def _offsetModel(kd1, kd2, maxShift, rotOrigin, precision):
"""
Factory function to return MCMC model for coordinate offset calculation
:param kd1:
:param kd2:
:param maxShift:
:param rotOrigin:
:param precision:
:return:
"""
rotOrigin = np.array(rotOrigin)
maxShift = np.array(maxShift)
shift = Uniform('shift', lower=-maxShift, upper=maxShift)
xyError = Uniform('xyerror', lower=precision / 1000, upper=precision * 1000)
@deterministic(plot=False)
def coordResiduals(kd1=kd1, kd2=kd2, shift=shift):
c1Xform, c2Xform = _transformCoords(kd1.data, kd2.data,
shift, rotOrigin)
d12, ind1near2 = kd1.query(c2Xform)
d21, ind2near1 = kd2.query(c1Xform)
msk1 = ind1near2[ind2near1] == np.arange(ind2near1.size)
resid = c1Xform - kd2.data[ind2near1]
resid[~msk1] = 0
return resid
opt = Cauchy('data', value=np.zeros_like(kd1.data),
alpha=coordResiduals, beta=xyError, observed=True)
return [shift, xyError, coordResiduals, opt]
def model(prob):
#observations
obs = prob.get_observation_values()
#priors
variables = []
sig = Uniform('sig', 0.0, 100.0, value=1.)
variables.append(sig)
a1 = Uniform('a1', 0.0, 5.0)
variables.append(a1)
k1 = Uniform('k1', 0.01, 2.0)
variables.append(k1)
a2 = Uniform('a2', 0.0, 5.0)
variables.append(a2)
k2 = Uniform('k2', 0.01, 2.0)
variables.append(k2)
#model
@deterministic()
def response(pars=variables, prob=prob):
values = []
for par in pars:
values.append(par)
values = array(values)
prob.set_parameters(values)
prob.forward()
return prob.get_simvalues()
#likelihood
y = Normal('y', mu=response, tau=1.0 / sig**2, value=obs, observed=True)
variables.append(y)
return variables
def Model_twostage_fit_v2(n_TF, n_gene, p_gene_array, p_TF_gene_array, num_iter, num_burn, num_thin, prior_T, prior_T_method, r_TF_gene, a_TF_gene_h1, a_TF_gene_h0, a_gene):
"""
Assumptions: We allow learning of parameters
"""
a_gp = float(a_gene)
if a_TF_gene_h0 == 'None':
a_tg0 = Uniform('a_tg0', lower=0.5, upper=1)
else:
a_tg0 = float(a_TF_gene_h0)
if a_TF_gene_h1 == 'None':
a_tg1 = Uniform('a_tg1', lower=0, upper=0.5)
else:
a_tg1 = float(a_TF_gene_h1)
p_T = float(prior_T)
if r_TF_gene == 'None':
r_tg = Uniform('r_tg', lower=0, upper=1)
else:
r_tg = float(r_TF_gene)
p_gene = np.zeros(n_gene, dtype=object) #the ovserved variables
T = np.zeros((n_TF, n_gene), dtype=object) #variables showing TF-gene-pheno relationship
T_sum = np.zeros(n_gene, dtype=object)
p_TF_gene = np.zeros((n_TF, n_gene), dtype=object) #p-value of correlation of gene TF
for j in range(n_gene):
for i in range(n_TF):
T[i, j] = Bernoulli('T_%i_%i' %(i, j), p=p_T)
#If T[i, j] = 0: then p_TF_gene is coming from a mixture of beta and uniform (r is the mixture param)
@pymc.stochastic(name='p_TF_gene_%i_%i' %(i, j), dtype=float, observed=True)
def temp_p_TF_gene(value=p_TF_gene_array[i, j], TF_gene_ind=T[i, j], a0=a_tg0, a1=a_tg1, r=r_tg) :
if TF_gene_ind:
out = pymc.distributions.beta_like(value, alpha=a1, beta=1)
else:
out = np.log(r * np.exp(pymc.distributions.beta_like(value, alpha=a1, beta=1))
+ (1 - r) * np.exp(pymc.distributions.beta_like(value, alpha=a0, beta=1)))
return out
p_TF_gene[i, j] = temp_p_TF_gene
#we define a deterministic function to find values of T
@pymc.deterministic(name='T_sum_%i' %j, plot=False)
def temp_T_sum(ind_vec=T[:,j]):
return (np.sum(ind_vec)>0)
T_sum[j] = temp_T_sum
#If T_sum[j] == 0: then p_TF_gene is coming from a uniform; else, beta
@pymc.stochastic(name='p_gene_%i' %j, dtype=float, observed=True)
def temp_p_gene(value=p_gene_array[j], ind=T_sum[j], a=a_gp):
if ind:
out = pymc.distributions.beta_like(value, alpha=a, beta=1)
else:
out = pymc.distributions.uniform_like(value, 0, 1)
return out
p_gene[j] = temp_p_gene
if a_gene == None and a_TF_gene_h0 == None and a_TF_gene_h1 == None:
M5 = pymc.MCMC([T, T_sum, a_gp, a_tg0, a_tg1])
else:
M5 = pymc.MCMC([T, T_sum])
M5.sample(iter=int(num_iter), burn=int(num_burn), thin=int(num_thin))
return(M5)
def createSignalModelExponential(data):
"""
Toy model that treats the first ~10% of the waveform as an exponential. Does a good job of finding the start time (t_0)
Since I made this as a toy, its super brittle. Waveform must be normalized
"""
print "Creating model"
switchpoint = DiscreteUniform('switchpoint', lower=0, upper=len(data))
noise_sigma = HalfNormal('noise_sigma', tau=sigToTau(.01))
exp_sigma = HalfNormal('exp_sigma', tau=sigToTau(.05))
#Modeling these parameters this way is why wf needs to be normalized
exp_rate = Uniform('exp_rate', lower=0, upper=.1)
exp_scale = Uniform('exp_scale', lower=0, upper=.1)
timestamp = np.arange(0, len(data), dtype=np.float)
@deterministic(plot=False, name="test")
def uncertainty_model(s=switchpoint, n=noise_sigma, e=exp_sigma):
''' Concatenate Poisson means '''
out = np.empty(len(data))
out[:s] = n
out[s:] = e
return out
@deterministic
def tau(eps=uncertainty_model):
return np.power(eps, -2)
## @deterministic(plot=False, name="test2")
## def adjusted_scale(s=switchpoint, s1=exp_scale):
## out = np.empty(len(data))
## out[:s] = s1
## out[s:] = s1
## return out
#
# scale_param = adjusted_scale(switchpoint, exp_scale)
@deterministic(plot=False)
def baseline_model(s=switchpoint, r=exp_rate, scale=exp_scale):
out = np.zeros(len(data))
out[s:] = scale * (np.exp(r * (timestamp[s:] - s)) - 1.)
# plt.figure(fig.number)
# plt.clf()
# plt.plot(out ,color="blue" )
# plt.plot(data ,color="red" )
# value = raw_input(' --> Press q to quit, any other key to continue\n')
return out
baseline_observed = Normal("baseline_observed",
mu=baseline_model,
tau=tau,
value=data,
observed=True)
return locals()
def make_model(self, data):
assert len(data) == 2, 'There must be exactly two data arrays'
name1, name2 = sorted(data.keys())
y1 = np.array(data[name1])
y2 = np.array(data[name2])
assert y1.ndim == 1
assert y2.ndim == 1
y = np.concatenate((y1, y2))
mu_m = np.mean(y)
mu_p = 0.000001 * 1 / np.std(y)**2
sigma_low = np.std(y) / 1000
sigma_high = np.std(y) * 1000
# the five prior distributions for the parameters in our model
group1_mean = Normal('group1_mean', mu_m, mu_p)
group2_mean = Normal('group2_mean', mu_m, mu_p)
group1_std = Uniform('group1_std', sigma_low, sigma_high)
group2_std = Uniform('group2_std', sigma_low, sigma_high)
nu_minus_one = Exponential('nu_minus_one', 1 / 29)
@deterministic(plot=False)
def nu(n=nu_minus_one):
out = n + 1
return out
@deterministic(plot=False)
def lam1(s=group1_std):
out = 1 / s**2
return out
@deterministic(plot=False)
def lam2(s=group2_std):
out = 1 / s**2
return out
group1 = NoncentralT(name1,
group1_mean,
lam1,
nu,
value=y1,
observed=True)
group2 = NoncentralT(name2,
group2_mean,
lam2,
nu,
value=y2,
observed=True)
return Model({
'group1': group1,
'group2': group2,
'group1_mean': group1_mean,
'group2_mean': group2_mean,
'group1_std': group1_std,
'group2_std': group2_std,
})
def compute(var_LB, var_UB, num_samples=10):
from pymc import Uniform, MCMC
X = Uniform('X', var_LB, var_UB)
mc = MCMC([X])
mc.sample(num_samples)
#import matplotlib.pyplot as plt
#plt.plot(X.trace()[:,0], X.trace()[:,1],',')
#plt.show()
return X.trace()
