def chi2_test_statistic(M, Obs, K, num_M, num_Obs):
#Getting frequencies from observations
Ns = T.dot(Obs, T.ones((K, 1)))
p = Obs / Ns
#Find the zeros so we can deal with them later
pZEROs = T.eq(p, 0)
mZEROs = T.eq(M, 0)
#log probabilities, with -INF as log(0)
lnM = T.log(M + mZEROs) - INF * mZEROs
lnp = T.log(p + pZEROs) - INF * pZEROs
#Using kroneker products so every row of M hits every row of P in the difference klnM - kln
O_ones = T.ones((num_Obs, 1))
M_ones = T.ones((num_M, 1))
klnM = kron(lnM, O_ones)
klnP = kron(M_ones, lnp)
klnP_M = klnP - klnM
kObs = kron(M_ones, Obs)
G = 2.0 * T.dot(klnP_M, kObs.T)
G = G * T.identity_like(G)
G = T.dot(G, T.ones((num_M * num_Obs, 1)))
G = T.reshape(G, (num_M, num_Obs))
#The following quotient improves the convergence to chi^2 by an order of magnitude
#source: http://en.wikipedia.org/wiki/Multinomial_test
#numerator = T.dot(- 1.0/(M + 0.01),T.ones((K,1))) - T.ones((num_M,1))
#q1 = T.ones((num_M,num_Obs)) + T.dot(numerator,1.0/Ns.T/6.0)/(K-1.0)
return G #/q1
def chi2_test_statistic(M, Obs, K, num_M, num_Obs):
#Getting frequencies from observations
Ns = T.dot(Obs,T.ones((K,1)))
p = Obs/Ns
#Find the zeros so we can deal with them later
pZEROs = T.eq(p, 0)
mZEROs = T.eq(M, 0)
#log probabilities, with -INF as log(0)
lnM = T.log(M + mZEROs) - INF*mZEROs
lnp = T.log(p + pZEROs) - INF*pZEROs
#Using kroneker products so every row of M hits every row of P in the difference klnM - kln
O_ones = T.ones((num_Obs,1))
M_ones = T.ones((num_M,1))
klnM = kron(lnM,O_ones)
klnP = kron(M_ones, lnp)
klnP_M = klnP - klnM
kObs = kron(M_ones, Obs)
G = 2.0*T.dot(klnP_M ,kObs.T)
G = G*T.identity_like(G)
G = T.dot(G,T.ones((num_M*num_Obs,1)))
G = T.reshape(G,(num_M,num_Obs))
#The following quotient improves the convergence to chi^2 by an order of magnitude
#source: http://en.wikipedia.org/wiki/Multinomial_test
#numerator = T.dot(- 1.0/(M + 0.01),T.ones((K,1))) - T.ones((num_M,1))
#q1 = T.ones((num_M,num_Obs)) + T.dot(numerator,1.0/Ns.T/6.0)/(K-1.0)
return G#/q1
def sample_h_given_v_2wise(v, W, Wh, bh, nh):
phi = T.dot(v, W) + bh
ephi = T.exp(phi)
adder = np.zeros((nh/2, nh), dtype=theano.config.floatX)
for i in xrange(len(adder)):
adder[i, 2*i] = 1
adder[i, 2*i+1] = 1
adder = theano.shared(adder)
