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**Spatial statistics in practice**

Lecture #3: Modeling spatial autocorrelation in normal, binomial/logistic, and Poisson variables: autoregressive and spatial filter specifications Spatial statistics in practice Center for Tropical Ecology and Biodiversity, Tunghai University & Fushan Botanical Garden

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**Topics for today’s lecture**

Autoregressive specifications and normal curve theory (PROC NLIN). Auto-binomial and auto-Poisson models: the need for MCMC. Relationships between spatial autoregressive and geostatistical models Spatial filtering specifications and linear and generalized linear models (PROC GENMOD). Autoregressive specifications and linear mixed models (PROC MIXED). Implications for space-time datasets (PROC NLMIXED)

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What is an auto- model? Y is on both sides of the = sign

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**The auto-normal (auto-Gaussian) model**

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**Popular autoregressive equations for the normal probability model**

2nd-order models 1st-order model M is diagonal, and often is I A normality assumption usually is added to the error term.

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**The simultaneous autoregressive (SAR) model**

spatial autoregression The workhorse of classical statistics is linear regression; the workhorse of spatial statistics is nonlinear regression. The simultaneous autoregressive (SAR) model where denotes the spatial autocorrelation parameter

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**Georeference data preparation**

Concern #1: the normalizing factor Rule: probabilities must integrate/sum to 1 Both a spatially autocorrelated and unautocorrelated mathematical space must satisfy this rule Jacobian term for Gaussian RVs – a function of the eigenvalues of matrix W (or C) non- symmetric set of eigenvalues symmetric set of eigenvalues

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**Calculation of the Jacobian term**

Step 1 extract the eigenvalues from n-by-n matrix W (or C) - eigenvalues are the n solutions to the equation det(W – I) = 0 - eigenvectors are the n solutions to the equation (W - I)E = 0. Step 2 (from matrix determinant) compute ; J2 is

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**Relative plots (in z scores)**

Minimizing SSE MIN OLS: MIN with Jacobian, which is a weight: worst case scenario 0.9795 Relative plots (in z scores) 1.0542

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**Gaussian approximations allow an evaluation of redundant information**

Houston (n=690) Syracuse (n=208) % redundant information n* population density 61 66 72 15 % male 32 156 18 78 black/white ratio 63 62 57 27 % widowed 52 91 16 84 % with university degree 70 49 43 37 % Chinese 51 93 34 48 effective sample size

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**The auto-binomial/logistic model**

NOTE: a data transformation does not exist that enables binary 0-1 responses to conform closely to a bell-shaped curve

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**Primary sources of overdispersion: binomial extra variation [Var(Y) = np(1-p) , and >1]**

misspecification of the mean function nonlinear relationships & covariate interactions presence of outliers heterogeneity or intra-unit correlation in group data inter-unit spatial autocorrelation choosing an inappropriate probability model to represent the variation in data excessive counts (especially 0s)

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**The auto-binomial/logistic model**

By definition, a percentage/binary response variable is on the left-hand side of the equation, and some spatial lagged version of this response variable also is on the right-hand side of the equation. Unlike the auto-Gaussian model, whose normalizing constant (i.e., its Jacobian term) is numerically tractable, here the normalizing constant is intractable. A specific relationship tends to hold between the logistic model’s intercept and autoregressive parameters.

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**Pseudo-likelihood estimation**

Maximum pseudo-likelihood treats areal unit values as though they are conditionally independent, and is equivalent to maximum likelihood estimation when they are independent. Each areal unit value is regressed on a function of its surrounding areal unit values. Statistical efficiency is lost when dependent values are assumed to be independent.

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**Quasi-likelihood estimation**

Maximum quasi-likelihood treats the variation of Y values as though it is inflated, and estimates of the variance term np(1-p) for the purpose of rescaling when testing hypotheses. This approach is equivalent to maximum likelihood estimation when = 1, and most log-likelihood function asymptotic theory transfers to the results.

