def __init__(self,
sf0,
ell0,
Z0,
U0,
whiten=False,
summ=False,
fix_ell=True,
fix_sf=True,
fix_Z=True,
fix_U=False):
with tf.name_scope('Brownian'):
Zg = Param(Z0, name="Z", summ=False, fixed=fix_Z)
Ug = Param(U0, name="U", summ=False, fixed=fix_U)
self.kern = OperatorKernel(sf0=sf0,
ell0=ell0,
ktype="id",
name='Kernel',
summ=summ,
fix_ell=fix_ell,
fix_sf=fix_sf)
self.Zg = Zg()
self.Ug = Ug()
self.jitter = 1e-6
self.whiten = whiten
self.fix_Z = fix_Z
self.fix_U = fix_U
def load_model(fname,sess):
with open(fname, 'rb') as f:
pars = pickle.load(f)
if len(pars) == 12:
sf,ell,ktype,fix_ell,fix_sf,Z,U,sn,whiten,fix_Z,fix_U,fix_sn = pars
kern = OperatorKernel(sf0=sf, ell0=ell, ktype=ktype, fix_ell=fix_ell, fix_sf=fix_sf)
npde = NPODE(Z0=Z, U0=U, sn0=sn, kern=kern, whiten=whiten, \
fix_Z=fix_Z, fix_U=fix_U, fix_sn=fix_sn)
else:
sf,ell,ktype,fix_ell,fix_sf,Z,U,sn,whiten,fix_Z,fix_U,fix_sn,\
sfg,ellg,Ug,Zg,whiteng,fix_sfg,fix_ellg,fix_Zg,fix_Ug = pars
kern = OperatorKernel(sf0=sf, ell0=ell, ktype=ktype, fix_ell=fix_ell, fix_sf=fix_sf)
diffus = BrownianMotion(sf0=sfg, ell0=ellg, U0=Ug, Z0=Zg, whiten=whiten,
fix_sf=fix_sfg, fix_ell=fix_ellg, fix_Z=fix_Zg, fix_U=fix_Ug)
npde = NPSDE(Z0=Z, U0=U, sn0=sn, kern=kern, diffus=diffus, whiten=whiten,
fix_Z=fix_Z, fix_U=fix_U, fix_sn=fix_sn)
sess.run(tf.global_variables_initializer())
return npde
class BrownianMotion:
def __init__(self,
sf0,
ell0,
Z0,
U0,
whiten=False,
summ=False,
fix_ell=True,
fix_sf=True,
fix_Z=True,
fix_U=False):
with tf.name_scope('Brownian'):
Zg = Param(Z0, name="Z", summ=False, fixed=fix_Z)
Ug = Param(U0, name="U", summ=False, fixed=fix_U)
self.kern = OperatorKernel(sf0=sf0,
ell0=ell0,
ktype="id",
name='Kernel',
summ=summ,
fix_ell=fix_ell,
fix_sf=fix_sf)
self.Zg = Zg()
self.Ug = Ug()
self.jitter = 1e-6
self.whiten = whiten
self.fix_Z = fix_Z
self.fix_U = fix_U
def g(self, X, t):
""" generates state dependent brownian motion
Args:
X: current states (in rows)
t: current time (used if diffusion depends on time)
Returns:
A tensor of the same shape as X
"""
Ug = self.Ug
Zg = self.Zg
kern = self.kern
if not kern.ktype == "id":
raise NotImplementedError()
M = tf.shape(Zg)[0]
D = tf.shape(X)[1]
if kern.ktype == "id":
Kzz = kern.K(Zg) + tf.eye(M, dtype=float_type) * self.jitter
else:
Kzz = kern.K(Zg) + tf.eye(M * D, dtype=float_type) * self.jitter
Lz = tf.cholesky(Kzz)
Kzx = kern.K(Zg, X)
A = tf.matrix_triangular_solve(Lz, Kzx, lower=True)
if not self.whiten:
A = tf.matrix_triangular_solve(tf.transpose(Lz), A, lower=False)
g = tf.matmul(A, Ug, transpose_a=True)
dw = tf.random_normal(tf.shape(X), dtype=float_type)
return g * dw
def __str__(self):
rep = '\ndiff signal variance: ' + str(self.kern.sf.eval()) + \
'\ndiff lengthscales: ' + str(self.kern.ell.eval())
return rep
def build_prior(self):
if self.whiten:
mvn = tfd.MultivariateNormalDiag(loc=tf.zeros_like(self.Ug))
else:
mvn = tfd.MultivariateNormalFullCovariance(
loc=tf.zeros_like(self.Ug),
covariance_matrix=self.kern.K(self.Zg, self.Zg))
return tf.reduce_sum(mvn.log_prob(self.Ug))
def build_model(sess,t,Y,model='sde',sf0=1.0,ell0=[2,2],sfg0=1.0,ellg0=[1e5],
W=6,ktype="id",whiten=True,
fix_ell=False,fix_sf=False,fix_Z=False,fix_U=False,fix_sn=False,
fix_ellg=False,fix_sfg=False,fix_Zg=True,fix_Ug=False):
"""
Args:
sess: TensowFlow session needed for initialization and optimization
t: Python array of numpy vectors storing observation times
Y: Python array of numpy matrices storing observations. Observations
are stored in rows.
