Managing Crowded Museums: Visitors Flow Measurement, Analysis, Modeling, and Optimization
Pietro Centorrino, Alessandro Corbetta, Emiliano Cristiani, Elia Onofri
MMANAGING CROWDED MUSEUMS: VISITORS FLOWMEASUREMENT, ANALYSIS, MODELING, AND OPTIMIZATION
Pietro Centorrino
Department of Physics, Sapienza Universit`a di Roma, Rome, Italy [email protected]
Alessandro Corbetta
Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands [email protected]
Emiliano Cristiani
Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Rome, Italy [email protected]
Elia Onofri
Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Rome, Italy [email protected]
Abstract.
We present an all-around study of the visitors flow in crowded museums:a combination of Lagrangian field measurements and statistical analyses enable us tocreate stochastic digital-twins of the guest dynamics, unlocking comfort- and safety-driven optimizations. Our case study is the Galleria Borghese museum in Rome (Italy),in which we performed a real-life data acquisition campaign.We specifically employ a Lagrangian IoT-based visitor tracking system based on Rasp-berry Pi receivers, displaced in fixed positions throughout the museum rooms, and onportable Bluetooth Low Energy beacons handed over to the visitors. Thanks to twoalgorithms: a sliding window-based statistical analysis and an MLP neural network, wefilter the beacons RSSI and accurately reconstruct visitor trajectories at room-scale. Viaa clustering analysis, hinged on an original Wasserstein-like trajectory-space metric, weanalyze the visitors paths to get behavioral insights, including the most common flowpatterns. On these bases, we build the transition matrix describing, in probability, theroom-scale visitor flows. Such a matrix is the cornerstone of a stochastic model capa-ble of generating visitor trajectories in silico . We conclude by employing the simulatorto increase the number of daily visitors while respecting numerous logistic and safetyconstraints. This is possible thanks to optimized ticketing and new entrance/exit man-agement.
Contents
1. Introduction 21.1. Relevant literature 31.2. Paper contributions 51.3. Paper organization 6
Key words and phrases.
IoT, machine learning, clustering, tracking system, museum simulator, museumoptimization. a r X i v : . [ phy s i c s . s o c - ph ] J u l P. CENTORRINO, A. CORBETTA, E. CRISTIANI, AND E. ONOFRI
2. Case study: the Galleria Borghese in Rome 63. Data collection: IoT visitors tracking system 74. Trajectory reconstruction and filtering 84.1. Sliding window approach 94.2. Machine learning approach 104.3. Handling not-detected beacons 125. Trajectory analysis and clustering 135.1. Basic statistics 135.2. Measuring the distance among trajectories: a Wasserstein-inspired metric 145.3. Clustering algorithms 175.4. Clustering results 196. Model and calibration 206.1. Time Varying Markov Model (TVMM) 216.2. Simulation results 227. Museum control and optimization 237.1. Entrance strategy optimization 257.2. Removing the finite time horizon of the visits 268. Conclusions and future work 27Acknowledgements 28Funding 28REFERENCES 281.
Introduction.
The analysis of the behavior of museum visitors has a long-standing tra-dition [45] and grows daily in importance as tourist flows increase and digital technologiesget ubiquitous [20, 53]. The outstanding issue of visitors management demands for multidis-ciplinary skills connected to, among others, psychology, computer science, statistics, physicsof complex systems as well as modeling and optimization theory.Museums curators are expected to achieve three complex and seemingly contradictoryobjectives: increasing the visitors number, enhancing the experience quality, and preservingthe artworks [56]. Accurately measuring and analyzing the visitors trajectories is an essentialcomponent towards these objectives and, specifically, when aiming at efficient organizationof the exhibits [2, 47], the determination of adequate ticketing strategies, and also to verifyif visitors experience complies with managers’ intents [49].A complete workflow enabling the full control of visitors in a museum consists of severalchallenging steps, that we here summarize.
Visitors tracking: the first goal is to understand the behavior of visitors in terms ofpaths followed in the museum. Not all museums have predefined paths and sometimes morethan one choice is possible [33]. Moreover, in large museums it is rare that visitors see thewhole exhibition [39]. A number of technologies exist for indoor tracking that are character-ized by a trade-off between deployment complexity, invasiveness and accuracy. Radio-basedapproaches, as considered in this work, enable room-level positioning accuracy: visitors tra-jectories are rendered into sequences of visited rooms and related permanence times. Atthe price of more invasive and complex deployments, sometimes impossible in the contextof cultural heritage, centimeter-level individual positioning can be also accomplished, e.g.via distributed grids of 3D scanners or video-cameras.Besides, psychological and sociological variables can be observed on side of paths, such asheart rate, skin conductance, emotional and aesthetic evaluations of specific artworks [29,
ANAGING CROWDED MUSEUMS 3
47, 48], interactions with groupmates, degree of attention, boredom or fatigue.Automatic systems could be complemented with manual activities, like paper-and-pencilannotations and questionnaires [28, 34, 47]. From questionnaires one can estimate demo-graphic related and museum visit related features [34] like age, gender, educational level,number of visits per year to museums, etc. After the visit, one can measure the degree ofsatisfaction, the relationship between perceived and real time spent in the museum [4], etc.
Behavior understanding: a number of variables can be estimated from visitors tra-jectories: busy hours, movement patterns, length of visits, permanence times in each room,number of stops. Two indicators are generally considered to quantify the importance of aspecific exhibit, the attraction power (relative amount of people who has stopped in frontof an artwork during their visit) and the holding power (average time spent in front of anartwork) [33].Clustering and AI-based algorithms can be used for inferring, from the whole trajectoriesdataset, the typical paths or, equivalently, the typical individual behaviors inside the mu-seum. Another interesting question regards the predictability of visitors behaviors [9, 27, 36]:can a person who starts visiting the museum in a certain manner be immediately labeled asa visitor of a certain type?
Social behavior can be observed too. For example, one can wonder, e.g., if people belongingto the same group follow the same path or they split, or whether individuals are attractedor repelled by crowding.
Museums digital twin: once statistics about visitors trajectories and behavior areavailable, it is possible to create an algorithm capable of generating real-like visits paths inthe museum [1]. This is done by reproducing the movements of people from one room toanother, duly determining their transition probability. Moreover, herding behavior in socialgroups or the response to congestion and fatigue could be taken into account. A digitaltwin is able to reproduce virtual visitors moving in the museum with a realistic behavior,possibly in new (i.e. unexperienced) conditions. It is also possible to forecast the visitorsflow from some initial conditions, like, e.g. the visitor inflow at a given time.
Visitors flow optimization: in order to use the museum digital twin as managing tool,curators and organizers have to identify relevant control variables and objectives : regardingthe former, one can, e.g., regulate the entrance flows, limit the maximum occupancy ofselected rooms, increase the number of entrances or exits, set a maximal duration of thevisit. The ticket price can obviously be controlled too.Regarding objectives instead, one can aim at maximizing the number of visitors, the pleas-antness of the visit, the amount of information conveyed, or keeping the environmentalparameters (e.g., temperature and humidity) in a given range, for best conservation of thecollection.Once this is done, a museum digital twin can be profitably used to simulate different scenar-ios, aiming at matching the objectives while varying the control variables. Here optimizationalgorithms like gradient-based or PSO methods can be used to automatize the search for asolution.1.1.
Relevant literature.
The first step (Visitors tracking) is the one that has receivedmost attention in the literature, as it relates to pedestrian dynamics in general, i.e. beyondthe museum context.Focusing on (indoor) tracking systems, all kind of technologies have been exploited, suchas RFID [33], Wi-Fi [22, 26], Bluetooth [8, 10, 17, 36, 38, 40, 41, 43, 44, 52, 54, 56], videocameras [30], 3D scanners [11, 46]. An exhaustive review of these methods is out of thescope of the paper; we refer the interested reader to the papers [21, 38] for more references.
