Elsevier

European Journal of Control

Volume 33, January 2017, Pages 43-51
European Journal of Control

Optimal smoothing for spherical Gauss–Markov Random Fields with application to weather data estimation

https://doi.org/10.1016/j.ejcon.2016.11.001Get rights and content

Abstract

This paper considers the smoothing problem for inhomogeneous Gauss–Markov Random Fields on a spherical lattice. Various observation models are considered, such as the case of noisy, possibly correlated, observations available only on a subset of sites, or a variable number of process components being measured. A 2D recursive optimal smoothing algorithm is derived, with computational complexity of O(N2) where N is the number of sites, in line with known more common algorithms for inhomogeneous fields on rectangular lattices. An application of the method in weather forecasting using real data is presented, showing the capability of the proposed method.

Introduction

A Markov Random Field (MRF) is a statistical model for a class of random variables defined on an index set which can be regarded as the vertices of a graph (see e.g. [7], [2]). In this paper, the focus is on MRFs defined on a spherical lattice. This type of graph can be thought of as having vertices which are a set of discrete values lying on the surface of a sphere and where there is only a very sparse level of interconnection between the vertices as represented by the set of edges of the graph. Such graphs are not planar,1 even though the vertices form a (finite) subset of Z2. In particular, the case will be considered where all random variables are jointly Gaussian, with zero mean, and a covariance function having a specific form which will be described. Essentially, a MRF exhibits a nearest-neighbour property; this means that the distribution of a given set of random variables, conditioned on all the others, depends actually of those among the conditioning variables that are “nearest” to the set. This concept will be clarified in the sequel.

There has been a substantial amount of work dealing with MRF defined on rectangular lattices. Some of this work is reviewed in Section 1.2. These models have found particular application in image processing for example. One of the most important areas of scientific endeavour in modern human history has been the study of phenomena associated with physical quantities defined on the surface of our planet. There are so many such examples that it is far beyond the scope of our work to address them in specificity. However, in recent times, the issue of climate modelling and estimation has come to the forefront in connection with the observed variations in the earth׳s surface temperature and associated inference problems. Indeed, it would be highly significant the development of new and/or different statistical models, which possess sufficient generality to allow modelling of such physical phenomena, and which are computationally cheap enough to permit robust computational solutions to related inference problems. This is the subject of our paper It has to be pointed out that our method is here applied to just the spatial feature of the process; it does not incorporate temporal structure. Indeed, the estimation of (spatial) covariances forms part of the numerical examples presented in the paper, and this estimation requires averaging over time thus removing any temporal information present. Nonetheless, even sampled cross-covariances between points at different times could be calculated by averaging at different times, and used as process statistics, thus our method could in principle be extended in order to solve prediction problems (weather forecasts).

In accordance with the above motivation, this work considers a class of statistical models which can be used to address inference problems for data collected at various points on the surface of a sphere. Gaussian MRF (GMRF) are of special interest for the usual reasons: (i) they are often well justified in terms of observed data statistics and (ii) they lead to linear models which permit tractable solutions, and possess a priori characterisation of estimator performance, at least in terms of estimation error covariance. Our approach is significant because it builds on the approach presented in [5] which deals with the stochastic realisation and optimal smoothing problems for MRFs defined on rectangular and toroidal lattices. This paper considers GMRFs defined on a true spherical lattice where the special role of the two “poles” is highlighted. Causal realisations and optimal estimation given noisy and/or incomplete measurements for such models are considered A detailed application for estimation of temperature on the earth׳s surface using available measurement data is presented and the inference algorithm developed in this paper is compared to several well-known methods.

MRF models are inherently non-causal in the usual sense of the data index set having a total ordering. In the one dimensional (1D) case, MRF models are termed reciprocal, and have been studied in some detail since the earliest part of the twentieth century. Readers are referred to [11], which was the first to address realisation and optimal estimation for Gaussian reciprocal processes. References contained therein give the reader an appreciation of the history of the study of such processes dating back to the 1930s. The approach of [11] is also relevant to two dimensional (2D) case considered here. In the 2D case, the first systematic study of MRF from a stochastic realisation viewpoint is given in [10] (and several reference given therein). This work is important because it highlights (i) the transformation of a 2D MRF defined on a rectangular lattice, to a 1D model and (ii) the representation of the resulting non-causal 1D model as a stable composition of two “causal” models; one operating forwards and one operating backwards in terms of the index variable. Thus, the modelling and estimation of the 2D GMRF model on a rectangular lattice can be essentially reduced to the 1D case. A similar approach was described in [13] which generalised the definition of the neighbourhood of a vertex in the MRF. The optimal estimation problem for the rectangular lattice was considered in [10] as well as [1], which also considered an image processing application. It is less clear that the regular properties of the rectangular lattice studied in [13], [1] hold in the case of a spherical 2D lattice. However, the 2D realisation problem for both the rectangular and toriodal lattice GMRF models was studied in [5] which also developed recursive equations for 2D quantities and did not rely on “stacking” all elements of the 2D field into a single vector as in [13]. For the rectangular case, different neighbourhood class to those used in [10], [13] was proposed. In the toroidal case, the rectangular field was “wrapped” onto a torus, but the case of a true sphere was not included. This is the subject of this paper.

