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replace scipy math functions by numpy
This commit is contained in:
Martin Bubel
parent
4e5bdae893
commit
4b552cb5f5
3 changed files with 288 additions and 201 deletions
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@ -13,213 +13,267 @@ import warnings
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from scipy import special as special
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from scipy import special as special
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class sde_StdPeriodic(StdPeriodic):
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class sde_StdPeriodic(StdPeriodic):
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"""
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"""
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Class provide extra functionality to transfer this covariance function into
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Class provide extra functionality to transfer this covariance function into
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SDE form.
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SDE form.
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Standard Periodic kernel:
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Standard Periodic kernel:
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.. math::
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.. math::
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k(x,y) = \theta_1 \exp \left[ - \frac{1}{2} {}\sum_{i=1}^{input\_dim}
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k(x,y) = \theta_1 \exp \left[ - \frac{1}{2} {}\sum_{i=1}^{input\_dim}
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\left( \frac{\sin(\frac{\pi}{\lambda_i} (x_i - y_i) )}{l_i} \right)^2 \right] }
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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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"""
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# TODO: write comment to the constructor arguments
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# TODO: write comment to the constructor arguments
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def __init__(self, *args, **kwargs):
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def __init__(self, *args, **kwargs):
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"""
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"""
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Init constructior.
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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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Two optinal extra parameters are added in addition to the ones in
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StdPeriodic kernel.
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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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:param approx_order: approximation order for the RBF covariance. (Default 7)
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:type approx_order: int
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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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: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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:type balance: bool
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"""
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"""
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#import pdb; pdb.set_trace()
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# import pdb; pdb.set_trace()
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if 'approx_order' in kwargs:
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if "approx_order" in kwargs:
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self.approx_order = kwargs.get('approx_order')
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self.approx_order = kwargs.get("approx_order")
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del kwargs['approx_order']
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del kwargs["approx_order"]
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else:
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else:
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self.approx_order = 7
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self.approx_order = 7
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if "balance" in kwargs:
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if 'balance' in kwargs:
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self.balance = bool(kwargs.get("balance"))
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self.balance = bool( kwargs.get('balance') )
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del kwargs["balance"]
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del kwargs['balance']
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else:
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else:
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self.balance = False
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self.balance = False
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super(sde_StdPeriodic, self).__init__(*args, **kwargs)
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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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def sde_update_gradient_full(self, gradients):
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"""
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"""
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Update gradient in the order in which parameters are represented in the
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Update gradient in the order in which parameters are represented in the
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kernel
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kernel
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"""
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"""
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self.variance.gradient = gradients[0]
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self.variance.gradient = gradients[0]
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self.period.gradient = gradients[1]
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self.period.gradient = gradients[1]
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self.lengthscale.gradient = gradients[2]
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self.lengthscale.gradient = gradients[2]
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def sde(self):
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def sde(self):
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"""
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"""
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Return the state space representation of the standard periodic 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 0.2. (independently of approximation order)
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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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After this Bessel functions of the first becomes NaN. Rescaling
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time variable might help.
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time variable might help.
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! Note: one must keep period 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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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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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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300 data points the low limit is 0.15.
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"""
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"""
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#import pdb; pdb.set_trace()
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# import pdb; pdb.set_trace()
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# Params to use: (in that order)
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# Params to use: (in that order)
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#self.variance
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# self.variance
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#self.period
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# self.period
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#self.lengthscale
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# self.lengthscale
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if self.approx_order is not None:
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if self.approx_order is not None:
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N = int(self.approx_order)
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N = int(self.approx_order)
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else:
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else:
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N = 7 # approximation order
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N = 7 # approximation order
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p_period = float(self.period)
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p_period = float(self.period)
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p_lengthscale = 2*float(self.lengthscale)
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p_lengthscale = 2 * float(self.lengthscale)
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p_variance = float(self.variance)
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p_variance = float(self.variance)
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w0 = 2*np.pi/p_period # frequency
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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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# lengthscale is multiplied by 2 because of different definition of lengthscale
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[q2,dq2l] = seriescoeff(N, p_lengthscale, p_variance)
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[q2, dq2l] = seriescoeff(N, p_lengthscale, p_variance)
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dq2l = 2*dq2l # This is because the lengthscale if multiplied by 2.
