Tropical deforestation in Madagascar: analysis using...

Ecological Modelling 185 (2005) 105–131

Tropical deforestation in Madagascar: analysis using hierarchical,spatially explicit, Bayesian regression models

Deepak K. Agarwala,∗, John A. Silander Jr.b, Alan E. Gelfandc,Robert E. Deward, John G. Mickelson Jr.e

a AT&T Shannon Research Labs, Florham Park, NJ 07932-0791, USAb Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs, CT 06269-3043, USA

c Department of Statistics and Decision Sciences, Duke University, Durham, NC 27708-0251, USAd Department of Anthropology, University of Connecticut, Storrs, CT 062692176, USA

e Center for International Earth Science Information Network, Columbia University, Palisades, NY 10964, USA

Received 4 August 2003; received in revised form 17 August 2004; accepted 29 November 2004

Abstract

Establishing cause–effect relationships for deforestation at various scales has proven difficult even when rates of deforestationappear well documented. There is a need for better explanatory models, which also provide insight into the process of deforesta-tion. We propose a novel hierarchical modeling specification incorporating spatial association. The hierarchical aspect allows usto accommodate misalignment between the land-use (response) data layer and explanatory data layers. Spatial structure seemsappropriate due to the inherently spatial nature of land use and data layers explaining land use. Typically, there will be missing

g approachot”. UsingtationallyAlso, west of our

esses. Wele.

a; Human

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values or holes in the response data. To accommodate this we propose an imputation strategy. We apply our modelinto develop a novel deforestation model for the eastern wet forested zone of Madagascar, a global rain forest “hot spfive data layers created for this region, we fit a suitable spatial hierarchical model. Though fitting such models is compumuch more demanding than fitting more standard models, we show that the resulting interpretation is much richer.employ a model choice criterion to argue that our fully Bayesian model performs better than simpler ones. To the beknowledge, this is the first work that applies hierarchical Bayesian modeling techniques to study deforestation procconclude with a discussion of our findings and an indication of the broader ecological applicability of our modeling sty© 2005 Elsevier B.V. All rights reserved.

Keywords:Tropical deforestation; Bayesian statistical hierarchical models; Spatially explicit regression models; Misaligned spatial datpopulation pressure; Land-use classification; Statistical model comparison; Multiple imputation

∗ Corresponding author. Tel.: +1 9733607154;fax: +1 9733608178.

E-mail address:[email protected] (D.K. Agarwal).

1. Introduction

The demise of the world’s tropical rain forests hbeen of central concern to conservation biologists

0304-3800/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2004.11.023

106 D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131

at least 20 years (Singh, 2001; Whitmore, 1998). Butcognizance of the problem was apparent decades ear-lier (cf. Jarosz, 1993; Perrier de la Bathie, 1921). Yetit has proven surprisingly difficult and controversial toobtain accurate estimates of deforestation rates or in-deed to relate current forest cover to potential or histor-ical extent of forest cover. This remains true even withthe advent of satellite imagery from the 1970s (Apan,1999). The difficulties arise from issues of definitionand benchmarks, spatial and temporal resolution, anddata adequacy (Singh, 2001; Angelsen and Kaimowitz,1999). As a consequence even the most recent esti-mates of deforestation rates, or more generally land-use change, vary considerably, and debate continuesover whether the rates of deforestation in the tropicsare declining or increasing (FAO, 2001; Matthews etal., 2000). Only for specific, narrowly defined regionscan one obtain reasonably reliable data on deforesta-tion rates over specific time periods (e.g.,Mertens andLambin, 2000).

Establishing cause–effect relationships for defor-estation or land use at various scales has also provendifficult even when the process appears well docu-mented (Angelsen and Kaimowitz, 1999; Irwin andGeoghegan, 2001; Barbier, 2001). The standard expla-nation for deforestation in the tropics has been rapidpopulation growth, associated poverty, and consequen-tial environmental destruction (Leach and Mearns,1988; Richards and Tucker, 1988; Mercier, 1991;Brown and Pearce, 1994; Sponsel et al., 1996). In se-l cut-t ,1 estl beenq ead-i er-s na-t ses,l it isn rva-t ess.M rayo ,2f .g.,O dr urals

by human management practices (e.g.,Lamb et al.,1997).

In this paper, we focus on deforestation processesand patterns of land use in the eastern wet tropicalforests of Madagascar (Fig. 1). The prevailing percep-tion is that the patterns and processes here are wellunderstood. In fact the land-use change maps ofGreenand Sussman (1990)are commonly reproduced in text-books as standard examples of deforestation processes.Nevertheless current and historical patterns of de-forestation in Madagascar remain poorly understood,and controversial (e.g.,Jarosz, 1993versusGreen andSussman, 1990). Current estimates of forest cover inMadagascar vary by at least five-fold, from 42,000 km2

for “closed forest,” up to 158,000 km2 for “natural for-est,” to 232,000 km2 for “total forest and woodland,”respectively, about 7, 27 and 40% of the land areaof Madagascar (UNEP, 1998; UNEP-WCMC, 2001).This underscores the pervasive difficulty of estimatingforest cover, which in part reflects differing definitionsof forest (cf. Silander, 2000). In contrast, the forestcover in 1902 was estimated at about 20% of the landarea (Pelet, 1902). In 1921, based on extensive fieldreconnaissance from 1900 to 1915, Perrier de la Bathieestimated that about 19% (110,000 km2) of Madagas-car was in forest or woodland of one sort or another. Butof this only about 35–70,000 km2 (6–12%) was closedforest. In 1934, forest cover was estimated at only 10%of the land area (Grandidier, 1934). These figures con-flict with estimates of forest cover in the 1960s: 33%f sDe ateso thea car-t 70so sti-m allya s oft loudst f for-e dy.I cur-r them ya 0w rrentf ce)

ected areas, commercial exploitation and clearing are also important causes of deforestation (Torsten992). However, conventional explanations for for

oss and environmental degradation have recentlyuestioned as too simplistic, general, or even misl

ng (Barbier, 2001). In the absence of a clear undtanding of the role that various contributing explaory variables may play in deforestation proceset alone establishing cause effect relationships,ot surprising that reforestation or forest conse

ion schemes have had relatively little or no succost of the reports on failures are buried in the gr secondary literature (e.g.,Bloom, 1998; IUFRO001; Sharma et al., 1994; cf. Oates, 1999); relatively

ew are to be found in the primary literature (elson, 1984; Elster, 2000); and rarely does one fin

eports of successful reforestation either via natuccession (e.g.,Guariguata and Ostertag, 2001) or

orested in the early to mid 1960s (Humbert and Courarne, 1965) versus 21% a few years later (Le Bourdiect al., 1969). The fact that these contemporary estimf forest cover are so different is surprising, givenvailability of aerial photography and more preciseographic methodology. Satellite data from the 19nward have been able to provide only limited eates of forest cover or deforestation rates. Virtull of the available images of the wet tropical forest

he eastern Madagascar were too obscured by chrough the 1990s to provide adequate estimates ost cover, especially for the focal region of this stu

n fact, the most recent available maps reproduce “ent” forest cover for most of the east coast fromaps published in 1965 (Faramalala, 1995; Du Pund Moat, 1999; see alsoGreen and Sussman, 199)ith no new data. These sources thus indicate cu

orest cover for our focal region (Taomasina Provin

D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131 107

Fig. 1. Color composite AVHRR satellite image of Madagascar. The box encompassing Taornasina Province shows the focal area for the study.The light and dark green areas in along the east coats of Madagascar correspond roughly to potential forest and mature, continuous forest,respectively.

at 66%, while the center of our area (Fenerive Prefec-ture) would be 95% forest cover. Clearly this is a grossoverestimate of actual forest cover (see results below),and a marked departure from all earlier estimates in thetwentieth century, and is indeed at odds with historicalaccounts from the 17th through 19th centuries. EvenGreen and Sussman’s seminal 1990 study of deforesta-tion rates in Madagascar are based largely on the sameset of maps provided byHumbert et al. (1964–1965).

The confusion and contradiction described above,underscores the need for a more critical, in depth ex-

amination of deforestation or more generally land-usepatterns and processes. It also points up the need forevaluating land-use patterns in a more thorough histo-riography context. Moreover, if there is any indicationthat patterns of land-use change spatially from one re-gion to another, this needs to be addressed or one mayrisk obscuring pattern and process across scales.

