Fast Estimation of Spatially Dependent Temporal Trends ... · Intro Statistical Model Results Fast Estimation of Spatially Dependent Temporal Trends using Gaussian Markov Random Fields

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.



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



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)



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.



Normalised Difference Vegetation Index

Figure: NDVI data for the Sahel 1983


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



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).



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.



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.



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 .



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]



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.



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.



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.




� 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 .

� Under the assumption of Gaussian measurements:

E((Y − AK)2jk |∗) = (Yjk − A(jk ,•)μK |•)2 + A(jk ,•)Q−1K |•A

�(jk ,•),

E(K�QK|∗) = tr(QQ−1K |•) + μ�

K |•QμK |•.

where A(jk ,•) is row jk of A.




� Utilizing the sparsity of Q, these calculations arecomputationally feasible even for large fields:

1. A(jk ,•)Q−1K |Y A�

(jk ,•) = tr(A�(jk ,•)A(jk ,•)Q

−1K |Y ).

2. Q and A�(jk ,•)A(jk ,•) are at least as sparse as QK |Y , thus to

calculate the traces we will at most need the elements of Q−1K |Y

that corresponds to neighboring points in the GMRF.3. Given the Cholesky factor, these elements can be calculated

without calculating the entire inverse.



Testing for Significant Trends

� For OLS regression, the (pointwise) significant trends can beobtained by standard hypothesis testing, we use a similarapproach here.

� Given the ML parameter estimates, we have the conditionalposterior K|Y,Σε, κ ∈ N(μK |•,Q−1

K |•)� Let k j

i be the estimated coefficient for trend f j at pixel xi , and

let σji denote the corresponding standard deviation given by

the square root of the relevant diagonal element in Q−1K |Y .

� reject the null hypothesis H0 : k ji = 0 against H1 : k j

i �= 0, at a95% confidence level if∣∣∣∣∣

k ji

σji

∣∣∣∣∣ > t0.025(f ),

where t0.025(f ) is the 2.5%-quantile of the Student’st-distribution with f degrees of freedom.



Testing for Significant Trends

� To perform the test, f has to be determined.

� It is resonable to assume that f is dependent on κ. It is,however, not easy to determine an analytic expression for f .

� We use a simulation based algorithm for estimating f .

� Simulation studies indicate that this test gives correctsignificance levels.


Intro Statistical Model Results Simulated Data Sahel Data

Simulated Data

� To compare this model with simple independent OLSregression for each pixel, we create simulated data as

Yt = K1 + tK2 + ε, t = 0 . . . , 16, ε ∈ N(0,Σεt )

with pixel variances drawn independently from U(0, α).

� For spatially dependent data, we generate K1 and K2 fromK|κ ∈ N(0, (κ(F�F) ⊗QX )−1) with κ = 0.5.

� For random data, K1 and K2 is generated by independentlydrawing values from N(0, 1).

� We use images of size 40 × 40 pixels in four different cases:

D1 Spatially dependent data with α = 1.D10 Spatially dependent data with α = 10.R1 Random data with α = 1.

R10 Random data with α = 10.



� Example: simulated data with κ = 0.5 and α = 10.



K ε1 K ε

2 Σε Cov (%) Sig (%)

D1 OLS 6.84 1.39 8.40 95.01 94.73GMRF 3.63 0.74 8.09 94.87 97.12

D10 OLS 21.66 4.42 84.43 95.06 84.40GMRF 6.27 1.28 79.86 94.85 95.41

R1 OLS 6.83 1.40 8.38 94.96 94.43GMRF 6.81 1.43 11.92 94.88 94.42

R10 OLS 21.63 4.45 83.75 94.90 82.78GMRF 21.02 4.57 84.57 94.44 82.58

Table: Results for simulated data. Each value is the average over 100different runs. Here, K ε

1 , K ε2 , and Σε show the difference, measured in

the Frobenius norm, of the true and estimated K1- , K2- and Σε-fields,respectively. Cov is the estimated coverage percentage for the K2

confidence intervals, which should be close to the nominal 95%. Sig isthe percentage of correctly labeled significant trends.



Sahel Data

� Slope (upper) and Intercept (lower) estimates.



Results

� Significance estimates for the slope of the linear trends



Extensions and Further work

� Allow spatial dependencies in the measurement noise.

� Allow for varying strength in the spatial dependencies.

� Introduce non-Gaussian observations to handle heavy tailedresiduals.

� Correct multiple hypothesis testing.



Figure: Contour plot of the estimated slope field.


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Fast Estimation of Spatially Dependent Temporal Trends ... · Intro Statistical Model Results Fast Estimation of Spatially Dependent Temporal Trends using Gaussian Markov Random Fields