Random fields as stochastic differential equations

Precision vs covariance, fight!

October 12, 2020 — March 1, 2021

dynamical systems
edge computing
linear algebra
signal processing
state space models
stochastic processes
time series

\[ \renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\sinc}{\operatorname{sinc}} \]

Figure 1

The representation of certain random fields, especially Gaussian random fields as stochastic differential equations. This is the engine that makes filtering Gaussian processes go, and is also a natural framing for probabilistic spectral analysis.

I do not have much to say right now about this, but I am using it so watch this space.

1 Creating a stationary Markov SDE with desired covariance

The Gauss-Markov Random Field approach.

Warning: I’m taking crib notes for myself here, so I lazily switch between signal processing filter terminology and probabilist termonology. I assume Bochner’s and Yaglom’s Theorems as comprehensible methods for analysing covariance kernels.

Let’s start with stationary kernels. We consider an SDE \(f: \mathbb{R}\to\mathbb{R}\) at stationarity. We will let its driving noise to be some Wiener process. We care concerned with deriving the parameters of the SDE such that it has a given stationary covariance function \(k\).

If there are no zeros in the spectral density, then there are no poles in the inverse transfer function, and we can model it with an all-pole SDE. This includes all the classic Matérn functions. This is covered in J. Hartikainen and Särkkä (2010), and Lindgren, Rue, and Lindström (2011). Worked examples starting from a discrete time formulation are given in a tutorial introduction Grigorievskiy and Karhunen (2016).

More generally, (quasi-)periodic covariances have zeros and we need to find a full rational function approximation. Särkkä, Solin, and Hartikainen (2013) introduces one such method. Bolin and Lindgren (2011) explores a sligtly different class

Solin and Särkkä (2014) has a fancier method employing resonators a.k.a. filter banks, to address a concern of Steven Reece et al. (2014) that atomic spectral peaks in the Fourier transform are not well approximated by rational functions.

Bolin and Lindgren (2011) consider a general class of realisable systems, given by \[ \mathcal{L}_{1} X(\mathbf{s})=\mathcal{L}_{2} \mathcal{W}(\mathbf{s}) \] for some linear operators \(\mathcal{L}_{1}\) and \(\mathcal{L}_{2} .\)

In the case that \(\mathcal{L}_{1}\) and \(\mathcal{L}_{2}\) commute, this may be put in hierarchical form: \[\begin{aligned} \mathcal{L}_{1} X_{0}(\mathbf{s})&=\mathcal{W}(\mathbf{s})\\ X(\mathbf{s})&=\mathcal{L}_{2} X_{0}(\mathbf{s}). \end{aligned}\]

They explain

\(X(\mathbf{s})\) is simply \(\mathcal{L}_{2}\) applied to the solution one would get to if \(\mathcal{L}_{2}\) was the identity operator.

They call this a nested PDE, although AFAICT you could also say ARMA. They are particularly interested in equations of this form: \[ \left(\kappa^{2}-\Delta\right)^{\alpha / 2} X(\mathbf{s})=\left(b+\mathbf{B}^{\top} \nabla\right) \mathcal{W}(\mathbf{s}) \]

The SPDE generating this class of models is \[ \left(\prod_{i=1}^{n_{1}}\left(\kappa^{2}-\Delta\right)^{\alpha_{i} / 2}\right) X(\mathbf{s})=\left(\prod_{i=1}^{n_{2}}\left(b_{i}+\mathbf{B}_{i}^{\top} \nabla\right)\right) \mathcal{W}(\mathbf{s}) \]

They show that spectral density for such an \(X(\mathbf{s})\) is given by \[ S(\mathbf{k})=\frac{\phi^{2}}{(2 \pi)^{d}} \frac{\prod_{j=1}^{n_{2}}\left(b_{j}^{2}+\mathbf{k}^{\top} \mathbf{B}_{j} \mathbf{B}_{j}^{\top} \mathbf{k}\right)}{\prod_{j=1}^{n_{1}}\left(\kappa_{j}^{2}+\|\mathbf{k}\|^{2}\right)^{\alpha_{j}}}. \]

2 Convolution representations

See stochastic convolution or pragmatically, assume Gaussianity and see Gaussian convolution processes.

