def normalization(X): P = X.shape[1] for p in xrange(P): mean= np.mean(X[:,p]) variance = np.variance(X[:,p]) X[:,p]= (X[:,p] - mean)/variance return X
def normalize_audio_feature(audio_feature: np.ndarray, per_frame=False):
""" Mean and variance normalization """
axis = 1 if per_frame else None
mean = np.mean(audio_feature, axis=axis)
std_dev = np.sqrt(np.variance(audio_feature, axis=axis) + 1e-9)
normalized = (audio_feature - mean) / std_dev
return normalized
def normalization(X):
P = X.shape[1]
for p in xrange(P):
mean = np.mean(X[:, p])
variance = np.variance(X[:, p])
X[:, p] = (X[:, p] - mean) / variance
return X
def delta_mf_variance_pooled(delta_mf, bootstrap_weight_array):
"""
Get a bootstrap estimator of the variance of delta_mf, pooling equally across all temporal differences rather than
treating each (S, A) pair differently.
:param delta_mf: array of temporal differences
:param bootstrap_weight_array: number_of_bootstrap_replicates x len(td) - size array of bootstrap multipliers
:return:
"""
n = len(delta_mf)
bootstrapped_deltas = np.zeros((0, n))
for multiplier in bootstrap_weight_array:
delta_b = np.multiply(delta_mf, multiplier)
bootstrapped_deltas = np.vstack((bootstrapped_deltas, delta_b))
elementwise_variances = np.variance(bootstrapped_deltas, axis=0)
return np.mean(elementwise_variances), bootstrapped_deltas
def covariates_variance(self):
r"""Variance explained by the covariates.
It is defined as
.. math::
\sigma_a^2 = \sum_{s=1}^p \left\{ \sum_{i=1}^n \left(
\mathrm M_{i,s}\beta_s - \sum_{j=1}^n
\frac{\mathrm M_{j,s}\beta_s}{n} \right)^2 \Big/ n
\right\}
where :math:`p` is the number of covariates and :math:`n` is the number
of individuals. One can show that it amounts to
:math:`\sum_s \beta_s^2` whenever the columns of :math:`\mathrm M`
are normalized to have mean and standard deviation equal to zero and
one, respectively.
"""
return fsum(variance(self.M * self.beta, axis=0))
def delta_mb_bias_and_variance_pooled(delta_mb, q_fn, env, gamma, X, Sp1,
transition_model_fitter,
bootstrap_weight_array):
"""
Estimate sampling variance of delta_mb by fitting model to bootstrapped samples of the data (and pooling).
"""
n = X.shape[0]
transition_model = transition_model_fitter()
bootstrapped_deltas = np.zeros((0, n))
for multiplier in bootstrap_weight_array:
X_b = np.multiply(multiplier, X_b)
transition_model_fitter.fit(X_b, Sp1)
R_b = transition_model_fitter.expected_reward(X)
expected_q_max_array, _ = expected_q_max(q_fn, X, env,
transition_model)
delta_b = R_b + gamma * expected_q_max_array - q_fn(X)
bootstrapped_deltas = np.vstack((bootstrapped_deltas, delta_b))
elementwise_variances = np.variance(bootstrapped_deltas, axis=0)
elementwise_means = np.mean(bootstrapped_deltas, axis=0)
return np.mean(elementwise_variances
), bootstrapped_deltas, delta_mb - elementwise_means
def variance_intensity(self):
return np.variance(self.intensity_image[self.image])
def oneIteration(self):
#*****************
# Decision phase
#*****************
# Compute change in likelihood
deltaML = np.zeros((self.MFull, 1))
action = self.actionReestimate * np.ones((self.MFull, 1))
usedFactor = self.factor[self.Used]
N, M = self.PHI.shape
# Reestimation
iu = np.where(usedFactor > self.controls['ZeroFactor'])[0]
index = self.Used[iu]
newAlpha = self.SOut[index]**2 / self.factor[index]
delta = 1.0 / newAlpha - 1.0 / self.Alpha[iu]
# Quick computation of change in log-likelihood given all re-estimations
deltaML[index] = 0.5 * (delta * self.QIn[index]**2 /