P. CENTORRINO, A. CORBETTA, E. CRISTIANI, AND E. ONOFRI
Different technologies require different degree of visitors involvement. For example,video cameras, 3D scanners, Wi-Fi or Bluetooth mass scans require no collaboration, whileBluetooth-based apps and RFID tags usually require some degree of visitors interaction.Measuring personal data like heart rate or skin conductance requires instead total involve-ment [29, 48]. Moreover, convincing people to participate in an experiment, for exampleby downloading and installing a smartphone app, can be difficult and time consuming [43].Sometimes free tickets could yield a good incentive [57].The second step (Behavior understanding) has also been investigated in great detail inconnection with museums. Regarding individual behavior , the predominant idea is to classifyvisitors into four categories based on the way they interact with the artworks: ‘Ants’ (tendto follow a specific path and observe extensively almost all the exhibits); ‘Butterflies’ (donot follow a specific path but are guided by the physical orientation of the exhibits; stopfrequently to acquire more information); ‘Fish’ (most of the time move around in the centerof the room and usually avoid looking at exhibits details); ‘Grasshoppers’ (seem to have aspecific preference for some pre-selected exhibits and focus their time on them, while tendingto ignore the rest), see, e.g., [27] or [50] for the origin of this taxonomy.Regarding social behavior instead, the idea is to label visitors in six categories based onhow they interact with group mates: ‘Doves’ (interested in other visitors while ignoringthe environment); ‘Meerkats’ (stand side by side, expressing great interest in the exhibits);‘Parrots’ (share their attention between exhibits and group members); ‘Geese’ (advancetogether, however one visitor appears in the lead); ‘Lone wolves’ (enter the museum togetherand then separate); ‘Penguins’ (cross the space together while ignoring the exhibits), see,e.g., [15, 33].Clustering techniques (e.g., k -means, hierarchical clustering, sequence alignment) havebeen used to assign every visitor trajectory (spanning from room-scale to continuum) to oneof the groups described above, or to some given typical movement patterns [8, 15, 17, 27,34, 36, 38, 41, 44, 55, 57]. This enables one to quantify the percentage of visitors belongingto each group. Note that typically the number of clusters is assigned a priori and thiscan be an important limitation. One crucial point for cluster investigation is the definitionof a suitable metric , to measure the distance between trajectories, and aggregate (cluster)trajectories close to each other. Examples of such metrics devised at room scale can befound in [8, 36, 41]. In particular, [8, 41] propose a combination of well known metricsdefined in the space of characters strings (as trajectories can be suitably represented assequences of characters), which is further corrected to take into account the differences intime of permanence in each room.Regarding trajectory comparisons, let us mention also other two papers: [55] comparesmeasured trajectories with those coming from a random walk simulator in order to under-stand which kind of visitors exhibits stronger patterns. The work [32] compares trajectoriesof visitors with and without audio-guides in order to measure the impact of the transmittedinformation.The third and fourth steps are also related to the rich pedestrian flow modeling literature:if one considers the museum as a continuous space as in [31], one can refer to differential(agent-based, kinetic, fluid-dynamic) or nondifferential (discrete choice, cellular automata)models. See, e.g., [3, 12, 13, 18, 19, 25, 35] for some reviews, books’ chapters and booksabout this topic.If one instead considers the museum as a graph – where the nodes represent the rooms ofthe museum and the edges represent connections among rooms – one can refer to some clas-sical tools like transition matrices and deterministic/stochastic Markov chains with/without ANAGING CROWDED MUSEUMS 5 memory [26, 40, 51] in order to simulate a room-level walk in the museum (i.e. a trajectoryon the graph).Although many mathematical tools are available, examples of actual museums digitaltwins developed with the aim of reproducing, understanding and optimizing visitors behaviorare largely missing. This fact holds despite the fact that the path followed by visitors isevidently conditioned by the design of the exhibition galleries [2, 47]. An interesting attemptcan be found in [23, 24]: the author describes a museum simulator and uses it to show thatchanges in the layout design of an exhibition result in different visitor circulation patterns.Unfortunately, that simulator can be hardly used in a museum with a very high density ofartworks exposed, since it requires a complex calibration of many artwork-scale parameterswhich usually show a high variance between visitors. See also [1] for a rudimentary simulatoron graph and [42] for a simulator developed under the NetLogo software environment.1.2.
Paper contributions.
In this paper we perform an all-around investigation whichinclude contributions to all four steps described above. Covering the whole process allowsus to reach an unprecedented level of understanding and control of the museum, whichunleashes the capability of improving deep modifications to the ticketing strategy as well asto the museum access management. Our results are based on real visitors data acquired inthe Galleria Borghese museum (Rome, Italy). In more details, the research unfolds alongthe following lines:1. We describe a cheap and easily reproducible data collection system, hinged on a IoT-based room-scale Lagrangian tracking system of the museum visitors. Each visitor is pro-vided with a portable Bluetooth Low Energy (BLE) beacon, whose signal is received byantennas (realized by means of common Raspberry Pi’s) displaced in fixed positions withinthe museum rooms. We employ this system for an extended data collection campaign whichprovides the high statistics measurements employed in this work.2. We employ and filter the Received Signal Strength Indicator (RSSI) of the beacons toreconstruct individual visitors trajectories. Due to the restricted space and the numerous ar-chitectural and historical constraints, each beacon is often captured by multiple antennas atthe same time. Accurately reconstructing the trajectories in this settings defines a challenge per se . We propose a new machine learning approach which outperforms standard slidingwindow processing, especially, when it comes to estimate the correct time of permanence inrooms.3. In order to get insights about visitors behavior, including the most common move-ment patterns, we analyze trajectories via statistical and clustering techniques. Inspired bythe Wasserstein distance , we introduce a new ad hoc trajectory clustering metric, whichrespects the geometrical properties of the museum. Our metric, in fact, builds upon thephysical distance among rooms. By using a hierarchical cluster analysis, only based onsuch metric (and with no a priori hypothesis on the number of clusters nor on their size),we can unveil automatically hard-to-see movement patterns that go well beyond the stan-dard animal-inspired classification (see Section 1.1). As a by-product we can also identifyanomalous behaviors.4. We employ statistical tools to build a probability transition matrix among museumrooms, which provides us with building blocks for a model capable of simulating in silico the museum visits. In particular, this enables us to forecast the path of visitors entering themuseum from any room. Unlike the simulator presented in [23, 24], our simulator leverages The Wasserstein distance was first introduced by Kantorovich in 1942 and then rediscovered manytimes. Nowadays, it is also known as
Lip (cid:48) -norm, earth mover’s distance, ¯ d -metric, Mallows distance. Animportant characterization is also given by the Kantorovich–Rubinstein duality theorem. P. CENTORRINO, A. CORBETTA, E. CRISTIANI, AND E. ONOFRI a. b.
Figure 1.
The two largest rooms in Galleria Borghese. a. Ratto di Proserpina locatedon the first floor. b. Main area on the second floor, part of the Pinacoteque. The mapof the museum and the room names will be shown later in Figure 4 and Table 1. on the measured permanence time in each room on side of the probability of transition fromone room to any other. This results in a tool easier to calibrate.5. Finally, we employ the simulator to significantly increase the efficiency of the ticketingstrategy and entrance/exit management. Our results suggest a way to increase the numberof daily visitors while keeping the numerous constraints within the limits.1.3.
Paper organization.
We present our methods and original contributions alongsideour field activity at Galleria Borghese museum, our case study. In Section 2 we introduceGalleria Borghese, and outline its floor plan and the current ticketing strategy. In Section 3we describe our tracking system and the dataset we collected in the museum. In Section 4we discuss our trajectory reconstruction methods. In Section 5 we analyze the trajectoriescollected, fit their statistical observable with known distributions, and introduce our clus-tering approach. In Section 6 we introduce the model which allows us to create a completedigital twin of the museum, and simulate in silico the visitors flow. In Section 7 we employthe model to find optimal strategies for ticketing and museum management. The discussionin Section 8 closes the paper.2.
Case study: the Galleria Borghese in Rome.