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dq2l = 2 * dq2l # This is because the lengthscale if multiplied by 2.
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eps = 1e-12
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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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if (
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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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np.any(np.isfinite(q2) == False)
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or np.any(np.abs(q2) > 1.0 / eps)
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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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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 dq2l :".format(eps) + q2.__format__("") )
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):
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warnings.warn(
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"sde_Periodic: Infinite, too small, or too large (eps={0:e}) values in q2 :".format(
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F = np.kron(np.diag(range(0,N+1)),np.array( ((0, -w0), (w0, 0)) ) )
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eps
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L = np.eye(2*(N+1))
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)
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Qc = np.zeros((2*(N+1), 2*(N+1)))
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+ q2.__format__("")
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P_inf = np.kron(np.diag(q2),np.eye(2))
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)
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H = np.kron(np.ones((1,N+1)),np.array((1,0)) )
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if (
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np.any(np.isfinite(dq2l) == False)
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or np.any(np.abs(dq2l) > 1.0 / eps)
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or np.any(np.abs(dq2l) < eps)
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):
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warnings.warn(
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"sde_Periodic: Infinite, too small, or too large (eps={0:e}) values in dq2l :".format(
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eps
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)
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+ q2.__format__("")
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)
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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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Qc = np.zeros((2 * (N + 1), 2 * (N + 1)))
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P_inf = np.kron(np.diag(q2), np.eye(2))
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H = np.kron(np.ones((1, N + 1)), np.array((1, 0)))
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P0 = P_inf.copy()
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P0 = P_inf.copy()
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# Derivatives
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# Derivatives
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dF = np.empty((F.shape[0], F.shape[1], 3))
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dF = np.empty((F.shape[0], F.shape[1], 3))
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dQc = np.empty((Qc.shape[0], Qc.shape[1], 3))
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dQc = np.empty((Qc.shape[0], Qc.shape[1], 3))
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dP_inf = np.empty((P_inf.shape[0], P_inf.shape[1], 3))
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dP_inf = np.empty((P_inf.shape[0], P_inf.shape[1], 3))
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# Derivatives wrt self.variance
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# Derivatives wrt self.variance
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dF[:,:,0] = np.zeros(F.shape)
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dF[:, :, 0] = np.zeros(F.shape)
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dQc[:,:,0] = np.zeros(Qc.shape)
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dQc[:, :, 0] = np.zeros(Qc.shape)
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dP_inf[:,:,0] = P_inf / p_variance
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dP_inf[:, :, 0] = P_inf / p_variance
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# Derivatives self.period
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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)) ) / p_period );
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dF[:, :, 1] = np.kron(
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dQc[:,:,1] = np.zeros(Qc.shape)
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np.diag(range(0, N + 1)), np.array(((0, w0), (-w0, 0))) / p_period
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dP_inf[:,:,1] = np.zeros(P_inf.shape)
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)
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dQc[:, :, 1] = np.zeros(Qc.shape)
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# Derivatives self.lengthscales
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dP_inf[:, :, 1] = np.zeros(P_inf.shape)
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dF[:,:,2] = np.zeros(F.shape)
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dQc[:,:,2] = np.zeros(Qc.shape)
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# Derivatives self.lengthscales
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dP_inf[:,:,2] = np.kron(np.diag(dq2l),np.eye(2))
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dF[:, :, 2] = np.zeros(F.shape)
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dQc[:, :, 2] = np.zeros(Qc.shape)
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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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dP0 = dP_inf.copy()
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if self.balance:
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if self.balance:
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# Benefits of this are not very sound.