def toy_model(tau=10000, prior='Beta0.5'):
b_obs = 200
f_AB = 400
f_CB = 1000
f_CA = 600
A = np.array([0, f_AB, f_CA, 0, f_CB, 0])
if prior == 'Normal':
ABp = Normal('ABp', mu=0.5, tau=100, trace=True)
CBp = Normal('CBp', mu=0.5, tau=100, trace=True)
CAp = Normal('CAp', mu=0.5, tau=100, trace=True)
elif prior == 'Uniform':
ABp = Uniform('ABp', lower=0.0, upper=1.0, trace=True)
CBp = Uniform('CBp', lower=0.0, upper=1.0, trace=True)
CAp = Uniform('CAp', lower=0.0, upper=1.0, trace=True)
elif prior == 'Beta0.25':
ABp = Beta('ABp', alpha=0.25, beta=0.25, trace=True)
CBp = Beta('CBp', alpha=0.25, beta=0.25, trace=True)
CAp = Beta('CAp', alpha=0.25, beta=0.25, trace=True)
elif prior == 'Beta0.5':
ABp = Beta('ABp', alpha=0.5, beta=0.5, trace=True)
CBp = Beta('CBp', alpha=0.5, beta=0.5, trace=True)
CAp = Beta('CAp', alpha=0.5, beta=0.5, trace=True)
elif prior == 'Beta2':
ABp = Beta('ABp', alpha=2, beta=2, trace=True)
CBp = Beta('CBp', alpha=2, beta=2, trace=True)
CAp = Beta('CAp', alpha=2, beta=2, trace=True)
elif prior == 'Gamma':
ABp = Gamma('ABp', alpha=1, beta=0.5, trace=True)
CBp = Gamma('CBp', alpha=1, beta=0.5, trace=True)
CAp = Gamma('CAp', alpha=1, beta=0.5, trace=True)
AB1 = ABp
AB3 = 1 - ABp
CB4 = CBp
CB5 = 1 - CBp
CA42 = CAp
CA52 = 1 - CAp
b = Normal('b',
mu=f_AB * AB3 + f_CB * CB4 + f_CA * CA42,
tau=tau,
value=b_obs,
observed=True,
trace=True)
# print [x.value for x in [ABp,CBp,CAp]]
# print b.logp
return locals()
def test_simple(self):
# Priors
mu = Normal('mu', mu=0, tau=0.0001)
s = Uniform('s', lower=0, upper=100, value=10)
tau = s**-2
# Likelihood with missing data
x = Normal('x', mu=mu, tau=tau, value=m, observed=True)
# Instantiate sampler
M = MCMC([mu, s, tau, x])
# Run sampler
M.sample(10000, 5000, progress_bar=0)
# Check length of value
assert_equal(len(x.value), 100)
# Check size of trace
tr = M.trace('x')()
assert_equal(shape(tr), (5000, 2))
sd2 = [-2 < i < 2 for i in ravel(tr)]
# Check for standard normal output
assert_almost_equal(sum(sd2) / 10000., 0.95, decimal=1)
def _model(data, robust=False):
# priors might be adapted here to be less flat
mu = Normal('mu', 0, 0.000001, size=2)
sigma = Uniform('sigma', 0, 1000, size=2)
rho = Uniform('r', -1, 1)
# we have a further parameter (prior) for the robust case
if robust == True:
nu = Exponential('nu', 1 / 29., 1)
# we model nu as an Exponential plus one
@pymc.deterministic
def nuplus(nu=nu):
return nu + 1
@pymc.deterministic
def precision(sigma=sigma, rho=rho):
ss1 = float(sigma[0] * sigma[0])
ss2 = float(sigma[1] * sigma[1])
rss = float(rho * sigma[0] * sigma[1])
return inv(np.mat([[ss1, rss], [rss, ss2]]))
if robust == True:
# log-likelihood of multivariate t-distribution
@pymc.stochastic(observed=True)
def mult_t(value=data.T, mu=mu, tau=precision, nu=nuplus):
k = float(tau.shape[0])
res = 0
for r in value:
delta = r - mu
enum1 = gammaln((nu + k) / 2.) + 0.5 * log(det(tau))
denom = (k / 2.) * log(nu * pi) + gammaln(nu / 2.)
enum2 = (-(nu + k) /
2.) * log(1 + (1 / nu) * delta.dot(tau).dot(delta.T))
result = enum1 + enum2 - denom
res += result[0]
return res[0, 0]
else:
mult_n = MvNormal('mult_n',
mu=mu,
tau=precision,
value=data.T,
observed=True)
return locals()
def set_priors(self):
"""set parameters prior distributions.
Hardcoded behavior for now, with non committing prior knowledge.
:return: None
"""
for group in ['control', 'variant']:
self.stochastics[group + '_p'] = Uniform(group + '_p', 0, 1)
def CreateBGModel(data, ene, flat, tritSpec, ge68Peak, fe55Peak, zn65Peak, flatSpec):