The world-renowned Galleria Borghe-se museum (Rome, Italy), is a relatively small, two-floor museum with 3 entrances and 21exhibition areas. Its sculptures and paintings attract visitors from all over the world, seeFigure 1. On the main floor, the exhibition area is circular, while on the second floor(Pinacoteque) it is U-shaped. Rooms are numbered but no obligatory exhibition path isassigned, so many people do not visit the rooms in their natural order. Moreover, thedensity of exhibits is so high that people often come back to already visited rooms multipletimes to admire artworks missed during the previous passages. Congestion is frequent issome rooms, like the one which host Caravaggio’s paintings. Audio-guides are available ondemand and guided tours are subject to quota (both in number and size).To cope with the many historical, artistic and architectural constraints, museum curatorsestablished to schedule the visits: tickets must be booked in advance and give access to themuseum for a slot of 2h. Five slots per day are granted. The maximum number of visitorsallowed in each slot is 360. Additionally, 30 tickets, called “last minute”, are sold 30 minutesafter the beginning of each time slot. People can also decide which floor to start the visitfrom, within some limits. At the end of each time slot, people are invited to leave, andthe museum empties completely. Let us also note that many visitors enter without theirsmartphone since they must leave their personal bags in the wardrobe.
ANAGING CROWDED MUSEUMS 7 a. b.
Figure 2. a.
Sample visitor wearing the BLE beacon. b. Raspberry Pi used asBluetooth antenna to receive beacon signals and measure their RSSI. Data collection: IoT visitors tracking system.
As many cultural heritage sitesworldwide, Galleria Borghese is covered by frescoes and paintings, which heavily limit thepossibility of displacing (electrical) devices. To cope with these historical and architecturalconstraints, we developed a noninvasive radio-based IoT measurement solution deliveringroom-level visitors trajectories.Figure 2 shows the main components of the tracking system, which consists of:
Transmitters: we gave a small BLE beacon to each visitor to track who was briefed aboutthe experiment, see Figure 2 a . The beacon transmitted a signal at +4dB with iBeaconstandard encoding [37], which carries a unique identifier (UUID). Fleet of receiving antennas: we employed RaspberryPi 3B+ (RPi) as receivers. A RPiis a single-board computer with embedded Bluetooth and Wi-Fi modules, see Figure2 b . RPi’s were located along the museum in fixed positions, see Figure 4. A Pythoncode running on the RPi’s was used to scan continuously the surrounding area listeningfor beacons signals. Each signal is stored as a tuple containing the beacon’s identity,the RSSI and the timestamp of reception. Every 5s a data packet was created (withonly one occurrence of each beacon detected), it got signed with the RPi identity andposted on a central server via an internet connection. Central server: the server received data packets from all RPi’s and stored them in a SQLdatabase along with the reception timestamp. Such couple of timestamps allows us toquantify the duration of the whole process.The data presented in this paper come from a measurement campaign lasted betweenJune and August 2019. The central SQL server received 1 , ,
617 records correspondingto 900 visitors trajectories surveyed during 13 2h-long visit slots. The percentage of trackedvisitors w.r.t. the total number was about 1:5. As it usually happens, the vast majority ofvisitors came in groups (family, friends, guided tours, etc.). In this case, apart from a fewexceptions, we tracked only one member of each group, thus losing the ability to detect theinteractions within social groups.We have also tested that collecting dozens of beacons at the same time within a smallphysical space does not impact on the reliability of the system.Figure 3 shows the history of a single beacon’s RSSI (i.e., a single visitor) during a visit.Beside the fact that signals are not uniformly sampled in time, the analysis of the raw dataimmediately confirms that RSSI signal suffers from high fluctuations (see also [5]). Thismeans that a beacon fixed in the middle of a room is not received with a constant RSSI,and the RSSI of two equidistant beacons might not be the same. In addition, we encounteredother two important difficulties:
P. CENTORRINO, A. CORBETTA, E. CRISTIANI, AND E. ONOFRI − − − − − − −
40 0 20 40 60 80 100 120 R SS i[ d B ] time [min]A01A02 A03A04 A05A06 A07A08 A09A10 A11A12 A13A14 − − −
80 29 29 . . . Figure 3.
Typical raw RSSI throughout a 120 minute visit as recorded by the 14 RPi’sreceiving antennas. (Sampling every ∆ t = 10 s; RPi’s antennas are distinguished bymarkers). Inset: between minute 29 and 32 the visitor is detected by both RPi4 andRPi5, and the maximal RSSI strongly oscillates between the two. As a consequence, thesignal strength is insufficient to associate unambiguously a visitor to a location.
1. A single beacon can be detected by multiple antennas at the same time. RSSI is usedto resolve the ambiguity but high fluctuations makes such task rather hard.2. Some areas of the museum could not be covered at all (e.g., staircase between the twofloors, due to lack of electrical outlets).Finally, let us also mention the possibility – which we consider very rare – that visitorswearing beacons could be influenced by the fact that they feel tracked, cf. [47, 56].In the next section we describe how the raw RSSI data from the transmitters is processedto estimate individual trajectories.4.
Trajectory reconstruction and filtering.
We use the raw data collected by the track-ing systems to reconstruct the sequence of visited rooms and the time of permanence in eachroom.First of all, we achieve uniform temporal sampling of the signals through a re-samplingin bins of fixed time length ∆ t = 10s. ∆ t is to be tuned according both to the resolutionneeded and the signal granularity. We employed a -120 dB threshold for those antennas notdetecting the beacon. The output of this procedure is a A × T matrix R for each beacon,where A is the number of antennas and T is the number of time bins (duration of the visitdivided by ∆ t ). In other words, for a given beacon, the element R a,t is the RSSI of thesignal received by a -th antenna and the t -th time bin.In order to simplify our room-scale tracking, we merge the 21 exhibition areas of themuseum into R = 9 (radio) rooms in which we deploy our A = 14 receiving antennas (seeFigure 4 and Table 1 for antennas positions and antenna-room assignments). We remarkthat signal readings in a given room do not imply that the emitting beacon is located in thesame room.The most natural way to reconstruct visitor trajectories is to compute the argmax ofthe RSSI history of each beacon: at each time bin one retains the antenna that receivesthe highest RSSI. Finally, one defines the current visitor location as the room associated tothe antenna. The result of this procedure is shown in Figure 5 ab . Unfortunately, when a ANAGING CROWDED MUSEUMS 9
Main floorPinacoteque (R9)A11 A1 A10 A5A4A7A6 A2A8 A9 A14A12 A13A3R6 R4 R3R2R1R5R8R7 S SE
Figure 4.
Floor plan of Galleria Borghese. Rooms (R) and receiving antennas (A)are reported (cf. Table 1 for room-antennas relations). Red lines represent closed pas-sages/doors. Visitors admittance happens both at the main entrance on the first floor(E) and by the stairs (S) at either floors.
Number Room nickname AntennaR1 Paolina A2R2 David A4R3 Apollo e Dafne A5R4 Ratto di Proserpina A1, A10R5 Portico A8, A9R6 Enea e Anchise A11R7 Satiro su delfino A7R8 Caravaggio A6R9 Pinacoteque A3, A12, A13, A14
Table 1.
Match among rooms (R) and RPi antennas (A) in Galleria Borghese museum. visitor is at approximately equidistant from two or more RPi’s, the maximal RSSI quicklybounces back and forth among the antennas. In signal terms, the visitor appears to performextremely rapid and unrealistic room changes.Building upon [7], we consider two data refinement methods: one based on a neuralnetwork and another, more standard, relying on a sliding window approach.4.1.
Sliding window approach.
The first method aims at smoothing the noise in theRSSI data by applying a low-pass filter, implemented as a weighted moving average , and a normalization . RSSI’s gathered close in time should have close values; besides, the closerthe bins, the higher is the correlation.In particular, we convolve the RSSI signals gathered by each antenna (i.e. the matrixrows R · ,t ) with a (symmetric triangular) kernel with size 2 δ + 1 and weights w , w , . . . , w δ .In formulas, this approach generates a new matrix ˜ R defined as˜ R a,t = t + δ (cid:88) d = t − δ R a,d · w δ − t + d , ≤ a < A, δ ≤ t < T − δ . (1) − − − − − − − − − a . a n t e nn a s ’ R SS I [ d B ] /946785123 b . r oo m − − − − − − − −
75 0 10 20 30 40 50 60 70 80 90 antennas c . a n t e nn a s ’ R SS I [ d B ] time [min]A01A02 A03A04 A05A06 A07A08 A09A10 A11A12 A13A14/946785123 d . r oo m time [min] . . . . . . . . . rooms e . r oo m p r o b a b ili t y time [min]R01R02 R03R04 R05R06 R07R08 R09//946785123 f . r oo m time [min] Figure 5.