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**Preliminary estimation (pseudo- and quasi-likelihood) results: F/P (%)**

model intercept SA seSA dispersion Deviance binomial -1.10 ***** 945.96 auto-binomial -1.11 0.89 0.001 384.51 quasi-auto-binomial 0.015 19.74 0.99 auto-logistic -2.03 0.80 0.032 0.93

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**What is the alternative to pseudo-likelihood?**

MCMC maximum likelihood estimation! exploits the sufficient statistics based upon Markov chain transition matrices converging to an equilibrium exploits marginal probabilities, and hence can begin with pseudo-likelihood results based upon simulation theory

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**Properties of estimators: a review**

Unbiasedness Efficiency Consistency Robustness BLUE BLUP Sufficiency

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**MCMC maximum likelihood estimation**

MCMC denotes Markov chain Monte Carlo Pseudo-likelihood works with the conditional marginal models MCMC is needed to compute the simultaneous likelihood result MCMC exploits the conditional models

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**The theory of Markov chains was developed by Andrei Markov at the beginning of the 20th century.**

A Markov chain is a process consisting of a finite number of states and known probabilities, pij, of moving from state i to state j. Markov chain theory is based on the Ergodicity Thm: irreducible, recurrent non-null, and aperiodic. If a Markov chain is ergodic, then a unique steady state distribution exists, independent of the initial state: for transition matrix M, ; P(Xt+1 = j| X0=i0, …, Xt=it) = P(Xt+1 = j| Xt=it) = tpij

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**Example transition matrix convergence:**

B C D E F 0.15 0.20

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Monte Carlo simulation is named after the city in the Monaco principality, because of a roulette, a simple random number generator. The name and the systematic development of Monte Carlo methods date from about 1944. The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments with a computer using pseudo-random numbers.

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**MCMC has been around for about 50 years.**

MCMC provides a mechanism for taking dependent samples in situations where regular sampling is difficult, if not completely impossible. The standard situation is where the normalizing constant for a joint or a posterior probability distribution is either too difficult to calculate or analytically intractable.

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**What is MCMC? A definition**

MCMC is used to simulate from some distribution p known only up to a constant factor, C: pi = Cqi where qi is known but C is unknown and too horrible to calculate. MCMC begins with conditional (marginal) distributions, and MCMC sampling outputs a sample of parameters drawn from their joint (posterior) distribution.

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**The distribution qi is called the proposal distribution.**

Starting with any Markov chain having transition matrix M over the set of states i on which p is defined, and given Xt = i, the idea is to simulate a random variable X* with distribution qi: qij = P(X* = j| Xt = i). The distribution qi is called the proposal distribution. po = 0.5 After a burn-in set of simulations, a chain converges to an equilibrium p=0.2

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Gibbs sampling is a MCMC scheme for simulation from p where a transition kernel is formed by the full conditional distributions of p. a stochastic process that returns a different result with each execution; a method for generating a joint empirical distribution of several variables from a set of modelled conditional distributions for each variable when the structure of data is too complex to implement mathematical formulae or directly simulate. a recipe for producing a Markov chain that yields simulated data that have the correct unconditional model properties, given the conditional distributions of those variables under study. its principal idea is to convert a multivariate problem into a sequence of univariate problems, which then are iteratively solved to produce a Markov chain.

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**A Gibbs sampling algorithm**

(1) t = 0; set initial values 0x = (0x1, …, 0xn)’ (2) obtain new values tx = (tx1, …, txn)’ from t-1x: tx1 ~ p(x1|{t-1x2, …, t-1xn) tx2 ~ p(x2|{tx1, t-1x3, …, t-1xn) … txn ~ p(x1|{tx1, …, txn-1) (3) t = t+1; repeat step (2) until convergence.

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**Monitoring convergence**

MCMC exploits the sufficient statistics, which should be monitored with a time-series plot for randomness. After removing burn-in iteration results, a chain should be weeded (i.e., only every kth output is retained). These weeded values should be independent; this property can be checked by constructing a correlogram. Convergence of m chains can be assessed using ANOVA: within-chain variance pooling is legitimate when chains have converged.

