def mean_from_output(output):
"""Given output of the form [params,ll], return mean parameter
vector"""
param_vectors,lls = transpose(output)
z = sum(mpmath.exp(ll) for ll in lls)
return map(sum,transpose([[(mpmath.exp(ll)/z)*p for p in param_vector]
for param_vector,ll in output]))
def integrate( self, a, b ):
halfk = self.halfforceconstant
x0 = self.reference
beta = self.beta
avalue = exp( -beta*halfk*(a-x0)**2 )
bvalue = exp( -beta*halfk*(b-x0)**2 )
return 0.5*(b-a)*( avalue + bvalue )
def sigma(self,z): # A+S 18.10. from mpmath import pi, jtheta, exp, mpc, sqrt, sin Delta = self.Delta e1, _, _ = self.__roots om = self.__periods[0] / 2 omp = self.__periods[1] / 2 if self.__ng3: z = mpc(0,1) * z if Delta > 0: tau = omp / om q = (exp(mpc(0,1) * pi() * tau)).real eta = -(pi()**2 * jtheta(n=1,z=0,q=q,derivative=3)) / (12 * om * jtheta(n=1,z=0,q=q,derivative=1)) v = (pi() * z) / (2 * om) retval = (2 * om) / pi() * exp((eta * z**2)/(2 * om)) * jtheta(n=1,z=v,q=q)/jtheta(n=1,z=0,q=q,derivative=1) elif Delta < 0: om2 = om + omp om2p = omp - om tau2 = om2p / (2 * om2) q = mpc(0,(mpc(0,1) * exp(mpc(0,1) * pi() * tau2)).imag) eta2 = -(pi()**2 * jtheta(n=1,z=0,q=q,derivative=3)) / (12 * om2 * jtheta(n=1,z=0,q=q,derivative=1)) v = (pi() * z) / (2 * om2) retval = (2 * om2) / pi() * exp((eta2 * z**2)/(2 * om2)) * jtheta(n=1,z=v,q=q)/jtheta(n=1,z=0,q=q,derivative=1) else: g2, g3 = self.__invariants if g2 == 0 and g3 == 0: retval = z else: c = e1 / 2 A = sqrt(3 * c) retval = (1 / A) * sin(A*z) * exp((c*z**2) / 2) if self.__ng3: return mpc(0,-1) * retval else: return retval
def fermihalf(x, sgn):
""" Series approximation to the F_{1/2}(x) or F_{-1/2}(x)
Fermi-Dirac integral """
f = lambda k: mp.sqrt(x ** 2 + np.pi ** 2 * (2 * k - 1) ** 2)
# if x < -100:
# return 0.0
if x < -9 or True:
if sgn > 0:
return mp.exp(x)
else:
return mp.exp(x)
if sgn > 0: # F_{1/2}(x)
a = np.array((1.0 / 770751818298, -1.0 / 3574503105, -13.0 / 184757992,
85.0 / 3603084, 3923.0 / 220484, 74141.0 / 8289, -5990294.0 / 7995))
g = lambda k: mp.sqrt(f(k) - x)
else: # F_{-1/2}(x)
a = np.array((-1.0 / 128458636383, -1.0 / 714900621, -1.0 / 3553038,
27.0 / 381503, 3923.0 / 110242, 8220.0 / 919))
g = lambda k: -0.5 * mp.sqrt(f(k) - x) / f(k)
F = np.polyval(a, x) + 2 * np.sqrt(2 * np.pi) * sum(map(g, range(1, 21)))
return F
def dedekind(tau, floatpre):
"""
Algorithm 22 (Dedekind eta)
Input : tau in the upper half-plane, k in N
Output : eta(tau)
"""
a = 2 * mpmath.pi / mpmath.mpf(24)
b = mpmath.exp(mpmath.mpc(0, a))
p = 1
m = 0
while m <= 0.999:
n = nearest_integer(tau.real)
if n != 0:
tau -= n
p *= b ** n
m = tau.real * tau.real + tau.imag * tau.imag
if m <= 0.999:
ro = mpmath.sqrt(mpmath.power(tau, -1) * 1j)
if ro.real < 0:
ro = -ro
p = p * ro
tau = (-p.real + p.imag * 1j) / m
q1 = mpmath.exp(a * tau * 1j)
q = q1 ** 24
s = 1
qs = mpmath.mpc(1, 0)
qn = 1
des = mpmath.mpf(10) ** (-floatpre)
while abs(qs) > des:
t = -q * qn * qn * qs
qn = qn * q
qs = qn * t
s += t + qs
return p * q1 * s
def genenergies(fnR,fnQ,seqsR,seqsQ,gamma,sQ,sR,R0): #Parses seqs and model type then calculates and returns energies R is transcription factor, Q is RNAP
ematR = np.genfromtxt(fnR,skiprows=1)
ematQ = np.genfromtxt(fnQ,skiprows=1)
fR = open(fnR)
fQ = open(fnQ)
mattype = fR.read()[:6] #mattype must be the same
#mattypeQ = fQ.read()[:6]
energies = np.zeros(len(seqsQ))
N = len(seqsQ)
mut_region_lengthQ = len(seqsQ[0])
mut_region_lengthR = len(seqsR[0])
if mattype == '1Point':
for i,s in enumerate(seqsR):
seq_matR = seq2mat(s)
seq_matQ = seq2mat(seqsQ[i])
RNAP = (seq_matQ*ematQ).sum()*sQ
TF = (seq_matR*ematR).sum()*sR + R0
energies[i] = -RNAP + mp.log(1 + mp.exp(-TF - gamma)) - mp.log(1 + mp.exp(-TF))
'''
elif mattype == '2Point':
for i,s in enumerate(seqs):
seq_mat = np.zeros(round(sp.misc.comb(mut_region_length,2))*16)
seq_mat[seq2mat2(s)] = 1
energies[i] = (seq_mat*(emat.ravel())).sum()
elif mattype == '3Point':
for i,s in enumerate(seqs):
seq_mat = np.zeros(round(sp.misc.comb(mut_region_length,3))*64)
seq_mat[seq2mat3(s)] = 1
energies[i] = (seq_mat*(emat.ravel())).sum()
'''
return energies
def test_stoch_eig_high_prec():
n = 1e-100
with mp.workdps(100):
P = mp.matrix([[1-3*(mp.exp(n)-1), 3*(mp.exp(n)-1)],
[mp.exp(n)-1 , 1-(mp.exp(n)-1)]])
run_stoch_eig(P, verbose=VERBOSE)
def dfdy (y,x,b,c):
global FT,XO,XT,XF
ft=FT
v=x[0]
i=y[0]
iss=IS=b[0]
n=N=b[1]
ikf=IKF=b[2]
isr=ISR=b[3]
nr=NR=b[4]
vj=VJ=b[5]
m=M=b[6]
rs=RS=b[7]
#fh = iss**mpm.mpf(2)*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1))))*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1))*mpm.exp((-i*rs + v)/(ft*n))/(mpm.mpf(2)*ft*n*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1)))) - iss*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1))))*mpm.exp((-i*rs + v)/(ft*n))/(ft*n)
#sh = isr*m*rs*(mpm.mpf(1) - (-i*rs + v)/vj)*((mpm.mpf(1) - (-i*rs + v)/vj)**mpm.mpf(2) + mpm.mpf('0.005'))**(m/mpm.mpf(2))*(mpm.exp((-i*rs + v)/(ft*nr)) - mpm.mpf(1))/(vj*((mpm.mpf(1) - (-i*rs + v)/vj)**mpm.mpf(2) + mpm.mpf('0.005'))) - isr*rs*((mpm.mpf(1) - (-i*rs + v)/vj)**mpm.mpf(2) + mpm.mpf('0.005'))**(m/mpm.mpf(2))*mpm.exp((-i*rs + v)/(ft*nr))/(ft*nr)
fh = iss**XT*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(mpm.exp((-i*rs + v)/(ft*n)) - XO)*mpm.exp((-i*rs + v)/(ft*n))/(XT*ft*n*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))) - iss*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*mpm.exp((-i*rs + v)/(ft*n))/(ft*n)
sh = isr*m*rs*(XO - (-i*rs + v)/vj)*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*(mpm.exp((-i*rs + v)/(ft*nr)) - XO)/(vj*((XO - (-i*rs + v)/vj)**XT + XF)) - isr*rs*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*mpm.exp((-i*rs + v)/(ft*nr))/(ft*nr)
return mpm.matrix ([[fh+sh]])
def test_gth_solve_high_prec():
n = 1e-100
with mp.workdps(100):
P = mp.matrix([[-3*(mp.exp(n)-1), 3*(mp.exp(n)-1)],
[mp.exp(n)-1 , -(mp.exp(n)-1) ]])
run_gth_solve(P, verbose=VERBOSE)
def bimax_integrand(self, z, wc, l, n, t):
"""
Integrand of electron-noise integral.