With these objectives in mind, we present a novel hi-erarchical Bayesian model for explaining deforestationprocesses applied to deforestation of Madagascar. Ourmethod is able to address several issues that arise in the


study of deforestation processes, which, to the best ofour knowledge, has not been addressed before. First, weprovide a framework to connect data layers that are spa-tially misaligned. In our example, we explain land-usepatterns (measured at a resolution of 1 km× 1 km) interms of population counts available at a much coarserresolution (by townships). We show that naive schemeslike allocating population uniformly to all pixels withina given township perform poorly compared to the mul-tilevel hierarchical Bayesian model we suggest. Next,we provide a novel multiple imputation method to dealwith missing data when modeling spatial processes likedeforestation. Note that “filling up holes” created dueto missing land-use classification using a crude methodlike “majority vote due to neighbors” might fail to pro-vide correct estimates of uncertainty for model param-eters. Finally, our model can explicitly incorporate spa-tial structure, which has hitherto not been addressed inthe context of explanatory models that study deforesta-tion processes. Apart from providing better statisticalinference by modeling excess heterogeneity present inthe data, maps of estimated spatial structure are infor-mative and can often lead to new ecological insights.However, this comes at a price of being able to fit com-plex hierarchical Bayesian models, which is a com-putationally challenging undertaking and may involvewriting special purpose code as is the case here. Hi-erarchical modeling is just beginning to receive atten-tion in the ecology literature (see our brief review ofthis work at the end of Section2). Beyond the obvi-o hicalm pat-t e is-s lains ss usi ate,s iss-i ers,a ausew ub-r re-g raten prett icalh del-i om-p linga

2. Model background and modeling issues

The goal of this paper is thus to develop, fit, andinterpret stochastic models to clarify spatial processesthat can explain patterns of deforestation or more gen-erally land-use patterns. This is a rather challengingundertaking on several accounts.

First, there are numerous factors which have beenlinked to land use and deforestation. These factors canbe socio-economic, e.g., population growth or eco-nomic growth; physical factors, e.g., topography orproximity of rivers and roads; policy-driven govern-ment intervention, e.g., agriculture and/or forestry poli-cies; external factors, e.g., demand for exports or fi-nancing conditions. For any given region, typicallyonly some of this information is available or selectivelyavailable at different spatial scales.

Second, there is no well-accepted notion of a re-sponse variable. In some cases deforestation rates orforest areas may be used. Deforestation rates are typi-cally calculated over limited time spans. Forested areasare usually obtained from cross-sectional studies of dif-ferent countries or different regions within countries.The alternative, which we adopt is to partition the studyregion into disjoint areal units and then attach an or-dinal land-use classification variable to each unit. Thisclassification represents successive departure from apotential, fully forested landscape, which all evidencepoints to for eastern Madagascar. Of course, such avariable is not uniquely defined in terms of number ofa

nsev nits.F onsev non-fe and,p realu hip”l iablec esul-t datac realu d toe

ntlys lana-t ying

us applications to land change problems, hierarcodeling has broad application to other ecological

erns and processes which present many of the samues listed above. For example, in attempting to exppecies presence–absence, abundance or richneng ecological explanatory variables, such as climoil geology or topography, one must address mng data, spatially misaligned explanatory data laynd spatial structure in the ecological pattern. Bece had some a priori indication that there were s

egional differences in land use within our studyion, we subdivided the analysis spatially into sepaorthern and southern sub-regions. Finally we inter

he results of the model in the context of the critistorical information available. Because the mo

ng approach we take is novel and technically clex, we start with detailed discussion of the modepproaches.

-

nd definition of classifications.Third, the explanatory variables and the respo

ariables are typically measured in different areal uor instance, in the dataset we investigate, the respariable is land-use classification (e.g., forested,orested, etc.), which is ascribed to 1 km× 1 km pix-ls derived from satellite images. On the other hopulation is recorded at various administrative anits. In our case, we use the equivalent of “towns

evel data, which are considered to be the most relensus data at the finest spatial resolution. The rant challenge is to develop a regression model forollected on spatially misaligned areal units, i.e., anits (pixels, polygons, etc.) that do not corresponach other or line up as GIS data layers.

Finally, land use and deforestation are inherepatial processes. So too, are many of the expory variables, such as population counts. Satisf


stochastic, as well as mechanistic objectives, modelingshould capture association between measurements onareal units in terms of proximity of these units. Oneway to accomplish this is to introduce appropriate setsof spatial random effects, which can serve as spatial sur-rogates for unmeasured or unavailable covariates thatare inevitable in any modeling protocol.

Little formal modeling of deforestation was at-tempted until the 1980s (Granger, 1998). Descriptivestatistical summaries of land use or forest area obtainedfor certain regions at certain time points were custom-ary. Most of the ensuing statistical work, which hasappeared has been based on standard multiple linearregression models relating deforestation rates or for-est area to a long list of potential explanatory variables(Granger, 1998). In fact, Granger lists at least 28 differ-ent variables, which have been linked directly or indi-rectly to deforestation or land-use change in a forestedlandscape, while Anglesen and Kaimowitz list 140different models developed to explain just economiccauses of deforestation. In these studies it appears thatthe major objective has been simply to maximizeR2;and models with eight or more explanatory variableshave been put forward. This is no longer consideredsufficient or useful in statistical modeling these days.Moreover, none of these models are explicitly spatialin nature. The little work with an explicitly spatial fla-vor, which does exist has arisen from an econometricperspective introduced byPalloni (1992)andChomitzand Gray (1996), and more recently summarized byI et theh s toi hes ricest atiala

tan-t torym ro-cGi d ap-p uc-c eter-m ula-t od-e

In general, stochastic models provide considerableadvantage over deterministic approaches in situationswhere substantial variability is always present, as inland-use change data. Stochastic modeling allows oneto resolve the signal and to clarify the nature of thenoise, which obscures it. The development of satisfy-ing stochastic models does raise a number of method-ological issues. Here, we offer an overview of a criticalsubset of these issues. We address the specific detailsof these issues later as part of the development section.

First, there is a matter of modeling style. Do weemploy simulation modeling, say as is frequently donewith cellular automata models, attempting to capturemechanistic aspects of the process? Or do we employphenomenological modeling to attempt to relate dif-ferent data layers, each measuring different variables?Where there is theory to guide us, can we be somewhatmechanistic in relating data layers? Finally, do we takea static or dynamic view of the process? There is nosimple right or wrong answer to these choices.

If we seek to explain land use, then we first haveto formally define the response variable. Even within aspecific context there is not usually a well-accepted no-tion. Also, we have to identify which factors we seek tolink to land-use. Do we seek a socio-economic explana-tion? Do we seek a physical or ecological explanation?Here we must recognize that no proposed explanatorymodel iscorrect, but that some are moreusefulthanothers and that no proposed explanatory model estab-lishes causality, merely relationship. Moreover, mod-e lainc tivet con-n

n bei fica-t pro-c el y isr ere,f deli tionm ap-p nga m-p lingi col-l apers

rwin and Geoghegan (2001). These models assumhat land use will be devoted to the activity yieldingighest “rent”. The modeling connects output price

nput prices through “production” functions with tpatial aspect introduced solely by relating these po market distance. These models remain quasi-spnd econometric.

Hence, we claim that despite the already subsial literature, there is a need for better, explanaodels, which may provide better insight into the p

ess of deforestation or land-use change (cf.Irwin andeoghegan, 2001; Veldkamp and Lambin, 2001). Var-

ous modeling lineages have been developed anlied to explain deforestation with questionable sess and little attempt at synthesis. These include dinistic and stochastic models, statistical and sim

ion models, phenomenological and mechanistic mls, spatial and non-spatial models, etc.

ls employing different subsets of variables can expomparably well. One must decide which perspeco explore and which subset of possible factors toect.

When data layers are to be interrelated, they cantroduced at various levels of a hierarchical speciion, which becomes an explanatory model for theess (cf.Wu and David, 2002). In this specification thowest level is the response. This modeling strategeferred to as hierarchical or multilevel modeling. Hor example, the hierarchical levels of the full monclude a population model, a land-use classfica

odel, and a model specifying latent variables. Thisroach allows for considerable flexibility in modelind the possibility of incorporating mechanistic coonents in linking certain levels. Hierarchical mode

s achieving increased utilization in analyzing dataected from complex processes. See the recent p


byWikle (2003), byClark et al. (2004), and byGelfandet al. (2003, 2005).

Full inference for hierarchical models is most easilyachieved (and often only possible) within a Bayesianframework. In particular, this framework enables a pos-terior distribution (i.e., a conditional distribution giventhe observed data) for all model unknowns. This dis-tribution allows any desired inference about any of theunknowns. Such models are fitted using simulation-based methods, e.g., Gibbs sampling and Markov chainMonte-Carlo methods. SeeCarlin and Louis (2000)andGelman et al. (1995)for an introductory presentationof this methodology.

In the context of explaining land use, the data layersare inherently spatial. Some are observed at particularpoint locations, and some are obtained through satelliteimagery as raster pixel values at some degree of reso-lution. Other data may be associated with formal gov-ernmental units such as towns or other municipal sub-divisions, stored as polygons. Data, which is obviouslyspatial should be modeled in an explicitly spatial fash-ion. Only rarely has this actually been done in modelingland-use patterns, and where spatial models have beendeveloped, the analysis is not entirely spatially contin-uous (Irwin and Geoghegan, 2001). Developing spatialprocess models that lack spatial contiguity is analogousto developing models of time series processes that lackknowledge of the time order of the processes—not veryuseful (Irwin and Geoghegan, 2001). While there maybe issues of model choice in providing spatial structurea lingi h arec nglya atiald iffer-e rents odel-i nceo rar-c

ribedt caseo meds to ar of ar spa-t e.g.,a e).

But there will always be omitted, perhaps unobserv-able, explanatory variables, which also provide spatialexplanation. In the absence of these variables, spatialassociation in the residuals will be expected, and hencemodels introducing spatial dependence are needed.