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**Sufficient statistics for normal, binomial, and Poisson models**

A sufficient statistic (established with the Rao-Blackwell factorization theorem) is a statistic that captures all of the information contained in a sample that is relevant to the estimation of a population parameter.

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**Implementation of MCMC for the autologistic model**

drawings from the binomial distribution is the Monte Carlo part Y1 Y2 … Y20 Y21 Y22 Y40 . Y381 Y382 Y400 MCMC-MLEs are extracted from the generated chains

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**MCMC results alpha rho df F prob iter-ation 44 1.0 0.52 0.47 chain 2**

25, ,000/100 burn-in + weeded alpha rho df F prob iter-ation 44 1.0 0.52 0.47 chain 2 0.1 0.91 0.92 inter-action 88 0.56 0.54 error 6615

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**Some prediction comparisons**

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**The (modified) auto-Poisson model**

NOTE: the auto-Poisson model can only capture negative spatial autocorrelation NOTE: excessive zeroes is a serious problem with empirical Poisson RVs

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Spatial autoregression: the auto-Poisson model The workhorse of spatial statistical generalized linear models is MCMC For counts, y, in the set of integers {0, 1, 2, 3, … }

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**c-1 is an intractable normalizing factor**

MCMC is initiated with pseudo-likelihood estimates c-1 is an intractable normalizing factor positive spatial autocorrelation can be handled with Winsorizing, or binomial approximation

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**When VAR(Y) > overdispersion (extra Poisson variation) is encountered**

Detected when deviance/df > 1 Often described as VAR(Y) = Leads to the Negative Binomial model Conceptualized as the number of times some phenomenon occurs before a fixed number of times (r) that it does not occur.

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**Preliminary estimation (pseudo- and quasi-likelihood) results: B/D**

model SA seSA dispersion Deviance Poisson ***** auto-Poisson 0.02 <0.001 822.23 quasi-auto-Poisson 0.006 0.96 auto-negative binomial 0.007 0.0626 1.01

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MCMC results 25, ,000/100 burn-in + weeded typical correlogram

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**Some prediction comparisons**

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**Geographic covariation: n-by-n matrix V**

autoregression works with the inverse covariance matrix & geostatistics works with the covariance matrix itself

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**Relationships between the range parameter and rho for an ideal infinite surface**

modified Bessel function for CAR Bessel function for SAR

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**YT(I – 11T/n)C (I – 11T/n)Y/ YT(I – 11T/n)Y**

Constructing eigenfunctions for filtering spatial autocorrelation out of georeferenced variables: MC = (n/1T C1)x YT(I – 11T/n)C (I – 11T/n)Y/ YT(I – 11T/n)Y the eigenfunctions come from (I – 11T/n)C (I – 11T/n)

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**C versus (I – 11T/n)C(I – 11T/n) = MCM**

2.06 2.07 * 0.00 -1.10 -1.09 -1.98 5.51 1.92 -0.10 -1.21 -2.02 5.19 5.21 1.78 1.79 -0.13 -0.12 -1.24 -1.23 -2.08 4.91 4.99 1.57 1.59 -0.15 -1.33 -2.12 4.35 1.32 1.35 -0.29 -0.28 -1.38 -2.15 4.01 4.06 1.23 1.26 -0.33 -1.44 -2.23 3.96 1.05 1.06 -0.49 -0.46 -1.54 -1.52 -2.24 3.84 3.88 0.93 0.94 -0.53 -1.56 -1.55 -2.33 -2.32 3.42 3.43 0.80 -0.59 -1.60 -1.59 -2.40 -2.39 3.35 0.78 0.79 -0.63 -1.64 -2.41 2.90 2.91 0.58 0.61 -0.80 -1.74 -2.43 -2.42 2.65 2.72 0.38 -0.89 -0.88 -1.84 -2.54 2.53 2.59 0.27 -0.92 -1.87 -2.62 2.35 2.40 0.17 0.19 -0.95 -1.90 -2.67 2.20 2.24 0.12 -1.07 -1.06 -1.96 -2.70

