def pdf(self, x, k, n, p):
'''distribution of success runs of length k or more
Parameters
----------
x : float
count of runs of length n
k : int
length of runs
n : int
total number of observations or trials
p : float
probability of success in each Bernoulli trial
Returns
-------
pdf : float
probability that x runs of length of k are observed
Notes
-----
not yet vectorized
References
----------
Muselli 1996, theorem 3
'''
q = 1 - p
m = np.arange(x, (n + 1) // (k + 1) + 1)[:, None]
terms = (-1)**(m-x) * comb(m, x) * p**(m*k) * q**(m-1) \
* (comb(n - m*k, m - 1) + q * comb(n - m*k, m))
return terms.sum(0)
def pdf(self, x, k, n, p):
'''distribution of success runs of length k or more
Parameters
----------
x : float
count of runs of length n
k : int
length of runs
n : int
total number of observations or trials
p : float
probability of success in each Bernoulli trial
Returns
-------
pdf : float
probability that x runs of length of k are observed
Notes
-----
not yet vectorized
References
----------
Muselli 1996, theorem 3
'''
q = 1-p
m = np.arange(x, (n+1)//(k+1)+1)[:,None]
terms = (-1)**(m-x) * comb(m, x) * p**(m*k) * q**(m-1) \
* (comb(n - m*k, m - 1) + q * comb(n - m*k, m))
return terms.sum(0)
def hypergProb(k, N, m, n):
""" Wikipedia: There is a shipment of N objects in which m are defective. The hypergeometric distribution describes the probability that in a sample of n distinctive objects drawn from the shipment exactly k objects are defective. """
#return float(choose(m, k) * choose(N-m, n-k)) / choose(N, n)
hp = float(scipy.comb(m, k) * scipy.comb(N-m, n-k)) / scipy.comb(N, n)
if scipy.isnan(hp):
stderr.write("error: not possible to calculate hyperg probability in util.py for k=%d, N=%d, m=%d, n=%d\n" %(k, N, m,n))
stdout.write("error: not possible to calculate hyperg probability in util.py for k=%d, N=%d, m=%d, n=%d\n" %(k, N, m,n))
return hp
def mv_hypergeometric(x, m):
"""
x : number of draws for each category.
m : size of each category.
"""
x = np.asarray(x)
m = np.asarray(m)
return log(comb(m, x).prod() / comb(m.sum(), x.sum()))
def prob(c, n, j, k):
pi = k - j # how many new marbles to pick
po = c - pi # how many old marbles to pick
old_count = j
new_count = n - j
return comb(old_count, po) * comb(new_count, pi) / comb(n, c)
def mv_hypergeometric(x,m):
"""
x : number of draws for each category.
m : size of each category.
"""
x = np.asarray(x)
m = np.asarray(m)
return log(comb(m,x).prod()/comb(m.sum(), x.sum()))
def runs_prob_odd(self, r):
n0, n1 = self.n0, self.n1
k = (r + 1) // 2
tmp0 = comb(n0 - 1, k - 1)
tmp1 = comb(n1 - 1, k - 2)
tmp3 = comb(n0 - 1, k - 2)
tmp4 = comb(n1 - 1, k - 1)
return (tmp0 * tmp1 + tmp3 * tmp4) / self.comball
def runs_prob_odd(self, r):
n0, n1 = self.n0, self.n1
k = (r+1)//2
tmp0 = comb(n0-1, k-1)
tmp1 = comb(n1-1, k-2)
tmp3 = comb(n0-1, k-2)
tmp4 = comb(n1-1, k-1)
return (tmp0 * tmp1 + tmp3 * tmp4) / self.comball
def hypergeom(x, fgTotal, bgMatching, bgTotal, log=True):
if log:
return logchoose(fgTotal, x) + logchoose(bgTotal - fgTotal,
bgMatching - x) - logchoose(
bgTotal, bgMatching)
else:
return scipy.comb(fgTotal, x) * scipy.comb(
bgTotal - fgTotal, bgMatching - x) / scipy.comb(
bgTotal, bgMatching)
def dumb_factor(goal, n):
# Assumes factors are not equal to each other and bounded above by
# upper_bound.
comb0 = comb(n, 2)
for i in range(0, n):
comb1 = comb(n - i, 2)
for j in range(i + 1, n):
if goal == round(comb0 - comb1 + (j - i - 1)):
return i, j
def hypergProb(k, N, m, n):
""" Wikipedia: There is a shipment of N objects in which m are defective. The hypergeometric distribution describes the probability that in a sample of n distinctive objects drawn from the shipment exactly k objects are defective. """
#return float(choose(m, k) * choose(N-m, n-k)) / choose(N, n)
hp = float(scipy.comb(m, k) * scipy.comb(N - m, n - k)) / scipy.comb(N, n)
if scipy.isnan(hp):
stderr.write(
"error: not possible to calculate hyperg probability in util.py for k=%d, N=%d, m=%d, n=%d\n"
% (k, N, m, n))
stdout.write(
"error: not possible to calculate hyperg probability in util.py for k=%d, N=%d, m=%d, n=%d\n"
% (k, N, m, n))
return hp
def hypergeometric(x, n, m, N):
"""
x : number of successes drawn
n : number of draws
m : number of successes in total
N : successes + failures in total.
"""
if x < max(0, n - (N - m)):
return 0.
elif x > min(n, m):
return 0.
else:
return comb(N - m, x) * comb(m, n - x) / comb(N, n)
def hypergeometric(x, n, m, N):
"""
x : number of successes drawn
n : number of draws
m : number of successes in total
N : successes + failures in total.
"""
if x < max(0, n-(N-m)):
return 0.
elif x > min(n, m):
return 0.
else:
return comb(N-m, x) * comb(m, n-x) / comb(N,n)
def _calculate_orders(self):
k = self.k
n = self.n
m = self.m
dim = self.dim
# Calculate the length of each order
self.order_idx = np.zeros(n+2, dtype=int)
self.order_length = np.zeros(n+1, dtype=int)
self.row_counter = 0
for ordi in xrange(n+1):
self.order_length[ordi] = (sp.comb(n, ordi+1, exact=1) *
((m-1)**(ordi+1)))
self.order_idx[ordi] = self.row_counter
self.row_counter += self.order_length[ordi]
self.order_idx[n+1] = dim+1
# Calculate nnz for A
# not needed for lil sparse format
x = (m*np.ones(n))**np.arange(n-1,-1,-1)
x = x[:k]
y = self.order_length[:k]
self.Annz = np.sum(x*y.T)
def triple(g, P, T, r, family): #if not g.index_set_leq(T,P): if g.intersection_matrix[g.schubert_list.index(P)][g.schubert_list.index(T)] == 0: return 0 else: delta = 1 quad, lin = num_equations(g,P, T) subspace_eqns = g.m + r - 1 if g.OG and r > g.k: subspace_eqns += 1 if quad > 0: quad -= 1 if g.type == 'D' and r == g.k: subspace_eqns = g.n+1 #if quad == 0 or single_ruling(g,P,T): if quad == 0: #subspace_eqns -= int((family + h(g,P,T)))%2 delta = int((family + h(g,P,T)))%2 subspace_eqns = g.n if quad > 0: quad -= 1 triple_list = [] for j in range(int(g.N - quad - lin - subspace_eqns)): triple_list.append((-1)**j * 2**(quad - j) * sp.comb(quad,j)) return delta*sum(triple_list)
def backward_difference_formula(k):
r"""
Construct the k-step backward differentiation method.
