Spatial Statistics with Image Analysis. Outline. A Statistical Approach. Johan Lindström 1. Lund October 6, 2016

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1 Spatial Statistics Spatial Examples More Spatial Statistics with Image Analysis Johan Lindström 1 1 Mathematical Statistics Centre for Mathematical Sciences Lund University Lund October 6, 2016 Johan Lindström - johanl@maths.lth.se Spatial Statistics 1/31 Spatial Statistics Spatial Examples More Outline Spatial Statistics with Image Analysis Bayesian statistics Hierarchical modelling Estimation Procedures Spatial Statistics Stochastic Fields Gaussian Markov Random Fields Image Reconstruction Examples Environmental Data Corrupted Pixels NDVI Learn more A Statistical Approach Johan Lindström - johanl@maths.lth.se Spatial Statistics 2/31 Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation All measurements contain random measurement errors/variation. Most natural phenomena have natural random variation. Often the uncertainty of an estimate is, at least, as important as the estimate itself. We need to describe and model random variation and uncertainties! Johan Lindström - johanl@maths.lth.se Spatial Statistics 3/31

2 Bayesian modelling Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation We assume that there is some unknown truth, that we would like to find out about. This reality can be measured, usually with measurement variation, and often only partially. Bayesian modelling A Bayesian model consists of A prior, a priori, model for reality, x, given by the probability density π(x). A conditional model for data, y, given reality, with density p(y x). The prior can be expanded into several layers creating a Bayesian hierarchical model. Johan Lindström - johanl@maths.lth.se Spatial Statistics 4/31 Bayes Formula Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation How should the prior and data model be combined to make statements about the reality x, given observations of y? Bayes Formula p(x y) = p(y x)π(x) p(y) p(y x)π(x) = x Ω p(y x )π(x ) dx p(x y) is called the posterior, or a posteriori, distribution. Often, only the proportionality relation p(x y) p(x, y) = p(y x)π(x) is needed, when seen as a function of x. Johan Lindström - johanl@maths.lth.se Spatial Statistics 5/31 Hierarchical Models Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation We often have some prior knowledge of the reality. Given knowledge of the true reality, what can we say about images and other data? Construct a model for observations given that we know the truth. Given data, what can we say about the unknown reality? This is the inverse problem. Johan Lindström - johanl@maths.lth.se Spatial Statistics 6/31

3 Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation Bayesian hierarchical modelling (BHM) A hierarchical model is constructed by systematically considering components/features of the data, and how/why these features arise. Bayesian hierarchical modelling A Bayesian hierarchical model typically consists of (at least) Data model, p(y x): Describing how observations arise assuming known latent variables x. Latent model, p(x θ): Describing how the latent variables (reality) behaves, assuming known parameters. Parameters, π(θ): Describing our, sometimes vauge, prior knowledge of the parameters. Estimation Procedures Johan Lindström - johanl@maths.lth.se Spatial Statistics 7/31 Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation Maximum A Posteriori (MAP): Maximise the posterior distribution p(x y) with respect to x. Standard optimisation methods Specialised procedures, using the model structure Simulation: Simulate samples from the posterior distribution p(x y). Estimate statistical properties from these samples. The samples can be seen as representative possible realities, given the available data. Markov chain Monte Carlo (MCMC) Gibbs sampling Image Reconstruction Johan Lindström - johanl@maths.lth.se Spatial Statistics 8/31 Spatial Interpolation Given observations at some locations (pixels), y(u i ), i = 1... n we want to make statements about the value at unobserved location(s), x(u 0 ). The typical model consists of a latent Gaussian field x N (μ, Σ), observed at locations u i, i = 1,..., n, with additive Gaussian noise (nugget or small scale variability) ( ) y i = x(u i ) + ε i ε i N 0, σ 2 ε. Johan Lindström - johanl@maths.lth.se Spatial Statistics 9/31

4 Stochastic Fields To perform the reconstruction (interpolation) we need a model for the spatial dependence between locations (pixels). 1. Assume a latent Gaussian field x N (μ, Σ). 2. Assume a regresion model for μ = Bβ. 3. Assume a parametric (stationary) model for the dependence (covariance) Σ i,j = C(x(u i ), x(u j )) = r(u i, u j ; θ) = r( ui u j ; θ). r(u i, u j ; θ) is called the covariance function. Johan Lindström - johanl@maths.lth.se Spatial Statistics 10/31 Matérn covariances functions One of the most common families of covariance functions is named after Bertil Matérn, who worked for Statens Skogsforskningsinstitut (Forest Research Institute of Sweden). Variance σ 2 > 0, scale parameter κ > 0 and shape parameter ν > 0 r M (h) = σ 2 Γ(ν) 2 ν 1 (κ h )ν K ν (κ h ), h R d, A measure of the range is given by ρ = 8ν/κ. Johan Lindström - johanl@maths.lth.se Spatial Statistics 11/ nu= nu= nu= nu= Johan Lindström - johanl@maths.lth.se Spatial Statistics 12/31