For at least some data layers there will be missingdata. For example, land-use classifications obtained byremotely sensed satellite data, particularly for wet trop-ical regions, will have portions obscured (hence miss-ing) by cloud cover. Typically missing data are treatedin an ad hoc manner. Models of spatial association forgridded data assume there are no missing observations.To accommodate suchholesin the dataset, imputation(i.e., statistical simulation of missing values) providesan alternative to subjective assignments of values. Thatis to say, one imputes observations at the missing loca-tions according to the likelihood of the different pos-sible values. Such imputation must be done randomlyand, moreover,multiple, hence variable, imputationsare necessary in order to capture the effect of the uncer-tainty associated with imputation and to better reflectvariability associated with inference in the presence ofmissing values.

Lastly, the multilevel hierarchical approach allowsfor the specification of many different models; thereis flexibility at each level. But, with many possiblemodels to consider, tools for model comparison arerequired. Simple notions likeR2 are clearly inappro-priate for multilevel models with categorical or countvariables as response. Criteria, which reflect the util-i etters nsi-b dataa bled isonsw

ess-f ingt ,2 000;C -t t ofo pliedt elim-i iter-a so exts if-

s well as the scale or resolution at which the modes done, it seems clear that measurements whicloser to each other in space should be more strossociated than those farther from each other. Spependence should be incorporated, but since dnt spatial data layers are often collected at diffecales, one needs a formal (rather than ad hoc) mng strategy to carry out full inference in the presef such consequential misalignment. Bayesian hiehical modeling provides this framework.

In this regard, some spatial features can be deschrough the mean structure of the model (or in thef categorical or count data, the mean on a transforcale). For instance, one can introduce distanceoad or to a town center with a coefficient as partegression structure. Or one can attempt to explainial features through a trend surface in the mean (

polynomial of low order in latitude and longitud

ty for the model emerge as more attractive and buited. However, such comparison is really only sele across models employing the same responsend with explanatory objectives with regard to availaata layers. So, these are the only model compare can provide.Hierarchical Bayes models have a long and succ

ul history by now in several areas but are just beginno be applied to problems in ecology (e.g.,Borsuk et al.004; Qian et al., 2003; Borsuk et al., 2001; Prato, 2lark, 2003). SeeBanerjee et al. (2003)for a general in

roduction to the methodology. However, to the besur knowledge, the methodology has not been ap

o study deforestation processes other than a prnary paper we have published in the statistical lture (Agarwal et al., 2002). Before providing detailn our model development and specification in nection, we would like to comment briefly on the d


ference between our approach and the simulation mod-eling approach that has been widely used to study de-forestation (see, for example,Soares-Filho et al., 2002;Duffy et al., 2001; Kok and Winograd, 2002). We re-mark the latter is really a different philosophic approachto the problem compared to the explanatory approachwe adopt. While simulation-based modeling seeks topredict observed land-use behavior by capturing theknown mechanistic aspects of the problem, our ex-planatory approach attempts to explain a response vari-able (in this case land use) in terms of other explanatoryvariables. Moreover, simulation models are dynamic,seeking to compare the evolution of deforestation. Ourmodel is static seeking to link land use to explana-tory variables, in particular to population pressure. Assuch, the two approaches are not directly comparableto understanding deforestation patterns and processes.In fact, the two approaches can in some cases comple-ment each other—insights into the mechanistic aspectsof the problems can be incorporated when buildingan explanatory model. However, this has rarely beenattempted.

3. Overview of model development andspecification

Our approach is to formulate a model for the jointdistribution of local human population attributes andforest exploitation, given other explanatory variablest e-d rs(e ineda thet wnb lit d-e ci-fi tionc useg tentp rvedc iatedw xell delr

We show how such models can be fitted using Gibbssampling (Gelfand and Smith, 1990), thus allowing fullinference of all of the modeling levels (cf.Gelfand etal., 2005). The misalignment problem extends recentwork of Mugglin and Carlin (1998), Mugglin et al.(2000)andGelfand et al. (2003). In the present con-text the other explanatory variables we employ are alsomeasured at the (l km× 1 km) pixel-level. However, ifthis were not the case, a general strategy is as follows.Identify the data layer at the finest spatial resolution.For each remaining layer create an individual overlayon this one. For each overlay, carry out a rasterizationso that each areal unit of the latter is contained on oneand only one unit of the former. Then model the jointdistribution of all the variable layers at this highest res-olution. Such an approach yields a model for each datalayer at its observed scale. Note that it is not neces-sary to align all pairs of data layers. Rather, we needonly align each layer with the one at the finest spatialresolution.

Two related implementation issues arise under ourmodeling approach. First, for the response data wework with, due to the presence of cloud cover, land-use classification is missing for approximately 8% ofthe roughly 47,000 pixels. Thus, we introduce an im-putation approach, i.e., a spatial multiple imputation(Schafer, 1997), to handle this missing data. Second,our modeling requires commitment of considerablecomputational effort (tailored programs for the indi-vidual application) and computer run time. To justifyt eal-i tos oree

4

odela ateda . Iti thew etf rest“ da-g ndoust 0;S e

hat are explicitly spatial. In so doing we are immiately faced with incompatibility of the data layeland use is obtained at the scale of 1 km× 1 km pix-ls (see below) while population counts are obtat irregular town-level polygons). We then overlay

own-level map on the pixel-level map and modify tooundaries, usingmajority pixel rule, so that each pixe

s contained in one and only one town. Upon suchras-erizationof the town data, we conduct the joint moling at the pixel level. This joint distribution is speed by modeling the unobserved pixel-level populaounts and, then the conditional distribution of landiven the associated count. The model for the laopulation counts leads to a model for the obseounts. With two sets of spatial effects, one associth the town population counts, the other with pi

evel land-use model, a multilevel hierarchical moesults.

his commitment we offer model comparison revng the inferential benefits of our modeling relativeimpler model specifications, which can be fitted masily and quickly.

. The dataset

The data used in developing the deforestation mre from Madagascar, an area of the world designs particularly high priority for conservation efforts

s recognized as 1 of 25 mega-diversity countries inorld (Meyers et al., 2000) and the eastern tropical w

orest in Madagascar is globally one of 12 rain fohot spots”. At the same time, the forests in Maascar have been characterized as under treme

hreat from deforestation (USDA Forest Service, 200ussman et al., 1994; Oxby, 1985). As a consequenc


many of the most well-known plant and animal speciesare listed by the International Union for the Conserva-tion of Nature (IUCN) as globally threatened.

There is considerable difference of opinion as to howmuch of the original forest in Madagascar is left as wepointed out earlier, but estimates range down to as littleas 10%, and perhaps only 25% of what is left is con-sidered primary or undisturbed forest (Sussman et al.,1994; USDA Forest Service, 2000). Of these systems,the coastal forests are the most threatened. Estimateddeforestation rates for Madagascar are quite variable:less than 1% per year to 5% or more. Some have esti-mated that within 20 years, little or no forest will remainoutside protected areas (currently about 1.85% of thetotal land surface). But, as mentioned earlier, most ofthese estimates of forest cover and deforestation ratesare speculative and controversial. Concerns over defor-estation and attempts to estimate rates is nothing new.In 1921 Perrier de la Bathie estimated that between1900 and 1920, 49,000 km2 of forest had been lost.Early in the 19th century, the first King of Madagascarproscribed the clearing and burning of forest (Verin andGriveaud, 1968).

The focal area for this study is the wet tropical for-est biome within Toamasina (or Tamatave) Province,Madagascar. This province is located along the EastCoast of Madagascar, and encompasses the great-est extent of tropical rain forest on the island nation(cf. Fig. 1). The aerial extent of Toamasina Provinceis roughly 75,000 km2. To model forest cover fort dig-i ries( leva-t 93)a

4

s-i inedf .c lusa I)( er-v r-i re.T erei the

cover types, their composition, distribution and loca-tion within the region. Two Landsat 5 Thematic Map-per TM images (September 15, 1993 and November 21,1994), a set of aerial photographs from the 1940s and1960s together with extensive independently collectedground-truthing data were used to aid fine scale patterndelineation. The 30 classes were iteratively assessed,reclassified with a supervised, maximum-likelihoodprocess and reduced to the final 5 used for the modeling:mature forest, secondary patch forest, cleared forest,non-forest and missing values (for small patch areasobscured by clouds or cloud shadows, haze, etc.). Webelieve this provides the most accurate characterizationof the landscape at this scale available. The vector mapsof forest cover currently available (Faramalala, 1995;Du Puy and Moat, 1999) are overly smoothed and inmany cases report forested and non-forested areas thatbased on our ground truth information, are misclassi-fied.

4.2. DEM generation

The quality of existing global topographic informa-tion (Gtopo30, Digital Chart of the World) we found tobe inadequate, containing numerous artifacts and noise.Instead we use vector elevation contour data (100 m)digitized by the Madagascar geodetic survey office (In-titut National de Geodesie et Cartographie, or FoibenTaosarin-tanin ’I Madagasikara (FTM)). These wereused to generate a continuous, 1 km× 1 km grid sur-f enS t v.2 ign-m pro-d isl r, us-i

4

onw inap ffice( itedN ctione 993.V ea-t ”),

he province, we obtained or constructed fivetal geospatial layers representing: town boundawith asociated 1993 population census data), eion, slope, road and transportation networks (19nd land cover.