# wobble = 1 + exp(phi_2i) + exp(phi_{2i+1}) + exp(phi_2i + phi_{21+1} + Wh_i)
# p(h_2i = 1 | v) = (exp(phi_2i) + exp(phi_2i + phi_{21+1} + Wh_i ) / wobble
# p(h_{2i+1} = 1 | v) = (exp(phi_2i) + exp(phi_2i + phi_{2i+1} + Wh_i )) / wobble
# the second term is the same in both - the pair term. but it must be broadcasted (the kron!)
# dotting by adder returns a vector of half the size of sums of pairs of elements
pairsum = T.dot(ephi, adder.T)
first = ephi.T[T.arange(0, nh, 2)].T
pairprod = pairsum*first - first**2
pairterm = pairprod*T.exp(Wh)
wobble = 1 + pairsum + pairterm
pairterm_broadcast = kron(pairterm.dimshuffle(0, 'x'), T.ones(2))
wobble_broadcast = kron(wobble.dimshuffle(0, 'x'), T.ones(2))
prop_up = (ephi + pairterm_broadcast) / wobble_broadcast
h = theano_rng.binomial(n=1, p = prop_up, dtype=theano.config.floatX, size=(nh,), ndim=1)
return h
def underfit_choice(Agression,
num_alpha,
Models,
Obs,
K,
num_M,
num_Obs,
pvalues,
alpha,
Preds=None):
#Switch for pvalue algorithm.. first is fast and an approximation, second slow and exact
if Preds == None:
Preds = predictiveness_profiles(Models, K, num_M)
# Letting P denote the whole predictiveness matrix, we will tile num_alpha*num_Obs of them together as such:
# PPPPPPPPPPPPPPPPPP..
#Then the pvalues will be broadcastand tiled so that if we number blocks of pvalues their ordering will be
# 123412341234,, with each block having the pvalues of generating models behind the predictiveness profiles
#Then alpha will be tiled so that there's a new alpha for each time the pvalues repeat:
#111111222222333333... The idea is to get every triple (alpha, observation, P)
#
#This visualization might help:
#
#PPPPPPPPPPPP
#123412341234
#111122223333
#
#Finally the code:
Preds = kron(T.ones((1, num_alpha * num_Obs)), Preds)
pvalues = kron(pvalues, T.ones((1, num_M)))
pvalues = kron(T.ones((1, num_alpha)), pvalues)
alpha = kron(T.reshape(alpha, (1, num_alpha)),
T.ones((num_M, num_M * num_Obs)))
# This "knocks out" rejected universes from predictiveness profiles
Scores = Preds + INF * (pvalues < alpha)
# The worst case predictiveness for each alpha is found
Scores = T.min(Scores, axis=0)
# Rearranging to put vary the alpha between the rows
Scores = T.reshape(Scores, (num_alpha, num_M * num_Obs))
# Integrating out the information/significance levels with agression
Scores = T.dot(Agression.T, Scores)
# Finally get the num_M by num_Obs matrix of scores for each observation
Scores = T.reshape(Scores, (num_Obs, num_M)).T
# Return the model choice
Choice = T.argmax(Scores, axis=0)
return Choice
def min_risk_choice(Posterior):
#The Loss function is a function of the predictiveness profiles
Preds = predictiveness_profiles(Models, K, num_M)
Loss = ifelse(T.eq(Choice_type, 1), T.pow(1.0 - Preds,2), ifelse(T.eq(Choice_type, 2), T.abs_(1.0 - Preds), - Preds))
#Kroneckering Loss up num_Obs times (tile Loss, making it num_M by num_M*num_Obs)
Loss = kron(T.ones((1,num_Obs)), Loss)
#Kroneckering up the Posterior, making it num_M by num_Obs*numM
Posterior = kron(Posterior, T.ones((1,num_M)))
#Dotting and reshaping down to give num_M by num_Obs expected loss matrix
Expected_Loss = T.dot(T.ones((1,num_M)),Posterior*Loss)
Expected_Loss = T.reshape(Expected_Loss, (num_Obs,num_M)).T
#Choice minimizes risk
Choice = T.argmin(Expected_Loss, axis = 0)
return Choice
def test_perform(self):
x = tensor.dmatrix()
y = tensor.dmatrix()
f = function([x, y], kron(x, y))
for shp0 in [(8, 6), (5, 8)]:
for shp1 in [(5, 7), (3, 3)]:
a = numpy.random.rand(*shp0)
b = numpy.random.rand(*shp1)
out = f(a, b)
assert numpy.allclose(out, scipy.linalg.kron(a, b))
def test_perform(self):
x = tensor.dmatrix()
y = tensor.dmatrix()
f = function([x, y], kron(x, y))
for shp0 in [(8, 6), (5, 8)]:
for shp1 in [(5, 7), (3, 3)]:
a = numpy.random.rand(*shp0)
b = numpy.random.rand(*shp1)
out = f(a, b)
assert numpy.allclose(out, scipy.linalg.kron(a, b))
def underfit_choice(Agression, num_alpha, Models, Obs, K, num_M, num_Obs, pvalues, alpha, Preds = None):
#Switch for pvalue algorithm.. first is fast and an approximation, second slow and exact
if Preds == None:
Preds = predictiveness_profiles(Models, K, num_M)