"""
return f1(wc*l/z/mp.sqrt(2)) * z * \
(mp.exp(-z**2) + n/mp.sqrt(t)*mp.exp(-z**2 / t)) / \
(mp.fabs(BiMax.d_l(z, wc, n, t))**2 * wc**2)
def DedekindEtaA4(tau):
''' Compute the derivative of the Dedekind Eta function for imaginary argument tau.
Numerically. '''
try:
import mpmath as mp
mpmath_loaded = True
except ImportError:
mpmath_loaded = False
return mp.cbrt(0.5*mp.jtheta(2,0,mp.exp(-mp.pi*tau))*mp.jtheta(3,0,mp.exp(-mp.pi*tau))*mp.jtheta(4,0,mp.exp(-mp.pi*tau)))
def fl1(x0, seta2, seff2):
coeff = 1./np.sqrt(np.pi)
xm = x0/np.sqrt(2.*seta2)
tau = lamda1*seff2/np.sqrt(2.*seta2)
intgr1 = coeff*exp(-(tau-xm)**2) -(tau-xm)*erfc(tau-xm)
intgr2 = coeff*exp(-(tau+xm)**2) -(tau+xm)*erfc(tau+xm)
return .5*(np.sqrt(2.*seta2)/(1.+lamda2*seff2))*(intgr1+intgr2)
def DedekindEtaA2(tau):
''' Compute the derivative of the Dedekind Eta function for imaginary argument tau.
Numerically. '''
try:
import mpmath as mp
mpmath_loaded = True
except ImportError:
mpmath_loaded = False
return mp.exp(-mp.pi/12.0)*mp.jtheta(3,mp.pi*(mp.j*tau+1.0)/2.0,mp.exp(-3.0*mp.pi))
def fl2(x0, seta2, seff2):
coeff = 1./np.sqrt(np.pi)
xm = x0/np.sqrt(2.*seta2)
tau = lamda1*seff2/np.sqrt(2.*seta2)
intgr1 = -coeff*(tau-xm)*exp(-(tau-xm)**2) +(.5+(tau-xm)**2)*erfc(tau-xm)
intgr2 = -coeff*(tau+xm)*exp(-(tau+xm)**2) +(.5+(tau+xm)**2)*erfc(tau+xm)
return (seta2/(1.+lamda2*seff2)**2)*(intgr1+intgr2)
def temp_t(t, x=mpf(1), Q=mpf(1), A=mpf(1), Ti=mpf(10)):
u'''
Definição da derivada da temperatura em relação ao tempo, temp_t = T_t(t, x)
'''
t = mpf(t)
termo1 = sqrt(A/(t*pi)) * exp(-x**2/(t*4*A))
termo2 = sqrt(A/(t*pi)) * (x**2/(t**2 * 2 * A)) * exp(-x**2/(t*4*A))
termo3 = (x**2/(sqrt(t**3 * A) * 4)) * erfc_z(x / (mpf(2) * sqrt(t * A)))
return Q * (termo1 + termo2 + termo3)
def sph_i2n_exact(n, z):
"""Return the value of i^{(2)}_n computed using the exact formula.
The expression used is http://dlmf.nist.gov/10.49.E10 .
"""
zm = mpmathify(z)
s1 = sum(mpc(-1,0)**k * _a(k, n)/zm**(k+1) for k in xrange(n+1))
s2 = sum(_a(k, n)/zm**(k+1) for k in xrange(n+1))
return exp(zm)/2 * s1 + mpc(-1,0)**n*exp(-zm)/2 * s2
def gaussian_total(self,offset):
factor = mpmath.sqrt(1.0/(self._period*self._tau))
factor_exponent = -1.0/(2*self._period)
exponent_factor = mpmath.mpf(offset)*offset
exponent_interval = self._tau*self._tau
exponent_offset = 2*self._tau*offset
factor_full = factor*mpmath.exp(exponent_factor*factor_exponent)
q = mpmath.exp(factor_exponent*exponent_interval)
z = factor_exponent*exponent_offset/(2*mpmath.j)
theta = mpmath.jtheta(3,z,q).real
return factor_full*theta
def sum_gaussian_theta(variance, offset, interval):
factor = mpmath.sqrt(1.0/(variance*tau))
factor_exponent=mpmath.mpf(-1.0)/(2*variance)
exponent_factor = mpmath.mpf(offset)*offset
factor_full = factor*mpmath.exp(exponent_factor*factor_exponent)
exponent_interval = interval*interval
exponent_offset = 2*interval*offset
q = mpmath.exp(factor_exponent*exponent_interval)
z = factor_exponent*exponent_offset/(2*mpmath.j)
theta = mpmath.jtheta(3,z,q)
return factor_full*theta
def fq(x0, seta2, seff2):
coeff = 1./np.sqrt(np.pi)
xm = x0/np.sqrt(2.*seta2)
tau = lamda1*seff2/np.sqrt(2.*seta2)
psi = lamda2*seff2*x0/np.sqrt(2.*seta2)
intgr1 = coeff*(-tau+2*psi+xm)*exp(-(tau+xm)**2) +.5*(1.+2.*(tau-psi)**2)*erfc(tau+xm)
intgr2 = coeff*(-tau-2*psi-xm)*exp(-(tau-xm)**2) +.5*(1.+2.*(tau+psi)**2)*erfc(tau-xm)
intgr3 = .5*x0**2*(erf(tau+xm)+erf(tau-xm))
return (seta2/(1.+lamda2*seff2)**2)*(intgr1+intgr2)+intgr3
def energy(self, clustering):
energy = mpmath.mpf(0.0)
new_vertex_distributions = _combine_vertex_distributions_given_clustering(
self.vertex_distributions, clustering)
# likelihood
likelihood_energy = -self._log_likelihood(clustering, new_vertex_distributions)