.1. Forest classification

A single NOAA-11 AVHRR, 10-day compote image for September 21, 1993 was obtarom the USGS EROS Data Center (http://edcwwwr.usgs.gov). Three reflective bands (1, 2 and 3) p

Normalized Difference Vegetation Index (NDVbands 3− 2\3 + 2), were presented to an unsupised Iterative Self Organizing (ISODATA) clusteng algorithm, within ERDAS Imagine v.8.4 softwahis clustering output 30 initial classes, which w

nterpreted using expert personal knowledge of

ace, utilizing a combination of Surfer v. 6.04 (Goldoftware, Golden CO) and Arc View Spatial Analys.0 (ESRI, Redlands CA) software. Spatial misalents and misregistration errors were corrected touce the final 1 km2 elevation grid layer. In turn, th

ayer was used to calculate a slope (degrees) layeng Arc View Spatial Analyst v. 2.0.

.3. Town, population and other cartographic data

Township (i.e., “firaisana”) boundary informatias digitized for all 162 firaisana in Taomasrovince by the Madagascar geodetic survey oFTM). Census data were obtained from the Unations via the Madagascar census bureau (Diret Statistique Sociales) for each firaisana for 1ector point data for various population center f

ures (i.e., village/hamlet localities or “fokontany

http://edcwww.cr.usgs.gov/

http://edcwww.cr.usgs.gov/


firaisana “headquarter” villages, fivindronana head-quarter towns, and the provincial capitol) weredigitized and supplied by FTM. Road and tracks werealso digitized and supplied by FTM (primary routes(sealed roads) and railroads, motorable (in dry season)tracks, rough ox cart tracks, and primary footpaths).These vector layers were all rasterized to 1 km2 grids.Grid cells fell along town boundaries were assignedto a particular town by majority area rule. Grid cellswith one type of road were assigned the appropriatevalue. When two or more road/path network types fellin a particular grid cell, the more developed road valuewas assigned to the pixel.

The final layers were rasterized to 1 km and clippedto the extent of the Town boundaries in the province. Allislands were clipped as was the one town that coveredless than 1 km2. All data were obtained in, or projectedto, Laborde (m), WGS84 coordinate values.

Following clipping, 159 townships and 74,6071 km× 1 km pixels remained in our data set.Fig. 2shows the town level map for the towns in the studyregion. In the western portion of the province an es-carpment demarcates the eastern wet forest or potentialforest biome from the western, seasonally dry grass-land/savannas mosaic. Visual inspection of the datalayers prior to analyses revealed some differences be-tween the northern and southern parts of the province.The North had fewer population centers most of whichtend to be clustered near the coast, and larger forestpatches, while the South had more populations cen-t allerf ent,i esto pro-c in-t cre-a edt tro-d n atl rtha andS d4 lp wnse ation6

ub-r ene-

Fig. 2. Northern and southern regions defined within the study re-gion. All town boundaries are shown. Population size categories areshown for northern and southern towns studied.

ity of spatial pattern of the sub-regions. Rather, dueto the fairly simple mean specification of the modeldescribed below, we were concerned that omitted orunobserved variables which carry spatial informationmay differentially affect land use in the sub-regions.This implies that the spatial model for random effects(which are introduced as surrogates for these variables)may differ among sub-regions. Moreover, the regres-sion coefficients of the observed explanatory variablesmay also vary among sub-regions. In general, the mes-sage is that if, a priori, one suspects pattern or pro-cess is operating differentially in portions of the study

ers scattered across the landscape, with many smorest patches and more extensive road developmncluding primary routes to the national capital, wf the study region. With concern that pattern andess might differ between these regions and witherest in making comparisons between them, weted two disjoint regions. In particular, we exclud

he western (non-forested majority) towns and inuced, between North and South, a buffer regio

east two towns wide (to provide separation of Nond South spatial effects). This yielded the Northouth regions identified inFig. 2. The North include6 towns encompassing 22,347 km2 pixels and totaopulation 707,786; in the South there are 66 toncompassing 24,623 such pixels and total popul64,066.

We note that in creating these two disjoint segions, the concern was not to strive for homog


Fig. 3. Histogram for population for the 159 towns in the full study area.

region, it is prudent to fit individual models to theseportions.

The GIS data layers used in the modeling are shownin Figs. 2–7. In Fig. 2, town population data forNorth versus South is overlaid on the town boundariesfor the entire province.Fig. 3 provides a histogramof the population data for all towns included in themodel.

The ordinal land-use classifications for the north-ern and southern region are shown inFig. 4: the mostdegraded land is shown as non-forest (class 1), fol-lowed by cleared and degraded (potential) forest lands(class 2), secondary patch forest (class 3) and matureforest (class 4) and the missing value class. Relativecontribution (in percentage) of each class is providedin Table 1. Fig. 5provides grey scale maps for elevationwhile Fig. 6provides grey scale maps for slope.

Road classification was modified prior to full modeldevelopment. Initially, we had five road types assignedto pixels, viz. no roads, footpath, rough track, mo-torable track and primary route or rail road (Fig. 7).However, due to the sparseness of counts in some ofthe categories, we needed to reduce this to a binary

road classifications. In fact, because the sparseness pat-terns differed between the North and the South, weused differently defined binary classifications for thetwo regions. In the North, no roads was one of the bi-nary road classifications while the other four road typesformed the second classification. In the South, no roadsand footpath formed one category while the other threeroad types were lumped into a second category.

5. Model details

We model the joint distribution of land use (L) andpopulation count (P) at the pixel level (1 km2). Let Lijdenote the land-use value for thejth pixel in the ithtownship and letPij denote the population count forthe jth pixel in theith township. TheLij are observedbut onlyPi =

∑jPij are observed at the town level. We

collect theLij andPij into town level vectorsL i andPi

and overall vectorsL andP. We also observe at eachpixel elevation,Eij , slope,Sij and road classificationRij .Lastly, letδi denote a spatial random effect for townshipi. We elaborate these effects below.

Table 1Contribution of land-use classes in percentage

Class 1(non-forest)

Class 2 (cleared/degraded forest)

Class 3 (secondarypatch forest)

Class 4 (mature forest) Missing

North 2.5 30.7 17.2 43.5 6.1South 10.7 42.5 11.1 26.2 9.5


Fig. 4. Ordinal land-use classifications for the study regions.

In specifying the joint distribution,f(L, P|{Eij},{Sij}, {Rij}, {δi}), we factor this joint distribution as

f (P |{Eij}, {Sij}, {Rij}, {δi})f (L|P, {Eij}, {Sij}, {Rij}).(1)

We condition in this fashion since one of our ob-jectives is to see if population provides a significanteffect for land use. Such explanation is evidentlynot causal; we could equally well-conditionPonL.


Fig. 5. Grey scale elevation maps for the study regions displayed in a limited number of classes here.

Turning to the first distribution in(1), the populationmodel, we assume that thePij are conditionally inde-pendent given theE’s, S’s, R’sandδ’s, i.e., we write

f (P |{Eij}, {Sij}, {Rij}, {δi})=∏i

∏j

f (Pij|Eij, Sij, Rij, δi). (2)

Moreover, sincePij is a population count and sincemany of these (unobserved) counts will be sparse forthis region at this scale, we assumePij : Poisson (λij )where

logλij = β0 + β1Eij + β2Sij + β3Rij + δi (3)

Since thePij ’s are conditionally independent Pois-son variables, summing overj yields Pi : Poisson

(λi) where logλi = log∑

jλij = log∑

jexp(β0 +β1Eij +β2Sij +β3Rij + δi).

In (3), the β”s are regression coefficients while,again, theδi ’s are township level spatial random ef-fects. They are intended to capture anticipated spatialsimilarity for neighboring townships with regard topopulation. Unlike usual random effects which areassumed to be independent and identically distributednormal random variables, the distributions of theδi ’sare specified conditionally given all of their neighbors.That is,

δi|δj, j = i∼N

(∑jωijδj∑jωij

,τ2∑jωij

)


Fig. 6. Grey scale slope maps for the study regions.

whereωij = 1 or 0 according to whether or not townj is contiguous with towni andτ2 is the spatial vari-ance component. In other words,δi is expected to varyabout the average of its neighbors. Such models are re-ferred to as conditionally autoregressive (CAR) mod-els. See, for example,Besag (1974)or Cressie (1993)for further details. We note that since thePij are not ob-served we cannot introduce pixel level spatial effectsinto the population model. We cannot adjust the meanof population counts we have not seen; unstable modelfitting results. So, spatial adjustment toλij will be iden-

tical for all pixels within a town and will be similar forpixels arising from adjacent towns. Also, since thePijare conditionally independent random variables with∑

jPij =Pi , we have{Pij}|Pi : multinomial(Pi ; {γ ij})whereγ ij =λij /λi .