The methods are implicit and have order k.
They have the form:
`\sum_{j=0}^{k} \alpha_j y_{n+k-j+1} = h \beta_j f(y_{n+1})`
They are generated using equation (1.22') from Hairer & Wanner III.1,
along with the binomial expansion.
.. note::
Accuracy is lost when evaluating the order conditions
for methods with many steps. This could be avoided by using
SAGE rationals instead of NumPy doubles for the coefficient
representation.
**References**:
#.[hairer1993]_ pp. 364-365
"""
from scipy import comb
alpha=np.zeros(k+1)
beta=np.zeros(k+1)
beta[k]=1.
gamma=np.zeros(k+1)
gamma[0]=1.
alphaj=np.zeros(k+1)
for j in range(1,k+1):
gamma[j]= 1./j
for i in range(0,j+1):
alphaj[k-i]=(-1.)**i*comb(j,i)*gamma[j]
alpha=alpha+alphaj
name=str(k)+'-step BDF method'
return LinearMultistepMethod(alpha,beta,name=name)
def triple(g, P, T, r, family): if g.intersection_matrix[g.schubert_list.index(P)][g.schubert_list.index(T)] == 0: return 0 else: delta = 1 quad, lin = num_equations(g,P, T) subspace_eqns = g.m + r - 1 #if g.type == 'D' and quad == 0 and r == g.k: # delta = int(family + h(g,P,T))%2 #if g.type == 'B': # if quad > 0: # quad -=1 # if r > g.k: # subspace_eqns +=1 if g.OG: if r > g.k: subspace_eqns += 1 if quad > 0: quad -= 1 if r == g.k and g.type == 'D': if quad == 0: delta = int(family + h(g,P,T))%2 if quad > 0: subspace_eqns +=1 quad -=1 #subspace_eqns += (1-delta) triple_list = [] for j in range(int(g.N - quad - lin - subspace_eqns)): triple_list.append((-1)**j * 2**(quad - j) * sp.comb(quad,j)) return delta*sum(triple_list)
def similarity_matrix(db, sim_func, check_func):
matrix = numpy.zeros((len(db["genomes"]), len(db["genomes"])))
sys.stderr.write("populating matrix\n")
num_work_to_do = int(scipy.comb(len(db["genomes"]), 2))
work_done = 0
for nameA, nameB in itertools.combinations(db["genomes"], 2):
nameA_num = db["genomes"].index(nameA)
nameB_num = db["genomes"].index(nameB)
sizeA = db["cds_counts"][nameA_num]
sizeB = db["cds_counts"][nameB_num]
hits_count = check_func((db["hit_matrix"][nameA_num][nameB_num],
db["hit_matrix"][nameB_num][nameA_num]))
matrix[nameA_num][nameB_num] = sim_func(sizeA, sizeB, hits_count)
matrix[nameB_num][nameA_num] = sim_func(sizeB, sizeA, hits_count)
work_done += 1
if work_done % 100 == 0:
sys.stderr.write("\r %s/%s completed" % (work_done, num_work_to_do))
sys.stderr.write("\r %s/%s completed\n" % (work_done, num_work_to_do))
return matrix
def triple(self, P, T, r, family): #if not self.index_set_leq(T,P): if self.intersection_matrix[self.schubert_list.index(P)][self.schubert_list.index(T)] == 0: return 0 else: delta = 1 quad, lin = self.outsourced_num_equations(P, T) subspace_eqns = self.m + r - 1 if self.OG and r > self.k: subspace_eqns += 1 if quad > 0: quad -= 1 if self.type == 'D' and r == self.k: subspace_eqns = self.n+1 #if quad == 0 or self.single_ruling(P,T): if quad == 0: #subspace_eqns -= int((family + self.h(P,T)))%2 delta = int((family + self.h(P,T)))%2 subspace_eqns = self.n if quad > 0: quad -= 1 triple_list = [] for j in range(int(self.N - quad - lin - subspace_eqns)): triple_list.append((-1)**j * 2**(quad - j) * sp.comb(quad,j)) return delta*sum(triple_list)
def Test():
# Double check stirlings approximation implementation
test_vals = [(30, 5), (18, 7), (91, 32)]
for n, k in test_vals:
print np.log(scipy.comb(n, k))
print ln_stirling_binomial(n, k)
m = np.matrix([[50, 39], [66, 5]])
print CalcPValues(m)
prob_m = np.matrix([[0.25, 0.25], [0.25, 0.25]])
print CalcPValues(m, prob_m)
def beta_binomial(k, n, a, b):
"""The pmf/pdf of the Beta-binomial distribution.
Computation based on beta function.