3 Covariance representation

Figure 2

Suppose there is a linear SDE on domain \(\mathbb{R}^d\) whose measure has the desired covariance structure, and ignore all questions of existence and convergence for now. We define terms of the driving noise \(\varepsilon\) and a linear differential operator \(\mathcal{L}\) such that \[ \mathcal{L}f(\mathbf{x})=\varepsilon(\mathbf{x}). \]

Assume there is a Green’s function for the PDE, i.e. that for any \(\mathbf{s} \in\mathbb{R}^d\) we may find a function \(G_\mathbf{s}(\mathbf{x})\) such that \[ \mathcal{L}G_\mathbf{s}(\mathbf{x})=\delta_\mathbf{s}(\mathbf{x}). \]

The solutions of the SDE, ignoring a whole bunch of existence stuff, are then given by the convolution of these Green’s functions with the driving noise, i.e. \(f(\mathbf{x}_p) \overset{\text{sorta}}{=}\int G_\mathbf{s}(\mathbf{x}_p)\varepsilon(\mathbf{s}) d \mathbf{s}.\) We use this to find the covariance of the solutions in terms of inner products of these fundamental solutions. \[\begin{align*} k(\mathbf{x}_p, \mathbf{x}_q) &=\mathbb{E}[f(\mathbf{x}_p)f(\mathbf{x}_q)] \\ &=\mathbb{E}\left[\int G_\mathbf{s}(\mathbf{x}_p)\varepsilon(\mathbf{s}) d \mathbf{s} \int G_\mathbf{t}(\mathbf{x}_q)\varepsilon(\mathbf{t}) d \mathbf{t} \right] \\ &=\mathbb{E}\left[\iint G_\mathbf{s}(\mathbf{x}_p) G_\mathbf{t}(\mathbf{x}_q) \varepsilon(\mathbf{s}) \varepsilon(\mathbf{t}) d \mathbf{t} d \mathbf{s} \right] \\ &=\iint G_\mathbf{s}(\mathbf{x}_p) G_\mathbf{t}(\mathbf{x}_q) \mathbb{E}[\varepsilon(\mathbf{s}) \varepsilon(\mathbf{t})] d \mathbf{t} d \mathbf{s} \\ &=\iint G_\mathbf{s}(\mathbf{x}_p) G_\mathbf{t}(\mathbf{x}_q) \sigma^2_\varepsilon \delta_\mathbf{s} (\mathbf{t}) d \mathbf{t} d \mathbf{s} &\text{ whiteness}\\ &=\sigma^2_\varepsilon \int G_\mathbf{s}(\mathbf{x}_p) G_\mathbf{s}(\mathbf{x}_q) d \mathbf{s}\\ &=\sigma^2_\varepsilon \langle G_\cdot(\mathbf{x}_p), G_\cdot(\mathbf{x}_q)\rangle \end{align*}\]

After that, the question is, given a Greens function can you produce a linear operator that realises it?

For example, the arc-cosine kernel of order \(1\) corresponding to the ReLU is \[\begin{align*} k(\mathbf{x}_p, \mathbf{x}_q) &= \frac{\sigma_\varepsilon^2 \Vert \mathbf{x}_p \Vert \Vert \mathbf{x}_q \Vert }{2\pi} \Big( \sin |\theta| + \big(\pi - |\theta| \big) \cos\theta \Big) \end{align*}\] so for Green’s functions inducing this to exist we would want \[\begin{align} \int G_\mathbf{s}(\mathbf{x}_p) G_\mathbf{s}(\mathbf{x}_q) d \mathbf{s} &=\frac{\Vert \mathbf{x}_p \Vert \Vert \mathbf{x}_q \Vert }{2\pi} \Big( \sin |\theta| + \big(\pi - |\theta| \big) \cos\theta \Big) \end{align}\] For this to work we would need \(G_\mathbf{s}(\mathbf{x})\propto\Vert \mathbf{x} \Vert.\)

