Paul M. Goggans Department of Electrical Engineering, University of Mississippi, Anderson Hall, University, Mississippi 38677

Transcription

1 Evaluation of decay ties in coupled spaces: Bayesian decay odel selection a),b) Ning Xiang c) National Center for Physical Acoustics and Departent of Electrical Engineering, University of Mississippi, 1 Coliseu Drive, University, Mississippi Paul M. Goggans Departent of Electrical Engineering, University of Mississippi, Anderson Hall, University, Mississippi Received 8 May 00; revised 3 Noveber 00; accepted 4 January 003 This paper applies Bayesian probability theory to deterination of the decay ties in coupled spaces. A previous paper N. Xiang and P. M. Goggans, J. Acoust. Soc. A. 110, discussed deterination of the decay ties in coupled spaces fro Schroeder s decay functions using Bayesian paraeter estiation. To this end, the previous paper described the extension of an existing decay odel N. Xiang, J. Acoust. Soc. A. 98, to incorporate one or ore decay odes for use with Bayesian inference. Bayesian decay tie estiation will obtain reasonable results only when it eploys an appropriate decay odel with the correct nuber of decay odes. However, in architectural acoustics practice, the nuber of decay odes ay not be known when evaluating Schroeder s decay functions. The present paper continues the endeavor of the previous paper to apply Bayesian probability inference for coparison and selection of an appropriate decay odel based upon easured data. Following a suary of Bayesian odel coparison and selection, it discusses selection of a decay odel in ters of experientally easured Schroeder s decay functions. The present paper, along with the Bayesian decay tie estiation described previously, suggests that Bayesian probability inference presents a suitable approach to the evaluation of decay ties in coupled spaces. 003 Acoustical Society of Aerica. DOI: / PACS nubers: Br, Ka MK I. INTRODUCTION A previous paper 1 discussed estiation of the decay ties in coupled spaces using a Bayesian paraeter estiation approach given that the nuber of decay rates is already known. However, in ost decay tie deterinations, the nuber of decay rates odes is not always readily known. Eyring pointed out that energy decay functions on a logarithic scale are not generally linear for coupled spaces having different natural reverberation ties or even for a single roo with nonuniforly distributed absorption and no diffusing schee. In effect, the sound energy decay in a singlespace roo can be of ulti-rate character. Therefore, the practical application of the proposed decay tie evaluations is not only the analysis of sound decay in coupled spaces but also in single spaces, such as in those reverberation chabers which often yield inconsistent absorption data, probably due to lack of diffusion. In contrast, single-rate energy decay can also be observed in coupled spaces, depending on the size of the coupling aperture, the position of the sound receiver, and the natural reverberation tie of each space. Often acousticians have to answer the question, How any decay rates are present in the collected data? before the relevant decay ties can be properly estiated. As discussed in previous a Portions of this work were presented at the 141st and 143rd eetings of the Acoustical Society of Aerica J. Acoust. Soc. A. 109, 83A 001; 111, 33A 00. b Dedicated to Prof. Dr. Jens Blauert on the occasion of his 65th birthday. c Electronic ail: nxiang@oleiss.edu works, 1,3,4 decay functions obtained fro easured roo ipulse responses using Schroeder s backward integration 5 contain inherent characteristic curvature towards the upper liit of the integration. This characteristic curvature ipedes identification of different decay odes. Visual inspection of Schroeder decay functions will not always reveal the nuber of decay rates. An algorithic deterination of decay order nuber of decay rates is needed in practice. The subject of the present paper is the application of Bayesian probability theory to the proble of estiating the nuber of decay rates present in Schroeder decay functions. The Bayesian ethod essentially calculates the probability of decay odels with different decay order based on the experientally easured data. The Bayesian literature refers to this as odel coparison and selection MCS. A odel-based approach using a generalized least square LS principle 4 can also estiate decay ties. The LS approach and all other paraeter estiations, including Bayesian decay tie estiation, 1 are all subject to the question of how any decay odes are present in the easured data. In resolving this question, the Bayesian approach proves to be ore coprehensive than the LS approach, since a Bayesian fraework can provide quantitative tools for both the odel selection and the paraeter estiation. Siilar to the decay tie estiation proble described in Ref. 1, systeatic developent of a Bayesian foralis for MCS starts with application of the Bayes theore, followed by incorporation of prior inforation and then arginalization defined in Ref. 1. Any interest in paraeter J. Acoust. Soc. A. 113 (5), May /003/113(5)/685/13/$ Acoustical Society of Aerica 685