(delta * self.SIn[index] + 1.0) -
np.log(1.0 + self.SIn[index] * delta))
# Deletion
iu = np.where(usedFactor <= self.controls['ZeroFactor'])[0]
index = self.Used[iu]
anyToDelete = (not index.size == 0) and M > 1
if (anyToDelete):
deltaML[index] = -0.5 * (
self.QOut[index]**2 / (self.SOut[index] + self.Alpha[iu]) -
np.log(1.0 + self.SOut[index] / self.Alpha[iu]))
action[index] = self.actionDelete
# Addition
GoodFactor = self.factor > self.controls['ZeroFactor']
GoodFactor[self.Used] = 0
GoodFactor[self.alignedOut] = 0
index = np.where(GoodFactor)[0]
anyToAdd = (not index.size == 0)
if (anyToAdd):
# Quick computation of change in log-likelihood
quot = self.QIn[index]**2 / self.SIn[index]
deltaML[index] = 0.5 * (quot - 1.0 - np.log(quot))
action[index] = self.actionAdd
if (anyToAdd and self.controls['PriorityAddition']) or (
anyToDelete and self.controls['PriorityDeletion']):
# We won't perform re-estimation this iteration
deltaML[action == self.actionReestimate] = 0
# We should enforce add if preferred and delete
if (anyToAdd and self.controls['PriorityAddition']
and (not self.controls['PriorityDeletion'])):
deltaML[action == self.actionDelete] = 0
if (anyToDelete and self.controls['PriorityDeletion']
and (not self.controls['PriorityAddition'])):
deltaML[action == self.actionAdd] = 0
# Finally, choose the action that results in the greatest change in likelihood
nu = np.atleast_1d(np.argmax(deltaML))
self.deltaLogMarginal = deltaML[nu]
self.selectedAction = action[nu]
anyWorthWhileAction = self.deltaLogMarginal > 0
# If basis nu is already in the model, find its index
j = []
if (self.selectedAction
== self.actionReestimate) or (self.selectedAction
== self.actionDelete):
j = np.where(self.Used == nu)[0]
self.Phi = np.atleast_2d(self.basis[:, nu])
newAlpha = self.SOut[nu]**2 / self.factor[nu]
change = np.abs(np.log(newAlpha) - np.log(self.Alpha[j]))
if (not anyWorthWhileAction) or (
(self.selectedAction == self.actionReestimate) and
(change < self.controls['MinDeltaLogAlpha']) and
(not anyToDelete)):
self.selectedAction = self.actionTerminate
# Alignment checks for addition
if (self.selectedAction == self.actionAdd):
p = np.dot(self.Phi.T, self.PHI)
findAligned = np.where(p > self.controls['AlignmentMax'])[0]
numAligned = findAligned.size
if (numAligned > 0):
# The added basis function is effectively indistinguishable from one present already
self.selectedAction = self.actionAlignmentSkip
self.alignDeferCount += 1
# Take note not to try this again
self.alignedOut = np.append(self.alignedOut,
nu * np.ones(
(numAligned, 1))).astype(int)
self.alignedIn = np.append(self.alignedIn,
self.Used[findAligned])
# Alignment checks for deletion
if (self.selectedAction == self.actionDelete):
findAligned = np.where(self.alignedIn == nu)[0]
numAligned = findAligned.size
if (numAligned > 0):
reinstated = self.alignedOut[findAligned]
self.alignedIn = np.delete(self.alignedIn, findAligned, 0)
self.alignedOut = np.delete(self.alignedOut, findAligned, 0)
#*****************
# Action phase
#*****************
updateRequired = False
if (self.selectedAction == self.actionReestimate):
# Basis function nu is already in the model and we're reeestimatig its alpha
oldAlpha = self.Alpha[j]
self.Alpha[j] = newAlpha
S_j = self.Sigma[:, j]
deltaInv = 1.0 / (newAlpha - oldAlpha)
kappa = 1.0 / (self.Sigma[j, j] + deltaInv)
tmp = kappa * S_j
newSigma = self.Sigma - np.dot(tmp, S_j.T)