5 nu=0.5 nu= nu=2.0 nu= A local model Johan Lindström - johanl@maths.lth.se Spatial Statistics 13/31 Instead of specifying the covariance function we could consider the local behaviour of pixels. A popular model is the conditional autoregressive, CAR(1) model. x ij = 1 ( ) xi 1,j 4 + κ 2 + x i+1,j + x i,j 1 + x i,j+1 + ε, ε N (0, 1τ ) 2. This corresponds to a model for x where ( x N 0, Q 1), where Q is called the precision matrix Matérn covariances Johan Lindström - johanl@maths.lth.se Spatial Statistics 14/31 The Matérn covariance family The covariance between two points at distance h is r M (h) = σ 2 Γ(ν) 2 ν 1 (κ h )ν K ν (κ h ) Fields with Matérn covariances are solutions to a Stochastic Partial Differential Equation (SPDE) (Whittle, 1954), ( κ 2 Δ) α/2 x(u) = W(u). Johan Lindström - johanl@maths.lth.se Spatial Statistics 15/31

6 Lattice on R 2 Order α = 1 (ν = 0): κ 2 1 } {{ } (C) } 1 {{ } Δ (G) Order α = 2 (ν = 1): 1 1 κ κ }{{}}{{} 1 (C) Δ (G) }{{} Δ 2 (G 2 =GC 1 G) Spatial models for data Johan Lindström - johanl@maths.lth.se Spatial Statistics 16/31 GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 Δ)(τ x(u)) = W(u), u R d Spatial models for data Johan Lindström - johanl@maths.lth.se Spatial Statistics 17/31 GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 Δ)(τ x(u)) = W(u), u Ω Johan Lindström - johanl@maths.lth.se Spatial Statistics 17/31

7 Spatial models for data GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 e iπθ Δ)(τ x(u)) = W(u), u Ω Spatial models for data Johan Lindström - johanl@maths.lth.se Spatial Statistics 17/31 GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 u + m u M u )(τ u x(u)) = W(u), u Ω Johan Lindström - johanl@maths.lth.se Spatial Statistics 17/31 Image Reconstruction II Model with observations, y, and latent field, x, ( ) ( y x N Ax, σ 2 I x N μ, Q 1). and Q = κ 2 C + G or Q = κ 4 C + 2κ 2 G + GC 1 G. Interpolation using a GMRF E (x y) = μ + 1 σ 2 Q 1 x y A (y Aμ) V (x y) = Q 1 x y = (Q + 1 σ 2 A A ) 1 Johan Lindström - johanl@maths.lth.se Spatial Statistics 18/31

8 Spatial Statistics Spatial Examples More Fields GMRF Reconstruction Image Reconstruction Johan Lindstro m - johanl@maths.lth.se Spatial Statistics Spatial Examples More Johan Lindstro m - johanl@maths.lth.se Spatial Statistics Spatial Examples More Spatial Statistics 19/31 Environmental Corrupted Pixels NDVI Spatial Statistics 20/31 Environmental Corrupted Pixels NDVI Global Temperature Data January 2003 Johan Lindstro m - johanl@maths.lth.se July 2003 Spatial Statistics 21/31

9 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Global Temperature Reconstruction Global mean: 15 C. Johan Lindström - johanl@maths.lth.se Spatial Statistics 22/31 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Satellite Data Vegetation January 1999 July 1999 Johan Lindström - johanl@maths.lth.se Spatial Statistics 23/31 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Satellite Data Trend in Vegetation K 2 Estimate Independent estimates K 2 Estimate Correlated estimates Johan Lindström - johanl@maths.lth.se Spatial Statistics 24/31

10 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Image Reconstruction Corrupted Pixels Typically we don t know which pixels that are bad. A better model is then Assume an underlying image, x. Assume an indicator image for bad pixels, z. Given the indicator we either observe the correct pixel value from x or noise. Use Bayes formula to compute the distribution for the unknown image (and indicator) given observations and parameters. Johan Lindström - johanl@maths.lth.se Spatial Statistics 25/31 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Image Reconstruction Corrupted pixels image reconstruction bad pixels bad pixels estimate Johan Lindström - johanl@maths.lth.se Spatial Statistics 26/31 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Normalized difference vegetation index (NDVI) NDVI = R NIR R RED R NIR + R RED R RED is the amount of reflected red light ( μm) R NIR is the amount of reflected near-infrared light ( μm) Johan Lindström - johanl@maths.lth.se Spatial Statistics 27/31

11 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Smoothed version of the NDVI Data Smooth the data to fill in missing values and remove noise due to cloud cover, etc Important ecological questions: Plant phenology (start and end of season) Plant productivity (integral) Johan Lindström - johanl@maths.lth.se Spatial Statistics 28/31 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Smoothing of Satellite Based Vegetation Measurements NDVI May 1993 Smooth Johan Lindström - johanl@maths.lth.se Spatial Statistics 29/31 Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI Smoothing of Satellite Based Vegetation Measurements 240 Obs Gaussian Gaussian 1D 220 Nig 1D Johan Lindström - johanl@maths.lth.se Spatial Statistics 30/31

12 Learn more! Spatial Statistics Spatial Examples More What? Spatial statistics with image analysis, FMSN20 When? HT2-2016, October December Where? Information and Matlab files will be available at Who? Lecturer: Johan Lindström MH:319 Johan Lindström - johanl@maths.lth.se Spatial Statistics 31/31