STATISTICAL AND ACOUSTICAL ROOM ANALYSIS
THROUGH FACTOR MIXTURE MODELS
A Degree Thesis Submitted to the Faculty of the
Escola Tècnica d'Enginyeria de Telecomunicació de Barcelona
Universitat Politècnica de Catalunya by
Joan Pallarès Sadó
In partial fulfilment of the requirements for the degree in Audiovisual Systems ENGINEERING
Advisor: Stefan Weinzierl
Berlin, February 2016
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Abstract
The acoustics of music rooms, theatre halls and venues in general depend greatly on the materials with which they have been built and designed. Yet the size, dimensions and morphology are factors that play a fundamental roll in the acoustical properties of the rooms. 300 concert venues from all over the world have been analysed and documented in dependence on these factors in order to respond the following aspects. On the one hand, the aim of the research is to prove if there is a correspondence between the morphological shape of rooms and their acoustical properties. In other words it is intended, by statistical analysis, to carry out a classification in classes in order to verify if such assumption is true or not. Moreover, a set of new acoustical variables is exposed as a result of the combination of original (primitive) acoustical parameters. For such achievements a statistical mixture model combining Latent Profile Analysis (LPA), Exploratory Factor Analysis (EFA) and Common Factor Analysis (CFA) has been developed.
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Resum
L’acústica de les sales de concerts, teatres i espais escènics en general depèn en gran mesura dels materials a partir dels quals s’han dissenyat i construït. No obstant això, les dimensions i la morfologia són factors que condicionen també les propietats acústiques de les sales. En aquest sentit, un total de 300 sales d’arreu del món han estat documentades i analitzades, d’acord amb els seus paràmetres acústics, per tal de donar resposta als següents aspectes. En primer terme, l’objectiu de la recerca és estudiar si existeix una correspondència entre la morfologia de les diferents sales i l’acústica d’aquestes. En altres paraules es pretén, mitjançant l’anàlisi estadística, dur a terme una classificació per classes amb la finalitat de comprovar si, efectivament, es verifica o no el plantejament. En segon lloc, es presenta un seguit de noves variables acústiques les quals són fruit de la combinació de paràmetres acústics originals (primitius). Per tal d’assolir ambdós objectius s'ha emprat un model que combina eines d'anàlisi estadística com són LPA (Latent Profile Analysis), EFA (Exploratory Factor Analysis) i CFA (Common Factor Analysis).
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Resumen
La acústica de las salas de conciertos, teatros y espacios escénicos en general depende en gran medida de los materiales a partir de los cuales se han diseñado y construido. Sin embargo, las dimensiones y la morfología son factores que condicionan también las propiedades acústicas de las salas. En este sentido, un total de 300 salas de todo el mundo han sido documentadas y analizadas, de acuerdo con sus parámetros acústicos, para dar respuesta a los siguientes aspectos. En primer término, el objetivo de la investigación es estudiar si existe una correspondencia entre la morfología de las diferentes salas y la acústica de éstas. En otras palabras se pretende, mediante el análisis estadístico, llevar a cabo una clasificación con el fin de comprobar si, efectivamente, se verifica o no dicho planteamiento. En segundo lugar, se presenta una serie de nuevas variables acústicas las cuales son fruto de la combinación de parámetros acústicos originales (primitivos). Para alcanzar ambos objetivos se ha empleado un modelo que combina herramientas de análisis estadístico como son LPA (Latent Profile Analysis), EFA (Exploratory Factor Analysis) y CFA (Common Factor Analysis).
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To my parents Toni and Mª Antònia,
my sister Anna and my brother Jordi.
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Acknowledgements
I’d like to express my gratitude and gratefulness to Dr.Stefan Weinzierl, head of TU-‐Berlin Audiokommunikation Department and advisor of the project, for giving me the opportunity of carrying out my thesis alongside with the team he leads.
Secondly, to Dr. Steffen Lepa, member of the Audiokommunikation Department and advisor of the research, for providing me with the appropriate statistical background that enabled the mathematical analysis and the consequent achievement of the final results.
Moreover, to the staff members and fellow students of the Audiokommunikation Research Group in the collection of data, hints and endorsement.
Likewise, my thankfulness to Professor Antoni Carrión Isbert for his support as co-‐tutor of the project at home university, Telecom Barcelona-‐UPC.
Finally, to Technische Universität of Berlin for hosting me during this period and for providing me with the necessary tools and infrastructures in order to develop the research.
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Revision history and approval record
Revision Date Purpose
0 02/01/2016 Document creation
1 02/02/2016 Document revision
DOCUMENT DISTRIBUTION LIST
Name e-‐mail
Joan Pallarès Sadó [email protected]
Stefan Weinzierl stefan.weinzierl@tu-‐berlin.de
Steffen Lepa steffen.lepa@tu-‐berlin.de
Written by: Reviewed and approved by:
Date 02/01/2016 Date 02/02/2016
Name Joan Pallarès Sadó Name Stefan Weinzierl
Position Project Author Position Project Supervisor
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Table of contents
Abstract ............................................................................................................................................................................ 1
Resum ............................................................................................................................................................................... 2
Resumen .......................................................................................................................................................................... 3
Acknowledgements ..................................................................................................................................................... 5
Revision history and approval record ................................................................................................................ 6
Table of contents .......................................................................................................................................................... 7
List of Figures ................................................................................................................................................................ 8
List of Tables .................................................................................................................................................................. 9
1. Introduction ...................................................................................................................................................... 10
1.1. Requirements and Specifications ................................................................................................... 10
1.2. Work Plan ................................................................................................................................................. 11
2. State of the art .................................................................................................................................................. 16
3. Methodology / project development: .................................................................................................... 17
4. Results .................................................................................................................................................................. 25
5. Budget .................................................................................................................................................................. 36
6. Conclusions and future development: ................................................................................................... 37
Bibliography: .............................................................................................................................................................. 38
Appendices (optional): ........................................................................................................................................... 40
Glossary ......................................................................................................................................................................... 50
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List of Figures
Figure 1 page 19
Figure 2 page 19
Figure 3 page 20
Figure 4 page 20
Figure 5 page 20
Figure 6 page 20
Figure 7 page 20
Figure 8 page 20
Figure 9 page 21
Figure 10 page 21
Figure 11 page 21
Figure 12 page 21
Figure 13 page 22
Figure 14 page 22
Figure 15 page 22
Figure 16 page 23
Figure 17 page 24
Figure 18 page 24
Figure 19 page 26
Figure 20 page 29
Figure 21 page 30
Figure 22 page 30
Figure 23 page 32
Figure 24 page 32
Figure 25 page 33
Figure 26 page 33
Figure 27 page 34
Figure 28 page 34
Figure 29 page 34
Figure 30 page 35
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List of Tables
Table 1 page 22
Table2 page 26
Table 3 page 27
Table 4 page 27
Table 5 page 27
Table 6 page 29
Table 7 page 31
Table 8 page 31
Table 9 page 31
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1. Introduction
Music rooms, theatre halls, opera houses and venues in general are daily frequented by thousands of people around the world looking forward to a vibrant artistic experience. The acoustics play an essential roll in it and engineers are conscious that an appropriate acoustical design can bring the spectator’s sensorial enjoyment to another level. This project has been developed under the willingness to investigate the acoustics of 300 rooms from an overall perspective. That is, considering all those factors that have a significant impact on the acoustical behaviour of a room. By this, variables such as volume, height, width, surface of the stage or the number of seats-‐ among many others-‐ have been included as part of the acoustical analysis. The research aims to accomplish two principal goals. Yet, a previous work of data acquisition has been carried out in order to satisfy such purposes. The resulting database contains a total number 300 rooms with 34 acoustical variables defining each sample. The first objective is to carry out a classification by classes of the whole database by using a factor mixture statistical model that combines CFA, EFA and LPA (Latent Profile Analysis). With that, the purpose is to test whether such clustering responds to a morphological pattern and, therefore, prove that there exists a correspondence between the shape and the acoustical parameters that describe a certain room. The second part of the research is focused on the existence of a new set of acoustical parameters. The goal is to define new variables as a result of applying Exploratory Factor Analysis to the 34 acoustical original (primitive) parameters that are in the database. To do so, the document consists on a theoretical part divided in two main blocks. On the one hand, the generation of the database and the acoustical parameters taking part in it. As far as the second block is concerned, the subsequent treatment of the data, the methodology and statistical analysis used for the obtaining of the results are detailed. Besides, the study is part of a larger project within the TU-‐Berlin Audiokommunikation research framework. Moreover, the software and methods that are presented in this document have been used in some projects within the Akustik Department as part of other researches. Basically, it is two mathematical tools that take part in the development of the results. On the one hand, Latent Profile Analysis and on the other hand Factor Analysis. Further on the document both concepts are widely detailed. More, a combination of these two, resulting in the so-‐called Factor Mixture Model (FMM) has been used for the obtaining of the results.
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1.1. REQUIREMENTS AND SPECIFICATIONS
In regards with the acquisition of the database the following is required:
-‐ A total number of 300 sample rooms in order to get to perform a solid statistical analysis and so achieve satisfactory results
-‐ The values within it are obtained according to standardized measurements [1] As for the statistical analysis:
-‐ MPLUS software is required for the procedure
-‐ Ill models need to be avoided since they do not provide reliable results
-‐ LPA models are acceptable if they have been replicated a number of times with different random start values, for instance 10 times.
-‐ The optimum LPA/LFA mixture model corresponds to the one with lowest BIC
value
-‐ The number starting values of the model must be a multiple of the number of processors selected for such procedure
1.2. WORK PLAN
1.2.1. WORK BREAKDOWN STRUCTURE
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1.2.2. WORK TASKS & MILESTONES
Project: Room acoustical statistical analysis WP ref: (WP1)
Major constituent: LPA/EFA statistical background Sheet 1 of 4
Short description:
To get familiar with the LPA and EFA statistical models as a tool to interpret, compute the final database.
Planned start date: 09/10/2015
Planned end date: 07/11/2015
Start event: 09/10/2015
End event: 07/11/2015
Project: Room acoustical statistical analysis WP ref: (WP2)
Major constituent: Generation of the database Sheet 2 of 4
Short description: To gather information regarding theatres & opera halls from all over the world based on their acoustic design specifications. About 85-‐90 samples in total so that a first statistical analysis can be performed.
Planned start date: 09/10/2015
Planned end date: 21/12/2015
Start event: 09/10/2015
End event: 22/12/2015
Internal task T1: To get an initial database approach by making use of Leo Beranek Book “Theaters & Opera Halls”, which includes a total number of 87 rooms.
Internal task T2: To expand the database through research within journals, publications from AES Journal, JASA, Acta Acustica & Audio Engineering Society in order to expand the database.
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Project: Room acoustical statistical analysis WP ref: (WP3)
Major constituent: Mplus Software Sheet 3 of 4
Short description:
To get familiar with Mplus software in order to execute and compute all the calculations required with the database obtained.
Planned start date: 09/10/2015
Planned end date: 28/12/2015
Start event: 24/10/2015
End event: 10/01/2016
Internal task T3: Estimate Exploratory Factor analysis (EFA) to determine latent factor structure for the indicators.
Internal task T4: Fit simple “Mixture-‐Factor-‐Models” (LPA with underlying factors) for all numbers of classes with homogenous variances across classes and latent Factors (introducing BY-‐statements to model specification with regards to results from step 5) until reaching a turning point for BIC.
Internal task T5: Fit complex “Mixture-‐Factor-‐Models” (LPA with underlying factors) for all numbers of classes with heterogeneous variances across classes and partly correlated indicators until reaching a turning point for BIC.
Internal task T6: Choose final model from all 7 model-‐variants with the best BIC and produce graphics of standardized class profiles (incl. Ds) and also tables with non-‐standardized class profiles (incl. SDs).
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Project: Room acoustical statistical analysis WP ref: (WP4)
Major constituent: Results Sheet 4 of 4
Short description: To obtain, interpret and expose the final results in relation to the initial proposals. Furthermore, to come up with the corresponding conclusions.
Planned start date: 07/10/2015
Planned end date: 28/12/2015
Start event: 10/01/2016
End event: 19/01/2016
Internal task T7: To obtain the final model classification as a result of the mixture model analysis. For each class, to make a table including the probability of each sample. More, the name of the auditorium/room and their shape.
Internal task T8: To display the standardized class profiles in dependence on the factors that have been achieved. Likewise, the same for the variables included in each of the factors.
Internal task T9: To display the non-‐standardized class profiles in dependence on the factors that have been achieved. Likewise, the same for the variables included in each of the factors.
Internal task T10: To expose the linear dependence across a factor and the variables within it. That is, the achievement of the “new” variables as a result of linear combination of primitive ones.
Internal task T11: To discuss and expose the final conclusions according to the results achieved.
1.2.3. INCIDENCES & WORK PLAN MODIFICATIONS
After performing a first statistical analysis approach with the MPlus software, we realized we had to expand the database dimensions in order to obtain more satisfactory results. That is, not only the number of samples, but also the number of variables per sample. Hence, we increased the database from 90 sample rooms per 10 variables each to 300 rooms with 34 variables per sample. On the other hand, the MPlus software calculations required a great amount of time due to the increase of the data. For that reason, we had to make use of the high performance computer provided by the Akustik TU-‐Berlin Department. However, it was not always
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accessible and some time-‐planning modifications had to be made. Furthermore, for initial analysis we considered all the possible LPA and LFA statistical models (that is, simple and complex models) with the purpose of getting the most suitable classification of the database. However, we realized that for this project in particular, there is significant correlation across some of the parameters. That is e.g the reverberation time (RT) with the early decay (EDT) time and the clarity (C80). In this context, we considered that basic LPA models (from 1 to 8 classes) didn’t contribute to the database classification and for this reason we decided to compute Mixture Models as a priority. Under these assumptions, we realized that better results were achieved. That is, best entropy and BIC values.
1.2.4 GANTT DIAGRAM
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2. State of the art
The development of clustering methodology has been a truly interdisciplinary endeavour. Taxonomists, social scientists, psychologists, biologists, statisticians, mathematicians, engineers, computer scientists, medical researchers, and others who collect and process real data have all contributed to clustering methodology. The appliance of clustering in general to all these sciences have led to relevant investigations [2] [3]. In regards with the motivation of our project, statistical methods such as EFA, LPA and FMM are today being implemented in countless number of studies. Especially FMM, they are considered to be mathematically more powerful as they arise from a combination of EFA and LPA and so offer more reliable results. As it is exposed in the introduction, these models are the tools through which we are developing the research. In this respect, acoustics is the subject that is under investigation. Acoustical engineering is a field that comprehends a wide range of sub disciplines: from urban sound legislation to acoustical conditioning in restaurants… Yet, in our case we are focused on the acoustical parameters that define the behaviour of concert halls and music rooms in general. With this, we can fit this study in the sub discipline of architectural acoustics. Back to the ancient times architectural acoustics were already a matter of important relevance. Greek classical theatres like Epidaurus from IV b. C or Roman Classical Theatres like Aspendos (Turkey) are a good example of that. They have emerged as paradigms of the perfect harmony among acoustics and architecture. Throughout the centuries and to the very present, such conjunction is still a current element of discussion among engineers as the marriage of these two is often difficult to achieve. That is, to provide a room-‐ with its predefined characteristic shape-‐ with the proper acoustical design as so art performances can be correctly developed. In the recent times there has been a great deal of research in this respect, starting from the discovery of the RT formula by Wallace C. Sabine. Over the 20th century many acoustical parameters were discovered with the purpose of expressing the behaviour of rooms and, what’s more, the feeling that such rooms produce to the listener. For instance, Leo Beranek, considered to be the father of concert halls acoustics, is the introducer of parameters such as Initial Time Delay Gap (ITDG) [4] [5], measurement to express the subjective impression of intimacy of the listener. As can be seen, the goal of the investigators is to exhaustively characterize all the phenomena that occur in interior spaces. That is, to numerically express the aspects taking part in it. To do so, the contribution of mathematics to this science is fundamental and this is exactly what this project aims.
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3. Methodology
3.1 DATABASE
The development of the project starts with the obtaining of the database. In this respect, the data was collected from the following sources: Concert Halls & Opera Houses [4] [5], Pop & Rock Venues [6], TU-‐Berlin Akustik Department Simulations [7] and finally a set of paper publications within journals such as Acta Acustica [8], Audio Engineering Society Journal [9-‐20] and Acoustical Society of America [21] [22]. A part from this, measurements from Nagata Acoustics Company involving a large number of rooms are part of it too. In total, 300 venues from all over the world are documented. Moreover, each of them is characterized by 34 acoustical parameters. Though, there are missing values in the great majority of the cases. This is due to the fact that the samples are extracted from different publications in which certain measurements are not considered. Next, the acoustical parameters included in the database are exposed. Reverberation Time (RT in sec): time that it takes for the sound in a hall to decay from 0dB to -‐60dB. Likewise, as the time, multiplied by 2, that it takes for the sound to decay from -‐5dB to -‐35dB. It expresses the degree of liveliness or brightness of the room. An enclosure with a large RT is called “bright” or “alive”, while it is known to be “off” in case RT is small. In the database both RT measured under occupied and unoccupied conditions are considered.
Early Decay Time (EDT in sec): time that it takes for a signal to decay from 0dB to -‐10dB relative to its steady state value. EDT is more related to the subjective impression of liveliness than the RT. In the database both EDT measured under occupied and unoccupied conditions are considered.
Clarity Factor (c80 in dB): ratio of the energy in the first 80msec of an impulse sound arriving at a listener’s position divided by the energy in the sound after 80msec. Such parameter indicates the degree of separation of the individual sounds taking part in a musical performance. Moreover, C80 is highly correlated with RT. Rooms with large RT’s (bright) result in low C80 values. Both C80 occupied and C80 unoccupied included in the database.
Strength Factor (G in dB): ratio of the sound energy at a seat in a hall that comes from a non-‐directional source to the sound energy from the same source when measured in an anechoic room at a distance of 10m. It corresponds to the degree of amplification produced by a room. Both G occupied and G unoccupied included in the database.
GLow(dB): low frequency strength factor. Average of the G’s measured in the 125Hz and 250Hz bands.
Bass Ratio (Bas): ratio of the average of RT’s at 125Hz and 250Hz to the average of RT’s at 500Hz and 1KHz. Bass ratio is related to the warmth of the room. Both BR occupied and BR unoccupied included in the database.
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Brilliance Ratio (Bri): ratio of the average of RT’s at 2KHz and 4KHz to the average of RT’s at 500Hz and 1KHz. It associates with the high frequencies. Br occupied and Br unoccupied included in the database.
Stage Support (ST1 or STEarly in dB): difference between the impulse sound energy from an omnidirectional sound source that arrives at a player’s position at the stage within the first 10msec (measured at a distance of 1 m from the sound source) and that one which arrives in the time interval between 20 and 100msec at the same position. It is meant to express the capacity of the musicians of listening to the orchestra and to themselves when performing.
ST2 or STTotal (dB): parameter used to describe the amount of support of the room. Difference between the impulse sound energy from an omnidirectional sound source that arrives at a player’s position at the stage within the first 10msec (measured at a distance of 1 m from the sound source) and the one that arrives in the time interval between 20 and 1000msec at the same position.
IACCE: interaural cross correlation coefficient determined for a time period of 0 to 80msec where 0msec the time at which the direct impulse sound from the omnidirectional source reaches the tiny microphones. It correlates with the ASW (Apparent Source Width) parameter, that is, with the spaciousness impression that the listener has in relation to the room.
IACCA: interaural cross correlation coefficient. Measurement of the difference in the sounds arriving at the two ears of a listener facing the performing entity in a hall.
IACCL: interaural cross correlation coefficient determined for a time period of 80msec to 750 msec. It is related to the degree of dissemination of the sound.
Definition (D50): ratio of the sound energy in the first 50msec after arrival of the direct sound at a listener’s position to the total sound energy arriving.
Initial-‐Time-‐Delay Gap (ITDG in msec): the time interval between the arrival, at a seat in the hall, of the direct sound from a source on stage to the arrival of the first reflection. It correlates with the subjective impression of “intimacy”. This measurement is defined in Leo Beranek publications [4] [5].
Lateral Fraction (LFEarly): it covers the time period of 0 to 80 msec. It is the ratio of the energy in the sound at a listener’s position that does not come from the direction of the source to that which comes from all the directions including that of the source. It correlates with the ASW (Apparent Source Width) parameter, that is, with the spaciousness impression that the listener has in relation to the room. Like in IACC measurement, it is related to the spaciousness impression of the room.
Volume (V in m3): total volume of the hall.
N: number of seats in the hall.
V/N: ratio between volume and number of seats.
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So (in m2): area of the stage.
H (in m): average room height, measured from main floor to ceiling in that part of the main-‐floor audience area not covered by balconies.
W (in m): average width, measured between sidewalls in the audience area on the main floor, disregarding any balcony overhang.
L (in m): average room length, measured from the stage front to the average of the back wall positions at all levels.
L/W: ratio between length and width.
H/W: ratio between height and width.
D (in m): distance from the front of the stage to the most remote listener.
Sa (in m2): area of the audience occupied by the seats.
SA (in m2): total area of the audience including the area occupied by the seats.
Sa/SA: ratio between Sa and SA.
A part from these, the shape of the rooms is also documented. In relation to this, 7 different typologies have been proposed [23]. Thus, each sample can be classified in one of the following:
1) Shoe Box
Rooms regarding this shape are relatively narrow with narrow balconies. More, parallel sidewalls (figure 1) assure early reflections to the audience on the main floor, essential to the desired acoustical attribute “spaciousness”. Furthermore, high acoustical intimacy and diffusivity associated.
Examples: Boston Symphony Hall (figure 2), Amsterdam Concertgebouw (figure 3), Kyoto Concert Hall (figure 4).
Figure 1. Shoe Box characteristic shape
Figure 2. Boston Symphony Hall
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2) Vineyard
The design is based on the overlap of several audience terraces, offering a large capacity (figure 5). Though it may seem complex in terms of shape it offers remarkable acoustical properties such as good spaciousness impression and acoustical intimacy.
Examples: Philharmonic Berlin (figure 6).
3) Hexagonal
Acoustically, they offer a high spaciousness impression and a large quantity of first sound reflections. In terms of capacity, they can hold a large number of spectators.
Examples: Bunka Kaikan, Tokyo (Figure 7 & 8).
Figure 3. Amsterdam Concertgebouw
Figure 4. Kyoto Concert Hall
Figure 5. Vineyard characteristic shape Figure 6. Philharmonie Berlin
Figure 7. Bunka Kaikan room map Figure 8. Bunka Kaikan, Tokyo
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4) Horse Shoe
Room profile used principally in Opera Houses. As for the acoustics, they offer low energy early reflections. In terms of the capacity, they hold a large audience.
Examples: Gran Teatre del Liceu, Barcelona (figure 9), Carnegie Hall, New York (figure 10).
5) Fan Shaped
No early sound reflections at the central part of these rooms, low spaciousness impression and low acoustical intimacy.
Example: Deutsche Oper, Berlin (figure 11 & 12).
6) Elliptical
Huge pavilions, arenas and large concert stadia are included in this group (up to 21.000 spectators).
Example: Palau Sant Jordi Barcelona (figure 13 & 14).
Figure 9. Gran Teatre del Liceu, Barcelona Figure 10. Carnegie Hall, New York
Figure 11. Deutsche Oper room map Figure 12. Deutsche Oper, Berlin
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7) Church
High ceilings (up to 20 meters), wide surfaces and the lack of absorptive materials result in reverberant and echoic chambers.
Examples: Jesus Christus Kirche Berlin (figure 15).
Figure 15. Jesus Christus Kirche, Berlin
Table 1 shows the final distribution of the database in relation to the number of samples according to their shape. Yet, as can be noticed, 33 rooms could not be appropriately fitted in any group because of lack of information.
Table 1. Database distribution according to the morphological shape
Figure 14. Palau Sant Jordi, Barcelona Figure 13. Palau Sant Jordi room map
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3.2 STATISTIC ANALYSIS: FACTOR MIXTURE MODELS To identify room subtypes we used a FMM, a statistical method that combines principles of Factor Analysis and LPA. While Factor Analysis is a variable centered approach that models the data in a set of latent continuous factors, LPA uses a case-‐centered approach that allows latent categorical factors to be identified and assigns cases to these classes. In other words, Factor Models serve to cluster items and Latent Profile Models, on the other hand, serve to cluster participants. The models obtained through FMM provide a class-‐factor classification of the items/samples that are in a database. Figure 16 shows an overview in regards with the methodology adopted. At this point, once we come up with the proper model, the question that arises is the following. Which is the criterion to determine the number of classes? There are many measures of model fit (the model reproduces the empirical data obtained), the most common ones being the AIC (Akaike Information Criterion) and BIC (Bayesian information Criterion). For this research we took BIC value as the measure for our choice. The stepwise is simple. We apply our model starting from 1 class on (usually up to 7-‐8 classes depending on the number of samples). We stop when we come across a BIC turning point. That will indicate that lowest BIC value has been reached [24]. So to exemplify this, see figure 17. In this case, 3 classes is the optimum choice.
Figure 16. Methodology stepwise scheme
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1930,000
1935,000
1940,000
1945,000
1950,000
1955,000
1 2 3 4
BIC
BIC
In the current study, we adopted the Mplus Editor v.6 software [25]. The samples were submitted to EFA, CFA and LPA using mixture modeling procedure with the robust maximum likelihood (MLR) estimator. Mplus is a statistical modeling program that provides researches with a flexible tool to analyze their data. It offers a wide choice of models, estimators, and algorithms in a program that has an easy-‐to-‐use interface and graphical displays of data and analysis results. 3.2.1 EXPLORATORY FACTOR ANALYSIS For receiving a proper factor model for the FMM, we need to find out about the proper factor structure, referring to number of factors and the grouping of items on them. This is done by EFA. Accordingly, the goal is to investigate common content among the items [26] by seeing if items group together on continuous latent variables called factors. Because the latent variable is continuous, there is no assumption of different subpopulations of individuals. It is assumed that all individuals in the sample are from the same homogenous population and that differences among individuals arise because of differences on the factor. The factor model is appropriate for data from a single homogeneous population. The model is designed to investigate the common content of observed scores such as questionnaire items. Continuous latent variables called factors are used to model the common content of the observed variables. Thus, each factor results in the combination of different variables that are accordingly weighted. See figure 18.
Figure 18. EFA scheme
Figure 17. BIC turning point curve example
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Like in the FMM, the criterion to determine the number of factors depends on the BIC values. That is, the lowest BIC indicates the best choice. However, it is not the only one. In this respect, the Kaiser criterion (eigenvalues <1) or the scree criterion are good indicators as well as. In our case, we have applied the scree criterion, which says to drop all further components after the one starting the elbow (see figure 19). After we have found out about a meaningful factor structure, the next step is to simplify it into a common factor model (CFA). This means converting the model into a so-‐called “simple structure” where cross-‐loadings and minor loadings are eliminated as far as possible. The reason for this is twofold: on the one hand, it eases interpretation of the factors. On the other hand it decreases the computational costs for the later FMM model estimation. 3.2.2 LATENT PROFILE ANALYSIS LPA identifies classes of individuals based on similarities in responses to a set of observable indicators. Unlike cluster analysis (such as K-‐means), which also classifies individuals, LPA is based on a statistical model, provides objective tests to determine model identification and does not require decisions regarding scaling of observed variables. LPA is useful when you want to reduce a large number of continuous variables to a few subgroups. They can also help experimenters in situations where the treatment effect is different for different people, but we do not know which people. It’s a technique whose aim is to recover hidden groups from observed data [27]. Indeed, a statistical analysis by just using LPA models is perfectly plausible. However, FMM are preferred, if applicable, because they normally achieve lower BIC values.
4. RESULTS
4.1 EXPLORATORY FACTOR ANALYSIS
Following the scree criterion a maximum of 3 factors were extracted from the database (see figure 19). In addition, some acoustical parameters were eventually omitted from the analysis due to errors in the modelling. We attribute such errors to two aspects. On the one hand, in reference to St2, LF (lateral fraction), IACCA, IACCL, IACCE and D50 (definition) the errors aroused because of the lack of information. That is, missing values for a significant part of the database. On the other hand, in relation to parameters such as EDT occupied, c80 occupied, G occupied, Volume, V/N, L/W, H/W and Sa/SA errors aroused because they don’t contribute with new information as they are totally correlated with other parameters (L/W for instance with L and W, Volume with height, surface and width…). Hence, the dimension of the database was reduced to 19 variables. Table 2 shows the grouping in factors after EFA (oc terminology stands for occupied). The Mplus code used for this model is specified in the Appendices (pages 40-‐41).
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Figure 19. Screeplot of Eigenvalues from Exploratory Factor Analysis
Table 2. Results from Exploratory Factor Analysis (loading matrix, standardized loadings)
From the eigenvalues scree criterion chart (figure 19) we establish that the number of factors is 3. Next, we look at the factor structure table (table 2) to see the loadings of each variable within each factor. Nevertheless, as we have explained before, EFA gives an orientation of the structure but it needs to be simplified because there might be some ambiguities in variables with similar loadings in more than one factor. For instance, if we
27
FAC2%Amplification00000 Loadings SE Wald p
G "0,861 0.043 19.989 0.000ST1 "0,560 0.145 3.862 0.000
GLOW "0,751 0.074 10.184 0.000ITDG 0.503 0.108 "4.640 0.000N 0.919 0.023 "39.523 0.000H 0.736 0.053 "13.861 0.000W 0.686 0.058 "11.786 0.000L 0.568 0.073 "7.767 0.000D 0.864 0.071 "12.132 0.000
look at Brioc (brilliance occupied) see that the weightings in factor 2 and factor 3 are similar. Another example might be So (surface of stage), which is not clearly fitted in any of the factors as the loadings turn out to be weak. Therefore, we have to simplify the structure by removing cross-‐loadings and minor loadings. This is done through CFA. In table 3, 4 and 5 the simplified Factor Structure is shown. Note that the geometrical parameters were allowed to keep their cross-‐loadings on either factor in order to check for their respective determining influence. Further, note that Brioc was left in the 3rd factor in spite a non-‐significant loading. This was necessary to prevent a breakdown of the Factor3 measurement model due to a plethora of missing data in the room database concerning the variables forming factor 3.Parameter So has been removed from the final structure due to errors resulting from its low loading on either factor (see table 2).
Table 4. Variable loadings after CFA for Factor2-Amplification
Table 3. Variable loadings after CFA for Factor1-Reverberation
28
FAC3%&Warmth/Color&& Loadings SE Wald p
BAS 0.859 0.058 14.760 0.000BASOC 0.913 0.046 19.918 0.000BRI 20.472 0.120 23.933 0.000
BRIOC 20.389 0.257 21.515 0.130SA 0.505 0.106 4.773 0.000H 0.127 0.043 2.916 0.004L 0.380 0.064 5.954 0.000W 0.101 0.049 2.063 0.039
The factor classification seems reasonable after a first interpretation as three acoustical aspects are well differentiated: Reverberation (Factor 1), Amplification (Factor 2), and Warmth/Color (Factor 3). One first thing to outline is that geometrical variables appear to play a determining role for all three factors, indicating their influence on the acoustical parameters. In other words, it denotes that geometrical measures have their importance in each factor.
On the other hand, by just looking at the acoustical variables within the factors we see that each of them stands for a particular acoustical aspect. To start with, factor 1 comprises either objective or subjective properties that have to do with the reverberation/echoing of the room. Therefore, we categorize Factor 1 as representative for “Reverberation”.
As far as Factor 2 is concerned, we see it contains acoustical parameters referred to the strength or amplification of the rooms such as G and Glow. Moreover, Stage Support (ST1) and ITDG (Initial Time Delay Gap) take part in it with remarkable significance. ITDG for instance is to express the intimacy of the room perceived by the listener and has to do with the distance between source and listener. Long ITDG values in large concert halls result in the impression of a close acoustic source whereas short ITDG will result in the impression of a distant acoustic source in a large hall. For small halls the influence of ITDG is pretty much the same: short or null ITDG values result in the impression of a distant source whereas longer values result in the impression of nearby sources. Therefore, in a way ITDG is reasonable to be part of this factor as a distance-‐amplification component is associated.
Another aspect to remark is that the weightings of the geometrical properties in Factor 2 are much higher than in the other two factors. So to compare, if we look at parameters L, H and W we see how large their respective influence is. For factor 1, for instance, width appears to be rather unimportant. With that we can state that “Amplification” as a general concept is strongly associated with the volume. Yet again, this hints to the factor solution and its interpretation being reasonable.
Factor 3 is formed by bass ratio and brilliance (occupied and unoccupied measures). It is representative for both warmth and brilliance. Warmth is a term used to describe a cozy
Table 5. Variable loadings after CFA for Factor3-Warmth/Color
29
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!SIMPLE!FACTOR!MIXTURE!MODEL
Variables)used:)rtoc%rt%c80%edt%g%st1%basoc%bas%bri%brioc%glow%itdg%n%h%w%l%d%sa%sa2Factor)1%by%rtoc%rt%edt%c80%Sa%l%h%w.%Factor)2)by%g%st1%glow%itdg%n%h%w%l%d.%Factor)3 %by%bas%basoc%bri%brioc%SA.Correlation)assumed:%rt%with%rtoc,%c80%with%rtoc,%glow%with%g,%c80%with%edt,%l%with%w,%l%with%h,%w%with%h.Number!of!classes Nº!Successful!Replications! LIKELIHOOD BIC AIC ADJUSTED!BIC ENTROPY Sc.!FACTOR Nº!FREE!PARAMETERS
1 10 #8534,000 17485,000 17215,000 17254,000 # 2,422 732 10 #8328,000 17097,000 16811,000 16852,000 1,000 2,346 773 10 #8276,000 17014,000 16714,000 16757,000 0,953 2,358 814 10 #8245,000 16975,000 16660,000 16705,000 0,961 2,317 855 # # # # # # # #
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Factor%correlation%from%EFA%analysis%&%same%indicator%variances%across%classes%
16700,000&
16800,000&
16900,000&
17000,000&
17100,000&
17200,000&
17300,000&
17400,000&
17500,000&
17600,000&
1& 2& 3& 4&
BIC$
BIC&
smoothness to the music. Its counterpart may be considered to be brilliance, which refers to a bright, clear, ringing sound. If a sound field is too warm, the hall can be undesirably “dark.” With too much brilliance, the sound is harsh, brittle, and metallic sounding. Hence, Factor 3 is representative for “Color/Warmth”.
4.2 FACTOR MIXTURE MODELS
The CFA model obtained in step 4.1 was now extended to a factor mixture model and class solutions from 1-‐5 were estimated. Following the BIC criterion a maximum number of 4 classes was extracted from the database, since 5 or more classes produced inadmissible results (so called "ill models"). Table 6 shows the FMM used in this case and figure 20 the BIC value curve. Notice that it decreases until 4 classes, when the value is the lowest. Certainly, there’s no turning point after that since an “ill” model occurs when it comes to 5 classes. For that, the assumption is that the lowest BIC is achieved when dealing with 4 classes. The Mplus code used for this model is specified in the Appendices (pages 40-‐41).
Table 6. Factor Mixture Model analysis scheme
Figure 20. BIC curve obtained through the FMM from 1-4 classes
30
!10$
!5$
0$
5$
10$
15$
20$
FAC1%&Reverbera-on&
FAC2%Amplifica-on&
FAC3%Warm
th/Color&
STANDARDIZED&MEAN&FACTOR&VALUES&
C1$
C2$
C3$
C4$
!10$
0$
10$
20$
30$
40$
50$
60$
70$
80$
FAC1%Reverbera,on/ FAC2%Amplifica,on/ FAC3%Warmth/Color/
Non%standardized/mean/factor/values/
C1$
C2$
C3$
C4$
The classification of the samples in the corresponding classes is included in the appendices in a table format (pages 42-‐49), where we specify the name of the halls, their shape and the probability in the class (the id column is not relevant). Besides, the results obtained through this model are detailed next. To start with, in figure 21 and figure 22 the standardized and non-‐standardized mean factor values of the classes are presented. From this we can get a first general idea about the content of the class typologies. On the one hand, C1 rooms turn out to be non-‐reverberant, soft in terms of amplification and cold/dark in terms of the FAC3. A similar trace is described by the C4 group. In this respect they turn out to be non-‐reverberant rooms with little amplification but warm instead. On the other hand, C2 and C3 rooms seem to be totally the opposite. C2 for instance corresponds to reverberant, midi-‐loud cold venues whereas C3 comprehends a group of reverberant, loud and midi-‐warm halls.
Figure 21. Standardized mean factor values
Figure 32. Non-standardized mean factor values
31
FAC1%Reverberation
Rtoc RT EDT c80 Sa H W L
C1%Studio<Halls !0,070 !0,110 !0,120 0,080 0,030 !0,060 !0,030 !0,190C2%Small<Arenas 0,850 1,050 1,240 !1,060 ! ! ! !C3%Big<Arenas 1,010 1,380 1,350 !0,710 ! ! ! !C4%Church<Halls 0,100 0,390 0,410 !0,140 !1,090 0,540 0,320 1,970
FAC2%Amplification
G ST1 N Glow ITDG D H W L
C1%Studio=Halls 0,003 0,000 $0,210 0,001 0,000 0,000 $0,060 $0,030 $0,190C2%Small=Arenas $ $ 2,960 $ $ $ $ $ $C3%Big=Arenas $ $ 4,740 $ $ $ $ $ $C4%Church=Halls $0,040 $ $0,250 $ $ 0,540 0,320 1,970
FAC3%Warmth/Colour
Basoc Bri Brioc Bas SA H W L
C1%Studio9Halls !0,010 !0,017 0,030 !0,220 !0,250 !0,060 !0,030 !0,190C2%Small9Arenas ! ! ! 0,400 ! ! ! !C3%Big9Arenas ! ! ! 0,910 ! ! ! !C4%Church9Halls ! ! ! 3,270 1,980 0,540 0,320 1,970
Having exposed this, we need to get into detail in some more aspects. For instance, it is important to know which variables stand behind these factors in reference to the classes. For this, we have to look at the standardized mean values of the variables that are in each factor in relation to every class. This is shown in tables 7, 8 and 9.
Table 9. Profile of obtained classes in terms of acoustical variables of Factor3-Warmth/Color
Grey marked columns correspond to those variables with non-‐missing values for any of the classes. When it comes to comparing them all we have to make use of these 6 acoustical properties (RToc, RT, EDT, c80, N, Bas). An important issue to outline is that for classes C2 and C3 the factors F2-‐Amplification and F3-‐Warmth/Colour are characterized just by parameters N and Bas (number of seats and bass ratio) accordingly. In this respect, the best-‐represented factor is FAC1-‐Reverberation since less missing variables conform it. Yet again, such downsides issue from the existence of a large number of missing values in the database. The FMM-‐approach helps to deal with this, since it imputes missing values by drawing on the factor measurement model. In figure 23 we can see the standardized mean values of the 6 variables depending on each class.
Table 7. Profile of obtained classes in terms of acoustical variables of Factor1-Reverberation
Table 8. Profile of obtained classes in terms of acoustical variables of Factor2-Amplification
32
!4# !3# !2# !1# 0# 1# 2# 3# 4# 5# 6#
Non$standarized-mean-c80$unoccupied$values-C4# C3# C2# C1#
!2,00%
!1,00%
0,00%
1,00%
2,00%
3,00%
4,00%
5,00%
6,00%
Rtoc% RT% EDT% c80% N% Bas%
Standardized*Class*Profiles***
C1%
C2%
C3%
C4%
FAC1!Reverbera>on% FAC2!Amplifica>on%FAC3!Warmth/Color%
Notice that the curve described by each class is practically the same as the one shown in figures 21 and 22. As said before, one could argue that C1 and C4 have aspects in common and on the other hand C2 with C3 too. From a general perspective, we stated that C2 and C3 were to be the most reverberant. However, we might still see some differences regarding FAC1-‐Reverberation. For instance, if we take a look at parameter c80, standing for clarity, in figure 24 (error bars correspond to the standard deviations), we can see that, while C4 and C1 are practically alike, C3 turns out to represent clear rooms in comparison with C2, which we could regard as the fuzziest.
Figure 23. Standardized mean variables for each class
Figure 24. Non-standardized mean C80 unoccupied values
33
0" 5000" 10000" 15000" 20000" 25000"
Non$standardized$mean$Number'of'seats'values'C4" C3" C2" C1"
0" 2" 4" 6" 8" 10" 12"
Non$standardized$mean$G"unoccupied"values$C4" C1"
As far as FAC2-‐Amplification is concerned, we can still outline some differences between the classes. Back to variable N, in figure 25 we perceive another remarkable difference between C2 and C3. While C4 and C1 hold similar audiences (relatively small), C3 rooms are the biggest in terms of the number of spectators, with significant difference in comparison with C2 halls.
Although N is the only representative variable in the 4 classes, there are some parameters within this factor that can be still analysed. So far, we have encountered differences among C2 and C3 but hardly any between C1 and C4. For this, we take a look at the rest of variables that are in this factor (recall that C2 and C3 are represented in FAC2 just by N). See figures 25-‐28.
Figure 25. Non-standardized mean Number of seats values
Figure 26. Non-standardized mean G unoccupied values
34
0" 5" 10" 15" 20" 25" 30"
Non$standardized$mean$Height'values$$
C4" C1"
0" 5" 10" 15" 20" 25" 30" 35" 40" 45"
Non$standardized$mean$Width&values&C4" C1"
0" 10" 20" 30" 40" 50" 60" 70"
Non$standardized$mean$Length'values$C4" C1"
From these we cannot really perceive notorious differences in the variables but in length and height. Notice for example that C4 and C1 are practically identical in terms of G (room strength) or, on the other hand, in regards with the W (width). From looking at figure 25 we determine that, while C1 and C4 are similar in terms of audience capacity (N), C4 venues are remarkably bigger in terms of length and height. Not only this, they turn out to be larger in terms of total surface, parameter SA (see figure 30). In other words, it results in that the V/N ratio in C4 is higher than in C1. A good example of it is the Basilica of Eberbach Monastery, classified in C4, which is about 16 m high, 75 meters long and the V/N ratio being 34. On the
Figure 27. Non-standardized mean Height values
Figure 28. Non-standardized mean Width values
Figure 29. Non-standardized mean Length values
35
0" 1000" 2000" 3000" 4000" 5000" 6000" 7000" 8000" 9000"
Non$standardized$mean$SA#values$C4" C1"
contrary, the Concertgebouw, gr. Saal, Amsterdam, classified in C1, is 17 m high, 26 meters long and the V/N ratio is 9.
Figure 30. Non-standardized mean SA values
With all that, we can finally compact these results into the following.
C1: Little-‐reverberant, soft, cold rooms with small capacity either in terms of seats and dimensions. This class represents the big majority of the database samples and includes shapes of all kind. However, by taking into account the previous properties, we can say it is representative for a category that we name “Studio Halls”.
C2: Reverberant, fuzzy, midi-‐loud cold rooms with capacity of holding a considerable audience. By looking at the properties we fit them in a category that we name “Small Arenas”.
C3: Reverberant, loud and strong rooms, midi-‐warm with large number of spectators. This category is representative for big arenas, pavilions and stadia. The shape in this case turns out to be mainly elliptical (see page 48 in the appendices). We name this class as “Big Arenas”.
C4: Midi-‐reverberant, soft/weak, warm rooms with small capacity in terms of seats though remarkably bigger in terms of dimensions. This class could be categorized as “Church Halls”.
In result, we obtained 4 well interpretable classes, with the overall classification solution obtaining very good relative entropy of 0,960. This means, that it would be easy to classify new rooms very unambiguously into a belonging class if the database is extended in the future.
36
5. Budget
PROJECT STAFF
Worker 1 Name: Stefan Weinzierl Position: Project leader and supervisor *Euros/hour (gross salary): 14 €
Worker 2 Name: Steffen Lepa Position: Project contributor and supervisor *Euros/hour (gross salary): 12 €
Worker 3 Name: Joan Pallarès Position: Student Average hours/day: 8 h *Euros/hour (gross salary): 8 €
Start date: 09/10/2015 End date: 28/01/2016
SOFTWARE Mplus Editor v.6 software license: 595 €
TOTAL COST (approximate) => 280 € + 480 € + 5120 € + 595 € = 6.475 €
Staff Working days Total hours Total Salary*
Worker 1 80 20 280 €
Worker 2 80 40 480 €
Worker 3 80 640 5120 €
37
6. Conclusions and future development
From the results we come to several conclusions. On the one hand, that it has been possible to group variables in the so-‐called factors in a coherent way. With this, we have introduced “Reverberation”, “Amplification” and “Colour/Warmth” as global properties to characterize rooms. Moreover, we have seen that these factors are certainly tied to geometrical measures such as height, length or width. Especially in the case of “FAC2-‐Amplification”, where their influence is strongest.
Second, we have conducted a classification of the rooms in 4 groups. It does not certainly respond to a clear separate shape clustering but in a way it helps to get an approach of the general properties of the classes. Yet again, we have seen that except from “C3-‐Big Arenas”, where the majority of the rooms are elliptical, the other classes are heterogeneous in terms of the shape. In this respect, we are conscious about the limitation that we have faced from the very beginning of the project and that is the number of missing values on a lot of the acoustical measures, which lead to the necessity to perform a lot of model based data imputation. As it is mentioned at some part of the theoretical exposure, the information was retrieved from different sources and so, depending on the author, certain variables are not considered. That is the case of variables like IACC (among others), which are documented only in Leo Beranek publications.
Is the database representative for all the existing rooms of this type in the world (concert venues and music halls)? In regards with this we have to say that the rooms documented in the sources respond generally to a criterion of popularity (emblematic worldwide rooms) as they turn out to be more deeply analysed and investigated by researchers. Another issue to discuss is the size of the database. That is, how many samples would be enough as so to have a better classification. It seems logical to think that the bigger the better, but more samples with still missing values would not certainly help. Instead, we could state that the more parameters the better. That would not only reinforce the thickness of the database but would also be important for the Factor Analysis. That is, it would empower the loadings of the variables within the factors resulting in more robust structures. Consequently, the FMM would be more robust too.
In order to test the validity of the classification, we propose an empirical test in which a group of listeners would experience different concerts in some of the rooms that have been documented in this project. Then, we would ask them to write down their acoustical sensation in each room and we would see if their assessments match our FMM classification or not.
Moreover, for further similar investigations, we encourage acousticians and engineers in general to contribute to the research and discovery of the acoustical properties that have been missing for this project. With it, a more suitable performance could be done and therefore, things could adjust even more to the reality.
38
Bibliography
[1] ISO 3382-‐1:2009: Acoustics -‐ Measurement of room acoustic parameters -‐ Part 1: Performance spaces. International Organization for Standardization, Geneva, 2009. [2] Blevins, C.A., Weathers, F.W., Witte, T.K. Dissociation and posttraumatic stress disorder: a latent profile analysis. J Trauma Stress. 2014. [3] Pattyn T, Van Den Eede F, Lamers F, Veltman D, Sabbe BG, Penninx BW. Identifying panic disorder subtypes using factor mixture modeling. Depress Anxiety, 2015. [4] L. L. Beranek: Concert and opera halls. How they sound. American Institute of Physics, Woodbury, 2004. [5] L. L. Beranek: Concert and opera halls. How they sound. American Institute of Physics, Woodbury, 1996. [6] Adelman-‐Larsen, Niels Werner: Pop & Rock Venues. Acoustical and Architectural Design. Springer-‐Verlag. Berlin Heidelberg, 2014. [7] David Ackermann, Maximilian Ilse: The simulation of Monaural and binaural Transfer Function for a Ground Truth for Room Acoustical Analysis and Perception. TU-‐Berlin, 2015. [8] Stefan Weinzierl, Paolo Sanvito, Frank Schultz, Clemens Büttner: The Acoustics of Renaissance Theatres in Italy. Acta Acustica United with Acustica, 2015. [9] Weihwa Chiang, Chingtsung Hwang, Yenkun Hsu: Acoustical Renovation of Tainan Municipal Cultural Center Auditorium. Audio Engineering Society, 2003. [10] Yenkun Hsu, Weihwa Chiang, JinJaw Tsai, Jiqing Wang: Acoustical Measurements of Courtyard-‐Type Traditional Chinese Theatre in East China. Audio Engineering Society, 2002. [11] Yenkun Hsu, Weihwa Chiang, JinJaw Tsai, Linping Xue: Acoustical Measurements of Traditional Theatres Integrated with Chinese Gardens. Audio Engineering Society, 2003. [12] Marco Facondini, Daniele Ponteggia: Acoustics of the Restored Petruzzelli Theater. Audio Engineering Society, 2010. [13] Michael Lannie, Leonid Makrinenko: Acoustics of Russian Classical Opera Houses. Audio Engineering Society, 1997. [14] Ernst-‐Joachim Voelker, Wolfgang Teuber: Room acoustics for rehearsals and concerts-‐The new Festhalle in Landau, Germany. Audio Engineering Society, 2003. [15] Ernst-‐Joachim Völker, Wolfgang Teuber: Quality criteria for the sound system in the new Weimar Hall in Weimar, Germany. Audio Engineering Society, 2002. [16] Henrik Möller, Tapio Lahti, Anssi Ruusuvuori: The acoustic conditions in Finnish concert spaces. Audio Engineering Society, 2001. [17] Weihwa Chiang, Liangkuang Yang, Wenling Jih: Acoustics Design of the Music Suite in Taipei National University of Arts. Audio Engineering Society, 2002. [18] Jan Voetmann, Lise-‐Lotte W. Tjellesen, José Mª Pérez Lacorzana: The New Symphony Hall in Las Palmas, Gran Canaria. Audio Engineering Society, 2002 [19] Weihwa Chiang, Chingtsung Hwang, Yenkun Hsu: Acoustical Renovation of Tainan Municipal Cultural Center Auditorium. Audio Engineering Society, 2003. [21] Takayuki Hidaka, Noriko Nishihara: Relation of acoustical parameters with and without audiences in concert halls and a simple method for simulating the occupied state. Acoustical Society of America, 2001. [22] Cyril M. Harris: Acoustical design of Benaroya Hall, Seattle. Columbia University, New York. Acoustical Society of America, 2001.
39
[23] Antoni Carrión Isbert: Diseño Acústico de Espacios Arquitectónicos. Edicions UPC, 1998. [24] Lubke, G., & Muthén, B: Investigating Population Heterogeneity With Factor Mixture Models, 2005. [25] Muthén, L.K. and Muthén, B.O. (1998-‐2012). Mplus User’s Guide. Seventh Edition. Los Angeles, CA: Muthén & Muthén. [26] Oberski, D. L: Mixture models: latent profile and latent class analysis. In Robertson, J., & Kaptein, M. (Eds.), In Modern statistical methods for HCI: a modern look at data analysis for HCI research. [27] Collins LM, Lanza ST. Latent class and latent transition analysis: With applications in the social, behavioral, and health sciences. New York: Wiley; 2010.
40
Appendices
EFA MODEL MPLUS CODE:
FACTOR MIXTURE MODEL MPLUS CODE:
41
42
!!!!!!!!!!!!!!!!!!!!!!!CLASS!1ID ROOM!NAME PROBABILITY! SHAPE
1 "Concertgebouw, kl. Saal, Amsterdam" 100,00% Fan Shaped2 "Concertgebouw, gr. Saal, Amsterdam" 100,00% Shoe Box3 "Music Theater, Amsterdam" 100,00% NaN4 "Megaron, Athen" 100,00% Fan Shaped5 "Festspielhaus, Baden-Baden" 100,00% Shoe Box6 "Joseph Meyerhoff Symphony Hall, Baltimore" 100,00% vineyard7 "Stadtcasiono, Basel" 100,00% Shoe Box8 "Deutsche Oper, Berlin" 100,00% Fan Shaped10 "Komische Oper, Berlin" 100,00% Horse Shoe11 "Konzerthaus kl. Saal, Berlin" 100,00% Shoe Box12 "Konzerthaus Schauspielha, gr. Saal, Berlin" 100,00% Shoe Box13 "Philharmonie, Kammermusiksaal, Berlin" 100,00% vineyard14 "Philharmonie, gr. Saal, Berlin" 100,00% vineyard15 "Symphony Hall, Boston" 100,00% Shoe Box16 "Palais des Beaux Arts, Bruessel" 100,00% Horse Shoe18 "Staatsoper, Budapest" 100,00% Horse Shoe19 "Teatro Colón, Buenos Aires" 100,00% Horse Shoe20 "Kleinhans Music Hall, Buffalo" 100,00% Fan Shaped21 "St. David's Hall, Cardiff" 100,00% vineyard22 "Civic Opera, Chicago" 100,00% Fan Shaped23 "Town Hall, Christchurch" 100,00% vineyard24 "Radiohuset Studio 1, Copenhagen" 100,00% Fan Shaped25 "Segerstrom Hall, Costa Mesa" 100,00% vineyard26 "Boettcher Hall, Denver" 100,00% vineyard27 "Semper Oper, Dresden" 100,00% Horse Shoe28 "Bass Performance Hall, Fort Worth" 100,00% Horse Shoe29 "Royal Concert Hall, Glasgow" 100,00% vineyard30 "Konserthus, Goeteborg" 100,00% Fan Shaped31 "Staatsoper, Hamburg" 100,00% Fan Shaped32 "Binyanei Ha'oomah, Jerusalem" 100,00% Fan Shaped33 "Fredric R. Mann Auditorium, Jerusalem" 100,00% Fan Shaped34 "Higashitotsuka Hall, Kanagawa" 100,00% NaN35 "Miyama Conceru, Kirishima" 100,00% Fan Shaped36 "Concert Hall, Kyoto" 100,00% Shoe Box37 "Koussevitzky Music Shed, Lenox" 100,00% Fan Shaped38 "Barbican, gr. Saal, London" 100,00% vineyard39 "Royal Albert Hall, London" 100,00% vineyard40 "Royal Opera House, London" 100,00% Horse Shoe41 "Auditorio Nacional de Musica, Madrid" 100,00% vineyard42 "Salle Wilfrid-Pelletier, Montreal" 100,00% Fan Shaped43 "Philharmonie am Gasteig, Muenchen" 100,00% vineyard44 "Teatro alla Scala, Mailand" 100,00% Horse Shoe45 "Avery Fisher Hall, New York" 100,00% Shoe Box
43
46 "Metropolitan Opera, New York" 100,00% Fan Shaped47 "Koncerthus Nielsen Hall, Odense" 100,00% Shoe Box48 "Symphony Hall, Osaka" 100,00% Shoe Box49 "Opera Bastille Paris" 100,00% Fan Shaped51 "Salle Pleyel, Paris" 100,00% Fan Shaped52 "Academy of Music, Philadelphia" 100,00% Horse Shoe53 "Martine Hall, Prag" 100,00% Fan Shaped54 "Staatsoper, Prag" 100,00% Horse Shoe55 "Eastman Theater, Rochester" 100,00% Fan Shaped56 "De Doelen Concertgewouw, Rotterdam" 100,00% vineyard57 "Abravanel Symphony Hall, Salt Lake City" 100,00% Shoe Box58 "Festspielhaus, Salzburg" 100,00% Fan Shaped59 "Mozarteum, Wiener Saal, Salzburg" 100,00% Shoe Box61 "Davies Symphony Hall, San Francisco" 100,00% vineyard62 "Concert Hall, Sapporo" 100,00% vineyard63 "Liederhalle, gr. Saal, Stuttgart" 100,00% vineyard64 "Cultural Centre, Taipei" 100,00% Fan Shaped65 "Bunka Kaikan, Tokyo" 100,00% Hexagonal66 "Casals Hall, Tokyo" 100,00% Shoe Box67 "Dai-Ichi Seimei Hall, Tokyo" 100,00% Fan Shaped68 "Hamarikyu Asahi Hall, Tokyo" 100,00% Shoe Box69 "Ishibashi Memorial Hall, Tokyo" 100,00% Hexagonal70 "Metropolitan Art Space, Tokyo" 100,00% Fan Shaped71 "Mitaka Arts Center, Tokyo" 100,00% NaN72 "New National Theatre, Tokyo" 100,00% Fan Shaped73 "Nissei Theater, Tokyo" 100,00% Fan Shaped74 "Opera City Concert Hall, Tokyo" 100,00% Shoe Box75 "Sumida Small Hall, Tokyo" 100,00% NaN76 "Suntory Hall, Tokyo" 100,00% vineyard77 "Tsuda Hall, Tokyo" 100,00% Shoe Box78 "Roy Thompson Hall, Toronto" 100,00% vineyard79 "Palau de la Musica, Valencia" 100,00% vineyard80 "JFK Center Opera House, Washington" 100,00% Fan Shaped81 "Brahmssaal, Wien" 100,00% Shoe Box82 "Konzerthaus, gr. Saal, Wien" 100,00% Shoe Box83 "Musikvereinssaal, gr. Saal, Wien" 100,00% Shoe Box84 "Staatsoper, Wien" 100,00% Horse Shoe85 "Mechanics Hall, Worcester" 100,00% Shoe Box86 "Tonhalle, kl. Saal, Zuerich" 100,00% Shoe Box87 "Tonhalle, gr. Saal, Zuerich" 100,00% Shoe Box88 Seminar Room HFT616 TU Berlin 100,00% Shoe Box89 Chamber Music Hall of Konzerthaus Berlin 100,00% Shoe Box90 Renaissance Theatre 100,00% Horse Shoe
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100 Wigmore Hall 100,00% Shoe Box101 Eroicasaal Palais Lobkowitz 100,00% Shoe Box102 Kammersaal_1 100,00% Shoe Box103 Kammersaal 2 100,00% Shoe Box111 Yachiyo-Za 100,00% Shoe Box112 Marakuni-Za 100,00% Shoe Box113 Hako-Gekijo 100,00% Shoe Box114 Kanamaru-Za 100,00% Shoe Box115 Houou-Za 100,00% Shoe Box120 Frosinone 100,00% Shoe Box121 G4 100,00% Shoe Box123 Kammermusiksaal 100,00% Shoe Box125 San Cebrian de Mazote 100,00% Church126 San Juan de Banos 100,00% Shoe Box129 Seminarraum_ITA 100,00% Shoe Box130 Studio 100,00% NaN131 "Bloomington, Indiana" 100,00% Fan shaped132 "Severance Hall, Cleveland" 100,00% Horse Shoe133 "Mc Dermott Concert Hall, Dallas" 100,00% Horse Shoe134 "Minneapolis Orchestra Hall" 100,00% Shoe Box135 "Carnegie Hall, New York" 100,00% Horse Shoe136 "Rochester Eastman Theatre, New York" 100,00% Fan Shaped137 "Concert Hall,Sydney" 100,00% vineyard138 "Northern Alberta Jubilee, Edmonton" 100,00% Fan Shaped139 "Tivoli Koncertsal, Copenhagen" 100,00% Fan Shaped140 "Colston Hall, Bristol" 100,00% Shoe Box141 "Usher Hall, Edinburgh" 100,00% Horse Shoe142 "Philharmonic Hall, Liverpool" 100,00% Fan Shaped143 "Free Trade Hall, Manchester" 100,00% Shoe Box144 Gewandhaus Leipzig 100,00% Vineyard145 "Amager Bio, Copenhaguen" 100,00% Fan Shaped146 "Forbraendingen, Copenhaguen" 100,00% Shoe Box147 "Godset, Kolding" 100,00% Shoe Box148 "Vega, Lille" 100,00% Shoe Box149 "Loppen, Copenhaguen" 100,00% Shoe Box151 "Palletten, Viborg" 100,00% Hexagonal152 "Pumpehuset, Copenhaguen" 100,00% Shoe Box153 "Rytmeposten, Odense" 100,00% Shoe Box154 "Musikhozet, Ronne" 100,00% Fan Shaped156 "Slagelse Musikhus, Älborg" 100,00% Shoe Box
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157 "Stars, Vordinborg" 100,00% Shoe Box158 "Store Vega, Copenhaguen" 100,00% Shoe Box159 "Sonderborghus, Sonderborg" 100,00% Shoe Box161 "Torvehallerne, Vejle" 100,00% Shoe Box162 "Train, Aarhus" 100,00% Fan Shaped164 "Voxhall, Aahrus" 100,00% Shoe Box165 "AB, Brussels" 100,00% Shoe Box167 "L'Alcatraz" 100,00% Shoe Box168 "Apolo, Barcelona" 100,00% Shoe Box169 "Apolo la [2] 100,00% Shoe Box170 "Astra, Berlin" 100,00% Shoe Box171 "Bikini" 100,00% NaN172 "The Cavern Club, Liverpool" 100,00% Shoe Box173 "Le Chabada, Angers" 100,00% NaN174 "Cirkus" 100,00% NaN176 "La Coopeérative de Mai" 100,00% Shoe Box182 "Grosse Freiheit" 100,00% Fan Shaped183 "Kaiser Keller" 100,00% NaN185 "Heineken Hall" 100,00% Shoe Box186 "HMV Hammersmith Apollo" 100,00% Horse Shoe188 "Live Music Club" 100,00% NaN190 "Melkweg" 100,00% Shoe Box195 "O13 Tilburg" 100,00% Shoe Box198 "Sala Barcelona" 100,00% NaN199 "Paradiso" 100,00% NaN200 "Porsche Arena" 100,00% Eliptical201 "Razzmatazz" 100,00% Shoe Box202 "Razzmatazz 2" 100,00% Shoe Box206 "Rote Fabrik 2" 100,00% NaN207 "Hans Martin Schleyer Halle" 100,00% Eliptical210 "Tunnel" 100,00% NaN213 "Werk Backstage" 100,00% Horse Shoe214 "Zeche" 100,00% NaN218 "Vega" 100,00% NaN252 "Benaroya Hall, Seattle" 100,00% Shoe Box253 "Euskalduna Jauregia Main Saal, Bilbao" 100,00% Vineyard254 "Sala A1 600, Bilbao" 100,00% Shoe Box257 "Benedict Music Tent, Aspen" 100,00% Vineyard258 "SanShan Clubhouse in Shanghai" 100,00% Shoe Box259 "KunShan" 100,00% Shoe Box260 "Quanjin Clubhouse, SuZhou" 100,00% Shoe Box261 "Jixiao, Yang Zhou" 100,00% Shoe Box262 "DaGuanYuan, Shanghai" 100,00% Shoe Box263 "Tianyi, NingBo" 100,00% Shoe Box264 "Pavilion for Drama" 100,00% Shoe Box
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265 "Fangchien Studio" 100,00% Shoe Box266 "Petruzzelli Theatre, Bari" 100,00% Horse Shoe269 "The Maily Theatre, Moscow" 100,00% Horse Shoe270 "MHAT, Moscow" 100,00% Horse Shoe271 "The Bolshoy Drama Theatre, Moscow" 100,00% Horse Shoe272 "Ostankino Palace, Moscow" 100,00% Horse Shoe273 "Festhalle, Landau" 100,00% Shoe Box274 "Wiemar Hall, Germany" 100,00% Shoe Box275 "Tapiola Hall, Espoo" 100,00% Fan Shaped276 "Helsinki Konservatorio" 100,00% NaN277 "Järvenpää" 100,00% Shoe Box278 "Kaukamestä" 100,00% Fan Shaped279 "Kaustinen" 100,00% NaN280 "Kuopio" 100,00% Fan Shaped281 "Laurentius" 100,00% Fan Shaped282 "Mikaeli" 100,00% NaN283 "Promenaadi" 100,00% Fan Shaped284 "Tampere Large Hall" 100,00% Fan Shaped285 "Tampere Chamber Hall" 100,00% NaN286 "Sigyn" 100,00% Shoe Box287 "Martinus" 100,00% Shoe Box290 "Galina Vishnevskaja's Theatre, Moscow" 100,00% Horse Shoe291 "The State Ballet Large Hall, Moscow" 100,00% Horse Shoe292 "The State Ballet Small Hall, Moscow" 100,00% Horse Shoe293 "Moscow Novaja Opera" 100,00% Horse Shoe294 "Tainan Municipal Cultural Center Auditorium" 100,00% Fan Shaped295 "Gran Teatre el Liceu, Barcelona" 100,00% Horse Shoe296 "Auditorium Barcelona" 100,00% Fan Shaped297 "Kursaal Auditorium, San Sebastian" 100,00% Fan Shaped298 "Auditorio Ciudad de León" 100,00% Fan Shaped299 "Auditorio Baluarte, Navarra" 100,00% Fan Shaped300 "Palau Congressos Catalunya" 100,00% Shoe Box268 "The Mariinsky Theatre, St. Petersuburg" 99,90% Horse Shoe17 "Erkel Theater, Budapest" 99,90% Fan Shaped
203 "Rockefeller" 99,90% NaN177 "Debaser Medis" 99,90% Fan Shaped178 "Elysée Montmartre" 99,90% NaN
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189 "LKA Langhorn" 99,90% NaN50 "Opera Garnier, Paris" 99,90% Horse Shoe93 Audimax TU-Berlin 99,90% Fan Shaped94 Kammersaal_Auditorio_Nacional 99,90% vineyard
124 Santa Maria de Melque 99,90% Church150 "Magasinet, Odense" 99,90% Shoe Box216 "Zenith Paris" 99,80% Fan Shaped228 "Concert Hall ATM, Tokyo" 99,80% Fan Shaped240 "Toppan Hall" 99,80% Shoe Box116 Aula 1 99,80% Shoe Box267 "The Bolshoy Theatre, Moscow" 99,80% Horse Shoe109 Teatro Olimpico 1 99,70% NaN256 "Teatro Olimpico, Sabbioneta" 99,50% Horse Shoe222 "The Bay Side Pocket" 99,50% Shoe Box223 "Calarts Theatre, LA" 99,40% Shoe Box250 "Asahikawa Taisetsu Crystal Hall" 98,40% Shoe Box196 "Olympia" 97,50% NaN92 Konzertsaal_Charkow 97,00% Shoe Box95 Cloitre du Couvent des Cordeliers 96,90% Shoe Box
106 Konzertsaal 2 96,80% Shoe Box205 "Rote Fabrik" 96,70% NaN288 "Taipei National University of Arts, Taiwan" 96,10% Shoe Box243 "The Harmony Hall, Matsumoto" 96,00% Shoe Box96 Cultuurzentrum 95,70% Fan Shaped
249 "Akiyoshidai International Art Village Hall" 95,00% NaN220 "Art Sphere, Tokyo" 94,40% Horse Shoe242 "Kioi Hall" 93,50% Shoe Box236 "Calderwood Hall" 93,50% NaN224 "Muriel Kauffman Theatre, Kansas" 92,70% Horse Shoe227 "Nagaoka Lyric Hall" 91,30% Eliptical255 "Teatro Olimpico, Vicenza" 90,80% Horse Shoe235 "Museo del Violino" 90,60% Vineyard233 "Auditorium Maison de la Radio" 88,70% Vineyard60 "Mozarteum, gr. Saal, Salzburg" 88,60% Shoe Box
239 "Danish Radio Concert Hall" 88,40% Vineyard248 "Walt Disney Concert Hall" 88,40% Vineyard289 "Symphony Hall, Las Palmas" 88,00% Fan Shaped232 "Shanghai Symphony Hall" 87,90% Shoe Box
9 "Jesus Christus Kirche, Berlin" 87,80% Church234 "Nospr Katwoice Hall" 86,60% Vineyard246 "Helzberg Hall" 86,20% Horse Shoe
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!!!!!!!!!!!!!!!!!!!!!!!CLASS!2ID ROOM!NAME PROBABILITY! SHAPE180 "Mediolanum Forum, Milano" 100% Shoe Box209 "Oslo Spectrum Arena" 100% Fan Shaped211 "Forest National" 100% Horse Shoe212 "Wembley Arena" 100% Eliptical217 "Zenith Strasbourg" 100% Horse Shoe184 "Hallen Stadion" 99% Eliptical179 "Festhalle" 99% Eliptical
!!!!!!!!!!!!!!!!!!!!!!!CLASS!3ID ROOM!NAME PROBABILITY! SHAPE191 "Men Arena" 100,0% Eliptical192 "O2 World Hamburg" 100,0% Eliptical193 "O2 World Berlin" 100,0% Eliptical194 "O2 World London" 100,0% Eliptical197 "Palau Sant Jordi" 100,0% Eliptical181 "Globe Arenas" 99,9% Eliptical208 "Scala, Milano" 97,5% Shoe Box187 "Jyske Bank Boxen" 63,6% Fan Shaped
226 "Nara Centennial Hall" 85,40% Shoe Box238 "Helsinki Music Concert Hall" 85,30% Vineyard99 Teatre Jean Vilar 84,60% Hexagonal
237 "New World Center Concert Hall" 83,70% Fan Shaped119 Eurogress 83,60% Fan Shaped98 Gulbenkian Hall 82,20% Fan Shaped
245 "Stanford Hall" 80,90% Eliptical215 "Zeche Carl" 77,20% Shoe Box110 Teatro Olimpico 2 72,40% NaN231 "Katsushika Symphony Hills" 70,30% Fan Shaped244 "Fukushima Concert Hall" 61,50% Shoe Box
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!!!!!!!!!!!!!!!!!!!!!!!CLASS!4ID ROOM!NAME PROBABILITY! SHAPE91 Basilica of Eberbach Monastery 100,00% Church
108 Teatro Farnese 100,00% Horse Shoe128 Sejong Concert Hall 100,00% Fan Shaped163 "Biften, Rodovre" 100,00% Shoe Box219 "Notsuri" 100,00% NaN247 "Shenzhen Concert Hall" 100,00% Vineyard105 Konzertsaal 1 99,90% vineyard225 "Ishikawa Ongakudo Concert Hall" 99,90% Shoe Box127 Kursaal 99,90% NaN175 "Le Confort Moderne, 99,80% Shoe Box166 "L'Aeronef, Lille" 99,50% Shoe Box107 Oper 99,40% Nan117 Dortmund 99,20% Shoe Box104 Kirche 99,10% Church229 "Kumamoto Prefectural Theatre" 98,40% Fan Shaped241 "Harmony Hall, Fukui" 98,20% Shoe Box230 "Sumida Triphony Hall" 98,00% Shoe Box251 "Okoyama Symphony Hall" 97,60% Shoe Box155 "Skráen, Älborg" 96,00% Shoe Box160 "Tobakken, Esbjerg" 95,90% Shoe Box97 Eglise du College St.Michel 93,90% Church
204 "Rockhall" 93,10% NaN221 "Yokosuka Arts Theatre" 88,00% Horse Shoe122 Haus fuer Musik 82,60% Shoe Box118 Elmia 76,20% Fan Shaped
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Glossary
AIC: Akaike Information Criterion
BIC: Bayesian Information Criterion
CFA: Common Factor Analysis
EFA: Exploratory Factor Analysis
FMM: Factor Mixture Model
LPA: Latent Profile Analysis.