# prior on similarity:
# We prefer the cluster whose minimum similarity is large.
# - the similarity of a pair of vertexes is measured by the similarity
# of top 10 words in the distribution. (measure each word type
# respectively and take average)
intra_cluster_energy = mpmath.mpf(0.0)
for cluster_id, cluster_vertex_set in enumerate(clustering):
min_similarity_within_cluster = self._min_similarity_within_cluster(cluster_vertex_set, new_vertex_distributions[cluster_id])
intra_cluster_energy += -mpmath.log(mpmath.exp(min_similarity_within_cluster - 1))
# Between cluster similarity:
# - For each pair of clusters, we want to find the pair of words with maximum similarity
# and prefer this similarity value to be small.
inter_cluster_energy = mpmath.mpf(0.0)
if len(clustering) > 1:
for i in range(0, len(clustering)-1):
for j in range(i+1, len(clustering)):
max_similarity_between_clusters = self._max_similarity_between_clusters(clustering[i], clustering[j])
inter_cluster_energy += -mpmath.log(mpmath.exp(-max_similarity_between_clusters))
# prior on clustering complexity: prefer small number of clusters.
length_energy = -mpmath.log(mpmath.exp(-len(clustering)))
# classification: prefer small number of categories.
class_energy = 0.0
if self._classifier is not None:
num_classes = self._calculate_num_of_categories(clustering, new_vertex_distributions)
class_energy = -mpmath.log(mpmath.exp(-(abs(num_classes-len(clustering)))))
# classification confidence: maximize the classification confidence
confidence_energy = 0.0
for cluster_id, cluster_vertex_set in enumerate(clustering):
(category, confidence) = self._predict_label(new_vertex_distributions[cluster_id])
confidence_energy += -mpmath.log(confidence)
energy += (0.5)*likelihood_energy + intra_cluster_energy + inter_cluster_energy + 30.0*length_energy + 20.0*class_energy + confidence_energy
logging.debug('ENERGY: {0:12.6f}\t{1:12.6f}\t{2:12.6f}\t{3:12.6f}\t{4:12.6f}\t{5:12.6f}'.format(
likelihood_energy.__float__(),
intra_cluster_energy.__float__(),
inter_cluster_energy.__float__(),
length_energy.__float__(),
class_energy.__float__(),
confidence_energy.__float__()))
return energy
def test_talbot():
"""test for Talbot numerical inverse Laplace with mpmath"""
a = Talbot(f=f1, n=24, shift=0.0, dps=None)
#t=0 raise error:
assert_raises(ValueError, a, 0)
#single value of t:
ok_(mpallclose(a(1), mpmath.exp(mpmath.mpf('-1.0'))))
#2 values of t:
ans = np.array([mpmath.exp(mpmath.mpf('-1.0')),
mpmath.exp(mpmath.mpf('-2.0'))])
ok_(mpallclose(a([1,2]),ans))
def DispersionRelation(w):
def Disp(w, kp, b, eta):
zeta = w / kp
Z = Z_PDF(zeta)
kyp = ky/kp
return - (1. - eta/2. * (1. + b))*kyp * Z \
- eta * kyp * (zeta + zeta**2 * Z) + zeta * Z + 1.
# proton
sum1MG0 = lambda_D2 * b + (1. - G0) + (1. - G0/m_ie)
#return -mp.exp(-b) * Disp(w, kp, b, eta) + mp.exp(-b/m_ie) * Disp(w,kp * mp.sqrt(m_ie), b/m_ie, eta*mp.sqrt(m_ie)) + sum1MG0
return mp.exp(-b) * Disp(w, kp, b, eta) - mp.exp(-b/m_ie) * Disp(w,kp * mp.sqrt(m_ie), b/m_ie, eta) + sum1MG0
def GetAnalyticalWaitingtimes(kon,koff,ksyn):
""" Get analytical waiting times """
import mpmath
mpmath.mp.pretty = True
A = mpmath.sqrt(-4*ksyn*kon+(koff + kon + ksyn)**2)
x = []
for i in np.linspace(-20,5,5000):
x.append(mpmath.exp(i))
y = []
for t in x:
B = koff + ksyn - (mpmath.exp(t*A)*(koff+ksyn-kon))-kon+A+ mpmath.exp(t*A)*A
p01diff = mpmath.exp(-0.5*t*(koff + kon + ksyn+A))*B/(2.0*A)
y.append(p01diff*ksyn)
return (x,y)
def step(array):
global _INFILE
global _BETA
global _NSTEP
beta = _BETA
old_positions, Ntides, is_3prime = array
internal = Aptamer("leaprc.ff12SB",_INFILE)
identifier = Ntides.replace(" ","")
internal.sequence(identifier,Ntides.strip())
internal.unify(identifier)
internal.command("saveamberparm union %s.prmtop %s.inpcrd"%(identifier,identifier))
time.sleep(2)
#print("WhereamI?")
print("Identifier: "+Ntides)
volume = (2*math.pi)**5
aptamer_top = app.AmberPrmtopFile("%s.prmtop"%identifier)
aptamer_crd = app.AmberInpcrdFile("%s.inpcrd"%identifier)
# print("loaded")
# if is_3prime == 1:
en_pos = [mcmc_sample(aptamer_top, aptamer_crd, old_positions, index, Nsteps=_NSTEP) for index in range(10)]
# else:
# en_pos_task = [mcmc_sample_five(aptamer_top, aptamer_crd, old_positions, index, Nsteps=200) for index in range(20)]
# barrier()
# en_pos = value(en_pos_task)
en = []
positions = []
positions_s = []
for elem in en_pos:
en += elem[0]
#print(elem[2], elem[1])
positions_s.append([elem[2], elem[1]])
positions = min(positions_s)[1]
fil = open("best_structure%s.pdb"%Ntides,"w")
app.PDBFile.writeModel(aptamer_top.topology,positions,file=fil)
fil.close()
del fil
Z = volume*math.fsum([math.exp(-beta*elem) for elem in en])/len(en)
P = [math.exp(-beta*elem)/Z for elem in en]
S = volume*math.fsum([-elem*math.log(elem*volume) for elem in P])/len(P)
print("%s : %s"%(Ntides,S))
return positions, Ntides, S
def rate_cov_strate_sleeping_v3_uniform(k_matrix,alpha,rate_th1,lamda_u1,bw):
# define the distribution of the activity
first_int = (1/3) # correspond to the integral in first term
second_int = (1/2) - (1/3) # correspond to the integral in second term
#---------------------------- calibrated -----------------------------------------
expected_activity = (1/2) # expected value of a # this changes for optimization hance should be calibratable
expected_strategic_function = (1/2) # expected value of s
#---------------------------- calibrated -----------------------------------------
noise_var = 1
# preprocessing - define empty elements and get information about the inputs
k_mat = k_matrix.copy()
num_tiers = k_mat.shape[0]
density_org = k_mat[:,2].copy()
power = gb.db2pow(k_mat[:,1])
density_update = np.array(density_org*([1]*(num_tiers-1)+[expected_strategic_function]))
# define necessary values
area_org = np.zeros(num_tiers,float) # original association probability