A sample beacon RSSI elaborated by argmax ( a. & b. ), sliding window ( c. & d. ), and machine learning ( e. & f. ) approaches. The left column reports the maxof RSSI for the argmax and sliding window approaches (antennas located in the sameroom are labeled with same color but different markers), and the maximum among therooms probabilities for the machine learning approach. The right column reports thecorresponding reconstructed trajectories as sequence of rooms. Not-detected statusesare marked by green crosses ( × ). Secondly a normalization is applied across the signals acquired by the different antennasin order to make them comparable. This produces a third matrix ¯ R as¯ R a,t = ˜ R a,t − µ t σ t , ≤ a < A, δ ≤ t < T − δ , (2)where µ t and σ t are respectively the mean and the standard deviation of ¯ R by time bin (i.e.by column, thus µ t = µ ( ¯ R · ,t ), σ t = σ ( ¯ R · ,t )). Figure 5 cd shows the result of this procedure.4.2. Machine learning approach.
To improve performance of the sliding window method,we propose a trajectory reconstruction approach based on neural networks. At any time bin t , we cast the localization of a visitor in one among the R rooms as a classification problem. ANAGING CROWDED MUSEUMS 11
INPUTTRAJECTORY a (1)2 a (1)56 t − δt + δt − δt + δt − δt + δ a (1)1 r R r r out (cid:96) = 1 (hidden) (cid:96) = 2 (cid:96) = 0 ROOMPROBABILITIES a a a A bias layer Figure 6.
The three layers ( L = 2) neural network employed to process the trajectoriescollected in Galleria Borghese. For each time bin the neural network predicts the visitorposition by considering the RSSI within the previous and the next minute. The inputlayer is made of (2 δ + 1) × A = 182 neurons, where δ = 6 is the semi-amplitude (one-minute long) of the sliding time interval for each of the A = 14 antennas. The outputlayer is made of R = 10 neurons, one for each room of the museum plus the “out”condition. The single hidden layer is composed by 56 = 14 × Our neural network processes the R matrix (in time windows) and returns the probabilityvector whose r -th component is the probability that the visitor is located in the room r .4.2.1. Building the neural network.
We consider a neural network made of L + 1 layers, with L = 2. The data is injected in the first layer and flows “forward” in the network throughthe hidden layer to the output layer. Each layer (cid:96) is build out of a different number n (cid:96) of nodes a ( (cid:96) ) , that represent the calculus units of the network, or artificial neurons, where a ( (cid:96) ) j , ≤ j ≤ n (cid:96) represents the j -th neuron of layer (cid:96) .The specific network that we employ, also known as Multi Layer Perceptron (MLP), isbuilt as a complete weighted directed graph between the nodes within layer 0 ≤ (cid:96) < L andthe nodes of the next layer (cid:96) + 1 (cf. Figure 6). A spare node with fixed value 1 ( bias ) andindex 0 is added to each layer but the last, that is a ( (cid:96) )0 = 1 , ≤ (cid:96) < L . We denote by Θ ( (cid:96) ) s,d the weight of the edge directed from the s -th node of the (cid:96) -th layer to the d -th node of the( (cid:96) + 1)-th layer. We employ the sigmoid function g : R → [0 , , g ( x ) = 11 + e − x (3)as activation function. Hence, data propagate through the network as a ( (cid:96) +1) d = g (cid:32) n (cid:96) (cid:88) s =0 Θ ( (cid:96) ) s,d · a ( (cid:96) ) s (cid:33) , ≤ d ≤ n (cid:96) +1 , ≤ (cid:96) < L (4)being { a (0) s , s = 1 , . . . } the input values.Specifically, at time t our input values are the (2 δ + 1) A values obtained by restrictingthe matrix R to the δ columns before and after t (analogous notation to Section 4.1), i.e.the column block R · ,t − δ ··· t + δ . The network output are R numbers in [0 ,
1] that we interpret- after ( L ) normalization - as the instantaneous probability of being in the r -th room.We train the network parameters via gradient descent such that the network output fitshand annotated data. In particular we consider a dataset of 5427 manually labelled inputsamples, 80% of which are used to effectively set the weights, while the remaining 20% isemployed for testing (i.e. checking for generalization and absence of overfit).4.2.2. Estimating a trajectory via neural networks.
Applying the neural network to the R matrix yields a R × T (cid:48) ( T (cid:48) = T − δ ) matrix P whose columns contain the probability offinding the considered visitor in room r at time t .Almost always, the network selects a room with overall majority (probability > .
5, seeFigure 5 e ). However, it can happen – particularly during a room transition – that no classreaches such majority. We apply therefore an ad hoc adjacency filter: probabilities valuessmaller than a fixed threshold, χ = 0 .
15, are removed from the candidates; then, unfeasibletransitions are either penalised or removed (e.g. transitions that implies wall crossings orthat cover more than two rooms). After data re-normalization, in the unlikely event thatno room is selected with overall majority, we assume the visitor in closest room or in thesame room.4.2.3.
Comparing with the sliding window approach.
Looking at Figure 5 d and Figure 5 f we observe two interesting features: the sliding window approach often prevents spikes fromarising (see minutes ∼ , ∼ .
858 compared to 0 .
734 obtainedby the sliding windows approach. Both of them however overcome results obtained via theargmax approach, which has an accuracy of 0 . Handling not-detected beacons.
Due to (small) areas uncovered by antennas orbecause of random signal losses, it may happen that a beacon remains not detected. This isalso what (correctly) happens before and after a visit or when the visitor leaves the museumduring the visit, for example to reach the toilet.Before performing statistical analyses, we amend for not-detected statuses whenever thiscan be done unambiguously. Although we notice that such a process might require mainlymuseum-specific solutions, we report two corrections which we deem of general interest.1. If the “blind period” is less than 3 minutes (10 minutes for the Pinacoteque), andthe visitor is detected in the same room before and after the blind period, the visitorassociated that room for the whole period.
ANAGING CROWDED MUSEUMS 13
2. If the blind period is less than 30 seconds, and the visitor is detected in two differentrooms before and after the blind period, the visitor is supposed to be in one betweenthe two rooms (at random).Whenever the overall not-detected status exceeds 25 minutes, we remove such trajectoryfrom the dataset. Performing such pre-processing on the measurements collected during ourfield campaign, we obtain a dataset of N = 848 trajectories, which will be the object of theanalysis in the next sections.5. Trajectory analysis and clustering.
In this section we analyze the two datasets(including 848 trajectories) reconstructed via the two methods described before.5.1.
Basic statistics.
We consider three basic illustrative statistics - that we also employin Section 6 to calibrate our digital twin:
Time of Permanence: we denote by ToP( v, r ) the total time spent by visitor v ∈ { , . . . , N } in room r ∈ { , . . . , R } during their visit. Returning visitors: we denote by RET( v, r ) the number of times visitor v stopped byroom r . People per Room: we denote by PpR( r, t ) the number of visitors in room r ∈ { , . . . , R } during the time bin t ∈ { , . . . , T } .Note that the first two indicators are Lagrangian, while the third is Eulerian.5.1.1. Time of Permanence.
To estimate ToP( v, r ) we employ the trajectories as estimatedby the neural network (Section 4.2), since the reconstruction accuracy resulted higher.For each room r , the distribution of { ToP( v, r ) } v ∈{ ,...,N } is well fit by a Weibull distribu-tion with r -dependent parameters λ and k . The Weibull distribution performed best amongthe tested ones according to the Akaike Information Criterion. In Figure 7, we report ToPempirical distributions and their Weibull fit for selected individual rooms as well as for thewhole museum.The Weibull distribution is related to the “time-to-failure” of a system, which, in ourcontext, is to be interpreted as the “time-to-exit” a room (more precisely as the “time-to-exit-and-do-not-return”, since we consider the ToP as the total time spent in a room). Theparameter λ (characteristic time of visit) gives information about the room holding power,while parameter k (Weibull slope) characterizes the decision to leave the room. In particular,for all rooms of Galleria Borghese k > k = 1 indicates that the exit rate is constant over time while k < Returning visitors.