See: http://en.wikipedia.org/wiki/Beta-binomial_distribution
and http://mathworld.wolfram.com/BetaBinomialDistribution.html
k = a vector of non-negative integers <= n
n = an integer
a = an array of non-negative real numbers
b = an array of non-negative real numbers
"""
return (comb(n, k) * beta(k+a, n-k+b) / beta(a,b)).prod(0)
def mnc2cum_g(mnc_):
'''convert non-central moments to cumulants
recursive formula produces as many cumulants as moments
http://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments
'''
mnc = [1] + list(mnc_)
kappa = [1]
for nn,m in enumerate(mnc[1:]):
n = nn+1
kappa.append(m)
for k in range(1,n):
kappa[n] -= scipy.comb(n-1,k-1,exact=1) * kappa[k]*mnc[n-k]
return kappa[1:]
def mnc2cum(mnc_):
'''convert non-central moments to cumulants
recursive formula produces as many cumulants as moments
http://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments
'''
mnc = [1] + list(mnc_)
kappa = [1]
for nn,m in enumerate(mnc[1:]):
n = nn+1
kappa.append(m)
for k in range(1,n):
kappa[n] -= scipy.comb(n-1,k-1,exact=1) * kappa[k]*mnc[n-k]
return kappa[1:]
def Test():
# Double check stirlings approximation implementation
test_vals = [(30, 5),
(18, 7),
(91, 32)]
for n, k in test_vals:
print np.log(scipy.comb(n, k))
print ln_stirling_binomial(n, k)
m = np.matrix([[50, 39],
[66, 5]])
print CalcPValues(m)
prob_m = np.matrix([[0.25, 0.25],
[0.25, 0.25]])
print CalcPValues(m, prob_m)
def mc2mnc_g(mc_):
'''convert central to non-central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(mc_)
mean = mc_[0]
mc = [1] + list(mc_) # add zero moment = 1
mc[1] = 0 # define central mean as zero for formula
mnc = [1, mean] # zero and first raw moments
for nn, m in enumerate(mc[2:]):
n = nn + 2
mnc.append(0)
for k in range(n + 1):
mnc[n] += scipy.comb(n, k, exact=1) * mc[k] * mean**(n - k)
return mnc[1:]
def mnc2mc(mnc_, wmean = True):
'''convert non-central to central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(mnc_)
mean = mnc_[0]
mnc = [1] + list(mnc_) # add zero moment = 1
mu = [] #np.zeros(n+1)
for n,m in enumerate(mnc):
mu.append(0)
#[scipy.comb(n-1,k,exact=1) for k in range(n)]
for k in range(n+1):
mu[n] += (-1)**(n-k) * scipy.comb(n,k,exact=1) * mnc[k] * mean**(n-k)
if wmean:
mu[1] = mean
return mu[1:]
def mc2mnc(mc_):
'''convert central to non-central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(mc_)
mean = mc_[0]
mc = [1] + list(mc_) # add zero moment = 1
mc[1] = 0 # define central mean as zero for formula
mnc = [1, mean] # zero and first raw moments
for nn,m in enumerate(mc[2:]):
n=nn+2
mnc.append(0)
for k in range(n+1):
mnc[n] += scipy.comb(n,k,exact=1) * mc[k] * mean**(n-k)
return mnc[1:]
def mnc2mc_g(mnc_, wmean = True):
'''convert non-central to central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(mnc_)
mean = mnc_[0]
mnc = [1] + mnc_ # index numbering starting at 1, or zero moment
mu = [] #np.zeros(n+1)
for n,m in enumerate(mnc):
mu.append(0)
#[scipy.comb(n-1,k,exact=1) for k in range(n)]
for k in range(n+1):
mu[n] += (-1)**(n-k) * scipy.comb(n,k,exact=1) * mnc[k] * mean**(n-k)
if wmean:
mu[1] = mean
return mu[1:]
def mnc2mc(_mnc, wmean=True):
'''convert non-central to central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(_mnc)
mean = _mnc[0]
mnc = [1] + list(_mnc) # add zero moment = 1
mu = [] #np.zeros(n+1)
for n, m in enumerate(mnc):
mu.append(0)
#[scipy.comb(n-1,k,exact=1) for k in range(n)]
for k in range(n + 1):
mu[n] += (-1)**(n - k) * scipy.comb(
n, k, exact=1) * mnc[k] * mean**(n - k)
if wmean:
mu[1] = mean
return mu[1:]
def cum2mc(kappa_):
'''convert non-central moments to cumulants
recursive formula produces as many cumulants as moments
References
----------
Kenneth Lange: Numerical Analysis for Statisticians, page 40
(http://books.google.ca/books?id=gm7kwttyRT0C&pg=PA40&lpg=PA40&dq=convert+cumulants+to+moments&source=web&ots=qyIaY6oaWH&sig=cShTDWl-YrWAzV7NlcMTRQV6y0A&hl=en&sa=X&oi=book_result&resnum=1&ct=result)
'''
mc = [1,0.0] #kappa_[0]] #insert 0-moment and mean
kappa = [1] + list(kappa_)
for nn,m in enumerate(kappa[2:]):
n = nn+2
mc.append(0)
for k in range(n-1):
mc[n] += scipy.comb(n-1,k,exact=1) * kappa[n-k]*mc[k]
mc[1] = kappa_[0] # insert mean as first moments by convention
return mc[1:]
def cum2mc_g(kappa_):
'''convert non-central moments to cumulants
recursive formula produces as many cumulants as moments
References
----------
Kenneth Lange: Numerical Analysis for Statisticians, page 40
(http://books.google.ca/books?id=gm7kwttyRT0C&pg=PA40&lpg=PA40&dq=convert+cumulants+to+moments&source=web&ots=qyIaY6oaWH&sig=cShTDWl-YrWAzV7NlcMTRQV6y0A&hl=en&sa=X&oi=book_result&resnum=1&ct=result)
'''
mc = [1, 0.0] #kappa_[0]] #insert 0-moment and mean
kappa = [1] + list(kappa_)
for nn, m in enumerate(kappa[2:]):
n = nn + 2
mc.append(0)
for k in range(n - 1):
mc[n] += scipy.comb(n - 1, k, exact=1) * kappa[n - k] * mc[k]
mc[1] = kappa_[0] # insert mean as first moments by convention
return mc[1:]
def diallelic_approximation_d(N_small, g, m0, m1):
"""
This is experimental.
The numerical integration should be replaced
by a call to the confluent hypergeometric function hyp1f1.
See also
http://functions.wolfram.com/HypergeometricFunctions/
Hypergeometric1F1/03/01/04/01/ .
Also
www.cs.unc.edu/Research/Image/MIDAG/p01/biostat/Digital_1.pdf .
Also a gsl implementation gsl_sf_hyperg_1F1_int in hyperg_1F1.c
specifically hyperg_1F1_ab_posint for positive integers a and b.
Also
http://mathworld.wolfram.com/
ConfluentHypergeometricFunctionoftheFirstKind.html
"""
hist = np.zeros(N_small + 1)
for n0 in range(1, N_small):
n1 = N_small - n0
prefix = scipy.comb(n0+n1, n0) * special.beta(n0, n1)
hist[n0] += m0 * prefix * diallelic_d_helper(n0, n1, g)
hist[n0] += m1 * prefix * diallelic_d_helper(n1, n0, -g)
return hist[1:-1] / np.sum(hist[1:-1])
def diallelic_approximation_d(N_small, g, m0, m1):
"""
This is experimental.
The numerical integration should be replaced
by a call to the confluent hypergeometric function hyp1f1.
See also
http://functions.wolfram.com/HypergeometricFunctions/
Hypergeometric1F1/03/01/04/01/ .
Also
www.cs.unc.edu/Research/Image/MIDAG/p01/biostat/Digital_1.pdf .
Also a gsl implementation gsl_sf_hyperg_1F1_int in hyperg_1F1.c
specifically hyperg_1F1_ab_posint for positive integers a and b.
Also
http://mathworld.wolfram.com/
ConfluentHypergeometricFunctionoftheFirstKind.html
"""
hist = np.zeros(N_small + 1)
for n0 in range(1, N_small):
n1 = N_small - n0
prefix = scipy.comb(n0 + n1, n0) * special.beta(n0, n1)
hist[n0] += m0 * prefix * diallelic_d_helper(n0, n1, g)
hist[n0] += m1 * prefix * diallelic_d_helper(n1, n0, -g)
return hist[1:-1] / np.sum(hist[1:-1])
def Adams_Moulton(k):
r"""
Construct the k-step, Adams-Moulton method.