2 values will be pushed into the background of the current proble through arginalization. Marginalization allows attention to be focused on the probabilities for different decay in a preselected set of decay odels. In Sec. III of the present paper, we develop the MCS foralis toward evaluation of the Schroeder decay functions in a step-by-step anner for the convenience of architectural acousticians. This developent of the MCS relies heavily on other works, 6 1 particularly Ref. 6. In Sec. IV we discuss results obtained using the Bayesian MCS with experientally easured decay functions fro coupled spaces. Additional detailed derivations involved in the Bayesian MCS of the present paper are given in the Appendix. II. MULTI-RATE DECAY A previous paper 1 established a decay odel for Schroeder decay functions with ultiple decay odes. In the present paper, a set of these decay odels F F 1,F,...,F M is under consideration with s1 F s A s,b s,t k j1 A sj G j B sj,t k, 0t k L and 1sM, 1 and G j B sj,t k expb sj t k for 1 js, Lt k for js1, where s is the decay order nuber of exponential decay odes, s1 is the nuber of additive ters of the odel, M is the total nuber of different odels under exaination, A s A s1,a s,...,a s contains linear paraeters, B s B s1,b s,...,b ss contains nonlinear paraeters, and L represents the upper liit of the Schroeder backward integration. 5 The decay ties are related to the nonlinear paraeter by the expression T sj 13.8/B sj, for 1 js. In this paper, tie will be treated as a discrete variable t k. The following derivations drop the subscript s of A s, B s and all other paraeters given odel F s for siplicity. As a specific case of a double-rate Schroeder decay function, the decay odel with s, 3 reads F A,B,t k A 1 expb 1 t k A expb t k A 3 Lt k. 3 The third ter on the right-hand side in Eq. 3 is associated with background noise in experientally easured roo ipulse responses fro which Schroeder decay functions are calculated. It results in the characteristic curvature that occurs towards the upper liit of the integration in a logarithic plot of Schroeder decay functions e.g., Fig. 1a. This curvature is well docuented in Refs. 1, 3, and 4 and, as previously entioned, it ipedes the deterination of the decay order s. The decay odel in Eq. 1 is in the for of a general linear odel. 1 It describes sound energy decay in enclosed spaces after a steady-state sound excitation in the spaces is switched off. Therefore, only positive-valued linear paraeters AA 1,A,...,A are of priary interest for decay FIG. 1. Schroeder decay functions siulated using a double-rate decay odel Eq. 3. Noralized tie scale is used for siplicity. a Decay paraeters fulfill the conditions in Eq. 4 (0.5T 1 T 1.0 and 10 log 10 A 1 0 db; 10 log 10 A 6dB).b Decay paraeters break the conditions in Eq. 4 (1.0T 1 T 0.5 and 10 log 10 A 1 0dB; 10 log 10 A 6 db). A single-rate decay curve with T is also plotted for coparison. The double-rate decay curve in b is very close to the single-rate curve. tie deterination in architectural acoustics practice, although there ay exist other acoustical situations or other kinds of systes without this restriction. In addition, architectural acousticians are priarily concerned with the conditions T 1 T T s and A 1 A A 0, 4 as pointed out in Ref. 13. Figure 1 illustrates two opposite exaples siulated using the double-rate decay odel in Eq. 3. A noralized tie scale is used for siplicity. In Fig. 1a, the Schroeder decay function fulfills the conditions in Eq. 4 with T 1 0.5, T 1.0 and with 10 log 10 A 1 0 db, 10 log 10 A 6 db, and 10 log 10 A 3 40 db. In Fig. 1b, the decay paraeters of the decay function break the conditions in Eq. 4 with the sae linear paraeters as those in Fig. 1a but T and T 0.5. This double-rate decay function is very close to the single-rate one with T as plotted in Fig. 1b for coparison. The saller the paraeter A is relative to A 1, the closer the double-rate decay function will be to a single-rate function. 686 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces

3 Often the decay function in Fig. 1b represents a sound decay process in coupled spaces sensed by a sound receiver in the priary space when the secondary space possesses a shorter natural reverberation tie than that of the priary space. In this situation, the coupling aperture acts as an absorption area resulting in a shorter decay tie than the natural reverberation tie in the priary space. It is, therefore, reasonably to treat the energy decay in a statistical sense as a single-rate decay. III. BAYESIAN MODEL SELECTION This section develops Bayesian foralis for odel coparison and selection MCS. Bayesian MCS relies on the easured Schroeder decay functions to copare and select the ost likely decay odels aong a specified odel set of copeting decay odels. The developent of Bayesian MCS foralis begins with calculating the posterior probability for a copeting odel by applying the Bayes theore. Bayesian MCS focuses on the posterior probabilities of the copeting odels rather than specific odel paraeters. Therefore, any interest in paraeter values will be pushed into the background of the current proble through arginalization. The developent needs to introduce soe additional paraeters when dealing with residual errors between the easured data and odels and dealing with prior probabilities. Marginalization will also reove these additional paraeters. Eventually the MCS is accoplished by evaluating the posterior probability for each copeting odel. The decay odel F s with s decay odes aong F F 1,F,...,F M approxiates the Schroeder decay function: 5 d k F s A,B,t k k, 1kK, 5 in such a way that F s (A,B,t k ) odels the Schroeder decay function d k with a residual error k. As stated in Ref. 1, this error is defined as the difference between the easured data and the odel and so is a function of s, A, and B. It includes easureent errors, fluctuations in the decay function data, and odeling errors. The current proble is specified by the background inforation I including the hypothesis that