Galleria Borghese has a circular structure with no fixed or sug-gested path for visitors (cf. Section 2). Besides, the density of artworks is so remarkablyhigh that visitors easily miss a fraction of the pieces during the first passage of a room.Therefore, we investigate the amount of times a person visits the same room, on average.In this analysis, we neglect quick returns (less than a minute of permanence).We observe that each guest visits a room, on average, 1.3 times (1.5 times, for Room 8,
Caravaggio ), while entrance rooms have 2.7 passages. On the other hand, 25% of the visitorsskips at least one room (especially room 7,
Satiro su delfino ). The time of permanence duringthe first passage by the a room is generally the longest, in comparison to the next ones (thishowever does not hold for entry rooms). The time of first return (time interval between the . . . . . a . pd f . . . . . b . pd f . . . . .
05 0 20 40 60 80 100 c . pd f T/max(T) [%] . . . . .
05 0 20 40 60 80 100 d . pd f T/max(T) [%]ToPWeibull reference linequantile-quantile
Figure 7.
Four ToP distributions and their Weibull fit. a. Satiro su delfino ( r , k = 1 . λ = 17). b. Apollo e Dafne ( r , k = 2, λ = 36). c. Pinacoteque ( r , k = 2 . λ = 221). d. Whole museum ( k = 4 . λ = 572). In the last case the Weibull distribution does notfit correctly due to the forced exit after 2h. This problem will be solved later in Section7.2, by censoring the last 5 minutes of the visit. Inset: related Quantile-Quantile plotsthat depict the Real vs. Weibull quantile relation. moment a visitor leaves a room and the moment they return) appears consistent throughoutthe museum rooms and is between 25 and 30 minutes.Finally, we highlight that the occurrence of fast returns (less than 5 minutes), which inour case are about 10% of all returns, could indicate that visitors frequently get lost orchange direction of visit (clockwise vs. counterclockwise, cf. museum map in Figure 4).5.1.3. People per room.
The number of people per room, PpR, is probably the most relevantindicator as well as that of largest interest for museum curators, as it connects connectionwith safety (hyper-congestion), comfort, and attractiveness to the audience (under-usedrooms could indicate scarce interest). We calculate the PpR( r, t ) by counting the numberof visitors of each turn who are in room r in time bin t . To amend for the fact that we gavebeacons to a sample of visitors, and only to one member of each social group, we consistentlyreplicate each trajectory q times, where the integer q is uniformly distributed between 1 and6. We compare the PpR time series of one room and of the whole museum with our simu-lations in Section 6.2.5.2. Measuring the distance among trajectories: a Wasserstein-inspired metric.
Defining a suitable metric in the space of trajectories is an essential step to quantify how‘close’ (or “similar”) are distinct paths followed by visitors. We define a new, ad hoc , metric
ANAGING CROWDED MUSEUMS 15 inspired by the Wasserstein distance. The Wasserstein distance is usually employed toquantify the distance between two abstract measures or two density functions. Analogously,we quantify how much it costs to transform one visitor trajectory, time bin-by-time bin,into another, until they are identical.First of all, we model the map of the museum as a graph with R nodes. Edges betweenpairs of nodes represent viable connections between rooms. We also consider the possibilitythat the museum is organized in different wings : wings are independent areas, which are sofar from each other that it is natural to assume that visitors rarely visit the same wing twice(it is usually the case when a museum has multiple floors or comes in different buildings).For technical reason, we always assume that there exists a wing called ‘Out’. Visitors inthis wing are waiting to enter or have already left the museum. Thus, we represent GalleriaBorghese in three wings: the main floor, the Pinacoteque, and the Out wing.Second, we introduce a distance function on the graph , i.e. between room-nodes. Wedefine the distance D ( r , r ) between room r and room r ( r , r ∈ { , . . . , R } ) by D ( r , r ) := (cid:26) , r = r , − α + α T r ( r , r ) + β T w ( r , r ) , r (cid:54) = r , (5)where T r ( r , r ) is the minimum number of room transitions (i.e. graph edges to hopthrough) necessary to traverse the graph from room r to room r , whereas T w ( r , r ) is thenumber of wing transitions, and α, β > − α is motivatedby the need of decreasing the weight of short transitions (each room transition counts α butthe first one which, instead, counts α ), which often happen as many visitors stand still atthe interface/door between two rooms, without actually moving in either direction. Notethat the distance D is not necessarily commutative. This can happen e.g. if a museum comeswith some one-way room transitions.Third, all trajectories in the database are extended in order to have the same number oftime bins. We achieve this by fixing a maximal theoretical duration of the visit (2h in ourcase) and then exploiting the fictitious wing Out, where people are placed before and afterthe actual visit.We have now all the ingredients to define the distance between two trajectories t , t ∈{ , . . . , R } T W ( t , t ) := T (cid:88) t =1 D ( t t , t t ) , (6)which represents the sum, time bin-by-time bin, of the distances between the rooms (ac-cording to (5)) occupied by the two visitors at each given instant. As an example, in Figure8 we report the distribution of the pairwise distances between all trajectories, computedusing metric (6).Figure 9 shows instead the most and least common trajectory. The higher the numberof trajectories ‘close’ to a given one, the more common the trajectory is. Hence, we reportthe single trajectory having the highest number of other trajectories within distance µ − σ (Figure 9 ab , cf. definition of µ and σ in Figure 8) and the single trajectory having the leastnumber of other trajectories within distance µ + σ (Figure 9 cd ).Finally, we mention the capability of finding automatically members of social groups (itcould happen that elements of the same social group went to the ticket office separately,thus were assigned more than one beacon). Indeed, two or more trajectories very close toone another likely belong to visitors in company. Figure 10 reports a sample of trajectoriesat different distance from a given reference trajectory. . . . . . . .
035 0 20 40 60 80 10000 . . . . . . .
035 0 20 40 60 80 100 µ ± σ pd f W / max( W )[%] µ Figure 8.
Distribution of mutual distances between trajectories. The x -axis is normal-ized w.r.t. the longest measured distance. The mean pairwise distance µ is reported inred while the shaded area denotes the range µ ± σ ( σ being the standard deviation ofthe distribution). . . . . . . . . b . pd f a . r oo m . . . . . . . .
04 0 20 40 60 80 100 d . pd f W / max( W )[%] c . r oo m time [min] Figure 9. a.
Most common trajectory and, b. , distribution of the distances betweensuch trajectory and all the others. The visitor performs a circular visit following theroom numbering in the main floor, then they reach the Pinacoteque upstairs. c. & d. Analogous plot for the least common trajectory in our dataset. The visitor entersthe museum via the Pinacoteque, then they visit the main floor twice, once clockwiseand once counterclockwise. x -axis in b. & d. is normalized w.r.t. the longest measureddistance among all the trajectories. ANAGING CROWDED MUSEUMS 17 a . r oo m time [min] . . . . .
05 0 10 20 30 40 50 b . pd f W / max( W )[%] Figure 10. a.
Sample of measured trajectories. b. Distribution of the distances be-tween the trajectory marked by “blue plus signs” (+) in a. and all the others. Distancesare reported in percentage w.r.t. the longest distance measured. Trajectories closer than0 . .
15% (red bins) are slightly time shifted; In trajectories closer than 0 .
30% (purple bins)relations are still identifiable; Trajectories farther away than 0 .
30% (green ones) arecompletely unrelated. Trajectories in a. are random sampled from corresponding colorpercentile sets in b. Clustering algorithms.