model: 'sde' or 'ode'
sf0: Integer initial value of the signal variance of drift GP
ell0: Python/numpy array of floats for the initial value of the
lengthscale of drift GP
sfg0: Integer initial value of the signal variance of diffusion GP
ellg0: Python/numpy array of a single float for the initial value of the
lengthscale of diffusion GP
W: Integer denoting the width of the inducing point grid. If the problem
dimension is D, total number of inducing points is W**D
ktype: Kernel type. We have made experiments only with Kronecker kernel,
denoted by 'id'. The other kernels are not supported.
whiten: Boolean. Currently we perform the optimization only in the
white domain
fix_ell: Boolean - whether drift GP lengthscale is fixed or optimized
fix_sf: Boolean - whether drift GP signal variance is fixed or optimized
fix_Z: Boolean - whether drift GP inducing locations are fixed or optimized
fix_U: Boolean - whether drift GP inducing vectors are fixed or optimized
fix_sn: Boolean - whether noise variance is fixed or optimized
fix_ellg: Boolean - whether diffusion GP lengthscale is fixed or optimized
fix_sfg: Boolean - whether diffusion GP signal variance is fixed or optimized
fix_Zg: Boolean - whether diffusion GP inducing locations are fixed or optimized
fix_Ug: Boolean - whether diffusion GP inducing vectors are fixed or optimized
Returns:
npde: A new NPDE model
"""
print('Model being initialized...')
def init_U0(Y=None,t=None,kern=None,Z0=None,whiten=None):
Ug = (Y[1:,:] - Y[:-1,:]) / np.reshape(t[1:]-t[:-1],(-1,1))
with tf.name_scope("init_U0"):
tmp = NPODE(Z0=Y[:-1,:],U0=Ug,sn0=0,kern=kern,jitter=0.25,whiten=False,
fix_Z=True,fix_U=True,fix_sn=True)
U0 = tmp.f(X=Z0)
if whiten:
Lz = tf.cholesky(kern.K(Z0))
U0 = tf.matrix_triangular_solve(Lz, U0, lower=True)
U0 = sess.run(U0)
return U0
D = len(ell0)
Nt = len(Y)
x0 = np.zeros((Nt,D))
Ys = np.zeros((0,D))
for i in range(Nt):
x0[i,:] = Y[i][0,:]
Ys = np.vstack((Ys,Y[i]))
maxs = np.max(Ys,0)
mins = np.min(Ys,0)
grids = []
for i in range(D):
grids.append(np.linspace(mins[i],maxs[i],W))
vecs = np.meshgrid(*grids)
Z0 = np.zeros((0,W**D))
for i in range(D):
Z0 = np.vstack((Z0,vecs[i].T.flatten()))
Z0 = Z0.T
tmp_kern = OperatorKernel(sf0,ell0,ktype="id",fix_ell=True,fix_sf=True)
U0 = np.zeros(Z0.shape,dtype=np.float64)
for i in range(len(Y)):
U0 += init_U0(Y[i],t[i],tmp_kern,Z0,whiten)
U0 /= len(Y)
sn0 = 0.5*np.ones(D)
Ug0 = np.ones([Z0.shape[0],1])*0.01
ell0 = np.asarray(ell0,dtype=np.float64)
ellg0 = np.asarray(ellg0,dtype=np.float64)
kern = OperatorKernel(sf0=sf0, ell0=ell0, ktype=ktype, fix_ell=fix_ell, fix_sf=fix_sf)
if model is 'ode':
npde = NPODE(Z0=Z0, U0=U0, sn0=sn0, kern=kern, whiten=whiten, fix_Z=fix_Z, fix_U=fix_U, fix_sn=fix_sn)
sess.run(tf.global_variables_initializer())
return npde
elif model is 'sde':