4 Input measures

Warning: this is just a dump of some notes from a paper I was writing; It does not make much sense RN. The essential idea I want to get at is considering different enveloping strategies for the SDE; Enveloping the input noise, for example

Suppose \(\mathbf{x}_p, \mathbf{x}_q \in \mathbb{R}^d\). The kernel satisfies \[\begin{aligned} k(\mathbf{x}_p, \mathbf{x}_q) = \sum_{j=1}^d \frac{\partial k}{\partial x_{pj}} x_{pj}. \end{aligned}\] Let \(f\) denote the Gaussian process with covariance function \(k\) and let \(\mathcal{F}_{\mu}[f]\) denote the Fourier transform of \(f\) with respect to the finite measure \(\mu\). Let the Fourier transform of \(\mu\) be denoted \(\mathcal{F}[\mu](\mathbf{\omega})=\int e^{-i \mathbf{\omega}^\top \mathbf{x}} \, \mu(\dd\mathbf{x})\), so that \(\mathcal{F}_{\mu}[f]=\mathcal{F}[f(x)\partial_x \mu(x)]=\mathcal{F}[f(x)]\ast\mathcal{F} [ \mu].\)

We have \[\begin{aligned} \mathbb{E} | \mathcal{F}_{\mu}[f](\mathbf{\omega})|^2 &= \iint \mathbb{E}\big[ f(\mathbf{x}_p) f(\mathbf{x}_q) \big]\, e^{-i\mathbf{\omega}^\top(\mathbf{x}_p - \mathbf{x}_q)} \mu(\dd\mathbf{x}_p) \mu(\dd\mathbf{x}_q) \\ &= \iint k(\mathbf{x}_p, \mathbf{x}_q) \, e^{-i\mathbf{\omega}^\top(\mathbf{x}_p - \mathbf{x}_q)}\,\mu(\dd\mathbf{x}_p) \mu(\dd\mathbf{x}_q) \\ &= \iint \sum_{j=1}^d \frac{\partial k}{\partial x_{pj}} x_{pj} e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \mu(\dd\mathbf{x}_p) \, e^{i \mathbf{\omega}^\top \mathbf{x}_q} \,\mu(\dd\mathbf{x}_q) \\ &= \int \sum_{j=1}^d i \frac{\partial}{\partial \omega_j} \Bigg( \int \frac{\partial k}{\partial x_{pj}} e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p)\Bigg) e^{i \mathbf{\omega}^\top \mathbf{x}_q} \mu(\dd\mathbf{x}_q)\\ &= \mathcal{F}_{\mu}^{\mathbf{x}_q} \left[ \sum_{j=1}^d i \frac{\partial}{\partial \omega_j} \Bigg( \int \frac{\partial k}{\partial x_{pj}} e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p)\Bigg) \right]\\ &= \mathcal{F}_{\mu}^{\mathbf{x}_q} \left[ \sum_{j=1}^d i \frac{\partial}{\partial \omega_j} \Bigg( \mathcal{F}_{\mu}^{\mathbf{x}_p}\left[ \frac{\partial k}{\partial x_{pj}} \right]\Bigg) \right].\end{aligned}\] Then \[\begin{aligned} \mathbb{E} | \mathcal{F}_{\mu}[f](\mathbf{\omega})|^2 &= \int \sum_{j=1}^d i \frac{\partial}{\partial \omega_j} \Bigg( \int \frac{\partial k}{\partial x_{pj}} e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p) \, \Bigg) e^{i \mathbf{\omega}^\top \mathbf{x}_q}\mu(\dd\mathbf{x}_q)\\ &= -\int \sum_{j=1}^d \frac{\partial}{\partial \omega_j} \Bigg( \omega_j \int k(\mathbf{x}_p, \mathbf{x}_q) e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p) \, \Bigg)e^{i \mathbf{\omega}^\top \mathbf{x}_q}\mu(\dd\mathbf{x}_q) \\ &= -\sum_{j=1}^d \int \Bigg( \int k(\mathbf{x}_p, \mathbf{x}_q) e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p) \, \Bigg) e^{i \mathbf{\omega}^\top \mathbf{x}_q}\mu(\dd\mathbf{x}_q)\\ &\phantom{{}={}}-\int \Bigg( \omega_j \frac{\partial}{\partial \omega_j} \int k(\mathbf{x}_p, \mathbf{x}_q) e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p) \, \Bigg) e^{i \mathbf{\omega}^\top \mathbf{x}_q}\mu(\dd\mathbf{x}_q) \\ (d+1)\mathbb{E} | \mathcal{F}_{\mu}[f](\mathbf{\omega})|^2 &= -\int \Bigg( \omega_j \frac{\partial}{\partial \omega_j} \int k(\mathbf{x}_p, \mathbf{x}_q) e^{-i \mathbf{\omega}^\top \mathbf{x}_p} \, \mu(\dd\mathbf{x}_p) \, \Bigg) e^{i \mathbf{\omega}^\top \mathbf{x}_q}\mu(\dd\mathbf{x}_q) \end{aligned}\]