# Letting P denote the whole predictiveness matrix, we will tile num_alpha*num_Obs of them together as such:
# PPPPPPPPPPPPPPPPPP..
#Then the pvalues will be broadcastand tiled so that if we number blocks of pvalues their ordering will be
# 123412341234,, with each block having the pvalues of generating models behind the predictiveness profiles
#Then alpha will be tiled so that there's a new alpha for each time the pvalues repeat:
#111111222222333333... The idea is to get every triple (alpha, observation, P)
#
#This visualization might help:
#
#PPPPPPPPPPPP
#123412341234
#111122223333
#
#Finally the code:
Preds = kron(T.ones((1,num_alpha*num_Obs)),Preds)
pvalues = kron(pvalues,T.ones((1,num_M)))
pvalues = kron(T.ones((1,num_alpha)),pvalues)
alpha = kron(T.reshape(alpha, (1, num_alpha)),T.ones((num_M,num_M*num_Obs)))
# This "knocks out" rejected universes from predictiveness profiles
Scores = Preds + INF*(pvalues < alpha)
# The worst case predictiveness for each alpha is found
Scores = T.min(Scores, axis = 0)
# Rearranging to put vary the alpha between the rows
Scores = T.reshape(Scores,(num_alpha,num_M*num_Obs))
# Integrating out the information/significance levels with agression
Scores = T.dot(Agression.T,Scores)
# Finally get the num_M by num_Obs matrix of scores for each observation
Scores = T.reshape(Scores,(num_Obs,num_M)).T
# Return the model choice
Choice = T.argmax(Scores, axis = 0)
return Choice
def min_risk_choice(Posterior):
#The Loss function is a function of the predictiveness profiles
Preds = predictiveness_profiles(Models, K, num_M)
Loss = ifelse(
T.eq(Choice_type, 1), T.pow(1.0 - Preds, 2),
ifelse(T.eq(Choice_type, 2), T.abs_(1.0 - Preds), -Preds))
#Kroneckering Loss up num_Obs times (tile Loss, making it num_M by num_M*num_Obs)
Loss = kron(T.ones((1, num_Obs)), Loss)
#Kroneckering up the Posterior, making it num_M by num_Obs*numM
Posterior = kron(Posterior, T.ones((1, num_M)))
#Dotting and reshaping down to give num_M by num_Obs expected loss matrix
Expected_Loss = T.dot(T.ones((1, num_M)), Posterior * Loss)
Expected_Loss = T.reshape(Expected_Loss, (num_Obs, num_M)).T
#Choice minimizes risk
Choice = T.argmin(Expected_Loss, axis=0)
return Choice
def test_perform(self):
assert imported_scipy, (
"Scipy not available. Scipy is needed for TestKron")
x = tensor.dmatrix()
y = tensor.dmatrix()
f = function([x, y], kron(x, y))
for shp0 in [(8, 6), (5, 8)]:
for shp1 in [(5, 7), (3, 3)]:
a = numpy.random.rand(*shp0)
b = numpy.random.rand(*shp1)
out = f(a, b)
assert numpy.allclose(out, scipy.linalg.kron(a, b))
def test_numpy_2d(self):
for shp0 in [(2, 3)]:
for shp1 in [(6, 7)]:
if len(shp0) + len(shp1) == 2:
continue
x = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp0))
y = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
a = numpy.asarray(self.rng.rand(*shp0)).astype(floatX)
b = self.rng.rand(*shp1).astype(floatX)
out = f(a, b)
assert numpy.allclose(out, numpy.kron(a, b))
def test_perform(self):
if not imported_scipy:
raise SkipTest('kron tests need the scipy package to be installed')
for shp0 in [(2,), (2, 3), (2, 3, 4), (2, 3, 4, 5)]:
for shp1 in [(6,), (6, 7), (6, 7, 8), (6, 7, 8, 9)]:
if len(shp0) + len(shp1) == 2:
continue
x = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp0))
y = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
a = numpy.asarray(self.rng.rand(*shp0)).astype(floatX)
b = self.rng.rand(*shp1).astype(floatX)
out = f(a, b)
assert numpy.allclose(out, scipy.linalg.kron(a, b))