The methods are implicit and have order k+1.
They have the form:
`y_{n+1} = y_n + h \sum_{j=0}^{k} \beta_j f(y_n-k+j+1)`
They are generated using equation (1.9) and the equation in
Exercise 3 from Hairer & Wanner III.1, along with the binomial
expansion.
.. note::
Accuracy is lost when evaluating the order conditions
for methods with many steps. This could be avoided by using
SAGE rationals instead of NumPy doubles for the coefficient
representation.
References:
[hairer1993]_
"""
from scipy import comb
alpha=np.zeros(k+1)
beta=np.zeros(k+1)
alpha[k]=1.
alpha[k-1]=-1.
gamma=np.zeros(k+1)
gamma[0]=1.
beta[k]=1.
betaj=np.zeros(k+1)
for j in range(1,k+1):
gamma[j]= -sum(gamma[:j]/np.arange(j+1,1,-1))
for i in range(0,j+1):
betaj[k-i]=(-1.)**i*comb(j,i)*gamma[j]
beta=beta+betaj
name=str(k)+'-step Adams-Moulton method'
return LinearMultistepMethod(alpha,beta,name=name)
def _munp(self, n, c):
k = np.arange(0, n + 1)
val = (1.0 / c)**n * np.sum(comb(n, k) * (-1)**k / (1.0 + c * k),
axis=0)
return where(c * n > -1, val, inf)
print 'zpkToCoeffs: ', scipy.allclose(rb, sb) and scipy.allclose(ra, sa) # Test the lpTolp functions a = scipy.array([1, -0.96, 0.80]) b = scipy.array([1, 0.5, 0.5]) freq = 0.23 rb, ra = loudia.lowPassToLowPass(b, a, freq) sb, sa = scipy.signal.lp2lp(b, a, freq) print 'lpTolp: ', scipy.allclose(rb, sb) and scipy.allclose(ra, sa) # Test the comb function rc = loudia.combination(5, 3) sc = scipy.comb(5, 3) print 'combination: ', round(sc) == round(rc) # Test the bilinear function a = scipy.array([10, -0.96, 0.80]) b = scipy.array([156, 0.5, 0.5]) rb, ra = loudia.bilinear(b, a, 1.0) sb, sa = scipy.signal.bilinear(b, a) # The loudia bilinear function does not return the coefficients normalized rb = rb / ra[:, 0] ra = ra / ra[:, 0] print 'bilinear: ', scipy.allclose(rb, sb) and scipy.allclose(ra, sa)
lines = []
headings = [
r"N_{\uparrow}", r"U/\mu B", r"M/N\mu", r"\Omega", r"S/k", r"kT/\mu B",
r"C/Nk"
]
nel = len(headings)
header = r"$" + r"$ & $".join(headings) + r"$ \\ \midrule \addlinespace[5pt]"
lines.append(r"\begin{tabular}{rrrcrrc}")
lines.append(header)
tableRows = []
for Nup in all_Nup:
Ndown = np.float96(Ntot - Nup)
U_by_mu_B = np.float96(Ntot - 2 * Nup)
M_by_N_mu = np.float96(Nup - Ndown) / np.float96(Ntot)
Omega = sp.comb(Ntot, Nup, exact=True)
S_by_k = np.log(np.float96(Omega))
tableRows.append([Nup, U_by_mu_B, M_by_N_mu, Omega, S_by_k])
rowNum = 0
for row in tableRows:
if rowNum == 0 or rowNum == Ntot:
kT_by_mu_B = 0
else:
kT_by_mu_B = (tableRows[rowNum+1][1]-tableRows[rowNum-1][1]) / \
(tableRows[rowNum+1][4]-tableRows[rowNum-1][4])
tableRows[rowNum].append(kT_by_mu_B)
rowNum += 1
rowNum = 0
for row in tableRows:
def runs_prob_even(self, r):
n0, n1 = self.n0, self.n1
tmp0 = comb(n0 - 1, r // 2 - 1)
tmp1 = comb(n1 - 1, r // 2 - 1)
return tmp0 * tmp1 * 2. / self.comball
def bin_pdf(n,p,k): return comb(n,k) * p**k * (1-p)**(n-k) def bin_cdf(n,p,m): return sum( (bin_pdf(n,p,k) for k in range(m+1)) )
def runs_prob_even(self, r):
n0, n1 = self.n0, self.n1
tmp0 = comb(n0-1, r//2-1)
tmp1 = comb(n1-1, r//2-1)
return tmp0 * tmp1 * 2. / self.comball
def __init__(self, n0, n1):
self.n0 = n0
self.n1 = n1
self.n = n = n0 + n1
self.comball = comb(n, n1)
#funcion que simula mil votaciones 100 veces para calcular un promedio de las mediciones
p = 0.0
a = 0.0
b = 0.0
c = 0.0
p1 = 0.0
pdf_x = []
pdf_y = []
for r in range(0,1000):
a = comb (33,18)*((float(r*0.001))**18)*((1-float(r*0.001))**15)
p+=a
pdf_x.append(r*0.001)
r+=1
for r in range(0,1000):
c = comb (33,18)*((float(r*0.001))**18)*((1-float(r*0.001))**15)
p2 = c/p
pdf_y.append(p2)
p1+=p2
r+=1
#hasta aca solo se definieron los ejes del primer grafico de la pdf
print(max(pdf_y))
for i in range (0,len(pdf_y)):
def kuiper_FPP(D, N):
"""Compute the false positive probability for the Kuiper statistic.
Uses the set of four formulas described in Paltani 2004; they report
the resulting function never underestimates the false positive probability
but can be a bit high in the N=40..50 range. (They quote a factor 1.5 at
the 1e-7 level.
Parameters
----------
D : float
The Kuiper test score.
N : float
The effective sample size.
Returns
-------
fpp : float
The probability of a score this large arising from the null hypothesis.
Reference
---------
Paltani, S., "Searching for periods in X-ray observations using
Kuiper's test. Application to the ROSAT PSPC archive", Astronomy and
Astrophysics, v.240, p.789-790, 2004.
"""
if D < 0. or D > 2.:
raise ValueError("Must have 0<=D<=2 by definition of the Kuiper test")
if D < 2. / N:
return 1. - factorial(N) * (D - 1. / N)**(N - 1)
elif D < 3. / N:
k = -(N * D - 1.) / 2.
r = sqrt(k**2 - (N * D - 2.) / 2.)
a, b = -k + r, -k - r
return 1. - factorial(N - 1) * (b**(N - 1.) * (1. - a) - a**(N - 1.) *
(1. - b)) / float(N)**(N - 2) * (b - a)
elif (D > 0.5 and N % 2 == 0) or (D > (N - 1.) / (2. * N) and N % 2 == 1):
def T(t):
y = D + t / float(N)
return y**(t - 3) * (y**3 * N - y**2 * t * (3. - 2. / N) / N - t *
(t - 1) * (t - 2) / float(N)**2)
s = 0.