deltaMu = -self.Mu[j] * tmp
self.Mu += deltaMu
self.SIn += kappa * np.dot(self.betaBasisPHI, S_j)**2
self.QIn -= np.dot(self.betaBasisPHI, deltaMu)
self.updateCount += 1
updateRequired = True
elif (self.selectedAction == self.actionAdd):
self.basisPhi = np.dot(self.basis.T, self.Phi)
self.basisPHI = np.hstack((self.basisPHI, self.basisPhi))
self.BPhi = self.beta * self.Phi
self.BASISBPhi = self.beta * self.basisPhi
tmp = np.dot(np.dot(self.BPhi.T, self.PHI), self.Sigma).T
self.Alpha = np.vstack((self.Alpha, newAlpha))
self.PHI = np.hstack((self.PHI, self.Phi))
s_ii = 1.0 / (newAlpha + self.SIn[nu])
s_i = -s_ii * tmp
TAU = -np.dot(s_i, tmp.T)
t1 = np.hstack((self.Sigma + TAU, s_i))
t2 = np.hstack((s_i.T, s_ii))
newSigma = np.vstack((t1, t2))
mu_i = s_ii * self.QIn[nu]
deltaMu = np.vstack((-mu_i * tmp, mu_i))
self.Mu = np.vstack((self.Mu, 0)) + deltaMu
mCi = self.BASISBPhi - np.dot(self.betaBasisPHI, tmp)
self.SIn -= s_ii * mCi**2
self.QIn -= mu_i * mCi
self.Used = np.hstack((self.Used, nu))
self.addCount += 1
updateRequired = True
elif (self.selectedAction == self.actionDelete):
self.basisPHI = np.delete(self.basisPHI, j, 1)
self.PHI = np.delete(self.PHI, j, 1)
self.Alpha = np.delete(self.Alpha, j, 0)
s_jj = self.Sigma[j, j]
s_j = self.Sigma[:, j]
tmp = s_j / s_jj
newSigma = self.Sigma - np.dot(tmp, s_j.T)
newSigma = np.delete(newSigma, j, 0)
newSigma = np.delete(newSigma, j, 1)
deltaMu = -self.Mu[j] * tmp
mu_j = self.Mu[j]
self.Mu += deltaMu
self.Mu = np.delete(self.Mu, j, 0)
jPm = np.dot(self.betaBasisPHI, s_j)
self.SIn += jPm**2 / s_jj
self.QIn += jPm * mu_j / s_jj
self.Used = np.delete(self.Used, j, 0)
self.deleteCount += 1
updateRequired = True
M = len(self.Used)
if (updateRequired):
self.SOut[:] = self.SIn
self.QOut[:] = self.QIn
tmp = self.Alpha / (self.Alpha - self.SIn[self.Used])
self.SOut[self.Used] = tmp * self.SIn[self.Used]
self.QOut[self.Used] = tmp * self.QIn[self.Used]
self.factor = (self.QOut * self.QOut - self.SOut)
self.Sigma = newSigma
self.Gamm = 1.0 - self.Alpha * np.atleast_2d(np.diag(self.Sigma)).T
self.betaBasisPHI = self.beta * self.basisPHI
self.logML += self.deltaLogMarginal[0, 0]
self.countLoop += 1
self.logMarginalLog = np.append(self.logMarginalLog, self.logML)
# Something went wrong. Recompute statistics
if (np.sum(self.Gamm) < 0):
self.fullStatistics()
# Recompute noise if not given
if (self.noise is None) and (
(self.selectedAction == self.actionTerminate) or
(self.loop <= self.controls['BetaUpdateStart']) or
(self.loop % self.controls['BetaUpdateFrequency'] == 0)):
# Gaussian noise estimate
betaZ1 = beta
y = np.dot(self.PHI, self.Mu)
e = self.targets - y
beta = (N - np.sum(self.Gamm)) / np.dot(e.T, e)
# Work-around zero-noise issue
beta = np.amin([beta, 1.e6 / np.variance(self.targets)])
deltaLogBeta = np.log(beta) - np.log(betaZ1)
# Full re-computation of statistics after beta update
if (np.abs(deltaLogBeta) > 1.e-6):
self.fullStatistics()
self.countLoop += 1
self.logMarginalLog = np.append(self.logMarginalLog,
self.logML)
if (self.selectedAction == self.actionTerminate):
self.selectedAction = self.actionNoiseOnly
print("Noise update. Termination deferred")
#self.AInv = np.diag(1.0/self.Alpha[:,0])
#Sigma = 1.0/self.beta * np.identity(self.N) + np.dot(np.dot(self.PHI, self.AInv), self.PHI.T)
#CInv, logD = cholInvert(Sigma)
#logL = -0.5*logD - 0.5*np.dot(np.dot(self.targets.T,CInv),self.targets)
print("{0:4d} - L={1:10.7f} - Gamma={2:10.7f} (M={3:4d}) - s={4:6.4f}".
