1 - f)
s = -n - 2 * s
a2 = 0
for j in range(15):
a2 += (-1)**j * g(j + .5) * (4*j + 1) / g(.5) * g(j + 1) * \
np.exp((4*j + 1)**2 * np.pi**2 / (-8 * s)) * \
integrate.quad(lambda y: np.exp(s / (8 * (y**2 + 1)) - (4*j + 1)**2 \
* np.pi**2 * y**2 / (8 * s)), 0, np.inf)[0]
a2 *= np.sqrt(2 * np.pi) / s
p = 1 - a2
is_rejected = p <= alpha
if verbose:
print(f'Significance level S = {s}')
print(f'a2 = {a2}')
print(f'p = {p}')
return is_rejected
def costFunction(features, target, weights, reg_parameter):
hypothesis = np.array(g(features.__matmul__(weights)))
I = np.ones((m, 1))
cost = (-1) * (target.T.__matmul__(np.log(hypothesis)) +
(I - target).T.__matmul__(np.log(I - hypothesis))) / m
reg_func = (reg_parameter / (2 * m)) * (sum(weights**2) - weights[0][0]**2)
return cost + reg_func
def costFunction(features, target, weights, reg_parameter):
hypothesis = np.array(g(np.matmul(features, weights)))
I = np.ones(target.shape)
a = np.matmul(target.T, np.log(hypothesis))
b = np.matmul((I - target).T, np.log(I - hypothesis))
cost = (-1) * (sum(a) + sum(b)) / m
reg_func = (reg_parameter / (2 * m)) * (sum(weights**2) - weights[0][0]**2)
a = max(cost)
return a + max(reg_func)
def trainGradientDescent(features,target,num_steps,learning_rate,add_intercept=False): if add_intercept: intercept=np.ones((features.shape[0],1)) features =np.hstack([intercept,features]) weights = np.zeros((features.shape[1],1)) for step in range(int(num_steps)): score = np.matmul(features,weights) hypothesis = g(score) output_error_signal= target - hypothesis gradient = np.matmul(features.T,output_error_signal) weights += (learning_rate)*gradient return weights
def calcGradient(features,
target,
weights,
reg_parameter,
add_intercept=False):
if add_intercept:
intercept = np.ones((features.shape[0], 1))
features = np.hstack([intercept, features])
score = np.matmul(features, weights)
hypothesis = g(score)
output_error_signal = hypothesis - target
gradient = np.matmul(features.T, output_error_signal)
weights_zero = copy.deepcopy(weights)
weights_zero[0][0] = 0
reg_gradient = gradient / m + (reg_parameter / m) * weights_zero
return reg_gradient
def h(x):
shape = x.shape
d = shape[1] * 1.
N = shape[0]
e = 0.00001
rho = np.zeros((N, N))
for i in range(N):
for j in range(N):
if i == j:
continue
else:
rho[i, j] = np.linalg.norm(x[i, :] - x[j, :])
rho = np.sort(rho, axis=0)
rho_min = rho[1, :]
h = d * np.average(rho)
h = h + np.log((N - 1) * np.pi**(d / 2) / g(1 + d / 2))
h = h + 0.577
return h
def normalGamma(m, l, a, b):
C = (b**a) * np.sqrt(l) / (g(a) * np.sqrt(2 * np.pi))
return lambda x, tau: C * tau**(a - 0.5) * np.exp(-b * tau - 0.5 * l * tau
* (x - m)**2)
def gamma(a, b):
C = b**a / g(a)
return lambda tau: C * tau**(a - 1) * np.exp(-b * tau)
def lnlike_ian(theta, data):