# NOTE: the upper limit of this sum is taken from Stephens 1965
for t in xrange(int(floor(N * (1 - D))) + 1):
term = T(t) * comb(N, t) * (1 - D - t / float(N))**(N - t - 1)
s += term
return s
else:
z = D * sqrt(N)
S1 = 0.
term_eps = 1e-12
abs_eps = 1e-100
for m in itertools.count(1):
T1 = 2. * (4. * m**2 * z**2 - 1.) * exp(-2. * m**2 * z**2)
so = S1
S1 += T1
if abs(S1 - so) / (abs(S1) +
abs(so)) < term_eps or abs(S1 - so) < abs_eps:
break
S2 = 0.
for m in itertools.count(1):
T2 = m**2 * (4. * m**2 * z**2 - 3.) * exp(-2 * m**2 * z**2)
so = S2
S2 += T2
if abs(S2 - so) / (abs(S2) +
abs(so)) < term_eps or abs(S1 - so) < abs_eps:
break
return S1 - 8 * D / (3. * sqrt(N)) * S2
def CalcPValues(count_mat, prob_mat=None, exact=False):
"""Calculates p-values for the co-incidence matrix of two variables.
Computes the probability we would observe a greater (i,j) value in mat
by randomly sampling from the global distribution of row and column values.
Suppose the rows are oxygen requirements (aerobe, anaerobe) and the
columns are genotypes (ed, emp). Overall, there are 75 aerobes and 75 anaerobes,
while there are 100 emp genotypes and 50 ed. If the genes and oxygen requirements
were randomly distributed among organisms independently then the probability
of randomly choosing an anaerobe with ed genes would be (50/150) * (75/150) =
1/6 (meaning there should be 25 such organisms). However, suppose we observe
that 1/4 of organisms (39) are anaerobes with ed genes. The probability of
observing a more extreme value than 1/4 by drawing randomly is
sum_{i > 39} ((150 choose i) * 1/6^i * (5/6)^(150-i))
which is the sum of the probability of randomly choosing any single value
greater than 39.
Args:
count_mat: matrix with independent variable values on rows, dependents on columns.
prob_mat: the matrix with probabilities of each (i,j) point. If None, will
computed according to counts.
exact: if exact binomial coefficients should be computed.
Returns:
A matrix with the same shape as mat with a p-value at each i,j position.
"""
total = np.sum(count_mat)
ceil_total = int(np.ceil(total))
total2 = float(total**2)
pval_mat = np.matrix(np.zeros(count_mat.shape))
rows, cols = count_mat.shape
for i in xrange(rows):
for j in xrange(cols):
# Get the total count
observed_count = count_mat[i, j]
# Compute or fetch the probability of this pair.
if prob_mat is not None:
prob = prob_mat[i, j]
else:
total_ind = np.sum(count_mat[i, :])
total_dep = np.sum(count_mat[:, j])
prob = float(total_ind * total_dep) / total2
# Calculate a p-value for each more extreme case.
pvals = []
floor_count = int(np.floor(observed_count))
for higher_count in xrange(floor_count + 1, ceil_total + 1):
comb = scipy.comb(ceil_total, higher_count, exact=exact)
# If we can't compute the actual value of the binomial coefficient
# then approximate the log using stirlings approximation
if not np.isfinite(comb):
comb = ln_stirling_binomial(ceil_total, higher_count)
pval = np.log(comb)
pval += (higher_count * np.log(prob))
pval += ((total - higher_count) * np.log(1 - prob))
pvals.append(np.exp(pval))
# our p-value is the sum over all more extreme cases
pval_mat[i, j] = np.sum(pvals)
return pval_mat
def kmers2int():
work_queue = Queue()
result_queue = Queue()
work_to_do = itertools.combinations(range(len(db["genomes"])), 2)
num_work_to_do = int(round(scipy.comb(len(db["genomes"]), 2)))
while True:
batch = list(itertools.islice(work_to_do, 1000))
if not batch:
break
work_queue.put(batch)
for i in range(options.num_tasks):
work_queue.put(False)
tasks = [Process(target=kmers2int_worker,
args=(work_queue, result_queue)) for i in range(options.num_tasks)]
for t in tasks:
t.start()
sys.stderr.write("populating matrix\n")
true_s_time = time.time()
s_time = time.time()
status_change = True
work_done = 0
delta_time = []
while True:
for _ in range(10):
try:
data = result_queue.get(block=False)
db["hit_matrix"][data[0]][data[1]] = data[2]
work_done += 1
status_change = True
except Empty:
break
if sum([1 for t in tasks if t.is_alive()]) == 0:
break
if status_change and work_done % 100 == 0:
try:
work_per_time = float(100 * len(delta_time)) / float(sum(delta_time))
except ZeroDivisionError:
work_per_time = 0
delta_time.append(time.time() - s_time)
try:
time_to_finish = (num_work_to_do - work_done) / work_per_time
except ZeroDivisionError:
time_to_finish = 0
sys.stderr.write("\r%-79s" % (" %s/%s completed [%.2f/s, %s remaining]" % (work_done,
num_work_to_do, work_per_time, formattime(time_to_finish)),))
if len(delta_time) == 100:
delta_time = delta_time[1:]
s_time = time.time()
status_change = False
for t in tasks:
t.join()
sys.stderr.write("\r%-79s\n" % (" %s/%s completed in %s" % (work_done,
num_work_to_do, formattime(time.time() - true_s_time)),))
#-- Table header
lines = []
headings = [r"N_{\uparrow}", r"U/\mu B", r"M/N\mu", r"\Omega", r"S/k",
r"kT/\mu B", r"C/Nk"]
nel = len(headings)
header = r"$" + r"$ & $".join(headings) + r"$ \\ \midrule \addlinespace[5pt]"
lines.append(r"\begin{tabular}{rrrcrrc}")
lines.append(header)
tableRows = []
for Nup in all_Nup:
Ndown = np.float96(Ntot - Nup)
U_by_mu_B = np.float96(Ntot - 2*Nup)
M_by_N_mu = np.float96(Nup-Ndown)/np.float96(Ntot)
Omega = sp.comb(Ntot, Nup, exact=True)
S_by_k = np.log(np.float96(Omega))
tableRows.append( [Nup, U_by_mu_B, M_by_N_mu, Omega, S_by_k] )
rowNum = 0
for row in tableRows:
if rowNum == 0 or rowNum == Ntot:
kT_by_mu_B = 0
else:
kT_by_mu_B = (tableRows[rowNum+1][1]-tableRows[rowNum-1][1]) / \
(tableRows[rowNum+1][4]-tableRows[rowNum-1][4])
tableRows[rowNum].append( kT_by_mu_B )
rowNum += 1
rowNum = 0
for row in tableRows:
def learn(A,K,net0={},opts={}):
"""
runs variational bayes for inference of network modules
(i.e. community detection) under a constrained stochastic block
model.