the data consist of a systeatic part F s ( ) and an additive error part k, and that the error has a finite-valued variance see Sec. III B in Ref. 1 and a set F of M utually exclusive, exhaustive copeting odels specified by Eqs. 1 and. A. Posterior probability of copeting odels The probability p(f s D,I) for odel F s given the Schroeder decay function Dd 1,d,...,d K and the relevant background inforation I can be expressed using Bayes theore: pf s D,I pf sipdf s,i, lsm. 6 PDI In exaining utually exclusive, exhaustive decay odels in the odel set FF 1,F,...,F M, p(di) is the probability of the data given only the background inforation I: M pdi s1 pf s IpDF s,i. In effect, p(di) is a noralization constant over all odels. Equation 7 is the consequence of the copeting odels being utually exclusive and exhaustive. For a utually exclusive and exhaustive set of odels M s1 pf s D,I1. p(f s I) is prior probability for odel F s given only the inforation denoted by I. The proposition I represents one s state of knowledge about the odels before obtaining the data D or before analyzing the data once the data are obtained. The global likelihood for the data given odel F s and the inforation I is denoted p(df s,i). The likelihood indicates how well the specified odel fits the data. The posterior probability p(f s D,I) is so-called because it applies after the data and the prior inforation have been taken into account. The calculation of the posterior probability for odels according to Eqs. 6 and 7 requires assignent of the prior probability for the odel F s and the calculation of the global likelihood. An appropriate uninforative prior probability given M possible odels is the unifor prior probability 1/M that expresses no preference for any odel in the odel set F. Using this prior and Eq. 7 the posterior probability for the odel becoes pf s D,I 7 8 pdf s,i M pdf s,i. 9 s1 Equation 9 is of central iportance for the decay odel selection and will be discussed in ore detail in the following sections. B. Global likelihood The global likelihood for the data is vital to the calculation of the posterior probability for odels as indicated in Eq. 9. The joint probability for the data and the odel paraeters p(d,a,bf s,i) can yield the global likelihood for the data p(df s,i) in ters of pdf s,i da db pd,a,bf s,i 10 in which the doain of the integral spans the ultidiensional paraeter space of the odel F s. Applying the product rule to the joint probability p(d,a,bf s,i) yields pd,a,bf s,,i pbf s,ipab,f s,ipda,b,f s,,i 11 so that the global likelihood given the variance for the odel F s reads J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces 687

4 pdf s,,i K db pbf s,i da pab,f s,i exp 1 K d k F s A,B,t k. k1 1 Equation 1 incorporates the likelihood function expression for p(da,b,f s,,i) given in the previous paper see Eq. 9 in Ref. 1. Since the task is to deterine the posterior probability for the decay odels, both the linear paraeters A and the nonlinear B paraeters can be treated as nuisance paraeters and the explicit dependence of the likelihood on A and B can then be reoved by considering all their possible values through arginalization. 1 The following section pursues the arginalization over these paraeters. C. Marginalization over aplitude A p(ab,f s,i) is the prior probability of paraeters A assuing that the nonlinear paraeters B of a decay odel F s are given. No specific inforation is assued about the values of the paraeters A expect that their values are finite. Application of the principle of axiu entropy to assign the prior probability, 7,8 given only the finite value assuption, results in the Gaussian assignent of pab,f s,,i 1 exp 1 K k1 l1 j1 A l A j G l G j. 13 In Eq. 13 the dependence of p(ab,f s,,i) on B is through the function G. The variance is associated with the uncertainty of the odel paraeter A, and j is the jth eigenvalue of the atrix g ij of Eq. 10 in Ref. 1. With the addition of the new paraeter, the global likelihood of the data in Eq. 1 reads pdf s,,,i K K da db 1 pbf s,i exp 1 K k1 l1 j1 A l A j G l G j exp 1 K d k F s A,B,t k. k1 14 The global likelihood of the data can further be rewritten as pdf s,,,i K K d db pbf s,i exp j j1 1 j1 j q j j1 j, 15 using Eqs. 13 and 17 in a previous paper 1 for j and q j, respectively. In Eq. 15, 1,,..., and K k1 d k. 16 Perforing the integral over the aplitude paraeters see Appendix A, the global likelihood siplifies to pdf s,,,i K db pbf s,i exp 1, 17 where 1 q j. 18 j1 The function represents a noralized for of q defined in Eq. 0 in Ref. 1, and plays the role of sufficient indicator in evaluating the values of the nonlinear paraeter B. 6 D. Marginalization over paraeter B When an appropriate odel is eployed, the data deterines the nonlinear paraeters well, the posterior probability around the global axiu Bˆ falls off rapidly, and there exists a sall region in paraeter space around Bˆ where the noralized sufficient indicator in Eq. 18 can be approxiated by BBˆ. 19 The posterior density function for the nonlinear paraeters B is ulti-odal with extrees of identical value when s is greater than one. Taking the approxiation in Eq. 19 and s! ulti-odal extrees see Ref. 1 into account and assuing that these extrees do not overlap to any significant degree, the global likelihood in Eq. 17 becoes pdf s,,,i K s! exp 1 BBˆ db pbf s,i. 0 The further developent requires assignent of the prior probability p(bf s,i) for the nonlinear paraeters B given odel F s. Here little prior inforation about the values of the paraeters B is assued, only that they have finite values. Under this assuption, application of the principle of axiu entropy results in the assignent of a Gaussian prior probability as follows: B pbf s,,i exp s j j1. 1 The variance expresses the uncertainty of the nonlinear paraeters B. Equation 1 does not express a strong opin- s 688 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces

5 ion about the paraeters provided that is large copared to. This eans that Eq. 1 effectively represents a constant over the range in which the likelihood function of the data is strongly peaked. Substitution of this prior probability into Eq. 0 results in the global likelihood of the data: pdf s,,,,i K s! where s j1 B j. s E. Marginalization over variances exp 1 BBˆ, 3 The three variances,, are referred to as hyperparaeters in Bayesian literature. Although their values are unknown, arginalization can reove the fro the proble. Marginalization over these hyperparaeters requires ultiplication of appropriate noralized prior probabilities with the global likelihood in Eq., followed by integration over these paraeters. Appropriate prior probabilities for these scale paraeters 10,, and are Jeffreys priors. 9 However, the Jeffreys prior presents an iproper probability distribution, the integral of which is not noralizable. 10 Nevertheless, the current proble can handle a bounded Jeffreys prior. Application of a noralized, bounded Jeffreys prior 6 for,, and see Appendix B approxiates the global likelihood of the data as pdf s, a, b, a, b, a, b,i Ks s! / s/ K/ log b / a log b / a log b / a / s/ 1 K/ BBˆ 4 with a, b, a, b, and a, b being the bounds of the paraeters,, and, respectively. In Eq. 4, (x) is the gaa function of x. So long as odels with both linear and nonlinear paraeters are copared using the sae data as in the case of ulti-rate decay functions, the posterior probability for the odel in Eq. 9 is independent of a, b, a, b, and a, b because log( b / a ), log( b / a ), and log( b / a ) will appear in both the nuerator and the denoinator see Eqs. 7 and 8. Hence the global likelihood of the data can be siplified to with pdf s,ic / s/ 1 K/ BBˆ 5 C s s! s K. 6 F. Model coparison and selection To copare which one of the two odels F x and F y is favored by the data, the ratio of the posterior probability of the two odels according to Eq. 9 is evaluated: O pf yd,i pf x D,I pdf y,i pdf x,i. 7 Equation 7 is referred to as the odds. Its logarithic for is conventionally calculated using pdf y,i E yx 10 log 10 8 pdf x,i to indicate which odel is preferred. E with db as the unit in Eq. 8 is often referred to in relevant Bayesian literature as Bayesian evidence. In addition to the odel coparison, the odel selection is accoplished by evaluating the posterior probability in Eq. 9 for each odel throughout the specified odel set F. Eventually, the data should favor a odel with a clearly higher posterior probability. This section has developed a Bayesian foralis of odel coparison and selection MCS. Bayesian MCS relies on the easured Schroeder decay functions to copare and select the ost likely decay odels aong a specified odel set containing M copeting decay odels. The developent of Bayesian MCS foralis begins with calculating the posterior probability for a copeting odel by applying the Bayes theore. Expressing no preference for any particular decay odel in advance of analyzing the data leads to the assignent of a unifor prior probability for the odels. In exaining a decay odel within a specified, utually exclusive, exhaustive odel set, the posterior probability for a decay odel relies solely on the global likelihood function. It can be proven 6 that the MCS described in this section still reains valid if soething else outside the odel set is not taken into account, since the MCS relies on a relative evaluation of posterior probabilities over the specified odel set. Marginalization has reoved all linear and nonlinear odel paraeters and variances arising fro residual errors, fro the assignent of prior probability on both linear and nonlinear paraeters during the developent. Eventually the odel coparison and selection can be accoplished by evaluating Bayesian evidence or the posterior probability of each odel by calculating the global likelihood for the data at the peak position in the nonlinear paraeter space. The current paper eploys Gibbs sapler as done in Ref. 1 to deterine the peak position in the nonlinear paraeter space. The analytical developent arrives at the approxiation of p(df s,i) in Eqs. 5 and 6 by assuing that the nonlinear paraeters associated with the peak position of the global likelihood are already well deterined. Besides the ean squared value of the nonlinear paraeters and soe gaa function values deterined by the decay order and the nuber of data points, the noralized sufficient indicator at the peak position ust be calculated to yield the global likelihood of the data. The developent of the MCS foral- J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces 689