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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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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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(F, L, Qc, H, P_inf, P0, dF, dQc, dP_inf, dP0) = ssm.balance_ss_model(
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F, L, Qc, H, P_inf, P0, dF, dQc, dP_inf, dP0
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)
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return (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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def seriescoeff(m=6, lengthScale=1.0, magnSigma2=1.0, true_covariance=False):
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def seriescoeff(m=6,lengthScale=1.0,magnSigma2=1.0, true_covariance=False):
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"""
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"""
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Calculate the coefficients q_j^2 for the covariance function
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Calculate the coefficients q_j^2 for the covariance function
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approximation:
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approximation:
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k(\tau) = \sum_{j=0}^{+\infty} q_j^2 \cos(j\omega_0 \tau)
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k(\tau) = \sum_{j=0}^{+\infty} q_j^2 \cos(j\omega_0 \tau)
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Reference is:
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Reference is:
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[1] Arno Solin and Simo Särkkä (2014). Explicit link between periodic
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[1] Arno Solin and Simo Särkkä (2014). Explicit link between periodic
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covariance functions and state space models. In Proceedings of the
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covariance functions and state space models. In Proceedings of the
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Seventeenth International Conference on Artifcial Intelligence and
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Seventeenth International Conference on Artifcial Intelligence and
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Statistics (AISTATS 2014). JMLR: W&CP, volume 33.
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Statistics (AISTATS 2014). JMLR: W&CP, volume 33.
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Note! Only the infinite approximation (through Bessel function)
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Note! Only the infinite approximation (through Bessel function)
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is currently implemented.
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is currently implemented.
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Input:
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Input:
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----------------
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----------------
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m: int
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m: int
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Degree of approximation. Default 6.
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Degree of approximation. Default 6.
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lengthScale: float
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lengthScale: float
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Length scale parameter in the kerenl
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Length scale parameter in the kerenl
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magnSigma2:float
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magnSigma2:float
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Multiplier in front of the kernel.
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Multiplier in front of the kernel.
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Output:
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Output:
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-----------------
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-----------------
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coeffs: array(m+1)
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coeffs: array(m+1)
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Covariance series coefficients
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Covariance series coefficients
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coeffs_dl: array(m+1)
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coeffs_dl: array(m+1)
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Derivatives of the coefficients with respect to lengthscale.
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Derivatives of the coefficients with respect to lengthscale.
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"""
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"""
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if true_covariance:
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if true_covariance:
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bb = lambda j,m: (1.0 + np.array((j != 0), dtype=np.float64) ) / (2**(j)) *\
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bb = (
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sp.special.binom(j, sp.floor( (j-m)/2.0 * np.array(m<=j, dtype=np.float64) ))*\
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lambda j, m: (1.0 + np.array((j != 0), dtype=np.float64))
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np.array(m<=j, dtype=np.float64) *np.array(sp.mod(j-m,2)==0, dtype=np.float64)
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/ (2 ** (j))
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* sp.special.binom(
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M,J = np.meshgrid(range(0,m+1),range(0,m+1))
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j, sp.floor((j - m) / 2.0 * np.array(m <= j, dtype=np.float64))
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)
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coeffs = bb(J,M) / sp.misc.factorial(J) * sp.exp( -lengthScale**(-2) ) *\
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* np.array(m <= j, dtype=np.float64)