In the second term in(1), the land-use classificationmodel, we assume conditional independence of theLijgiven theP’s, E’s, S’s andR’s. To handle the ordinalnature of theLij ’s, we follow Albert and Chib (1993),introducing a latent variableWij associated with eachLij . Wij is conceived as a continuous random variable


Fig. 7. Vector road classification for the entire province.

on the real line, interpreted as the extent of forestcover. We then imagine that the four ordinal land-useclassifications cut the real line into four intervals sothat Lij = l if Wij ∈ (γ l−1, γ l), l = 1, . . ., 4 with regardto degradation whereγ0 =−∞, γ4 =∞. Hence,πijl ≡ P(Lij = l) = P(Wij ∈ (γl−1, γl)), l = 1, . . . ,4,provides the four land-use classification probabilitiesat each pixel in each town. In other words,Lij is amultinomial trial with possible outcomesl = 1, . . ., 4having respective probabilitiesπijl , l = 1, . . ., 4. TheWijs can only be identified up to translations. That is, theWij ’s are locatedrelativeto each other on this imaginary

scale but the data cannotabsolutelyplace theWij ’s. Todo so, without loss of generality, we can set the cut pointγ1 = 0 butγ2 andγ3 are unknown. Hence, givenWij andtheγ ’s, Lij is determined, i.e., the conditional distribu-tion degenerates to a particularl givenWij . Conversely,given Lij and theγ ’s, Wij is restricted to an interval.Thus, the land-use classification model becomesf(L,W|{Eij}, {Sij}, {Rij}, {Pij}, γ2, γ3) which we canwrite as

f (L|W, γ2, γ3) · f (W |{Eij}, {Sij}, {Rij}, {Pij}). (4)


Again, due to the conditional independence, the firstdistribution in(4) can be written as�i�j f(Lij /Wij |Eij ,γ2, γ3). The second becomes�i�j f(Wij |Eij , Sij , Rij ,Pij ) where we use a linear regression model,

E(Wij) = α0 + α1Eij + α2Sij + α3Rij + α4Pij. (5)

However, note that because theWij ’s are not observed,their latent scale is not identifiable. We could multiplyeach by a constant, multiply theα’s by this constantand multiply the standard deviation ofWij by this con-stant and the modeling would be unaffected. Hence,without loss of generality, we set var(Wij ) = 1. Notefurther that we cannot introduce spatial random effects(in fact any random effects at the pixel level) into theland-use model. Were we to propose including effectssayϕij in (5), sincePij is not observed, the data wouldnot be able to separatePij andϕij in the componentα4Pij +ϕij . Nonetheless, theδi ’s in (3) induce spatialsmoothing in explaining thePij ’s which, in turn,produces some spatial smoothness in explaining theWij , henceLij . A simpler version of the model in(1)–(5) is presented with some analysis along with amore technical perspective inAgarwal et al. (2002).

We conclude with several remarks:

Remark 1. A more customary modeling approach toexplain theLij would be a non-hierarchical categoricalregression. For instance, we could model logit

πij,l−1πij,l

,

l = 2, 3, 4 as a linear model inEij , Sij andRij . Suchm tisti-cw fi ree ever,t wn-s -tu intyi

R nge d am efi

R esE n

of Lij andPij . However, with regard to conditional andmarginal specifications, if we use these covariates toexplainPij , should we also use them to explainLijgivenPij (or perhaps, vice versa)? In familiar normallinear modeling, we would introduce redundantparameters in the marginal model forLij . The marginalmodel is over-parametrized, the regression coefficientsare not identified. Within the Bayesian framework, noproblem arises. We would find little learning about thecomponents of the coefficient vector but no problemwith the overall coefficient. In the present situation,with a Poisson model for thePij ’s, the identifiabilityproblem does not arise; in the marginal model forLij ,we have both an additive form and a multiplicativeform in the covariates. Perhaps more importantly, themarginal model forLij would not involvePij . We areexplicitly interested in a conditional explanation forLij givenPij and the other covariates.

6. Fitting and inference for the model

The model defined by(1)–(5)is referred to as a mul-tilevel or hierarchical specification. In fact, at the lowestlevel we have the observed land-use classifications. Atthe level above we have the latentW’s. At the top levelwe have the latent pixel level populations constrainedby the observed township populations. This model is ofhigh dimension with unknowns,{Wij}, {Pij}, {δi}, α,β, γ2, γ3 andτ2. Fitting of this entire model is feasibleo veag tteru istri-b fica-t allt intd as-pd -p sedi s too l in.W ons,w ossi-b nce)w thep

odels can be routinely fitted using standard staal software packages but since we do not havePij ,hat should we do? SetPij =Pi? SetPij =Pi (area oj th pixel)/(area of townshipi)? These spcifications aquivalent since all pixels are the same size. How

he assumption of a uniform distribution across tohips is inappropriate asFig. 9 below reveals. By inroducing the latentPij ’s, theEij , Sij , Rij , andδi allows to learn about them while reflecting our uncerta

n their actual values.

emark 2. More generally, note how our modelixplicitly handles the misalignment issue. To builodel at the town level for thePi , what should we us

or Ei , Si , andRi? To build a model forLij , we shouldntroduce aPij .

emark 3. It is perfectly fine to use the covariatij , Sij , andRij , to characterize the joint distributio

nly within a Bayesian framework. That is, we halready provided the joint distribution forL, W andPiven the model parameters. If we view these lanknowns as random and add a so-called prior dution for them we have a complete model speciion, i.e., a specification for the joint distribution ofhe variables in the hierarchical model. With this joistribution we can provide inference regarding anyect of the model. The prior distribution for theδ’s wasescribed above. Forα,β,γ2,γ3 andτ2, the general aproach is to use whatever prior or partially data-ba

nformation we may have regarding these unknownbtain some idea of what range they are likely to fale use this range to develop proper prior distributihich are as non-informative (i.e., as vague) as ple (in order to let the observed data drive the inferehile still retaining stable computation. Details ofrior specifications are supplied in Appendix A.


The resulting model is fitted using simulation-basedmethods, i.e., Gibbs sampling (Gelfand and Smith,1990) and Markov chain Monte-Carlo methods (see,for example,Gilks et al., 1996). The output from suchsimulation is sampled from the joint posterior distribu-tion of all model unknowns. Unfortunately, such modelfitting requires considerable effort and time. (In fact,we summarize the required full conditional distribu-tions and provide some pseudo-code for implement-ing the sampling in Appendix B.) The reward is exactinference (without relying on possibly inappropriateasymptotics) and more accurate measurement of vari-ability by capturing the uncertainty regarding the modelunknowns, which is not obtained using classical statis-tical approaches. In fact, we know of no other way tofit models(1)–(5). Simplified versions may be fittedusing the E–M algorithm (Dempster et al., 1977) toobtain likelihood-based inference. But then we haveconcerns regarding associated asymptotic variance es-timates. It may also be possible to fit the model stagesseparately rather than simultaneously. Again, we areconcerned that this will misrepresent the extent of vari-ability. These relate to issues of model choice, whichwe discuss below.

7. Imputing missing land-use classification

Table 1shows the distribution of land-use classi-fications for the North and South regions. Note that6 toc uth.O thate thep wn,g notd omo hocd ct toi om-p thei usec dis-t ulds l tob dos ourr sess

the variability in the model unknowns. Various impu-tation models could be adopted; we employed a conve-nient Potts model (Potts, 1952; Green and Richardson,2002). Details are supplied in Appendix C. We usedthree imputations, finding negligible differences in theresulting parameter estimates across these imputations.

8. Model determination

The model formalized in(1)–(5) is elaborate andchallenging to fit. Is the effort justified? Here we con-sider three simplifications resulting in easier to fit mod-els and, using a model choice criterion, show that theyare not as good.

For instance, a crude model could ignore the pop-ulation model(3). Instead, we could fit the land-usemodel in(5) inserting for the unknownPij ’s an areallyallocated portion of the total township population toeach pixel within the township. In other words, if thereareni pixels in townshipi, we setPij =Pi /ni . Hence wehave only the one stage model in(4) and(5) to fit.

