def __init__(self, X, Y, optimize=True, theta=None):
self.X = X
self.Y_mean = np.mean(Y)
self.Y = Y - self.Y_mean
self.cov = Covariance(self.X, self.Y, theta=theta, optimize=optimize)
def __init__(self, X, Y, optimize=True, theta=None):
self.X = X
self.Y_mean = np.mean(Y)
self.Y = Y - self.Y_mean
self.cov = Covariance(self.X, self.Y, theta=theta, optimize=optimize)
self.method = method # 1 - Girard, 2 - Deisenroth, 3 - Candela
class GaussianProcess(object):
def __init__(self, X, Y, optimize=True, theta=None):
self.X = X
self.Y_mean = np.mean(Y)
self.Y = Y - self.Y_mean
self.cov = Covariance(self.X, self.Y, theta=theta, optimize=optimize)
def predict(self, x):
k = self.cov.Gcov(x, x)
# k_vector = self.cov.cov_matrix_ij(x.reshape(1,-1), self.X) - self.cov._get_vt() # Slower
k_vector = np.empty((self.X.shape[0], 1))
for i in range(self.X.shape[0]):
k_vector[i] = self.cov.Gcov(x, self.X[i, :])
k_vector = k_vector.reshape(1, -1)
# Implementation from Girard, Pg. 22
mean = np.dot(k_vector, np.dot(self.cov.Kinv, self.Y)) + self.Y_mean
var = k - np.dot(k_vector, np.dot(self.cov.Kinv, k_vector.T))
# Implementation from Girard, Pg. 32
# beta = np.dot(self.cov.Kinv, self.Y)
# mean = 0
# s = 0
# for i in range(len(beta)):
# mean += beta[i] * k_vector[0][i]
# for j in range(len(beta)):
# s += self.cov.Kinv[i,j] * k_vector[0][i] * k_vector[0][j]
# var = k - s
# mean += self.Y_mean
return mean, var
def batch_predict(self, Xs):
Xs = np.array(Xs)
k = self.cov.cov_matrix_ij(Xs, Xs)
kv = self.cov.cov_matrix_ij(Xs, self.X, add_vt=False)
mean = np.dot(kv, np.dot(self.cov.Kinv, self.Y)) + self.Y_mean
variance = np.diag(np.diag(k)) - np.dot(kv, np.dot(
self.cov.Kinv,
kv.T)) # + self.cov._get_vt() #code variance + aleatory variance
return mean, variance
class UncertaintyPropagation(object):
def __init__(self, X, Y, optimize = True, theta = None, method = 3):
self.X = X
self.Y_mean = np.mean(Y)
self.Y = Y - self.Y_mean
self.cov = Covariance(self.X, self.Y, theta = theta, optimize = optimize)
self.method = method # 1 - Girard, 2 - Deisenroth, 3 - Candela
self.T = np.zeros(10)
def predict(self, mu_x, sigma_x):
d = self.cov.d
self.beta = np.dot(self.cov.Kinv, self.Y)
self.W = scipy.diag(self.cov._get_w())
self.Winv = scipy.linalg.inv(self.W)
self.Winv2 = scipy.linalg.inv(self.W/2)
self.Sigma_x = np.diag(sigma_x)
if len(self.Sigma_x) == 1:
self.Sigma_x = [self.Sigma_x]
if self.method == 1:
# Girard Eq. 3.40, 3.41
self.DeltaInv = self.Winv - scipy.linalg.inv(self.W + self.Sigma_x) # Based on the Girard Thesis
self.M = 1 / np.sqrt( np.linalg.det( np.eye(d) + np.dot(self.Winv, [self.Sigma_x]) ) )
self.DeltaInv2 = 2*self.Winv - scipy.linalg.inv(0.5*self.W + self.Sigma_x)
self.M2 = 1 / np.sqrt( np.linalg.det( np.dot(self.Winv2, self.Sigma_x) + np.eye(d) ) )
if self.method == 2:
# Deisenroth Eq. 2.34
self.DeltaInv = scipy.linalg.inv(self.W + self.Sigma_x)
self.M = 1 / np.sqrt( np.linalg.det( np.dot([self.Sigma_x], self.Winv) + np.eye(d) ) )
self.DeltaInv2 = np.dot(np.dot(scipy.linalg.inv(self.Sigma_x + 0.5*self.W), self.Sigma_x), self.Winv)
self.M2 = 1 / np.sqrt( np.linalg.det( 2*np.dot([self.Sigma_x], self.Winv) + np.eye(d) ) )
if self.method == 3:
# Candela Eq. 22
self.DeltaInv = scipy.linalg.inv(self.W + self.Sigma_x)
self.M = 1 / np.sqrt( np.linalg.det( np.dot(self.Winv, self.Sigma_x) + np.eye(d) ) )
self.DeltaInv2 = np.dot(np.dot(self.Winv, scipy.linalg.inv(2*self.Winv + scipy.linalg.inv(self.Sigma_x))), self.Winv)
self.M2 = 1 / np.sqrt( np.linalg.det( 2*np.dot(self.Winv, self.Sigma_x) + np.eye(d) ) )
mean = self.predict_mean(mu_x)
variance = self.predict_variance(mu_x, mean)
print self.T[:4]
return mean, variance
def _C_corr(self, x, xi):
diff = x - xi
if self.method == 1:
# Girard Eq. 3.40
return self.M * np.exp( 0.5 * np.dot( diff.T, np.dot( self.DeltaInv, diff ) ) )
if self.method == 2:
# Deisenroth Eq. 2.34
v = self.cov._get_v()
return v * self.M * np.exp( - 0.5 * np.dot( diff, np.dot( self.DeltaInv, diff ) ) )
if self.method == 3:
# Candela Eq. 22
v = self.cov._get_v()
return v * self.M * np.exp( - 0.5 * np.dot( diff, np.dot( self.DeltaInv, diff ) ) )
def _C_corr_2(self, x, x_):
diff = x - x_
if self.method == 1:
# Girard Eq. 3.41
return self.M2 * np.exp( 0.5 * np.dot( diff.T, np.dot( self.DeltaInv2, diff ) ) )
if self.method == 2:
# Deisenroth Eq. 2.42
return self.M2 * np.exp( np.dot( diff.T, np.dot( self.DeltaInv2, diff ) ) )
if self.method == 3:
# Candela Eq. 24
return self.M2 * np.exp( 2 * np.dot( diff.T, np.dot( self.DeltaInv2, diff ) ) )
def predict_mean(self, mu_x):
if self.method == 1:
# Girard Eq. 3.40
mean = 0.