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(lengthScale**(-2))**J *magnSigma2
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* np.array(sp.mod(j - m, 2) == 0, dtype=np.float64)
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)
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coeffs_dl = np.sum( coeffs*lengthScale**(-3)*(2.0-2.0*J*lengthScale**2),0)
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M, J = np.meshgrid(range(0, m + 1), range(0, m + 1))
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coeffs = np.sum(coeffs,0)
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coeffs = (
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bb(J, M)
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/ sp.misc.factorial(J)
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* np.exp(-(lengthScale ** (-2)))
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* (lengthScale ** (-2)) ** J
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* magnSigma2
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)
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coeffs_dl = np.sum(
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coeffs * lengthScale ** (-3) * (2.0 - 2.0 * J * lengthScale**2), 0
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)
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coeffs = np.sum(coeffs, 0)
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else:
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else:
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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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coeffs = (
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if np.any( np.isfinite(coeffs) == False):
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2
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* magnSigma2
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* np.exp(-(lengthScale ** (-2)))
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* special.iv(range(0, m + 1), 1.0 / lengthScale ** (2))
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)
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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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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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coeffs[0] = 0.5 * coeffs[0]
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#print(coeffs)
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# print(coeffs)
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# Derivatives wrt (lengthScale)
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# Derivatives wrt (lengthScale)
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coeffs_dl = np.zeros(m+1)
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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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coeffs_dl[1:] = (
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(-4*special.iv(range(0,m),lengthScale**(-2)) + 4*(1+np.arange(1,m+1)*lengthScale**(2))*special.iv(range(1,m+1),lengthScale**(-2)) )
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magnSigma2
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* lengthScale ** (-3)
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* np.exp(-(lengthScale ** (-2)))
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* (
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-4 * special.iv(range(0, m), lengthScale ** (-2))
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+ 4
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* (1 + np.arange(1, m + 1) * lengthScale ** (2))
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* special.iv(range(1, m + 1), lengthScale ** (-2))
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)
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)
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# The first element
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# The first element
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coeffs_dl[0] = magnSigma2*lengthScale**(-3) * np.exp(-lengthScale**(-2))*\
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coeffs_dl[0] = (
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(2*special.iv(0,lengthScale**(-2)) - 2*special.iv(1,lengthScale**(-2)) )
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magnSigma2
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* lengthScale ** (-3)
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||||||
|
* np.exp(-(lengthScale ** (-2)))
|
||||||
|
* (
|
||||||
|
2 * special.iv(0, lengthScale ** (-2))
|
||||||
|
- 2 * special.iv(1, lengthScale ** (-2))
|
||||||
|
)
|
||||||
|
)
|
||||||
|
|
||||||
return coeffs.squeeze(), coeffs_dl.squeeze()
|
return coeffs.squeeze(), coeffs_dl.squeeze()
|
||||||
|
|
|
||||||
|
|
@ -11,12 +11,14 @@ from .stationary import RatQuad
|
||||||
|
|
||||||
import numpy as np
|
import numpy as np
|
||||||
import scipy as sp
|
import scipy as sp
|
||||||
|
|
||||||
try:
|
try:
|
||||||
from scipy.linalg import solve_continuous_lyapunov as lyap
|
from scipy.linalg import solve_continuous_lyapunov as lyap
|
||||||
except ImportError:
|
except ImportError:
|
||||||
from scipy.linalg import solve_lyapunov as lyap
|
from scipy.linalg import solve_lyapunov as lyap
|
||||||
import warnings
|
import warnings
|
||||||
|
|
||||||
|
|
||||||
class sde_RBF(RBF):
|
class sde_RBF(RBF):
|
||||||
"""
|
"""
|
||||||
|
|
||||||
|
|
@ -30,37 +32,35 @@ class sde_RBF(RBF):
|
||||||
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} }
|
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} }
|
||||||
|
|
||||||
"""
|
"""
|
||||||
|
|
||||||
def __init__(self, *args, **kwargs):
|
def __init__(self, *args, **kwargs):
|
||||||
"""
|
"""
|
||||||
Init constructior.
|
Init constructior.
|
||||||
|
|
||||||
Two optinal extra parameters are added in addition to the ones in
|
Two optinal extra parameters are added in addition to the ones in
|
||||||
RBF kernel.
|
RBF kernel.
|
||||||
|
|
||||||
:param approx_order: approximation order for the RBF covariance. (Default 10)
|
:param approx_order: approximation order for the RBF covariance. (Default 10)
|
||||||
:type approx_order: int
|
:type approx_order: int
|
||||||
|
|
||||||
:param balance: Whether to balance this kernel separately. (Defaulf True). Model has a separate parameter for balancing.
|
:param balance: Whether to balance this kernel separately. (Defaulf True). Model has a separate parameter for balancing.
