modelling spatio-temporal movement of tourists using finite markov chains

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 79 (2009) 1544–1553

Modelling spatio-temporal movement of touristsusing finite Markov chains

Jianhong (Cecilia) Xia a,∗, Panlop Zeephongsekul b, Colin Arrowsmith b

a Department of Spatial Sciences, Curtin University of Technology, GPO Box U1987, Perth, WA, Australiab School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476V, Melbourne, Victoria 3001, Australia

Received 24 June 2007; received in revised form 23 June 2008; accepted 30 June 2008Available online 23 July 2008

Abstract

This paper presents a novel method for modelling the spatio-temporal movements of tourists at the macro-level using Markovchains methodology. Markov chains are used extensively in modelling random phenomena which results in a sequence of eventslinked together under the assumption of first-order dependence. In this paper, we utilise Markov chains to analyse the outcome andtrend of events associated with spatio-temporal movement patterns. A case study was conducted on Phillip Island, which is situatedin the state of Victoria, Australia, to test whether a stationary discrete absorbing Markov chain could be effectively used to modelthe spatio-temporal movements of tourists. The results obtained showed that this methodology can indeed be effectively used toprovide information on tourist movement patterns. One significant outcome of this research is that it will assist park managers indeveloping better packages for tourists, and will also assist in tracking tourists’ movements using simulation based on the modelused.© 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Modelling; Spatial and temporal; Finite Markov chains; Movements; Tourists

1. Introduction

The spatio-temporal movement of tourists is a complex process. Nevertheless, it is vital for a park manager tohave some knowledge of these movements as well as locations of popular sites in the park. Agent modelling softwaresuch as RBSim3 [15] can be used to model these movements under different management scenarios. Modeling canalso be undertaken based on the desired level of spatial and temporal accuracy and purpose of trips. For example,spatio-temporal movements can be modeled as continuous processes with high resolution or as discrete processes withlow resolution [13]. Resolution can be measured in both spatial term (recording movements to the nearest centimeter,meter, or kilometer) or in temporal term (recording movements to the nearest second, minute, hour, day, or year). Ifthe objective of the exercise is to understand detailed movements, then modelling will need to be at the micro-level[3,16,24,27]. Otherwise, it should be modeled at the macro-level [10,29].

A variety of techniques are available to observe tourist movements. The traditional tracking techniques are based onobservations and interviews that require the researcher to follow an individual tourist and records his or her movements

∗ Corresponding author. Tel.: +61 8 92667563.E-mail address: [email protected] (J. Xia).

0378-4754/$32.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2008.06.007

mailto:[email protected]

dx.doi.org/10.1016/j.matcom.2008.06.007

J. Xia et al. / Mathematics and Computers in Simulation 79 (2009) 1544–1553 1545

[8]. Participants are also asked to trace or retrace their spatial movements on a cartographic map using self-administeredquestionnaires [9,27]. Recently, new tracking techniques such as global positioning systems (GPS) [1], timing systems[21], camera-based systems [12], personal digital assistants (PDAs) tracking devices [11,20], mobile phone tracking[26] have been utilised to record movement information of tourists with resulting high resolution.

Before they will yield useful information on significant tourist movement patterns, the tourist track data must be firstsimplified and analysed. Data mining methods, such as expectation maximisation (EM) clustering, are used to identifythe spatio-temporal movement patterns of tourists for different tourist topologies [19,28,30]. Statistical methods, suchas log-linear models, logistic-regression and Markov chains are also applied to analyse movement data. For example,the hidden movement patterns of tourists could be identified by the log-linear method [2]. Markov chains techniqueshave been used to explain human patterns of migration [5,25], to modelling human random walks [22] and through topredicting population distribution within a given area for a certain time interval [7].