4.1 \(\mu\) is a hypercube

We assume that \(\mu\) is invariant with respect to permutation of coordinates. If we aren’t being silly, that means a cartesian product of intervals \(I\), \(\mu(A):=\operatorname{Leb}(A\cap I^d).\) Let us go with \(I=[-1,1].\) Then \[\begin{aligned} \mathcal{F}[\mu](\mathbf{\omega}) &=\prod_{j=1}^d \sinc \left( \frac{\omega_j}{4\pi}\right)\\ &=\prod_{j=1}^d \frac{\sin (\omega_j/2)}{\omega}\end{aligned}\] Also \[\begin{aligned} \sinc'x &=\frac{\cos \pi x - \sinc x}{x}.\end{aligned}\]

4.2 \(\mu\) is the unit sphere


4.3 \(\mu\) is an isotropic Gaussian

Suppose \(\mu\) is an isotropic Gaussian of variance \(I\sigma^2\) so that \(\dd \mu(\mathbf{x})=(2\pi)^{-d/2}\sigma^{-d}e^{-\sigma^2\mathbf{x}^\top\mathbf{x}/2}\) and \(\mathcal{F}[\mu]=e^{-\sigma^2\mathbf{\omega}^\top\mathbf{\omega}/2}=(2\pi)^{-d/2}\sigma^{-d}\dd \mu(\mathbf{\omega}).\)

5 Without stationarity via Green’s functions

Suppose our SDE may be specified in terms of a Gaussian white driving noise with variance \(\sigma_w^2\) and an impulse response function/Green’s function, \(g\) such that \[\begin{aligned} f(x):=\int g(\mathbf{u},\mathbf{x})\dd w(\mathbf{u}).\end{aligned}\] We know that the kernel is an inner product kernel and therefore invariant to rotation about \(\mathbf{0},\) i.e. for orthogonal \(Q\), \(k(Q\mathbf{x}_p, Q\mathbf{x}_q)=k(\mathbf{x}_p, \mathbf{x}_q).\) It follows that \(g(Q\mathbf{u}, Q\mathbf{x})=g(\mathbf{u}, \mathbf{x}).\) In fact, we may write each in dot-product form, i.e. \(k(\mathbf{x}_p, \mathbf{x}_q)=k(\mathbf{x}_p\cdot \mathbf{x}_q)\) and \(g(\mathbf{u}, \mathbf{x})=g(\mathbf{u}\cdot \mathbf{x}).\) The kernel satisfies \[\begin{aligned} k(\mathbf{x}_p, \mathbf{x}_q) &= \mathbb{E}\left[\int g(\mathbf{u},\mathbf{x}_p)\dd w(\mathbf{u})\int g(\mathbf{v},\mathbf{x}_q)\dd w(\mathbf{v})\right]\\ &= \mathbb{E}\left[\iint g(\mathbf{u},\mathbf{x}_p) g(\mathbf{v},\mathbf{x}_q)\dd w(\mathbf{u})\dd w(\mathbf{v})\right]\\ &= \iint g(\mathbf{u},\mathbf{x}_p) g(\mathbf{v},\mathbf{x}_q) \sigma_w^2\delta(\mathbf{u},\mathbf{v})\dd \mathbf{v}\dd \mathbf{u}\\ &= \sigma_w^2\int g(\mathbf{u},\mathbf{x}_p) g(\mathbf{u},\mathbf{x}_q) \dd\mathbf{u}\end{aligned}\] Up to a scaling factor, the green’s function is simply the covariance kernel under the assumption that the driving noise is white.