diffus = BrownianMotion(sf0=sfg0, ell0=ellg0, U0=Ug0, Z0=Z0, whiten=whiten,\
fix_sf=fix_sfg, fix_ell=fix_ellg, fix_Z=fix_Zg, fix_U=fix_Ug)
npde = NPSDE(Z0=Z0, U0=U0, sn0=sn0, kern=kern, diffus=diffus, whiten=whiten,\
fix_Z=fix_Z, fix_U=fix_U, fix_sn=fix_sn)
sess.run(tf.global_variables_initializer())
return npde
else:
raise NotImplementedError("model parameter should be either 'ode' or 'sde', not {:s}\n".format(model))
def npde_fit(sess,
t,
Y,
model='sde',
sf0=1.0,
ell0=[2, 2],
sfg0=1.0,
ellg0=[1e5],
W=6,
ktype="id",
whiten=True,
Nw=50,
fix_ell=False,
fix_sf=False,
fix_Z=False,
fix_U=False,
fix_sn=False,
fix_ellg=False,
fix_sfg=False,
fix_Zg=True,
fix_Ug=False,
num_iter=500,
print_int=10,
eta=5e-3,
dec_step=20,
dec_rate=0.99,
plot_=True):
""" Fits the NPDE model to a dataset and returns the fitted object
Args:
sess: TensowFlow session needed for initialization and optimization
t: Python array of numpy vectors storing observation times
Y: Python array of numpy matrices storing observations. Observations
are stored in rows.
model: 'sde' or 'ode'
sf0: Integer initial value of the signal variance of drift GP
ell0: Python/numpy array of floats for the initial value of the
lengthscale of drift GP
sfg0: Integer initial value of the signal variance of diffusion GP
ellg0: Python/numpy array of a single float for the initial value of the
lengthscale of diffusion GP
W: Integer denoting the width of the inducing point grid. If the problem
dimension is D, total number of inducing points is W**D
ktype: Kernel type. We have made experiments only with Kronecker kernel,
denoted by 'id'. The other kernels are not supported.
whiten: Boolean. Currently we perform the optimization only in the
white domain
Nw: Integer number of samples used for optimization in SDE model
fix_ell: Boolean - whether drift GP lengthscale is fixed or optimized
fix_sf: Boolean - whether drift GP signal variance is fixed or optimized
fix_Z: Boolean - whether drift GP inducing locations are fixed or optimized
fix_U: Boolean - whether drift GP inducing vectors are fixed or optimized
fix_sn: Boolean - whether noise variance is fixed or optimized
fix_ellg: Boolean - whether diffusion GP lengthscale is fixed or optimized
fix_sfg: Boolean - whether diffusion GP signal variance is fixed or optimized
fix_Zg: Boolean - whether diffusion GP inducing locations are fixed or optimized
fix_Ug: Boolean - whether diffusion GP inducing vectors are fixed or optimized
num_iter: Integer number of optimization steps
num_iter: Integer interval of optimization logs to be printed
eta: Float step size used in optimization, must be carefully tuned
dec_step: Float decay interval of the step size
dec_rate: Float decay rate of the step size
plot_: Boolean for plotting the model fit. Valid only for demo
Returns:
npde: Fitted model
"""
print('Model being initialized...')
