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STATE-SPACE: Recent modifications to state-space inference, including bug fixes in state-space kernels.
This commit is contained in:
Alex Grigorievskiy
parent
5ce5c5d8ce
commit
dfe32266b6
6 changed files with 533 additions and 91 deletions
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@ -9,6 +9,7 @@ from .standard_periodic import StdPeriodic
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import numpy as np
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import scipy as sp
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import warnings
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from scipy import special as special
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@ -26,6 +27,38 @@ class sde_StdPeriodic(StdPeriodic):
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\left( \frac{\sin(\frac{\pi}{\lambda_i} (x_i - y_i) )}{l_i} \right)^2 \right] }
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"""
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# TODO: write comment to the constructor arguments
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def __init__(self, *args, **kwargs):
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"""
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Init constructior.
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Two optinal extra parameters are added in addition to the ones in
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StdPeriodic kernel.
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:param approx_order: approximation order for the RBF covariance. (Default 7)
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:type approx_order: int
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:param balance: Whether to balance this kernel separately. (Defaulf False). Model has a separate parameter for balancing.
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:type balance: bool
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"""
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#import pdb; pdb.set_trace()
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if 'approx_order' in kwargs:
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self.approx_order = kwargs.get('approx_order')
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del kwargs['approx_order']
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else:
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self.approx_order = 7
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if 'balance' in kwargs:
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self.balance = bool( kwargs.get('balance') )
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del kwargs['balance']
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else:
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self.balance = False
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super(sde_StdPeriodic, self).__init__(*args, **kwargs)
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def sde_update_gradient_full(self, gradients):
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"""
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Update gradient in the order in which parameters are represented in the
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@ -38,40 +71,47 @@ class sde_StdPeriodic(StdPeriodic):
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def sde(self):
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"""
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Return the state space representation of the covariance.
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Return the state space representation of the standard periodic covariance.
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! Note: one must constrain lengthscale not to drop below 0.25.
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After this bessel functions of the first kind grows to very high.
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! Note: one must constrain lengthscale not to drop below 0.2. (independently of approximation order)
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After this Bessel functions of the first becomes NaN. Rescaling
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time variable might help.
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! Note: one must keep wevelength also not very low. Because then
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! Note: one must keep period also not very low. Because then
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the gradients wrt wavelength become ustable.
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However this might depend on the data. For test example with
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300 data points the low limit is 0.15.
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"""
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#import pdb; pdb.set_trace()
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# Params to use: (in that order)
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#self.variance
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#self.period
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#self.lengthscale
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N = 7 # approximation order
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if self.approx_order is not None:
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N = int(self.approx_order)
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else:
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N = 7 # approximation order
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p_period = float(self.period)
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p_lengthscale = 2*float(self.lengthscale)
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p_variance = float(self.variance)
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w0 = 2*np.pi/self.period # frequency
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lengthscale = 2*self.lengthscale
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w0 = 2*np.pi/p_period # frequency
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# lengthscale is multiplied by 2 because of different definition of lengthscale
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[q2,dq2l] = seriescoeff(N,lengthscale,self.variance)
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# lengthscale is multiplied by 2 because of slightly different
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# formula for periodic covariance function.
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# For the same reason:
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[q2,dq2l] = seriescoeff(N, p_lengthscale, p_variance)
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dq2l = 2*dq2l
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dq2l = 2*dq2l # This is because the lengthscale if multiplied by 2.
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if np.any( np.isfinite(q2) == False):
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raise ValueError("SDE periodic covariance error 1")
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eps = 1e-12
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if np.any( np.isfinite(q2) == False) or np.any( np.abs(q2) > 1.0/eps) or np.any( np.abs(q2) < eps):
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warnings.warn("sde_Periodic: Infinite, too small, or too large (eps={0:e}) values in q2 :".format(eps) + q2.__format__("") )
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if np.any( np.isfinite(dq2l) == False) or np.any( np.abs(dq2l) > 1.0/eps) or np.any( np.abs(dq2l) < eps):
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warnings.warn("sde_Periodic: Infinite, too small, or too large (eps={0:e}) values in dq2l :".format(eps) + q2.__format__("") )
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if np.any( np.isfinite(dq2l) == False):
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raise ValueError("SDE periodic covariance error 2")
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F = np.kron(np.diag(range(0,N+1)),np.array( ((0, -w0), (w0, 0)) ) )
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L = np.eye(2*(N+1))
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@ -88,10 +128,10 @@ class sde_StdPeriodic(StdPeriodic):
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# Derivatives wrt self.variance
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dF[:,:,0] = np.zeros(F.shape)
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dQc[:,:,0] = np.zeros(Qc.shape)
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dP_inf[:,:,0] = P_inf / self.variance
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dP_inf[:,:,0] = P_inf / p_variance
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# Derivatives self.period
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dF[:,:,1] = np.kron(np.diag(range(0,N+1)),np.array( ((0, w0), (-w0, 0)) ) / self.period );
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dF[:,:,1] = np.kron(np.diag(range(0,N+1)),np.array( ((0, w0), (-w0, 0)) ) / p_period );
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dQc[:,:,1] = np.zeros(Qc.shape)
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dP_inf[:,:,1] = np.zeros(P_inf.shape)
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@ -101,6 +141,11 @@ class sde_StdPeriodic(StdPeriodic):
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dP_inf[:,:,2] = np.kron(np.diag(dq2l),np.eye(2))
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dP0 = dP_inf.copy()
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if self.balance:
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# Benefits of this are not very sound.
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import GPy.models.state_space_main as ssm
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(F, L, Qc, H, P_inf, P0, dF, dQc, dP_inf,dP0) = ssm.balance_ss_model(F, L, Qc, H, P_inf, P0, dF, dQc, dP_inf, dP0 )
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return (F, L, Qc, H, P_inf, P0, dF, dQc, dP_inf, dP0)
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@ -164,9 +209,9 @@ def seriescoeff(m=6,lengthScale=1.0,magnSigma2=1.0, true_covariance=False):
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coeffs = 2*magnSigma2*sp.exp( -lengthScale**(-2) ) * special.iv(range(0,m+1),1.0/lengthScale**(2))
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if np.any( np.isfinite(coeffs) == False):
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raise ValueError("sde_standard_periodic: Coefficients are not finite!")