area_sc_update = np.zeros(num_tiers,float) # association probability of disconnected cell
N_k_u1 = np.zeros(num_tiers,float) #number of users in tier K BS
N_k_sc = np.zeros(num_tiers,float) #number of users in tier K BS
N_k_total = np.zeros(num_tiers,float)
threshold_u1 = np.zeros(num_tiers,float) #threshold for users in tier K BS
t_func_main = np.zeros(num_tiers,float)
t_func_sc = np.zeros(num_tiers,float)
for i in range(num_tiers):
area_org[i] = A_k(density_org,power,alpha,i)
area_sc_update[i] = A_k(density_update,power,alpha,i)
N_k_u1 = 1 + 1.28*lamda_u1*(area_org/density_org)
N_k_sc = expected_activity*(1-expected_strategic_function)*density_org[-1]*N_k_u1[-1]*(area_sc_update/density_update) #for binary optimization
N_k_total = N_k_u1 + N_k_sc
threshold_u1 = 2**((rate_th1/bw)*N_k_total) -1
for i in range(num_tiers):
first_exp_term = -(threshold_u1[i]*noise_var/power[i])
z_term = 0 if (threshold_u1[i]==0) else (threshold_u1[i])**(2/alpha) *mp.quad(lambda u: 1/ (1+u**(alpha/2)),[(1/threshold_u1[i])**(2/alpha),mp.inf])
second_exp_term = -mp.pi*z_term* sum(density_update*(power/power[i])**(2/alpha))
third_exp_term_main = -mp.pi * sum(density_org*(power/power[i])**(2/alpha))
third_exp_term_sc = -mp.pi * sum(density_update*(power/power[i])**(2/alpha))
t_func_main[i] = mp.quad(lambda y: y * mp.exp(first_exp_term*y**alpha) * mp.exp(second_exp_term*y**2) * mp.exp(third_exp_term_main*y**2),[0,mp.inf])
t_func_sc[i] = mp.quad(lambda y: y* mp.exp(first_exp_term*y**alpha) * mp.exp(second_exp_term*y**2) * mp.exp(third_exp_term_sc*y**2), [0,mp.inf])
temp_second_sum = sum(2*mp.pi*density_update*t_func_sc)
temp_third_sum = sum(2*mp.pi*density_org[0:-1]*t_func_main[0:-1])
#rate_coverage = (2*mp.pi*density_org[-1]/expected_activity)*(t_func_main[-1]*first_int + temp_second_sum*second_int) + temp_third_sum
rate_coverage = (area_org[-1]/expected_activity)*((2*mp.pi*density_org[-1]/area_org[-1])*t_func_main[-1]*first_int + temp_second_sum*second_int) + temp_third_sum
#print((sum(2*mp.pi*density_update*t_func_sc)*second_int+t_func_main[-1]*first_int)*(2*mp.pi*density_org[-1]))
#print(t_func_main[-1]*first_int)
#print(temp_second_sum*second_int)
#print((2*mp.pi*density_org[-1]/expected_activity)*(t_func_main[-1]*first_int) + temp_third_sum)
#print(temp_second_sum)
return (rate_coverage)
def ThermionicEmissionCurrent(self, Va, phi_bn, debug=False):
kT = to_numeric(k * self.T)
q_n = to_numeric(q)
A = self.Area
Rs = self.Rs
if self.Semiconductor.dop_type == 'n':
Ar = self.Semiconductor.reference['A_R_coeff_n'] * constants['A_R']
else:
Ar = self.Semiconductor.reference['A_R_coeff_p'] * constants['A_R']
Js = Ar * (self.T ** 2) * mp.exp(-q_n * phi_bn / kT)
if debug: print 'Js, Is =', Js, A * Js
J = -Js + kT / (q_n * A * Rs) * mp.lambertw((q_n * A * Rs * Js / kT) * mp.exp(q_n * (Va + A * Js * Rs) / kT))
if debug: print 'J, I =', J, A * J
Vd = Va - A * J * Rs
return np.float(Vd), np.float(J)
def cost(q, alpha):
b = alpha * sum(q)
if b == 0:
return 0
mx = max(q)
a = sum(exp((x - mx) / b) for x in q)
return mx + b * log(a)
def polyfit_erfc(nroots, x, low):
t = x
if t > 19682.99:
t = 19682.99
if t > 1.0:
tt = mpmath.log(t) / mpmath.log(3) + 1.0 # log3(t) + 1
else:
tt = mpmath.sqrt(t)
it = int(tt)
tt = tt - it
tt = 2.0 * tt - 1.0 # map [0, 1] to [-1, 1]
u = low * 2 - 1 # map [0, 1] to [-1, 1]
tab_rs, tab_ws = tabulate_erfc(nroots, it)
im = clenshaw_d1(tab_rs.astype(float), u, nroots)
rr = clenshaw_d1(im, tt, nroots)
rr = [r / (1 - r) for r in rr]
im = clenshaw_d1(tab_ws.astype(float), u, nroots)
ww = clenshaw_d1(im, tt, nroots)
if x * low**2 < DECIMALS * .7:
factor = mpmath.exp(-x * low**2)
ww = [w * factor for w in ww]
return rr, ww
def calc_model_evidence(self):
vval = 0
mp.mp.dps = 50
for action in range(self.hparams.num_actions):
# val=1
# aa = self.a[action]
# for i in xrange(int(self.a[action]-self.a0)):
# aa-=1
# val*=aa
# val/=(2.0*math.pi)
# val/=self.b[action]
# val*=gamma(aa)
# val/=(self.b[action]**aa)
# val *= np.sqrt(np.linalg.det(self.lambda_prior * np.eye(self.hparams.context_dim + 1)) / np.linalg.det(self.precision[action]))
# val *= (self.b0 ** self.a0)
# val/= gamma(self.a0)
# vval += val
#val= 1/float((2.0 * math.pi) ** (self.a[action]-self.a0))
#val*= (float(gamma(self.a[action]))/float(gamma(self.a0)))
#val*= np.sqrt(float(np.linalg.det(self.lambda_prior * np.eye(self.hparams.context_dim + 1)))/float(np.linalg.det(self.precision[action])))
#val*= (float(self.b0**self.a0)/float(self.b[action]**self.a[action]))
val= mp.mpf(mp.fmul(mp.fneg(mp.log(mp.fmul(2.0 , mp.pi))) , mp.fsub(self.a[action],self.a0)))
val+= mp.loggamma(self.a[action])
val-= mp.loggamma(self.a0)
val+= 0.5*mp.log(np.linalg.det(self.lambda_prior * np.eye(self.hparams.context_dim + 1)))
val -= 0.5*mp.log(np.linalg.det(self.precision[action]))
val+= mp.fmul(self.a0,mp.log(self.b0))
val-= mp.fmul(self.a[action],mp.log(self.b[action]))
vval+=mp.exp(val)
vval/=float(self.hparams.num_actions)
return vval
def PiP0(self, gamma):
U = 4 * self.theta_f * self.Lf / (2. * self.NN)
R = 2 * self.Lf * self.rho / (2. * self.NN)
return self.GammaDist(gamma) * mpmath.exp(-(self.GammaDist(gamma) * U /
(2. * self.NN)) /
(gamma / (self.NN + 0.) + R /
(2. * self.NN)))
def integrand(self, eta, l, p, n, kind="A"):
if kind not in ("A", "B", "C", "D", "E"):
raise NotImplementedError("Integrand types supported \
go only from A to E")
if kind == "C":
return (mpmath.sqrt(self.k ** 2 - eta ** 2) / eta \
* self.integrand(eta, l, p, n, kind="A"))
if kind == "D":
return (mpmath.sqrt(self.k ** 2 - eta ** 2) / eta \
* self.integrand(eta, l, p, n, kind="B"))
pm = 1
exponent = n - 1
if kind == "B":
pm = -1
elif kind == "E":
pm = 0
exponent = n
if p == 0:
lgr_factor = 1
else:
lgr_factor = (self.a(eta) ** 2 \
* mpmath.laguerre(p - 1, l, self.a(eta) ** 2))
kz = mpmath.sqrt(self.k**2 - eta**2)
