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Subject:

Responses to space-time correlation in WinBUGS

From:

ERIK E OSNAS <[log in to unmask]>

Reply-To:

ERIK E OSNAS <[log in to unmask]>

Date:

Tue, 1 Aug 2006 10:37:52 -0500

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (597 lines)

Bugs Group,

Thanks for all the responses to my question about using the car.normal 
function to have a random area effect that is simultaneously 
correlated in space and time.  I will summarize the responses below.  

I have been working on this for some time and couldn’t quite figure it 
out.  Many people pointed out that I should look at the book “Disease 
Mapping with WinBUGS and MLwiN.”  I have looked at this book but not 
for a long time.  Many people pointed out the model of Bernardinelli 
(1995) and of Waller (1997) discussed in this book (starting on page 
128).  Bernardinelli’s model is a “random slope” type of model where 
the time trend is linear but area-specific with some spatial 
correlation determined by the inputs to car.normal.  Waller’s model is 
a nested type model that has different area-specific spatially 
correlated effects for each separate year in the time series, but no 
temporal correlation.  This allows more freedom for the time trend as 
it does not force a linear trend with time (pointed out to me by 
Rodney Sparapani).  Anyhow, it turns out that I had been fitting a 
model similar to Waller’s model but realized the correlation across 
time was important.  It also turns out
 that I had fit a “random slope” type model like Bernardinelli long 
ago but had big stability problems with the random effect variance and 
was getting very unrealistic (I thought) maps.  Hence, my plea for 
help for the Bugs group.
	I had been trying to figure out an indexing scheme with the 
adjacency matrix input to the car.normal but couldn't quite get my 
head around it.  Rodney Sparapani pointed out the trick that made 
solving this problem quite easy (see below).  I ended up writing an R 
function that takes the spatial vectors num[] and adj[] and the number 
of time points T, then outputs the temp.num[] and temp.adj[] vector 
that refer to the number of neighbors and the index which refers to 
the adjacent neighbors in space-time.  The way that I have it now, any 
given position in space-time is correlated to all the neighbors in the 
3x3x3 space-time cube with equal weight.  This seems like a reasonable 
and simple first step for my situation, but I’m not convinced that 
equal weights for the whole cube are the best a priori expectation.  
I’m dealing with fairly sparse data in a small-scale disease mapping 
problem, so using the whole cube as neighbors (as I see it) helps deal 
with the sparseness of the da
ta.  
	I have included the R code for the function that I describe.  
I’m new to R programming and there are probably more stylish ways to 
do the programming.  I would be interested in people’s comments on the 
program, on its use, and on any problems that are discovered.  I have 
tested this program on small (2x2 and 3x3) spatial maps, which I found 
to work fine.  The function also works well and fast for my space-
time:  3600 locations X 4 years.  
	When I use the space-time correlated random effect to make my 
map in WinBUGS, the iterations go fairly fast and convergence for the 
random-effect variance and fixed effect parameters are good (at least 
as good as the Waller model).  The map produced is also much more 
reasonable given my prior beliefs based on past explorations of the 
data and theoretical expectations.  I should also mention that I am 
not fitting the same Poisson disease mapping models as Bernardinelli 
and Waller.  Instead of the Poisson distribution as the likelihood, I 
am using the Bernoulli, and instead of the log-link function, I am 
using the complimentary-log-log link function.  (Just for those who 
are interested.)  
	I would appreciate any general comments on the use of this 
Space-time correlation effect for disease mapping as I have described 
it.  I know I haven’t been very thorough, but the details will have to 
wait until the manuscript is finished.  Thanks for all the help!

Here is the R function:
######################################################################
spacetime <- function(num,adj,T)
{
	if(sum(num)!=length(adj))
		return('Error: num[] vector does not match adj[] 
vector');

	sumnum <- cumsum(num);
	N <- length(num);
	
	temp_adj <- c();
	temp_num <- c();


	cnter <- 1;
	for(i in 1:T)
	{
		for(j in 1:N)
		{	######Make temp_num vector######
			if(i==1|i==T)
			{
				temp_num[cnter] <- num[j]*2+1;
			}
			else
			{
				temp_num[cnter] <- num[j]*3+2;
			}
			cnter <- cnter+1;

			#####Make temp_adj vector####
			if(i==1)
			{
				if(j==1)from <- 1;
				if(j>1)from <- sumnum[j-1]+1;
				to <- sumnum[j];
				part <- c(adj[from:to],adj[from:to]
+N,j+N);
			}
			else if(i==T)
			{
				if(j==1)from <- 1;
				if(j>1)from <- sumnum[j-1]+1;
				to <- sumnum[j];
				part <- c(adj[from:to]+N*(T-1),
adj[from:to]+N*(T-2),j+N*(T-2));
			}
			else
			{
				if(j==1)from <- 1;
				if(j>1)from <- sumnum[j-1]+1;
				to <- sumnum[j];
				part <- c(adj[from:to]+N*(i-1),
adj[from:to]+N*(i-2),
adj[from:to]+N*i,j+N*(i-2),j+N*i);
			}
			temp_adj <- c(temp_adj,part);
		}
	}
	result <- list(temp_num,temp_adj);
	return(result);
#####################################################################

Here is the original request:

-----Original Message-----
From: (The BUGS software mailing list) [mailto:[log in to unmask]] On 
Behalf
Of ERIK E OSNAS
Sent: Tuesday, July 18, 2006 11:20 AM
To: [log in to unmask]
Subject: space-time correlation in WinBUGS

Bugs Group,

I am interested in using the car.normal function in WinBUGS for 
mapping an emerging infectious disease in a deer population.  I have 
successfully used this function to implement the Besag, York et al. 
model of spatial correlation across adjacent locations.  However, I 
have several years of data and am interested in modeling the dynamics 
across years as well as spatial locations.  I would like to be able to 
implement both space and time correlations where the random-effect for 
any area is correlated to its adjoining spatial neighbors and to its 
temporal neighbors (including itself) both one time step before and 
after.  Is it possible to use car.normal in WinBUGS to do this?  I 
have looked at many examples but have only found examples for temporal 
or spatial correlation but not simultaneous space and time 
correlations.  If this is not possible in WinBUGS, can someone please 
point me to a resource where this could be modeled using MCMC.

Thanks,

Erik


Erik Osnas
Assistant Scientist

Department of Wildlife Ecology
University of Wisconsin
218 Russell Labs
1630 Linden Dr.
Madison, WI 53706

608-262-1984

Here is the email correspondence (chronological order):

From 	"Best, Nicky G" <[log in to unmask]> 

Sent 	Monday, July 24, 2006 7:51 am
To 	ERIK E OSNAS <[log in to unmask]> 

Cc 	 

Bcc 	 

Subject 	RE: space-time correlation in WinBUGS
Attachments 	winmail.dat
5K

Dear Erik
 
We have recently published an article on space-time modelling of two 
related diseases (all done in WinBUGS) which might help you. See 
 
Richardson S; Abellan JJ; Best N. (2006) Bayesian spatio-temporal 
analysis of joint patterns of male and female lung cancer risks in 
Yorkshire (UK) STAT METHODS MED RES. 15: 385-407

 

Nicky

From 	Sujit Ghosh <[log in to unmask]> 

Sent 	Monday, July 24, 2006 10:47 am
To 	ERIK E OSNAS <[log in to unmask]> 

Cc 	Lee Hyeyoung <[log in to unmask]> 

Bcc 	 

Subject 	Re: space-time correlation in WinBUGS
Hi Erik,

We have a paper on using spatial DLM to model spatio-temporal data and 
we have implemented it in winbugs using a reparamerization approach. 
You 
may find more details in our tech report #2587 available at the 
following site: http://www.stat.ncsu.edu/library/mimeo.html

Hope it helps. If you want to collaborate and need hep with winbugs 
programming, let me know.

-SG

From 	[log in to unmask] 

Sent 	Monday, July 24, 2006 10:49 am
To 	ERIK E OSNAS <[log in to unmask]> 

Cc 	 

Bcc 	 

Subject 	Re: [BUGS] space-time correlation in WinBUGS

Hi Erik, 

Hope all's well in the CWD modeling world.  I don't have a good answer 
for you, however I can offer a bit of advice on incorporating space 
and time simultaneously.  I found (Farnsworth et al. 2006, Eco. Apps 16
(3)) that incorporating two random effects in the form of spatial and 
non-spatial heterogeneity was, not surprisingly, sensitive to the 
choice of priors on the precisions and that the choice of grid-cell 
size had a large impact on subsequent inference, which I imagine may 
be the same for your work, unless of course you're only interested in 
a single spatial scale (e.g., management units).  I imagine that 
incorporating temporal correlation into a spatial model will be quite 
data hungry, although for a large enough unit size and given the 
amount of data you're working with you might be able to partition out 
signal from noise. I'm sure you're well aware of all this.   Anyway, 
if you can post to the group with useful replies I, and others I'm 
sure, will appreciate it.  One
 person who comes to mind is Chris Wikle's work on house finch 
invasion across the U.S.  If I remember correctly he used integro-
difference models in a hierarchical framework to look at space time 
invasion dynamics.  Beyond that, you might try looking at Alan 
Gelfand's recent work on modeling plant diversity in South Africa, 
although I can't recall if he considers both space and time 
simultaneously.  You're squarely in an area that is at the cutting 
edge of research, which of course makes your efforts challenging, at 
least by my standards.  I think some folks have used RJMCMC 

cheers, 
matt 

Matthew Farnsworth, Ph.D.
Research Biologist
USDA/APHIS/WS
National Wildlife Research Center
4101 LaPorte Ave.
Fort Collins, CO. 80521
(970) 266-6129
(970) 266-6138 (fax)


From 	Rudy Banerjee <[log in to unmask]> 

Sent 	Monday, July 24, 2006 5:03 pm
To 	[log in to unmask] 

Cc 	 

Bcc 	 

Subject 	Re: [BUGS] space-time correlation in WinBUGS
Hi Eric,

Have you tried the Bernardinelli et al (1995) model. You can find
examples of space time models based on car.normal autocorr (see below).
'Disease Mapping with WinBUGS and MLWin' by Lawson et al. 2003 (pages
128-133) shows some examples. I have implemented space-time models with
my data successfully. I will be happy to share the code... 

Best of luck,

Rudy


model-->

model
{

for (i in 1:m)
{
for (k in 1:T)
{
# Poisson likelihood for observed counts; y is the observed
value of the dependent variable and e is the expected value
y[i,k]~dpois(mu[i,k])

log(mu[i,k])<-log(e[i,k])+alpha+ alpha1*variable1+...(your other
variables)..+
         +u[i]+v[i]+beta*t[k]+delta[i]*t[k] #u is the structured
spatial heterogeneity or spatial autocorr and v is the unstrsuctured or
random effects, beta is the time random component and delta is the 
space
time interaction effect.

# Relative Risk in each area and period of time
theta[i,k]<-exp(alpha 
+ alpha1*variable1[i,k] +...
                 +u[i]+v[i]+beta*t[k]+delta[i]*t[k])
}
TT[i]<-exp(beta+delta[i])
}

# CAR prior distribution for spatial correlated heterogeneity
u[1:m]~car.normal(adj[],weights[],num[],tau.u)
delta[1:m]~car.normal(adj[],weights[],num[],tau.delta)

# Prior distributions for the Uncorrelated Heterogeneity
for(i in 1:m)
{
v[i]~dnorm(0,tau.v)
}

# Weights
for(k in 1:sumNumNeigh)
{
weights[k]<-1
}

alpha~dflat()
alpha1~dflat()
.....

# Hyperprior distributions on inverse variance parameter of random
effects
beta~dnorm(0,1.0E-5)
tau.v~dgamma(0.5,0.0005)
tau.u~dgamma(0.5,0.0005)
tau.delta~dgamma(0.5,0.0005)

sigma.v <- 1/sqrt(tau.v)
sigma.u <- 1/sqrt(tau.u)
sigma.delta <- 1/sqrt(tau.delta)

}


NOTE:  theta[i,k] the Relative Risk and TT will give the time period
RR.


DATA-->
list(y=structure(.Data=c(),.Dim=c(#of areal units,# of time units)),


m = 304, T=5,t=c(1,2,3,4,5),
sumNumNeigh = 1860,
num = c(),
adj = c())


INITS-->

list(alpha=0,
alpha1=0,
...,

beta=0,
tau.v=1,
tau.u=1,
tau.delta=1,
u=c(), # zeores
delta=c(), # zeores
v=c()) # zeores

From 	Bridget Freisthler <[log in to unmask]> 

Sent 	Monday, July 24, 2006 6:12 pm
To 	[log in to unmask] 

Cc 	 

Bcc 	 

Subject 	space-time correlation in WinBUGS
In case no one else has responded, I have seen two different ways of 
modeling space-time variation.  Details can be found in:

Bernardinelli, L., Clayton, D., Pascutto, C., Montomoli, C., 
Ghislandi, M., & Songini, M. (1995).  Bayesian-analysis of space-time 
variation in disease risk. Statistics in Medicine, 14(21-22), 2433-43.

Waller, L.A., Carlin, B.P., Xia, H., and Gelfand, A. 
(1997). "Hierarchical spatio-temporal mapping of disease rates". 
Journal of the American Statistical Association 92, 607-617.  

Best,
Bridget


Bridget Freisthler, Ph.D., Assistant Professor
Department of Social Welfare, UCLA School of Public Affairs
3250 Public Policy Building, Box 951656, Los Angeles, CA 90095
(Phone) 310/206-1602 (Fax) 310/206-7564
(e-mail) [log in to unmask]

From 	Rodney Sparapani <[log in to unmask]> 

Sent 	Tuesday, July 25, 2006 4:12 pm
To 	ERIK E OSNAS <[log in to unmask]> 

Cc 	 

Bcc 	 

Subject 	Re: space-time correlation
Hi Erik:

This sounds like a very interesting project that I'd love to hear more 
about. 
I think this can be done, but you'd need to do some book-keeping.  
Suppose
that we have N areas and T times, letting M=TxN.  If N=4 and T=2, then 
it
should be simple enough to display if adjacency is only nearest 
neighbors and
the weights are all 1:

S[8] ~ car.normal(adj[], weight[], num[], tau);

Areas
1   2
3   4

Index      Area       Time       adj[1]    adj[2]    adj[3]   num[]
1               1               1            2            3            
0         2
2               2               1            1            4            
0         2
3               3               1            1            4            
0         2
4               4               1            2            3            
0         2
5               1               2            1            6            
7         3
6               2               2            2            5            
8         3
7               3               2            3            5            
8         3
8               4               2            4            6            
7         3

Rodney

From 	Rodney Sparapani <[log in to unmask]> 

Sent 	Wednesday, July 26, 2006 12:15 pm
To 	ERIK E OSNAS <[log in to unmask]> 

Cc 	 

Bcc 	 

Subject 	Re: space-time correlation
ERIK E OSNAS wrote:
> Thanks for your response.  I have been trying to think of an 
indexing 
> scheme like what you propose, but I haven't been able to get my head 
> around it.  
>
> I do think that there will have to be N x T "locations" in the 
> adjacency matrix and I have been able to code that in for the num[] 
> vector.  The adj[] vector is more problematic.  Your example helps.  
I 
> will have to think more about it to see if it solves my problem.  
> The "trick" you propose is to refer to an adjacent neighbor in the 
new 
> NT index rather than to the original index of N areas.  That makes 
> sense to me.  If I can write reasonable code for that problem and it 
> works, I'll let you know.   
>
> Do you think that explicitly coding the spatial and temporal 
neighbors 
> into the car.normal function is different than using a model of 
> Bernardinelli (1995)?  The model is:
>
> x(ij) = a + u(i) + B t(j) + v(i) t(j)
>
> i is an index for spatial location, j is an index for time t, a is 
the 
> average effect (intercept), u(i) is the car.normal random effect (+ 
an 
> unstructured effect as in the BYM model that I left out for 
> simplicity), B is a regression effect for time, and v(i) is a area 
> specific car.normal "random slope" for time.  
>
> Is this different than the indexing model that I asked my original 
> question about?  I think so, but I'm not sure exactly how to think 
> about this.  It seem to me that the Bernardinelli model allows the 
> spatially correlated time effect v(i) to be independent of the 
> spatially correlated area-specific random effect u(i); whereas, the 
> model that I propose forces simultaneous correlation across time and 
> space for each location.  
>
> Thanks for any help,
>
> Erik
>   
Hi Erik:

Well, I haven't seen your model, but I suspect the two models are 
different.  The Bernardinelli
model has a random slope which seems to imply that there is a linear 
relationship with time.
These are called "growth models" and I don't care for them.  They make 
sense only
in circumstances where a specific relationship with time makes sense:  
although, that sounds like it is
obvious, these models are often misused, especially in psychiatric 
clinical trials that I am familiar
with.  So, in my opinion, your model makes alot more sense.  Of 
course, 
you can try both models
and see which one is a better fit for your data.  However, it is nice 
to 
have a hypothesis that one
will be better than another before trying it out.  I think this would 
lead to a nice test via a
Bayes Factor.

Rodney


From 	Darren Mayne <[log in to unmask]> 

Sent 	Wednesday, July 26, 2006 8:50 pm
To 	ERIK E OSNAS <[log in to unmask]> 

Cc 	 

Bcc 	 

Subject 	RE: space-time correlation in WinBUGS
Hi Erik

I am certain you have already received plenty of advise from people 
much
more qualified than me; however, you can find examples of space x time
models of the formulation you describe in Disease Mapping with WinBUGS 
and
MLwiN (Lawson et al, 2003, pp 128-133). The programs for implementing 
the
examples in this book are also available from
http://www.sph.sc.edu/alawson/. I have not implemented these models 
myself;
however, a college has a paper that may be of interest: Spatio-temporal
analysis of acute admissions for Ischemic Heart Disease in NSW, 
Australia.
Burden S, GuhaS, Morgan G, Ryan L, Sparks R, Young L. Environmental and
Ecological Statistics 2005;12(4):427-448.

Cheers
Darren



Erik Osnas
Assistant Scientist

Department of Wildlife Ecology
University of Wisconsin
218 Russell Labs
1630 Linden Dr.
Madison, WI 53706

608-262-1984

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