for i in range(self.cov.N):
mean += self.beta[i] * self.cov.Gcov(mu_x, self.X[i,:]) * self._C_corr(mu_x, self.X[i,:])
else:
# Deisenroth Eq. 2.34, Candela Eq. 21
q = np.empty((self.cov.N,1))
for i in range(self.cov.N):
q[i] = self._C_corr(mu_x, self.X[i,:])
mean = np.dot(self.beta, q)
self.q = q
return mean + self.Y_mean
def predict_variance(self, mu_x, mean):
if self.method == 1:
# Girard Eq. 3.43
S = 0.
for i in range(self.cov.N):
for j in range(self.cov.N):
x_ = (self.X[i,:] + self.X[j,:]) / 2.
Ci = self.cov.Gcov(mu_x, self.X[i,:])
Cj = self.cov.Gcov(mu_x, self.X[j,:])
C_corr_2 = self._C_corr_2(mu_x, x_)
S += (self.cov.Kinv[i,j] - self.beta[i]*self.beta[j]) * Ci * Cj * C_corr_2
var = self.cov.Gcov(mu_x, mu_x) - S - mean**2 # Girard 3.43
if self.method == 2:
# Deisenroth Eq. 2.41
Q = np.empty((self.cov.N, self.cov.N))
for i in range(self.cov.N):
for j in range(self.cov.N):
x_ = (self.X[i,:] + self.X[j,:]) / 2.
Ci = self.cov.Gcov(mu_x, self.X[i,:])
Cj = self.cov.Gcov(mu_x, self.X[j,:])
C_corr_2 = self._C_corr_2(mu_x, x_)
Q[i,j] = Ci * Cj * C_corr_2
k_vector = np.empty((self.X.shape[0],1))
for i in range(self.X.shape[0]):
k_vector[i] = self.cov.Gcov(mu_x, self.X[i,:])
# sigma2_GP = self.cov._get_v() #self.cov.Gcov(mu_x, mu_x)
sigma2_GP = self.cov.Gcov(mu_x, mu_x) - np.dot( k_vector.reshape(1,-1), np.dot(self.cov.Kinv, k_vector.reshape(-1,1) ) ) # Candela Eq. 3 = self.cov.Gcov(mu_x, mu_x)?
var = sigma2_GP - np.trace( np.dot(self.cov.Kinv, Q) ) + np.dot( self.beta, np.dot(Q, self.beta)) - mean**2
if self.method == 3:
# Candela Eq. 24
k_vector = np.diag(self.cov.cov_matrix_ij(np.tile(mu_x.reshape(1,-1), (self.X.shape[0],1)), self.X, add_vt=False))
L = np.empty((self.cov.N, self.cov.N))
for i in range(self.cov.N):
for j in range(self.cov.N):
st = time.time()
x_ = (self.X[i,:] + self.X[j,:]) / 2.
self.T[0] += time.time() - st
st = time.time()
ki = k_vector[i]#self.cov.Gcov(mu_x, self.X[i,:])
kj = k_vector[j]#self.cov.Gcov(mu_x, self.X[j,:])
self.T[1] += time.time() - st
st = time.time()
C_corr_2 = self._C_corr_2(mu_x, x_)
self.T[2] += time.time() - st
st = time.time()
L[i,j] = ki * kj * C_corr_2
self.T[3] += time.time() - st
# k_vector = np.empty((self.X.shape[0],1))
# for i in range(self.X.shape[0]):
# k_vector[i] = self.cov.Gcov(mu_x, self.X[i,:])
kk = np.dot(k_vector.reshape(-1,1), k_vector.reshape(1,-1))
L = L - kk
bb = np.dot(self.beta.reshape(-1,1), self.beta.reshape(1,-1))
ll = np.dot(self.q.reshape(-1,1), self.q.reshape(1,-1))
sigma2_GP = self.cov.Gcov(mu_x, mu_x) - np.dot( k_vector.reshape(1,-1), np.dot(self.cov.Kinv, k_vector.reshape(-1,1) ) ) # Candela Eq. 3 = self.cov.Gcov(mu_x, mu_x)?
var = sigma2_GP + np.trace( np.dot(L, bb - self.cov.Kinv) ) + np.trace( np.dot( kk - ll, bb ) )
return var