This paper presents a new methodology for modelling the spatio-temporal movement of tourists at the macro-levelusing Markov chain analysis. The theory of Markov chains provides a powerful technique for analysing a series ofevents that are linked together by the first-order dependence and are used widely in various disciplines to analysetrend and outcomes of events. In this paper it is demonstrated how Markov chains can be applied to calculate theprobabilities of tourists visiting a number of attractions in a well known island. These probabilities are calculated usingdata, which were collected over a period of time. From such analysis it should be possible to equip park managers andother interested parties (e.g. tour operators) to pick out the movement pattern of tourists. A significant outcome of thiswork is that park managers can use the results to develop suitable tour packages and arrange appropriate times andactivities for tourists at each attraction.

This paper is organised as follows. In Section 2 we give a brief review of Markov chain theory whilst in Section 3,the spatio-temporal movement of tourists is described and coded, and the probabilities of movement patterns calculatedaccordingly. Section 4 describes the results of a case study conducted at Phillip Island, Victoria, Australia based onmethods developed in the previous sections. We validate the model in Section 5. Finally, some conclusions of ourstudies are drawn in Section 6.

2. Markov chains

A stochastic process is defined as a family of random variables {Xt}t ∈ T defined on a given probability space andindexed by t belonging to a parameter set T. The set T is often regarded as the time sequence of the process and iseither discrete or continuous. In the discrete case T = {0, 1, 2, . . .} and in the continuous case T = [0, ∞). The rangeof Xt produces a state space S which could again either be discrete or continuous. When S is finite or countable, theprocess is referred to as a discrete stochastic process; otherwise, the process is a continuous stochastic process.

A discrete stochastic process is referred to as a Markov chain (MC) if the future evolution of {Xt}t ∈ T is dependenton its present state. Mathematically, this is defined formally as follows:

A stochastic process {Xt}t ∈ T taking values in a discrete set S, which we may take as the set of positive integersfor convenience, is a Markov process if, for any sequence t0 < t1 < t2 < · · · < tn < t, positive integer n and valuesi0, i1, . . . , in and j ∈ S, the following identity involving conditional probabilities holds:

Pr(Xt = j|Xtn = in, Xtn−1 = in−1, . . . , Xt1 = i1, Xt0 = i0) = Pr(Xt = j|Xtn = in) (1)

A comprehensive coverage of the theory of Markov processes and its diverse applications can be obtained from excellentsources such as [14,17,23].

When T = {1, 2, . . .}, the set of conditional probabilities

Pr(Xn+1 = j|Xn = i) = pij(n) (2)

for i, j ∈ S and n = 1, 2, . . . is called the set of one-step transition probabilities of the MC. One-step transitionprobabilities satisfy

(i) pij(n) ≥ 0 for all pairs i, j ∈ S.(ii)

∑j ∈ S pij(n) = 1.

1546 J. Xia et al. / Mathematics and Computers in Simulation 79 (2009) 1544–1553

If the distribution of the initial state, i.e. X1, is known, say

Pr(X1 = i) = vi

i ∈ S, then, applying the product rule of probability and identity (1), the joint probability of the event

{X1 = i1, X2 = i2, X3 = i3, . . . , Xn = in}is given by

Pr(X1 = i1, X2 = i2, X3 = i3, . . . , Xn = in) = vi1 pi1i2 (1) pi2i3 (2) pi3i4 (3) . . . pin−1in (n − 1) (3)

A MC is stationary if its one-step transition probabilities pij(n) are independent of n. In other words, the probabilityof moving from one state to another is invariant of the epoch at which the transition takes place. In this case, we writepij(n) = pij for all i, j and the joint probability (3) can be simplified to

Pr(X1 = i1, X2 = i2, X3 = i3, . . . , Xn = in) = vi1 pi1i2 pi2i3 pi3i4 . . . pin−1in (4)

One-step transition probabilities of a stationary MC can be arranged in a matrix, denoted by P, called the one-steptransition probability matrix, where

P =

⎛⎜⎜⎜⎜⎝

p11 p12 p13 p14 . . .

p21 p22 p23 p24 . . .

p31 p32 p33 p34 . . .