return ( mpmath.power(eta, np.abs(l) + 1) / mpmath.sqrt(kz) \
* mpmath.exp(-self.a(eta) ** 2 / 2) * (1 + pm * kz / self.k) \
* mpmath.power(eta / self.k, exponent) * lgr_factor)
def fmt1(t, m, low=None):
#
# F[m] = int u^{2m} e^{-t u^2} du
# = 1/(2m+1) int e^{-t u^2} d u^{2m+1}
# = 1/(2m+1) [e^{-t u^2} u^{2m+1}]_0^1 + (2t)/(2m+1) int u^{2m+2} e^{-t u^2} du
# = 1/(2m+1) e^{-t} + (2t)/(2m+1) F[m+1]
# = 1/(2m+1) e^{-t} + (2t)/(2m+1)(2m+3) e^{-t} + (2t)^2/(2m+1)(2m+3) F[m+2]
#
half = mpmath.mpf('.5')
b = m + half
e = half * mpmath.exp(-t)
x = e
s = e
bi = b + 1
while x > .1**DECIMALS:
x *= t / bi
s += x
bi += 1
f = s / b
out = [f]
for i in range(m):
b -= 1
f = (e + t * f) / b
out.append(f)
return np.array(out)[::-1]
def fmt2_erfc(t, m, low=0):
half = mpmath.mpf('.5')
tt = mpmath.sqrt(t)
low = mpmath.mpf(low)
low2 = low * low
f = mpmath.sqrt(
mpmath.pi) / 2 / tt * (mpmath.erf(tt) - mpmath.erf(low * tt))
e = mpmath.exp(-t)
e1 = mpmath.exp(-t * low2) * low
b = half / t
out = [f]
for i in range(m):
f = b * ((2 * i + 1) * f - e + e1)
e1 *= low2
out.append(f)
return np.array(out)
def mobility(self, z=1000, E=0, T=300, pn=None):
if pn is None:
Eg = self.band_gap(T, symbolic=False, electron_volts=False)
# print Eg, self.__to_numeric(-Eg/(k*T)), mp.exp(self.__to_numeric(-Eg/(k*T)))
pn = self.Nc(T, symbolic=False) * self.Nv(T, symbolic=False) * mp.exp(
self.__to_numeric(-Eg / (k * T))) * 1e-12
# print pn
N = 0
for dopant in self.dopants:
N += dopant.concentration(z)
N *= 1e-6
# print N
mobility = {'mobility_e': {'mu_L': 0, 'mu_I': 0, 'mu_ccs': 0, 'mu_tot': 0},
'mobility_h': {'mu_L': 0, 'mu_I': 0, 'mu_ccs': 0, 'mu_tot': 0}}
for key in mobility.keys():
mu_L = self.reference[key]['mu_L0'] * (T / 300.0) ** (-self.reference[key]['alpha'])
mu_I = (self.reference[key]['A'] * (T ** (3 / 2)) / N) / (
mp.log(1 + self.reference[key]['B'] * (T ** 2) / N) - self.reference[key]['B'] * (T ** 2) / (
self.reference[key]['B'] * (T ** 2) + N))
try:
mu_ccs = (2e17 * (T ** (3 / 2)) / mp.sqrt(pn)) / (mp.log(1 + 8.28e8 * (T ** 2) * (pn ** (-1 / 3))))
X = mp.sqrt(6 * mu_L * (mu_I + mu_ccs) / (mu_I * mu_ccs))
except:
mu_ccs = np.nan
X = 0
# print X
mu_tot = mu_L * (1.025 / (1 + ((X / 1.68) ** (1.43))) - 0.025)
Field_coeff = (1 + (mu_tot * E * 1e-2 / self.reference[key]['v_s']) ** self.reference[key]['beta']) ** (
-1 / self.reference[key]['beta'])
mobility[key]['mu_L'] = mu_L * 1e-4
mobility[key]['mu_I'] = mu_I * 1e-4
mobility[key]['mu_ccs'] = mu_ccs * 1e-4
mobility[key]['mu_tot'] = mu_tot * 1e-4 * Field_coeff
return mobility
def nPDF(self, x):
p = np.zeros(x.size)
for i, xx in enumerate(x):
gil_pelaez = lambda t: mp.re(
self.char_fn(t) * mp.exp(-1j * t * xx))
cutoff = self.find_cutoff(1e-30)
# Instead of finding roots, break up quadrature into degrees proportional to the
# expected number of oscillations of e^(i xx t) within t = [0, cutoff]
nosc = cutoff / (1 / max(10, np.abs(xx - self.mean())))
# roots = self.find_roots(gil_pelaez, cutoff)
# if np.abs(xx - self.mean()) < 3 * np.sqrt(self.variance()):
I = mp.quad(gil_pelaez, np.linspace(0, cutoff, nosc), maxdegree=10)
# I = mp.quadosc(gil_pelaez, (0, cutoff), zeros=roots)
# else:
# For now, do not trust any results out greater than 3sigma
# I = 0
# if np.abs(xx - self.mean()) >= 2 * np.sqrt(self.variance()):
# I = self.asymptotic_expansion(xx)
p[i] = 1 / np.pi * float(I)
print(i)
return p
def convert_rh_to_q(rh, temp_abs, pressure):
esat = 611.2 * mpmath.exp(17.67 * (temp_abs - 273.16) /
(temp_abs - 29.66)) # Stull
q_abs = (ratio_rmm / (R_dry * temp_abs)) * rh / 100 * esat
r = R_dry / (1 - q_abs * temp_abs / pressure * (R_vapour - R_dry))
specific_humidity = q_abs * r * temp_abs / pressure
return specific_humidity
def q_eit_standing_wave(Delta, Deltac, Omega, g1d, periodLength, phaseShift):
if Delta == Deltac:
return 0
gprime = 1 - g1d
kd = pi / periodLength
Mcell = eye(2)
for i in range(periodLength):
OmegaAtThisSite = Omega * cos(kd * i + pi * phaseShift)
beta3 = (g1d * (Delta - Deltac)) / (
(-2.0j * Delta + gprime) *
(Delta - Deltac) + 2.0j * OmegaAtThisSite**2)
M3 = matrix([[1 - beta3, -beta3], [beta3, 1 + beta3]])
Mf = matrix([[exp(1j * kd), 0], [0, exp(-1j * kd)]])
Mcell = Mf * M3 * Mcell
ret = (1.0 / periodLength) * acos(-0.5 * (Mcell[0, 0] + Mcell[1, 1]))
return ret
def fx_mmse(s, r):
x = np.zeros_like(s)
x_var = np.zeros_like(r)
px = 0.5
for i in range(2 * NUM_ANT):
sum_n1 = 0
sum_n2 = 0
sum_norm = 0
s_i = float(s[i, 0])
r_i = float(r[i, 0])
for x_cand in [-1 / mpmath.sqrt(2), 1 / mpmath.sqrt(2)]:
tmp = mpmath.exp(-0.5 * (x_cand - r_i)**2 / s_i)
pr_xcand = tmp / mpmath.sqrt(2 * mpmath.pi * s_i)
norm = px * pr_xcand
n1 = x_cand * norm
n2 = 0.5 * norm
sum_norm += norm
sum_n1 += n1
sum_n2 += n2
x_i = float(sum_n1 / sum_norm)
x_var_i = float(sum_n2 / sum_norm - x_i**2)
x[i, 0] = x_i
x_var[i, 0] = x_var_i
return x, x_var
def logistic_gaussian(m, v):
if m == oo:
if v == oo:
return oo
return Float('1.0')
if v == oo:
return Float('0.5')
mpmath.mp.dps = 500
mmpf = m._to_mpmath(500)
vmpf = v._to_mpmath(500)
# The integration routine below is obtained by substituting x = atanh(t)
# into the definition of logistic_gaussian
#
# f = lambda x: mpmath.exp(-(x - mmpf) * (x - mmpf) / (2 * vmpf)) / (1 + mpmath.exp(-x))
# result = 1 / mpmath.sqrt(2 * mpmath.pi * vmpf) * mpmath.quad(f, [-mpmath.inf, mpmath.inf])
#
# Such substitution makes mpmath.quad call much faster.
tanhm = mpmath.tanh(mmpf)
# Not really a precise threshold, but fine for our data
if tanhm == mpmath.mpf('1.0'):
return Float('1.0')
f = lambda t: mpmath.exp(-(mpmath.atanh(t) - mmpf) ** 2 / (2 * vmpf)) / ((1 - t) * (1 + t + mpmath.sqrt(1 - t * t)))
coef = 1 / mpmath.sqrt(2 * mpmath.pi * vmpf)
int, err = mpmath.quad(f, [-1, 1], error=True)
result = coef * int
if mpmath.mpf('1e50') * abs(err) > abs(int):
print(f"Suspiciously big error when evaluating an integral for logistic_gaussian({m}, {v}).")