As we recalled in Section 1.1, clustering algorithms can beused for inferring, from the whole trajectories data set, the typical paths or, equivalently,the typical individual behaviors inside the museum.Here we employ algorithms which do not require to define a priori the number, k , of clus-ters, nor to assign predefined reference trajectories around which clusters are agglomerated(as typically happens with, e.g., k -means approaches). Moreover, we do not use the typicaltaxonomy (ant, butterfly, fish, grasshopper ,cf. Section 1.1) to guide the clustering, aimingat other, possibly hybrid, behaviors. To this end, we employ an agglomerative hierarchicalclustering (AHC) approach (see, e.g. [14]). These techniques consider a bottom-up clustertree (dendrogram), that, step by step, gathers trajectories according to their mutual likeli-hood. At the beginning, each trajectory is considered to be the only element of a distinctcluster. Then, at each iteration, the two closest clusters get merged into a cluster. Theprocess is deterministic, unless we have two couples of clusters at exactly the same distance,and it always ends with one single cluster after N − N is the number of initialtrajectories. Note that cutting the dendrogram at the (cid:96) -th layer from the tree leaves pro-vides exactly N − (cid:96) clusters. Finding an adequate cutting layer is an issue which we discussin the following.To measure the distance between two clusters, we leverage on (6). We consider, inparticular, three common methods: C-LINK: in Complete Linkage, the distance between two clusters C and C is the maxi-mum amongst the distances between all the trajectories within the two clusters: W ( C , C ) = max {W ( t , t ) : t ∈ C , t ∈ C } . (7) S-LINK: in Single Linkage, the distance between two clusters C and C is the minimumamongst the distances between all the trajectories of the two clusters: W ( C , C ) = min {W ( t , t ) : t ∈ C , t ∈ C } . (8) a . CLINK:SLINK:UPGMC: p - s i g n i fi c a n t c l u s t e r s Dendrogram depth555 151515 b . p - s i g n i fi c a n t c l u s t e r s Dendrogram depth p = 5 p = 15 p = 30 p = 50 plateau Figure 11. a.
Number of 5- (filled markers) and 15- (empty markers) significant clustersas a function of the dendrogram depth for C-LINK, S-LINK and UPGMC methods. b. Number of p = 5 , , ,
50 significant clusters obtained via UPGMC method. Thedendrogram is cut in correspondence to the plateau at depth 67.
UPGMC: in Unweighted Pair Group Method with Mean Centroid, each cluster C is iden-tified by a representative trajectory ¯ t C , and the distance between two clusters is eval-uated as the distance between representative trajectories: W ( C , C ) = W (¯ t C , ¯ t C ) . (9)Determining a representative trajectory ¯ t C in a trajectory set C is useful in general, andmandatory to employ UPGMC. To do so, we compute a mode among all the trajectories:for each time bin t , our representative trajectory reports the most visited room among theelements of C : (¯ t C ) t := arg max r ∈{ ,...,R } (cid:40)(cid:88) t ∈C t t = r (cid:41) , < t ≤ T . (10)Note that the centroids found with a specific cut may also be employed to clusterize adifferent set of trajectories. This also means that, if new trajectories are gathered, the samecentroids may be used in order to get a clustering. This may reveal that habits have beenbroken or new paths have been discovered.
Cutting the dendrogram.
In order to find the right cutting threshold for the dendrogram,we consider the number of the p - significant clusters, i.e. the clusters with more than p elements, while traversing the tree from the leaves to the root. Having a high variation inthe number of significant clusters in the proximity of the root often implies that clustersare unstable, i.e. they merge randomly in the process, preventing valuable interpretations.Having instead a very small number of significant clusters, say one or two, often means thateach cluster contains very dishomogeneous elements, thus resulting practically useless forcategorization.Figure 11 a reports the number of 5- and 15-significant clusters as a function of thedendrogram depth, for the three methods described before. C-LINK yields many smallunstable clusters joining together, with no meaningful interpretation, towards the end ofthe process. S-LINK offers, on the other hand, a poor set of typical clusters to which all thetrajectories converge quickly throughout the clustering process. Conversely, UPGMC leadsto relatively small amount of consistent stable clusters. ANAGING CROWDED MUSEUMS 19 a . r oo m b . r oo m c . r oo m time [min] 946785123 d . r oo m time [min] Figure 12.
Four representative trajectories (centroids) of clusters joining respectively a. b. c. d.
1% of the trajectory data set. Representative trajectoriesmay show spikes (see, e.g., c ., ≈
75 minute). According to (10), this phenomenon ariseswhenever rooms have approximately the same number of visitors within the same intervalof time.
In particular, the UPGMC dendrogram shows a plateau around layer ¯ (cid:96) ≈
67, for manyvalues of p , see Figure 11 b . We adopt such a cutting layer since it ensures the maximumamount of highly significant clusters ( p = 30 ,
50 have the last absolute maximum there)without trading-off too much information in smaller clusters.5.4.
Clustering results.
We consider here the representative trajectories of each clusterobtained after a dendrogram cut at layer ¯ (cid:96) . Although none of the representative trajectoriesstrictly coincides with any among the trajectories observed, they all appear real (i.e. conformwith potential visit). This emphasizes that clusters indeed aggregate similar trajectories.Figure 12 shows four representative trajectories related to four clusters of different size.The two most common patterns are related to visitors who follow the natural numbering ofthe rooms, starting or ending the visit in the Pinacoteque, which is visited once. This iden-tifies the most typical visit pattern for the curators. Nevertheless, clustering investigationbrings to light other, less expected, patterns: the one which does not include the visit atthe Pinacoteque (possibly visitors who did not find the staircase) and patterns where thePinacoteque comes amidst the visit. Note that both patterns have been observed by themuseum managers and are discouraged.
Filtering by clustering.
Clustering can be also used to detected unfeasible/unreal tra-jectories coming from system malfunctioning, since those trajectories tend to gather in asingle cluster. This powerful feature helps to design filters to clean up the data during thepreprocessing phase. a . r oo m time [min] 946785123 b . r oo m time [min] Figure 13.
Two anomalies detected. a. A rare pattern where the Pinacoteque andmain floor are both visited twice. b. A strange pattern with many changes of direction(clockwise/counterclockwise).
Anomaly detection.
Trajectories which remain isolated in the last layers (close to the rootof the tree) are, by definition, far from all the other centroids and therefore very atypical.We claim that these trajectories are anomalies detected during the process. If they donot come from system malfunctioning, they belong to people who behave abnormally orsuspiciously and deem additional checks. Figure 13 shows some of the anomalies detectedin our study.6.
Model and calibration.
In this section we develop a digital twin of the museum, i.e. analgorithm which is capable of generating new trajectories, (statistically) indistinguishablefrom measured ones.In order to represent the complex visitor behaviour, we employ a stochastic approachbased on Markov Chains (MC). We design our simulator to generate visiting paths withrelevant observable features such as guests skipping one or more rooms and/or returningmultiple times to the same room.The model is based on two important assumptions:
Visitors are independent from each other: the decision to leave or remain in a roomdoes not depend on the number of people in that room. This assumption is certainlyreasonable up to mild congestion levels. On the other hand, hyper-congestion hassurely an impact on visitors choices, however our current data collection seems stillinsufficient to quantify such a challenging aspect. We suspect that congestion caneither increase or decrease the ToP, depending on the perceived importance and fameof the room content.
Social groups behave as one individual: social groups visit the museum remaining to-gether, i.e. following the same trajectory and thus spending the same time in eachroom. This assumption, which is an important limitation, is consistent with the factthat beacons were given almost always to a single member of each social group. There-fore, we are not capable of disentangling interactions and differences within socialgroups.In a standard MC, the transition probability from a state (room) to the next dependsonly on the current state. However, in our context it is an intuitive idea that the visitorschoices depend, in some way, on the rooms that they have previously visited. Furthermore,
ANAGING CROWDED MUSEUMS 21
Out Wing 1 Wing 2 r r r r r r r r r r out . . . . . h ( t ) 0.56 . h ( t ) 0.16 . .
09 0 . . h ( t ) Figure 14.
Transition probabilities between rooms in Galleria Borghese (the probabilityto remain in the same room is not included). We can see that the counterclockwise pathis preferred and fast transitions from rooms 5, 2 to rooms 2, 4, respectively, exist. Theprobability of leaving the museum is sampled as a hazard function h ( t ). since we are assuming that there is no predefined visit path, a standard MC creates a bouncephenomenon among rooms, (i.e. 1 → → → → → → → → → → → memory in the MarkovChain, to represent the visitors knowledge of the visited rooms. Moreover, we reasonablyassume that visitors also remember the time spent in each room. We use a non homogeneoustransition matrix, which is time dependent through a weight function S . Henceforth, werefer to this model as Time Varying Markov Model (TVMM).6.1. Time Varying Markov Model (TVMM).