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#import pdb; pdb.set_trace()
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#import pdb; pdb.set_trace()
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coeffs[0] = 0.5*coeffs[0]
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#print(coeffs)
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# Derivatives wrt (lengthScale)
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coeffs_dl = np.zeros(m+1)
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coeffs_dl[1:] = magnSigma2*lengthScale**(-3) * sp.exp(-lengthScale**(-2))*\
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@ -177,4 +222,4 @@ def seriescoeff(m=6,lengthScale=1.0,magnSigma2=1.0, true_covariance=False):
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(2*special.iv(0,lengthScale**(-2)) - 2*special.iv(1,lengthScale**(-2)) )
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return coeffs, coeffs_dl
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return coeffs.squeeze(), coeffs_dl.squeeze()
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@ -11,6 +11,7 @@ from .stationary import RatQuad
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import numpy as np
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import scipy as sp
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import warnings
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class sde_RBF(RBF):
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"""
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@ -25,6 +26,37 @@ class sde_RBF(RBF):
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k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} }
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"""
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def __init__(self, *args, **kwargs):
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"""
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Init constructior.
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Two optinal extra parameters are added in addition to the ones in
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RBF kernel.
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:param approx_order: approximation order for the RBF covariance. (Default 10)
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:type approx_order: int
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:param balance: Whether to balance this kernel separately. (Defaulf True). Model has a separate parameter for balancing.
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:type balance: bool
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"""
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if 'balance' in kwargs:
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self.balance = bool( kwargs.get('balance') )
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del kwargs['balance']
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else:
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self.balance = True
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if 'approx_order' in kwargs:
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self.approx_order = kwargs.get('approx_order')
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del kwargs['approx_order']
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else:
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self.approx_order = 6
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super(sde_RBF, self).__init__(*args, **kwargs)
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def sde_update_gradient_full(self, gradients):
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"""
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Update gradient in the order in which parameters are represented in the
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@ -37,23 +69,43 @@ class sde_RBF(RBF):
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def sde(self):
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"""
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Return the state space representation of the covariance.
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"""
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N = 10# approximation order ( number of terms in exponent series expansion)
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Note! For Sparse GP inference too small or two high values of lengthscale
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lead to instabilities. This is because Qc are too high or too low
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and P_inf are not full rank. This effect depends on approximatio order.
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For N = 10. lengthscale must be in (0.8,8). For other N tests must be conducted.
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N=6: (0.06,31)
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Variance should be within reasonable bounds as well, but its dependence is linear.
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The above facts do not take into accout regularization.
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"""
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#import pdb; pdb.set_trace()
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if self.approx_order is not None:
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N = self.approx_order
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else:
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N = 10# approximation order ( number of terms in exponent series expansion)
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roots_rounding_decimals = 6
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fn = np.math.factorial(N)
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kappa = 1.0/2.0/self.lengthscale**2
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p_lengthscale = float( self.lengthscale )
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p_variance = float(self.variance)
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kappa = 1.0/2.0/p_lengthscale**2
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Qc = np.array((self.variance*np.sqrt(np.pi/kappa)*fn*(4*kappa)**N,),)
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Qc = np.array( ((p_variance*np.sqrt(np.pi/kappa)*fn*(4*kappa)**N,),) )
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pp = np.zeros((2*N+1,)) # array of polynomial coefficients from higher power to lower
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eps = 1e-12
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if (float(Qc) > 1.0/eps) or (float(Qc) < eps):
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warnings.warn("""sde_RBF kernel: the noise variance Qc is either very large or very small.
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It influece conditioning of P_inf: {0:e}""".format(float(Qc)) )
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pp1 = np.zeros((2*N+1,)) # array of polynomial coefficients from higher power to lower
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for n in range(0, N+1): # (2N+1) - number of polynomial coefficients
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pp[2*(N-n)] = fn*(4.0*kappa)**(N-n)/np.math.factorial(n)*(-1)**n
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pp1[2*(N-n)] = fn*(4.0*kappa)**(N-n)/np.math.factorial(n)*(-1)**n
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pp = sp.poly1d(pp)
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pp = sp.poly1d(pp1)
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roots = sp.roots(pp)
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neg_real_part_roots = roots[np.round(np.real(roots) ,roots_rounding_decimals) < 0]
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@ -69,6 +121,7 @@ class sde_RBF(RBF):
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H[0,0] = 1
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# Infinite covariance:
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#import pdb; pdb.set_trace()
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Pinf = sp.linalg.solve_lyapunov(F, -np.dot(L,np.dot( Qc[0,0],L.T)))
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Pinf = 0.5*(Pinf + Pinf.T)
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# Allocating space for derivatives
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@ -79,17 +132,17 @@ class sde_RBF(RBF):
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# Derivatives:
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dFvariance = np.zeros(F.shape)
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dFlengthscale = np.zeros(F.shape)
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dFlengthscale[-1,:] = -aa[-1:0:-1]/self.lengthscale * np.arange(-N,0,1)
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dFlengthscale[-1,:] = -aa[-1:0:-1]/p_lengthscale * np.arange(-N,0,1)
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dQcvariance = Qc/self.variance
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dQclengthscale = np.array(((self.variance*np.sqrt(2*np.pi)*fn*2**N*self.lengthscale**(-2*N)*(1-2*N,),)))
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dQcvariance = Qc/p_variance
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dQclengthscale = np.array(( (p_variance*np.sqrt(2*np.pi)*fn*2**N*p_lengthscale**(-2*N)*(1-2*N),),))
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dPinf_variance = Pinf/self.variance
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dPinf_variance = Pinf/p_variance
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lp = Pinf.shape[0]
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coeff = np.arange(1,lp+1).reshape(lp,1) + np.arange(1,lp+1).reshape(1,lp) - 2