print(f"Integral: {int}")
print(f"integral error estimate: {err}")
print(f"Coefficient: {coef}")
print(f"Result (Coefficient * Integral): {result}")
return Float(result)
def gauss_warp_arb(X, l1, l2, lw, x0):
r"""Warps the `X` coordinate with a Gaussian-shaped divot.
.. math::
l = l_1 - (l_1 - l_2) \exp\left ( -4\ln 2\frac{(X-x_0)^2}{l_{w}^{2}} \right )
Parameters
----------
X : :py:class:`Array`, (`M`,) or scalar float
`M` locations to evaluate length scale at.
l1 : positive float
Global value of the length scale.
l2 : positive float
Pedestal value of the length scale.
lw : positive float
Width of the dip.
x0 : float
Location of the center of the dip in length scale.
Returns
-------
l : :py:class:`Array`, (`M`,) or scalar float
The value of the length scale at the specified point.
"""
if isinstance(X, scipy.ndarray):
if isinstance(X, scipy.matrix):
X = scipy.asarray(X, dtype=float)
return l1 - (l1 - l2) * scipy.exp(-4.0 * scipy.log(2.0) *
(X - x0)**2.0 / (lw**2.0))
else:
return l1 - (l1 - l2) * mpmath.exp(-4.0 * mpmath.log(2.0) *
(X - x0)**2.0 / (lw**2.0))
def normal_cdf_moment_ratio(n, x):
mpmath.mp.dps = 500
xmpf = x._to_mpmath(500)
nmpf = n._to_mpmath(500)
if x < 0:
return Float(mpmath.power(2, -0.5 - nmpf / 2) * mpmath.hyperu(nmpf / 2 + 0.5, 0.5, xmpf * xmpf / 2))
return Float(mpmath.exp(xmpf * xmpf / 4) * mpmath.pcfu(0.5 + nmpf, -xmpf))
def ila_integrand_lp1_b(self, eta, rloc):
k, p, l = self.k, self.p, self.l
kz = mpmath.sqrt(k**2 - eta**2)
res = mpmath.power(eta, np.abs(l) + 1) / mpmath.sqrt(kz) \
* mpmath.laguerre(p, l, self.a(eta) ** 2) * mpmath.exp(-self.a(eta) ** 2 / 2) \
* (1 + kz / k) * mpmath.besselj(l, eta * rloc / k)
return res
def test_tklmbda_zero_shape(self):
# When lmbda = 0 the CDF has a simple closed form
one = mpmath.mpf(1)
assert_mpmath_equal(
lambda x: sp.tklmbda(x, 0),
lambda x: one/(mpmath.exp(-x) + one),
[Arg()], rtol=1e-7)
def test_weighted_logsumexp():
x = [1.0, 0.5, -1.0, -2.0]
w = [3.5, 0.0, 1.0, 3.0]
y = logsumexp(x, weights=w)
wsum = mpmath.fsum([wi * mpmath.exp(xi) for xi, wi in zip(x, w)])
expected = mpmath.log(wsum)
assert mpmath.almosteq(y, expected)
def pdf(x, p, b, loc=0, scale=1):
"""
Probability density function of the generalized inverse Gaussian
distribution.
The PDF for x > loc is:
z**(p - 1) * exp(-b*(z + 1/z)/2))
---------------------------------
s * K_p(b)
where s is the scale, z = (x - loc)/s, and K_p(b) is the modified Bessel
function of the second kind. For x <= loc, the PDF is zero.
"""
x = mpmath.mpf(x)
p = mpmath.mpf(p)
b = mpmath.mpf(b)
loc = mpmath.mpf(loc)
scale = mpmath.mpf(scale)
if x <= loc:
return mpmath.mp.zero
z = (x - loc) / scale
return (mpmath.power(z, p - 1) * mpmath.exp(-b * (z + 1 / z) / 2) /
(2 * mpmath.besselk(p, b)) / scale)
def __call__(self, Xi, Xj, sigmaf, l1, l2, lw, x0):
"""Evaluate the covariance function between points `Xi` and `Xj`.
Parameters
----------
Xi, Xj : :py:class:`Array`, :py:class:`mpf` or scalar float
Points to evaluate covariance between. If they are :py:class:`Array`,
:py:mod:`scipy` functions are used, otherwise :py:mod:`mpmath`
functions are used.
sigmaf : scalar float
Prefactor on covariance.
l1, l2, lw, x0 : scalar floats
Parameters of length scale warping function, passed to
:py:attr:`warp_function`.
Returns
-------
k : :py:class:`Array` or :py:class:`mpf`
Covariance between the given points.
"""
li = self.warp_function(Xi, l1, l2, lw, x0)
lj = self.warp_function(Xj, l1, l2, lw, x0)
if isinstance(Xi, scipy.ndarray):
if isinstance(Xi, scipy.matrix):
Xi = scipy.asarray(Xi, dtype=float)
Xj = scipy.asarray(Xj, dtype=float)
return sigmaf**2.0 * (scipy.sqrt(2.0 * li * lj /
(li**2.0 + lj**2.0)) *
scipy.exp(-(Xi - Xj)**2.0 / (li**2 + lj**2)))
else:
return sigmaf**2.0 * (mpmath.sqrt(2.0 * li * lj /
(li**2.0 + lj**2.0)) *
mpmath.exp(-(Xi - Xj)**2.0 /
(li**2 + lj**2)))
def zp(x):
"""
plasma dispersion function
using complementary error function in mpmath library.
"""
return -mp.sqrt(mp.pi) * mp.exp(-x**2) * mp.erfi(x) + mpc(0, 1) * mp.sqrt(mp.pi) * mp.exp(-x**2)
def gauss_warp_arb(X, l1, l2, lw, x0):
r"""Warps the `X` coordinate with a Gaussian-shaped divot.
.. math::
l = l_1 - (l_1 - l_2) \exp\left ( -4\ln 2\frac{(X-x_0)^2}{l_{w}^{2}} \right )
Parameters
----------
X : :py:class:`Array`, (`M`,) or scalar float
`M` locations to evaluate length scale at.
l1 : positive float
Global value of the length scale.
l2 : positive float
Pedestal value of the length scale.
lw : positive float
Width of the dip.
x0 : float
Location of the center of the dip in length scale.
Returns
-------
l : :py:class:`Array`, (`M`,) or scalar float
The value of the length scale at the specified point.