format(self.loop, self.logML[0] / N, np.sum(self.Gamm), M,
np.sqrt(1.0 / self.beta)))
if (self.selectedAction == self.actionTerminate):
print("Stopping at iteration {0} - max_delta_ml={1}".format(
self.loop, self.deltaLogMarginal[0, 0]))
print("L={0} - Gamma={1} (M={2}) - s={3}".format(
self.logML[0] / N, np.sum(self.Gamm), M,
np.sqrt(1.0 / self.beta)))
iterationLimit = self.loop == 200
self.lastIteration = iterationLimit
def variance(array):
return np.variance(array)
import numpy as np
import matplotlib.pyplot as plt
mean = np.zeros(10)
index = np.arange(10)
for i in range(len(index)):
s = np.random.poisson(3,1000)
mean[i] = np.mean(s)
print mean
a = np.mean(mean)
a_v = np.variance(mean)
print a, a_v
plt.figure()
plt.title('10 Times')
plt.hist(mean, normed=True)
plt.show()
#############
import numpy as np
import matplotlib.pyplot as plt
mean = np.zeros(100)
index = np.arange(100)
for i in range(len(index)):
s = np.random.poisson(3,1000)
mean[i] = np.mean(s)
print mean
b = np.mean(mean)
def oneIteration(self):
#*****************
# Decision phase
#*****************
# Compute change in likelihood
deltaML = np.zeros((self.MFull, 1))
action = self.actionReestimate * np.ones((self.MFull,1))
usedFactor = self.factor[self.Used]
N, M = self.PHI.shape
# Reestimation
iu = np.where(usedFactor > self.controls['ZeroFactor'])[0]
index = self.Used[iu]
newAlpha = self.SOut[index]**2 / self.factor[index]
delta = 1.0 / newAlpha - 1.0 / self.Alpha[iu]
# Quick computation of change in log-likelihood given all re-estimations
deltaML[index] = 0.5 * (delta * self.QIn[index]**2 / (delta * self.SIn[index]+1.0) - np.log(1.0+self.SIn[index]*delta))
# Deletion
iu = np.where(usedFactor <= self.controls['ZeroFactor'])[0]
index = self.Used[iu]
anyToDelete = (not index.size == 0) and M > 1
if (anyToDelete):
deltaML[index] = -0.5 * (self.QOut[index]**2 / (self.SOut[index] + self.Alpha[iu]) - np.log(1.0+self.SOut[index] / self.Alpha[iu]))
action[index] = self.actionDelete
# Addition
GoodFactor = self.factor > self.controls['ZeroFactor']
GoodFactor[self.Used] = 0
GoodFactor[self.alignedOut] = 0
index = np.where(GoodFactor)[0]
anyToAdd = (not index.size == 0)
if (anyToAdd):
# Quick computation of change in log-likelihood
quot = self.QIn[index]**2 / self.SIn[index]
deltaML[index] = 0.5 * (quot - 1.0 - np.log(quot))
action[index] = self.actionAdd
if (anyToAdd and self.controls['PriorityAddition']) or (anyToDelete and self.controls['PriorityDeletion']):
# We won't perform re-estimation this iteration
deltaML[action == self.actionReestimate] = 0
# We should enforce add if preferred and delete
if (anyToAdd and self.controls['PriorityAddition'] and (not self.controls['PriorityDeletion'])):
deltaML[action == self.actionDelete] = 0
if (anyToDelete and self.controls['PriorityDeletion'] and (not self.controls['PriorityAddition'])):
deltaML[action == self.actionAdd] = 0
# Finally, choose the action that results in the greatest change in likelihood
nu = np.atleast_1d(np.argmax(deltaML))
self.deltaLogMarginal = deltaML[nu]
self.selectedAction = action[nu]
anyWorthWhileAction = self.deltaLogMarginal > 0
# If basis nu is already in the model, find its index
j = []