Recalling \(k(\mathbf{x}_p, \mathbf{x}_q) = \mathbb{E}\big[ \psi(\mathbf{W}^\top \mathbf{x}_q) \psi(\mathbf{W}^\top \mathbf{x}_p) \big]= \mathbb{E}\big[ \psi(Z_p) \psi(Z_q) \big]\) the Green’s function thus must satisfy \[\begin{aligned} \sigma_w^2\int g(\mathbf{u}\cdot\mathbf{x}_p) g(\mathbf{u}\cdot \mathbf{x}_q) \dd\mathbf{u} &= \mathbb{E}\big[ \psi(\mathbf{W}^\top \mathbf{x}_q) \psi(\mathbf{W}^\top \mathbf{x}_p) \big]\end{aligned}\] Now we need to see how this works for individual kernels.

6 References

Aasnaes, and Kailath. 1973. An Innovations Approach to Least-Squares Estimation–Part VII: Some Applications of Vector Autoregressive-Moving Average Models.” IEEE Transactions on Automatic Control.
Álvarez, Luengo, and Lawrence. 2013. Linear Latent Force Models Using Gaussian Processes.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Antoulas, ed. 1991. Mathematical System Theory: The Influence of R. E. Kalman.
Bakka, Rue, Fuglstad, et al. 2018. Spatial Modeling with R-INLA: A Review.” WIREs Computational Statistics.
Bart, Gohberg, and Kaashoek. 1979. Minimal Factorization of Matrix and Operator Functions. Operator Theory, Advances and Applications, v. 1.
Berry, Giannakis, and Harlim. 2020. Bridging Data Science and Dynamical Systems Theory.” arXiv:2002.07928 [Physics, Stat].
Bolin. 2014. Spatial Matérn Fields Driven by Non-Gaussian Noise.” Scandinavian Journal of Statistics.
Bolin, and Kirchner. 2020. The Rational SPDE Approach for Gaussian Random Fields With General Smoothness.” Journal of Computational and Graphical Statistics.
Bolin, and Lindgren. 2011. Spatial Models Generated by Nested Stochastic Partial Differential Equations, with an Application to Global Ozone Mapping.” The Annals of Applied Statistics.
Borovitskiy, Terenin, Mostowsky, et al. 2020. Matérn Gaussian Processes on Riemannian Manifolds.” arXiv:2006.10160 [Cs, Stat].
Bruinsma, and Turner. 2018. Learning Causally-Generated Stationary Time Series.” arXiv:1802.08167 [Stat].
Chang, Wilkinson, Khan, et al. 2020. “Fast Variational Learning in State-Space Gaussian Process Models.” In MLSP.
Curtain. 1975. Infinite-Dimensional Filtering.” SIAM Journal on Control.
Dowling, Sokół, and Park. 2021. Hida-Matérn Kernel.”
Dutordoir, Hensman, van der Wilk, et al. 2021. Deep Neural Networks as Point Estimates for Deep Gaussian Processes.” In arXiv:2105.04504 [Cs, Stat].
Duttweiler, and Kailath. 1973a. RKHS Approach to Detection and Estimation Problems–IV: Non-Gaussian Detection.” IEEE Transactions on Information Theory.
———. 1973b. RKHS Approach to Detection and Estimation Problems–V: Parameter Estimation.” IEEE Transactions on Information Theory.
E. 2017. A Proposal on Machine Learning via Dynamical Systems.” Communications in Mathematics and Statistics.
Friedlander, Kailath, and Ljung. 1975. Scattering Theory and Linear Least Squares Estimation: Part II: Discrete-Time Problems.” In 1975 IEEE Conference on Decision and Control Including the 14th Symposium on Adaptive Processes.
Gevers, and Kailath. 1973. An Innovations Approach to Least-Squares Estimation–Part VI: Discrete-Time Innovations Representations and Recursive Estimation.” IEEE Transactions on Automatic Control.
Grigorievskiy, and Karhunen. 2016. Gaussian Process Kernels for Popular State-Space Time Series Models.” In 2016 International Joint Conference on Neural Networks (IJCNN).
Grigorievskiy, Lawrence, and Särkkä. 2017. Parallelizable Sparse Inverse Formulation Gaussian Processes (SpInGP).” In arXiv:1610.08035 [Stat].
Hartikainen, J., and Särkkä. 2010. Kalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models.” In 2010 IEEE International Workshop on Machine Learning for Signal Processing.