The existence of graphical models used in machine learning, many of which are similar to MRFs, has been drawn to our attention (see e.g. [8], [16], [3], [4]). Related inference algorithms, such as the junction tree algorithm (JTA) (see e.g. [6]) or the loopy belief propagation (LBP), have been suggested as applicable in the present problem. Whilst a detailed analysis of the application of the JTA and LBP to the present problem is outside the scope of this paper, we just point out that LPB is not an exact algorithm (even though it may converge to the exact solution, see for instance [3] and the reference therein for details on this approach), whereas the method we are proposing is exact, as it exploits the particular underlying graph structure. We also point out that it is commented in [9] (based on references therein) that JTA is appropriate for sparse, irregular and causal models, and not lattice-type structures. In the present problem, our models are not sparse, irregular nor causal and are clearly lattices. Also most work on JTA uses discrete states, not Gaussian models where the (inverse) covariance structure can be easily exploited (as in [5], where the complexity is further reduced by the assumption of homogeneity, which does not hold in the present formulation). Thus it would appear that JTA is not likely to be an efficient algorithm to apply in our problem.

This paper makes two significant contributions. Firstly, it extends the rectangular and toroidal GMRF models of [5] to permit a uniform covering of the sphere and also to properly account for the two poles in a precise way. These issues are not considered in [5], and need to be addressed separately as the poles have a different nearest neighbour structure. There are many ways of accounting for the poles, so we have chosen the one corresponding to a lattice exactly reproducing the intersections between meridians and parallels on the earth surface because of the particular example of application considered. As a matter of fact, using latitude and longitude is the most common way of locating points on the earth surface, and most of the real data available are indeed collected in this way (and we use real data for the experimental part of the present paper). Moreover, an interesting feature of this kind of spherical lattice is that, even though it introduces a different and more complex nearest neighbour structure for the poles with respect to the other nodes, nevertheless, as we will see, our smoothing algorithm results to be unaffected by that in terms of overall computational complexity. Noncausal 2D, 1D, and causal 1D stochastic realisations are developed as well as the optimal smoother formulation. The smoother, which also generalises [5] has computational complexity O(N2), which is the same order as for the most efficient algorithms already known for the rectangular case (cf. supra).

Secondly, the paper applies the approach to the analysis of temperature data collected on the earth׳s surface. In particular, the paper examines the performance of the method in terms of estimating the temperature on a large lattice given noisy measurements on a sub-lattice. Since “ground truth” is known, performance analysis as well as comparison with other methods is reported. Since our model is able to properly include the poles, the application section investigates the effectiveness of the approach in estimating the temperature at the north pole based on temperature data gathered in several European cities. The issues of model estimation and efficient, stable numerical implementations are highlighted.

The layout of the paper is as follows: Section 2 introduces the structure of the class of spherical MRF models which are the basis for subsequent work and reviews the general formulation for a 2D stochastic realisation of a GMRF. Reciprocal and causal 1D realisations are developed. Section 3 describes the optimal smoother for the spherical GMRF when observed in additive Gaussian noise (which itself can be a GMRF). Missing measurements can also be included. This is important for the study on weather data which is presented in Section 4. Finally, some remarks and possible directions for future related work are drawn.

Section snippets

Stochastic realisations for a spherical GMRF

This section introduces the graphical structure of the spherical MRF and reviews the general formulation for a stochastic realisation of a GMRF. Then, it introduces a set of 1D processes obtained by “scanning” the field (along lines of constant latitude), and derives a reciprocal model for these processes with the poles playing the role of boundary conditions. The presence of the poles yields a somewhat different causal “forward–backward” 1D realisation from the rectangular or toroidal fields,

Optimal smoothing on a spherical lattice

Now suppose that in addition to the GMRF variables X, which will be referred to subsequently as state variables, there is another GMRF Y={Yv:vV¯} with each YvRm(v),m(v)n for all vV¯, given byYv=CvXv+DvWvwhere CvRm(v)×n are known. Note that the dimension of the observation vector Yv may depend on v itself, i.e. there may be a different number m(v) of measurements made at each vertex v. If there are no measurements made at a vertex v one formally defines correspondingly m(v)=0. The process W=

Application to the smoothing of climate data

In this section, the application of the described techniques to a weather data smoothing problem is presented. Hereafter, the CRU TS 2.1 data-set [12] is considered, which is a database (available for download) of monthly climate observations of the period 1901–2002 constructed from meteorological stations on a large portion of a 0.5° spaced grid covering the earth surface. Due to the constant angular spacing of parallels and meridians in the data-set, an appropriate 2D map representation of

Concluding remarks

A large amount of data has been collected over the last century concerning the earth׳s climate, and at an even increasing rate given the advances in storage and measurement capability. This large amount of data makes the issue of developing climate-related models based on a purely statistical knowledge a relevant one. This has been our basic motivation for the present research. The specific technical issue is how to perform a smoothing on a “true” spherical lattice, namely a grid reflecting

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