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coeff[np.mod(coeff,2) != 0] = 0
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dPinf_lengthscale = -1/self.lengthscale*Pinf*coeff
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dPinf_lengthscale = -1/p_lengthscale*Pinf*coeff
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dF[:,:,0] = dFvariance
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dF[:,:,1] = dFlengthscale
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@ -101,10 +154,11 @@ class sde_RBF(RBF):
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P0 = Pinf.copy()
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dP0 = dPinf.copy()
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# Benefits of this are not very sound. Helps only in one case:
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# SVD Kalman + RBF kernel
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import GPy.models.state_space_main as ssm
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(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf,dP0, T) = ssm.balance_ss_model(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0 )
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if self.balance:
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# Benefits of this are not very sound. Helps only in one case:
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# SVD Kalman + RBF kernel
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import GPy.models.state_space_main as ssm
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(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf,dP0) = ssm.balance_ss_model(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0 )
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return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
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@ -432,6 +432,8 @@ cdef class AQcompute_batch_Cython(Q_handling_Cython):
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(self.reconstruct_indices.nbytes if (self.reconstruct_indices is not None) else 0)
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self.Q_svd_dict = {}
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self.Q_square_root_dict = {}
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self.Q_inverse_dict = {}
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self.last_k = 0
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# !!!Print statistics! Which object is created
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# !!!Print statistics! Print sizes of matrices
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@ -478,18 +480,53 @@ cdef class AQcompute_batch_Cython(Q_handling_Cython):
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cdef np.ndarray[DTYPE_t, ndim=1] S
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cdef np.ndarray[DTYPE_t, ndim=2] Vh
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if matrix_index in self.Q_svd_dict:
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square_root = self.Q_svd_dict[matrix_index]
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if matrix_index in self.Q_square_root_dict:
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square_root = self.Q_square_root_dict[matrix_index]
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else:
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U,S,Vh = sp.linalg.svd( self.Qs[:,:, matrix_index],
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if matrix_index not in self.Q_svd_dict
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U,S,Vh = sp.linalg.svd( self.Qs[:,:, matrix_index],
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full_matrices=False, compute_uv=True,
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overwrite_a=False, check_finite=False)
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self.Q_svd_dict[matrix_index] = (U,S,Vh)
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else:
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U,S,Vh = self.Q_svd_dict[matrix_index]
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square_root = U * np.sqrt(S)
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self.Q_svd_dict[matrix_index] = square_root
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self.Q_suqare_root_dict[matrix_index] = square_root
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return square_root
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cpdef Q_inverse(self, int k, float jitter=0.0):
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"""
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Square root of the noise matrix Q
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"""
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cdef int matrix_index = <int>self.reconstruct_indices[k]
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cdef np.ndarray[DTYPE_t, ndim=2] square_root
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cdef np.ndarray[DTYPE_t, ndim=2] U
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cdef np.ndarray[DTYPE_t, ndim=1] S
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cdef np.ndarray[DTYPE_t, ndim=2] Vh
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if matrix_index in self.Q_inverse_dict:
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Q_inverse = self.Q_inverse_dict[matrix_index]
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else:
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if matrix_index not in self.Q_svd_dict
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U,S,Vh = sp.linalg.svd( self.Qs[:,:, matrix_index],
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full_matrices=False, compute_uv=True,
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overwrite_a=False, check_finite=False)
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self.Q_svd_dict[matrix_index] = (U,S,Vh)
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else:
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U,S,Vh = self.Q_svd_dict[matrix_index]
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Q_inverse = Q_inverse = np.dot( Vh.T * ( 1.0/(S + jitter)) , U.T )
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self.Q_inverse_dict[matrix_index] = Q_inverse
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return Q_inverse
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# def return_last(self):
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# """
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# Function returns last available matrices.
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@ -12,6 +12,8 @@ import numpy as np
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import scipy as sp
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import scipy.linalg as linalg
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import warnings
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try:
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from . import state_space_setup
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setup_available = True
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@ -41,6 +43,10 @@ if print_verbose:
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else:
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print("state_space: cython is NOT used")
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# When debugging external module can set some value to this variable (e.g.)
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# 'model' and in this module this variable can be seen.s
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tmp_buffer = None
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class Dynamic_Callables_Python(object):
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@ -227,7 +233,7 @@ class R_handling_Python(Measurement_Callables_Class):
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self.R_square_root = {}
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def Rk(self, k):
|
||||
return self.R[:, :, self.index[self.R_time_var_index, k]]
|
||||
return self.R[:, :, int(self.index[self.R_time_var_index, k])]
|
||||
|
||||
def dRk(self, k):
|
||||
if self.dR is None:
|
||||
|
|
@ -305,7 +311,7 @@ class Std_Measurement_Callables_Python(R_handling_Class):
|
|||
P: parameter for Jacobian, usually covariance matrix.
|
||||
"""
|
||||
|
||||
return self.H[:, :, self.index[self.H_time_var_index, k]]
|
||||
return self.H[:, :, int(self.index[self.H_time_var_index, k])]
|
||||
|
||||
def dHk(self, k):
|
||||
if self.dH is None:
|
||||
|
|
@ -2303,6 +2309,8 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
self.v_dQk = None
|
||||
|
||||
self.square_root_computed = False
|
||||
self.Q_inverse_computed = False
|
||||
self.Q_svd_computed = False
|
||||
# !!!Print statistics! Which object is created
|
||||
|
||||
def f_a(self, k,m,A):
|
||||
|
|
@ -2337,7 +2345,10 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
self.v_Qk = v_Qk
|
||||
self.v_dAk = v_dAk
|
||||
self.v_dQk = v_dQk
|
||||
|
||||
self.Q_square_root_computed = False
|
||||
self.Q_inverse_computed = False
|
||||
self.Q_svd_computed = False
|
||||
else:
|
||||
v_Ak = self.v_Ak
|
||||
v_Qk = self.v_Qk
|
||||
|
|
@ -2359,8 +2370,11 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
self.last_k = 0
|
||||
self.last_k_computed = False
|
||||
self.compute_derivatives = compute_derivatives
|
||||
self.Q_square_root_computed = False
|
||||
|
||||
self.Q_square_root_computed = False
|
||||
self.Q_inverse_computed = False
|
||||
self.Q_svd_computed = False
|
||||
self.Q_eigen_computed = False
|
||||
return self
|
||||
|
||||
def Ak(self,k,m,P):
|
||||
|
|
@ -2381,12 +2395,19 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
|
||||
def Q_srk(self,k):
|
||||
"""
|
||||
Check square root, maybe rewriting for Spectral decomposition is needed.
