# Markov analysis of land use dynamics A Case Study in Madagascar

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2 2 Volume 22 opulation pressure is one o the major causes o deorestation in tropical countries. In the region o ianarantsoa (Madagascar), two national parks Ranomaana and ndringitra are connected by a orest corridor, which is o critical importance to maintain the regional biodiversity, see igure. The need or cultivated land pushes people to encroach on the corridor. To reconciling orest conservation with agricultural production, it is important to understand and model the dynamic o post-orest land use o these parcels. We will use a data set developed by IRD, see igure, corresponding to 42 parcels and 22 s, rom 985 to 26. The states are : annual crop (), allow (), perennial crop () and natural orest (). The state will be omitted in the inal observation series as it is transient and it provides no inormation. The state is systemically ollowed by the state that will be considered as the irst observation. The observation series will be denoted (e (p) :N p ) p=:42 where e (p) n belongs to the state space E = de {,, } and N p is the length o the observation series o the parcel p. Here the notation n = n : n 2 stands or n = n,..., n 2 or n n 2.. irst model derived rom the maximum likelihood estimate We make the ollowing hypothesis : (H ) The dynamics o the parcels are independent and identical, they are Markovian and time-homogeneous. This means that (e (p) :N p ) p=:42, are 42 independent realizations o a same process (X n ) n and that this process is Markov and time-homogeneous. This assumption is o course simplistic as the dynamics o a given parcel may depends on : armer decisions ; exposition, slope and distance rom the orest, that means properties o the same plot ; neighboring parcels. ssumption (H ) however will prove interesting in this irst study, it will allow us to build a simple model that nevertheless lead us to interesting results. To identiy the 3 3 transition matrix Q (6 parameters) we irst use the maximum likelihood approach, this consists simply in calculating the empirical transition matrix, i.e. the the relative occurrences o all the transitions (see [3] or details) : ˆQ ij = #{i j}/ j E #{i j } where #{i j} is the number o transitions rom i to j in the dataset. The resulting matrix is depicted as graph in igure 2 (let : model ). Naturally as the transitions, and are not in the initial data set, they do not appear in the model. The main implication is that the state is absorbing and the other states are transient so that the limit distribution is δ. So a irst question is to describe the

3 4 igure Let : The study area bordering the orest corridor between Ranomaana and ndringitra. Right : nnual states corresponding to 42 parcels and 22 s. These data were collected between the s 985 and 26. The parcels are located on the slopes and lowlands on the edge o the orest corridor o Ranomaana-ndringitra, Madagascar, see igure. The states are : annual crop (), allow (), perennial crop () and natural orest (). The state, that will be omitted, is systemically ollowed by the state that will (p) be considered as the irst observation. Hence the observation series (en=:np )p=:42 will be the sequence o states {,, } and Np will be the length o the observation series o the parcel p, e.g. N = 7, N2 = 6 etc. orest corridor protected area district boundary township boundary study area ianarantsoa Volume 22 LEGEND 5 parcel number 3 Kilomètres ndringitra National ark ³ Ranomaana National ark Tableau nnual states corresponding to 42 parcels and 22 s. These data were collected between the s 985 and 26. The parcels are located on the slopes and lowlands on the edge o the orest corridor o Ranomaana-ndringitra, Madagascar, see igure. The states are : annual crop (), allow (), perennial crop () and natural orest (). The state, that will be omitted, is systemically ollowed by the state that (p) will be considered as the irst observation. Hence the observation series (en=:np )p=:42 will be the sequence o states {,, } and Np will be the length o the observation series o the parcel p, e.g. N = 7, N2 = 6 etc. The dynamics o the parcels are independent and identical, they are Markovian and time(h ) homogeneous. Markov analysis o land use dynamics 3