Since the museum comes with R rooms,we consider a R × R transition matrix K . Following the frequentist definition of probability, K r ,r is computed by first counting, from all the measured trajectories, the number oftransitions from room r to room r , where r = r holds if the visitor remains in the sameroom. K r ,r = N (cid:88) n =1 k n ( r , r ) , (11)where k n ( r , r ) denotes the number of r → r transitions along the n -th trajectory of thedata set. The sum over columns of K r ,r represents the total time, in time bin, spent by alltracked visitors in room r . If we normalize K by row, so that (cid:80) r p r ,r = 1, we obtain atransition matrix M where the new element p r ,r represents the probability to move fromroom r to room r , see Figure 14.In order to avoid the room bouncing phenomenon, we make the transition matrix M time-dependent. More precisely, we consider the matrix˜ M r ,r ( t ) = M r ,r S r ( t ) , r , r ∈ { , . . . , R } , (12) where S r ( t ) is the survival function associated to ToP( · , r ) via its Weibull fit parameters( λ r , k r ), S r ( t ) = e − ( t/λ r ) kr . (13)In other words, S r ( t ) quantifies the probability that a guest visits room r for a time intervallonger than t . S r ( t ) is a decreasing function such that S r (0) = 1 and S r ( t max ) = 0, where t max is the largest measured ToP( · , r ).At each time step of the simulation, the function S r ( t ) must be updated on the basis ofthe time spent in each room, and the transition matrix ˜ M has to be normalized by rows inorder to have a correct definition of the transition probability.In the following, we detail the exceptions to the transition dynamics in (12) to cope withthe access and exit conditions. Beginning of a visit: in Galleria Borghese visitors enter all together at the beginningof the visit turn. However, due to some delay (ticket control, late arrival, queueat wardrobe), the entrance process is completed in about 20 minutes. We simulatethese dynamics extracting the delay at random from the set of measured delay. Inaddition, we use another probability distribution function to assign the entrance room(we recall that Galleria Borghese has three entrances:
Ratto di Proserpina (room 4),
Portico (room 5), and Pinacoteque (room 9)).
Conclusion of a visit: the wing Out is conceptually different from the other wings. There-fore, we manage the exit time in a distinct manner. The exit is not controlled by ˜ M ,instead it is managed via the hazard function h of the Weibull distribution with pa-rameters ( λ ∗ , k ∗ ) associated to the total time of visit, see Figure 7 d . More precisely,at every time bin t of the simulation, the exit probability is given by P ( r t +1 = Out | r t = r exit ) = h ( t ; k ∗ , λ ∗ ) , (14)where r exit is an exit room and h ( t ; k ∗ , λ ∗ ) = k ∗ λ k ∗ ∗ t k ∗ − . (15)6.2. Simulation results.
Before presenting the results of our model, we explain how wecompare simulated trajectories with the meaurements in our data set. We perform suchcomparison to quantify the accuracy of the simulation. The observables of interest are thefollowing:
ToP: we evaluate the mean and coefficient of variation of the ToP distributions, for bothreal and simulated visits. We consider, as accuracy measure, the relative ToP differencebetween real and simulated trajectories.
PpR: we consider 100 statistically independent simulated turns, each including a total of400 generated trajectories (similarly to Section 5.1.3, simulated visits are replicated q times, where q is a uniform integer random variable between 1 and 6, to mimic socialgroups). We compute the PpR at each time bin as an ensemble average across such100 realizations. Clusters: we use the same clustering technique presented in Section 5.3 to aggregate sim-ulated trajectories. The aim of this analysis is to check if the most numerous clusteris sufficiently close to the measurements; this guarantees that the simulator creates asufficient amount of plausible trajectories.Figure 15 shows two simulated trajectories, which indeed share typical features with mea-surements: the Pinacoteque is visited once and the visit path follows the natural numberingof rooms. At times, people come back to rooms already visited, as in real life.
ANAGING CROWDED MUSEUMS 23 a . r oo m time [min] 946785123 b . r oo m time [min] Figure 15.
Two simulated trajectories: a. A long trajectory which begins from thePinacoteque (room 9) and then moves to the main floor according the room enumeration. b. A short trajectory that begins from room 5, traverses the main floor according to theroom enumeration, and finally reaches the Pinacoteque.
Room 1 2 3 4 5 6 7 8 9 Museum δµ
11% 10% 2% 8% 12% − −
2% 7% − − δ VC − − − − −
20% 13% 12% 3% 31% − Table 2.
Relative error between mean and coefficient of variation of ToP distributionevaluated for real trajectories and simulated ones. δx = ( x sim /x real − x iseither µ or V C . Table 2 compares the real and simulated ToP distibutions, by considering the relativedifferences in ToP averages ( µ ) and variation coefficient (VC = σ/µ , σ being the standarddeviation of the ToP distribution), respectively δµ = µ sim µ real − δ VC = VC sim VC real − . (16)The mean values of the distributions are well approximated, despite the simulations tend toslightly overestimate the ToP in the main floor and to underestimate it in the Pinacoteque.The δ VC indicator highlights instead some differences between model and data: real visitorsare more unpredictable than simulated ones, which yields negative δ VC values. On thecontrary, the dynamics in the Pinacoteque appears predictable and even more consistentthan in simulations. This most likely relates with the fact that the Pinacoteque is the areawith the weakest antenna coverage: amending not detected data diminishes the variance ofthe measured ToP distribution.In Figure 16, we compare measurements and simulations considering the PpR as a func-tion of time. Simulations are reported in terms of ensemble statistics among 100 realizations,in particular we consider ensemble PpR average and ensemble PpR standard deviation.In Figure 17 we finally report the representative trajectories of the two most numerousclusters obtained by gathering real and simulated trajectories.7.
Museum control and optimization.
We are now ready to employ the digital twinintroduced in the previous section as a tool to improve the museum experience. More pre-cisely, we simulate different scenarios and observe visitors behavior in virtual environments, a . p e o p l e p e rr oo m time [min] µ ± σ real simulated b . p e o p l e p e rr oo m time [min] Figure 16.
Comparison of the average PpR of real visits (red line) and the ensemble-average PpR of simulated visits (blue line) in a. Ratto di Proserpina and in b. thewhole museum. The shaded area corresponds to the interval [ µ − σ, µ + σ ]. We note thatthe blue line is almost entirely contained in the shaded area, as expected. r oo m time [min]realsim Figure 17.
The two most numerous clusters obtained gathering real and simulatedtrajectories. The real case joins 16% of real trajectories, whereas the simulated one 18%.We note that they share a number features, e.g., the ToP in each room, the total timeof visit, the entry room (
Portico , room 5), and the final room (Pinacotque, room 9).The main difference is the behavior after completing the visit of the main floor. Realvisitors come back counterclockwise, while simulated visitors keep walking clockwise.This could be explained by the fact that many visitors ask for information in room 5and are sent backwards to the staircases. The model does not include the interactionswith the museum staff, hence cannot catch this feature. aiming at supporting curators decisions. In this regard, it is useful to remark that changingthe ticketing strategy or the duties of security staff can require weeks of training in real life.For our case study, we identify the following control variables and objectives.
Control variables:
C1: considering that Galleria Borghese has three entrances (
Ratto di Proserpina , Portico ,Pinacoteque), museum managers can assign a certain percentage of visitors to eachentrance (currently they are 15%, 60%, and 25% respectively). Operationally, suchcontrol can be implemented by introducing a tag (e.g. name or color) in the ticketwhich specifies the entrance.
ANAGING CROWDED MUSEUMS 25 , , , ,
40 60 , ,
20 80 , , , ,
40 40 , ,
20 60 , , , ,
40 20 , ,
20 40 , , , ,
20 20 , , , , r , r , r T o T [ m i n ] visitors per entrance [%] r r { r , . . . , r } Figure 18.