if (self.selectedAction == self.actionReestimate) or (self.selectedAction == self.actionDelete):
j = np.where(self.Used == nu)[0]
self.Phi = np.atleast_2d(self.basis[:,nu])
newAlpha = self.SOut[nu]**2 / self.factor[nu]
change = np.abs(np.log(newAlpha) - np.log(self.Alpha[j]))
if (not anyWorthWhileAction) or ((self.selectedAction == self.actionReestimate) and (change < self.controls['MinDeltaLogAlpha']) and (not anyToDelete)):
self.selectedAction = self.actionTerminate
# Alignment checks for addition
if (self.selectedAction == self.actionAdd):
p = np.dot(self.Phi.T, self.PHI)
findAligned = np.where(p > self.controls['AlignmentMax'])[0]
numAligned = findAligned.size
if (numAligned > 0):
# The added basis function is effectively indistinguishable from one present already
self.selectedAction = self.actionAlignmentSkip
self.alignDeferCount += 1
# Take note not to try this again
self.alignedOut = np.append(self.alignedOut, nu*np.ones((numAligned,1))).astype(int)
self.alignedIn = np.append(self.alignedIn, self.Used[findAligned])
# Alignment checks for deletion
if (self.selectedAction == self.actionDelete):
findAligned = np.where(self.alignedIn == nu)[0]
numAligned = findAligned.size
if (numAligned > 0):
reinstated = self.alignedOut[findAligned]
self.alignedIn = np.delete(self.alignedIn, findAligned, 0)
self.alignedOut = np.delete(self.alignedOut, findAligned, 0)
#*****************
# Action phase
#*****************
updateRequired = False
if (self.selectedAction == self.actionReestimate):
# Basis function nu is already in the model and we're reeestimatig its alpha
oldAlpha = self.Alpha[j]
self.Alpha[j] = newAlpha
S_j = self.Sigma[:,j]
deltaInv = 1.0 / (newAlpha - oldAlpha)
kappa = 1.0 / (self.Sigma[j,j] + deltaInv)
tmp = kappa * S_j
newSigma = self.Sigma - np.dot(tmp, S_j.T)
deltaMu = -self.Mu[j] * tmp
self.Mu += deltaMu
self.SIn += kappa * np.dot(self.betaBasisPHI, S_j)**2
self.QIn -= np.dot(self.betaBasisPHI, deltaMu)
self.updateCount += 1
updateRequired = True
elif (self.selectedAction == self.actionAdd):
self.basisPhi = np.dot(self.basis.T, self.Phi)
self.basisPHI = np.hstack((self.basisPHI, self.basisPhi))
self.BPhi = self.beta * self.Phi
self.BASISBPhi = self.beta * self.basisPhi
tmp = np.dot(np.dot(self.BPhi.T, self.PHI), self.Sigma).T
self.Alpha = np.vstack((self.Alpha,newAlpha))
self.PHI = np.hstack((self.PHI, self.Phi))
s_ii = 1.0 / (newAlpha + self.SIn[nu])
s_i = -s_ii * tmp
TAU = -np.dot(s_i, tmp.T)
t1 = np.hstack((self.Sigma+TAU, s_i))
t2 = np.hstack((s_i.T, s_ii))
newSigma = np.vstack((t1,t2))
mu_i = s_ii * self.QIn[nu]
deltaMu = np.vstack((-mu_i*tmp, mu_i))
self.Mu = np.vstack((self.Mu, 0)) + deltaMu
mCi = self.BASISBPhi - np.dot(self.betaBasisPHI, tmp)
self.SIn -= s_ii * mCi**2
self.QIn -= mu_i * mCi
self.Used = np.hstack((self.Used, nu))
self.addCount += 1
updateRequired = True
elif (self.selectedAction == self.actionDelete):
self.basisPHI = np.delete(self.basisPHI, j, 1)
self.PHI = np.delete(self.PHI, j, 1)
self.Alpha = np.delete(self.Alpha, j, 0)
s_jj = self.Sigma[j,j]
s_j = self.Sigma[:,j]
tmp = s_j / s_jj
newSigma = self.Sigma - np.dot(tmp, s_j.T)
newSigma = np.delete(newSigma, j, 0)
