def full_gen_data(self, fbar, nSamples, sigma, density):
totalSize = self.ndc * nSamples
ss = sigma * sigma
denom = 2.0 * self.k * pi * ss * density + self.g0
phi = np.zeros((self.ndc))
phi_bar = 0
for i, d in enumerate(self.dist2):
if d == 0:
p = self.g0 / denom
else:
p = sy.nsum(lambda t: exp(self.mu2 * t) *
exp(-d / (4.0 * ss * t)) / (2.0 * t), [
1, sy.inf], error=False, verbose=False,
method='euler-maclaurin', steps=[1000])
p = p / denom
phi_bar += p * nSamples
phi[i] = p
phi_bar /= float(totalSize)
r = (phi - phi_bar) / (1.0 - phi_bar)
pIBD = fbar + (1.0 - fbar) * r
pIBD = np.array(pIBD, dtype=float)
# simulate values from binomial distribution
# np.random.seed(1209840)
counts = np.random.binomial(nSamples, pIBD)
return counts
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 4*prec
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method='richardson')
return v._mpf_
def full_likelihood(self):
phi = np.zeros((self.ndc))
phi_bar = 0
denom = 2 * self.k * self.ss * pi * self.de + self.g0
for i, d in enumerate(self.dist2):
if d == 0:
p = self.g0 / denom
else:
p = sy.nsum(lambda t: exp(self.mu2 * t) *
exp(-d / (4.0 * self.ss * t)) / (2.0 * t), [
1, sy.inf], error=False, verbose=False,
method='euler-maclaurin', steps=[100])
p = p / denom
phi_bar += p * self.sz[i]
phi[i] = p
phi_bar /= float(self.tsz)
cml = 0
for i in xrange(self.ndc):
r = (phi[i] - phi_bar) / (1.0 - phi_bar)
pIBD = self.fhat + (1 - self.fhat) * r
if pIBD <= 0:
print("WARNING: Prabability of IBD has fallen "
"below zero for distance class {}.").format(self.dist[i])
print("This marginal likelihood will not be "
"included in composite likelihood.")
continue
cml += self.data[i] * log(pIBD) +\
(self.sz[i] - self.data[i]) * log(1 - pIBD)
return cml
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1 / g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 4 * prec
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method="richardson")
return v._mpf_
def cramerVonMises(x, y): try: x = sorted(x); y = sorted(y); pool = x + y; ps, pr = _sortRank(pool) rx = array ( [ pr[ind] for ind in [ ps.index(element) for element in x ] ] ) ry = array ( [ pr[ind] for ind in [ ps.index(element) for element in y ] ] ) n = len(x) m = len(y) i = array(range(1, n+1)) j = array(range(1, m+1)) u = n * sum ( power( (rx - i), 2 ) ) + m * sum ( power((ry - j), 2) ) t = u / (n*m*(n+m)) - (4*n*m-1) / (6 * (n+m)) Tmu = 1/6 + 1 / (6*(n+m)) Tvar = 1/45 * ( (m+n+1) / power((m+n),2) ) * (4*m*n*(m+n) - 3*(power(m,2) + power(n,2)) - 2*m*n) / (4*m*n) t = (t - Tmu) / power(45*Tvar, 0.5) + 1/6 if t < 0: return -1 elif t <= 12: a = 1-mp.nsum(lambda x : ( mp.gamma(x+0.5) / ( mp.gamma(0.5) * mp.fac(x) ) ) * mp.power( 4*x + 1, 0.5 ) * mp.exp ( - mp.power(4*x + 1, 2) / (16*t) ) * mp.besselk(0.25 , mp.power(4*x + 1, 2) / (16*t) ), [0,100] ) / (mp.pi*mp.sqrt(t)) return float(mp.nstr(a,3)) else: return 0 except Exception as e: print e return -1
size = 50
XX, YY = np.mgrid[0:d.magnitude:complex(0, size),
0:d.magnitude:complex(0, size)]
Pressure = np.empty((size, size))
dmag = d.magnitude
def somando(x, y, r, s):
return (cos((2*r + 1)*pi)*x/dmag)*(cos((2*s + 1)*pi)*y/dmag)/((2*r + 1)**2 + (2*s + 1)**2)
for i in xrange(size):
for j in xrange(size):
somatorio = nsum(lambda ri, si: somando(XX[i, j], YY[i, j], ri, si),
[0, inf], [0, inf], maxterms=30)
pres = Pi - (16. / pi**2) * ((muo * qo_std * Bo) / (k * h)) * somatorio
Pressure[i, j] = pres.magnitude
fig1 = plt.figure()
fig1.patch.set_facecolor('white')
plt.rcParams['legend.loc'] = 'best'
plt.title(u'Pressão em curvas de nível - padrão five-spot')
plt.xlabel(u'x [ft]')
plt.ylabel(u'y [ft]')
pressure_levels = np.linspace(500, 2200, 9).tolist()
CS = plt.contour(XX, YY, Pressure, levels=pressure_levels, colors='k')
plt.clabel(CS, fmt='%.1f', fontsize=12, inline=1)
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify
if prec == float('inf'):
raise NotImplementedError('does not support inf prec')
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
term = expr.subs(n, 0)
if not term.is_Rational:
raise NotImplementedError("Non rational term functionality is not implemented.")
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
vold = None
ndig = prec_to_dps(prec)
while True:
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process; we check the answer to
# make sure that the desired precision has been reached, too.
prec2 = 4*prec
term0 = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term0]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method='richardson')
vf = C.Float(v, ndig)
if vold is not None and vold == vf:
break
prec += prec # double precision each time
vold = vf
return v._mpf_