|
||||
Square root of the noise matrix Q
|
||||
"""
|
||||
|
||||
if ((self.last_k == k) and (self.last_k_computed == True)):
|
||||
if not self.Q_square_root_computed:
|
||||
(U, S, Vh) = sp.linalg.svd( self.v_Qk, full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
if not self.Q_svd_computed:
|
||||
(U, S, Vh) = sp.linalg.svd( self.v_Qk, full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
self.Q_svd = (U, S, Vh)
|
||||
self.Q_svd_computed = True
|
||||
else:
|
||||
(U, S, Vh) = self.Q_svd
|
||||
|
||||
square_root = U * np.sqrt(S)
|
||||
self.square_root_computed = True
|
||||
self.Q_square_root = square_root
|
||||
|
|
@ -2397,6 +2418,55 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
|
||||
return square_root
|
||||
|
||||
def Q_inverse(self, k, p_largest_cond_num, p_regularization_type):
|
||||
"""
|
||||
Function inverts Q matrix and regularizes the inverse.
|
||||
Regularization is useful when original matrix is badly conditioned.
|
||||
Function is currently used only in SparseGP code.
|
||||
|
||||
Inputs:
|
||||
------------------------------
|
||||
k: int
|
||||
Iteration number.
|
||||
|
||||
p_largest_cond_num: float
|
||||
Largest condition value for the inverted matrix. If cond. number is smaller than that
|
||||
no regularization happen.
|
||||
|
||||
regularization_type: 1 or 2
|
||||
Regularization type.
|
||||
|
||||
regularization_type: int (1 or 2)
|
||||
|
||||
type 1: 1/(S[k] + regularizer) regularizer is computed
|
||||
type 2: S[k]/(S^2[k] + regularizer) regularizer is computed
|
||||
"""
|
||||
|
||||
#import pdb; pdb.set_trace()
|
||||
|
||||
if ((self.last_k == k) and (self.last_k_computed == True)):
|
||||
if not self.Q_inverse_computed:
|
||||
if not self.Q_svd_computed:
|
||||
(U, S, Vh) = sp.linalg.svd( self.v_Qk, full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
self.Q_svd = (U, S, Vh)
|
||||
self.Q_svd_computed = True
|
||||
else:
|
||||
(U, S, Vh) = self.Q_svd
|
||||
|
||||
Q_inverse_r = psd_matrix_inverse(k, 0.5*(self.v_Qk + self.v_Qk.T), U,S, p_largest_cond_num, p_regularization_type)
|
||||
|
||||
self.Q_inverse_computed = True
|
||||
self.Q_inverse_r = Q_inverse_r
|
||||
|
||||
else:
|
||||
Q_inverse_r = self.Q_inverse_r
|
||||
else:
|
||||
raise ValueError("""Inverse of Q can not be computed, because Q has not been computed.
|
||||
This requires some programming""")
|
||||
|
||||
return Q_inverse_r
|
||||
|
||||
|
||||
def return_last(self):
|
||||
"""
|
||||
Function returns last computed matrices.
|
||||
|
|
@ -2463,6 +2533,9 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
(self.reconstruct_indices.nbytes if (self.reconstruct_indices is not None) else 0)
|
||||
|
||||
self.Q_svd_dict = {}
|
||||
self.Q_square_root_dict = {}
|
||||
self.Q_inverse_dict = {}
|
||||
|
||||
self.last_k = None
|
||||
# !!!Print statistics! Which object is created
|
||||
# !!!Print statistics! Print sizes of matrices
|
||||
|
|
@ -2503,17 +2576,66 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
Square root of the noise matrix Q
|
||||
"""
|
||||
matrix_index = self.reconstruct_indices[k]
|
||||
if matrix_index in self.Q_svd_dict:
|
||||
square_root = self.Q_svd_dict[matrix_index]
|
||||
if matrix_index in self.Q_square_root_dict:
|
||||
square_root = self.Q_square_root_dict[matrix_index]
|
||||
else:
|
||||
(U, S, Vh) = sp.linalg.svd( self.Qs[:,:, matrix_index],
|
||||
if matrix_index in self.Q_svd_dict:
|
||||
(U, S, Vh) = self.Q_svd_dict[matrix_index]
|
||||
else:
|
||||
(U, S, Vh) = sp.linalg.svd( self.Qs[:,:, matrix_index],
|
||||
full_matrices=False, compute_uv=True,
|
||||
overwrite_a=False, check_finite=False)
|
||||
self.Q_svd_dict[matrix_index] = (U,S,Vh)
|
||||
|
||||
square_root = U * np.sqrt(S)
|
||||
self.Q_svd_dict[matrix_index] = square_root
|
||||
self.Q_square_root_dict[matrix_index] = square_root
|
||||
|
||||
return square_root
|
||||
|
||||
def Q_inverse(self, k, p_largest_cond_num, p_regularization_type):
|
||||
"""
|
||||
Function inverts Q matrix and regularizes the inverse.
|
||||
Regularization is useful when original matrix is badly conditioned.
|
||||
Function is currently used only in SparseGP code.
|
||||
|
||||
Inputs:
|
||||
------------------------------
|
||||
k: int
|
||||
Iteration number.