Total time duration in which the overcrowding threshold is exceeded (ToT)in room 9 (Pinacoteque), in room 8 (
Caravaggio ) and in all other rooms (sum of eachToT is considered), for 13 triplets ( E , E , E ). We observe that the overall ToT exceedsat least 120 min over a day of visit regardless the entrance system. C2: the scheduled entry times in the museum can be tuned (currently visitors enters at09:00, 11:00, 13:00, 15:00 and 17:00, after the museum empties).
C3: the number of visitors allowed in each turn can be modified (currently it is 360 reservedin advance plus 30 last-minute).
C4: the fixed duration of a visit turn can be either (C4a) modified or (C4b) totally removed(currently it is set to 2h slots).
Objectives:
O1: keeping the PpR below a certain room-dependent treshold. Historically, our studybegan precisely to control the number of visitors in the Pinacoteque, which has a verylow admittance limit for safety reasons.
O2: keeping the PpR, in any room at any time, approximately constant. This would reducestrong variations of relative humidity which can damage the artworks [6, Chapter 2].
O3: decreasing the queue at the entrance.
O4: increasing the number of visitors per day.Note that O1, considering the emergency situation caused by the COVID-19 virus pan-demic, can be employed to respect the imposed social distances legislation.Among the many possibilities, we focused on two improvements: C1 aiming at O1, andC2, C3 & C4b aiming at O1 & O2.7.1.
Entrance strategy optimization.
Keeping the existing conditions regarding thenumber of visitors and the time horizon, we explore the effects of a different visitor partitionamong the three entrances (C1). We aim at a PpR as low as possible in all rooms (O1),especially in the Pinacoteque, which is the room with the most stringent safety constraints.We fix an overcrowding threshold for each room, representing a PpR limit the curatorsdo not want to exceed. Then, we pursue a brute force attack to the optimization problem,trying all the possible triplets ( E , E , E ) ∈ [0 , , (cid:80) e =1 E e = 100, which indicate thepercentage of visitors starting the visit from each entrance e = 1 ( Ratto di Proserpina ), e = 2 ( Portico ), and e = 3 (Pinacoteque).Figure 18 shows the results of the optimization process evaluating the total time the PpRexceeds the overcrowding threshold (ToT), for room 9 (Pinacoteque), room 8 ( Caravaggio ),and for all the remaining rooms. The best triplet for the Pinacoteque is (20 , , a . c h f time [min] b . c h f time [min]rawcensored empiricalend of visit Figure 19.
Cumulative hazard function associated to the Weibull distribution of thewhole museum. Empirical values are calculated with Kaplan-Meier method. a. Withoutcensoring (cf. Figure 7 d. ) and b. after censoring the last 5 minutes of visit (newparameters are k ∗ = 3 . λ ∗ = 596). This method allows us to get a better fit of thereal distribution between 0 and 2h, i.e. the visit interval. The uncensored fit, instead, isnegatively influenced by the forced exit. the best triplet for Caravaggio is (40 , , , , Removing the finite time horizon of the visits.
The full elimination of the currentfinite time horizon allowed for the visits is a challenging improvement for the museumexperience. The idea is to keep the reservation mandatory, with entry interval fixed every30, 60 or 120 minutes (C2), but, unlike current setting, remove the requirement to leaveafter 2h (C4b). The immediate advantage is that the museum staff does not have to shutthe museum down at the end of the visit turn, thus saving about 5-7 minutes during whichthe museum remains completely empty. Moreover, this would also be a great advantage forthe (few) visitors who want to stay very long time inside the museum.Unfortunately, as it happens for every mathematical model, simulation results are reli-able only in the conditions in which the simulator was developed and calibrated. In ourmeasurements, less than 1/4 of visitors are still inside the museum when the time limit isreached (and are forced to exit); for these a (negative) influence of the time limit certainlyoccurs. Nevertheless, such influence possibly exists also for the other 3/4, that might havescheduled their visit according to the existing time constraints.We attacked the problem by censoring the Weibull distribution of the time of visit ofthe whole museum (cf. Section 5.1.1). This statistical procedure allows us to deal with dataset in which the event of interest is not observed during the study. We obtain the newdistribution as a maximum likelihood estimate censoring the last 5 minutes of visit, seeFigure 19. We use the estimated parameters to modify the hazard function which controlsthe conclusion of the visit.We simulated an entire day (9 a.m. – 7 p.m., corresponding to total time span of the 5visit turns currently implemented). This is necessary as after removing the time limit, visitturns overlap and museum never empties. Figure 20 shows the result of the optimizationprocess. The best strategy is to let 100 visitors (C3) enter from the main floor (C1) every 30
ANAGING CROWDED MUSEUMS 27 a . p e o p l e p e rr oo m real simulated comfort b . m u s e u m v i s i t o r s time Figure 20.
PpR as a function of time in the current settings and considering the bestentrance strategy. The comparison includes a. room 8 ( Caravaggio ) and b. the wholemuseum. minutes (C2). These choices eliminate completely the peaks in the PpR indicator (congestionmoments, O1) and the PpR remains stable with small fluctuations during the whole visitday (O2). Having the system approximately at this thermodynamic-like equilibrium greatlyfacilitates the management since it allows to calculate – using the measured transition matrix– the average number of people in each room from the number of visitors allowed (i.e. soldtickets).8. Conclusions and future work.
This study aimed at measuring, analyzing, modelingand optimizing visitors behavior in museums or similar environments. The practical goalwas to provide suggestions to museum curators for efficiently managing visitors flows.The implemented measurement system is sustainable for the museum, being economicallyviable and well accepted by visitors. A free application to be installed on the smartphonecould serve as beacon as well, provided visitors find it useful (as an audio-guide, for exam-ple). Employing Raspberry Pi’s as fixed Bluetooth antennas appeared quite convenient andallowed the necessary development flexibility.A major issue surely comes from the noisiness of the Bluetooth signal, which must beovercome by suitable data post-processing. The sliding window approach has proven tobe more effective in measuring room transitions, while the machine learning approach per-formed better at estimating the permanence time in the various rooms.From the trajectory analysis we have identified some issues in the museum design andvisit experience that can be considered by curators: for example, rooms of the same size havedrastically different time of permanence, like
Caravaggio and
Satiro su delfino . This suggestsa different positioning of artworks, although this is not always possible due to historical or architectural constraints. In addition, rooms like
Paolina have an uneven distribution ofvisitors, being congested in the first half of the visit turn and under-used in the second half.The museum simulator allowed us to propose the implementation of a new ticketing andentrance system. The entry scheme identified is to let 100 people enter every 30 minutesfrom
Portico and
Ratto di Proserpina , while eliminating the 2h time limit, thus reducingcongestion and fluctuations of the number of people in each room.In the next future we plan to further improve the model presented here. In particular, weaim at including the internal dynamics of social groups (families, friends, guided tours), andat considering the impact of congestion on the individual behavior. This is to lift the currentstatistical independence of simulated trajectories, thus increasing the level of complexity.The impact of visitors on the local microclimate is also an outstanding issue to which weaim. On the basis of the present work and [16], one can achieve a coupled model for localand future temperature, humidity and crowding, on which basis one can program intelligentair conditioning systems.
Acknowledgements.
We would like to thank Sara Suriano, Massimiliano Adamo, FedericoRicci Tersenghi, Elisabetta Giani, and all the staff of Galleria Borghese for all their timeand support during this project.
Funding.
Results presented in this paper are achieved under the project
Management offlow of visitors inside the Galleria Borghese in Rome , supported by Ministry of CulturalHeritage and Activities and Tourism, Galleria Borghese, and Istituto per le Applicazioni delCalcolo of National Research Council of Italy. Project’s Principal Investigators are MarinaMinozzi (Galleria Borghese) and Roberto Natalini (IAC-CNR).E. Cristiani also acknowledges the Italian Minister of Instruction, University and Researchto support this research with funds coming from PRIN Project 2017 No. 2017KKJP4Xentitled
Innovative numerical methods for evolutionary partial differential equations andapplications .A. Corbetta also acknowledges the support of the Talent Scheme (Veni) research pro-gramme, through project number 16771, which is financed by the Netherlands Organizationfor Scientific Research (NWO).
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