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Spatial statistics in practice Center for Tropical Ecology and Biodiversity, Tunghai University & Fushan Botanical G

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### Pseudo-likelihood estimation

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

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

### What is MCMC? A definition

### Monitoring convergence

### Spatial autoregression: the auto-Poisson modelThe workhorse of spatial statistical generalized linear models is MCMC

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

### Interpretation of MIN-MC selections

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

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)

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.

spatial autoregressionThe 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

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

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

Minimizing SSE

MINOLS: 1.1486

MIN with Jacobian,

which is a weight: 1.8959

worst case scenario

0.9795

Relative plots (in z scores)

1.0542

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

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.

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.

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.

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

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
- Sufficiency

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

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

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 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.

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.

po = 0.5

After a burn-in

set of simulations,

a chain converges

to an equilibrium

p=0.2

Gibbs sampling is a MCMC scheme for simulation from pwhere 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.

(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

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.

MCMC exploits the sufficient statistics, which should be monitored with a time-series plot for randomness.

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.

Implementation of MCMC for the autologistic model

drawings

from the

binomial distribution

is the Monte Carlo

part

MCMC-MLEs are extracted from the generated chains

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

For counts, y, in the set

of integers {0, 1, 2, 3, … }

MCMC is initiated with pseudo-likelihood estimates

c-1 is an intractable normalizing factorpositive spatial autocorrelation can be handled with Winsorizing, or binomial approximation

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.

Geographic covariation:n-by-n matrix V

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

modified

Bessel function

for CAR

Bessel function

for SAR

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)

Eigenvectors of MCM

- (I – 11T/n) = M ensures that theeigenvector 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

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 =

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.

SF results

in 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.

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

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 Erand 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.

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)

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 pairs

dmax = 9.32478

Random intercept term

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.

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

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

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

learned today ?

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