"""
if isinstance(X, scipy.ndarray):
if isinstance(X, scipy.matrix):
X = scipy.asarray(X, dtype=float)
return l1 - (l1 - l2) * scipy.exp(-4.0 * scipy.log(2.0) * (X - x0)**2.0 / (lw**2.0))
else:
return l1 - (l1 - l2) * mpmath.exp(-4.0 * mpmath.log(2.0) * (X - x0)**2.0 / (lw**2.0))
def skewness(mu=0, sigma=1):
"""
Skewness of the lognormal distribution.
"""
_validate_sigma(sigma)
sigma2 = sigma**2
return (mpmath.exp(sigma2) + 2) * mpmath.sqrt(mpmath.expm1(sigma2))
def BSLaplace(S,K,T,t,r,sig,N,phi):
"""Solving the Black Scholes PDE in the Laplace domain"""
x = ln(S/K)
r = mpf(r);sig = mpf(sig);T = mpf(T);t=mpf(t)
S = mpf(S);K = mpf(K);x=mpf(x)
mu = r - 0.5*(sig**2)
tau = T - t
c1 = mpf('0.5017')
c2 = mpf('0.6407')
c3 = mpf('0.6122')
c4 = mpc('0','0.2645')
ans = 0.0
h = 2*pi/N
h = mpf(h)
for k in range(N/2): # Use symmetry
theta = -pi + (k+0.5)*h
z = N/tau*(c1*theta/tan(c2*theta) - c3 + c4*theta)
dz = N/tau*(-c1*c2*theta/(sin(c2*theta)**2) + c1/tan(c2*theta)+c4)
eps1 = (-mu + sqrt(mu**2 + 2*(sig**2)*(z+r)))/(sig**2)
eps2 = (-mu - sqrt(mu**2 + 2*(sig**2)*(z+r)))/(sig**2)
b1 = 1/(eps1-eps2)*(eps2/(z+r) + (1 - eps2)/z)
b2 = 1/(eps1-eps2)*(eps1/(z+r) + (1 - eps1)/z)
ans += exp(z*tau)*bs(x,b1,b2,eps1,eps2,z,r,phi)*dz
val = (K*(h/(2j*pi)*ans)).real
return 2*val
def __call__(self, Xi, Xj, sigmaf, l1, l2, lw, x0):
"""Evaluate the covariance function between points `Xi` and `Xj`.
Parameters
----------
Xi, Xj : :py:class:`Array`, :py:class:`mpf` or scalar float
Points to evaluate covariance between. If they are :py:class:`Array`,
:py:mod:`scipy` functions are used, otherwise :py:mod:`mpmath`
functions are used.
sigmaf : scalar float
Prefactor on covariance.
l1, l2, lw, x0 : scalar floats
Parameters of length scale warping function, passed to
:py:attr:`warp_function`.
Returns
-------
k : :py:class:`Array` or :py:class:`mpf`
Covariance between the given points.
"""
li = self.warp_function(Xi, l1, l2, lw, x0)
lj = self.warp_function(Xj, l1, l2, lw, x0)
if isinstance(Xi, scipy.ndarray):
if isinstance(Xi, scipy.matrix):
Xi = scipy.asarray(Xi, dtype=float)
Xj = scipy.asarray(Xj, dtype=float)
return sigmaf**2.0 * (
scipy.sqrt(2.0 * li * lj / (li**2.0 + lj**2.0)) *
scipy.exp(-(Xi - Xj)**2.0 / (li**2 + lj**2))
)
else:
return sigmaf**2.0 * (
mpmath.sqrt(2.0 * li * lj / (li**2.0 + lj**2.0)) *
mpmath.exp(-(Xi - Xj)**2.0 / (li**2 + lj**2))
)
def Asian(S, K, T, t, sig, r, N):
# Assigning multi precision
S = mpf(S)
K = mpf(K)
sig = mpf(sig)
T = mpf(T)
t = mpf(t)
r = mpf(r)
# Geman and Yor's variable
tau = mpf(((sig**2) / 4) * (T - t))
v = mpf(2 * r / (sig**2) - 1)
alp = mpf(sig**2 / (4 * S) * K * T)
beta = mpf(-1 / (2 * alp))
tau = mpf(tau)
N = mpf(N)
# Initiate the stepsize
h = 2 * pi / N
mp.dps = 100
c1 = mpf('0.5017')
c2 = mpf('0.6407')
c3 = mpf('0.6122')
c4 = mpc('0', '0.2645')
# The for loop is evaluating the Laplace inversion at each point theta which is based on the trapezoidal
# rule
ans = 0.0
for k in range(N / 2): # N/2 : symmetry
theta = -pi + (k + 0.5) * h
z = 2 * v + 2 + N / tau * (c1 * theta / tan(c2 * theta) - c3 +
c4 * theta)
dz = N / tau * (-c1 * c2 * theta / sin(c2 * theta)**2 +
c1 / tan(c2 * theta) + c4)
zz = N / tau * (c1 * theta / tan(c2 * theta) - c3 + c4 * theta)
mu = sqrt(v**2 + 2 * z)
a = mu / 2 - v / 2 - 1
b = mu / 2 + v / 2 + 2
G = (2 * alp)**(-a) * gamma(b) / gamma(mu + 1) * hyp1f1(
a, mu + 1, beta) / (z * (z - 2 * (1 + v)))
ans += exp(zz * tau) * G * dz
return 2 * exp(tau * (2 * v + 2)) * exp(
-r * (T - t)) * 4 * S / (T * sig**2) * h / (2j * pi) * ans
def rate_cov_random_switching_v2(k_matrix, alpha, rate_th, lamda_user, q_on,
bw):
k_mat = k_matrix.copy()
num_tiers = k_mat.shape[0]
#density = k_mat[:,2]
#density[-1] = q_on
density = np.array(
k_mat[:, 2] *
([1] * (num_tiers - 1) +
[q_on])) #hence last row always corresponds to small cells
#print(density)
power = gb.db2pow(k_mat[:, 1])
small_cell_idx = num_tiers - 1 #indicates the index of the small cell
#density[small_cell_idx] = q_on
# initialize the integration result matrix
tier_integ_result = np.zeros(num_tiers,
float) #integration results of each tier
area_tiers = np.zeros(num_tiers, float) #area of the tiers
threshold_tier = np.zeros(num_tiers, float) #threshold of the tiers
N_k = np.zeros(num_tiers, float) #number of users in each tier
first_exp_term = np.zeros(num_tiers,
float) #first exponential term in integral
second_exp_term = np.zeros(num_tiers,
float) #second exponential term in integral
third_exp_term = np.zeros(num_tiers,
float) #third exponential term in integral
for i in range(num_tiers):
area_tiers[i] = A_k(density, power, alpha, i)
#N_k[i] = 0 if density[i]==0 else 1.28*lamda_user*area_tiers[i]/density[i]
N_k[i] = 0 if density[
i] == 0 else 1 + 1.28 * lamda_user * area_tiers[i] / density[i]
threshold_tier[i] = mp.inf if (
bw == 0) else 2**(rate_th * N_k[i] / bw) - 1
first_exp_term[i] = threshold_tier[i] * 1 / power[i]
third_exp_term[i] = mp.pi * sum(density *
(power / power[i])**(2 / alpha))
Z_term = 0 if (threshold_tier[i] == 0) else threshold_tier[i]**(
2 / alpha) * mp.quad(lambda u: 1 / (1 + u**(alpha / 2)),
[(threshold_tier[i])**(-2 / alpha), mp.inf])
second_exp_term[i] = third_exp_term[i] * Z_term
for k in range(num_tiers):
tier_integ_result[k] = mp.quad(
lambda y: y * mp.exp(-first_exp_term[k] * y**alpha) * mp.exp(
-second_exp_term[k] * y**2) * mp.exp(-third_exp_term[k] * y**2
), [0, mp.inf])
rate_cov_prob = 2 * mp.pi * sum(density * tier_integ_result)
return (rate_cov_prob)
def compute_MI_origemcee(seq_matQ,seq_matR,batches,ematQ,ematR,gamma,R_0):
# preliminaries
n_seqs = len(batches)
n_batches = int(batches.max()) + 1 # assumes zero indexed batches
