Lecture #3:Lecture #3: Modeling spatial Modeling spatial autocorrelation in autocorrelation in

normal, normal, binomial/logistic, and binomial/logistic, and

Poisson variables: Poisson variables: autoregressive and autoregressive and

spatial filter spatial filter specificationsspecifications

Spatial statistics in Spatial statistics in practicepractice

Center for Tropical Ecology and Center for Tropical Ecology and Biodiversity, Tunghai University & Fushan Biodiversity, Tunghai University & Fushan

Botanical GardenBotanical Garden

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)

What is an auto- model?

Y is on both sidesof the = sign

The auto-normal (auto-Gaussian)

model

Popular autoregressive equations for the normal probability model

MCIDDεXβDYY

ερW)Xβ(IWYY

εXβWYY

εY

)ρ( , :

ρ:

ρ:

)μ,(:

T

CAR

SAR

AR

fationautocorrelpure

A normality assumption usually is added to the error term.

M is diagonal,and often is I

2nd-order

models

1st-order

model

spatial autoregressionThe workhorse of classical statistics is linear regression; the workhorse of

spatial statistics is nonlinear regression.

The simultaneous autoregressive (SAR) model

εXβWIWYY )ρ(ρwhere denotes the spatial autocorrelation parameter ρ

JJJ ee])ρ(ρ[e εXβWIWYY

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)

symmetricset of

eigenvalues

non-symmetric

set ofeigenvalues

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

n

1ii )ρλLN(1

n

1

n

1ii )ρλLN(1

n

2

λ

λ

Minimizing SSE

0.9795

1.0542

MIN OLS: 1.1486MIN with Jacobian,

which is a weight: 1.8959

Relative plots (in z scores)

worst case scenario

Gaussian approximations allow an evaluation of redundant information

Houston (n=690) Syracuse (n=208)

% redundant information

n* % redundant information

n*

population density 61 66 72 15% male 32 156 18 78black/white ratio 63 62 57 27% widowed 52 91 16 84% with university degree

70 49 43 37

% Chinese 51 93 34 48

effectivesample size

The auto-binomial/logist

ic model

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

to a bell-shaped curve

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)

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.

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.

]e[1ep

n

1jjiji

n

1jjiji ycρβXαycρβXα

i

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.

0.41Rpseudo 0.61;Rpseudo

Ic ;)F/P/P(Fc

2I

2pct

n

1jj25%,ij%25i

n

1jjjiji

25%

ICYC

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

model intercept SA seSA dispersion Deviance

binomial -1.10 0 ***** ***** 945.96

auto-binomial -1.11 0.89 0.001 0 384.51

quasi-auto-binomial -1.10 0.89 0.015 19.74 0.99

auto-logistic -2.03 0.80 0.032 ***** 0.93

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

Properties of estimators: a review

• Unbiasedness• Efficiency• Consistency• Robustness• BLUE• BLUP

• SufficiencySufficiency

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

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 ThmErgodicity 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

* lim k

kMM

Example transition matrix convergence:

3

1

3

10

3

100

4

1

4

1

4

10

4

10

03

1

3

100

3

13

100

3

1

3

10

04

10

4

1

4

1

4

1

003

10

3

1

3

134

15.020.015.015.020.015.0

15.020.015.015.020.015.0

15.020.015.015.020.015.0

15.020.015.015.020.015.0

15.020.015.015.020.015.0

15.020.015.015.020.015.0

A B C

D E F

0.15 0.20 0.15

0.15 0.20 0.15

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.

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.

MCMC has been around for about 50 years.

What is MCMCMCMC? 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.

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.

After a burn-inset of simulations,

a chain converges

to an equilibrium

po = 0.5

p=0.2

• 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.

Gibbs sampling is a MCMCMCMC scheme for simulation from p where a transition kernel is

formed by the full conditional distributions of p.

(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.

A Gibbs sampling algorithm

Monitoring convergenceMCMC 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.

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.

1yn

1ii

p..., 2, 1, j , xyn

1iiji

Implementation of MCMC for the autologistic model

Y1 Y2 … Y20

Y21 Y22 … Y40

.

.

.

.

.

.

.

.

.

.

.