|
||||||
:type balance: bool
|
:type balance: bool
|
||||||
"""
|
"""
|
||||||
|
|
||||||
if 'balance' in kwargs:
|
if "balance" in kwargs:
|
||||||
self.balance = bool( kwargs.get('balance') )
|
self.balance = bool(kwargs.get("balance"))
|
||||||
del kwargs['balance']
|
del kwargs["balance"]
|
||||||
else:
|
else:
|
||||||
self.balance = True
|
self.balance = True
|
||||||
|
|
||||||
|
if "approx_order" in kwargs:
|
||||||
if 'approx_order' in kwargs:
|
self.approx_order = kwargs.get("approx_order")
|
||||||
self.approx_order = kwargs.get('approx_order')
|
del kwargs["approx_order"]
|
||||||
del kwargs['approx_order']
|
|
||||||
else:
|
else:
|
||||||
self.approx_order = 6
|
self.approx_order = 6
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
super(sde_RBF, self).__init__(*args, **kwargs)
|
super(sde_RBF, self).__init__(*args, **kwargs)
|
||||||
|
|
||||||
def sde_update_gradient_full(self, gradients):
|
def sde_update_gradient_full(self, gradients):
|
||||||
"""
|
"""
|
||||||
Update gradient in the order in which parameters are represented in the
|
Update gradient in the order in which parameters are represented in the
|
||||||
|
|
@ -73,86 +73,111 @@ class sde_RBF(RBF):
|
||||||
def sde(self):
|
def sde(self):
|
||||||
"""
|
"""
|
||||||
Return the state space representation of the covariance.
|
Return the state space representation of the covariance.
|
||||||
|
|
||||||
Note! For Sparse GP inference too small or two high values of lengthscale
|
Note! For Sparse GP inference too small or two high values of lengthscale
|
||||||
lead to instabilities. This is because Qc are too high or too low
|
lead to instabilities. This is because Qc are too high or too low
|
||||||
and P_inf are not full rank. This effect depends on approximatio order.
|
and P_inf are not full rank. This effect depends on approximatio order.
|
||||||
For N = 10. lengthscale must be in (0.8,8). For other N tests must be conducted.
|
For N = 10. lengthscale must be in (0.8,8). For other N tests must be conducted.
|
||||||
N=6: (0.06,31)
|
N=6: (0.06,31)
|
||||||
Variance should be within reasonable bounds as well, but its dependence is linear.
|
Variance should be within reasonable bounds as well, but its dependence is linear.
|
||||||
|
|
||||||
The above facts do not take into accout regularization.
|
The above facts do not take into accout regularization.
|
||||||
"""
|
"""
|
||||||
#import pdb; pdb.set_trace()
|
# import pdb; pdb.set_trace()
|
||||||
if self.approx_order is not None:
|
if self.approx_order is not None:
|
||||||
N = self.approx_order
|
N = self.approx_order
|
||||||
else:
|
else:
|
||||||
N = 10# approximation order ( number of terms in exponent series expansion)
|
N = 10 # approximation order ( number of terms in exponent series expansion)
|
||||||
|
|
||||||
roots_rounding_decimals = 6
|
roots_rounding_decimals = 6
|
||||||
|
|
||||||
fn = np.math.factorial(N)
|
fn = np.math.factorial(N)
|
||||||
|
|
||||||
p_lengthscale = float( self.lengthscale )
|
p_lengthscale = float(self.lengthscale)
|
||||||
p_variance = float(self.variance)
|
p_variance = float(self.variance)
|
||||||
kappa = 1.0/2.0/p_lengthscale**2
|
kappa = 1.0 / 2.0 / p_lengthscale**2
|
||||||
|
|
||||||
|
Qc = np.array(((p_variance * np.sqrt(np.pi / kappa) * fn * (4 * kappa) ** N,),))
|
||||||
|
|
||||||
Qc = np.array( ((p_variance*np.sqrt(np.pi/kappa)*fn*(4*kappa)**N,),) )
|
|
||||||
|
|
||||||
eps = 1e-12
|
eps = 1e-12
|
||||||
if (float(Qc) > 1.0/eps) or (float(Qc) < eps):
|
if (float(Qc) > 1.0 / eps) or (float(Qc) < eps):
|
||||||
warnings.warn("""sde_RBF kernel: the noise variance Qc is either very large or very small.
|
warnings.warn(
|
||||||
It influece conditioning of P_inf: {0:e}""".format(float(Qc)) )
|
"""sde_RBF kernel: the noise variance Qc is either very large or very small.