|
||||
|
||||
p_largest_cond_num: float
|
||||
Largest condition value for the inverted matrix. If cond. number is smaller than that
|
||||
no regularization happen.
|
||||
|
||||
regularization_type: 1 or 2
|
||||
Regularization type.
|
||||
|
||||
regularization_type: int (1 or 2)
|
||||
|
||||
type 1: 1/(S[k] + regularizer) regularizer is computed
|
||||
type 2: S[k]/(S^2[k] + regularizer) regularizer is computed
|
||||
"""
|
||||
#import pdb; pdb.set_trace()
|
||||
|
||||
matrix_index = self.reconstruct_indices[k]
|
||||
if matrix_index in self.Q_inverse_dict:
|
||||
Q_inverse_r = self.Q_inverse_dict[matrix_index]
|
||||
else:
|
||||
|
||||
if matrix_index in self.Q_svd_dict:
|
||||
(U, S, Vh) = self.Q_svd_dict[matrix_index]
|
||||
else:
|
||||
(U, S, Vh) = sp.linalg.svd( self.Qs[:,:, matrix_index],
|
||||
full_matrices=False, compute_uv=True,
|
||||
overwrite_a=False, check_finite=False)
|
||||
self.Q_svd_dict[matrix_index] = (U,S,Vh)
|
||||
|
||||
Q_inverse_r = psd_matrix_inverse(k, 0.5*(self.Qs[:,:, matrix_index] + self.Qs[:,:, matrix_index].T), U,S, p_largest_cond_num, p_regularization_type)
|
||||
self.Q_inverse_dict[matrix_index] = Q_inverse_r
|
||||
|
||||
return Q_inverse_r
|
||||
|
||||
|
||||
def return_last(self):
|
||||
"""
|
||||
Function returns last available matrices.
|
||||
|
|
@ -3073,7 +3195,8 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
@classmethod
|
||||
def _cont_to_discrete_object(cls, X, F, L, Qc, compute_derivatives=False,
|
||||
grad_params_no=None,
|
||||
P_inf=None, dP_inf=None, dF = None, dQc=None):
|
||||
P_inf=None, dP_inf=None, dF = None, dQc=None,
|
||||
dt0=None):
|
||||
"""
|
||||
Function return the object which is used in Kalman filter and/or
|
||||
smoother to obtain matrices A, Q and their derivatives for discrete model
|
||||
|
|
@ -3110,7 +3233,14 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
threshold_number_of_unique_time_steps = 20 # above which matrices are separately each time
|
||||
dt = np.empty((X.shape[0],))
|
||||
dt[1:] = np.diff(X[:,0],axis=0)
|
||||
dt[0] = 0#dt[1]
|
||||
if dt0 is None:
|
||||
dt[0] = 0#dt[1]
|
||||
else:
|
||||
if isinstance(dt0,str):
|
||||
dt = dt[1:]
|
||||
else:
|
||||
dt[0] = dt0
|
||||
|
||||
unique_indices = np.unique(np.round(dt, decimals=unique_round_decimals))
|
||||
number_unique_indices = len(unique_indices)
|
||||
|
||||
|
|
@ -3161,6 +3291,9 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
|
||||
x_{k} = A_{k} * x_{k-1} + q_{k-1}; q_{k-1} ~ N(0, Q_{k-1})
|
||||
|
||||
TODO: this function can be redone to "preprocess dataset", when
|
||||
close time points are handeled properly (with rounding parameter) and
|
||||
values are averaged accordingly.
|
||||
|
||||
Input:
|
||||
--------------
|
||||
|
|
@ -3222,11 +3355,9 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
n = F.shape[0]
|
||||
|
||||
if not isinstance(dt, collections.Iterable): # not iterable, scalar
|
||||
|
||||
#import pdb; pdb.set_trace()
|
||||
# The dynamical model
|
||||
A = matrix_exponent(F*dt)
|
||||
if np.any( np.isnan(A)):
|
||||
A = linalg.expm3(F*dt)
|
||||
|
||||
# The covariance matrix Q by matrix fraction decomposition ->
|
||||
Phi = np.zeros((2*n,2*n))
|
||||
|
|
@ -3265,15 +3396,17 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
# The discrete-time dynamical model*
|
||||
if p==0:
|
||||
A = AA[:n,:n,p]
|
||||
Q_noise_2 = P_inf - A.dot(P_inf).dot(A.T)
|
||||
Q_noise = Q_noise_2
|
||||
Q_noise_3 = P_inf - A.dot(P_inf).dot(A.T)
|
||||
Q_noise = Q_noise_3
|
||||
#PP = A.dot(P).dot(A.T) + Q_noise_2
|
||||
|
||||
# The derivatives of A and Q
|
||||
dA[:,:,p] = AA[n:,:n,p]
|
||||
dQ[:,:,p] = dP_inf[:,:,p] - dA[:,:,p].dot(P_inf).dot(A.T) \
|
||||
- A.dot(dP_inf[:,:,p]).dot(A.T) - A.dot(P_inf).dot(dA[:,:,p].T) # Rewrite not ro multiply two times
|
||||
tmp = dA[:,:,p].dot(P_inf).dot(A.T)
|
||||
dQ[:,:,p] = dP_inf[:,:,p] - tmp \
|
||||
- A.dot(dP_inf[:,:,p]).dot(A.T) - tmp.T
|
||||
|
||||
dQ[:,:,p] = 0.5*(dQ[:,:,p] + dQ[:,:,p].T) # Symmetrize
|
||||
else:
|
||||
dA = None
|
||||
dQ = None
|
||||
|
|
@ -3283,6 +3416,10 @@ class ContDescrStateSpace(DescreteStateSpace):
|
|||
#Q_noise = Q_noise_1
|
||||
|
||||
# Return
|
||||
#import pdb; pdb.set_trace()
|
||||
#if dt != 0:
|
||||
# Q_noise = Q_noise + np.eye(Q_noise.shape[0])*1e-8
|
||||
Q_noise = 0.5*(Q_noise + Q_noise.T) # Symmetrize
|
||||
return A, Q_noise,None, dA, dQ
|
||||
|
||||
else: # iterable, array
|
||||
|
|
@ -3486,4 +3623,124 @@ def balance_ss_model(F,L,Qc,H,Pinf,P0,dF=None,dQc=None,dPinf=None,dP0=None):
|
|||
|
||||
# (F,L,Qc,H,Pinf,P0,dF,dQc,dPinf,dP0)
|
||||
|
||||
return bF, bL, bQc, bH, bPinf, bP0, bdF, bdQc, bdPinf, bdP0, T
|
||||
return bF, bL, bQc, bH, bPinf, bP0, bdF, bdQc, bdPinf, bdP0
|
||||
|
||||
#def psd_matrix_inverse(k,Q, U=None,S=None, p_largest_cond_num=None, regularization_type=2):
