base ==== .. py:module:: smolgp.kernels.base .. autoapi-nested-parse:: These kernels are compatible with :class:`smolgp.solvers.StateSpaceSolver`, which uses Bayesian filtering and smoothing algorithms to perform scalable GP inference. (see :mod:`smolgp.solvers` for more technical details). On GPU, a performance boost may be observed for large datasets by using the :class:`smolgp.solvers.parallel.ParallelStateSpaceSolver` class. Like the quasisep kernels, these methods are experimental, so you may find the documentation patchy in places. You are encouraged to `open issues or pull requests `_ as you find gaps. Classes ------- .. autoapisummary:: smolgp.kernels.base.StateSpaceModel smolgp.kernels.base.Sum smolgp.kernels.base.Product smolgp.kernels.base.Wrapper smolgp.kernels.base.Scale smolgp.kernels.base.Constant smolgp.kernels.base.SHO smolgp.kernels.base.Exp smolgp.kernels.base.Matern32 smolgp.kernels.base.Matern52 smolgp.kernels.base.Cosine smolgp.kernels.base.ExpSineSquared smolgp.kernels.base.Matern Functions --------- .. autoapisummary:: smolgp.kernels.base.extract_leaf_kernels Module Contents --------------- .. py:function:: extract_leaf_kernels(kernel, all=False) Recursively extract leaf kernels from a sum or product of kernels If all==True, extract from both Sum and Product nodes (useful for returning all kernel elements). Default is False, which only extracts from Sum nodes. (as used in decomposing multi-component models, only valid for sums) .. py:class:: StateSpaceModel Bases: :py:obj:`tinygp.kernels.base.Kernel` The base class for an instantaneous linear Gaussian state space model. .. attribute:: name Unique identifier used in multi-component models to select individual components from a sum or product kernel. :type: str .. attribute:: dimension Dimensionality :math:`d` of the state space model. :type: int A state space model is defined by the following components: 1. :meth:`design_matrix` — The feedback matrix :math:`F` 2. :meth:`stationary_covariance` — The stationary covariance :math:`\mathbf{P}_\infty` 3. :meth:`observation_matrix` — The observation matrix :math:`H` 4. :meth:`noise` — The spectral density :math:`Q_c` of the driving white noise 5. :meth:`noise_effect_matrix` — The noise effect matrix :math:`L` 6. :meth:`transition_matrix` — The transition matrix :math:`A_k` (optional; default uses :func:`jax.scipy.linalg.expm`) 7. :meth:`process_noise` — The process noise :math:`Q_k` (optional; default uses :math:`\mathbf{P}_\infty - A_k \mathbf{P}_\infty A_k^\top` or the Van Loan matrix exponential) As a subclass of :class:`tinygp.kernels.Kernel`, this class supports addition and multiplication with other kernels via :class:`Sum` and :class:`Product`, as well as direct kernel evaluation. .. py:attribute:: name :type: str .. py:method:: coord_to_sortable(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray A helper function used to convert coordinates to sortable 1-D values By default, this is the identity, but in cases where ``X`` is structured (e.g. multivariate inputs), this can be used to appropriately unwrap that structure. .. py:property:: dimension :type: int The dimension of the state space model, d .. py:method:: design_matrix() -> tinygp.helpers.JAXArray :abstractmethod: The design (also called the feedback) matrix for the process, :math:`F` .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray :abstractmethod: The stationary covariance of the process, :math:`\mathbf{P}_\infty` .. py:method:: observation_model(X: tinygp.helpers.JAXArray, component: str | None = None) -> tinygp.helpers.JAXArray The observation model for the process, :math:`H` .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :abstractmethod: The observation matrix for the process, :math:`H` .. py:method:: noise() -> tinygp.helpers.JAXArray :abstractmethod: The spectral density of the white noise process, :math:`Q_c` .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray :abstractmethod: The noise effect matrix, :math:`L` .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix :math:`A_k` between two states at coordinates X1 and X2. Default behavior uses jax.scipy.linalg.expm(self.design_matrix() * (X2 - X1)), which is appropriate for stationary kernels defined by a linear Gaussian SSM. Overload this method if you have a more general model or simply wish to define the transition matrix analytically. .