6 is in this section has incorporated soe approxiations to yield a practicable end foralis. In the following section, MCS calculations using experientally easured data will exaine both their validation and the liitation of these approxiations. IV. EXPERIMENTAL RESULTS This section eploys experientally easured Schroeder decay functions fro 1:8 scale-odel roos and real halls to deonstrate how the Bayesian MCS can be applied to decay odel deterination associated with the decay tie evaluation. All roo ipulse responses RIRs have been octave band-pass filtered before Schroeder decay functions 5 are calculated fro the. All evaluations being discussed below use the noralized decay function data fro the tie liit at 5 db to the upper liit of the integration. The upper liit of the integration is selected to be large enough to include a portion of the noise tail in each RIR. A. Double-rate decay functions Figure illustrates a RIR easured in 1:8 scale odel roos Fig. a, its energy decay curve Fig. b, and the Schroeder s decay function Fig. c. The peak-to-noise ratio of the RIR is 53 db. The priary roo of in diension, approxiately 19 3 in volue of real size, in which the sound source and receiver are located, possesses a natural reverberation tie of 0.5 s at 1 khz. The secondary roo of in diension, approxiately in volue coupled to the priary roo through a coupling aperture of 4, possesses a natural reverberation tie of 1.0 s. Care had to be given to arrange ost of the interior surface of both roos as diffuse as possible within the frequency range of interest. As stated previously, recognition of the nuber of decay odes fro the three fors of presentations in Fig. is not straightforward. When using a single-rate decay odel F 1, the posterior probability density function PPDF presents a sharp peak at 0.50 s as depicted in Fig. 3a. Figure c contains the Schroeder single-rate odel curve using this estiated paraeter. The noralized sufficient indicator NSI as given in Eq. 18 around the peak position presents a relatively flat shape, but still peaked at 0.50 s as depicted in a zooed presentation in Fig. 3b. When using a double-rate decay odel F, the PPDF presents two peaks with equal value over the paraeter space (T 1,T ) as depicted in Fig. 4. Either of the will serve when seeking the global axiu. 1 These! equal-valued peaks for diensionality of two are well separated, eeting the assuption required in Sec. III D when siplifying Eq. 17 to Eq. 0. Figure 5 shows the distribution of the NSI over the sae sub-space as given in Fig. 4c. The NSI peaks at the sae location as the PPDF with a relatively flat shape. The NSIs shown in Figs. 3 and 5 present such a flat shape as to clearly justify the approxiation undertaken in Eq. 19. A double-rate odel FIG.. Roo ipulse response and Schroeder decay function achieved fro an experientally easured roo ipulse response in two coupled scale-down odel roos. a Roo ipulse response, octave band-pass filtered at 1 khz, the peak-to-noise ratio aounts to 53 db. b Energy decay curve ETC. c Schroeder decay function. The decay odels with single-, double-, and three-rate odes are also depicted for coparison. Table I lists the odel paraeters associated with single-, double-, and three-rate odel. 690 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces

7 FIG. 3. Noralized posterior probability and noralized sufficient indicator when using a single-rate decay odel to analyze Schroeder s decay function easured in coupled scale-down odel roos. a Coparison between the noralized posterior probability and the noralized sufficient indicator. b. Noralized sufficient indicator in a zooed scale curve using the decay paraeters associated with one of peak positions is also depicted in Fig. c. Sharp peaks over the nonlinear paraeter space associated with both a double-rate and a single-rate odel suggest that these two decay odels are copeting candidates for the decay tie estiation. To deterine which candidate is favored by the data, Bayesian evidence E 1 for a double-rate decay odel F over a single-rate decay odel F 1 is deterined based upon the easured data. Table I contains soe relevant paraeter values for evaluating p(df 1,I), p(df,i), and p(df 3,I). The Gibbs sapler algorith is applied to deterine these paraeters. Figure c shows three decay odel curves along with the easured Schroeder decay function. The Bayesian evidence E db is estiated as listed in Table I, indicating that the data Schroeder s decay function as shown in Fig. c strongly favors odel F : a double-rate decay function. Using a triple-rate decay odel, the assuptions required in Sec. III D cannot be et. Investigation results illustrated in Fig. 6 ay yield an explanation. Given the estiated first decay tie (T s), which can be easily verified fro a sall portion at the beginning of the decay function, Fig. 6 shows the PPDF over two other decay ties (T,T 3 ). Two peaks are still recognizable over the two decay FIG. 4. Noralized posterior probability distribution evaluated over paraeter space of two decay ties. Two separate peaks of identical peak value are identified within this range. a 3-D presentation over the decay tie space. A grid of 600 by 600 is defined over the decay tie paraeter space between 0.35 and 0.9 s. b -D presentation over the decay tie paraeter space between 0.35 and 0.9 s. c Zooed presentation over the decay tie paraeter subspace of 0.45T s and 0.69T 0.71 s, a grid of 84 by 84 is defined to evaluate the posterior probability. tie paraeter space 0.5 st, T s). However, these two peaks, overlapped by each other in a significant degree, occur along the line of syetry T T 3. One of the peaks occurs at T T s and the other occurs at T T s with a lower peak value. In other words, the three-diensional decay tie space within a reasonable J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces 691