4 4 Volume 22 behavior o the process beore it reaches and a second question is to analyze the time length to reach the state, that is the distribution law o τ where : de τ e = in{n : X n = e} i n such that X n = e, + otherwise. () is the irst time to reach a given state e. The answer to the irst question is given by the so-called quasi-stationary distribution, it is roughly the limit distribution that the system reach beore being absorbed by. We consider the probability to be in e {, } beore reaching and starting rom, i.e. µ n (e) = (X n = e τ > n, X = ) where τ is the irst time to reach. Suppose that µ n (e) σ qs (e) as n or e {, } then the probability distribution (σ qs (e)) e {,} is called quasi-stationary probability distribution. This problem was originally solved in [6] : σ qs = [σ qs (), σ qs ()] exists and is solution o ( ) Qqs q σ qs Q qs = λ σ qs, with Q = (2) with σ qs (e), σ qs () + σ qs () =, and λ is the spectral radius o Q qs. In (2), Q qs is the submatrix o Q corresponding to the 2 irst states. rom (2) we can compute the quasi-stationary distribution σ qs associated with the MLE ˆQ o Q. We get : σ qs = (σ qs (), σ qs ()) = (.5794,.426). Hence, conditionally to the act that the process does not reach, it will spend 58% o its time in the state and 42% o its time in the state. The computation o the distribution law o τ is explicit, we get : time n The mean time to reach is 36 s with a standard deviation o 35 s. 2. second model derived rom a Bayesian approach The model derived rom the maximum likelihood estimate does not allow the transitions, and. There is no logic or the transitions and, but the transition can be observed on a time scale o several decades. So we suppose that : (H 2 ) The transitions and are not possible, all other transitions are possible. This hypothesis leads to a transition matrix Q with our parameters θ = (θ i ) i=,2,3,4. The maximum likelihood approach will lead to the model o igure 2 (let) where Q(, ) =. We thereore have to make use o Bayesian methods. Given a prior distribu-

5 Markov analysis o land use dynamics 5 tion law π(θ) on the parameter θ, ccording to the Bayes rule, the posterior distribution law π(θ) on θ given the observations X is : π post (θ) L 2 (θ) π prior (θ) (3) ( means that π post (θ) = L 2 (θ) π prior (θ)/ L 2 (θ ) π prior (θ ) dθ ) where L 2 (θ) is the likelihood unction, let l 2 (θ) be the corresponding log-likelihood unction. The derivation o these expressions are straightorward, details can be ound in [3]. The Bayes estimator θ o the parameter θ is the mean o the a posteriori distribution : θ de = Θ θ π post(θ) dθ = θ L2(θ) πprior(θ) dθ Θ Θ L2(θ) πprior(θ) dθ. (4) Numerical tests have been perormed and suggest that the Jereys prior is well adapted to the present situation. The Jereys prior distribution (non-inormative) is deined by : π prior (θ) ( det[i(θ)] where I(θ) = [E de θ 2 l 2(θ) θ k θ l, (5) )] k,l 4 I(θ) is the isher inormation matrix. gain, see [3] or a precise expression o π prior (θ). lthough the Jerey prior distribution is explicit, we cannot compute analytically the corresponding Bayes estimator (4). We use a Monte Carlo Markov chain (MCMC) method, namely a Metropolis-Hastings algorithm with a Gaussian proposal distribution. or the real data, the Bayes estimation o the transition matrix with the Jereys prior and the MCMC method leads to the transition matrix Q depicted as a graph in igure 2 (right : model 2). Note that the probability o the transition is now strictly positive. Moreover, as [ Q 2 ] ij > or all i, j, Q is a regular transition matrix (i.e. irreducible and aperiodic) and so there exist a unique invariant measure σ solution o σ = σ Q. ter computation : σ = (.427,.365,.288). 3. Discussion We derived two Markovian models, see igure 2. In the irst model, is an absorbing taste and the limit distribution will charge this state only. Model 2 is regular and admits a limit invariant measure which is strictly positive or the 3 states. The dierence is that or the irst model the probability o the transition is null and it is.2 or the second model. Starting rom the state, the evolution o the o parcels in state, and or the two models is given in igure 3 (let). These evolutions are almost identical in an horizon o 2 s, they appear to be dierent only ater ew decades. They are radically dierent on the long scale o centuries : or model almost all parcels are in the state, model 2 converges to an equilibrium ater s. In the irst model we saw that the mean time to reach the state is 36 s.

6 6 Volume annual crop.2 annual crop.2.24 allow.34 perennial crop.24 allow perennial crop.66 Model (Maximum likelihood estimation).68 Model 2 (Bayesian estimation) igure 2 Markov models derived by the direct maximum likelihood approach (let) and the Bayesian approach with a Jereys prior. In the second model, the equilibrium is reached ater one century and it is 4% o parcels in state, 3% in state, 28% in state. Model presents a quasistationary behavior. Indeed i we consider the evolution o the relative o parcels in state and in state, model and model 2 present quasi identical proiles, see igure 3 (right). Hence ater less than 5 s, the relative o parcels in state and in state is close to 58% and 42%. In [3] we assessed the adequacy o the model to real data. We tested i the empirical sojourn times correspond to a geometric distribution. We used a parametric bootstrap goodness-o-it on empirical distribution. Not that in the Model 2, the probabilities o transition and o transition are both equal to.2, still the variance on the estimation o the transition probability, which appears in the data set, is smaller than the variance on the estimation o the transition probability see details in [4]. 4. erspectives new database is currently being developed by the IRD. It will be or a longer period o time and a greater number o parcels, it will also allow to consider a more detailed state space comprising more than three states. art o the complexity o these agro-ecological temporal data comes rom the act that some transitions are natural, due to ecological dynamics, while others come rom human decisions (annual cropping, crop abandonment, planting perennial crops, etc.). It should also be interesting to study the dynamics o parcels conditionally on the dynamics o the neighbor parcels. This model could be more realistic but requires irst studying the armers practices in order to limit the number o unknown parameters in the model. Such spatio-temporal models taking into account the neighborhood dynamics have been already proposed, notably in the context o deorestation in Madagascar [].

7 Markov analysis o land use dynamics 7 model 2 model model 2 model annual crop allow annual crop allow perennial crop states restriction to the states and igure 3 Evolution o the o parcels in state, and or the two models. Let : with the three states ; right : restricted to the two irst states.

8 8 Volume 22 The present results are preliminary, still they allow to prove that the data set time scale permits to iner the addressed questions on the corridor dynamics. In the near uture we will relax the Markov hypothesis and consider semi-markov models. cknowledgment : This works was partially supported by the International Laboratory or Research in Computer Science and pplied Mathematics (LIRIM/INRI program), by the Cooperation and Cultural ction Oice (SCC) o the rance Embassy in Madagascar, and by gence universitaire de la rancophonie (U). 5. Bibliographie [] D. K. garwal, J.. Silander Jr.,. E. Geland, R. E. Deward and J. G. Mickelson Jr. Tropical deorestation in Madagascar : analysis using hierarchical, spatially explicit, Bayesian regression models. Ecological Modelling,85 :5 3, 25. [2] T. W. nderson and L.. Goodman. Statistical inerence about Markov chains. nnals o Mathematical Statistics, 28 :89 9, 957. [3]. Campillo, D. Hervé,. Raherinirina and R. Rakotozay. Markov model o land use dynamics. INRI Research Report, RR-767, 2. [4]. Campillo, D. Hervé,. Raherinirina and R. Rakotozay. Markov analysis o land use dynamics : Case Study in Madagascar. To appear, 22. [5] S. M. Carrière, D. Hervé,. ndriamaheazay, and. Méral. Corridors : Compulsory passages? The Malagasy example. In C. ubertin and E. Rodary, editors, rotected reas, Sustainable Land?, pages IRD Editions - shgate, 2. [6] J. N. Darroch and E. Seneta. On quasi-stationary distributions in absorbing discrete-time inite Markov chains. Journal o pplied robability, 2() :88, 965. [7] M. orrest Hill, Jon D. Witman, and Hal Caswell. Markov chain analysis o succession in a rocky subtidal community. The merican Naturalist, 64 :E46 E6, 24. [8] C. R. Rakotoasimbahoaka, V. Ratiarson, B. O. Ramamonjisoa, and D. Hervé. Modélisation de la dynamique d aménagement des bas-onds rizicoles en orêt. In roceedings o the th International rican Conerence on Research in Computer Science and pplied mathematics (CRI ), October 8 2, Yamoussoukro, Côte d Ivoire, 2. [9] V. Ratiarson, D. Hervé, C. R. Rakotoasimbahoaka, and J.-. Müller. Calibration et validation d un modèle de dynamique d occupation du sol postorestière à base d automate temporisé à l aide d un modèle markovien. Cahiers gricultures, 2(4) :274 9, 2. [] B. C. Tucker and M. nand. The application o Markov models in recovery and restoration. International Journal o Ecology and Environmental Sciences, 3 :3 4, 24. [] M. B. Usher. Markovian approaches to ecological succession. Journal o nimal Ecology, 48(2) :43 426, 979.

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