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Spatial Statistics Spatial Examples More

Spatial Statistics with Image Analysis

Johan Lindstrom1

1Mathematical StatisticsCentre for Mathematical Sciences

Lund University

LundOctober 1, 2015

Johan Lindstrom - [email protected] Spatial Statistics 1/31


Outline

Spatial Statistics with Image AnalysisBayesian statisticsHierarchical modellingEstimation Procedures

Spatial StatisticsStochastic FieldsGaussian Markov Random FieldsImage Reconstruction

ExamplesEnvironmental DataCorrupted PixelsNDVI

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Spatial Statistics Spatial Examples More Bayesian statistics Hierarchical Estimation

A Statistical Approach

◮ All measurements contain random measurementerrors/variation.

◮ Most natural phenomena have natural randomvariation.

◮ Often the uncertainty of an estimate is, at least, asimportant as the estimate itself.

◮ We need to describe and model random variation anduncertainties!



Bayesian modelling

We assume that there is some unknown truth, that wewould like to find out about. This “reality” can bemeasured, usually with measurement variation, and oftenonly 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 aBayesian hierarchical model.



Bayes’ Formula

How should the prior and data model be combined to makestatements 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.



Hierarchical Models

◮ 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.



Bayesian hierarchical modelling (BHM)

A hierarchical model is constructed by systematicallyconsidering components/features of the data, and how/whythese features arise.

Bayesian hierarchical modelling

A Bayesian hierarchical model typically consists of (at least)

Data model, p(y|x): Describing how observations ariseassuming 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, priorknowledge of the parameters.



Estimation Procedures

Maximum A Posteriori (MAP): Maximise the posteriordistribution p(x|y) with respect to x.

◮ Standard optimisation methods◮ Specialised procedures, using the model

structure

Simulation: Simulate samples from the posteriordistribution p(x|y). Estimate statisticalproperties from these samples. The samples canbe seen as representative “possible realities”,given the available data.

◮ Markov chain Monte Carlo (MCMC)◮ Gibbs sampling


Spatial Statistics Spatial Examples More Fields GMRF Reconstruction

Image Reconstruction

Spatial Interpolation

Given observations at some locations (pixels),y(ui), i = 1 . . .nwe want to make statements about the value atunobserved location(s), x(u0).

The typical model consists of a latent Gaussian field

x ∈ N (μ,Σ) ,

observed at locations ui, i = 1, . . . ,n, with additive Gaussiannoise (nugget or small scale variability)

yi = x(ui)+ εi εi ∈ N(

0,σ2ε

).



Stochastic Fields

To perform the reconstruction (interpolation) we need amodel 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 thedependence (covariance)

Σi,j = C(x(ui), x(uj)) = r(ui,uj;θ) = r(∥∥ui − uj

∥∥ ;θ).

r(ui,uj;θ) is called the covariance function.



Matern covariances functions

◮ One of the most common families of covariancefunctions is named after Bertil Matern, who worked forStatens Skogsforskningsinstitut (Forest Research Institute of Sweden).

◮ Variance σ2 > 0, scale parameter κ > 0 and shapeparameter ν > 0

rM(h) =σ2

Γ(ν)2ν−1 (κ‖h‖)νKν(κ‖h‖), h ∈ Rd,

◮ A measure of the range is given by ρ =√

8ν/κ.



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A local model

Instead of specifying the covariance function we couldconsider the local behaviour of pixels. A popular model is theconditional autoregressive, CAR(1) model.

xij =1

4 + κ2

(xi−1,j + xi+1,j + xi,j−1 + xi,j+1

)+ ε,

ε ∈ N(

0,1τ2

).

This corresponds to a model for x where

x ∈ N(

0,Q−1),

where Q is called the precision matrix



Matern covariances

The Matern covariance family

The covariance between two points at distance ‖h‖ is

rM(h) =σ2

Γ(ν)2ν−1 (κ‖h‖)νKν(κ‖h‖)

Fields with Matern covariances are solutions to a StochasticPartial Differential Equation (SPDE) (Whittle, 1954),

(κ2 −Δ

)α/2x(u) = W(u).



Lattice on R2

Order α = 1 (ν = 0):

κ2

1

︸︷︷︸(C)

+

−1−1 4 −1

−1

︸︷︷︸≈−Δ (G)

Order α = 2 (ν = 1):

κ4

1

︸︷︷︸(C)

+2κ2

−1−1 4 −1

−1

︸︷︷︸≈−Δ (G)

+

12 −8 2

1 −8 20 −8 12 −8 2

1

︸︷︷︸≈Δ2 (G2=GC−1G)



Spatial models for data

GMRF representations of SPDEs can be constructed forto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.

(κ2 −Δ)(τ x(u)) = W(u), u ∈ Rd





(κ2 −Δ)(τ x(u)) = W(u), u ∈ Ω





(κ2 eiπθ −Δ)(τ x(u)) = W(u), u ∈ Ω





(κ2u +∇ · mu −∇ · Mu∇)(τux(u)) = W(u), u ∈ Ω



Image Reconstruction II

Model with observations, y, and latent field, x,

y|x ∈ N(

Ax,σ2I)

x ∈ N(μ,Q−1

).

and Q = κ2C + G or Q = κ4C + 2κ2G + GC−1G.

Interpolation using a GMRF

E (x|y) = μ+1σ2 Q−1

x|y A⊤ (y − Aμ)

V (x|y) = Q−1x|y =

(Q +

1σ2 A⊤A

)−1



Image Reconstruction


Spatial Statistics Spatial Examples More Environmental Corrupted Pixels NDVI



Global Temperature — Data

January 2003 July 2003



Global Temperature — Reconstruction

Global mean: 15◦C.



Satellite Data — Vegetation

January 1999

July 1999



Satellite Data — Trend in Vegetation

K2 Estimate

−0.4

−0.2

0

0.2

0.4

Independent estimates

K2 Estimate

−0.3

−0.2

−0.1

0

0.1

Correlated estimates



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 theunknown image (and indicator) given observations andparameters.



Image Reconstruction — Corrupted pixels

image reconstruction

bad pixels − estimatebad pixels



Normalized difference vegetation index (NDVI)

NDVI =RNIR − RRED

RNIR + RRED

◮ RRED is the amount of reflected red light (0.58−0.68 μm)◮ RNIR is the amount of reflected near-infrared light

(0.72 − 1.00 μm)Johan Lindstrom - [email protected] Spatial Statistics 27/31


Smoothed version of the NDVI Data

Smooth the data to fill in missing values and remove noisedue to cloud cover, etc.

1986 1988 1990

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Important ecological questions:◮ Plant phenology (start and end of season)◮ Plant productivity (integral)



Smoothing of Satellite Based Vegetation Measurements

NDVI May 1993 Smooth

1991 1992 1993 1994 1995 1996140

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Smoothing of Satellite Based Vegetation Measurements

1991 1992 1993 1994 1995 1996 1997

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ObsGaussianGaussian 1DNig 1D



Learn more!

What?Spatial statistics with image analysis, FMSN20

When?HT2-2015, October–December

Where?Information and Matlab files will be available atwww.maths.lth.se/matstat/kurser/fmsn20masm25/

Who?Lecturer: Johan Lindstrom

MH:319


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