#print theta
W0, eta, Mt, rh, B = theta
#print theta
rho_dat = data[0]
sigma_dat = data[1]
#discrete_dat = data[2]
#try:
eta = eta
G = np.pi * 4.0 / 9.0
init = W0, 0.0
C = 1 - ((1 - B) / eta**2)
params = [W0, eta, Mt, rh]
if restart == 0:
m = SPES(1000, W0, eta, B, 1, 0, 1)
if restart == 1:
m = SPES(1000, W0, eta, B, 1, 1, 1)
r = m.r
phi = m.phi
minim = 0
rt = m.rt
dispersion = m.dispersion
density = m.density
count = 0
Mss = []
Mf = (4 * np.pi * np.trapz((r**2) * density, r))
Mcs = []
for l in range(len(phi) - 1):
Mc = (4 * np.pi * np.trapz(
(r[0:l + 1]**2) * density[0:l + 1], r[0:l + 1]))
if (Mc > 0.5 * Mf) & (count == 0):
count = 1
rs = rh / r[l]
break
Ms = Mt / (Mf)
vs = np.sqrt((Ms / rs) * (np.pi * 4.0 / 9.0))
density_all = np.array(density) * (Ms) / (rs**3)
disp_all = np.array(dispersion) * (vs)
r_us = r
r = r * rs
prob = 0
inf1 = (np.exp(W0 / (eta**2)) * gu(1.5, W0 / (eta**2))) * g(1.5)
A = (Ms / ((vs**3) * (rs**3))) * 1. / (4 * np.pi * np.sqrt(2) *
(np.exp(W0) * g(1.5) * gl(1.5, W0) -
(2.0 / 3.0) * W0**(1.5) * B) -
(4.0 / 15.0) * C * W0**(2.5) +
((1 - B) * eta**3 * inf1))
values = np.loadtxt('data/equalmass_32')
sigma_mod_interp = np.interp(sigma_dat[0], r, disp_all, right=-np.inf)
rho_mod_interp = np.interp(rho_dat[0], r, density_all, right=-np.inf)
loglike = 0
for i in range(len(rho_dat[0])):
loglike -= (rho_mod_interp[i] -
rho_dat[1][i])**2 / (2 * (rho_dat[2][i]**2))
loglike -= log(sqrt(2 * pi * rho_dat[2][i]**2))
# Likelihood: velocity dispersion
for i in range(len(sigma_dat[0])):
loglike -= (sigma_mod_interp[i] -
sigma_dat[1][i])**2 / (2 * (sigma_dat[2][i]**2))
loglike -= log(sqrt(2 * pi * sigma_dat[2][i]**2))
if numpy.isnan(loglike):
print "nan"
loglike = -np.inf
return (loglike)
def costFunction(features,target,weights): prediction =np.array(g(features.__matmul__(weights))) I=np.ones((m,1),order='F') cost_function=(-1)*(target.T.__matmul__(np.log(prediction)) + (I-target).T.__matmul__(np.log(I-prediction)))/m return cost_function
def __init__(self, rmax,phi0,eta,B,delta,res,mm):
self.r = np.array([0])
self.C = 1 - ((1-B)/eta**2)
self.mmb=1
self.mmpe=mm
#print self.C
#delta=rhm
# The 2 variables in the Poisson equation: phi, U, where U = -GM(<r)
self._y = [phi0, 0]
self.phi0 = phi0
init = [phi0,0]
self.G = 9/(4*pi)
# Solve
sol = ode(self._odes)
#sol.set_integrator('dopri5',nsteps=1e6,max_step=0.1)
if res == 0:
sol.set_integrator('dopri5')#,nsteps=1e6,atol=1e-10,rtol=1e-10)
if res==1:
sol.set_integrator('dopri5',max_step=0.1,nsteps=1e6,atol=1e-10,rtol=1e-10)
sol.set_f_params([phi0,eta,B])
sol.set_solout(self._logcheck)
sol.set_initial_value(numpy.array([phi0,0]),0.0001)
sol.integrate(rmax)
# Save phi, rho and M from Poisson solver
self.phi = self._y[0]
#print self.phi
phi_in = self.phi[self.phi>0]