net0 and opts inputs are optional. if provided, length(net0.a0)
must equal K.
inputs:
A: N-by-N undirected (symmetric), binary adjacency matrix w/o
self-edges (note: fastest for sparse and logical A)
K: (maximum) number of modules
net0: initialization/hyperparameter structure for network
net0['Q0']: N-by-K initial mean-field matrix (rows sum to 1)
net0['ap0']: alpha_{+0}, hyperparameter for prior on \theta_+
net0['bp0']: beta_{+0}, hyperparameter for prior on \theta_+
net0['am0']: alpha_{-0}, hyperparameter for prior on \theta_-
net0['bm0']: beta_{-0}, hyperparameter for prior on \theta_-
net0['a0']: alpha_{\mu0}, 1-by-K vector of hyperparameters for
prior on \pi
opts: options
opts['TOL_DF']: tolerance on change in F (outer loop)
opts['MAX_FITER']: maximum number of F iterations (outer loop)
opts['VERBOSE']: verbosity (0=quiet (default),1=print, 2=figures)
outputs:
net: posterior structure for network
net['F']: converged free energy (same as net.F_iter(end))
net['F_iter']: free energy over iterations (learning curve)
net['Q']: N-by-K mean-field matrix (rows sum to 1)
net['K']: K, passed for compatibility with vbmod_restart
net['ap']: alpha_+, hyperparameter for posterior on \theta_+
net['bp']: beta_+, hyperparameter for posterior on \theta_+
net['am']: alpha_-, hyperparameter for posterior on \theta_-
net['bm']: beta_-, hyperparameter for posterior on \theta_-
net['a']: alpha_{\mu}, 1-by-K vector of hyperparameters for
posterior on \pi
"""
# default options
TOL_DF=1e-2
MAX_FITER=30
VERBOSE=0
SAVE_ITER=0
#print "get options from opts struct"
if (type(opts) == type({})) and (len(opts) > 0):
if 'TOL_DF' in opts: TOL_DF=opts['TOL_DF']
if 'MAX_FITER' in opts: MAX_FITER=opts['MAX_FITER']
if 'VERBOSE' in opts: VERBOSE=opts['VERBOSE']
if 'SAVE_ITER' in opts: SAVE_ITER=opts['SAVE_ITER']
N=A.shape[0] # number of nodes
M=0.5*A.sum(0).sum(1) # total number of non-self edges
M=M[0,0]
C=comb(N,2) # total number of possible edges between N nodes
uk=mat(ones([K,1]))
un=mat(ones([N,1]))
#print "default prior hyperparameters"
ap0=2;
bp0=1;
am0=1;
bm0=2;
a0=ones([1,K]);
#print "get initial Q0 matrix and prior hyperparameters from net0 struct"
if (type(net0) == type({})) and (len(net0) > 0):
if 'Q0' in net0: Q=net0['Q0']
if 'ap0' in net0: ap0=net0['ap0']
if 'bp0' in net0: bp0=net0['bp0']
if 'am0' in net0: am0=net0['am0']
if 'bm0' in net0: bm0=net0['bm0']
if 'a0' in net0: a0=net0['a0']
#print "initialize Q if not provided"
try: Q
except NameError: Q=init(N,K)
Qmat=mat(Q)
#print "size of Q=", Q.shape
#Q=init(N,K)
# ensure a0 is a 1-by-K vector
assert(a0.shape == (1,K))
#print "intialize variational distribution hyperparameters to be equal to prior hyperparameters"
ap=ap0
bp=bp0
am=am0
bm=bm0
a=a0
n=Q.sum(0)
# get indices of non-zero row/columns
# to be passed to vbmod_estep_inline
# jntj: must be better way
(rows,cols)=A.nonzero()
# vector to store free energy over iterations
F=[]
for i in range(MAX_FITER):
####################
#VBE-step, to update mean-field Q matrix over module assignments
####################
# compute local and global coupling constants, JL and JG and
# chemical potentials -lnpi
psiap=digamma(ap)
psibp=digamma(bp)
psiam=digamma(am)
psibm=digamma(bm)
psip=digamma(ap+bp)
psim=digamma(am+bm)
JL=psiap-psibp-psiam+psibm
JG=psibm-psim-psibp+psip
lnpi=digamma(a)-digamma(sum(a))
estep_inline(rows,cols,Q,float(JL),float(JG),array(lnpi),array(n))
"""
# local update (technically correct, but slow)
for l in range(N):
# exclude Q[l,:] from contributing to its own update
Q[l,:]=zeros([1,K])
# jntj: doesn't take advantage of sparsity
Al=mat(A.getrow(l).toarray())
AQl=multiply((Al.T*uk.T),Q).sum(0)
nl=Q.sum(0)
lnQl=JL*AQl-JG*nl+lnpi
lnQl=lnQl-lnQl.max()
Q[l,:]=exp(lnQl)
Q[l,:]=Q[l,:]/Q[l,:].sum()
"""
####################
#VBM-step, update distribution over parameters
####################
# compute expected occupation numbers <n*>s
n=Qmat.sum(0)
# compute expected edge counts <n**>s
#QTAQ=mat((Q.T*A*Q).toarray())
#npp=0.5*trace(QTAQ)
npp=0.5*(Qmat.T*A*Qmat).diagonal().sum()
npm=0.5*trace(Qmat.T*(un*n-Qmat))-npp
nmp=M-npp
nmm=C-M-npm
# compute hyperparameters for beta and dirichlet distributions over
# theta_+, theta_-, and pi_mu
ap=npp+ap0
bp=npm+bp0
am=nmp+am0
bm=nmm+bm0
a=n+a0
#print ap, bp, am, bm, a
# evaluate variational free energy, an approximation to the
# negative log-evidence
F.append(betaln(ap,bp)-betaln(ap0,bp0)+betaln(am,bm)-betaln(am0,bm0)+sum(gammaln(a))-gammaln(sum(a))-(sum(gammaln(a0))-gammaln(sum(a0)))-sum(multiply(Qmat,log(Qmat))))
F[i]=-F[i]
print "iteration", i+1 , ": F =", F[i]
# F should always decrease
if (i>1) and F[i] > F[i-1]:
print "\twarning: F increased from", F[i-1] ,"to", F[i]
if (i>1) and (abs(F[i]-F[i-1]) < TOL_DF):
break
return dict(F=F[-1],F_iter=F,Q=Q,K=K,ap=ap,bp=bp,am=am,bm=bm,a=a)
def pmf(self, m,n):
perm = comb(m+n, m)
T2 = self.values * m * n / ((m+n)**2 * lcm(m,n)**2) # zeta -> T2
pmf = self.frequencies / perm
return T2, pmf
def __init__(self, n0, n1):
self.n0 = n0
self.n1 = n1
self.n = n = n0 + n1
self.comball = comb(n, n1)
def bin_pdf(n, p, k):
return comb(n, k) * p**k * (1 - p)**(n - k)
def pmf(self, m, n):
perm = comb(m + n, m)
T2 = self.values * m * n / ((m + n)**2 * lcm(m, n)**2) # zeta -> T2
pmf = self.frequencies / perm
return T2, pmf
def M_step(self, anneal, model_params, my_suff_stat, my_data):
""" BSC M_step
my_data variables used:
my_data['y'] Datapoints
my_data['candidates'] Candidate H's according to selection func.