|
||||
# """
|
||||
# Function inverts positive definite matrix and regularizes the inverse.
|
||||
# Regularization is useful when original matrix is badly conditioned.
|
||||
# Function is currently used only in SparseGP code.
|
||||
#
|
||||
# Inputs:
|
||||
# ------------------------------
|
||||
# k: int
|
||||
# Iteration umber. Used for information only. Value -1 corresponds to P_inf_inv.
|
||||
#
|
||||
# Q: matrix
|
||||
# To be inverted
|
||||
#
|
||||
# U,S: matrix. vector
|
||||
# SVD components of Q
|
||||
#
|
||||
# p_largest_cond_num: float
|
||||
# Largest condition value for the inverted matrix. If cond. number is smaller than that
|
||||
# no regularization happen.
|
||||
#
|
||||
# regularization_type: 1 or 2
|
||||
# Regularization type.
|
||||
# """
|
||||
# #import pdb; pdb.set_trace()
|
||||
## if (k == 0) or (k == -1): # -1 - P_inf_inv computation
|
||||
## import pdb; pdb.set_trace()
|
||||
#
|
||||
# if p_largest_cond_num is None:
|
||||
# raise ValueError("psd_matrix_inverse: None p_largest_cond_num")
|
||||
#
|
||||
# if U is None or S is None:
|
||||
# (U, S, Vh) = sp.linalg.svd( Q, full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
# if S[0] < (1e-4):
|
||||
# #import pdb; pdb.set_trace()
|
||||
# warnings.warn("""state_space_main psd_matrix_inverse: largest singular value is too small {0:e}.
|
||||
# condition number is {1:e} Maybe somethigng is wrong
|
||||
# """.format(S[0], S[0]/S[-1]))
|
||||
# S = S + (1e-4 - S[0]) # make the S[0] at least 1e-4
|
||||
#
|
||||
# current_conditional_number = S[0]/S[-1]
|
||||
# if (current_conditional_number > p_largest_cond_num):
|
||||
# if (regularization_type == 1):
|
||||
# regularizer = S[0] / p_largest_cond_num
|
||||
# # the second computation of SVD is done to compute more precisely singular
|
||||
# # vectors of small singular values, since small singular values become large.
|
||||
# # It is not very clear how this step is useful but test is here.
|
||||
# (U, S, Vh) = sp.linalg.svd( Q + regularizer*np.eye(Q.shape[0]),
|
||||
# full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
#
|
||||
# Q_inverse_r = np.dot( U * 1.0/S , U.T ) # Assume Q_inv is positive definite
|
||||
#
|
||||
# # In this case, RBF kernel we get complx eigenvalues. Probably
|
||||
# # for small eigenvalue corresponding eigenvectors are not very orthogonal.
|
||||
# ##########Q_inverse = np.dot( Vh.T * ( 1.0/(S + regularizer)) , U.T )
|
||||
# elif (regularization_type == 2):
|
||||
#
|
||||
# new_border_value = np.sqrt(current_conditional_number)/2
|
||||
# if p_largest_cond_num >= new_border_value: # this type of regularization works
|
||||
# regularizer = ( S[0] / p_largest_cond_num / 2.0 )**2
|
||||
#
|
||||
# Q_inverse_r = np.dot( U * ( S/(S**2 + regularizer)) , U.T ) # Assume Q_inv is positive definite
|
||||
# else:
|
||||
#
|
||||
# better_curr_cond_num = new_border_value
|
||||
# warnings.warn("""state_space_main psd_matrix_inverse: reg_type = 2 can't be done completely.
|
||||
# Current conditionakl number {0:e} is reduced to {1:e} by reg_type = 1""".format(current_conditional_number, better_curr_cond_num))