phi_out = self.phi[self.phi<0]
###Checks for infinitys in density and dispersion
inf1 = (np.exp(init[0]/(eta**2))*gu(1.5,init[0]/(eta**2)))*g(1.5)
if (inf1==np.inf) or (inf1!=inf1):
inf1=(init[0]/eta**2)**0.5
inf = np.exp(phi_in/(eta**2))*gu(1.5,phi_in/(eta**2))*g(1.5)
inf2 =np.exp(phi_in/(eta**2))*gu(2.5,phi_in/(eta**2))*g(2.5)
for i in range(len(inf)):
if (inf[i]==np.inf) or (inf[i]!=inf[i]):
inf[i] = (phi_in[i]/eta**2)**0.5
if (inf2[i]==np.inf) or (inf2[i]!=inf2[i]):
inf2[i] = (phi_in[i]/eta**2)**1.5
###Density and Velocity dispersion for r<r_t
self.rho0 = ((np.exp(init[0])*g(1.5)*gl(1.5,init[0])-(2.0/3.0)*init[0]**(1.5)*B) - (4.0/15.0)*self.C*init[0]**(2.5)+ ((1-B)*eta**3 *inf1))#*kv(0.25,eta**2/init[0])))
self.rho_b = self.mmb* (np.exp(phi_in)*g(1.5)*gl(1.5,phi_in)-(2.0/3.0)*phi_in**(1.5)*B - (4.0/15.0)*self.C*phi_in**(2.5))
self.rho_pe = self.mmpe* ((1-B)*eta**3 *inf)#*kv(0.25,eta**2/phi_in))
self.rho_in = self.rho_b + self.rho_pe
self.pressure_in = (((np.exp(phi_in)*g(2.5)*gl(2.5,(phi_in))-0.4*phi_in**(2.5)*B) - (4.0/35.0)*self.C*phi_in**(3.5)+ ((1-B)*eta**5 *inf2*kv(0.25,eta**2/phi_in))))
self.pressure_b = (((np.exp(phi_in)*g(2.5)*gl(2.5,(phi_in))-0.4*phi_in**(2.5)*B) - (4.0/35.0)*self.C*phi_in**(3.5)))
self.pressure_pe=((1-B)*eta**5 *inf2)#*kv(0.25,eta**2/phi_in))
self.sigma_b = np.sqrt( 2*self.pressure_b/self.rho_b)
self.sigma_pe = np.sqrt( 2*self.pressure_pe/self.rho_pe)
sigma_in = np.sqrt( 2*self.pressure_in/self.rho_in)
###Density and velocity dispersion for r>r_t
inf_neg = np.exp(phi_out/eta**2)*g(1.5)
inf2_neg = np.exp(phi_out/(eta**2))*g(2.5)
self.rho_out = self.mmpe*(((1-B)*eta**3 *inf_neg))#/self.rho0
sigma_out = np.sqrt( 2*eta**2*(inf2_neg/inf_neg))
###Total rho_hat and sigma
#print len(self.rho_in),len(self.rho_out)
self.rhohat = np.array(np.concatenate([self.rho_in,self.rho_out]))/np.array(self.rho0)
self.sigma = np.concatenate([sigma_in,sigma_out])
###dM/dE & f(E)
self.phii = np.linspace(0,-phi0,100)
self.N1=[0]*len(phi_in)
self.N2=[0]*len(self.phii)
self.fe = [0]*len(phi_in)
self.fe_1 = [0]*len(phi_in)
self.fe_2 = [0]*len(phi_in)
A = 1./(4*np.pi*np.sqrt(2)*(np.exp(init[0])*g(1.5)*gl(1.5,init[0])-(2.0/3.0)*init[0]**(1.5)*B) - (4.0/15.0)*self.C*init[0]**(2.5) + ((1-B)*eta**3 *inf1))
self.fe2=[0]*len(self.phii)
self.fe2_1=[0]*len(self.phii)
self.fe2_2=[0]*len(self.phii)
for i in range(len(phi_in)):
self.N1[i] = ((4*np.pi)*(np.exp(phi_in[i])-B-(self.C*phi_in[i]))*simps((self.r[0:(i+1)]**2 * np.sqrt((-phi_in[i] + phi_in[0:(i+1)]))),self.r[0:(i+1)])/self.rho0)
self.fe[i] = A*(np.exp(phi_in[i])-B-(self.C*phi_in[i]))
self.fe_1[i] = A*(np.exp(phi_in[i])-self.C)
self.fe_2[i] = A*(np.exp(phi_in[i]))
for i in range(len(self.phii)):