Annealing variables used:
anneal['T'] Temperature for det. annealing
anneal['N_cut_factor'] 0.: no truncation; 1. trunc. according to model
"""
comm = self.comm
H, Hprime = self.H, self.Hprime
gamma = self.gamma
W = model_params['W'].T
pies = model_params['pi']
sigma = model_params['sigma']
mu = model_params['mu']
# Read in data:
my_y = my_data['y'].copy()
candidates = my_data['candidates']
logpj_all = my_suff_stat['logpj']
all_denoms = np.exp(logpj_all).sum(axis=1)
my_N, D = my_y.shape
N = comm.allreduce(my_N)
# Joerg's data noise idea
data_noise_scale = anneal['data_noise']
if data_noise_scale > 0:
my_y += my_data['data_noise']
SM = self.state_matrix # shape: (no_states, Hprime)
# To compute et_loglike:
my_ldenom_sum = 0.0
ldenom_sum = 0.0
# Precompute factor for pi update
A_pi_gamma = 0
B_pi_gamma = 0
for gamma_p in range(gamma + 1):
A_pi_gamma += comb(H, gamma_p) * (pies**gamma_p) * (
(1 - pies)**(H - gamma_p))
B_pi_gamma += gamma_p * comb(H, gamma_p) * (pies**gamma_p) * (
(1 - pies)**(H - gamma_p))
E_pi_gamma = pies * H * A_pi_gamma / B_pi_gamma
# Truncate data
if anneal['Ncut_factor'] > 0.0:
tracing.tracepoint("M_step:truncating")
#alpha = 0.9 # alpha from ET paper
#N_use = int(alpha * (N * (1 - (1 - A_pi_gamma) * anneal['Ncut_factor'])))
N_use = int(N * (1 - (1 - A_pi_gamma) * anneal['Ncut_factor']))
cut_denom = parallel.allsort(all_denoms)[-N_use]
which = np.array(all_denoms >= cut_denom)
candidates = candidates[which]
logpj_all = logpj_all[which]
my_y = my_y[which]
my_N, D = my_y.shape
N_use = comm.allreduce(my_N)
else:
N_use = N
dlog.append('N', N_use)
# Calculate truncated Likelihood
L = H * np.log(1 - pies) - 0.5 * D * np.log(
2 * pi * sigma**2) - np.log(A_pi_gamma)
Fs = np.log(np.exp(logpj_all).sum(axis=1)).sum()
L += comm.allreduce(Fs) / N_use
dlog.append('L', L)
# Precompute
pil_bar = np.log(pies / (1. - pies))
corr_all = logpj_all.max(axis=1) # shape: (my_N,)
pjb_all = np.exp(logpj_all -
corr_all[:, None]) # shape: (my_N, no_states)
# Allocate
my_Wp = np.zeros_like(W) # shape (H, D)
my_Wq = np.zeros((H, H)) # shape (H, H)
my_pi = 0.0 #
my_sigma = 0.0 #
#my_mup = np.zeros_like(W) # shape (H, D)
#my_muq = np.zeros((H,H)) # shape (H, H)
my_mus = np.zeros(H) # shape D
data_sum = my_y.sum(axis=0) # sum over all data points for mu update
## Calculate mu
#for n in xrange(my_N):
#tracing.tracepoint("Calculationg offset")
#y = my_y[n,:] # length D
#cand = candidates[n,:] # length Hprime
#logpj = logpj_all[n,:] # length no_states
#corr = corr_all[n] # scalar
#pjb = pjb_all[n, :]
## Zero active hidden cause (do nothing for the W and pi case)
## this_Wp += 0. # nothing to do
## this_Wq += 0. # nothing to do
## this_pi += 0. # nothing to do
## One active hidden cause
#this_mup = np.outer(pjb[1:(H+1)],y)
#this_muq = pjb[1:(H+1)] * np.identity(H)
#this_mus = pjb[1:(H+1)]
## Handle hidden states with more than 1 active cause
#this_mup[cand] += np.dot(np.outer(y,pjb[(1+H):]),SM).T
#this_muq_tmp = np.zeros_like(my_muq[cand])
#this_muq_tmp[:,cand] = np.dot(pjb[(1+H):] * SM.T,SM)
#this_muq[cand] += this_muq_tmp
#this_mus[cand] += np.inner(SM.T,pjb[(1+H):])
#denom = pjb.sum()
#my_mup += this_mup / denom
#my_muq += this_muq / denom
#my_mus += this_mus / denom
## Calculate updated mu
#if 'mu' in self.to_learn:
#tracing.tracepoint("M_step:update mu")
#mup = np.empty_like(my_mup)
#muq = np.empty_like(my_muq)
#mus = np.empty_like(my_mus)
#all_data_sum = np.empty_like(data_sum)
#comm.Allreduce( [my_mup, MPI.DOUBLE], [mup, MPI.DOUBLE] )
#comm.Allreduce( [my_muq, MPI.DOUBLE], [muq, MPI.DOUBLE] )
#comm.Allreduce( [my_mus, MPI.DOUBLE], [mus, MPI.DOUBLE] )
#comm.Allreduce( [data_sum, MPI.DOUBLE], [all_data_sum, MPI.DOUBLE] )
#mu_numer = all_data_sum - np.dot(mus,np.dot(np.linalg.inv(muq), mup))
#mu_denom = my_N - np.dot(mus,np.dot(np.linalg.inv(muq), mus))
#mu_new = mu_numer/ mu_denom
#else:
#mu_new = mu
# Iterate over all datapoints
tracing.tracepoint("M_step:iterating")
for n in range(my_N):
y = my_y[n, :] - mu # length D
cand = candidates[n, :] # length Hprime
pjb = pjb_all[n, :]
this_Wp = np.zeros_like(
my_Wp) # numerator for current datapoint (H, D)
this_Wq = np.zeros_like(
my_Wq) # denominator for current datapoint (H, H)
this_pi = 0.0 # numerator for pi update (current datapoint)
# Zero active hidden cause (do nothing for the W and pi case)
# this_Wp += 0. # nothing to do
# this_Wq += 0. # nothing to do
# this_pi += 0. # nothing to do
# One active hidden cause
this_Wp = np.outer(pjb[1:(H + 1)], y)
this_Wq = pjb[1:(H + 1)] * np.identity(H)