n_bins = 1000
#energies = sp.zeros(n_seqs)
f = sp.zeros((n_batches,n_seqs))
# compute energies
# for i in range(n_seqs):
# energies[i] = sp.sum(seqs[:,:,i]*emat)
# alternate way
energies = np.zeros(n_seqs)
for i in range(n_seqs):
RNAP = (seq_matQ[:,:,i]*ematQ).sum()
TF = (seq_matR[:,:,i]*ematR).sum() + R_0
energies[i] = -RNAP + mp.log(1 + mp.exp(-TF - gamma)) - mp.log(1 + mp.exp(-TF))
# sort energies
inds = sp.argsort(energies)
for i,ind in enumerate(inds):
f[batches[ind],i] = 1.0/n_seqs # batches aren't zero indexed
# bin and convolve with Gaussian
f_binned = sp.zeros((n_batches,n_bins))
for i in range(n_batches):
f_binned[i,:] = sp.histogram(f[i,:].nonzero()[0],bins=n_bins,range=(0,n_seqs))[0]
#f_binned = f_binned/f_binned.sum()
f_reg = sp.ndimage.gaussian_filter1d(f_binned,0.04*n_bins,axis=1)
f_reg = f_reg/f_reg.sum()
# compute marginal probabilities
p_b = sp.sum(f_reg,axis=1)
p_s = sp.sum(f_reg,axis=0)
# finally sum to compute the MI
MI = 0
for i in range(n_batches):
for j in range(n_bins):
if f_reg[i,j] != 0:
MI = MI + f_reg[i,j]*sp.log2(f_reg[i,j]/(p_b[i]*p_s[j]))
print MI
return MI,f_reg
def __init__(self, number_of_layers, vector_d, vector_W, mass_vector,
exp_index, toll):
''' Инициализация входных данных'''
# Колличество уровней с постоянной эффективной массой
# в гетероструктуре
self.number_of_layers = number_of_layers
# Вектор размерности n с размерами каждого слоя
self.structure_width_vector = vector_d
# Вектор размерности n с потенциалом для каждого уровня
self.barriers_high_vector = vector_W
# Вектор размерности n для эффективных масс в каждом слое
self.mass_vector = mass_vector
self.toll = toll
self.exp_index_list = exp_index
self.roots = []
# Разбиение экспонециального барьера на ступеньки
new_D = np.zeros(self.toll * len(self.exp_index_list) +
(self.number_of_layers - len(self.exp_index_list)))
new_W = np.zeros(self.toll * len(self.exp_index_list) +
(self.number_of_layers - len(self.exp_index_list)))
k = 0
if not self.exp_index_list == []:
for i in range(self.number_of_layers):
if i not in self.exp_index_list:
new_D[i + k] = self.structure_width_vector[i]
new_W[i + k] = self.barriers_high_vector[i]
else:
for n in range(self.toll):
new_D[i + k +
n] = self.structure_width_vector[i] / self.toll
new_W[i + k + n] = self.barriers_high_vector[i] * (
1 - mp.exp(-(10 * n / self.toll)))
k = k + (self.toll - 1)
else:
new_D = self.structure_width_vector
new_W = self.barriers_high_vector
self.structure_width_vector = new_D
self.barriers_high_vector = new_W
self.x_max = 0
for ind in range(len(self.structure_width_vector)):
if not ind == 0:
self.x_max = self.x_max + self.structure_width_vector[ind]
self.a = np.zeros(len(self.structure_width_vector) + 1)
for num in range(len(self.a)):
if num == 0:
self.a[num] = self.structure_width_vector[num]
elif num == 1:
self.a[num] = 0
else:
self.a[num] = self.structure_width_vector[num - 1]
self.a[num] = self.a[num] + self.a[num - 1]
def mle(x, loc=None, scale=None):
"""
Maximum likelihood estimates for the Gumbel distribution.
`x` must be a sequence of numbers--it is the data to which the
Gumbel distribution is to be fit.
If either `loc` or `scale` is not None, the parameter is fixed
at the given value, and only the other parameter will be fit.
Returns maximum likelihood estimates of the `loc` and `scale`
parameters.
Examples
--------
Imports and mpmath configuration:
>>> import mpmath
>>> mpmath.mp.dps = 20
>>> from mpsci.distributions import gumbel_min
The data to be fit:
>>> x = [6.86, 14.8 , 15.65, 8.72, 8.11, 8.15, 13.01, 13.36]
Unconstrained MLE:
>>> gumbel_min.mle(x)
(mpf('12.708439639698245696235'), mpf('2.878444823276260896075'))
If we know the scale is 2, we can add the argument `scale=2`:
>>> gumbel_min.mle(x, scale=2)
(mpf('13.18226169025112165358'), mpf('2.0'))
"""
with mpmath.extradps(5):
x = [mpmath.mpf(xi) for xi in x]
if scale is None and loc is not None:
# Estimate scale with fixed loc.
loc = mpmath.mpf(loc)
# Initial guess for findroot.
s0 = stats.std([xi - loc for xi in x])
scale = mpmath.findroot(
lambda t: _mle_scale_with_fixed_loc(t, x, loc), s0)
return loc, scale
if scale is None:
scale = _solve_mle_scale(x)
else:
scale = mpmath.mpf(scale)
if loc is None:
ex = [mpmath.exp(xi / scale) for xi in x]
loc = scale * mpmath.log(stats.mean(ex))
else:
loc = mpmath.mpf(loc)
return loc, scale
def test_pow_E(self):
# E ^ x
expr = Expression("Power", Symbol("E"), Symbol("x"))
args = [CompileArg("System`x", real_type)]
cfunc = _compile(expr, args)
for _ in range(1000):
x = random.random()
self.assertAlmostEqual(cfunc(x), mpmath.exp(x))
def fmt2_erfc(t, m, low=0, factor=mpmath.mpf(1)):
tt = mpmath.sqrt(t)
low = mpmath.mpf(low)
low2 = low * low
f = factor * mpmath.sqrt(
mpmath.pi) / 2. / tt * (mpmath.erf(tt) - mpmath.erf(low * tt))
e = mpmath.exp(-t)
e1 = mpmath.exp(-t * low2) * low
e *= factor
e1 *= factor
b = mpmath.mpf('.5') / t
out = [f]
for i in range(m):
f = b * ((2 * i + 1) * f - e + e1)
e1 *= low2
out.append(f)
return np.array(out)
def get_prob_poisson(events, length, rate):
""" P(k, lambda = t * rate) = """
avg_events = mpmath.fmul(rate, length) # lambda
prob = mpmath.fmul((-1), avg_events)
for i in range(1, events + 1):
prob = mpmath.fadd(prob, mpmath.log(mpmath.fdiv(avg_events, i)))
prob = mpmath.exp(prob)
return prob
def target_evaluation_func(self, current_clustering, context=None):
#print(current_labeling)
energy = self.calculate_energy(current_clustering)
temperature = 1000
#print(energy)
if context is not None:
temperature = self.cooling_schedule(context.iteration_counter)
return mpmath.exp(-(energy / temperature))