# flat = Container([])
# flatSpec = Container([flat.append(1.0) for i in xrange(len(ene))])
# print type(flatSpec)
# flat = []
# for i in xrange(len(ene)):
# flat.append(1.0)
# flatSpec = np.asarray(flat)
# stochastic variables (not compleletely detmined by parent values; they have a prob. distribution.)
# need to float resolution, energy, and offset for each peak ...
tritScale = Uniform('tritScale', lower=0, upper=10)
flatScale = Uniform('flatScale', lower=0, upper=10)
fe55Scale = Uniform('fe55Scale', lower=0, upper=10)
zn65Scale = Uniform('zn65Scale', lower=0, upper=10)
ge68Scale = Uniform('ge68Scale', lower=0, upper=10)
# deterministic variables : given by values of parents
# set up the model for uncertainty (ie, the noise) and the signal (ie, the spectrum)
@deterministic(plot=False, name="uncertainty")
def uncertainty_model(s=tritScale):
out = s * np.ones(len(data))
return out
@deterministic
def tau(eps=uncertainty_model):
# pymc uses this tau parameter instead of sigma to model a gaussian. its annoying.
return np.power(eps, -2)
@deterministic(plot=False, name="BGModel")
def signal_model(t=tritScale, f=flatScale, fe=fe55Scale, zn=zn65Scale, ge=ge68Scale):
theModel = t * tritSpec + fe * fe55Peak + zn * zn65Peak + ge * ge68Peak
return theModel
# full model
baseline_observed = Normal("baseline_observed", mu=signal_model, tau=tau, value=data, observed=True)
return locals()
def __init__(self, name, role=None, groups=[], lam=None, env=None):
super(PoissonStudent, self).__init__(name, role, groups, env)
if lam is not None:
self.lam = lam
else:
self.lam = Uniform('lam_%s' % self.name, lower=0, upper=1)
self.expT = Exponential('tau_%s' % self.name, self.lam)
self.dt = []
self.timestamps = []
self.t = 0
self.params = [self.lam, self.expT]
def set_priors(self):
"""set parameters prior distributions.
Hardcoded behavior for now, with non committing prior knowledge.
:return: None
"""
obs = np.concatenate((self.control, self.variant))
obs_mean = np.mean(obs)
for group in ['control', 'variant']:
self.stochastics[group + '_mu'] = Uniform(group + '_mu', 0, 100 * obs_mean)
def set_priors(self):
"""set parameters prior distributions.
Hardcoded behavior for now, with non committing prior knowledge.
:return: None
"""
obs = np.concatenate((self.control, self.variant))
obs_mean, obs_sigma = np.mean(obs), np.std(obs)
for group in ['control', 'variant']:
self.stochastics[group + '_mean'] = Normal(group + '_mean', obs_mean, 0.000001 / obs_sigma ** 2)
self.stochastics[group + '_sigma'] = Uniform(group + '_sigma', obs_sigma / 1000, obs_sigma * 1000)
def test_find_MAP():
tol = 2.0**-11 # 16 bit machine epsilon, a low bar
data = np.random.randn(100)
# data should be roughly mean 0, std 1, but let's
# normalize anyway to get it really close
data = (data - np.mean(data)) / np.std(data)
with Model():
mu = Uniform("mu", -1, 1)
sigma = Uniform("sigma", 0.5, 1.5)
Normal("y", mu=mu, tau=sigma**-2, observed=data)
# Test gradient minimization
map_est1 = starting.find_MAP(progressbar=False)
# Test non-gradient minimization
map_est2 = starting.find_MAP(progressbar=False, method="Powell")
close_to(map_est1["mu"], 0, tol)
close_to(map_est1["sigma"], 1, tol)
close_to(map_est2["mu"], 0, tol)
close_to(map_est2["sigma"], 1, tol)
def test_find_MAP():
tol = 2.0**-11 # 16 bit machine epsilon, a low bar
data = np.random.randn(100)
# data should be roughly mean 0, std 1, but let's
# normalize anyway to get it really close
data = (data - np.mean(data)) / np.std(data)
with Model() as model:
mu = Uniform('mu', -1, 1)
sigma = Uniform('sigma', .5, 1.5)
y = Normal('y', mu=mu, tau=sigma**-2, observed=data)
# Test gradient minimization
map_est1 = starting.find_MAP()
# Test non-gradient minimization
map_est2 = starting.find_MAP(fmin=starting.optimize.fmin_powell)
close_to(map_est1['mu'], 0, tol)
close_to(map_est1['sigma'], 1, tol)
close_to(map_est2['mu'], 0, tol)
close_to(map_est2['sigma'], 1, tol)
def __model__(self, RAA_x, RAA_y, RAA_xerr, RAA_yerr):
'''RAA model for pymc '''
a = Uniform('a', lower=0, upper=10)
b = Uniform('b', lower=0, upper=10)
c = Uniform('c', lower=0, upper=1)
@deterministic(plot=False)
def mu(a=a, b=b, c=c):
'''compute the RAA from convolution:
int_d(Delta_pt) P(Delta_pT) * sigma(pt+Delta_pt) / sigma(pt)
'''
intg_res = np.zeros_like(RAA_x)
for i, x in enumerate(gala30x):
# integral DeltaPt from 0 to infinity
scale_fct = RAA_x / gala30x[-1]
x = x * scale_fct
shifted_pt = RAA_x + x
mean_dpt = self.mean_ptloss(shifted_pt, b, c)
alpha = a
beta = a / mean_dpt
pdpt = gamma_dist(x, alpha, beta)
if self.with_ppdata:
intg_res += scale_fct * gala30w[i] * pdpt * self.pp_fit(
shifted_pt)
else:
intg_res += scale_fct * gala30w[i] * pdpt
if self.with_ppdata:
return intg_res / self.pp_fit(RAA_x)
else:
return intg_res
likelihood_y = Normal('likelihood_y',
mu=mu,
tau=1 / (RAA_yerr)**2,
observed=True,
value=RAA_y)
return locals()
def set_priors(self):
"""set parameters prior distributions.