An improvement in this naive approach first fits themodel in(3) (with or without spatial effects). This fit-ting considers thePi ’s as the only data, thePij ’s arelatent variables, and estimates thePij ’s along withβ

(andδ if spatial effects are included). The resultant fit-ting provides posterior meansE(Pij |Pi) which couldthen be inserted into the land-use model(5). These ex-pected values are anticipated to provide better estimateso ur-tw ss,w otk int spa-t useo de a( us-t m.A tes,i icht esti-m ire.

sionoa n-

.1% of classifications are missing (primarily dueloud cover) in the North and 9.5% are in the Sone of our fundamental modeling assumptions isach pixel belongs to one and only one town andixel level population, when aggregated over a toives the known town level population. Thus, we canelete pixels with missing land-use classification frur analysis. Rather than relying on subjective or adetermination of these missing pixel values, we ele

mpute missing land-use values in order to have a clete set for fitting our hierarchical model. In fact,

mputation provides an entire set of missing land-lassifications using a neighbor-based joint spatialribution for the pixel values. Such a distribution shoupport (but not require) land use for a given pixee similar to that of its neighbors. Moreover, weeveral such imputations to see the sensitivity ofesulting inference to the imputation, to better as

f the truePij than those from the crude allocation. Fhermore, with the inclusion of theδi , theE(Pij |{Pi})ill tend to be smoother than without. But regardlee are still failing to capture the uncertainty in nnowingPij ; we will underestimate the variabilityhe overall model. We note that in the absence ofial effects, our fitting approach is analogous to thef the E–M algorithm to obtain maximum likelihoostimates (Dempster et al., 1977) with incomplete datthePij are missing). In fact, we can also obtain the comary likelihood inference using the E–M algorithgain, concern is with the resultant interval estima

.e., the approximate normality assumption on whhey are based and the goodness of the variabilityates (estimated standard errors) which they requTo compare the foregoing models we use a ver

f the posterior predictive loss approach inGelfandnd Ghosh (1998). Using squared error loss for co


Table 2Model comparison for the North and South regions

North South

Model G P D G P D

I (without population) 10949.68 11911.42 22861.10 15317.60 16788.42 32106.02II (without spatial) 10776.78 11489.25 22266.03 14667.41 15407.17 30074.58III (full bayesian) 10751.74 11460.42 22212.16 14670.59 15404.39 30074.98

Gi : goodness of fit term;Pi : penalty term;Di : combined.

venience this approach provides a criterion, which iscomposed of two parts. One measures goodness of fit(G) of the model; the other is a penalty term (P), penal-izing model complexity. In other words, more complexmodels will tend to fit better but should be penalizedfor their “size” in order to encourage parsimony.

Specifically,

G =159∑i=1

ni∑j=1

4∑l=1

(Lijl,obs− πijl)2

and

P =159∑i=1

ni∑j=1

4∑l=1

V (Lijl)

where Lijl ,obs= 1 if Lij ,obs= l, 0 otherwise,πijl =P(Lij = l|Lobs) and V(Lijl ) is the predictivevariability of Lij�, i.e., πijl(1 − πijl). Recall that foreachi, j, oneLijl ,obs= 1 and the remaining three are0. UnderG, a model is incrementally rewarded atan i, j pair if its πijl ’s (which also sum to 1) behavesimilarly to theLijl ,obs. UnderP, a model is rewardedat i, j if it produces “informative,” i.e., extremeπijl ’s. In calculatingG and P we could sum over only theobservedLij ’s or also over the imputedLij ’s. Since the

imputation was implemented apart from the modelfitting, we chose the latter.

It is useful to record bothG andP (in fact, in somecases it might be useful to plot the components ofGandP. However, if we seek to reduce model choice toa single number, then we choose the model which min-imizesD=G+P. Table 2presents the values ofD, GandP for both the North region and the South region forModel I (simple allocation, without population model)Model II (improved model, includes population model,but not spatial effects) and Model III (the full Bayesianmodel in(1)–(5)). Clearly, Models II and III are pre-ferred to Model I and, at least in the North, Model IIIis preferred to Model II.

9. Model results

Table 3provides point estimates and 95% credibleintervals for both the population and land-use mod-els. For comparison,Table 4presents point estimatesand associated interval estimates for Model II fitted us-ing an E–M algorithm. We see that with regard to thepopulation model, the two models provide very similarpoint estimates but that the approximate likelihood in-

Table 3Median and 95% credible interval (in parentheses) for the full Bayesian Model III

North South

P.412,−682, .051, 1.158, 2.4

L24, 4.2

.0198,

.6649,−

.0119,−

opulation model parametersβ1 (elevation)× 10−4 −8.210 (−8β2 (slope) .0704 (.0β3 (roads) 1.157 (1.1τδ2 1.503 (1.0

and-use model parametersα1 (elevation)× 10−3 4.144 (4.0α2 (slope) −.0089 (−α3 (roads) −.5931 (−α4 (population) −.0101 (−

8.038) −6.487 (−7.028,−5.986)725) −.0226 (−.0302,−.0150)63) .2124 (.1922, .2314)23) 1.791 (1.297, 2.455)

42) 1.942 (1.874, 2.018).0020) .0415 (.0306, .0526).5232) −.4587 (−.5373,−.3892).0081) −.0028 (−.0033,−.0024)


Table 4Point estimates and 95% interval estimates (in parentheses) obtained from the EM algorithm (Model II)

Region North South

Population model parametersβl (elevation)× 10−4 −8.217 (−8.287,−8.147) −6.397 (−6.467,−6.327)β2 (slope) .0703 (.0695,−0711) −.0241 (−.0253,−.0229)β3 (roads) 1.157 (1.154, 1.160) .2182 (.2151, .2213)

Land-use model parametersα1 (elevation)× 10−3 4.304 (4.292, 4.316) 2.084 (2.076, 2.092)α2 (slope) −.0075 (−.0191, .0041) .0621 (.0499,−0742)α3 (roads) −.7377 (−.8232,−.6522) −.4744 (−.5548,−.3940)α4 (population) −.0019 (−.0033,−.0005) −.0024 (−.0028,−.0020)

ference in Model II produces much tighter confidenceintervals revealing the significant underestimation ofuncertainty. For the land-use model, at least for eleva-tion, this continues to be the case. Thus, for the remain-der of this section we describe the results for Model III.

For this model, lower elevation and presence of roadnetworks are associated with population settlement inthe North as well as the South with the effects beingmore significant in the North. In the South, lower slopeis associated with higher expected population, which isconsistent with what we anticipate. In the North, how-ever, higher slope is associated with higher expectedpopulation, contrary to what one might expect. Turningto the land-use model, the coefficient for population al-though small is significantly negative for the North andthe South with the influence being more pronounced inthe North. In fact, the interval estimates for the Northand South do not overlap. Increased population pres-sure is associated with increased chances of deforesta-tion with the effect being less in the South. Elevationis associated with forest cover with the effect beingmore pronounced in the North. Here too, the intervalestimates for the two regions don’t overlap. Presenceof road networks significantly increase the chance ofdeforestation in the North as well as South. Slope ismixed with regard to significance. While it emerges asinsignificant in the North, it significantly increases thechances of forest cover in the South.

Recall that the spatial random effects, theδi ’s, wereintroduced to provide spatial smoothing to the popu-l g tot el,a ean0 this,i s of

theδi ’s.The figure clearly reveals that spatial proxim-ity encourages similarity in the expected spatial effectsof population processes. Here, lighter colors indicatea negative expected spatial effect, i.e., the adjustmentto the mean population for pixels in that township willtend to diminish population, vice versa for the darkercolors.

We have claimed that a model for pixel populationsuch as(3) is more effective than a naive areal allo-cation of township population to pixels. The results ofthe model comparison presented inTable 2certainlysupport this. Here, we present some further informalsupport. InFig. 9 we have created a map of the pos-terior mean populations (on the square root scale) atthe pixel level. The square root scale is employed sincethis is the customary symmetrizing and variance sta-bilizing transformation for count data. The smoothingof the population is evident at finer pixel resolutions.Overlaid on this map is the population center dot map.The dots locate the villages, town centers or larger ad-ministrative centers. The size of the dot roughly corre-sponds to population size. Though the dot map couldnot be reliably used to model pixel level population,it is evident fromFig. 9 that, generally, there is goodagreement between the maps.

10. Discussion

Returning to the model issues section, we sum-m havef ale.W dela enp tion

ation model which in turn induces such smoothinhe land-use model. A priori, under the CAR modll δi ’s have essentially the same distribution and m. A posteriori, spatial pattern emerges. To see

n Fig. 8 we present a map of the posterior mean

arize first what has been accomplished. Weormulated an ordinal land-use classification sc

e have built a phenomenological hierarchical mot the pixel level, which explains land use givopulation and other covariates as well as popula


Fig. 8. Population model spatial random effects shown for the study regions.

given other covariates. We have accommodated themisalignment between the population data layer (at thetownship level) and the land-use data layer (on a 1 km2

pixel grid). We have employed a spatial imputation(simulation) strategy to handle the missing land-usepixels in an objective fashion. We have introducedspatial smoothing for the population model, theonly feasible possibility given the data available. Wehave uncovered significant explanation in the meanstructure from the elevation, slope and roads datalayers in both the population and land-use models.In this regard, we have found a significant negative

coefficient for population in the land-use model.Increased population pressure increases the chanceof forest degradation. We have argued, through amodel comparison criterion, that a more sophisticated,hierarchical model, which includes spatial effectsexplicitly and a population model outperforms modelslacking these specifications. We have also argued anddemonstrated that our hierarchical model capturesuncertainty, which was underestimated by simplermodels. Lastly, we have shown the sort of statisticalinference that is possible with our hierarchical model,particularly with regard to the spatial smoothing.


Fig. 9. Expected population for the study regions (on the square root scale) at the pixel level with the population vector point map overlaid.

