Spatial smoothing over complex domain Alessandro Ottavi NTNU Department of Mathematical Science August 16-20 2010
Outline Spatial smoothing Complex domain The SPDE approch Manifold Some tests
Spatial smoothing Outline Spatial smoothing Complex domain The SPDE approch Manifold Some tests
Spatial smoothing What is smoothing To smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise. This technique is used in many different field as geology biology insurance medicine...
Spatial smoothing Standard approach: Gaussian Field A standard approach is trough Gaussian Field. We say that x(s) s D R 2 is a Gaussian Field (GF) if all finite collection x(s i ) are jointly Gaussian. A GF is usually specify trough: a mean function µ(.) that lead to a mean vector µ(s i ) a covariance function C(.,.) that lead to a covariance matrix Σ = C(s i, s j )
Spatial smoothing Advantages: Intuitive approach Easy to use in many cases Easy to parametrize Problems: Can be computationally heavy Not good results when far away from standard cases (for example when the shape of the domain is important)
Complex domain Outline Spatial smoothing Complex domain The SPDE approch Manifold Some tests
Complex domain Complex domain and boundaries Spatial smoothing methods often assume the region of interest is R 2 but we can be in different situations: We might have a river, lake or a mountain, which, at least physically, prevents smoothing over the river, lake or mountain. In other cases topological caracteristics of the domain may play an important role and is important to introduce this aspects in the model.
Spatial smoothing over complex domain Complex domain Figure: example of a complex domain Fig 1 Distribution of flower and non-flowering S. leprosula in the alluvial (blue) and mudstone (brown) forest Fig 2 as above for S. johorensis Fig 3 as above for P tomentella Fig 4 as above for S. xanthophylla 2007 In the figure on the left there is an example where introducing information about the domain can improve the quality of the smoothing procedure: is showed the distribution of flowering and non flowering for three different spices of flowers over 2 different possible regions: alluvial and mud stone areas.
Complex domain In the figure on the right there is another possible example of complex domain, when there is the necessity to smooth for example over see region strictly connected over earth region and when is not feasible to do that then... may be. Figure: Cost of Norway
Complex domain Proposed methodology Non isotropic/ non stationary Gaussian Fields can be used to mimic the presence of complex boundaries; the strategy is to consider the distance between two points as length of the minimum arc not crossing the constrain. More recently Simon Wood (2008) 1 proposed a method based on spline able to be incorporated in GAM or mixed models. 1 Wood, S. N., Bravington, M. V. and Hedley, S. L. (2008), Soap film smoothing. Journal of the Royal Statistical Society
The SPDE approach Outline Spatial smoothing Complex domain The SPDE approch Manifold Some tests
3.5" Spatial smoothing over complex domain The SPDE approach We want to use the relation between Gaussian Field, SPDE Gaussian Markov Random Field and 2. 9 2 " 2.92" Stochastic Differential Equation GF GMRF in order to include in our (Bayesian) models information about the physical constrain.
The SPDE approach Gaussian Markov Random Field A Gaussian Markov Random Field is a Gaussian field x(s i ) s i D R 2 where the full conditionals only depends on a set of neighbors: π(x i x i ) = π(x i x δi ) With the property that the precision matrix is sparse due to: Q ij 0 i δ j A GMRF is discretized version of GF over a set of location s i where Q 1 = Σ.
The SPDE approach Second order differential equation We define a second order partial differential equation: Where: (κ 2 )x(s) = ɛ(s) s D (1) κ : is a range parameter, : is the Laplacian operator, in R 2 it results as ɛ : is a White Noise on D D : is a smooth surface f = 2 f x 2 + 2 f y 2
The SPDE approach SPDE and GF Is possible to prove that: The solution x(s) of the differential equation is Gaussian Whittle in 1963 2 prove that if D = R 2 the solution is a Gaussian field with Matérn covariance function given by: r(h) σ 2 (κh) 3 K 3 (κh) where K is a Bessel function of second order and the relation between the parameter κ and the effective range ρ in the corresponding GF is given by ρ = 24/κ 2 Whittle P., Stochastic processes in several dimensions, Statistics in Physical Sciences 1963
The SPDE approach SPDE and GMRF Is it possible to obtain the precision matrix Q for the process x i directly from the solution of the SPDE. Proceed to a triangularization of the region Express the SPDE in weak formulation < φ, ( + χ 2 )x >=< φ, ɛ > Approximate x i as a combination of basic functions ˆx = n φ i w i i=1 Find the precision matrix of x i
Manifold Outline Spatial smoothing Complex domain The SPDE approch Manifold Some tests
z z Spatial smoothing over complex domain Manifold What is a Manifold? A manifold is a space D that behaves locally as R d. Property: D is locally planar. 8 6 4 2 10 Y 0 2 10 5 5 0 Y X X 0 5 5 10 10
Some tests Outline Spatial smoothing Complex domain The SPDE approch Manifold Some tests
Some tests Correlation around an obstacle Correlation around lake, alpha=2, chi=0.2 Correlation around lake, alpha=2, chi=0.2 Correlation around lake, alpha=2, chi=0.2 Correlation around lake, alpha=2, chi=0.2
Some tests correlation 0.4 0.6 0.8 1.0 The correlation function across the obstacle (dots line) decrease much faster then the correlation away from the obstacle (continuous line). 0 2 4 6 8 10 distance
Some tests Correlation across obstacle and bottleneck
Some tests
Some tests Correlation across bottleneck Correlation trough tunnel, alpha=2, chi=0.2 Correlation trough tunnel, alpha=2, chi=0.2 Correlation trough tunnel, alpha=2, chi=0.2 Correlation trough tunnel, alpha=2, chi=0.2
Some tests Last example