Hardcoded behavior for now, with non committing prior knowledge.
:return: None
"""
obs = np.concatenate((self.control, self.variant))
mean, sigma, med = np.mean(obs), np.std(obs), np.median(obs)
location = np.log(med)
scale = np.sqrt(2 * np.log(mean / med))
for group in ['control', 'variant']:
self.stochastics[group + '_location'] = Normal(group + '_location', location, 0.000001 / sigma ** 2)
self.stochastics[group + '_scale'] = Uniform(group + '_scale', scale / 1000, scale * 1000)
def test_fit(self):
statement = {
'actor': u'123456789-1234-1234-1234-12345678901234',
'verb': u'http://adlnet.gov/expapi/verbs/completed',
'timestamp': 1519862425.0
}
user_name = '123456789-1234-1234-1234-12345678901234'
lam = Uniform('lam', lower=0, upper=1)
s1 = PoissonStudent(user_name, lam=lam)
env = Environment([s1], [statement])
env.add_agent(s1)
res = env.fit([lam], method='mcmc')
self.assertIsInstance(res,
MCMC,
msg="The output of fit is not an PYMC instance")
def main2():
data = load_file("statements-brneac3-20180301-20180531.json")
statements = load_statements(data)
user_name = "2890ebd9-1147-4f16-8a65-b7239bd54bd0"
user_name = "2890ebd9-1147-4f16-8a65-b7239bd54bd0"
lam = Uniform('lam', lower=0, upper=1)
s1 = PoissonStudent(user_name, lam=lam)
## Creating environment and fitting data
env = Environment([s1], statements)
env.add_student(s1)
res = env.fit([lam], method='mcmc')
print(res)
## plotting. can be integrated in env
hist(res.trace('lambda_{}'.format(user_name))[:])
show()
def main():
s1 = PoissonStudent("arnaud", 1)
s2 = PoissonStudent("francois", 1)
s3 = PoissonStudent("david", 0.5)
students = [s1, s2, s3]
env = Environment(students)
statements = env.simulate(1000, verbose=True)
student_names = set(s['actor'] for s in statements)
lam = Uniform('lam', lower=0, upper=1)
students = [PoissonStudent(name=name, lam=lam) for name in student_names]
env = Environment(students, statements)
params = [lam]
for s in students:
params.extend(s.params)
m = MCMC(params)
m.sample(iter=10000, burn=1000, thin=10)
hist(m.trace('lambda_david')[:])
show()
def test_non_missing(self):
"""
Test to ensure that masks without any missing values are not imputed.
"""
fake_data = rnormal(0, 1, size=10)
m = ma.masked_array(fake_data, fake_data == -999)
# Priors
mu = Normal('mu', mu=0, tau=0.0001)
s = Uniform('s', lower=0, upper=100, value=10)
tau = s**-2
# Likelihood with missing data
x = Normal('x', mu=mu, tau=tau, value=m, observed=True)
# Instantiate sampler
M = MCMC([mu, s, tau, x])
# Run sampler
M.sample(20000, 19000, progress_bar=0)
# Ensure likelihood does not have a trace
assert_raises(AttributeError, x.__getattribute__, 'trace')
def generate_model(subset=CONCURRENT):
"""Generate a model for a given subset of data"""
data = get_data(subset)
obs_indiv = data["obs_indiv"]
obs_summ = data["obs_summ"]
# =========================
# = Individual-level data =
# =========================
if subset is HISTORICAL:
# Impute Phe for individuals with only range given
phe_isnan = np.isnan(obs_indiv['phe'])
phe_low_missing = obs_indiv['phe_low'][phe_isnan]
phe_high_missing = obs_indiv['phe_high'][phe_isnan]
phe_missing = Uniform('phe_missing',
lower=phe_low_missing,
upper=phe_high_missing,
plot=False)
else:
phe_missing = None
# Study unit random effect for intercept
# Mean intercept
mu_int = Normal('mu_int', mu=100, tau=0.01, value=100)
# SD of intercepts
sigma_int = Uniform('sigma_int', lower=0, upper=100, value=10.)
tau_int = Lambda('tau_int', lambda s=sigma_int: s**-2)
# Intecepts by study
beta0 = Normal('beta0',
mu=mu_int,
tau=tau_int,
value=np.zeros(len(data["unique_papers"])))
# Study unit random effect for slope
# Mean slope
mu_slope = Normal('mu_slope', mu=0, tau=0.1, value=-0.01)
# SD of slopes
sigma_slope = Uniform('sigma_slope', lower=0, upper=10, value=1.)