......

......

...

⎞⎟⎟⎟⎟⎠

When the state space S is finite with cardinality m, then P is a finite square matrix with dimension m.The n-step transition probabilities Pr(Xn−1 = j|X0 = i) of a stationary MC will be denoted by p

(n)ij , i, j ∈ S. Since

Pr(Xn−1 = j|X0 = i) =∑

in−2,...,i2,i1

Pr(Xn−1 = j, Xn−2 = in−2, . . . , X2 = i2, X1 = i1|X0 = i)

it is a routine exercise, on applying (4), to show that that p(n)ij is the (i, j) element of the matrix

Pn = P × P × · · · × P (5)

which is obtained by multiplying the matrix P by itself n times.

3. Modelling spatio-temporal movements of tourists using Markov chains

Spatio-temporal movements of tourists can be modelled at either the micro- or macro-level. The micro-level isconsidered to be relatively short distances, say of 1–2 km, for relatively short time intervals, say up to 1 h. The macro-level is considered to be larger distances, say up to hundreds of kilometres and time intervals of hours and possiblydays.

At the micro-level, movements of tourists can be modelled by a continuous stochastic process {Xt}t ∈ T , whereT = [0, ∞) taking values in a state space S consisting of various spatial points representing the locations of the personsundergoing movements [23]. If we consider a natural tourist destination such as Uluru (Ayers Rock), the AustralianAlps or Phillip Island, this can also be referred to as the space S. Within this space S, there will be a number of attractionssuch as lookouts, car parks, geomorphic features and these can be regarded as individual “state” of location throughwhich a tourist will visit or move. At the macro-level, they are represented by a discrete stochastic process {Xt}t ∈ T

where T = {0, 1, 2, . . .}. Here, the space S is usually specific tourist locations which could be located some distanceapart. For example, if there are nine attractions visited by tourists on Phillip Island, then space S (Phillip Island) willconsist of these nine attractions plus possibly some auxiliary states (refer to Section 4 for a description of these states).In this paper, we focus on the movement process at the macro-level which we will now describe in detail.

Let S = {A1, A2, . . . , Ak} where Ai, i = 2, 3, . . . , k are the tourist locations and A1 represents the state O ≡“OUT′′, i.e. the region exterior to the space S. The stochastic process representing the movement of tourists within S


will be modelled by a stationary discrete MC where each movement occurs after a unit time step. Thus the movementof a tourist over a sequence of times from t = 1 to t = m will be represented by the sequence

M(m) = (Ai1 , Ai2 , Ai3 , . . . , Aim )

where the last state reached is always A1, i.e. O. This is because the trip would always end up by a tourist leavingthe Island. In addition, this state will be an absorbing state since the process terminates once it reaches that state.We remark that due to the absorbing state, the other attractions are transient states, and, starting from any of theseattractions, and assuming that the process is allowed to go on for some time, it would eventually reach the absorbingstate [17].

By the Markov property (refer to (3))

Pr(M(m)) = Pr(Ai1 )Pr(Ai2 |Ai1 )Pr(Ai3 |Ai2 ) . . . Pr(A1|Aim−1 ) (6)

where we note that the last state reached is “OUT”. Further, the one-step transition probability matrix of the processis

P =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 . . . 0

Pr(A1(n + 1)|A2(n)) 0 Pr(A3(n + 1)|A2(n)) . . . Pr(Ak(n + 1)|A2(n))

Pr(A1(n + 1)|A3(n)) Pr(A2(n + 1)|A3(n)) 0 . . . Pr(Ak(n + 1)|A3(n))...

......

...

Pr(A1(n + 1)|Ak(n)) Pr(A2(n + 1)|Ak(n)) Pr(A3(n + 1)|Ak(n)) . . . 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

Since we have assumed that the MC is stationary, the one-step transition probabilities in the matrix P will be independentof n.