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray, use_van_loan=False) -> tinygp.helpers.JAXArray The process noise matrix :math:`Q_k` Default behavior computes :math:`Q_k = \mathbf{P}_\infty - A_k \mathbf{P}_\infty A_k^T` (see `Eq. 7 in Solin & Särkka (2014) `_). Pass ``use_van_loan=True`` to instead compute :math:`Q_k` via the `Van Loan (1978) `_ matrix exponential method using :math:`F`, :math:`L`, and :math:`Q_c`. Overload this method if you have a more general model or wish to define the process noise analytically. .. py:method:: reset_matrix(instid: int = 0) -> tinygp.helpers.JAXArray The reset matrix for an instantaneous state space model is trivially the identity matrix. .. py:method:: __add__(other: tinygp.kernels.base.Kernel | tinygp.helpers.JAXArray) -> tinygp.kernels.base.Kernel .. py:method:: __radd__(other: Any) -> tinygp.kernels.base.Kernel .. py:method:: __mul__(other: tinygp.kernels.base.Kernel | tinygp.helpers.JAXArray) -> tinygp.kernels.base.Kernel .. py:method:: __rmul__(other: Any) -> tinygp.kernels.base.Kernel .. py:method:: evaluate(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The kernel evaluated via the state space representation. See `Eq. 4 in Hartikainen & Särkkä (2010) `_. .. py:method:: evaluate_diag(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray For state space kernels, the variance is simple to compute .. py:method:: psd(omega: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The power spectral density (PSD) of the kernel. See `Eq. 8 in Solin & Särkka (2014) `_. .. py:class:: Sum(kernel1, kernel2) Bases: :py:obj:`StateSpaceModel` The sum of two :class:`StateSpaceModel` kernels. The joint state dimension is :math:`d = d_1 + d_2`, and all matrices are assembled as block-diagonal combinations of the two component kernels. .. py:attribute:: kernel1 :type: StateSpaceModel .. py:attribute:: kernel2 :type: StateSpaceModel .. py:attribute:: name .. py:property:: dimension :type: int The dimension of the summed state space model .. py:method:: coord_to_sortable(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray We assume that both kernels use the same coordinates .. py:method:: design_matrix() -> tinygp.helpers.JAXArray :math:`F = \mathrm{BlockDiag}(F_1,\, F_2)` .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray :math:`L = \mathrm{BlockDiag}(L_1,\, L_2)` .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray :math:`\mathbf{P}_\infty = \mathrm{BlockDiag}(\mathbf{P}_{\infty,1},\, \mathbf{P}_{\infty,2})` .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :math:`A_k = \mathrm{BlockDiag}(A_{k,1},\, A_{k,2})` .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :math:`Q_k = \mathrm{BlockDiag}(Q_{k,1},\, Q_{k,2})` .. py:method:: observation_model(X: tinygp.helpers.JAXArray, component=None) -> tinygp.helpers.JAXArray :math:`H = [H_1 \;\; H_2]`, with optional component masking. .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :math:`H = [H_1 \;\; H_2]` .. py:method:: reset_matrix(instid: int = 0) -> tinygp.helpers.JAXArray :math:`\mathrm{RESET} = \mathrm{BlockDiag}(\mathrm{RESET}_1,\, \mathrm{RESET}_2)` .. py:method:: noise() -> tinygp.helpers.JAXArray :math:`Q_c = \mathrm{BlockDiag}(Q_{c,1},\, Q_{c,2})` .. py:class:: Product(kernel1, kernel2) Bases: :py:obj:`StateSpaceModel` The product of two :class:`StateSpaceModel` kernels. The joint state dimension is :math:`d = d_1 \cdot d_2`, and all matrices are assembled using Kronecker products of the two component kernels. .. py:attribute:: kernel1 :type: StateSpaceModel .. py:attribute:: kernel2 :type: StateSpaceModel .. py:attribute:: name .. py:property:: dimension :type: int The dimension of the product state space model .. py:method:: coord_to_sortable(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray We assume that both kernels use the same coordinates .. py:method:: design_matrix() -> tinygp.helpers.JAXArray :math:`F = F_1 \otimes I + I \otimes F_2` .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray :math:`L` for products is not uniquely defined, only the combination :math:`L Q_c L^T` is. A convenient choice then for :math:`L` is the identity matrix, which we return here. Then, :math:`Q_c` is chosen so that :math:`L Q_c L^T` gives the correct process noise. .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray :math:`\mathbf{P}_\infty = \mathbf{P}_{\infty,1} \otimes \mathbf{P}_{\infty,2}` .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :math:`A_k = A_{k,1} \otimes A_{k,2}` .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :math:`Q_k` for a product is best determined via the identity :math:`Q_k = \mathbf{P}_\infty - A_k \mathbf{P}_\infty A_k^T`. .. py:method:: observation_model(X: tinygp.helpers.JAXArray, component=None) -> tinygp.helpers.JAXArray :math:`H = H_1 \otimes H_2`, with optional component masking. .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray :math:`H = H_1 \otimes H_2` .. py:method:: reset_matrix(instid: int = 0) -> tinygp.helpers.JAXArray :math:`\mathrm{RESET} = \mathrm{RESET}_1 \otimes \mathrm{RESET}_2` .. py:method:: noise() -> tinygp.helpers.JAXArray :math:`Q_c` for products is not uniquely defined. Returns :math:`Q_c = L_1 Q_{c,1} L_1^T \otimes \mathbf{P}_{\infty,2} + \mathbf{P}_{\infty,1} \otimes L_2 Q_{c,2} L_2^T`, with :math:`L = I`, so that :math:`L Q_c L^T` yields the correct process noise. .. py:class:: Wrapper Bases: :py:obj:`StateSpaceModel` A base class for wrapping kernels with some custom implementations .. py:attribute:: kernel :type: StateSpaceModel .. py:property:: dimension :type: int The dimension of the state space model, d .. py:method:: coord_to_sortable(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray A helper function used to convert coordinates to sortable 1-D values By default, this is the identity, but in cases where ``X`` is structured (e.g. multivariate inputs), this can be used to appropriately unwrap that structure. .