10.1. Historical context of landuse patterns inMadagascar

Before we discuss the patterns revealed in our anal-ysis, it is instructive to review the historical context ofland-use change in eastern Madagascar. Based on ourearlier summary of deforestation trends, controversysurrounds issues of rates and timing of forest loss inMadagascar. Claims that most of the forest loss has oc-curred in recent decades are clearly misleading. The

French colonial archives clearly shows concern overforest loss throughout French occupancy (Jarosz, 1993;Perrier de la Bathie, 1921). In fact the claim, certainlyexaggerated, was made that some 70% of the primaryforest was destroyed between 1895 and 1925 (Hornac,1943–1944, cited by Jarosz, 1993). In any case thestandard explanations offered for forest loss was theextensive nature of traditional slash-and-burn uplandrice culture practiced by the locals, forest concessionsencouraged by the French with associated destructive


logging practices, and the advocacy of selective forestconversion to plantation and cash crops (Jarosz, 1993;Perrier de la Bathie, 1921). It is likely that all of theseactivities would have been encouraged by any site fac-tors that eased access to forested blocks, i.e., adjacencyto villages and towns, access by road or even foot-path, and lower, shallower slopes. Land tenure systemschanged with the 1926 decree that all vacant land be de-clared state land. Local populations no longer held titleto ownership over ancestral lands that might currentlybe unoccupied (Esoavelomandroso, 1985). Moreover,during the late 19th and early 20th century several lawswere passed prohibiting the burning of forests (Jarosz,1993). Together these laws may well have acted to dis-enfranchise local populations with respect to traditionalland conservation practices, actually exacerbating landdegradation.

Yet despite these observations, there are many indi-cations that deforestation processes have been on go-ing for centuries. Few scholars have looked for olderrecords of forest extent in Madagascar. Such exist,in the form of historical accounts of the 17th–19thcentury (Ellis, 1858; Flacourt, 1658; Grandidier etal., 1920). These narratives make it clear that thedistribution of forests were patchy and geographi-cally patterned long before the French colonial pe-riod. It is likely that modern patterns of forest distri-bution reflect, in part, forest distributions of the rel-atively distant past. The patchy nature of the east-ern wet forests was evident to mid-17th century ex-pM ain-l rav-ea capef hisl andg s, andm ito longt notm iss wasf ringa era( 97c era(

The early accounts provide a picture of periodic cy-cles of landscape clearing for agriculture and abandon-ment with some evidence for forest regrowth. In partthis reflected local cycles of human population growthand decline (Campbell, 1991; Kent, 1992), with the re-sult being a very heterogeneous landscape. The histori-cal perspective is much clearer for the northern sectionof the region than the southern. This reflects the infre-quent visits of Europeans to the southern half of oureast coast focal region.

In the central northern area (around Fenerive), therewas a shift in human settlements evident in about the17th century (Wright and Dewar, 2000), with the aban-donment of coastal villages, and the establishment ofnew villages on defensible hilltops at a distance fromthe coast (cf.Flacourt, 1658: Map of Isle Ste. Marieand adjacent Madagascar mainland). This is also re-flected in the narratives of 19th century travelers (cf.Grandidier et al., 1920; Ellis, 1858). Many of thesenew villages were much larger than previous occupa-tions, so that local population densities were increas-ing, and may have led to concomitant increase in localanthropogenic effects on the countryside. All of thesechanges are most plausibly accounted for by an in-crease in insecurity, and the virtual absence of tradingor slaving along the southern coastal area (Kent, 1992),would have meant that population distributions andlocal environmental effects likely followed divergentpaths.

1

exto hasc pula-t um-b aturef om-p 60s,d(t tel-l is am lands,w hillc hestm ts ofc ecent

lorers (Grandidier et al., 1920, cf. Flacourt, 1658:ap of Isle Ste. Marie and adjacent Madagascar m

and), as well as the early and mid-19th century tlers (Hebert, 1980; Ellis, 1858). All of these earlyccounts describe large tracts of deforested lands

rom the sea up to 600 m on the mountains. Tandscape was clearly a patchwork of cultivatedrazed lands, secondary forests, abandoned landature forests.Ellis (1858) characterized the limf closed forest in the mid-1800s as a band a

he eastern mountains from 25 to 40 miles wide,uch different from what is found today. There

ome evidence that extraction of forest resourcesrequent and widespread, together with land clealong the east coast during the 7th–11th centuryDomenichini-Ramiaramanana, 1988; Dewar, 19)ontinuing up to the beginning of the colonialKent, 1992).

0.2. Forest changes over the past century

Putting the historical information into the contf current land use, it is clear that deforestationontinued up to the present, exacerbated by poion growth. The trend has tended to reduce the ner and extent of patches of secondary and m

orests in the landscape. This is easily seen on caring aerial photographs from the 1940s and 19etailed forest patches mapped in the 1960s byDandoy1973), and byBenoit de Coignac et al. (1973)abouthe same time, with recent ground truth and saite images (data not shown). The result today

uch more homogeneous landscape of degradedith any remaining patches of forest relegated torests and protected valleys. Only along the higountain slopes does one find continuous trac

losed, mature forest. These have retreated in r


decades, but maps of large forest tracts today are sur-prisingly similar to those of 1969, 1934, 1921 and1902.

The maps provided byHumbert et al. (1964–1965)have figured prominently in depicting current for-est cover for our region (Du Puy and Moat, 1999)and in estimating deforestation rates (Green andSussman, 1990). Original forest cover was taken fromthe thumbnail “Types de Vegetation” maps, which pur-port to show potential vegetation in the absence ofman (Humbert and Cours Darne, 1965; pp. 152–154).These are conjectured maps with little supporting in-formation, and are primarily constructed from the main“V egetation Naturelle” maps of the east coast. Theseare used as the baseline “original extent” of forest coverbyGreen and Sussman (1990), but these provide a mis-leading benchmark at best. The main “Vegetation Na-turelle” maps of Humbert and Cours Darne are usedto portray current forest cover in the most recent mapsof forest cover (Faramalala, 1995; Du Puy and Moat,1999), as well as forest cover in 1950 (Green andSussman, 1990). The Humbert and Cours Darne mapswere assembled by committee and the forest classi-fication was based on surrogate elevation, bioclimateand edaphic variables with uncertain use of aerial pho-tography, and little or no ground truthing. From ourown experiences with aerial photography of the region,vegetation classification without accompanied groundtruthing is difficult and misleading. Based on all avail-able information it is apparent that the forest coverm videa cur-r er-t n oft overo

1

pat-t emo-g LeB tht d in1 thep reada cen-t coast

Over that time, population densities have simply in-creased across the region.

10.4. Interpretation

How does this information integrate with the resultsof our spatial analysis of land-use patterns? Populationlevels are negatively associated with elevation andforested landscape are positively associated with ele-vation in both the North and the South. This obviouslyreflects the current and historical location of settlementarea at low to moderate elevation and the fact that mostof the remaining forest blocks are spatially restrictedto the highest mountain slope or locally on hill tops.The effect of elevation is much higher in the Norththan South. This reflects undoubtedly that fact thatlarge blocks of forest are associated with the highermountains of this section of the province. Slope is neg-atively associated with populated blocks in the Southbut positively associated in the North. This may reflectthe fact that settlements were shifted to fortified hilltops (but not at great elevation) in the North beginningin the 17th century (cf.Flacourt, 1658: Map of Isle Ste.Marie and adjacent Madagascar mainland—the centerof our North region; no comparable maps are extantfor the South). Pixels with hill tops would tend toreflect higher slopes. However, this trend slope was notobserved in the South. Rather settlements tend to bespread out more evenly across the landscape over timein areas of convenience, likely avoiding areas of slopea atiald . Int withf iths hillt s ands thatt withg lopec y bel apeo xt.W atialr tionp .T ngu-l ueso

aps produced by Humbert and Cours Darne progross over estimate of the extent of forest either

ently or for any point over the past half century. Cainly this is true based on even a cursory comparisohe Cartes Forestier with current or recent forest cn the ground.

0.3. Demographic trends

One can gain some insight on past populationserns across the landscape from the available draphic maps (de Martonne, 1911; Gourou, 1945;ourdiec et al., 1969). All show a trend North to Sou

hat is reflected in the population densities observe993 (cf.Fig. 8). In the South the populations overast 90 years have tended to be more evenly spcross the landscape. In the North the population

ers tend to be more restricted to areas nearer the
.
nd elevation. Slope shows a similar trend in the spistribution of forested blocks, North versus South

he South steep slopes are positively associatedorested blocks. In the North there is no trend wlope. This may reflect the trend of finding someops (and hence steep slopes) occupied by townome by forests. An alternative explanation ishe North may represent a rougher topographyreater elevational change but also with greater shange within and among pixels. The net result maittle effective signal of slope on most of the landscccupied by forest. We explore this possibility nehile there is no generally accepted measure of sp

oughness for an areal unit, a measure of local variaroposed byPhilip and Watson (1986)is widely usedhis measure is computed through automatic tria

ation at locations within the areal unit. Larger valf this index indicate greater roughness. InTable 5, we


Table 5Roughness comparison between North and South regions

QuantilesRegion .025 .25 .5 .75 .975North (21466) 0 .0002 .0033 .0072 .0403South (22950) 0 0 .0018 .0061 .0369

summarize the distribution of this index over 21,466pixels in the North region and 22,950 pixels in theSouth region. The table offers descriptive comparisonsince there is no stochastic modeling of this index.However, the quantiles for the distribution of rough-ness are somewhat larger in the North than the Southproviding quantitative support for a somewhat roughertopography.