tau_slope = Lambda('tau_slope', lambda s=sigma_slope: s**-2)
# Random slopes by study
alpha0 = Normal('alpha0',
mu=mu_slope,
tau=tau_slope,
value=np.zeros(len(data["unique_papers"])))
alpha1 = Normal('alpha1', mu=0, tau=0.01, value=0.0)
@deterministic
def beta1_indiv(a0=alpha0, a1=alpha1, crit=obs_indiv['critical_period']):
"""Calculate Phe effect (slope) for each individual"""
return a0[data["paper_id_indiv"]] + a1 * crit
@deterministic
def mu_iq(b0=beta0,
b1=beta1_indiv,
phe=obs_indiv['phe'],
phe_m=phe_missing):
"""Expected IQ"""
if subset is HISTORICAL:
# Insert values for missing phe
phe[phe_isnan] = phe_m
return b0[data["paper_id_indiv"]] + b1 * phe
# Process noise (variance of observations about predicted mean)
sigma_iq = Uniform('sigma_iq', lower=0, upper=100, value=1)
tau_iq = Lambda('tau_iq', lambda s=sigma_iq: s**-2)
# Data likelihood
iq_like = Normal('iq_like',
mu=mu_iq,
tau=tau_iq,
value=obs_indiv['iq'],
observed=True)
@deterministic
def iq_pred(mu=mu_iq, tau=tau_iq):
"""Simulated data for posterior predictive checks"""
return rnormal(mu, tau, size=len(obs_indiv['iq']))
# ====================================================
# = Infer slope from correlations of summarized data =
# ====================================================
# Means and stdevs of phe and IQ
stdev_phe = obs_summ['phe_sd']
stdev_iq = obs_summ['iq_sd']
mean_phe = obs_summ['phe']
mean_iq = obs_summ['iq']
@deterministic(cache_depth=0)
def beta1_summ(a0=alpha0, a1=alpha1, crit=obs_summ['critical_period']):
"""Calculate mean of slope for summarized data"""
return a0[data["paper_id_summ"]] + a1 * crit
@potential
def r_like(b1=beta1_summ, n=obs_summ['n']):
"""Likelihood for correlation coefficients of summarized data"""
# Convert slope to r
rho = b1 * stdev_phe / stdev_iq
# Fisher transformation to allow for normality assumption
eps = np.arctan(rho) - np.arctan(obs_summ['correlation'])
# Difference should be mean-zero
return normal_like(eps, mu=np.zeros(len(n)), tau=n - 3)
# Calculate probabilites of IQ<85
# Generate combinations of predictors
crit_pred, phe_pred = np.transpose([[crit, phe] for crit in [0, 1]
for phe in np.arange(200, 3200, 200)])
@deterministic(cache_depth=0)
def pred(a1=alpha1,
mu_int=mu_int,
tau_int=tau_int,
mu_slope=mu_slope,
tau_slope=tau_slope,
tau_iq=tau_iq,
values=(70, 75, 80, 85)):
"""Estimate the probability of IQ<85 for different covariate values"""
b0 = rnormal(mu_int, tau_int, size=len(phe_pred))
a0 = rnormal(mu_slope, tau_slope, size=len(phe_pred))
b1 = a0 + a1 * crit_pred
iq = rnormal(b0 + b1 * phe_pred, tau_iq)
return [iq < v for v in values]
return locals()
@stochastic
def theta(value=array([2., 5.])):
"""Slope and intercept parameters for a straight line.
The likelihood corresponds to the prior probability of the parameters."""
slope, intercept = value
prob_intercept = uniform_like(intercept, -10, 10)
prob_slope = log(1. / cos(slope)**2)
return prob_intercept + prob_slope
init_x = data_x.clip(min=0, max=50)
# Inferred true inputs.
x = Uniform('x', lower=0, upper=50, value=init_x)
@deterministic
def modelled_y(x=x, theta=theta):
"""Return y computed from the straight line model, given the
inferred true inputs and the model paramters."""
slope, intercept = theta
return slope * x + intercept
"""
Input error model.
Define the probability of measuring x knowing the true value.
"""
Zero-inflated Poisson example using simulated data.
