Intro Statistical Model Results Fast Estimation of Spatially Dependent Temporal Trends using Gaussian Markov Random Fields David Bolin 1 , Johan Lindstr¨ om 1 , Lars Eklundh 2 , Finn Lindgren 1 1 Centre for Mathematical Sciences, Lund University 2 GeoBiosphere Science Centre, Lund University Boulder September 17, 2008 David Bolin - [email protected]Fast Estimation of Spatially Dependent Temporal Trends
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Intro Statistical Model Results
Fast Estimation of Spatially Dependent TemporalTrends using Gaussian Markov Random Fields
David Bolin1, Johan Lindstrom1, Lars Eklundh2, Finn Lindgren1
1 Centre for Mathematical Sciences, Lund University2 GeoBiosphere Science Centre, Lund University
Boulder September 17, 2008
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results The Sahel NDVI Data
African Sahel
� A semi-arid region directly south of the Sahara desert.
� Rainfall has decreased severely since the 1960’s.
� Starting in the late 1960’s, the area suffered droughts for overtwenty years, culminating with a severe drought and famine in1983-1984.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results The Sahel NDVI Data
African Sahel
� Since the 1983-1984 famine, serveral authors have usedsatellite imagery to study vegetation in the sahel.
� Recently Eklundh and Olsson.1 analysed vegetation trends inthe Sahel region for the period 1982-1999.
� Results indicate a vegetation recovery.
1Eklundh, L. and Olsson, L., Vegetation index trends for the African Sahel1982-1999,Journal of Geophysical Research,30, 2003, 1430–1433
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results The Sahel NDVI Data
Normalised Difference Vegetation Index (NDVI)
� NDVI is calculated using satellite measurements of thereflectance from the Earth’s surface.
NDVI =Rnir − Rred
Rnir + Rred
� Rred is the measured reflectance of red light (0.58 − 0.68μm)� Rnir is the measured reflectance near-infrared light
(0.72 − 1.00μm)
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results The Sahel NDVI Data
Normalised Difference Vegetation Index
� Data taken from the NOAA/NASA Pathfinder AVHRR Land(PAL) data set.
� The PAL data set contains 36 measurements per year.� In order to generate annual data, we use the TIMESAT
processing scheme.
Figure: Example of an NDVI time series and the resulting fit usingTIMESAT.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results The Sahel NDVI Data
Normalised Difference Vegetation Index
Figure: NDVI data for the Sahel 1983
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Does the amount of vegetation increase?
� Model yearly vegetation using linear trends for each pixel:
y(t) = k · t + m + εt
� Test if the slopes, k, are significant.
� Pixels close to each other should have similar trends.
� Several ways of incorporating the spatial context into theanalysis:
� Direct smoothing� Bayesian Hierarchical model
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Our Approach
� Spatial model using a linear trend:
Y(si , t) = K(si ) · t + M(si ) + εit .
� Or more generally using m trends:
Y(si , t) =m∑
j=1
Kj(si ) · fj(t) + εit .
� Let the regression coefficients be spatially dependent.
� Priors for Ki will be chosen via priors for
Xt =m∑
i=1
Ki(si) · fi (t).
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Assumptions
� We assume that each observation, Yt , t ∈ [0,T − 1], isgenerated as
Yt |Xt ,Σεt ∈ N(AtXt ,Σεt ).
� Xt is a latent field with prior π(Xt), Xi ⊥ Xj , i �= j .� Σεt is a measurement noise covariance matrix.� At is an observation matrix determining which pixels that are
measured.
� Σεt is chosen to be diagonal with nt unique elements,σ2
1 , . . . , σ2nt
, representing the noise variances in the nt
observed pixels.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Gaussian Markov Random Fields
� A random variable x = (x1, . . . , xn)� ∈ N(μ,Q−1) is called a
GMRF if the joint distribution for x satisfies
π(xi |{xj : j �= i}) = π(xi |{xj : j ∈ Ni}) ∀ i .
Q = Σ−1 is called the precision matrix. It determines theneighborhood structure of the Markov field.
� Some important properties: if i �= j then
xi ⊥ xj |x−{i ,j} ⇐⇒ Qi ,j = 0 ⇐⇒ j /∈ Ni . (2.1)
� Fast algorithms that utilize the sparsity of Q exist.
� See ”Gaussian Markov Random Fields – Theory andApplications” by Rue & Held for details.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Intrinsic GMRFs
� An intrinsic GMRF is improper, that is, its precision matrix,Q, does not have full rank.
� We will use second order polynomial IGMRFsx ∈ N(μ, (κQX )−1) as priors for Xt .
� Neighborhood structure:
⎡⎢⎢⎢⎢⎣
◦ ◦ • ◦ ◦◦ • • • ◦• • � • •◦ • • • ◦◦ ◦ • ◦ ◦
⎤⎥⎥⎥⎥⎦
� κ determines the strength of the spatial dependencies.
� Invariant to addition of an arbitrary plane p(i , j) = a + bi + cj .
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Regression Model
� To estimate time varying trends in the observations, werestrict Xt to follow these trends:
Xt =m∑
i=1
Ki (si ) · fi(t)
� Given the IGMRF priors for X = [X�0 , . . . ,X�
T−1]�, the
distribution for the parameter fields K = [K�1 , . . . ,K�
m]� isthen obtained as
K|κ ∈ N(0, (κQ)−1)
where Q = (F�F) ⊗ QX , and F = [f1, . . . , fm]
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Regression Model
� The sparsity structure of Q is determined by Qx andorthogonality of the regression basis.
� Example: The sparsity structure of Q for a field with 400nodes and three regression basis vectors for orthogonalregression basis (left) and non-orthogonal regression basis(right) respectively.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Regression Model
� Using flat priors for Σε and κ, the posterior distribution for Kgiven Y = [Y�
0 , . . . ,Y�T−1]
� and the parameters is
K|Y,Σε, κ ∈ N(μK |•,Q−1K |•) , with
μK |• = Q−1K |•A
�Σ−1ε Y,
QK |• = κQ + A�Σ−1ε A.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Parameter Estimation
� The model depends on the parameters κ and σ21 , . . . , σ
2n, that
have to be estimated from data.
� The parameters can, potentially, be estimated using anMCMC based approach, but the large data-set makes thiscomputationally infeasible.
� A better alternative is to use the EM algorithm, which allowsfor much faster ML parameter estimates.
David Bolin - [email protected] Fast Estimation of Spatially Dependent Temporal Trends
Intro Statistical Model Results Assumptions GMRFs Regression Model Estimation
Parameter Estimation
� Under the assumptions of flat priors for κ and σ21, . . . , σ
2n, the
updating equations becomes:
σ2(i+1)j =
1
nj
nj∑k=1
E((Y − AK)2jk |∗), for 1 ≤ j ≤ n,
κ(i+1) =m(n − 3)
E(K�QK|∗) .
where, nj is the number of observations of pixel j , and jk isthe kth observation at pixel j .