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Eigenvectors of MCM (I – 11T/n) = M ensures that the eigenvector means are 0 symmetry ensures that the eigenvectors are orthogonal M ensures that the eigenvectors are uncorrelated replacing the 1st eigenvalue with 0 inserts the intercept vector 1 into the set of eigenvectors thus, the eigenvectors represent all possible distinct (i.e., orthogonal and uncorrelated) spatial autocorrelation map patterns for a given surface partitioning Legendre and his colleagues are developing analogous eigenfunction spatial filters based upon the truncated distance matrix used in geostatistics

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**Expectations for the Moran Coefficient for linear regression with normal residuals**

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**residual spatial autocorrelation =**

A spatial filtering counterpart to the auto-normal model specification. Y = Ekß + ε b = EkTY Only a single regression is needed to implement the stepwise procedure. MAX: R2; eigenvectors selected in order of their bivariate correlations residual spatial autocorrelation =

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**Selected demographic attributes of China**

# common to MAX-R2, MIN-MC # not truly redundant info (~MAX-R2, MIN-MC) # spatially structured (MAX-R2, ~MIN-MC) population density (|zres| = 7.5 → 6.3) 149 151 71 crude fertility rate (|zres| = 4.4 → 2.7) 229 105 % 100+ years old (|zres| = 0.4 → 0.0) 145 8 20 births/deaths ratio (|zres| = 2.7 → 0.6) 233 119

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**Overdispersion: binomial extra variation**

E(Y) = np and Var(Y) = np(1-p) , and >1 tends to have little impact on regression parameter point estimates (maximum likelihood estimator typically is consistent, although small sample bias might occur); but, regression parameter standard error estimates (variances/covariances) are underestimated may be reflected in the size of the deviance statistic difficult to detect in binary 0-1 data

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**Spatial structure and generalized linear modeling: “Poisson” regression**

CBR: the spatial filter is constructed with 199 of 561 candidate eigenvectors. SF results in green Poisson Negative binomial SF negative binomial deviance 1.02 1.10 mean 0.1241 0.1351 0.1308 dispersion 0.0933 0.0302 Pseudo-R2 (observed vs predicted births) 0.762 0.903 SF

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**Spatial structure and generalized linear modeling: “binomial” regression**

% population 100+ years old: the spatial filter is constructed with 92 of 561 candidate eigenvectors. binomial SF binomial deviance 4.76 1.00 Intercept (0.0124) (0.0276) scale 1 1.47 Pseudo-R2 (observed vs predicted births) 0.283 SF

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**Advantages of spatial filtering**

Do not need MCMC for GLM parameter estimation – conventional statistical theory applies Uncover distinct map pattern components of spatial autocorrelation that relate directly to the MC The eigenvectors are orthogonal and uncorrelated Can always calculate the necessary eigenvectors as long as the number of areal units does not exceed n ≈ 10,000

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**Interpretation of MIN-MC selections**

Matrix Ek contains three disjoint eigenvector subsets: Er, for those representing redundant locational information; Es, for those representing spatially structured random effects; and, Emisc, for those being unrelated to Y. Accordingly, the pure spatial autocorrelation model becomes Y = µ1 + Erßr + (Esßs + e) , where ßr and ßs respectively are regression coefficients defining relationships between Y and the sets of eigenvectors Er and Es, and the term (Esßs + e) behaves like a spatially structured random effect.

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Random effects model is a random observation effect (differences among individual observational units) is a time-varying residual error (links to change over time) The composite error term is the sum of the two.

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**Random effects model: normally distributed intercept term**

~ N(0, ) and uncorrelated with covariates supports inference beyond the nonrandom sample analyzed simplest is where intercept is allowed to vary across areal units (repeated observations are individual time series) The random effect variable is integrated out (with numerical methods) of the likelihood fcn accounts for missing variables & within unit correlation (commonality across time periods)

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**Random effects: mixed models**

Moving closer to a Bayesian perspective, spatial autocorrelation can be accounted for by introducing a (spatially structured) random effect into a model specification. SAS PROC MIXED supports this approach for linear modeling in which a map is treated as a multivariate sample of size 1. SAS PROC NLMIXED supports this approach for generalized linear modeling.