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the process, :math:`F` .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix, :math:`L` .. py:method:: noise() -> tinygp.helpers.JAXArray The spectral density of the white noise process, :math:`Q_c` .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the process, :math:`\mathbf{P}_\infty` .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation matrix for the process, :math:`H` .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix :math:`A_k` between two states at coordinates X1 and X2. Default behavior uses jax.scipy.linalg.expm(self.design_matrix() * (X2 - X1)), which is appropriate for stationary kernels defined by a linear Gaussian SSM. Overload this method if you have a more general model or simply wish to define the transition matrix analytically. .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise matrix :math:`Q_k` Default behavior computes :math:`Q_k = \mathbf{P}_\infty - A_k \mathbf{P}_\infty A_k^T` (see `Eq. 7 in Solin & Särkka (2014) `_). Pass ``use_van_loan=True`` to instead compute :math:`Q_k` via the `Van Loan (1978) `_ matrix exponential method using :math:`F`, :math:`L`, and :math:`Q_c`. Overload this method if you have a more general model or wish to define the process noise analytically. .. py:method:: reset_matrix(instid=0) The reset matrix for an instantaneous state space model is trivially the identity matrix. .. py:class:: Scale Bases: :py:obj:`Wrapper` The product of a scalar and a state space kernel. .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the process, :math:`\mathbf{P}_\infty` .. py:method:: noise() -> tinygp.helpers.JAXArray The spectral density of the white noise process, :math:`Q_c` .. py:class:: Constant(sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'Constant', **kwargs) Bases: :py:obj:`StateSpaceModel` A constant kernel .. math:: k(\mathbf{x}_i,\,\mathbf{x}_j) = \sigma^2 where :math:`\sigma` is a parameter. :param sigma: The parameter :math:`\sigma` in the above equation. .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: name :value: 'Constant' .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for a Constant process, F .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of a Constant process, Pinf .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for a Constant process .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for a Constant process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for a Constant process .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for a Constant process .. py:method:: process_noise(X1, X2, use_van_loan=False) The process noise Q_k for a Constant process .. py:class:: SHO(omega: tinygp.helpers.JAXArray | float, quality: tinygp.helpers.JAXArray | float, sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'SHO', **kwargs) Bases: :py:obj:`StateSpaceModel` The damped, driven stochastic harmonic oscillator kernel This form of the kernel was introduced by `Foreman-Mackey et al. (2017) `_, and it takes the form: .. math:: k(\Delta) = \sigma^2\,\exp\left(-\frac{\omega_0\,\Delta}{2\,Q}\right) \left\{\begin{array}{ll} 1 + \omega_0\,\Delta & \mbox{for } Q = 1/2 \\ \cosh(\eta\,\omega_0\,\Delta) + \frac{1}{2\eta Q} \sinh(\eta\,\omega_0\,\Delta) & \mbox{for } Q < 1/2 \\ \frac{1}{2\eta Q}\cos(\eta\,\omega_0\,\Delta) + \sin(\eta\,\omega_0\,\Delta) & \mbox{for } Q > 1/2 \end{array}\right. for :math:`\Delta = |x_i - x_j|`, :math:`\eta = \sqrt{|1 - 1/(4Q^2)|}`. :param omega: The parameter :math:`\omega_0`. :param quality: The parameter :math:`Q`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. Specifying the explicit value here provides a slight performance boost compared to independently multiplying the kernel with a prefactor. :type sigma: optional .. py:attribute:: omega :type: tinygp.helpers.JAXArray | float .. py:attribute:: quality :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: eta :type: tinygp.helpers.JAXArray | float .. py:attribute:: name :value: 'SHO' .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the SHO process, F .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the SHO process, Pinf .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for the SHO process .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for the SHO process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for the SHO process .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for the SHO process .