4. Experiment and results

In order to test whether a discrete-time MC could effectively be used to model the movement behaviors of tourists,a survey was conducted on Phillip Island with the aim of collecting data on tourist spatio-temporal movement, suchas tourists’ profiles, arrival times and duration for each attraction that they visited, and sequence of routes they usedto reach each destination. Using a self-administered questionnaire tourists marked their movements across PhillipIsland at the macro-level. Eight hundred questionnaires were distributed from the 6th to the 8th of March in 2004 andfrom 17th to the 20th January 2005 at the Phillip Island Information Centre, Churchill Island, the Koala ConservationCentre and the Penguin parade. In total, 500 questionnaires were completed and returned with 456 valid and completequestionnaires entered into the database. The remaining 44 questionnaires were incomplete and therefore discarded.

As part of the questionnaire, tourists were asked to write down their approximate arrival time and duration of stayat each attraction visited for the entire day and their socio-demographic information or profiles and travel informationsuch as travel mode, length of stay and with whom the tourist was travelling on that day.

4.1. Study area and sampling technique

Phillip Island, located at the mouth of Westernport Bay, is 140 km south-east of Melbourne, the capital city of theState of Victoria, Australia. There is a diverse natural flora and fauna on the island, including penguins, koalas, seals,shearwaters, mangroves, wetlands, as well as a range of geomorphic features including wide sandy beaches and ruggedrocky cliff faces. Major attractions include the Penguin Parade, the Koala Conservation Centre, the township of Cowes,Churchill Island, Rhyll Inlet, Cape Woolamai and the Nobbies (Fig. 1), where visitors can experience wildlife in itsnatural environment.

Codes used for the various attractions used in the analysis are listed below. These constitute the state space S of thestationary MC.

A Information CenterB Cape Woolamai


Fig. 1. Phillip Island map.

C Churchill IslandD Koala Conservation CentreE Rhyll InletF CowesG Penguin ParadeH The Nobbies/Seal RockJ VentnorO Outside the Island

4.2. The result

In our case study, we considered the movement of tourists on Phillip Island as moving around the nine attractionslisted above. A stationary discrete MC will be used to model the movements of these tourists between each attractionfrom the moment they enter the island until the moment they completed their visits.

The states of the chain are the nine attractions visited with an additional absorbing state which we labeled “O”which signals the completion of their daily trips. The maximum number of attractions visited by each visitor on a dailytrip was seven. The transition probability of the MC is the conditional probability of moving from one attraction toanother. For example, using the rule of conditional probability, the transition probability of going from attraction Ai

to attraction Aj, i /= j, between time n to n + 1 where the attractions belong to the above set is

Pr(Aj(n + 1)|Ai(n)) = Pr(Aj(n + 1) ∩ Ai(n))

Pr(Ai(n))(7)

where Pr(Aj(n + 1) ∩ Ai(n)) the probability of visiting both attraction Ai and Aj , with attraction Ai being visited firstthen followed by Aj . The event Ai(n) can be partitioned into mutually exclusive events, i.e.

Ai(n) = ∪kj=1 (Ai(n) ∩ Aj(n + 1)).

Hence, from (7)

Pr(Aj(n + 1)|Ai(n)) = Pr(Aj(n + 1) ∩ Ai(n))∑k

j=1Pr(Ai(n) ∩ Aj(n + 1))

. (8)

By the assumption of stationarity, the transition probabilities (7), and hence (8), will be independent of n.A graph tree (see Fig. 2) has been derived which summarizes all sequences of movements by tourists and frequency

of visits at each attraction as well as frequency of movements from one attraction to the next. The roots of the tree


Fig. 2. Graph tree.