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**SAS PROC MIXED and random effects: Y=XB + Zu**

The spatially correlated errors model is performed with PROC MIXED through the REPEATED statement. The SUBJECT=INTERCEPT option specifies that the correlation between units is essentially between experimental units that are different observations within the data set. The LOCAL option in the REPEATED statement tells PROC MIXED to include a nugget effect. EXAMPLE: density of workers across Germany’s 439 Kreises LN(density – 23.53) ~ N

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**A spatial covariance structure coupled with a random slope coefficient model**

192,721 distance pairs dmax =

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**PROC MIXED output: intercept**

estimate corre-lation -2log(L) nugget (partial) sill range 5.28 (0.06) none 1445.1 1.5782 5.01 (0.12) spherical 1348.4 0.9139 0.5542 1.3801 (0.18) exponential 1349.8 0.9154 0.5873 0.7824 (0.13) Gaussian 1344.7 0.9858 0.5194 0.7260 power 0.2786

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**Spherical semivariogram**

Random intercept term The spatial filter contains 27 (of 98) eigenvectors, with R2 = , P(S-Wresiduals) < measure No covariates Spatial filter Spherical semivariogram -2log(L) 1445.1 1179.3 1348.4 Intercept variance 0.9631 0.2538 0.5542 Residual variance 0.6116 0.6011 0.9139 Intercept estimate 5.2827 (0.0599) (0.0443) 5.0142 (0.1210)

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**Generalized linear mixed models**

One drawback of spatial filtering is that as the number of areal units increases, the number of eigenvectors needed to construct a spatial filter tends to increase, resulting in asymptotics being difficult or impossible to achieve. This situation can be remedied by resorting to a space-time data set, with time being repeated measures whose correlation can be captured by a random effects intercept term.

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**Unemployment in Germany: 1996-2002**

year year-specific eigenvectors common eigenvectors global regional local 1996 E9, E16, E21, E25, E41, E52, E53, E64 E89 E2 - E5 E6 - E8, E11, E18, E24, E28, E30, E39, E60 E74 1997 E1 E15, E19, E21, E34, E38, E64 E93 1998 E13, E15, E16, E19, E21, E34, E38, E42, E52, E66 E68, E93 1999 E9, E13, E15, E16, E19, E21, E34, E38, E42, E52, E66 2000 E9, E13, E15, E16, E19, E21, E25, E34, E38, E42, E51, E52, E66 E93, E97 2001 E9, E12, E13, E15, E16, E19, E34, E42, E52, E56, E65, E66 E68, E93, E97 2002 E9, E12, E13, E15, E16, E19, E20, E25, E38, E42, E52, E65, E66

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**Unemployment in Germany: annual spatial filters**

year # of eigenvecvtors scale adjusted pseudo-R2 1996 24 21.98 0.5929 1.0232 1997 23 24.38 0.6425 1.0412 1998 27 23.52 0.6846 1.0438 1999 23.25 0.7068 1.0364 2000 30 23.83 0.7483 1.0507 2001 25.18 0.7683 1.0489 2002 29 26.08 0.7549 1.0459

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**The composite spatial filter constructed with common vectors**

Dark red: very high Light red: high Gray: medium Light green: low Dark green: very low former east-west divide year SF residuals MC GR 1996 0.67 → 0.21 0.62 1997 0.73 → 0.20 0.66 1998 0.76 → 0.20 0.64 1999 0.79 → 0.21 0.61 2000 0.83 → 0.25 0.59 2001 0.85 → 0.27 0.57 2002 0.56 SF 1.14 0.15

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**Generated space-time predictions**

the lack of serial correlation information in 1996 is conspicuous the best fit is in the center of the space-time series

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**% urban in Puerto Rico: SF-logistic with a spatial structured random effect**

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What have we learned today ?

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