|
||||
#
|
||||
# regularizer = S[0] / better_curr_cond_num
|
||||
# # the second computation of SVD is done to compute more precisely singular
|
||||
# # vectors of small singular values, since small singular values become large.
|
||||
# # It is not very clear how this step is useful but test is here.
|
||||
# (U, S, Vh) = sp.linalg.svd( Q + regularizer*np.eye(Q.shape[0]),
|
||||
# full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
#
|
||||
# regularizer = ( S[0] / p_largest_cond_num / 2.0 )**2
|
||||
#
|
||||
# Q_inverse_r = np.dot( U * ( S/(S**2 + regularizer)) , U.T ) # Assume Q_inv is positive definite
|
||||
#
|
||||
# assert regularizer*10 < S[0], "regularizer is not << S[0]"
|
||||
# assert regularizer > 10*S[-1], "regularizer is not >> S[-1]"
|
||||
#
|
||||
## Old version ->
|
||||
## lamda_star = np.sqrt(current_conditional_number)
|
||||
## if 2*p_largest_cond_num >= lamda_star:
|
||||
## lamda = current_conditional_number / 2 / p_largest_cond_num
|
||||
##
|
||||
## regularizer = (S[-1] * lamda)**2
|
||||
##
|
||||
## Q_inverse_r = np.dot( U * ( S/(S**2 + regularizer)) , U.T ) # Assume Q_inv is positive definite
|
||||
## else:
|
||||
## better_curr_cond_num = (2*p_largest_cond_num)**2 / 2 # division by 2 just in case here
|
||||
## warnings.warn("""state_space_main psd_matrix_inverse: reg_type = 2 can't be done completely.
|
||||
## Current conditionakl number {0:e} is reduced to {1:e} by reg_type = 1""".format(current_conditional_number, better_curr_cond_num))
|
||||
##
|
||||
## regularizer = S[0] / better_curr_cond_num
|
||||
## # the second computation of SVD is done to compute more precisely singular
|
||||
## # vectors of small singular values, since small singular values become large.
|
||||
## # It is not very clear how this step is useful but test is here.
|
||||
## (U, S, Vh) = sp.linalg.svd( Q + regularizer*np.eye(Q.shape[0]),
|
||||
## full_matrices=False, compute_uv=True, overwrite_a=False, check_finite=False)
|
||||
##
|
||||
## lamda = better_curr_cond_num / 2 / p_largest_cond_num
|
||||
##
|
||||
## regularizer = (S[-1] * lamda)**2
|
||||
##
|
||||
## Q_inverse_r = np.dot( U * ( S/(S**2 + regularizer)) , U.T ) # Assume Q_inv is positive definite
|
||||
##
|
||||
## assert lamda > 10, "Some assumptions are incorrect if this is not satisfied."
|
||||
## Old version <-
|
||||
# else:
|
||||
# raise ValueError("AQcompute_batch_Python:Q_inverse: Invalid regularization type")
|
||||
#
|
||||
# else:
|
||||
# Q_inverse_r = np.dot( U * 1.0/S , U.T ) # Assume Q_inv is positive definite
|
||||
# # When checking conditional number 2 times difference is ok.
|
||||
# Q_inverse_r = 0.5*(Q_inverse_r + Q_inverse_r.T)
|
||||
#
|
||||
# return Q_inverse_r
|
||||
|
|
@ -23,7 +23,16 @@ from . import state_space_main as ssm
|
|||
from . import state_space_setup as ss_setup
|
||||
|
||||
class StateSpace(Model):
|
||||
def __init__(self, X, Y, kernel=None, noise_var=1.0, kalman_filter_type = 'regular', use_cython = False, name='StateSpace'):
|
||||
def __init__(self, X, Y, kernel=None, noise_var=1.0, kalman_filter_type = 'regular', use_cython = False, balance=False, name='StateSpace'):
|
||||
"""
|
||||
Inputs:
|
||||
------------------
|
||||
|
||||
balance: bool
|
||||
Whether to balance or not the model as a whole
|
||||
|
||||
"""
|
||||
|
||||
super(StateSpace, self).__init__(name=name)
|
||||
|
||||
if len(X.shape) == 1:
|
||||
|
|
@ -51,6 +60,7 @@ class StateSpace(Model):
|
|||
ss_setup.use_cython = use_cython
|
||||
|
||||
#import pdb; pdb.set_trace()
|
||||
self.balance = balance
|
||||
|
||||
global ssm
|
||||
#from . import state_space_main as ssm
|
||||
|
|
@ -58,8 +68,8 @@ class StateSpace(Model):
|
|||
reload(ssm)
|
||||
# Make sure the observations are ordered in time
|
||||
sort_index = np.argsort(X[:,0])
|
||||
self.X = X[sort_index]
|
||||
self.Y = Y[sort_index]
|
||||
self.X = X[sort_index,:]
|
||||
self.Y = Y[sort_index,:]
|
||||
|
||||
# Noise variance
|
||||
self.likelihood = likelihoods.Gaussian(variance=noise_var)
|
||||
|
|
@ -86,11 +96,12 @@ class StateSpace(Model):
|
|||
|
||||
#np.set_printoptions(16)
|
||||
#print(self.param_array)
|
||||
#import pdb; pdb.set_trace()
|
||||
|
||||
|
||||
# Get the model matrices from the kernel
|
||||
(F,L,Qc,H,P_inf, P0, dFt,dQct,dP_inft, dP0t) = self.kern.sde()
|
||||
|
||||
#Qc = Qc + np.eye(Qc.shape[0]) * 1e-8
|
||||
#import pdb; pdb.set_trace()
|
||||
# necessary parameters
|
||||
measurement_dim = self.output_dim
|
||||
grad_params_no = dFt.shape[2]+1 # we also add measurement noise as a parameter
|
||||
|
|
@ -112,8 +123,9 @@ class StateSpace(Model):
|
|||
dR[:,:,-1] = np.eye(measurement_dim)
|
||||
|
||||
# Balancing
|
||||
#(F,L,Qc,H,P_inf,P0, dF,dQc,dP_inf,dP0) = ssm.balance_ss_model(F,L,Qc,H,P_inf,P0, dF,dQc,dP_inf, dP0)
|
||||
|
||||
if self.balance:
|
||||
(F,L,Qc,H,P_inf,P0, dF,dQc,dP_inf,dP0) = ssm.balance_ss_model(F,L,Qc,H,P_inf,P0, dF,dQc,dP_inf, dP0)
|
||||
print("SSM parameters_changed balancing!")