.

Y381 Y382 … Y400

1- τand τof mixture a is * τwhere,

e1

e p

)p1,binomial(n ~Y

/n)y(or p :0τ

n

1j*τj,ij

n

1j*τj,ij

ycρ̂α̂

ycρ̂α̂

τi,

τi,τi,

n

1ii2

1i

drawingsfrom the

binomial distributionis the Monte Carlo

part

partchain Markov theis )1(

n

1i

n

1jτj,ijτi,2

1τ2,

n

1iτi,τ1,

ycyT

1yT :statistics sufficient

MCMC-MLEs are extracted from the generated chains

MCMC results 25,000 + 225,000/100burn-in + weeded

alpha rho

df F prob F prob

iter-ation

44 1.0 0.52 1.0 0.47

chain 2 0.1 0.91 0.1 0.92

inter-action

88 1.0 0.56 1.0 0.54

error 6615

Some prediction comparisons

The (modified) auto-Poisson

model

NOTE: the auto-Poisson model can only capture negative spatial

autocorrelationNOTE: excessive zeroes is a

serious problem with empirical Poisson RVs

Spatial autoregression: the auto-Poisson model

The workhorse of spatial statistical generalized linear models is MCMCMCMC

For counts, y, in the setof integers {0, 1, 2, 3, … }

μVAR(Y)

μE(Y)

y!μe)P(Y

yμ

y

XβCYμ ρ)(LN

!y)ycρβX(αeP i

yn

1jjiji

)ycρβX(α-1

i

i

n

1jjiji

c

c-1 is an intractable normalizing factor

ationautocorrel spatial negative :0ρbut

MCMC is initiated with pseudo-likelihood estimates

positive spatial autocorrelation can be handled with Winsorizing, or binomial approximation

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

μ

ημ)μ(1

Conceptualizedas the number of times some phenomenon occurs before a fixed number oftimes (r) that itdoes not occur. 2

ry

p)/p-r(1VAR(Y)

p)/p-r(1E(Y)

p)(1p1r

1ryC)P(Y

y

i

)μ̂LN(

ρ̂n

1j

c/dbi

2ratio

2counts

n

1jjjiji

DeeB̂

0.17Rpseudo 0.86;Rpseudo

)/d(bc

ijjj

YC

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

model SA seSA dispersion Deviance

Poisson 0 ***** ***** 1230.20

auto-Poisson 0.02 <0.001 0 822.23

quasi-auto-Poisson 0.02 0.006 29.2553 0.96

auto-negative binomial 0.02 0.007 0.0626 1.01

MCMC results

typical correlogram

25,000 + 500,000/100burn-in + weeded

Some prediction comparisons

Geographic covariation:n-by-n matrix V

IV

εV1Y

then ation,autocorrel spatial no if

,μ 1/2

RVV

CSIWIV

k )()( :icsgeostatist

)ρ - ( )ρ - ( :SAR1/2-T1/2-

-12/1

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

Relationships between the range parameter and rho for an ideal

infinite surface

modifiedBessel function

for CAR

0.00003RESS

e10.2298

rB10.25

ρ̂

1.8403 r 7229.62.1825

1

CAR

Bessel functionfor SAR

0.00041RESS

e10.2896

rB10.25

ρ̂

0.8879 r 6842.02.0130

1

SAR

b

ij

r

d

1

ij

r

d

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)

C versus (I – 11T/n)C(I – 11T/n) = MCMC MCM 2.06 2.07 * 0.00 -1.10 -1.09 -1.98 -1.98