this_pi = pjb[1:(H + 1)].sum()
this_mus = pjb[1:(H + 1)].copy()
# Handle hidden states with more than 1 active cause
this_Wp[cand] += np.dot(np.outer(y, pjb[(1 + H):]), SM).T
this_Wq_tmp = np.zeros_like(my_Wq[cand])
this_Wq_tmp[:, cand] = np.dot(pjb[(1 + H):] * SM.T, SM)
this_Wq[cand] += this_Wq_tmp
this_pi += np.inner(pjb[(1 + H):], SM.sum(axis=1))
this_mus[cand] += np.inner(SM.T, pjb[(1 + H):])
denom = pjb.sum()
my_Wp += this_Wp / denom
my_Wq += this_Wq / denom
my_pi += this_pi / denom
my_mus += this_mus / denom
# Calculate updated W
if 'W' in self.to_learn:
tracing.tracepoint("M_step:update W")
Wp = np.empty_like(my_Wp)
Wq = np.empty_like(my_Wq)
comm.Allreduce([my_Wp, MPI.DOUBLE], [Wp, MPI.DOUBLE])
comm.Allreduce([my_Wq, MPI.DOUBLE], [Wq, MPI.DOUBLE])
#W_new = np.dot(np.linalg.inv(Wq), Wp)
#W_new = np.linalg.solve(Wq, Wp) # TODO check and switch to this one
rcond = -1
if float(np.__version__[2:]) >= 14.0:
rcond = None
W_new = np.linalg.lstsq(
Wq, Wp, rcond=rcond)[0] # TODO check and switch to this one
else:
W_new = W
# Calculate updated pi
if 'pi' in self.to_learn:
tracing.tracepoint("M_step:update pi")
pi_new = E_pi_gamma * comm.allreduce(my_pi) / H / N_use
else:
pi_new = pies
# Calculate updated sigma
if 'sigma' in self.to_learn:
tracing.tracepoint("M_step:update sigma")
# Loop for sigma update:
for n in range(my_N):
y = my_y[n, :] - mu # length D
cand = candidates[n, :] # length Hprime
logpj = logpj_all[n, :] # length no_states
corr = logpj.max() # scalar
pjb = np.exp(logpj - corr)
# Zero active hidden causes
this_sigma = pjb[0] * (y**2).sum()
# Hidden states with one active cause
this_sigma += (pjb[1:(H + 1)] * ((W - y)**2).sum(axis=1)).sum()
# Handle hidden states with more than 1 active cause
SM = self.state_matrix # is (no_states, Hprime)
W_ = W[cand] # is (Hprime x D)
Wbar = np.dot(SM, W_)
this_sigma += (pjb[(H + 1):] *
((Wbar - y)**2).sum(axis=1)).sum()
denom = pjb.sum()
my_sigma += this_sigma / denom
sigma_new = np.sqrt(comm.allreduce(my_sigma) / D / N_use)
else:
sigma_new = sigma
# Calculate updated mu:
if 'mu' in self.to_learn:
tracing.tracepoint("M_step:update mu")
mus = np.empty_like(my_mus)
all_data_sum = np.empty_like(data_sum)
comm.Allreduce([my_mus, MPI.DOUBLE], [mus, MPI.DOUBLE])
comm.Allreduce([data_sum, MPI.DOUBLE], [all_data_sum, MPI.DOUBLE])
mu_new = all_data_sum / my_N - np.inner(W_new.T / my_N, mus)
else:
mu_new = mu
for param in anneal.crit_params:
exec('this_param = ' + param)
anneal.dyn_param(param, this_param)
dlog.append('N_use', N_use)
return {'W': W_new.T, 'pi': pi_new, 'sigma': sigma_new, 'mu': mu_new}
def avail(p, n, k):
q = 1 - p
return sum(comb(n, i) * (p**i) * (q**(n - i)) for i in xrange(k, n + 1))
def _munp(self, n, c):
k = np.arange(0,n+1)
val = (1.0/c)**n * np.sum(comb(n,k)*(-1)**k / (1.0+c*k),axis=0)
return where(c*n > -1, val, inf)
# Test the lpTolp functions a = scipy.array([1, -0.96, 0.80]) b = scipy.array([1, 0.5, 0.5]) freq = 0.23 rb, ra = loudia.lowPassToLowPass(b, a, freq) sb, sa = scipy.signal.lp2lp(b, a, freq) print 'lpTolp: ', scipy.allclose(rb, sb) and scipy.allclose(ra, sa) # Test the comb function rc = loudia.combination(5, 3) sc = scipy.comb(5, 3) print 'combination: ', round(sc) == round(rc) # Test the bilinear function a = scipy.array([10, -0.96, 0.80]) b = scipy.array([156, 0.5, 0.5]) rb, ra = loudia.bilinear(b, a, 1.0) sb, sa = scipy.signal.bilinear(b, a) # The loudia bilinear function does not return the coefficients normalized rb = rb / ra[:,0] ra = ra / ra[:,0] print 'bilinear: ', scipy.allclose(rb, sb) and scipy.allclose(ra, sa)
1], 'or 1 in', 1. / freq[1]
#('Prob of 0 sets for the independent case', 0.00038668213395202477, 'or 1 in', 2586.1034482758619)
nsolutions = []
n15 = []
for i in range(5000):
count15 = 0
for cards in one_game():
if cards.shape[1] > 12:
count15 += 1
nsolutions.append(len(find_sets2(cards)))
n15.append(count15)
pylab.gcf().set_size_inches(11, 6.4)
freq, bins, _ = pylab.hist(nsolutions, bins=range(-1, 14), normed=1, hold=0)
pylab.draw()
print 'Prob of 0 sets for 15 cards', freq[1], 'or 1 in', 1. / freq[1]
# ('Prob of 0 sets for 15 cards', 0.010681255698840693, 'or 1 in', 93.621951219512198)
freq, bins, _ = pylab.hist(n15, bins=range(-1, 14), normed=1, hold=0)
print 'number of triplets in 12 cards', scipy.comb(12, 3)
print 'number of new triplets when adding 3 cards', 3 * scipy.comb(
12, 2) + scipy.comb(3, 2) * 12 + 1
# save
import cPickle
f = open('more_set_results.pickle', 'w')
cPickle.dump((nsolutions_indep, nsolutions, n15), f)
f.close()