|
||||||
|
It influece conditioning of P_inf: {0:e}""".format(
|
||||||
|
float(Qc)
|
||||||
|
)
|
||||||
|
)
|
||||||
|
|
||||||
pp1 = np.zeros((2*N+1,)) # array of polynomial coefficients from higher power to lower
|
pp1 = np.zeros(
|
||||||
|
(2 * N + 1,)
|
||||||
|
) # array of polynomial coefficients from higher power to lower
|
||||||
|
|
||||||
for n in range(0, N+1): # (2N+1) - number of polynomial coefficients
|
for n in range(0, N + 1): # (2N+1) - number of polynomial coefficients
|
||||||
pp1[2*(N-n)] = fn*(4.0*kappa)**(N-n)/np.math.factorial(n)*(-1)**n
|
pp1[2 * (N - n)] = (
|
||||||
|
fn * (4.0 * kappa) ** (N - n) / np.math.factorial(n) * (-1) ** n
|
||||||
pp = sp.poly1d(pp1)
|
)
|
||||||
roots = sp.roots(pp)
|
|
||||||
|
|
||||||
neg_real_part_roots = roots[np.round(np.real(roots) ,roots_rounding_decimals) < 0]
|
pp = np.poly1d(pp1)
|
||||||
aa = sp.poly1d(neg_real_part_roots, r=True).coeffs
|
roots = np.roots(pp)
|
||||||
|
|
||||||
F = np.diag(np.ones((N-1,)),1)
|
neg_real_part_roots = roots[
|
||||||
F[-1,:] = -aa[-1:0:-1]
|
np.round(np.real(roots), roots_rounding_decimals) < 0
|
||||||
|
]
|
||||||
|
aa = np.poly1d(neg_real_part_roots, r=True).coeffs
|
||||||
|
|
||||||
L= np.zeros((N,1))
|
F = np.diag(np.ones((N - 1,)), 1)
|
||||||
L[N-1,0] = 1
|
F[-1, :] = -aa[-1:0:-1]
|
||||||
|
|
||||||
H = np.zeros((1,N))
|
L = np.zeros((N, 1))
|
||||||
H[0,0] = 1
|
L[N - 1, 0] = 1
|
||||||
|
|
||||||
|
H = np.zeros((1, N))
|
||||||
|
H[0, 0] = 1
|
||||||
|
|
||||||
# Infinite covariance:
|
# Infinite covariance:
|
||||||
Pinf = lyap(F, -np.dot(L,np.dot( Qc[0,0],L.T)))
|
Pinf = lyap(F, -np.dot(L, np.dot(Qc[0, 0], L.T)))
|
||||||
Pinf = 0.5*(Pinf + Pinf.T)
|
Pinf = 0.5 * (Pinf + Pinf.T)
|
||||||
# Allocating space for derivatives
|
# Allocating space for derivatives
|
||||||
dF = np.empty([F.shape[0],F.shape[1],2])
|
dF = np.empty([F.shape[0], F.shape[1], 2])
|
||||||
dQc = np.empty([Qc.shape[0],Qc.shape[1],2])
|
dQc = np.empty([Qc.shape[0], Qc.shape[1], 2])
|
||||||
dPinf = np.empty([Pinf.shape[0],Pinf.shape[1],2])
|
dPinf = np.empty([Pinf.shape[0], Pinf.shape[1], 2])
|
||||||
|
|
||||||
# Derivatives:
|
# Derivatives:
|
||||||
dFvariance = np.zeros(F.shape)
|
dFvariance = np.zeros(F.shape)
|
||||||
dFlengthscale = np.zeros(F.shape)
|
dFlengthscale = np.zeros(F.shape)
|
||||||
dFlengthscale[-1,:] = -aa[-1:0:-1]/p_lengthscale * np.arange(-N,0,1)
|
dFlengthscale[-1, :] = -aa[-1:0:-1] / p_lengthscale * np.arange(-N, 0, 1)
|
||||||
|
|
||||||
dQcvariance = Qc/p_variance
|
dQcvariance = Qc / p_variance
|
||||||
dQclengthscale = np.array(( (p_variance*np.sqrt(2*np.pi)*fn*2**N*p_lengthscale**(-2*N)*(1-2*N),),))
|
dQclengthscale = np.array(
|
||||||
|
(
|
||||||
dPinf_variance = Pinf/p_variance
|
(
|
||||||
|
p_variance
|
||||||
|
* np.sqrt(2 * np.pi)
|
||||||
|
* fn
|
||||||
|
* 2**N
|
||||||
|
* p_lengthscale ** (-2 * N)
|
||||||
|
* (1 - 2 * N),
|
||||||
|
),
|
||||||
|
)
|
||||||
|
)
|
||||||
|