Table 1Initial probabilities

Pr(A) 0.011Pr(B) 0.061Pr(C) 0.061Pr(D) 0.208Pr(E) 0.024Pr(F) 0.292Pr(G) 0.204Pr(H) 0.123Pr(J) 0.016

indicate the first attractions that tourists visited on entering Phillip Island and the number besides each attraction isthe number of visitors to that attraction. For example, 95 tourists visited attraction D (Koala Conservation Centre) inthe first stage of their visits. The edges of the tree represent successive tourists’ movements through the island. Wenote that the maximum number of attractions visited by each visitor on a daily trip was seven and therefore there area maximum of seven nodes in each branch of the tree. Based on this graph tree, the probability of any sequence ofmovements from the time a tourist enters the island until the time she leaves it can be calculated by using Eq. (3) oncewe are able to estimate the transition probability matrix and the initial probability vector. These probabilities will beestimated by the maximum likelihood estimation method using information contained in the graph tree.

The initial probability of visiting an attraction Ai, Pr(Ai(1)), is estimated by counting the number of visits to theattraction as a first destination divided by the total number of visits to all first attractions. For example, in the case ofattraction D

Pr(D(1)) = 95

456= 0.208

The initial probabilities of all nine attractions are tabulated in Table 1 below.The estimation of (8) is also straightforward using information contained in the graph tree. For example, to estimate

the conditional probability of going from D to C, we first calculate Pr(D(n) ∩ C(n + 1)) using the first two steps ofthe following steps:

(1) Count the number of movements that satisfy the profile D(n) ∩ C(n + 1) for n = 1, 2, . . . m − 1 where m is themaximum number of possible visits. For example, N(D(1) ∩ C(2)) is the number of movements where touristsbegan their trips at D and then move to C.

(2) Sum these frequencies, i.e.∑m−1

n=1 N(D(n) ∩ C(n + 1)).(3) Repeat Steps 1 and 2 for all states in S other than D and sum all these frequencies to obtain the total number of

one-step movement patterns starting in state D.(4) Divide the number obtained in (2) by that of (3).

Note that in Table 2, the rows are the source locations and the columns the destinations. The probabilities of movementpatterns can be computed using Eq. (3) once the one-step transition probabilities and the distribution of the initial statehave been estimated. For example, the probability of travel pattern D → F → G, i.e. the Koala Conservation Centre(D) is the first destination visited, followed by the township of Cowes (F) with Penguin Parade (G) being the lastdestination visited before exiting the island is calculated using Tables 1 and 2 above to equal

Pr(DFGO) = Pr(D) × Pr(F |D) × Pr(G|F ) × Pr(O|G) = 0.208 × 0.481 × 0.568 × 0.948 = 0.054.

Each movement pattern will also depend on the number of attractions visited. Therefore, in order to identify thesignificant patterns from the data collected, we first categorise these patterns according to the number of attractionsvisited. The most significant movement patterns will then be identified by the values of the probabilities of thesepatterns conditioned on the number of attractions. To obtain these probabilities, we simply divide the probabilityof each pattern consisting of n visits, n = 1, 2, . . . , 7 by the sum of the probabilities of all patterns consisting of nattractions. For example, for n = 1, the probabilities of the events G, D and C are estimated as 0.168, 0.01 and 0.008,


Table 2Transition probability matrix

OUT A B C D E F G H J

OUT 1 0 0 0 0 0 0 0 0 0A 0.000 0.000 0.000 0.000 0.200 0.000 0.200 0.200 0.400 0.000B 0.043 0.000 0.000 0.043 0.217 0.043 0.284 0.088 0.239 0.043C 0.103 0.000 0.102 0.000 0.333 0.026 0.308 0.102 0.026 0.000D 0.052 0.000 0.022 0.030 0.000 0.030 0.481 0.193 0.185 0.007E 0.000 0.000 0.077 0.000 0.115 0.000 0.577 0.193 0.038 0.000F 0.014 0.000 0.017 0.010 0.031 0.010 0.000 0.568 0.339 0.011G 0.948 0.000 0.000 0.002 0.000 0.002 0.022 0.000 0.026 0.000H 0.014 0.000 0.005 0.005 0.019 0.010 0.191 0.746 0.000 0.010J 0.000 0.000 0.133 0.000 0.000 0.133 0.200 0.400 0.134 0.000