11. Conclusions

To summarize, we have provided a new frameworkto explain deforestation patterns in the presence of mis-alignment of data layers, missing data arising fromcloud cover, and incorporating spatially explicit struc-ture through the use of a bayesian hierarchical model.Although illustrated in the context of Madagascar, themethods are more generally applicable to other eco-logical processes as we have noted at the end of Sec-tions 1 and 2. Returning to an examination of pop-ulation block effects, these are negatively associatedspatially with forested blocks, although the effect isfairly weak. This weak signal may reflect the perva-sive influence of slash-and-burn agriculture practicedthroughout the landscape even areas with low popula-tion densities. This agricultural practice may be moredetermined by accessibility than proximity to popula-tion centers (Oxby, 1985). Elevation obviously playsinto this, as do roads of any sort, or indeed even foot-p ly as-s as-s teso t thep asa withr ationp pa-t pe.B nuet u-

lation centers and forested patches observed today inthe landscape and appear to account for the differencesseen between the northern and southern sections of theprovince.

Generalizing from our specific results, it is evidentthat the structural hierarchical modeling style we haveemployed is applicable to the modeling of land-use pat-terns in many other contexts. Similarity in land use be-tween neighboring areal units would be expected, andspatial random effects could be used to capture such as-sociation. Misalignment problems among data layers,hence, between response and explanatory variables arealso likely to be common themes.

Our present application has led to a model for landuse, which is static, since we lack temporal informa-tion. However, were data available across time, wecould extend the modeling in(1)–(4). Formally, weneed only add a subscriptt to those measurementswhich change over time. Mechanistically, we mightthink of the land-use process as evolving in both spaceand time. Spatio-temporal random effects,δit could beintroduced to capture association across both space andtime.

Lastly, while the present setting has an ordinal cate-gorical response variable, in other applications the re-sponse could be binary, e.g., presence or absence of aspecies, or a count, e.g., abundance of a species. Thefirst stage model for the response would change to re-flect this but hierarchical modeling with spatial struc-ture could still be employed and would provide thes vail-a

A

SF( hnT .E.D ain,S o-v anal-y ed-b ess,a dexw andR them

aths. These transportation networks are negativeociated spatially with forest blocks and positivelyociated with inhabited blocks. Many of these rouf transport have been used for centuries. In facaths followed by Ellis in the mid-1800s continueccess routes used today. In sum elevation alongoad/path networks, and to a lesser extent populatterns play an important role in explaining the s

ial distribution of forested blocks in the landscaut in addition historical patterns of land use conti

o play a pervasive role in the distribution of pop

ame benefit in terms of richer inference than is able with standard methods.

cknowledgements

This project was supported by grants from NDMS 99-71206) to A.E. Gelfand and from the Jo. and Catherine D. MacArthur Foundation to Rewar and J.A. Silan-der, Jr. Roland de Gouventeven I. Citron-Pousty, Daniel L. Civco kindly prided assistance on some of the GIS componentses of this study. Thomas H. Meyer provided feack and discussion on spatial indices of roughnnd implemented software for estimating the ine use here. Andrew Latimer, Robin Chazdonichard Primack provided critical comments onanuscript.


Detail on prior specifications

To complete the specification of the hierarchicalmodel we require priors forα, γ, β, γ2, γ3 andτ2. Toobtain a well-behaved chain, we use priors forα andβ that are “data-centered” with large variances. Infer-ence is not sensitive to this centering (we could use saymean 0 centering) but our Markov chain Monte-Carloalgorithm burns in more rapidly with such centering. Inparticular, we assumeβ∼N(β, kDβ) whereβ is a pointestimate obtained from an expectation maximizationalgorithm (Dempster et al., 1977), which imputes pop-ulation at the pixel level andDβ is the correspondingdispersion matrix. Forα, we adoptN(α, kDα) obtain-ing the estimates by fitting a multinomial ordinal re-sponse model (using standard statistical packages likeSAS) that areally allocates the town population to eachpixel within the town. We experimented with severalvalues ofk and found little sensitivity fork> 10. Forthe two unknown cut-pointsγ2 andγ3, we assume auniform prior subject to the constraint 0 <γ2 <γ3 <∞.For τ2 we adopt an inverse gamma prior. In particularτ2 ∼ IG(2, .23). This specification has infinite variancewith mean roughly the sample variability in the logλi(whereλi = Pi.). As is customary, to ensure identifi-ability, i.e., a well-behaved posterior distribution, weimpose the constraints

∑iδi = 0. (Besag et al., 1995).

Brief detail on model fitting

ofe d) int yg lu

• nn

nor-

•

•-in

(5). Sampling from a multivariate distribution isstandard.

• Pi∼ multinomial(Pi ,λij /λi))∏

j exp(−.5(eij −α4Pij )2)where eij =Wij −α0 −α1Eij −α2Sij −α3Rij , i = 1,. . ., T where T is the number of townships. WeupdatePi using a Metropolis step with proposaldensity which is multinomial (Pi , λ∗

ij/λ∗i.) where

λ∗ij = λij exp(α4eij). We have found this to work

very well with acceptance rate of around 50%.• δi ∝ exp(−λi)λ

Pi

i N(∑

jωijδj/∑

jωij, τ2/∑

jωij),i = 1, . . . , T . This is log concave and updatedusing adaptive rejection sampling (Gilks and Wild,1992). Code is available fromhttp://www.mrc-bsu.cam.ac.uk/BSUsite/Research/ars.shtml

• βl ∝ (∏

i

∏j exp(−λij)λ

Pij

ij )N(β, kDβ), l = 0, 1, 2, 3.This is also log concave and up dated using adaptiverejection sampling. Take care with regard to under-flows when evaluating the log density.

τ2 ∼ Inverse gamma (shape =T/2+2, scale = . 23 +∑i–j(δi −δj)2/2) where the relationi–j meansi andj

are neighbors. This is easily sampled using standardroutines for simulating from the gamma distribution(seeDevroye, 1986).

Psuedo code to fit model HI

• Initialize θ to θ0. We setPij0 =Pi /ni and takingα0,γ20, �30 estimates obtained from fitting a multino-mial ordinal response model andβ0 estimates ob-

tial-

•• rate

nfer-

D

gl sh-i hatLt

We briefly describe the conditional distributionsach variable and how they get updated (simulate

he gibbs sampler. At timetwe denote this operation bibbsup-date(θt, θt+1) whereθ is a vector of all modenknowns that are to be updated.

We assume theW’s have a normal distributiowith mean given by(5) and variance 1. TheWij∼N(E(Wij),1)1(γLij−1 < Wij ≤ γLij ) and getsupdated using draws from univariate truncatedmals.γ l ∼ Uniform(a, b); a= max(�l−1, max(Wij s.t.Lij = l)); b= min(γ l+1, min(Wij s.t.Lij = l + 1)), l = 2,3, . . .α∼N((X′X + D−1

α /k)−1

(X′W + D−1α α/k),

(X′X + D−1α /k)

−1) where X is the design ma

trix that results from the linear regression

tained by fitting Poisson regression assumingP=P0.The other parameters would be automatically iniized by the operationgibbsupdate(θ0,θ1).for i in 1:(B+ Th× Nsamp)gibbsupdate(θi−1, θi)Discard the first B iterates, store every Thth itethereafter and use the Nsamp iterates to make ience

For the current application,B= 20000, Th = 50and Nsamp = 1000 were appropriate.

etails on the multiple imputation

The imputation for all of the pixels with missinand-use classification is carried out in an iterative faon using a joint spatial Potts distribution. Recall tij denotes the classification for thejth pixel in theith

own andL denotes the set of allLij . The joint density

http://www.mrc-bsu.cam.ac.uk/bsusite/research/ars.shtml


of L under the Potts model is

f (L) ∝ exp(φ > U(L)) (C.1)

whereU(L) =&1(Lij =Li′j′ ) and the summation is overpairs of neighboring contiguous pixels andφ is an as-sociation parameter. That is,φ = 0 indicates no spatialassociation whileφ > 0 encourages classification agree-ment between adjacent pixels. Hence the conditionaldistribution ofLij given all of the other pixels

f (Lij|remainingL′s) ∝ exp(φ∑

1(Lij = Li′j′ )

(C.2)

where the summation is over the neighbors ofLij . Inother words,Lij takes one of the values 1, 2, 3 or 4.From(C.2), the conditional probabilities of these val-ues reflect the relative agreement each one has with thevalues of the neighbors ofLij .

The iterative updating proceeds as follows: Givenφ,let L(0) denote all of the observed land-use classifica-tions with a random assignment of initial values for allof the unobserved pixels. We then update in any con-venient sequence, each missingLij to L

(1)ij using(C.2)

where the remainingL’s are fixed at current levels. Aniteration is completed after each missingLij is updated.We ran 10,000 iterations keeping the final set of iter-ated values as one imputation for the missingLij ’s. Toobtain a new imputation, we can either restart the it-eration at a new randomL(0) or else continue iteratingfor another 10,000 iterations retaining the values at thats

ti-m .F ofφ ardf e-lE ise aba vC ntedc onr ortha gionf ion.M .22a

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