|
||||||
|
dPinf_variance = Pinf / p_variance
|
||||||
|
|
||||||
lp = Pinf.shape[0]
|
lp = Pinf.shape[0]
|
||||||
coeff = np.arange(1,lp+1).reshape(lp,1) + np.arange(1,lp+1).reshape(1,lp) - 2
|
coeff = (
|
||||||
coeff[np.mod(coeff,2) != 0] = 0
|
np.arange(1, lp + 1).reshape(lp, 1)
|
||||||
dPinf_lengthscale = -1/p_lengthscale*Pinf*coeff
|
+ np.arange(1, lp + 1).reshape(1, lp)
|
||||||
|
- 2
|
||||||
|
)
|
||||||
|
coeff[np.mod(coeff, 2) != 0] = 0
|
||||||
|
dPinf_lengthscale = -1 / p_lengthscale * Pinf * coeff
|
||||||
|
|
||||||
dF[:,:,0] = dFvariance
|
dF[:, :, 0] = dFvariance
|
||||||
dF[:,:,1] = dFlengthscale
|
dF[:, :, 1] = dFlengthscale
|
||||||
dQc[:,:,0] = dQcvariance
|
dQc[:, :, 0] = dQcvariance
|
||||||
dQc[:,:,1] = dQclengthscale
|
dQc[:, :, 1] = dQclengthscale
|
||||||
dPinf[:,:,0] = dPinf_variance
|
dPinf[:, :, 0] = dPinf_variance
|
||||||
dPinf[:,:,1] = dPinf_lengthscale
|
dPinf[:, :, 1] = dPinf_lengthscale
|
||||||
|
|
||||||
P0 = Pinf.copy()
|
P0 = Pinf.copy()
|
||||||
dP0 = dPinf.copy()
|
dP0 = dPinf.copy()
|
||||||
|
|
@ -161,10 +186,14 @@ class sde_RBF(RBF):
|
||||||
# Benefits of this are not very sound. Helps only in one case:
|
# Benefits of this are not very sound. Helps only in one case:
|
||||||
# SVD Kalman + RBF kernel
|
# SVD Kalman + RBF kernel
|
||||||
import GPy.models.state_space_main as ssm
|
import GPy.models.state_space_main as ssm
|
||||||
(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf,dP0) = ssm.balance_ss_model(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0 )
|
|
||||||
|
(F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0) = ssm.balance_ss_model(
|
||||||
|
F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0
|
||||||
|
)
|
||||||
|
|
||||||
return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
|
return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
|
||||||
|
|
||||||
|
|
||||||
class sde_Exponential(Exponential):
|
class sde_Exponential(Exponential):
|
||||||
"""
|
"""
|
||||||
|
|
||||||
|
|
@ -195,30 +224,31 @@ class sde_Exponential(Exponential):
|
||||||
variance = float(self.variance.values)
|
variance = float(self.variance.values)
|
||||||
lengthscale = float(self.lengthscale)
|
lengthscale = float(self.lengthscale)
|
||||||
|
|
||||||
F = np.array(((-1.0/lengthscale,),))
|
F = np.array(((-1.0 / lengthscale,),))
|
||||||
L = np.array(((1.0,),))
|
L = np.array(((1.0,),))
|
||||||
Qc = np.array( ((2.0*variance/lengthscale,),) )
|
Qc = np.array(((2.0 * variance / lengthscale,),))
|
||||||
H = np.array(((1.0,),))
|
H = np.array(((1.0,),))
|
||||||
Pinf = np.array(((variance,),))
|
Pinf = np.array(((variance,),))
|
||||||
P0 = Pinf.copy()
|
P0 = Pinf.copy()
|
||||||
|
|
||||||
dF = np.zeros((1,1,2));
|
dF = np.zeros((1, 1, 2))
|
||||||
dQc = np.zeros((1,1,2));
|
dQc = np.zeros((1, 1, 2))
|
||||||
dPinf = np.zeros((1,1,2));
|
dPinf = np.zeros((1, 1, 2))
|
||||||
|
|
||||||
dF[:,:,0] = 0.0
|
dF[:, :, 0] = 0.0
|
||||||