def init_U0(Y=None, t=None, kern=None, Z0=None, whiten=None):
Ug = (Y[1:, :] - Y[:-1, :]) / np.reshape(t[1:] - t[:-1], (-1, 1))
tmp = NPODE(Y[0, :].reshape((1, -1)),
t,
Y,
Z0=Y[:-1, :],
U0=Ug,
sn0=0,
kern=kern,
jitter=0.2,
whiten=False,
fix_Z=True,
fix_U=True,
fix_sn=True)
U0 = tmp.f(X=Z0)
if whiten:
Lz = tf.cholesky(kern.K(Z0))
U0 = tf.matrix_triangular_solve(Lz, U0, lower=True)
U0 = sess.run(U0)
return U0
D = len(ell0)
Nt = len(Y)
x0 = np.zeros((Nt, D))
Ys = np.zeros((0, D))
for i in range(Nt):
x0[i, :] = Y[i][0, :]
Ys = np.vstack((Ys, Y[i]))
maxs = np.max(Ys, 0)
mins = np.min(Ys, 0)
grids = []
for i in range(D):
grids.append(np.linspace(mins[i], maxs[i], W))
vecs = np.meshgrid(*grids)
Z0 = np.zeros((0, W**D))
for i in range(D):
Z0 = np.vstack((Z0, vecs[i].T.flatten()))
Z0 = Z0.T
tmp_kern = OperatorKernel(sf0, ell0, ktype="id", fix_ell=True, fix_sf=True)
U0 = np.zeros(Z0.shape, dtype=np.float64)
for i in range(len(Y)):
U0 += init_U0(Y[i], t[i], tmp_kern, Z0, whiten)
U0 /= len(Y)
sn0 = 0.5 * np.ones(D)
Ug0 = np.ones([Z0.shape[0], 1]) * 0.01
ell0 = np.asarray(ell0, dtype=np.float64)
ellg0 = np.asarray(ellg0, dtype=np.float64)
kern = OperatorKernel(sf0=sf0,
ell0=ell0,
ktype=ktype,
fix_ell=fix_ell,
fix_sf=fix_sf)
if model is 'ode':
npde = NPODE(x0=x0,
t=t,
Y=Y,
Z0=Z0,
U0=U0,
sn0=sn0,
kern=kern,
whiten=whiten,
fix_Z=fix_Z,
fix_U=fix_U,
fix_sn=fix_sn)
with tf.name_scope("cost"):
X = npde.forward()
ll = []
for i in range(len(X)):
mvn = tf.contrib.distributions.MultivariateNormalFullCovariance(
loc=X[i], covariance_matrix=tf.diag(npde.sn))
ll.append(tf.reduce_sum(mvn.log_prob(Y[i])))
ll = tf.reduce_logsumexp(ll)
ode_prior = npde.build_prior()
cost = -(ll + ode_prior)
elif model is 'sde':
diffus = BrownianMotion(sf0=sfg0,
ell0=ellg0,
U0=Ug0,
Z0=Z0,
whiten=whiten,
fix_sf=fix_sfg,
fix_ell=fix_ellg,
fix_Z=fix_Zg,
fix_U=fix_Ug)
npde = NPSDE(x0=x0,
t=t,
Y=Y,
Z0=Z0,
U0=U0,
sn0=sn0,
kern=kern,
diffus=diffus,
whiten=whiten,
fix_Z=fix_Z,
fix_U=fix_U,
fix_sn=fix_sn)
with tf.name_scope("cost"):
Xs = npde.forward(Nw=Nw)
ll = 0
for i in range(len(Y)):
mvn = tf.contrib.distributions.MultivariateNormalFullCovariance(
loc=Y[i], covariance_matrix=tf.diag(npde.sn))
ll_i = tf.stack([
mvn.log_prob(Xs[i][j, :, :]) for j in range(Xs[i].shape[0])
]) # Nw x D
ll_i = tf.reduce_sum(
tf.log(tf.reduce_mean(tf.exp(ll_i), axis=0)))
ll += ll_i
sde_prior = npde.build_prior()
cost = -(ll + sde_prior)
else:
raise NotImplementedError(
"model parameter should be either 'ode' or 'sde', not {:s}\n".
format(model))
print('Adam optimizer being initialized...')
global_step = tf.Variable(0,
dtype=tf.int32,
trainable=False,
name='global_step')
expdec = tf.train.exponential_decay(eta,
global_step,
dec_step,
dec_rate,
staircase=True)
optimizer = tf.train.AdamOptimizer(expdec).minimize(cost, global_step)
sess.run(tf.global_variables_initializer())
print('Optimization starts.')
print('{:>16s}'.format("iteration") + '{:>16s}'.format("objective"))
for i in range(1, num_iter + 1):
_cost, _ = sess.run([cost, optimizer])
if i == 1 or i % print_int == 0 or i == num_iter:
print('{:>16d}'.format(i) + '{:>16.3f}'.format(_cost))
print('Optimization ends.')
if plot_:
print('Plotting...')
plot_model(npde, Nw=50)
return npde