self.N2[i] = ((1-B) * (4*np.pi/np.sqrt(2))*np.exp(self.phii[i]/eta**2)*simps(self.r[self.phi>0]**2 * np.sqrt(2*(-self.phii[i] + phi_in)),self.r[self.phi>0])/self.rho0)
self.fe2[i] = A*(1-B)*(np.exp(self.phii[i]/eta**2))
self.fe2_1[i] = A*(1-B)*(np.exp(self.phii[i]/eta**2))/eta**2
self.fe2_2[i] = A*(1-B)*(np.exp(self.phii[i]/eta**2))/eta**4
###Scale values
count = 0
Mss = []
self.Mf = (4*np.pi*simps((self.r[0:len(phi_in)]**2)*self.rhohat[0:len(phi_in)],self.r[0:len(phi_in)]))
for i in range(len(phi_in)-1):
Mc =(4*np.pi*simps((self.r[0:i+1]**2)*self.rhohat[0:i+1],self.r[0:i+1]))
if (Mc > 0.5*self.Mf) & (count==0):
count = 1
self.rhm = self.r[i]
break
self.rt = np.interp(0,-self.phi,self.r)
self.Mt = (4*np.pi*simps((self.r[self.r<self.rt]**2)*self.rhohat[self.r<self.rt],self.r[self.r<self.rt]))
#self.rs = (1.1*rlb)/(delta*self.rt)
#self.Ms = Mt/(Mf)
#self.vs = np.sqrt( (self.Ms/self.rs)*(0.0043*(np.pi*4.0/9.0)))
###Scaled Parameters
#self.rhohat_s = self.rhohat*(self.Ms)/(self.rs**3)
#self.rhob_s = (self.rho_b/self.rho0)*(self.Ms)/(self.rs**3)
#self.rhope_s = (self.rho_pe/self.rho0)*(self.Ms)/(self.rs**3)
self.Mb =(4*np.pi*simps((self.r[self.phi>0]**2)*self.rho_b/self.rho0,self.r[self.phi>0]))#-M_bp# (simps(N,-phi))
self.Mpe = 4*np.pi*simps((self.r[self.phi>0]**2)*self.rho_pe/self.rho0,self.r[self.phi>0])
self.Mfrac = self.Mpe/(self.Mb+self.Mpe)
#self.sigma_s = self.sigma*(self.vs)
#self.N1_s = np.array(self.N1)*(self.Ms/(self.vs**2))
#self.N2_s = np.array(self.N2)*(self.Ms/(self.vs**2))
#self.phi_s = self.phi*self.vs**2
#self.phii_s = self.phii*self.vs**2
#self.r_s = self.r*self.rs
#self.rt = self.rt*self.rs
###Surface Density and projected velocity dispersion
R = self.r[self.r<delta*self.rt]
self.density = self.rhohat[self.r<delta*self.rt]
self.dispersion = self.sigma[self.r<delta*self.rt]
self.surface_density = np.zeros(len(self.density)-1)
self.surface_dispersion = np.zeros(len(self.density)-1)
self.rall = self.r
self.phi = self.phi[self.r<delta*self.rt]
self.r = self.r[self.r<delta*self.rt]
#self.sigma_b = self.sigma_b[self.r<delta*self.rt]
#self.rho_b = self.rho_b[self.r<delta*self.rt]
#self.sigma_pe = self.sigma_pe[self.r<delta*self.rt]
#self.rho_pe = self.rho_pe[self.r<delta*self.rt]
for i in range(len(self.r)-1):
c = (self.r >= R[i])
r_2 = self.r[c]
z = sqrt(abs(r_2**2 - R[i]**2)) # avoid small neg. values
self.surface_density[i] = 2.0*abs(simps(self.density[c], x=z))
self.surface_dispersion[i] = np.sqrt((abs(2.0*(simps(self.density[c]*(1.0/3.0)*self.dispersion[c]**2,x=z))))/self.surface_density[i])
#print self.surface_dispersion
#self.surface_density = self.surface_density*(self.Ms)/(self.rs**2)
#self.surface_dispersion = self.surface_dispersion*(self.vs)
self.R = self.r[0:len(self.r)-1]
def acceptF(lambda_all,adv,p):
return sum(log(g(1.+adv*lambda_all)/g(adv*lambda_all)*(p**(lambda_all*adv))))