newSigma = np.delete(newSigma, j, 1)
deltaMu = -self.Mu[j] * tmp
mu_j = self.Mu[j]
self.Mu += deltaMu
self.Mu = np.delete(self.Mu, j, 0)
jPm = np.dot(self.betaBasisPHI, s_j)
self.SIn += jPm**2 / s_jj
self.QIn += jPm * mu_j / s_jj
self.Used = np.delete(self.Used, j, 0)
self.deleteCount += 1
updateRequired = True
M = len(self.Used)
if (updateRequired):
self.SOut[:] = self.SIn
self.QOut[:] = self.QIn
tmp = self.Alpha / (self.Alpha - self.SIn[self.Used])
self.SOut[self.Used] = tmp * self.SIn[self.Used]
self.QOut[self.Used] = tmp * self.QIn[self.Used]
self.factor = (self.QOut * self.QOut - self.SOut)
self.Sigma = newSigma
self.Gamm = 1.0 - self.Alpha * np.atleast_2d(np.diag(self.Sigma)).T
self.betaBasisPHI = self.beta * self.basisPHI
self.logML += self.deltaLogMarginal[0,0]
self.countLoop += 1
self.logMarginalLog = np.append(self.logMarginalLog, self.logML)
# Something went wrong. Recompute statistics
if (np.sum(self.Gamm) < 0):
self.fullStatistics()
# Recompute noise if not given
if (self.noise is None) and ((self.selectedAction == self.actionTerminate) or (self.loop <= self.controls['BetaUpdateStart']) or
(self.loop % self.controls['BetaUpdateFrequency'] == 0)):
# Gaussian noise estimate
betaZ1 = beta
y = np.dot(self.PHI, self.Mu)
e = self.targets - y
beta = (N - np.sum(self.Gamm)) / np.dot(e.T, e)
# Work-around zero-noise issue
beta = np.amin([beta, 1.e6 / np.variance(self.targets)])
deltaLogBeta = np.log(beta) - np.log(betaZ1)
# Full re-computation of statistics after beta update
if (np.abs(deltaLogBeta) > 1.e-6):
self.fullStatistics()
self.countLoop += 1
self.logMarginalLog = np.append(self.logMarginalLog, self.logML)
if (self.selectedAction == self.actionTerminate):
self.selectedAction = self.actionNoiseOnly
print("Noise update. Termination deferred")
#self.AInv = np.diag(1.0/self.Alpha[:,0])
#Sigma = 1.0/self.beta * np.identity(self.N) + np.dot(np.dot(self.PHI, self.AInv), self.PHI.T)
#CInv, logD = cholInvert(Sigma)
#logL = -0.5*logD - 0.5*np.dot(np.dot(self.targets.T,CInv),self.targets)
print("{0:4d} - L={1:10.7f} - Gamma={2:10.7f} (M={3:4d}) - s={4:6.4f}".format(self.loop,self.logML[0]/N, np.sum(self.Gamm), M, np.sqrt(1.0/self.beta)))
if (self.selectedAction == self.actionTerminate):
print("Stopping at iteration {0} - max_delta_ml={1}".format(self.loop, self.deltaLogMarginal[0,0]))
print("L={0} - Gamma={1} (M={2}) - s={3}".format(self.logML[0]/N, np.sum(self.Gamm), M, np.sqrt(1.0/self.beta)))
iterationLimit = self.loop == 200
self.lastIteration = iterationLimit
def StartingValue(self):
return np.array(np.mean(self._data), np.variance(self._variance))
import numpy as np
import matplotlib.pyplot as plt
mean = np.zeros(10)
index = np.arange(10)
for i in range(len(index)):
s = np.random.poisson(3, 1000)
mean[i] = np.mean(s)
print mean
a = np.mean(mean)
a_v = np.variance(mean)
print a, a_v
plt.figure()
plt.title('10 Times')
plt.hist(mean, normed=True)
plt.show()
#############
import numpy as np
import matplotlib.pyplot as plt
mean = np.zeros(100)
index = np.arange(100)
for i in range(len(index)):
s = np.random.poisson(3, 1000)
mean[i] = np.mean(s)
print mean
b = np.mean(mean)
b_v = np.variance(mean)