respectively, giving a total probability of 0.186 for a one-attraction pattern. The conditional probability of visiting asingle attraction G, D or C is therefore equal to 0.168/0.186 = 0.903, 0.01/0.186 = 0.054 or 0.008/0.186 = 0.043,respectively. Thus, G is the most significant pattern for a one attraction visit to the island. The same process was appliedto for n = 2, . . . , 7 visits and, for the sake of illustration, we tabulate the results for the case n = 4 in Table 3 below andnote that D → F → H → G was by far the most popular n = 4 sequence of attractions. Note that for all movementpatterns, the last attraction visited before exiting from the island, i.e. to O, is G i.e. Penguin Parade. This is becausethe highlight of this attraction, the parade of penguins, takes place after sunset and it is therefore the final destinationof tourists to the island.

The value of the above approach in predicting tourists’ movements using data from a graph tree should be fairlyapparent. Although a graph tree offers much useful information, one of which is the frequency of visits to eachattraction, it does not assist organisers such as a park manager, say, for marketing purposes, to select the most attractivesequences of attractions. However, as we have just illustrated, a model utilizing MC techniques will do preciselythat. In addition, the estimated MC model could be used to simulate movements process, by incorporating it within amulti-agent modelling system, such as RBSim3 [15], thereby assisting park managers to simulate sequence of visitsusing expert knowledge.

5. Model validation

In this section, we address the question of how well MC technique can be used to model tourists’ movements onPhillip Island. Our objective is to assess how close the observed frequencies are to the expected frequencies when (4) isused to calculate the probabilities of different movement patterns. This objective will be achieved using the chi-squaredgoodness-of-fit test [6].

We adopted a data mining approach by dividing the data set, consisting of 456 records, into two parts: trainingdata and test data. The selection of training data was implemented by a random sampling method called equalprobability selection method (EPSEM) [18]. This data was used to fit the Markov chain model, i.e. to estimate thetransition probability matrix P and initial probability vector. For a fixed number of visits n and corresponding numberof movement patterns N(n), the expected frequencies of movement patterns Ei obtained thereby were tested against

Table 3Probability of movement patterns (n = 4 visits)

Pattern DFHGO CDFGO CFHGO EFHGO BFHGO DHFGO BDFGOProbability 0.342 0.096 0.082 0.068 0.063 0.057 0.052Pattern FGHGO CDHGO BHFGO FDHGO FBHGO GFHGO HGFGOProbability 0.051 0.049 0.023 0.020 0.014 0.014 0.013Pattern DBHGO ADFGO AHFGO GHFGO FHJGO JEFGO HEFGOProbability 0.011 0.010 0.008 0.007 0.006 0.005 0.005Pattern DFBGO HFJGO FEDGO FDCGO GFDGO BDGHO HJFDOProbability 0.002 0.001 0.001 0.000 0.000 0.000 0.000


Table 4Chi-squared test statistics when 70% of data is used for training

n = 1 n = 2 n = 3 n = 4 n = 5

χ2 0.13 3.27 3.39 15.19 4.39p-Value 0.72 0.20 0.07 0.004 0.11

Table 5Chi-squared test statistics when 60% of data is used for training

n = 1 n = 2 n = 3 n = 4 n = 5

χ2 0.37 1.60 6.92 2.32 4.12p-Value 0.83 0.45 0.14 0.31 0.04

corresponding observed frequencies Oi, i = 1, 2, . . . N(n), from the test data using a chi-squared goodness-of-fit test.We selected 70% of the data (319 records) as training data, leaving the remainder as test data and then repeated theprocess by selecting 60% of the data (274 records) as training data. The objective here is to gauge the effect samplingbias has on the results obtained. Tables 4 and 5 summarize the results obtained from these two cases, respectively. Wenote that the chi-squared test statistic χ2 is obtained using the equation