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise Q_k for the SHO process .. py:class:: Exp(scale: tinygp.helpers.JAXArray | float, sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'Exp', **kwargs) Bases: :py:obj:`StateSpaceModel` A state space implementation of :class:`tinygp.kernels.quasisep.Exp` This kernel takes the form: .. math:: k(\Delta)=\sigma^2\,\exp\left(-\frac{\Delta}{\ell}\right) for :math:`\Delta = |x_i - x_j|`. Also known as the "Ornstein-Uhlenbeck" kernel, and is also equivalent to a Matérn-1/2 kernel. :param scale: The parameter :math:`\ell`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. :type sigma: optional .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: lam :type: tinygp.helpers.JAXArray | float .. py:attribute:: name :value: 'Exp' .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the Exp process, F .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the Exp process, Pinf .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for the Exp process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for the Exp process .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for the Exp process .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for the Exp process .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise Q_k for the Exp process .. py:class:: Matern32(scale: tinygp.helpers.JAXArray | float, sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'Matern32', **kwargs) Bases: :py:obj:`StateSpaceModel` A state space implementation of :class:`tinygp.kernels.quasisep.Matern32` This kernel takes the form: .. math:: k(\Delta)=\sigma^2\,\left(1+f\,\Delta\right)\,\exp(-f\,\Delta) for :math:`\Delta = |x_i - x_j|` and :math:`f = \sqrt{3} / \ell`. :param scale: The parameter :math:`\ell`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. :type sigma: optional .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: lam :type: tinygp.helpers.JAXArray | float .. py:attribute:: name :value: 'Matern32' .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the Matern-3/2 process, F .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the Matern-3/2 process, Pinf .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for the Matern-3/2 process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for the Matern-3/2 process .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for the Matern-3/2 process .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for the Matern-3/2 process .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise Q_k for the Matern-3/2 process .. py:class:: Matern52(scale: tinygp.helpers.JAXArray | float, sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'Matern52', **kwargs) Bases: :py:obj:`StateSpaceModel` A state space implementation of :class:`tinygp.kernels.quasisep.Matern52` This kernel takes the form: .. math:: k(\Delta)=\sigma^2\,\left(1+f\,\Delta + \frac{f^2\,\Delta^2}{3}\right) \,\exp(-f\,\Delta) for :math:`\Delta = |x_i - x_j|` and :math:`f = \sqrt{5} / \ell`. :param scale: The parameter :math:`\ell`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. :type sigma: optional .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: lam :type: tinygp.helpers.JAXArray | float .. py:attribute:: name :value: 'Matern52' .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the Matern-5/2 process, F .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the Matern-5/2 process, Pinf .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for the Matern-5/2 process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for the Matern-5/2 process .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for the Matern-5/2 process .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for the Matern-5/2 process .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise Q_k for the Matern-5/2 process .. py:class:: Cosine(scale, sigma=jnp.ones(()), name='Cosine', **kwargs) Bases: :py:obj:`StateSpaceModel` A state space implementation of :class:`tinygp.kernels.quasisep.Cosine` This kernel takes the form: .. math:: k(\Delta)=\sigma^2\,\cos(-2\,\pi\,\Delta/\ell) for :math:`\Delta = |x_i - x_j|`. :param scale: The parameter :math:`\ell`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. :type sigma: optional .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: omega :type: tinygp.helpers.JAXArray | float .. py:attribute:: name :value: 'Cosine' .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the Cosine process, F .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the Cosine process, Pinf .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for the Cosine process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for the Cosine process .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for the Cosine process .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for the Cosine process .