5.51 * 1.92 1.92 -0.10 -0.10 -1.21 -1.21 -2.02 -2.02

5.19 5.21 1.78 1.79 -0.13 -0.12 -1.24 -1.23 -2.08 -2.08

4.91 4.99 1.57 1.59 -0.15 -0.15 -1.33 -1.33 -2.12 -2.12

4.35 4.35 1.32 1.35 -0.29 -0.28 -1.38 -1.38 -2.15 -2.15

4.01 4.06 1.23 1.26 -0.33 -0.33 -1.44 -1.44 -2.23 -2.23

3.96 3.96 1.05 1.06 -0.49 -0.46 -1.54 -1.52 -2.24 -2.24

3.84 3.88 0.93 0.94 -0.53 -0.53 -1.56 -1.55 -2.33 -2.32

3.42 3.43 0.80 0.80 -0.59 -0.59 -1.60 -1.59 -2.40 -2.39

3.35 3.35 0.78 0.79 -0.63 -0.63 -1.64 -1.64 -2.41 -2.41

2.90 2.91 0.58 0.61 -0.80 -0.80 -1.74 -1.74 -2.43 -2.42

2.65 2.72 0.38 0.38 -0.89 -0.88 -1.84 -1.84 -2.54 -2.54

2.53 2.59 0.27 0.27 -0.92 -0.92 -1.87 -1.87 -2.62 -2.62

2.35 2.40 0.17 0.19 -0.95 -0.95 -1.90 -1.90 -2.67 -2.67

2.20 2.24 0.12 0.12 -1.07 -1.06 -1.96 -1.96 -2.70 -2.70

C

MCM

1.005λ0.011

λ̂

C1,λ

1ρ

C11MCM

T1,λn

MC

MCλn

T

n, C11MCM

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

Expectations for the Moran Coefficient for linear

regression with normal residuals

p1n

])TR[(nE(MC)

T1T

T

CXXXX

C11

C11TMC

2σ

A spatial filtering counterpart to the auto-normal model specification.•Y = Ekß + ε•b = Ek

TY• Only a single regression is needed to implement the stepwise procedure.

MAX: R2; eigenvectors selected in order of their bivariate correlations

residual spatial autocorrelation =

Selected demographic attributes of China

attribute

# 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 0

% 100+ years old (|zres| = 0.4 → 0.0)

145 8 20

births/deaths ratio (|zres| = 2.7 → 0.6)

233 119 0

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

Spatial structure and generalized linear modeling: “Poisson” regression

CBR: the spatial filter is constructed with 199 of 561 candidate eigenvectors.

Poisson Negative binomial

SF negative binomial

deviance 1377.31 1.02 1.10

mean 0.1241 0.1351 0.1308

dispersion 0 0.0933 0.0302

Pseudo-R2

(observed vs predicted births)

0.762 0.762 0.903

SF resultsin green

SF

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 binomialdeviance 4.76 1.00Intercept -12.0706

(0.0124)

-12.5000

(0.0276)scale 1 1.47Pseudo-R2

(observed vs predicted births)

0 0.283

SF

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

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.

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.

) , f( εξXβY ξ

ε

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)

2σξ

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.

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

A spatial covariance structure coupled with a random slope coefficient model

192,721 distance pairsdmax = 9.32478

PROC MIXED output: intercept

intercept

estimate

corre-lation

-2log(L) nugget (partial)

sill

range

5.28

(0.06)

none 1445.1 0 1.5782 0

5.01

(0.12)

spherical 1348.4 0.9139 0.5542 1.3801

5.01

(0.18)

exponential 1349.8 0.9154 0.5873 0.7824

5.01

(0.13)

Gaussian 1344.7 0.9858 0.5194 0.7260

5.01

(0.18)

power 1349.8 0.9154 0.5873 0.2786

Random intercept term

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)

5.2827

(0.0443)

5.0142

(0.1210)

The spatial filter contains 27 (of 98) eigenvectors, with R2 = 0.4542, P(S-Wresiduals) < 0.0001.

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.

Unemployment in Germany: 1996-2002

year year-specific eigenvectors common eigenvectors

global regional local 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

E93

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 E1 E9, E12, E13, E15, E16, E19, E20, E25, E38, E42, E52, E65,

E66

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 27 23.25 0.7068 1.0364

2000 30 23.83 0.7483 1.0507

2001 30 25.18 0.7683 1.0489

2002 29 26.08 0.7549 1.0459

SSPE/SSE

The composite spatial filter constructed with common vectors

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 0.85 → 0.27 0.56

SF 1.14 0.15

Dark red: very highLight red:

highGray:

mediumLight green:

lowDark green: very low

former east-westdivide

Generated space-time predictions

the lack of serialcorrelation informationin 1996 is conspicuous

the best fit is inthe center of the

space-time series

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