dF[:,:,1] = 1.0/lengthscale**2
|
dF[:, :, 1] = 1.0 / lengthscale**2
|
||||||
|
|
||||||
dQc[:,:,0] = 2.0/lengthscale
|
dQc[:, :, 0] = 2.0 / lengthscale
|
||||||
dQc[:,:,1] = -2.0*variance/lengthscale**2
|
dQc[:, :, 1] = -2.0 * variance / lengthscale**2
|
||||||
|
|
||||||
dPinf[:,:,0] = 1.0
|
dPinf[:, :, 0] = 1.0
|
||||||
dPinf[:,:,1] = 0.0
|
dPinf[:, :, 1] = 0.0
|
||||||
|
|
||||||
dP0 = dPinf.copy()
|
dP0 = dPinf.copy()
|
||||||
|
|
||||||
return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
|
return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
|
||||||
|
|
||||||
|
|
||||||
class sde_RatQuad(RatQuad):
|
class sde_RatQuad(RatQuad):
|
||||||
"""
|
"""
|
||||||
|
|
||||||
|
|
@ -238,12 +268,12 @@ class sde_RatQuad(RatQuad):
|
||||||
Return the state space representation of the covariance.
|
Return the state space representation of the covariance.
|
||||||
"""
|
"""
|
||||||
|
|
||||||
assert False, 'Not Implemented'
|
assert False, "Not Implemented"
|
||||||
|
|
||||||
# Params to use:
|
# Params to use:
|
||||||
|
|
||||||
# self.lengthscale
|
# self.lengthscale
|
||||||
# self.variance
|
# self.variance
|
||||||
#self.power
|
# self.power
|
||||||
|
|
||||||
#return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
|
# return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
|
||||||
|
|
|
||||||
|
|
@ -29,6 +29,7 @@ class TestGridModel:
|
||||||
self.dim = self.X.shape[1]
|
self.dim = self.X.shape[1]
|
||||||
|
|
||||||
def test_alpha_match(self):
|
def test_alpha_match(self):
|
||||||
|
self.setup()
|
||||||
kernel = GPy.kern.RBF(input_dim=self.dim, variance=1, ARD=True)
|
kernel = GPy.kern.RBF(input_dim=self.dim, variance=1, ARD=True)
|
||||||
m = GPy.models.GPRegressionGrid(self.X, self.Y, kernel)
|
m = GPy.models.GPRegressionGrid(self.X, self.Y, kernel)
|
||||||
|
|
||||||
|
|
@ -38,6 +39,7 @@ class TestGridModel:
|
||||||
np.testing.assert_almost_equal(m.posterior.alpha, m2.posterior.woodbury_vector)
|
np.testing.assert_almost_equal(m.posterior.alpha, m2.posterior.woodbury_vector)
|
||||||
|
|
||||||
def test_gradient_match(self):
|
def test_gradient_match(self):
|
||||||
|
self.setup()
|
||||||
kernel = GPy.kern.RBF(input_dim=self.dim, variance=1, ARD=True)
|
kernel = GPy.kern.RBF(input_dim=self.dim, variance=1, ARD=True)
|
||||||
m = GPy.models.GPRegressionGrid(self.X, self.Y, kernel)
|
m = GPy.models.GPRegressionGrid(self.X, self.Y, kernel)
|
||||||
|
|
||||||
|
|
@ -55,6 +57,7 @@ class TestGridModel:
|
||||||
)
|
)
|
||||||
|
|
||||||
def test_prediction_match(self):
|
def test_prediction_match(self):
|
||||||
|
self.setup()
|
||||||
kernel = GPy.kern.RBF(input_dim=self.dim, variance=1, ARD=True)
|
kernel = GPy.kern.RBF(input_dim=self.dim, variance=1, ARD=True)
|
||||||
m = GPy.models.GPRegressionGrid(self.X, self.Y, kernel)
|
m = GPy.models.GPRegressionGrid(self.X, self.Y, kernel)
|
||||||
|
|
||||||
|
|
|
||||||
Loading…
Add table
Add a link
Reference in a new issue