"""
import numpy as np
from pymc import Uniform, Beta, observed, rpoisson, poisson_like
# True parameter values
mu_true = 5
psi_true = 0.75
n = 100
# Simulate some data
data = np.array(
[rpoisson(mu_true) * (np.random.random() < psi_true) for i in range(n)])
# Uniorm prior on Poisson mean
mu = Uniform('mu', 0, 20)
# Beta prior on psi
psi = Beta('psi', alpha=1, beta=1)
@observed(dtype=int, plot=False)
def zip(value=data, mu=mu, psi=psi):
""" Zero-inflated Poisson likelihood """
# Initialize likeihood
like = 0.0
# Loop over data
for x in value:
if not x:
# coin_model.py
# Binomial model for coin tossing
import numpy as np
from pymc import Uniform, Cauchy, deterministic
from bayes import gen_lighthouse_data
# set true values of parameters
alpha_true = 1.0
beta_true = 1.5
# set dataset size
nflashes = 10000
# gather observed data from true parameters
x_obs = gen_lighthouse_data(nflashes, alpha_true, beta_true)
# parameter prior: distance along shore
alpha = Uniform("alpha", lower = 0.0, upper = 2.0)
#alpha = Normal("alpha", mu = 1.5, tau = (1 / (0.3)**2))
#alpha = Normal("alpha", mu = 1.5, tau = (1 / (0.1)**2))
# parameter prior: distance from shore
beta = Uniform("beta", lower = 0.0, upper = 2.0)
#beta = Normal("beta", mu = 1.5, tau = (1 / (0.3)**2))
#beta = Normal("beta", mu = 1.5, tau = (1 / (0.1)**2))
# model: flash arrival points
x = Cauchy("x", alpha = alpha, beta = beta, value = x_obs, observed = True)
ysim = odesolve(model, tspan)
# Add error to the underlying data
seed = 2
random = np.random.RandomState(seed)
sigma = 0.1
ydata = ysim['A_obs'] * (random.randn(len(ysim['A_obs'])) * sigma + 1)
solver = Solver(model, tspan)
# The prior distribution for our rate parameter, a stochastic variable
# with either:
# - a uniform distribution between 0 and 1, or
# - a lognormal distribution centered at 0.1, with a variance of 3
# log10 units on either side
k = Uniform('k', lower=0, upper=1)
#k = Lognormal('k', mu=np.log(1e-1), tau=(1/(np.log(10)*np.log(1e3))))
# Our "model" is a deterministic random variable that is determined by the
# value of k that is passed in as an argument
@deterministic(plot=False)
def decay_model(k=k):
# The solver object needs all of the parameters, including the initial
# condition parameter A_0 that is not being fit; the final parameter
# value of 1.0 is the __source_0 parameter which is required because
# we are using a degradation rule.
# NOTE: Make sure the parameter values passed to the solver are in
# the right order!
solver.run(np.array([A_0.value, k, 1.0]))
y = solver.yobs['A_obs']
def random(theta2, y, rho):
mean = y[0] + rho * (theta2 - y[1])
var = 1. - rho ^ 2
return rnormal(value, mean, 1. / var)
@stoch
def theta2(value, theta1, y, rho):
"""Conditional probability p(theta2|theta1, y, rho)"""
def logp(value, theta2, y, rho):
mean = y[1] + rho * (theta1 - y[0])
var = 1. - rho ^ 2
return normal_like(value, mean, 1. / var)
def random(theta2, y, rho):
mean = y[0] + rho * (theta2 - y[1])
var = 1. - rho ^ 2
return rnormal(value, mean, 1. / var)
rho = Uniform('rho', rseed=True, lower=0, upper=1)
@data
def y(value=(3, 6)):
return 0
G = GibbsSampler([theta1, theta2, rho])
import numpy as np
xs = np.array([2,3,4,5,6,7])
ys = np.array([1.5, 1.8, 1.9, 2.3, 2.5, 2.8])
data = load('data/bran_body_weight.txt', 33)
converted_data = [(float(x[0]), float(x[1]), float(x[2])) for x in data]
xs = [x[1] for x in converted_data]
ys = [x[2] for x in converted_data]
b0 = Normal("b0", 0, 0.0003)
b1 = Normal("b1", 0, 0.0003)
err = Uniform("err", 0, 500)
x_weight = Normal("weight", 0, 1, value=xs, observed=True)
@deterministic(plot=False)
def pred(b0=b0, b1=b1, x=x_weight):
return b0 + b1*x
y = Normal("y", mu=pred, tau=err, value=ys, observed=True)
model = Model([pred, b0, b1, y, err, x_weight])
m = MCMC(model)
m.sample(burn=2000, iter=10000)
bb0 = sum(m.trace('b0')[:])/len(m.trace('b0')[:])
"""
from pymc import rweibull, Uniform, Weibull
"""
First, we will create a fake data set using some
fixed parameters. In real life, of course, you
already have the data !
"""
alpha = 3
beta = 5
N = 100
dataset = rweibull(alpha, beta, N)
"""
Now we create a pymc model that defines the likelihood
of the data set and prior assumptions about the value
of the parameters.
"""
a = Uniform('a', lower=0, upper=10, value=5, doc='Weibull alpha parameter')
b = Uniform('b', lower=0, upper=10, value=5, doc='Weibull beta parameter')
like = Weibull('like', alpha=a, beta=b, value=dataset, observed=True)
pred = Weibull('like', alpha=a, beta=b, value=dataset)
if __name__ == '__main__':
from pymc import MCMC, Matplot
# Sample the parameters a and b and analyze the results
M = MCMC([a, b, like])
M.sample(10000, 5000)
Matplot.plot(M)
def create_model():
all_vars = []
sigma_qd = Uniform('sigma_qd', 0,0.15)
sigma_qd.value = 0.05
mu_sc = Uniform('mu_sc', 0,1)
mu_sc.value=0.5
sigma_sc = Uniform('sigma_sc', 0, 0.2)
sigma_sc.value = 0.1
sigma_bias = Uniform('sigma_bias', 0,0.1)
# sigma_bias.value = 0.01
sigma_student_handin_capabilities = Uniform('sigma_student_handin_capabilities', 0,0.2)
sigma_student_handin_capabilities.value = 0.05
sigma_student_question_capabilities = Uniform('sigma_student_question_capabilities', 0,0.2)
sigma_student_question_capabilities.value = 0.05
all_vars.append(sigma_qd)
all_vars.append(mu_sc)
all_vars.append(sigma_sc)
all_vars.append(sigma_bias)
all_vars.append(sigma_student_handin_capabilities)
all_vars.append(sigma_student_question_capabilities)
for i in xrange(num_assignments):
questions = []
for j in xrange(num_questions_pr_handin):
difficulty = Normal('difficulty_q_%i_%i'% (i,j), 0, pymc.Lambda('tau_%i_%i'% (i,j), lambda a=sigma_qd: 1/ (sigma_qd.value*sigma_qd.value)))
q = Question(difficulty)
questions.append(q)
all_vars.append(difficulty)
assignment = Assignment(questions)
assignments.append(assignment)
for i in xrange(num_students):
# student_capabilities = Normal('student_capabilities_s_%i'%i, mu_sc, 1/sigma_sc)
# student_capabilities = TruncNormal('student_capabilities_s_%i'%i, 0, 1, mu_sc, 1/sigma_sc)
tau = pymc.Lambda('tau1_%i'%i, lambda a=sigma_sc: 1/(sigma_sc.value*sigma_sc.value))
student_capabilities = TruncNormal('student_capabilities_s_%i'%i, 0, 1, mu_sc, tau)
# grading_bias = Normal('grading_bias_s_%i'%i, 0, 1/sigma_bias)
tau = pymc.Lambda('tau2_%i'%i, lambda a=sigma_bias: 1/ (sigma_bias.value*sigma_bias.value))
grading_bias = Normal('grading_bias_s_%i'%i, 0, tau)
all_vars.append(student_capabilities)
all_vars.append(grading_bias)
s = Student(student_capabilities, grading_bias)
students.append(s)
for j, assignment in enumerate(assignments):
# student_handin_capabilities = Normal('student_handin_capabilities_sh_%i_%i' % (i,j), 0, 1/sigma_student_handin_capabilities)
tau = pymc.Lambda('tau2_%i_i%i'%(i, j), lambda a=sigma_student_handin_capabilities: 1/ (sigma_student_handin_capabilities.value*sigma_student_handin_capabilities.value))
student_handin_capabilities = Normal('student_handin_capabilities_sh_%i_%i' % (i,j), 0, tau)
all_vars.append(student_handin_capabilities)
question_capabilities = []
for k, q in enumerate(assignment.questions):
tau = pymc.Lambda('tau2_%i_i%i_%i'%(i, j,k), lambda a=sigma_student_question_capabilities: 1/ (sigma_student_question_capabilities.value*sigma_student_question_capabilities.value))
student_question_capabilities = Normal('student_question_capabilities_shq_%i_%i_%i' % (i,j,k ), 0, tau)
all_vars.append(student_question_capabilities)
question_capabilities.append(student_question_capabilities)
handins.append(Handin(s, assignment, student_handin_capabilities, question_capabilities))
# assign grader
all_grades = []
for handin in handins:
potential_graders = range(0, len(students))
potential_graders.remove(students.index(handin.student))
idx = np.random.randint(0, len(potential_graders), num_graders_pr_handin)
graders = [students[i] for i in idx]
grades = handin.grade(graders)
all_grades.append(grades)
grade_list = sum(sum(all_grades, []),[])
grade_list_real = [g.value for g in grade_list]
# zip([str(g) for g in grade_list[0].extended_parents], [g.value for g in grade_list[0].extended_parents])
print len([f for f in grade_list_real if f >1 or f < 0])
grade_list_real = [min(max((g), 0), 1) for g in grade_list_real]
sigma_exo_grade = Uniform('mu_exo_grade', 0,0.2)
tau = pymc.Lambda('tau2_%i_i%i_%i'%(i, j,k), lambda a=sigma_exo_grade: 1/ (sigma_exo_grade.value*sigma_exo_grade.value))
all_vars.append(sigma_exo_grade)
print "Creating grade list"
# grade_list = [Normal('grade_%i'%i, g, 1/mu_exo_grade, value=g_real, observed=True) for i, (g_real, g) in enumerate(zip(grade_list_real, grade_list))]
grade_list = [TruncNormal('grade_%i'%i, 0, 1, g, tau, value=g_real, observed=True) for i, (g_real, g) in enumerate(zip(grade_list_real, grade_list))]
print "Grade list created"
# grade_list = []
# grade_list = [pymc.ScipyDistributions.cdists.truncnorm(a=a_truncnorm, b=b_truncnorm, loc=g, scale=1/mu_exo_grade, value=g_real, observed=True) for ]
# for g_real, g in zip(grade_list_real, grade_list):
# stok = pymc.stochastic_from_dist('grade', lambda x: pymc.ScipyDistributions.truncated_normal_like(x, a=a_truncnorm, b=b_truncnorm, mu=g, tau=1/mu_exo_grade))
all_vars += grade_list
all_vars = list(set(all_vars))
print len(all_vars)
# print [str(v) for v in all_vars]
return locals(), grade_list_real, all_vars