Hartikainen, Jouni, and Särkkä. 2011. “Sequential Inference for Latent Force Models.” In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence. UAI’11.
Hartikainen, Jouni, Seppänen, and Särkkä. 2012. State-Space Inference for Non-Linear Latent Force Models with Application to Satellite Orbit Prediction.” In Proceedings of the 29th International Coference on International Conference on Machine Learning. ICML’12.
Higdon, David. 1998. A Process-Convolution Approach to Modelling Temperatures in the North Atlantic Ocean.” Environmental and Ecological Statistics.
Higdon, Dave. 2002. Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues.
Hildeman, Bolin, and Rychlik. 2019. Joint Spatial Modeling of Significant Wave Height and Wave Period Using the SPDE Approach.” arXiv:1906.00286 [Stat].
Huber. 2014. Recursive Gaussian Process: On-Line Regression and Learning.” Pattern Recognition Letters.
Hu, and Steinsland. 2016. Spatial Modeling with System of Stochastic Partial Differential Equations.” WIREs Computational Statistics.
Kailath, Thomas. 1971. “The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures.” The Annals of Mathematical Statistics.
Kailath, T. 1971a. RKHS Approach to Detection and Estimation Problems–I: Deterministic Signals in Gaussian Noise.” IEEE Transactions on Information Theory.
———. 1971b. A Note on Least-Squares Estimation by the Innovations Method.” In 1971 IEEE Conference on Decision and Control.
———. 1974. A View of Three Decades of Linear Filtering Theory.” IEEE Transactions on Information Theory.
Kailath, T., and Duttweiler. 1972. An RKHS Approach to Detection and Estimation Problems– III: Generalized Innovations Representations and a Likelihood-Ratio Formula.” IEEE Transactions on Information Theory.
Kailath, T., and Geesey. 1971. An Innovations Approach to Least Squares Estimation–Part IV: Recursive Estimation Given Lumped Covariance Functions.” IEEE Transactions on Automatic Control.
———. 1973. An Innovations Approach to Least-Squares Estimation–Part V: Innovations Representations and Recursive Estimation in Colored Noise.” IEEE Transactions on Automatic Control.
Kailath, T., Geesey, and Weinert. 1972. Some Relations Among RKHS Norms, Fredholm Equations, and Innovations Representations.” IEEE Transactions on Information Theory.
Kailath, T., and Weinert. 1975. An RKHS Approach to Detection and Estimation Problems–II: Gaussian Signal Detection.” IEEE Transactions on Information Theory.
Karvonen, and Särkkä. 2016. Approximate State-Space Gaussian Processes via Spectral Transformation.” In 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP).
Lee, Higdon, Calder, et al. 2005. Efficient Models for Correlated Data via Convolutions of Intrinsic Processes.” Statistical Modelling.
Lindgren, Bolin, and Rue. 2021. The SPDE Approach for Gaussian and Non-Gaussian Fields: 10 Years and Still Running.” arXiv:2111.01084 [Stat].
Lindgren, and Rue. 2015. Bayesian Spatial Modelling with R-INLA.” Journal of Statistical Software.
Lindgren, Rue, and Lindström. 2011. An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Ljung, and Kailath. 1976. Backwards Markovian Models for Second-Order Stochastic Processes (Corresp.).” IEEE Transactions on Information Theory.
Ljung, Kailath, and Friedlander. 1975. Scattering Theory and Linear Least Squares Estimation: Part I: Continuous-Time Problems.” In 1975 IEEE Conference on Decision and Control Including the 14th Symposium on Adaptive Processes.
Meyer, Edwards, Maturana-Russel, et al. 2020. Computational Techniques for Parameter Estimation of Gravitational Wave Signals.” WIREs Computational Statistics.
Park. 1981. Representations of Gaussian Processes by Wiener Processes.” Pacific Journal of Mathematics.
Pluch. 2007. Some Theory for the Analysis of Random Fields - With Applications to Geostatistics.” arXiv:math/0701323.