|
||||
# Use the Kalman filter to evaluate the likelihood
|
||||
grad_calc_params = {}
|
||||
grad_calc_params['dP_inf'] = dP_inf
|
||||
|
|
@ -125,7 +137,7 @@ class StateSpace(Model):
|
|||
kalman_filter_type = self.kalman_filter_type
|
||||
|
||||
# The following code is required because sometimes the shapes of self.Y
|
||||
# becomes 3D even though is must be 2D. The reason is undescovered.
|
||||
# becomes 3D even though is must be 2D. The reason is undiscovered.
|
||||
Y = self.Y
|
||||
if self.ts_number is None:
|
||||
Y.shape = (self.num_data,1)
|
||||
|
|
@ -146,7 +158,7 @@ class StateSpace(Model):
|
|||
|
||||
if np.any( np.isfinite(grad_log_likelihood) == False):
|
||||
#import pdb; pdb.set_trace()
|
||||
print("State-Space: NaN valkues in the grad_log_likelihood")
|
||||
print("State-Space: NaN values in the grad_log_likelihood")
|
||||
#print(grad_log_likelihood)
|
||||
|
||||
grad_log_likelihood_sum = np.sum(grad_log_likelihood,axis=1)
|
||||
|
|
@ -159,7 +171,7 @@ class StateSpace(Model):
|
|||
def log_likelihood(self):
|
||||
return self._log_marginal_likelihood
|
||||
|
||||
def _raw_predict(self, Xnew=None, Ynew=None, filteronly=False, **kw):
|
||||
def _raw_predict(self, Xnew=None, Ynew=None, filteronly=False, p_balance=False, **kw):
|
||||
"""
|
||||
Performs the actual prediction for new X points.
|
||||
Inner function. It is called only from inside this class.
|
||||
|
|
@ -178,6 +190,9 @@ class StateSpace(Model):
|
|||
Use only Kalman Filter for prediction. In this case the output does
|
||||
not coincide with corresponding Gaussian process.
|
||||
|
||||
balance: bool
|
||||
Whether to balance or not the model as a whole
|
||||
|
||||
Output:
|
||||
--------------------
|
||||
|
||||
|
|
@ -211,6 +226,11 @@ class StateSpace(Model):
|
|||
(F,L,Qc,H,P_inf, P0, dF,dQc,dP_inf,dP0) = self.kern.sde()
|
||||
state_dim = F.shape[0]
|
||||
|
||||
# Balancing
|
||||
if (p_balance==True):
|
||||
(F,L,Qc,H,P_inf,P0, dF,dQc,dP_inf,dP0) = ssm.balance_ss_model(F,L,Qc,H,P_inf,P0, dF,dQc,dP_inf, dP0)
|
||||
print("SSM _raw_predict balancing!")
|
||||
|
||||
#Y = self.Y[:, 0,0]
|
||||
# Run the Kalman filter
|
||||
#import pdb; pdb.set_trace()
|
||||
|
|
@ -261,10 +281,23 @@ class StateSpace(Model):
|
|||
# Return the posterior of the state
|
||||
return (m, V)
|
||||
|
||||
def predict(self, Xnew=None, filteronly=False, include_likelihood=True, **kw):
|
||||
def predict(self, Xnew=None, filteronly=False, include_likelihood=True, balance=None, **kw):
|
||||
"""
|
||||
Inputs:
|
||||
------------------
|
||||
|
||||
balance: bool
|
||||
Whether to balance or not the model as a whole
|
||||
|
||||
"""
|
||||
|
||||
if balance is None:
|
||||
p_balance = self.balance
|
||||
else:
|
||||
p_balance = balance
|
||||
|
||||
# Run the Kalman filter to get the state
|
||||
(m, V) = self._raw_predict(Xnew,filteronly=filteronly)
|
||||
(m, V) = self._raw_predict(Xnew,filteronly=filteronly, p_balance=p_balance)
|
||||
|
||||
# Add the noise variance to the state variance
|
||||
if include_likelihood:
|
||||
|
|
@ -277,8 +310,22 @@ class StateSpace(Model):
|
|||
# Return mean and variance
|
||||
return m, V
|
||||
|
||||
def predict_quantiles(self, Xnew=None, quantiles=(2.5, 97.5), **kw):
|
||||
mu, var = self._raw_predict(Xnew)
|
||||
def predict_quantiles(self, Xnew=None, quantiles=(2.5, 97.5), balance=None, **kw):
|
||||
"""
|
||||
Inputs:
|
||||
------------------
|
||||
|
||||
balance: bool
|
||||
Whether to balance or not the model as a whole
|
||||
|
||||
"""
|
||||
if balance is None:
|
||||
p_balance = self.balance
|
||||
else:
|
||||
p_balance = balance
|
||||
|
||||
|
||||
mu, var = self._raw_predict(Xnew, p_balance=p_balance)
|
||||
#import pdb; pdb.set_trace()
|
||||
return [stats.norm.ppf(q/100.)*np.sqrt(var + float(self.Gaussian_noise.variance)) + mu for q in quantiles]
|
||||
|
||||
|
|
|
|||
|
|
@ -91,12 +91,14 @@ class StateSpaceKernelsTests(np.testing.TestCase):
|
|||
mean_compare_decimal=5, var_compare_decimal=5)
|
||||
|
||||
def test_RBF_kernel(self,):
|
||||
#import pdb;pdb.set_trace()
|
||||
|
||||
np.random.seed(234) # seed the random number generator
|
||||
(X,Y) = generate_sine_data(x_points=None, sin_period=5.0, sin_ampl=10.0, noise_var=2.0,
|
||||
plot = False, points_num=50, x_interval = (0, 20), random=True)
|
||||
X.shape = (X.shape[0],1); Y.shape = (Y.shape[0],1)
|
||||
|
||||
ss_kernel = GPy.kern.sde_RBF(1, 110., 1.5, active_dims=[0,])
|
||||
ss_kernel = GPy.kern.sde_RBF(1, 110., 1.5, active_dims=[0,], balance=True, approx_order=10)
|
||||
gp_kernel = GPy.kern.RBF(1, 110., 1.5, active_dims=[0,])
|
||||
|
||||
self.run_for_model(X, Y, ss_kernel, check_gradients=True,
|
||||
|
|
@ -375,7 +377,7 @@ if __name__ == "__main__":
|
|||
print("Running state-space inference tests...")
|
||||
unittest.main()
|
||||
|
||||
#tt = StateSpaceKernelsTests('test_periodic_kernel')
|
||||
#tt = StateSpaceKernelsTests('test_RBF_kernel')
|
||||
#import pdb; pdb.set_trace()
|
||||
#tt.test_Matern32_kernel()
|
||||
#tt.test_Matern52_kernel()
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue