Evacuation travel paths in virtual reality experiments for tunnel safety analysis

Fire Safety Journal 71 (2015) 257–267

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Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf

Evacuation travel paths in virtual reality experiments for tunnel safety analysis Enrico Ronchi a,n, Max Kinateder b, Mathias Müller c, Michael Jost b, Markus Nehfischer b, Paul Pauli b, Andreas Mühlberger d a

Department of Fire Safety Engineering, Lund University, Lund, Sweden Department of Psychology (Biological Psychology, Clinical Psychology, and Psychotherapy), University of Würzburg, Würzburg, Germany c VTplus, Würzburg, Germany d Department of Psychology, Chair of Clinical Psychology, and Psychotherapy, University of Regensburg, Regensburg, Germany b

art ic l e i nf o

a b s t r a c t

Article history: Received 11 April 2014 Received in revised form 3 September 2014 Accepted 23 November 2014

A case study on the analysis of evacuation travel paths in virtual reality (VR) tunnel fire experiments is presented to increase the understanding on evacuation behaviour. A novel method based on the study of the parametric equations of the occupants’ evacuation travel paths using vector operators inspired by functional analysis theory and the new concept of interaction areas (IAs) is introduced. IAs are presented and calculated in order to represent the distance of an occupant from a reference point (e.g., an emergency exit, the fire source, etc.) over time. The method allows comparisons of travel paths between experimental groups as well as comparisons with reference paths (e.g. user-defined paths, real-world paths, etc.). Results show that a common assumption employed by evacuation models (the use of a hypothetical path based on the shortest distance) may be an over-simplistic approximation of the evacuation paths. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Fire evacuation Travel paths Tunnel safety Virtual reality Functional analysis

1. Introduction A series of tragic events such as the Mont Blanc Tunnel fire in 1999 and the Saint Gotthard fire in 2001 attracted the attention of the transportation safety community on the study of tunnel fire evacuations. These accidents demonstrated that fire evacuation assumes particular importance among the accidents in a road network due to the possible serious consequences in terms of loss of lives [1]. In order to understand how the occupants’ behaviours affect tunnel safety, the study of evacuation movement became the focus of dedicated research (e.g., [2,3]). Different methods can be adopted for the study of human behaviour in fire accidents and the evacuation movement of the occupants. Research methods include case studies (i.e. the analysis and/or reconstruction of actual events [4,5]), modelling techniques [6–8], and evacuation experiments (i.e., hypothetical scenario experiments, laboratory experiments and field experiments) (see [9]). Among different types of evacuation experiments, the use of virtual reality (VR) is ever more frequent since it allows high experimental control, no ethical restrictions due to physical harm, logistic and economic advantages. VR has been employed in the n

Corresponding author. E-mail address: [email protected] (E. Ronchi).

http://dx.doi.org/10.1016/j.firesaf.2014.11.005 0379-7112/& 2014 Elsevier Ltd. All rights reserved.

field of fire safety research for different scopes such as the study of occupant training on tunnel evacuation [10,11], pre-evacuation behaviours [12], or the impact of way-finding installations [13]. To date, tunnel evacuation experiments (both at real settings [14–17] or VR [18] are generally focused on the study of the effects of different conditions or variables on the occupants' performance in terms of exit usage and its impact on total evacuation times. This information is crucial since it permits the evaluation of different design solutions, factors affecting human behaviour, emergency management strategies, etc. [9]. Nevertheless, this information does not allows a quantification of the impact of those variables on the occupants' movement and navigation in space over time, e.g. it does not address a quantitative evaluation of evacuation travel paths. VR experiments allow a detailed reconstruction of the evacuation travel paths in space and time. Data about exit usage and evacuation times are generally treated using quantitative methods (e.g., inferential statistics) [10]. In contrast, analysts often provide general qualitative comments on the paths of the occupants and the impact of different factors on their behaviour. The use of a quantitative method to analyse occupant movement would instead permit the comparison of travel paths (e.g., for validation purposes) and this may increase the use of the experimental results. The study of travel paths is particularly important in the context of fire safety science since the occupants' life safety is

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importantly affected by the exposure to toxic products [19] and therefore by the position of the occupants in space over time, in relation to safe and dangerous locations in a given scenario. Thus, it is important not only to study the time spent inside the tunnel (affected by emergency exit usage), but also the position of the occupants during the passage of time (which affects their exposure to hazardous conditions). The study of travel paths should also take into account their impact on total evacuation time, which includes the pre-evacuation phase. In this context, it should be noted that route choice may increase travel distances and cause longer movement times. The consequence could be an increased impact of movement times on evacuation times. In order to investigate evacuees' behaviour, the present work introduces a new method for the study of travel paths from tunnel evacuation experiments. Data on people movement in VR may have limitations in terms of their external validity (occupant behaviours in VR may differ in a real evacuation scenario). In fact, due to methodological, ethical, and practical reason, it is almost impossible to perform fully valid VR experiments [20]. On the other hand, previous studies showed that VR experiments are a useful tool to evaluate the causal relations between independent variables and human behaviour [21,22]. VR studies allow full experimental control which is very difficult to achieve with other experimental methods due to the complex nature of tunnel fire scenarios and even impossible to achieve in unannounced drills. A set of questions can be prompted by these issues. How do we investigate the limitations and the uncertainty associated with experimental VR travel paths? How do we quantitatively compare evacuation movement in a VR environment with other data (from the real world, other VR or modelling studies)? The overall objective of this paper is the definition and exemplary application of a quantitative method to analyse tunnel evacuation travel paths. The goal is not to exactly replicate the conditions of a given fire emergency, but to provide a tool for the study of evacuation behaviour, in particular studying the paths adopted by evacuees during their movement towards a safe place. The scope is therefore to provide a methodology which can be used for the evaluation of the assumptions used by computational modelling tools (i.e. evacuation models). The proposed method for the analysis is based on the study of the parametric equations of the evacuation movement of the occupants. The results of the proposed research will lead to an improved description of the participants'/occupants' behaviour and thus, contribute to a better understanding of evacuation behaviour. An illustrative case study consisting of a set of VR tunnel evacuation experiments performed in the 3D multisensory CAVE (i.e., Cave Automatic Virtual Environment) laboratory of the Department of Psychology of the University of Würzburg, Germany is presented. The case study employs the method to investigate evacuation travel paths towards an emergency exit during a simulated tunnel emergency. Finally, a discussion on the advantages and limitations associated with the use of the method is presented.

2. Method In order to study the evacuation travel paths in a quantitative way, the present work suggests the use of operators inspired by the concepts of functional analysis theory [23]. Functional analysis is a branch of mathematics using generalisations of measures of vectors applied to functions. In this paper, the term functional analysis is used to refer to generalisation of linear algebra, analysis and geometry. Evacuation travel paths are studied through the analysis of the coordinates where people are located in the VR environment over time (represented using parametric equations).

It should be noted that the present work makes use of the term sequence to refer to a finite sequence of points (representing the coordinates of pedestrian paths) or a finite number of pedestrians. The term convergence is used to refer to the average value that these finite sequences tend to. Tunnels are generally simple (generally straight) geometries without compartmentation in which the relative distance of the occupants to the hazard (i.e. the fire source and its characteristics) and the time spent inside the tunnel are generally the main factors affecting life safety. In addition, previous experiments [9,16] showed that the likelihood of occupants using an emergency exit and their movement speed is affected by their position in the cross section (i.e. their proximity to the wall). For these reasons, the present work suggests to perform a quantitative assessment of evacuation travel paths by investigating people movement as a two-dimensional problem where the average coordinates are calculated and studied. The method proposed for the study of VR evacuation travel paths can be summarised in three main steps: 1) The variables of interest (the parametric equations of the evacuation travel paths) describing the travel paths are identified. 2) The convergence of the variables of interest and acceptance criteria for the uncertainty associated with the estimation of the mean coordinates are identified. The analysis is performed by calculating a set of convergence measures (inspired by functional analysis theory) of the VR coordinates of the experimental travel paths towards the average. 3) A quantitative comparison of travel paths is performed. The method can be used to perform two main types of analysis: (1) Validation studies: A comparison between VR travel paths and reference paths (e.g., evacuation paths from the real world, paths obtained from other models which are already validated, etc.); (2) Analysis of different conditions: the comparison of different subsets of travel paths against each other permits the study of the impact of different variables on people movement. 2.1. Occupant trajectories Occupant trajectories are the travel paths that pedestrians follow through space as a function of time. In VR experiments the data about occupant position and orientation is measured and recorded with the VR simulation system. The data consists of a finite sequence of the coordinates in space over discrete timesteps. Thus, each occupant trajectory corresponds to a parametric equation. It is possible to obtain a set of n finite sequences for each jth occupant corresponding to n experimental travel paths:

(

)

Occ j = x j (ti ), y j (ti ), ti ,

for 0 ≤ ti ≤ texit

(1)

where:

x j (ti ) is the set of xj coordinates for each jth occupant during the experiments. y j (ti ) is the set of yj coordinates for each jth occupant during the experiments. ti is the time-step for a total of q time-steps, based on the experimental data. The trajectories of the occupants can be represented using sequences which use the x or y coordinates of the jth occupant in space as the dependent variable and the time-step ti as the independent variable. Each jth occupant experimental travel path will be represented using a set of sequences corresponding to its coordinates. For instance, Eq. (2) presents a sequence of values

E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

corresponding to the x coordinate.

Occ xj

=

(

x1j ,

j

…, xi , …,

x qj

)

for 0 < ti < texit

(2)

For all n occupants, texit corresponds to the time spent by the slowest occupant to reach the exit during the experiments. Hence, if a “faster” occupant reaches the exit at a time tp otexit , its coordinates remains constant for every t p rti rtexit and correspond to the coordinates of the exit. Using this assumption, the comparison between the trajectories of occupants having different texit can be made considering sequences with the same number of elements. During the experiments, participants may also be assigned to different experimental conditions. Thus, occupants can be split in different groups. Examples of such conditions may be demographic characteristics, environmental conditions, etc. Given a group of occupants, the position of each jth occupant in each timestep ti can be represented using matrices representing its coordinates (Occ x for the x coordinate and Occ y for the y coordinate). The rows in the matrix represent the x (or y) coordinate of the individual experimental travel paths. The columns of the matrices show the coordinates of all occupants for each individual timestep. For instance, Eq. (3) presents Occ x , the matrix representing the x coordinate.

⎛ Occ 1 ⎞ ⎛ x11 … x⎟ ⎜ ⎜ ⎜ ⋮ ⎟ ⎜⋮ ⋱ ⎜ Occ x = ⎜⎜Occ xj ⎟⎟ = ⎜ x1j … ⎜ ⋮ ⎟ ⎜⋮ ⋱ ⎜ ⎜ n⎟ ⎝Occ x ⎠ ⎜⎝ x1n …

x1i

…

x1q ⎞ ⎟

⋮⎟ ⎟ … x qj ⎟ ⋮ ⋱ ⋮⎟ ⎟ xin … x nq ⎟⎠

⇀ ∑in= 1 (ai bi ) a , b> <⇀ = ⇀ ∑in= 1 bi2 || b ||2

(5)

EPC calculates a factor to reduce the distance between the ⇀ vectors ⇀ a and b to its minimum, i.e. the best possible fit of the two curves represented by the vectors. The more similar the curves, the more EPC tends to 1. Nevertheless, curves are not necessarily similar if EPC is close to 1. This is because two curves may cross each other several times. To further study two curves, the concept of Secant Cosine (SC) is also introduced. It represents a measure of the differences of the shapes of two curves. This is investigated by analysing the first derivative of both curves. For n data points, a multi-dimensional set of n  1 vectors can be defined to approximate the derivative (see Eq. (6)) ⇀ 〈⇀ a , b〉 ⇀ = ||⇀ a || || b || ∑ni = s + 1 ((a i − a i − s )(b i − b i − s )/ s 2 (t i − t i − 1)) ∑ni = s + 1 ((a i

− a i − s )2/ s 2 (t i − t i − 1)) ∑ni = s + 1 ((b i − b i − s )2/ s 2 (t i − t i − 1))

(6)

where:

(3)

The concepts of functional analysis theory are used in this paper to study data using vector notations and define operations on these vectors. This permits the evaluation of the coordinates of occupant trajectories using vector operators. In fact, the sequences of occupant coordinates can be treated as sets of multi-dimensional vectors which are representative of the travel paths in the x and y coordinates. Three concepts of vector mathematics are introduced, namely the Euclidean Relative Difference (ERD), the Euclidean Projection Coefficient (EPC) and the Secant Cosine (SC). Further information on previous applications of these concepts in the field of fire science is discussed in Peacock et al. [24], Galea et al. [25] and Ronchi et al. [26]. The single comparison of two individual points in a curve can be made by finding the norm of the difference between the two vectors representing the data. A norm represents the length of a vector. The distance between two vectors corresponds to the length of the vector resulting from the difference of the two vectors. For a generic vector ⇀ a ||. This a , the norm is represented using the symbol ||⇀ concept can be extended to multiple dimensions. The distance be⇀ tween two generic multi-dimensional vectors ⇀ a and b is therefore ⇀ ⇀ the norm of the difference of these vectors || a − b ||. The Euclidean relative difference between two vectors can be normalised as a ⇀ relative difference to the vector b (see Eq. (4)).

t is the measure of the spacing of the data, i.e. in this case, t ¼1 since there is a data point for each travel path; s represents the number of data points in the interval (or the period of the noise in the data); n is the number of data points in the data-set. SC ¼1 means that two curves have the same shape. Nevertheless, SC ¼1 does not necessarily mean that the curves are identical since they may be translated of a constant offset. Further information about the selection of the parameters to be employed for the calculation of the Secant Cosine can be found in Ronchi et al. [26]. The above mentioned issues associated with the values of EPC and SC lead to the study of the values for ERD, EPC and SC together in order to correctly interpret the differences between two curves. The only case in which the sole analysis of ERD is sufficient is the case of ERD ¼0, since it ensures that two curves are identical. It ⇀ should also be noted that the chosen order of the vectors ⇀ a and b can affect the calculation of the measures. This is reflected in the use of one of the two vectors as reference for the normalisation. The chosen order of the values within each vector can also impact the measures, i.e., it may affect the shapes of the corresponding curves (and the subsequent values of the vector operators). For this reason, it is important that the vector operators are used after defining a consistent method for ordering the values within the vectors and the reference vector for the normalisation. 2.3. Analysis of average occupant paths

∑in= 1 (ai − bi )2 ∑in= 1 (bi )2

EPC =

=

xij

2.2. Vector operators

⇀ a − b || ||⇀ ERD = = ⇀ || b ||

the two vectors and the cosine of the angle between them. The inner product can be interpreted as the standard dot product. The Euclidean Projection Coefficient (EPC) is found by studying the minimum problem, i.e., studying when the derivative of the function is zero (see Peacock et al. [24] for the full solution of the minimum) and it is presented in Eq. (5).

SC =

⋮ ⋱

259

(4)

The Euclidean Relative Difference (ERD) represents, therefore, the overall agreement between two curves. If ERD ¼0, two curves are identical. ⇀ The concept of projection coeffictient is also introduced. 〈⇀ a , b〉 is the inner product of two vectors, i.e., the product of the length of

The average of the values of the observed x or y coordinates for each group corresponds to a set of n values, i.e., a sequence for every jth aggregate sample of occupants (see Eq. (7) for an example referring to the x coordinate).

(

j j j j X av j = X 1, … , X i , … , X q

where

X1j

= (1/j)

∑ij =< 1n

x1j ,

)

…,

(7) X ij

= (1/j)

∑ij =< 1n

x i , …, j

X qj

= (1/j)

∑ij =< 1n

x qj .

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E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

It is possible to produce matrices corresponding to the average experimental x or y coordinates over time for each additional jth occupant under consideration. The rows of the matrix represent the average of the x or y coordinates for j occupants. The first row corresponds to a single occupant, the last row corresponds to the average of all n occupants of a group. The columns of the matrix represent the average coordinates for an individual time-step ti (see Eq. (8) for an example referring to the x coordinate).

(

)

j j j X av = X av = 1 … X av i … X avq

⎛ X 1 = x1 1 ⎜ 1 ⎜ ⋮ ⎜ = ⎜ X1j ⎜ ⋮ ⎜ ⎜ Xn 1 ⎝

…

X1i

=

⋱

⋮

…

X ij

⋱ …

⋮ X ni

x1i

… X1q = x1q ⎞ ⎟ ⎟ ⋱ ⋮ ⎟ … X qj ⎟ ⎟ ⋱ ⋮ ⎟ n … X q ⎟⎠

j Xav i

(8)

j Yav i

where

X eq, j

X1e,

= (1/j)

j

= (1/j) ∑ij =< 1n x1e, j ,

∑ij =< 1n

j

)

(9) ….,

X ei , j = (1/j) ∑ij =< 1n x ie, j ,

j 2 ∑ni = 1 (X e, i )

(10)

ERDyei, j

The consecutive and can be interpreted as sequences convergent to the expected value equal to 0 (the case of two identical curves). For this reason, a measure of the con,j ,j and ERDyeconvi (see Eq. (11) vergence is possible by studying ERDxeconvi for an example referring to the x coordinate).

(11)

,j ,j The calculation of ERDxeconvi and ERDyeconvi permits estimation of the impact of the sample size on the overall differences between consecutive average paths.

paths respectively of the x and y coordinates of the experiments. This representation is useful for the application of the operators based on functional analysis theory. Each vector corresponds to a curve representing the average path of an aggregate sample of occupants (see Eq. (9) for an example of the vector corresponding to the x coordinate).

(

ERDxei, j

j j− 1 2 ∑ni = 1 (X e, − X e, ) i i

,j ERDxeconvi = |ERDex,i j − ERDex,i −j 1|

A set of consecutive and are obtained for each additional travel path. The set of all n consecutive average x and y coordinates is found for every time-step ti. This produces a total of j j n sequences or rows (one for each occupant) X av i and Y av i for q time-steps. ⎯⎯⎯⎯⎯⎯→ The sequences can also be considered as vectors X eav, jj and ⎯⎯⎯⎯⎯⎯→ Y eav, jj which are representative of the aggregate average travel

⎯⎯⎯⎯⎯⎯→ X eav, jj = X1e, j, …, X ie, j, …, X eq,

ERDxe, j

⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯→ || X eav, jj − X eav, jj − 1|| = = ⎯⎯⎯⎯⎯⎯→ || X eav, jj||

…,

x qe, j

2.4. Convergence measures This section introduces a set of measures to investigate the convergence of the coordinates of a single group of travel paths towards the average. The average coordinates corresponding to the e and entire sample can be found in the last row of the matrices Xav e . Yav 2.4.1. Convergence measures 1: Euclidean Relative Distance (ERD) In order to perform the analysis of the convergence of the experimental data towards the average, the operators based on functional analysis theory are used. A set of convergence measures are developed for the study of Xav and Yav . The first operator employed to study the convergence of the travel path is ERD. In this instance, this measure is used to study the consecutive average curves representing the coordinates of the VR experimental travel paths. This is performed considering the consecutive average coordinates calculated in Xav and Yav . The rows of those matrices represent the coordinates of the average travel paths of the occupants. A set of ERDxe, j and ERDye, j can be calculated for progressive pairs ⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→ of vectors X eav, jj , (x coordinates) and Y eav, jj (y coordinates), which represent the progressive average coordinates of travel paths (see Eq. (10) for an example referring to the x coordinate).

2.4.2. Convergence measures 2: Euclidean Projection Coefficient (EPC) The same type of convergence measures can be produced for EPC in order to identify the fit between consecutive average curves. The consecutive EPC xei, j and EPC yei, j can be interpreted as a sequence convergent to the expected value equal to 1 (the best possible agreement between two consecutive average curves representing the coordinates of the travel paths). For this reason, a measure of the convergence of the sequence can be performed by ,j ,j and EPC yeconvi (see Eq. (12) for an example referstudying EPC xeconvi ring to the x coordinate). ,j EPCxeconvi = |EPCex,i j − EPCex,i −j 1| ,j EPC xeconvi

(12)

,j EPC yeconvi

and permits the estimation of the impact of the sample size on the possible agreement between two consecutive average travel paths. 2.4.3. Convergence measures 3: Secant Cosine (SC) The last convergence measures can be developed for the Secant Cosine (SC). This permits the estimation of the agreement between the shape of two consecutive average curves. Also in this case, the consecutive SC xei, j and SC yei, j can be interpreted as a sequence convergent to the expected value equal to 1 (when the shapes of the two consecutive average curves representing the coordinates are equal). Hence, a measure of the convergence of the sequence can ,j ,j and SC yeconvi (see Eq. (13) for an be performed considering SC xeconvi example referring to the x coordinate). ,j SCxeconvi = |SCex,i j − SCex,i −j 1| ,j SC xeconvi

(13)

,j SC yeconvi

and allows the analysis of the impact of the sample size on the possible differences between the shapes of two consecutive average curves representing the coordinates over time of the average travel paths. This permits the study of the variability around the average of the coordinates representing the travel paths of the participants. In addition it allows assessing if the average observed travel path is stable (i.e. representative of the behaviour). It makes it possible to identify if an additional participant would modify significantly the average travel path observed. 2.5. Assessing the variability of the travel paths The measures presented in the previous sections are used to study the convergence of the experimental travel paths towards the average. A set of criteria can be employed to define if the variability associated with the average coordinates of travel paths (in relation to the experimental sample size) is deemed to be acceptable for the analysis under consideration. This consists of the definition of three thresholds

E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

261

corresponding to the convergence measures for each coordinates, i.e. the maximum accepted value for the convergence measures (the acceptance criteria are met if all thresholds are bigger than the corresponding convergence value). This permits understanding if the sample size of the VR tunnel evacuation experiments is representative of the evacuation travel paths. Six thresholds are identified:

 TRERDx and TRERDy: two thresholds for the Euclidean Relative Distance for the x and y coordinates.

 TREPCx and TREPCy: two thresholds for the Euclidean Projection Coefficient for the x and y coordinates.

 TRSCx and TRSCy: two thresholds for the Secant Cosine for the x and y coordinates. The comparison between the thresholds and the final values of the corresponding measures can be used to assess the convergence of the observed travel paths towards the average in relation to the sample size. An additional acceptance criteria needs to be assessed, i.e., a finite number of consecutive values b for which the acceptable thresholds must not be crossed. This needs to be assessed in order to verify that the convergence measures are stable under certain thresholds over a pre-defined number of additional test participants. This requirement is based on the assumptions that a higher value for b increases the confidence on the fulfilment of the acceptance criteria (in line with the law of large numbers). The selection of the acceptance criteria may depend on several factors such as the scope of the analysis, the evacuation scenario, etc. The use of the convergence measures corresponding to one of the two coordinates (either x or y) permits an independent analysis of the coordinates of the travel paths in relation to the variable of interest (e.g., the distance to the fire source or the tunnel walls). If the ERD is equal to 0, it would not be necessary to study the other two measures since the curves are identical. If ERD is different than 0, it is recommended to analyse the three convergence measures together (for each coordinate of interest), in order to investigate the correspondence between the curves. 2.6. Reference travel paths Previous studies demonstrated that the comparison between experimental paths and reference paths is a useful tool to perform validation studies and evaluate causal relations between different variables during evacuation movement [27–29]. For instance, it is possible to compare the VR travel paths with paths based on real world data, travel paths generated by evacuation models, userdefined paths, shortest paths, etc. Fig. 1 shows two examples of reference travel paths in relation to the starting position where an occupant is approaching the exit. The continous line is a user-defined hypothetical path. The dashed lines instead represent the shortest paths. The shortest paths may correspond to the paths adopted by evacuation simulation models, which are often based on the assumptions that agents walk the shortest paths [30,31]. The comparison with reference paths can be done considering a single reference path Occ r , or a set of different reference paths (if for example a distribution of different travel paths is taken into consideration). They correspond respectively to a single (a single reference path) or multiple (several reference paths) set of parametric equations (see Eq. (14)). r Occ r = (ti, x r (ti ), y r (ti ) ), for 0 ≤ ti ≤ texit

(14)

where:

Fig. 1. Possible reference paths. Continuous lines refer to hypothetical paths while dashed lines refer to shortest paths.

x r (ti ) is the set of xj coordinates for the hypothetical user-defined travel path. y r (ti ) is the set of yj coordinates for the hypothetical user-defined travel path. A set of additional information is necessary to describe the r reference paths. There is a need to identify the exit time texit and the coordinates of the reference paths. This information may be directly available if the reference path is derived from real-world experiments. If user-defined reference paths are employed, for example, a constant hypothetical speed v r can be used to calculate r . The coordinates of the reference travel path (x r (ti ) , y r (ti )) are texit calculated at every time-steps (based on the hypothetical userdefined trajectory). r e e may be different than texit (texit corresponds to the Since texit time of the slowest occupant of the VR sample), a set of assumpr e , x r (ti ) tions are needed to compare the travel paths. If texit < texit r e and y r (ti ) remain constant for every texit r ti r texit , and they corr e , x e (ti ) respond to the coordinates of the exit. Similarly, if texit > texit r e e and y (ti ) remain constant for every texit Z ti Z texit , and they correspond to the coordinates of the exit. This assumption is needed to perform the subsequent study of the travel paths using the concepts inspired by functional analysis theory. It is therefore possible to produce similar matrices and sequences as for the case of experimental data. For instance, it is possible to consider a multiple set of reference paths corresponding to all jth occupants for a total of m occupants and all the ith time-steps for a total q time-steps (see Eq. (15) for an example referring to the x coordinate).

⎛ Occ r, 1 ⎞ ⎛ x1r, 1 x ⎜ ⎟ ⎜ ⎜ ⋮ ⎟ ⎜ ⋮ ⎜ Occ xr = ⎜⎜ Occ xr, j ⎟⎟ = ⎜ x1r, j ⎜ ⋮ ⎟ ⎜ ⋮ ⎜ ⎜ r, m ⎟ ⎝Occ x ⎠ ⎜⎝ x1r, m

… xir, ⋱

1

⋮

… xir,

j

⋱ ⋮ … xir, m

… x rq, 1 ⎞ ⎟ ⋱ ⋮ ⎟ ⎟ … x rq, j ⎟ ⋱ ⋮ ⎟ ⎟ r, m ⎟ … xq ⎠

(15) r Xav

ti is the time-step for a total of q time-step. This is set in order to be the same as the time-step in the virtual reality experimental data.

Hence, similarly to the experimental data, the matrices and r corresponding to the consecutive average jth reference travel Yav paths (corresponding to a total of m occupants) can be obtained for the x and y coordinates.

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E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

Similarly to the case of the VR travel paths, it is possible to investigate the convergence of the reference paths adopting the procedure presented in Sections 2.3 and 2.4. In the case of a single reference travel path, the coordinates are represented using a single sequence of data for each coordinate and there is no need to perform any convergence study (see Eq. (16) for an example referring to the x coordinate). r X av = ( x1r , …, xir , …, x qr )

(16)

2.7. Comparison of travel paths The comparison between VR experimental travel paths and reference paths can now be performed. This type of analysis permits the comparison between different conditions, groups with different characteristics, the performance of validation studies, etc. The comparison between the observed average VR travel paths and the reference paths is made using ERD, EPC and SC. These operators are calculated considering the average coordinates of the experimental VR travel paths and the reference paths. The vectors corresponding to the average paths of the complete ex⎯⎯⎯⎯⎯→ perimental sample and X eav ¼ (X1e, n, … , X ie, n, … , X eq, n) ⎯⎯⎯⎯⎯→ e , n e, n e Y av ¼ (Y1 , … , Y ei , n, … , Y q ) are compared with the vectors re⎯⎯⎯⎯⎯→ presenting the reference paths X rav ¼ (X1r , n, … , X ri , n, … , X rq, n) and ⎯⎯⎯⎯⎯→ Y rav ¼ (Y1r , n, … , Y ir , n, … , Y rq, n). It should be noted that the vectors of the reference paths may correspond to either the average coordinates of a group of paths, or the coordinates of a single path. The comparison of the vectors results in a set of equations for ERDcx, EPCcx, SCcx, ERDcy, EPCcy and SCcy (see Eqs. (17)–(19) for an example of the operators referring to the x coordinate).

ERDcx

⎯⎯⎯⎯→ ⎯⎯⎯⎯→ r || X eav − X av || = = ⎯⎯⎯⎯→ r || X av ||

EPCcx =

∑in= 1 (X ie,

n

− X ir, n)2

∑in= 1 (X ir, n)2

(17)

⎯⎯⎯⎯→ ⎯⎯⎯⎯→ r ∑n (X e, nX ir, n) 〈 X eav , X av 〉 = i=1 i ⎯⎯⎯⎯→ 2 r 2 ∑in= 1 X ir, n || X av ||

(18)

⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→ 〈 X eav , X rav〉 SC cx = ⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→ = || X eav|||| X rav|| ∑n i=s+1

= ∑n i=s+1

((X ei ,

j IA e, x =

∫0

j IA e, y =

∫0

t eexit

t eexit

x e,j j (t) dt

(21)

y e,j j (t) dt

(22)

Similarly for the distance: j IA e, d =

∫0

t eexit

de,j j (t) dt

(23)

Given the availability of a discrete set of values (x and y coordinates and distances) for each time-step of the VR experiments, the integral can be calculated summing the areas corresponding to each time-step. In the case of multiple experimental groups, the e, h

e, h

e, h

average Interaction Area(s) IA x , IA y , IAd

and the corre-

sponding variances IA e2, h , IA e2, h , IA e2, h can be obtained for each σ x

σ y

σ d

experimental group h. Hence it is possible to compare an experimental group against a reference path(s) or different h groups against each other by performing standard statistical tests. This permits understanding if there are statistical significant differences between the interaction areas taking into account the repulsion/attraction force of an element of the geometry.

3. Case study

((X ei ,

)(

)

n − X e , n X r , n − X r , n /s 2 (t i − t i − 1)) i−s i i−s

n − X e, n 2 /s 2 (t i − t i − 1)) ∑n i=s+1 i −s

)

((X ri ,

n − X r , n 2 /s 2 (t i − t i − 1)) i −s

)

(19)

The obtained values permit an initial quantitative assessment of the differences between the VR experimental travel paths and reference paths or between different paths (e.g., difference between experimental groups). This analysis is followed by a further study of the differences of the travel paths. The analysis is based on the calculation of the areas below the curve(s) representing the x ej (t) and y ej (t) coordinates of each travel paths over time. This permits the study of a 1-dimensional problem such as the distance from a tunnel wall or the study of the distances over time of each to one element of the geometry (e.g. the exit, the fire source, etc.). However, it would be equally possible to study different problems such as the distance in the x and y dimensions combined over time to any given object in the geometry. Eq. (20) presents the sequence representing the distance of the jth occupants over time to a reference point.

(

system ofoordinates is located in correspondence to the point of interest (e.g., an emergency exit, the fire source or a tunnel wall) at the time-step 0, IA ej can be interpreted as representative for the attractive/repulsive force of a given element. Thus, it is possible to quantitatively compare the attractive /repulsion force of such elements. This area can be calculated performing Riemann integration of the area corresponding to the absolute value of the coordinates or the distance. For instance, if the origin of the Cartesian system corresponds to the fire source or an emergency exit, the area IA ej could correspond to a higher or lower risk exposure. Fig. 2 shows an example of the area corresponding to an Occ ej occupant of the tunnel for either the x(t) or y(t) coordinate or the distance d(t) to the origin of the system of coordinates (all variables are expressed as a function of the time). Hence for the single x and y coordinates, for each Occ ej occupant:

Occde, j = d1e, j , …, die, j , …, dqe, j

) for

e 0 < ti < texit

This section presents a virtual reality (VR) case study in which the method presented in the previous section is used to investigate evacuation paths during a simulated tunnel fire. During a tunnel fire evacuation, occupants' information processing and the selection of the appropriate evacuation strategy may be limited by unclear or conflicting information [9]. The physical environment

(20)

This area is here named Interaction Area (IA) and it is defined for each Occ ej occupant as IA ej . Assuming that the origin of the

Fig. 2. Representation of the attraction/repulsive area IA ej corresponding to one of the coordinates of the travel paths or distance to a reference point/element.

E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

may become very difficult to interpret (e.g., the impact of smoke on occupant interactions with exit signs [32]. The present case study investigates flight behaviour after tunnel occupants left their vehicle because of a fire. Specifically, the movement phase of the evacuation process is studied (i.e., pre-evacuation behaviours inside the cars are not taken into consideration). Participants were situated on foot in a simulated VR road tunnel. There, they saw a burning Heavy Goods Vehicle (HGV) and smoke started expanding towards the participants. In the present study the travel paths of those participants who went to the same emergency exit were investigated using the new method presented in Section 2 and compared to a reference travel path. 3.1. Apparatus The study was conducted in the 3D-multisensory CAVE laboratory of the Department of Psychology I of the University of Würzburg. The simulation system immerses participants into the virtual environment and elicits a strong experience of presence. The virtual scenes are based on an in-house written multiplayer modification (VrSessionMod 0.5) of the first-person game Half-Life 2 using the Source Engine (Valve, Bellevue, Washington, USA) (Fig. 3). Experimental control and data recording was established using the VR software CyberSession CS-Research 5.6 (VTplus GmbH, Würzburg, Germany; see www.cybersession.info for detailed information). The visual presentation of the rendered scenes was realized in a 5-sided Cave Automatic Virtual Environment (I Space by BARCO, Kuurne, Belgium (Fig. 4). Six projectors visualised the scenes at a size of 4  3  3 m3 with a resolution of 1920  1200 pixels (the 4 by 3 m front wall was equipped with two projectors and had a higher resolution of 2016  1486. Each projector used two PCs to produce stereoscopic images. The control software CyberSession as well as the control unit for the rendering was run on an additional PC. Stereoscopic images were created using passive interference-filtering-glasses (Infitec Premium, Infitec, Ulm, Germany). Audio was presented using a 7.1 Surround Soundsystem. Movement and orientation data were tracked using an active infrared LED tracking system using 4 cameras (PhaseSpace Impulse, PhaseSpace Inc., San Leandro, CA, USA). Navigation in the VR was possible through a wireless gamepad which was equipped with a tracking target. Visualisation of the 3D images was adapted according to the position and orientation of the head. 3.2. Evacuation scenarios and procedure The experiments consisted of an evacuation scenario in a twobore uni-directional road tunnel. The total size of the tunnel cross section was 9.50 m in accordance with German Standards [33], consisting of two lanes and sidewalks (see Fig. 5).

263

The emergency exit led to an emergency tunnel for pedestrian linking the two bores of the road tunnel. The exit is signposted by an illuminated panel (a standard European back-lit sign [33]). Standard European Emergency signage was available in the tunnel every 25 m [33]. A burning HGV blocked the road in the proximity of the emergency exit. Smoke was rendered in the simulated tunnel in order to produce reduced visibility conditions (visibility conditions range corresponds to approximately 8–15 m for lightreflecting objects [34]). The smoke moved towards the participants starting position (see Fig. 6). The evacuation scenario took place in one of the two bores of the tunnel. Fig. 6 shows the starting position of the participants at the beginning of each trial. Test participants performed the experiment one at a time, i.e., they performed the experiment in the CAVE individually. Trials did not include leaving the car but started standing in the tunnel assuming that participants already had left their vehicle. Other cars were present in the tunnel as shown in Fig. 6. Test participants have to navigate and find the emergency exit in this environment which successively became filled with smoke. Ethical clearance was granted from the ethics committee of the medical faculty of the University of Würzburg, Germany and the participants gave their informed consent before the start of the experimental session. 3.3. Sample Twenty-one participants were performing the experiment. Of these participants, fifteen went to the emergency exit during the experiment, five went to the emergency phone and one participant went to the entrance of the tunnel. Data from the 15 participants who went to the emergency exit (9 female) was used for the present case study. Since anxiety and fear may possibly influence evacuation behaviour, state and trait anxiety (STAI) [35,36] and tunnel anxiety (TAQ; [37]) were assessed. In addition, presence (the experience of being immersed into the VR) and simulator sickness (symptoms of nausea and dizziness associated with being immersed into a VR) were assessed after the experiment using the iGroup Presence Questionnaire (IPQ; [38]) and the Simulator Sickness Questionnaire (SSQ; [39]). Table 1 shows the descriptive statistics of the sample. 3.4. Results The participants who went to the emergency exit needed on average 34.6 s (standard deviation ¼13.2) to reach it. The authors argue that the large deviation is mostly associated with the variability in the evacuation paths adopted by test participants (see Fig. 7). Detailed tracking of the position of the participants in the space over time was performed (every 0.016 s). A qualitative graphical representation of the evacuation travel paths is presented in Fig. 7. 3.5. Analysis of the evacuation travel paths

Fig. 3. Screenshot of the virtual tunnel emergency situation. For improved illustration, the brightness and contrast of this screenshot is increased by 40% compared to the original scene.

An explanatory application of the new method presented in Section 2 is performed here. A hypothetical reference path (e.g. a path representing the assumptions employed by an evacuation model) is compared with the VR experimental travel paths. This type of comparison is useful to perform evacuation model validation studies. The reference path adopted here is based on the hypothesis that occupants walk with a fixed walking speed of 1.25 m/s along the shortest possible path (e.g., see the dashed lines in Fig. 1) and subsequently an occupant would spend 30.3 s to walk 37.9 m (the shortest distance to the emergency exit from their initial position) and reach the emergency exit. This reference path has

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E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

Fig. 4. The 5-sided Cave Automatic Virtual Environment (CAVE) (left) and schematic representation of a participant in the CAVE (right). The CAVE can be closed during the experiment with a slide door so that projection is possible on 4 walls and the floor.

evacuation model to simulate people movement in smoke of an adult [40]. This paper shows an explanatory example of the study of the position over time of the test participants and if they can be approximated with a hypothetical reference path as it is generally assumed by evacuation models. In the specific case of tunnel evacuation, the study of the coordinates of the occupants would permit to evaluate if the reference path adopted is able not only to calculate similar evacuation times, but also to approximate the position of the occupants over time along the tunnel. This is particularly important since it permits an indirect evaluation of the risk exposure of the occupants (i.e. their distance to the fire source).

Fig. 5. Schematic representation of the tunnel cross section.

Fig. 6. Schematic representation of the tunnel evacuation scenario. The arrows represent the origin of the system of coordinates (corresponding to the centre of the emergency exit).

Table 1 Descriptive Statistics of the sample.

Age STAI trait STAI state t1 STAI state t2 TAQ driver TAQ co-driver IPQ SSQ

M

SD

23.33 36.20 35.47 37.20 3.53 1.93 8.40 6.27

3.06 8.13 5.99 6.07 2.03 1.79 9.10 3.13

Note: Only data of participants who went to the emergency exit were analysed; 15 participants. M ¼ Mean, SD ¼Standard Deviation; STAI ¼State-Trait Anxiety Inventory; TAQ¼ Tunnel Anxiety Questionnaire; IPQ¼iGroup Presence Questionnaire; SSQ ¼ Simulator Sickness Questionnaire; Sumscores were calculated for each questionnaire.

been adopted since the simulation of people movement in evacuation models is often based on the assumptions that people walk the shortest path [30]. The walking speed of the reference path is based on the correlation employed by

3.5.1. Convergence towards the average The present study adopts operators inspired by functional analysis theory for the study of the distance to the emergency exit over time of the tunnel evacuation travel paths. The convergence of the coordinates representing the travel paths towards the average is calculated. A set of thresholds have been defined as acceptance criteria in line with the scope of the current analysis (see Section 2.5). The analysis of the convergence permits the evaluation of the suitability of the used sample size as representative of the average coordinates of travel paths. The threshold for the convergence measures of the consecutive aggregate sample of occupants was set to 10% over b ¼5 participants. The selection of these criteria is based on several factors, such as the scope of the analysis (in this case the evaluation of the distance to the emergency exit over time), the evacuation scenario (a tunnel) and the uncertainties associated with the data collection methods. Given the above mentioned factors, the authors argue that the values selected for the thresholds are deemed to permit an initial quantitative comparison of the travel paths with a sufficient degree of accuracy. It should be noted that acceptance criteria for significance testing are generally selected in a similar way. Convergence measures for the distance to the emergency exit of the VR evacuation travel paths are calculated. The values for the convergence measures are presented in Fig. 8. Convergence criteria are met for each of the convergence measures. That is, the sample size in both experiments is sufficient to meet the acceptance criterion of 10%. 3.5.2. Comparison with reference path The application of the method presented in Section 2 permits a quantitative study of occupant travel paths by comparing the average coordinates of the test participants against the reference path (see Eqs. (17)–(19)). An exemplary application is presented in Fig. 9 where two curves are available. One curve is obtained by calculating the average experimental path and showing the

E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

265

Fig. 7. Semitransparent movement paths/trajectories of the participants. The darker the shading the more frequent a participant or the more participants passed a coordinate. Single spots indicate where a participant stopped moving.

distance to the emergency exit over time of a person walking that path. The second curve represents a reference shortest path, i.e., the distance to the exit over time that a hypothetical person walking the shortest path would have. Fig. 9 is shown in order to give an example of the comparison between a hypothetical reference path (which may be for instance the shortest path assumed by an evacuation model) and an average path obtained by experimental data. This allows a direct comparison between modelling assumptions and experimental data. Results obtained employing vector operators refers to the relative differences of the experimental and reference paths in terms of the distance of the occupants over time to the emergency exits. Results are respectively ERDcd ¼0.24, EPCcd ¼ 1.13, and SCcd ¼0.88. The closer ERD is to 0, and EPC and SC to 1, the more similar are the vectors representing the two curves. That is, an initial assessment of the vectors representing the distance of the experimental and reference paths (with regard to the acceptance criterion of 10%) suggest that there are differences greater than the uncertainty (up to 24%). Those values quantitatively measure the likelihood of the participants in walking in different positions in the longitudinal dimension of the tunnel. In order to confirm the analysis performed using the operators inspired by functional analysis theory, a quantitative study of the Interaction Areas corresponding to the distances of the experimental (IA ed, j ) and reference paths (IA rd ) is performed. In a first step,

IA e,y j were calculated for each participant. A non-parametric onesample Shapiro–Wilk test shows that the experimental interaction areas IA ed, j are not normally distributed. The Interaction area of the reference path is calculated and it corresponds to IA rd ¼574.9. This value is then compared with the e

average experimental interaction area (IA d=767.7) using a twotailed Wilcoxon signed-rank test. The test showed that the mean experimental interaction areas are significantly different (α ¼.05, p ¼0.008, r ¼ 0.53). This analysis quantitatively confirms the observation that – in the example under consideration – the assumed reference path is not a sufficient approximation of the distance of the occupants to the emergency exit over time.

ERDdj

EPCdj

Fig. 9. Curves representing the distance to the emergency exit over time for the experimental and reference paths. The origin of the coordinates is the position of the emergency exit.

4. Discussion This paper investigates evacuation travel paths from VR tunnel evacuation experiments presenting a new quantitative method for their analysis. A case study based on the comparison of a set of VR experimental travel paths and a reference path (representative of the assumptions currently employed by evacuation models) is presented. The present study sought to identify a method for investigating limitations and the uncertainty associated with empirically acquired travel paths. The effectiveness of the proposed method relies on the combined study of the coordinates using ERD, EPC and SC to understand the variability of occupant trajectories as well as the analysis of IAs to take into account the attractive/repulsion force of an exit/fire source. In contrast, the sole use of statistical tests for the study of the IAs may have the problem to consider equal areas even if they correspond to different curves (the same value for the integral may be obtained for differently shaped curves). It could be possible to test the variance of the paths without using the IA concept employing a summation of squares during the application of the vector operators inspired by functional analysis theory. Nevertheless, this would not take into account the repulsion/attraction force of an element of the geometry (e.g., the study of a source of risk exposure). The use of

ERDdconvj

SCdj

EPCdconvj

SCdconvj

1.20 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

1.00 0.80 0.60 0.40 0.20 0.00 0

5 10 Test participants (n)

0

5 10 Test participants (n)

15

Fig. 8. Convergence measures for the distance to the emergency exit of the experimental travel paths; ERD ¼ Euclidian Relative Distance; EPC¼ Euclidian Projection Coefficient; SC ¼ Secant Cosine.

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E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

vector operators together with a statistical analysis of the IAs permits a complete quantitative assessment of the difference between experimental travel paths. ERD and EPC permit the evaluation of the agreement between the curves representing different trajectories (either from the same experimental group, different experimental groups or experiment vs reference paths), i.e. it is a quantitative measure of the differences between the coordinates of the trajectories. SC has instead the scope to capture the differences of the shapes of the curves representing the coordinates. Thus, vector operators allows obtaining an overall quantitative evaluation on the differences between the curves, while the statistical tests of the IAs permit understanding if those differences are statistically significant taking into account the repulsion/attraction force of an element of the geometry. The proposed method can be applied to various types of travel paths. For example, data derived from VR experiments can be compared with the evacuation movement obtained by other types of experimental data (e.g. field studies, laboratory experiments, etc.), actual accidents, and simulation studies. For instance, the use of the method for a comparison with data from the real world would permit a full quantitative validation of the evacuation movement observed in VR. The proposed methods exceeds the current state of validation possibilities, as to date, simplistic reductions of behavioural outcomes are mainly used (e.g. number of people evacuated, total evacuation time, etc.). In addition, this type of quantitative validation of the VR travel paths exceeds the possibilities of a qualitative evaluation of travel paths. The new method led to useful information derived from an analysis of the observed positions of the occupants over time. In fact, the method is designed to allow a detailed evaluation of the assumptions adopted in evacuation models. For example, the present case study demonstrates that the assumption of many evacuation models that tunnel users will always take the shortest way to the emergency exit may be over-simplistic. Two behavioural patterns could be identified from the descriptive analysis of the movement paths. First, the participants tend to perform short stops during their evacuation journey and do not choose the shortest route towards the emergency exit. Second, the participants choose different routes towards the emergency exit. The most frequent path chosen was along the right tunnel wall from the starting position. At the level of the emergency signage the participants changed for the other side of the tunnel (the side of the emergency signage and the exit) and then went to the exit along the tunnel wall. These findings are in line with other studies which showed that tunnel users walk along the tunnel wall through a smoke filled tunnel [16]. However, other participants chose a route on the middle of the road. Future studies can also employ the proposed method for the study of the position of the occupants in the cross section (or any other spatial dimension in a building). In fact, emergency exit usage in smoke-filled tunnels is affected by different factors such as 1) the physical visibility of the emergency exit, 2) the likelihood of an occupant actually noticing the exit, 3) the likelihood of the occupants paying attention to the exit and using the information provided (e.g., a sign which indicates the emergency exit), 4) the likelihood of the occupants using the information received, and 5) previous experience and training [10,32]. From a life safety perspective, the position of the occupants in the cross section may have an impact on the first four factors listed above, thus affecting the likelihood of using the exit and the subsequent risk exposure. This is another example of the increased understanding on evacuation behaviour given the application of the proposed method. This paper presents an explanatory example of a quantitative comparison of evacuation movement derived from a virtual environment with a reference path which is representative of the assumption used by many evacuation models. Nevertheless, this

paper presents a method to compare different experimental groups and quantitatively assess their differences. For example, future studies may focus on different factors affecting evacuation behaviour such as occupant demographics, social influence, environmental conditions, etc. It should be noted that the goal of the present research is not the exact replication of the conditions in VR of a given evacuation emergency, but to understand underlying conditions. This requires (among others) experimentally controlled empirical studies. Observations from evacuation drills may lack of experimental control, ergo causal interpretations about behavioural measures cannot be retrieved. To address this issue, this paper suggests the use of VR experimental studies, where a high level of experimental control can be achieved [18]. A comparison of VR experiments and the assumptions employed by evacuation simulations permitted the identification of an important limitation of one of the assumptions generally employed by evacuation models, i.e., they possibly oversimplify occupants’ movement. Further experiments are necessary to test this hypothesis. In this context, future studies are necessary to compare the present results to field and laboratory experiments. One of the limitations of the proposed method is that it adopts the concepts of convergence in mean and the central limit theorem rather than a statistical estimation of the expected mean values of the coordinates. Also, the choice of the index j in Eq. (7) can affect the calculation of the convergence measures since it impacts the order of the experimental travel paths and the subsequent calculation of the progressive average values. For instance, the study of the convergence can be made during the data collection stage to study the appropriateness of the sample size (after setting a priori the acceptance criteria) by adding progressively additional experimental travel paths to the calculation of the convergence. Hence, the choice of the requirement for the acceptance criteria should be carefully evaluated in relation to the scope of the analysis, and an analysis using inferential statistics should be applied together with the new method when using empirical data. Another limitation is the relatively small sample size of the VR study since the adequate sample sizes could not be calculated a priori as the variance of the dependent variables was not known in the population. Significant differences between the reference value in the IA and the empirical data have been found. The effect size corresponds to a mean effect size [41]. Nonetheless, future studies should investigate evacuation behaviour with larger sample sizes. The method presented here is designed for the case of simple straight plane geometries (e.g., tunnels) where the study of travel paths can be approximated to a two-dimensional problem where the information on the average coordinates is useful for the study of life safety issues (e.g. the proximity to the wall where the emergency exits are located or the distance to the fire source may have a direct impact on exit choice). More complex geometries and evacuation travel paths would need a detailed assessment of the suitability of this assumption (the study of the average travel paths may not be sufficient to study human behaviour). However, the method can be applied as well to more complex environments (e.g. evacuation from staircases). In summary, this paper presents a novel methodological approach to a better and more detailed understanding of evacuation travel paths. Functional analysis and inferential statistics can be applied to describe and analyse complex movement patterns. This exceeds the currently commonly employed methods (e.g. study of evacuation time only). The new method was applied to a case study and demonstrated that participants in a VR tunnel emergency study may not always use the shortest way to reach an emergency exit.

E. Ronchi et al. / Fire Safety Journal 71 (2015) 257–267

5. Conclusions The present study introduces a methodology for the analysis of evacuation travel paths during tunnel fire emergencies. The methodology is based on the study of vector operators inspired by functional analysis theory and the newly developed concept of Interaction Areas. The proposed method is a useful tool to study human behaviour during tunnel evacuations and the subsequent development and validation of the assumptions adopted by computational modelling tools (i.e. evacuation models). The development of robust models permits an increased accuracy in risk assessment in the case of tunnel fire evacuations and the definition of optimal design solutions for tunnel safety.

Acknowledgements This study was partially supported by the German Federal Ministry of Education and Research within the SKRIBTPlus project (13N9645A); the authors would like to thank the SKRIBTPlus Consortium.

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