Rackauckas, Ma, Martensen, et al. 2020. Universal Differential Equations for Scientific Machine Learning.” arXiv.org.
Reece, Steven, Ghosh, Rogers, et al. 2014. Efficient State-Space Inference of Periodic Latent Force Models.” The Journal of Machine Learning Research.
Reece, S., and Roberts. 2010. An Introduction to Gaussian Processes for the Kalman Filter Expert.” In 2010 13th International Conference on Information Fusion.
Rue, and Tjelmeland. 2002. Fitting Gaussian Markov Random Fields to Gaussian Fields.” Scandinavian Journal of Statistics.
Särkkä. 2011. Linear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression.” In Artificial Neural Networks and Machine Learning – ICANN 2011. Lecture Notes in Computer Science.
Särkkä, Álvarez, and Lawrence. 2019. Gaussian Process Latent Force Models for Learning and Stochastic Control of Physical Systems.” IEEE Transactions on Automatic Control.
Särkkä, and Piché. 2014. On Convergence and Accuracy of State-Space Approximations of Squared Exponential Covariance Functions.” In 2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP).
Särkkä, and Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10.
Särkkä, Solin, and Hartikainen. 2013. Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine.
Scharf, Hooten, Johnson, et al. 2017. Process Convolution Approaches for Modeling Interacting Trajectories.” arXiv:1703.02112 [Stat].
Segall, Davis, and Kailath. 1975. Nonlinear Filtering with Counting Observations.” IEEE Transactions on Information Theory.
Segall, and Kailath. 1976. Orthogonal Functionals of Independent-Increment Processes.” IEEE Transactions on Information Theory.
Sigrist, Künsch, and Stahel. 2015a. Spate : An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process.” Application/pdf. Journal of Statistical Software.
———. 2015b. Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Solin. 2016. Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.”
Solin, and Särkkä. 2013. Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E.
———. 2014. Explicit Link Between Periodic Covariance Functions and State Space Models.” In Artificial Intelligence and Statistics.
———. 2020. Hilbert Space Methods for Reduced-Rank Gaussian Process Regression.” Statistics and Computing.
Tompkins, and Ramos. 2018. Fourier Feature Approximations for Periodic Kernels in Time-Series Modelling.” Proceedings of the AAAI Conference on Artificial Intelligence.
Weinert, Howard L., and Kailath. 1974. Stochastic Interpretations and Recursive Algorithms for Spline Functions.” The Annals of Statistics.
Weinert, H. L., and Kailath. 1974. Minimum Energy Control Using Spline Functions.” In 1974 IEEE Conference on Decision and Control Including the 13th Symposium on Adaptive Processes.
Whittle. 1963. “Stochastic-Processes in Several Dimensions.” Bulletin of the International Statistical Institute.
Wilkinson, Andersen, Reiss, et al. 2019. Unifying Probabilistic Models for Time-Frequency Analysis.” In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Wilkinson, Särkkä, and Solin. 2021. Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees.”
Wilson, Borovitskiy, Terenin, et al. 2020. Efficiently Sampling Functions from Gaussian Process Posteriors.” In Proceedings of the 37th International Conference on Machine Learning.
Wilson, Borovitskiy, Terenin, et al. 2021. Pathwise Conditioning of Gaussian Processes.” Journal of Machine Learning Research.
Wolpert, and Ickstadt. 1998. Poisson/Gamma Random Field Models for Spatial Statistics.” Biometrika.
Yaglom. 1987a. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics.
———. 1987b. Correlation Theory of Stationary and Related Random Functions Volume I.
———. 2004. An Introduction to the Theory of Stationary Random Functions.