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise Q_k for the Cosine process .. py:class:: ExpSineSquared(period: tinygp.helpers.JAXArray | float, gamma: tinygp.helpers.JAXArray | float = jnp.ones(()), sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'ExpSineSquared', order: int | None = None, **kwargs) Bases: :py:obj:`Wrapper` The exponential sine squared or "periodic" kernel .. math:: k(\mathbf{x}_i,\,\mathbf{x}_j) = \sigma^2 \exp(-\Gamma\,\sin^2 \pi r) where, by default, .. math:: r = ||(\mathbf{x}_i - \mathbf{x}_j) / P||_1 In the state space representation, this kernel is approximated using a finite number of basis functions. The method was introduced by `Solin & Särkkä (2014) `_. See their Figure 2 for the number of basis functions required to reach desired accuracy. Default behavior will automatically select the number of basis functions based on the length scale :math:`\ell`. :param period: The parameter :math:`P`. :param gamma: The parameter :math:`\Gamma`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. .. py:attribute:: gamma :type: tinygp.helpers.JAXArray | float | None :value: None .. py:attribute:: period :type: tinygp.helpers.JAXArray | float .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: omega :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: order :type: int | None .. py:attribute:: kernel :type: StateSpaceModel .. py:attribute:: name :value: 'ExpSineSquared' .. py:method:: error_bound() An upper bound on the error in the covariance from the Taylor series approximation. See Sec 3.4 of Solin & Särkkä (2014). .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the process, :math:`\mathbf{P}_\infty` .. py:class:: PeriodicTerm(order, omega, scale, **kwargs) Bases: :py:obj:`StateSpaceModel` A single term in the state space representation of the exponential sine squared or "periodic" kernel See Solin & Särkkä (2014) for details. .. py:attribute:: order :type: int .. py:attribute:: omega :type: tinygp.helpers.JAXArray | float .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: name .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the PeriodicTerm, :math:`F` .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the PeriodicTerm process, :math:`\mathbf{P}_\infty` .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation matrix for the PeriodicTerm, :math:`H` .. py:method:: noise() -> tinygp.helpers.JAXArray The spectral density of the white noise for the PeriodicTerm, :math:`Q_c` .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix :math:`L` for the PeriodicTerm .. py:method:: Ij(j, x, terms=50) -> tinygp.helpers.JAXArray The modified Bessel function of the first kind, order j, at x. Approximated via a truncated Taylor series expansion. .. py:method:: transition_matrix(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The transition matrix A_k for the PeriodicTerm .. py:method:: process_noise(X1: tinygp.helpers.JAXArray, X2: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The process noise Q_k for the PeriodicTerm .. py:class:: Matern(nu: tinygp.helpers.JAXArray | float, scale: tinygp.helpers.JAXArray | float, sigma: tinygp.helpers.JAXArray | float = jnp.ones(()), name: str = 'Matern', **kwargs) Bases: :py:obj:`StateSpaceModel` A state space implementation of the generic half-integer Matérn kernel with smoothness parameter :math:`\nu = p + 1/2` for integer :math:`p \geq 0`, length scale :math:`\ell`, and amplitude :math:`\sigma`. The model is dimension :math:`d = \nu + 1/2` and has the form (see Eq. 12.34-35 of Särkkä and Solin 2019): .. math:: F = \begin{pmatrix} 0 & 1 & 0 & & \\ \vdots & 0 & 1 & & \\ & & \ddots & \ddots & \\ -a_1\lambda^d & -a_2\lambda^{d-1} & \cdots & -a_d\lambda \end{pmatrix}, L = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, where :math:`\lambda = \sqrt{2\nu}/\ell` and the coefficients :math:`a_i = \binom{d}{i-1}` are the binomial coefficients. The spectral noise density is given by: .. math:: Q_c = \sigma^2 \frac{[(d-1)!]^2}{(2d-2)!} (2\lambda)^{2d-1}. :param nu: The smoothness parameter :math:`\nu` (must be half-integer). :param scale: The parameter :math:`\ell`. :param sigma: The parameter :math:`\sigma`. Defaults to a value of 1. :type sigma: optional .. py:attribute:: nu :type: tinygp.helpers.JAXArray | float .. py:attribute:: scale :type: tinygp.helpers.JAXArray | float .. py:attribute:: sigma :type: tinygp.helpers.JAXArray | float .. py:attribute:: lam :type: tinygp.helpers.JAXArray | float .. py:attribute:: name .. py:property:: dimension :type: int The dimension of the state space for the Matérn process .. py:method:: design_matrix() -> tinygp.helpers.JAXArray The design (also called the feedback) matrix for the Matern process, F .. py:method:: observation_matrix(X: tinygp.helpers.JAXArray) -> tinygp.helpers.JAXArray The observation model H for the Matern process .. py:method:: noise_effect_matrix() -> tinygp.helpers.JAXArray The noise effect matrix L for the Matern process .. py:method:: noise() -> tinygp.helpers.JAXArray The scalar Qc for the Matern process .. py:method:: stationary_covariance() -> tinygp.helpers.JAXArray The stationary covariance of the Matérn process, :math:`\mathbf{P}_\infty`