χ2 =N(n)∑i=1

(Oi − Ei)2

Ei

The p-values given in the tables indicate the degree of significance in the results. Customarily, a p-value of 0.05 orless offers evidence against the model, i.e. the MC model provides a poor fit to the data. As is evident from the tables,in the majority of cases, the MC model did provided a good fit to the data. However, p-values generally decline withincreasing n, leading one to suspect that either the stationary assumption breaks down or higher-order Markov chainsare required for longer trips. Selection criteria, such as Bayes’ and Akaike information criteria [4], can be used to selectbetween alternative models based on the order of dependence.

6. Conclusion and future work

This paper presents a novel method for modelling the spatio-temporal movements of tourists at the macro-levelusing MC methodology. Markov chains provide a powerful tool for analysing trends and outcomes of a seriesof events that are linked together by first-order dependence. The technique was specifically used in this paper tocalculate the probabilities of movement patterns by tourists in an island. A case study was undertaken at PhillipIsland, from which an extensive survey was conducted to collect data for the purpose of testing whether or not astationary, absorbing, discrete MC could be used to model the spatio-temporal movements of tourists visiting theisland.

A fraction of the data was used to fit the model and the remainder to test the model. The model was then validatedusing a series of chi-squared goodness-of-fit tests, the results of which suggest that the MC model is appropriate aslong as the length of a visit is not too long. When it is, the fit is not as good and we conjecture that in this case, thestationary and first-order dependence assumptions underlying the model break down. However, we believe that thelevel of generality in this work is sufficient for many practical situations that could occur in a natural environment.

There is much scope in extending and relaxing the assumptions in the present work. For example, transitionprobabilities can be made to depend on duration of visits to sites as well as inter-site distances, thereby making themmore realistic. In future work, we will relax some of these assumptions such as stationarity and compare the results ofthese new studies with the results obtained in this paper. We also plan to model the continuous movement process atthe micro-level using continuous time Markov process.

Finally, we mention that a significant outcome of this study is that it will assist park managers in identifyingsignificant movement patterns of tourists in their parks. This will assist them in designing better tourist packages aswell as providing more attractive combinations of attractions. In addition, the probabilities obtained from employingthe approach of this paper could be used as input into agent-based simulation system such as RBSim3 [15] and then


used to evaluate tourist behavior under different management scenarios in order to isolate potential problems and obtainoptimum outcomes.

References

[1] C. Arrowsmith, D. Zanon, P. Chhetri, Monitoring visitor patterns of use in natural tourist destinations, in: C. Ryan, S. Page, M. Aicken (Eds.),Taking Tourism to the Limits: Issues, Concepts and Managerial Perspectives, Elsevier, The Netherlands, 2005, pp. 33–52.

[2] B. Ashtakala, A.S.N. Murthy, Sequential models to determine intercity commodity transportation demand, Transportation Research Part A:Policy and Practice 27 (1993) 373–382.

[3] M. Batty, Predicting where we walk, Nature 388 (1997) 19–20.[4] G.E.P. Box, G.M. Jenkins, G.C. Reinsel, Time Series Analysis: Forecasting and Control, third edition, Prentice Hall, New Jersey, 1994.[5] A. Constant, K. Zimmerman, The dynamics of repeat migration: a Markov chain analysis, Centre for Comparative Immigration Studies,

Working Papers, Paper wrkg 85, 2003.[6] J.L. Devore, Probability and Statistics for Engineering and the Sciences, seventh edition, Thomson-Brooks/Cole, Belmont, CA, 2008.[7] K.E. Donnelly, Markov chains, http://www.saintjoe.edu/∼karend/m244/NBK6–1.html, 2005.[8] B. Dumont, P. Roovers, H. Gulinck, Estimation of off-track visits in a nature reserve: a case study in central Belgium, Landscape and Urban

Planning 71 (2004) 311–321.[9] D.A. Fennell, A tourist space-time budget in the Shetland Islands, Annals of Tourism Research 23 (1996) 811–829.

[10] P. Forer, D. Simmons, Analysing and mapping tourist flows, Tourism, Recreation Research & Education Centre, Lincoln University, NewZealand, 1998.

[11] D. Hadley, R. Grenfell, C. Arrowsmith, Deploying location-based services for nature-based tourism in non-urban environments, in: SpatialSciences Coalition Conference, Canberra, 2003.

[12] I. Haritaoglu, D. Harwood, L. Davis, W4: who, when, where, what: a real time system for detecting and tracking people, in: InternationalConference on Face and Gesture Recognition Conference, Nara, Japan, 1998, pp. 222–227.

[13] K. Hornsby, M. Egenhofer, Modelling moving objects over multiple granularities, Annals of Mathematics and Artificial Intelligence 36 (2002)177–194.

[14] M. Iosifescu, Finite Markov Processes and their Applications, Wiley, Chichester, New York, 1980.[15] R.M. Itami, RBSim3: agent-based simulations of human behaviour in GIS environments using hierarchical spatial reasoning, in: ModSim 2003

International Congress on Modelling and Simulation, Townsille, Australia, 2003.[16] G.M. Jacquez, P. Goovaerts, P. Rogerson, Space-time intelligence systems: technology, applications and methods, Journal of Geographical

Systems 7 (2005) 1–5.[17] J.G. Kemeny, J.L. Snell, Finite Markov Chains, Springer-Verlag, New York, 1976.[18] L. Kish, Survey Sampling, John Wiley, New York, 1965.[19] R.J. Kuo, H.S. Wang, T.L. Hu, S.H. Chou, Application of ant K-means on clustering analysis, Computers and Mathematics with Applications

50 (2005) 1709–1724.[20] D. Loiterton, I.D. Bishop, Virtual environments and location-based questioning for understanding visitor movement in urban parks and gardens,

in: Real-time Visualisation and Participation, Dessau, Germany, 2005.[21] A. O’Connor, A. Zerger, B. Itami, Geo-temporal tracking and analysis of tourist movement, Mathematics and Computers in Simulation 69

(2005) 135–150.[22] F. Spitzer, Principles of Random Walk, Springer, New York, 2001.[23] W.J. Stewart, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, New Jersey, 1994.[24] S. Timpf, Map cube model—a model for multi-scale data, in: T.K. Poiker, N. Chrisman (Eds.), 8th International Symposium on Spatial Data

Handling (SDH’98), International Geographical Union, Vancouver, Canada, 1998, pp. 190–201.[25] W. Tobler, Movement modelling on the sphere, Geographical and Environmental Modelling 1 (1997) 97–103.[26] VeriLocation, GPS Personal Tracking. http://www.verilocation.com/default.aspx, London, 2004.[27] B. Wang, R.E. Manning, Computer simulation modelling for recreation management: a study on carriage road use in Acadia National Park,

Maine, USA, Environmental Management 23 (1999) 193–203.[28] Y. Wang, E.P. Lim, S.Y. Hwang, Efficient mining of group patterns from user movement data, Data & Knowledge Engineering 57 (2006)

240–282.[29] J. Xia, C. Arrowsmith, Managing scale issues in spatio-temporal movement of tourists modelling, in: MODSIM 05, Melbourne, Australia,

2005.[30] J. Xia, F. Basic, C. Arrowsmith, Techniques for tracking movement of tourists, in: SSC 2005 Spatial Intelligence, Innovation and Praxis: The

National Biennial Conference of the Spatial Sciences Institute, Melbourne, Australia, 2005.

http://www.saintjoe.edu/~karend/m244/NBK6--1.html

http://www.verilocation.com/default.aspx

modelling spatio-temporal movement of tourists using finite markov chains

Documents