8 TABLE I. The paraeters associated with single-, double-, and triple-rate decay odels for evaluation of a decay function easured in two scaledown coupled odel roos as depicted in Fig.. Peak-to-noise ratio of the roo ipulse response is 53 db. The decay paraeters evaluated using a triple-rate odel cannot fulfill the condition in Eq. 4. Single-rate odel F 1 Double-rate odel F Triple-rate odel F 3 Decay order s 1 3 No. of ters 3 4 No. of data K Sufficient indicator Linear paraeter A 1 (db) Decay tie T 1 (s) Linear paraeter A (db) Decay tie T (s) Linear paraeter A 3 (db) Decay tie T 3 (s) 0.87 Bayesian evidence E 1 (db) 38.6 Bayesian evidence E 3 (db). FIG. 5. Distribution of noralized sufficient indicator over paraeter subspace of two decay ties. In the paraeter subspace of 0.45T s and 0.69T 0.71 s, a grid of 84 by 84 is defined to evaluate the noralized sufficient indicator. a 3-D pseudo-color presentation. b -D presentation. range does not include 3! equal-valued, spatially wellseparated peaks as required when approxiations are used to siplify the global likelihood fro Eq. 17 to Eq. 0. It clearly shows that the data contain two distinct decay rates (T 1,T ) rather than three decay rates. The third exponential ter along with decay tie T 3, as required by the triple-rate odel, is redundant, and should have been reoved fro the current proble. When seeking peaks in a three-diensional paraeter space, any search algorith would converge to a point in the space associated with no reasonable paraeters (T 1,T,T 3 ). Therefore, the calculation of global likelihood at the peak position cannot be considered as reasonable. Table I lists the relevant paraeters for the triple-rate odel. The evaluated linear paraeters (A db, A 6.37 db, and A db) associated with these nonlinear paraeters break the condition in Eq. 4, especially A being so close to A 3, indicating that the third decay ter is redundant. It is not reasonable in this case to use the evaluated Bayesian evidence (E 3. db) to indicate that the data favor the odel F. Moreover, Fig. c shows that the decay paraeters evaluated using the triple-rate odel do not yield a reasonable decay function when copared with the easured one. In a siilar fashion, Fig. 7 depicts the Schroeder decay function easured at 1 khz in real coupled spaces Student Union, University of Mississippi, University, MS. A tall secondary roo of in diension, approxiately in volue, is coupled with a short priary roo of in diension, approxiately in volue, through a coupling aperture of in area. The secondary roo possesses a natural reverberation tie of. s at 1 khz while the priary roo has a natural reverberation tie of 0.95 s. Table II lists soe relevant paraeters evaluated using the single-, double-, and triple-rate odels, respectively. Sharp peaks of the PPDF over the paraeter space can be found using both single- and double-rate odels. Bayesian evidence for a double-rate odel over a single-rate one is E db. For the triplerate odel, however, given the estiated first decay tie (T s), the PPDF over two other decay tie spaces T,T 3 within a reasonable value range does not present two equal-valued peaks off the line of syetry T T 3 as required. This indicates that the third decay tie T 3 is redundant and should have been reoved fro the proble. For the triple-rate odel, calculation of the global likelihood using Eqs. 5 and 6 in a three-diensional paraeter space is not reliable and it is isleading to use the calculated evidence (E 3.9 db) for odel coparison and selection. A closer look at the linear paraeters in Table II evaluated based on the nonlinear paraeters reveals that A and A A 3, breaking the conditions in Eq. 4 as well. Figure 7 illustrates that the triple-rate odel evaluated with these decay paraeter values cannot yield a reasonable odel function. The exaples of double-rate decay functions discussed above show that the global likelihood of the data changes significantly when going fro using a single-rate to a double-rate odel. Aong these, Bayesian odel coparison and selection can provide a quantitative easure Bayesian evidence to indicate which odel is the ost probable one for proper decay tie estiation. The calculated value of Bayesian evidence E is liable to vary depending on the data 69 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces

9 FIG. 7. Schroeder decay function achieved fro an experientally easured roo ipulse response in real coupled spaces Student Union, University of Mississippi, USA. The decay odels with single, double, and triple rate are also depicted for coparison. Table II lists the odel paraeters associated with these odels. FIG. 6. Posterior probability distribution over decay tie (T,T 3 ) while keeping T s. A grid of 5151 is defined for evaluating the noralized posterior probability distribution using a triple-rate decay odel. One peak with higher peak value occurs along the syetrical line at T T s. The second peak with a lower peak value occurs at T T s. It indicates that third decay tie T 3 is redundant and should have been reoved fro the proble. and the nuber of data points. However, the assuptions for siplifying the global likelihood calculations cannot be et when the odel order becoes higher than that favored by the data, in this case a triple-rate odel. Moreover, the decay paraeters evaluated using a triple-rate odel break the conditions in Eq. 4, resulting in iproper estiation. With even higher decay order, the nonlinear paraeter space does not contain an adequate nuber of well-separated equalvalued peaks either. Of course, a (K1)-degree equation will fit K points of experiental data. Ockha s razor tells us, however, an explanation of the facts should be no ore coplicated than necessary. 15 B. Single-rate decay functions Figure 8a shows RIR at khz octave easured in the secondary roo of the 1:8 scale odel roos discussed in Sec. IV A when the coupling aperture is acoustically closed. For these data, the peak-to-noise ratio is 46 db. Its energy decay curve and the Schroeder s decay function are given in Figs. 8b and c, respectively. Evaluation of the PPDF using a single-rate odel yields a sharp peak at s as shown in Fig. 9a. In the sae range, the NSI presents a flat curve. Figure 9b shows the NSI function at a zooed scale. The peak of the NSI is in the sae position as that of the PPDF. The NSI shown in Fig. 9 validates the siplification undertaken in Eq. 19. Table III lists the relevant odel paraeters evaluated at the peak position. Evaluation of the PPDF using a double-rate odel over the decay tie space T 1,T within a reasonable value range between 0.98 and 1.4 s presents only a single peak along the line of syetry T 1 T as shown in Fig. 10, indicating that the second exponential decay ter of the odel along with the decay tie T is redundant and should have been reoved fro the proble. In this situation the MCS cannot use the Bayesian evidence at the peak posi tion although the search algorith will settle to two decay paraeters soewhere around the single peak in the decay tie space. Note that one of two linear paraeters at the peak position becoes negative as listed in Table III, breaking the conditions in Eq. 4. TABLE II. The paraeters associated with single-, double-, and triple-rate decay odels for evaluation of a decay function easured in two coupled halls as depicted in Fig. 7. Peak-to-noise ratio of the roo ipulse response is 51 db. The decay paraeters evaluated using a triple-rate odel cannot fulfill the condition in Eq. 4. Single-rate odel F 1 Double-rate odel F Triple-rate odel F 3 Decay order s 1 3 No. of ters 3 4 No. of data K Sufficient indicator Linear paraeter A 1 (db) linear Decay tie T 1 (s) Linear paraeter A (db) Decay tie T (s) Linear paraeter A 3 (db) Decay tie T 3 (s) 1.90 Bayesian evidence E 1 (db) Bayesian evidence E 3 (db).9 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces 693

10 FIG. 9. Noralized posterior probability and noralized sufficient indicator when using a single-rate decay odel to analyze Schroeder s decay function easured in one scale-down odel roo. a Coparison between the noralized posterior probability and the noralized sufficient indicator. b Noralized sufficient indicator in a zooed scale FIG. 8. Roo ipulse response and Schroeder decay function achieved fro an experientally easured roo ipulse response in one of the scale-down odel roos. a Roo ipulse response, octave band-pass filtered at khz, the peak-to-noise ratio aounts to 46 db. b Energy decay curve ETC. c Schroeder decay function. The decay odel with single rate is also depicted for coparison. Table III lists the odel paraeters associated with single- and double-rate odel. C. Discussion The present paper has described a siplified foralis for Bayesian odel coparison and selection. A close coparison between Eqs. 5 and 6 of this paper and Eq. 1 of the earlier paper 1 reveals that the end foralis in Eqs. 5 and 6 can be straightforwardly extended fro the Bayesian paraeter estiation described in Eq. 1 of the earlier paper 1 since one of the ultiplication factors (1 ) (K)/ in Eq. 5 is in the for of the Student t-distribution as given in Eq. 1 of the earlier paper. 1 Once the global optial position in the paraeter space is found, calculations of reaining factors in Eqs. 5 and 6 require insignificant effort. For this reason, both the MCS and the paraeter estiation using advanced optiization approaches 1 can be accoplished hand-in-hand. Soe necessary approxiations, particularly that given in Secs. III C and D, ake this foralis practicable in architectural acoustic applications. The exaples using experientally easured Schroeder s functions in both coupled roos and single-space roos confir the validation of the siplifying steps when an adequate nuber of well-separated peaks exist in the corresponding nonlinear paraeter space. The exploratory exaples restricted so far to the easured Schroeder functions show that the estiation of the 694 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces

11 TABLE III. The paraeters associated with single- and double-rate decay odels for evaluation of a decay function easured in one 1:8 scale-down single-space roo. The decay paraeters evaluated using the double-rate odel cannot fulfill the condition in Eq. 4. Single-rate odel F 1 Double-rate odel F Decay order s 1 No. of ters 3 No. of data K Sufficient indicator Linear paraeter A 1 (db) Decay tie T 1 (s) Linear paraeter A (db) linear a Decay tie T (s) 1.79 Linear paraeter A linear a Bayesian evidence E 1 db 75.3 a The linear paraeters A 0andA 3 0 are estiated using the double-rate odel. relevant paraeters, including the global likelihood value, becoes unreliable when the odel order is one ore than the optial one. Often this happens with an inadequate nuber of equal-valued, well-separated peaks in the given paraeter space since one decay ter in the odel is redundant and should have been reoved fro the proble. In this situation, the odel coparison and selection described in this paper cannot rely on the Bayesian evidence calculated in ters of the global likelihood. It has often been observed that the odel paraeters in this situation break the conditions in Eq. 4. In practice, the conditions expressed in Eq. 4 can FIG. 10. Noralized posterior probability distribution over decay tie (T 1,T ). A grid of 5656 is defined for evaluating the noralized posterior probability distribution using a double-rate decay odel. A single peak occurs along the syetrical line T 1 T. It indicates that second decay tie T is redundant and should have been reoved fro the proble. provide an additional indicator to reject a decay odel with a odel order one higher than the optial order. Checking the conditions in Eq. 4 requires deterination of all linear paraeters at the peak position. Equation 4 of the earlier paper 1 yields the expected value estiates of the linear paraeters without substantial coputational load. Optiization search algoriths play a crucial role in both Bayesian paraeter estiation described in Ref. 1 and odel coparison and selection in this paper. Advanced ethods can converge to the global extree over the paraeter space of given diensionality with a reasonable coputational load. Deterination of the diensionality of the paraeter space is again a MCS proble rather than an optiization proble. This is clearly the case because a search algorith, searching for the global extree over a three-diensional paraeter space, will converge soewhere in that space even if the actual extree should have been in a two-diensional paraeter space. This confirs what profound thinkers say: The trick lies not in finding the answer, but in asking the right question. 11 V. SUMMARY AND CONCLUSIONS Recently, acoustically coupled spaces have been drawing ore and ore attention in the architectural acoustics counity. As a result, deterination of decay ties in these coupled spaces fro easured Schroeder decay function data has becoe a ore significant proble. It has long been recognized that evaluations of ore than one decay tie in coupled spaces requires considerable effort. A previous work 1 proposed a Bayesian paraeter estiation approach to evaluating decay ties. Bayesian paraeter estiation can yield reasonable estiation results only when it uses an appropriate odel with the correct nuber of decay odes decay order for Schroeder decay functions. The decay order should be first deterined. This paper has described a odel coparison and selection MCS procedure using Bayesian probability theory. In effect, Bayesian evidence has to be calculated for coparison between two alternative odels. Given a set of copeting exclusive, exhaustive decay odels, the global likelihood for each odel over the odel set has to be evaluated based upon the easured data. All results discussed in this paper have been restricted to exploratory exaples with soe necessary approxiations. One needs to coe back to the original step beginning with the Bayesian theore if the assued approxiations cannot be ade in specific probles. The current work also reveals soe liitations associated with the described MCS applied in Schroeder s decay functions. Solutions of these probles reain for future efforts. ACKNOWLEDGMENTS The authors are very grateful to Professor M. R. Schroeder for his encourageent and Dr. Ch. Jaffe, Dr. W. T. Chu, and Dr. J. M. Sabatier for their valuable suggestions and constructive criticiss. Thanks are also due to Donghua Li who contributed to the data collection in the coupled scale-down odels as part of his Master s degree thesis. J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces 695

12 APPENDIX A Equation 15 can be expressed as pdf s,,,i K K pbf s,iv db, A1 where K pdf s,,,i pbf s,i which results in Eq. 17. exp 1 db, A6 V exp j1 exp 1 exp j 1 j1 1 exp 1 j1 q j d. j1 j q j j1 j j q j j1 d d j j A Here and are defined in Eqs. 16 and 18, respectively. A change of variable u j d j du j yields Vexp 1 j1 j q j, exp u j du j. 0 A3 According to Eq. A5 in Ref. 1 / V exp 1. A4 Substitution of V in Eq. A4 into K A1 yields pdf s,,,i pbf s,i exp 1 db. / A5 Assuing that prior uncertainty in the aplitudes associated with is uch greater than, p(df s,,,i) ay be approxiated 6 as APPENDIX B Assuing that there exist two bounds a, b, a, b, and a, b for,, and, respectively, the noralized Jeffreys prior becoes p,,, a, b, a, b, a, b,i 1 log b / a log b / a log b / a. The arginalization over these paraeters reads pdf s, a, b, a, b, a, b,i a b a b a bp,,,a, b, a, b, a, b,i B1 pdf s,,,,id d d. B Substitution of Eqs. B1 and into Eq. B yields pdf s, a, b, a, b, a, b,i s! Ks W Y log b / a log b / a Z B3 log b / a BBˆ with W b 1 a exp d, B4 Y b s1 a exp d, B5 and Z b K1 a exp 1 d. B6 Equations B4 B6 are all of the sae for. This for reduces to the following: a b x n expyx dx n 1 y n1/ B7 when a 0, b. Taking Eq. B4 as an exaple, by letting x1/ so that d dx and a 0asx and b as x 0, n 1 and y. Equation B4 can be approxiately deterined by 696 J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces

13 W / B8 given that the interval ( a, b ) is wide so as to approxiately fulfill the conditions a 0, b ). In siilar fashion, and Y s s/ B9 Z K 1 K/. B10 Substituting Eqs. B8 B10 into Eq. B3 results in Eq N. Xiang and P. M. Goggans, Evaluation of decay ties in coupled spaces: Bayesian paraeter estiation, J. Acoust. Soc. A. 110, F. Eyring, Reverberation tie easureents in coupled roos, J. Acoust. Soc. A. 3, W. T. Chu, Coparison of reverberation easureents using Schroeder s ipulse ethod and decay-curve averaging ethod, J. Acoust. Soc. A. 63, N. Xiang, Evaluation of reverberation ties using a nonlinear regression approach, J. Acoust. Soc. A. 98, M. R. Schroeder, New ethod of easuring reverberation tie, J. Acoust. Soc. A. 37, G. L. Bretthorst, Bayesian analysis. II. Signal detection and odel selection, J. Mag. Res. 88, E. T. Jaynes, Highly inforative priors, in Bayesian Statistics, edited by J. M. Bernardo, M, H. Degroot, D. V. Lindley, and A. F. M. Sith Elservier Science, Asterda, The Netherlands, 1985, p P. M. Woodward, Probability and Inforation Theory Pergaon, Oxford, UK, H. Jeffreys, Theory of Probability Oxford U.P., London, 1939, 3rd revised ed D. S. Sivia, Data Analysis: A Bayesian Tutorial Clarendon, Oxford, A. J. M. Garrett, Ockha s Razor, in Maxiu Entropy and Bayesian Methods, edited by W. T. Grandy and L. H. Schick Kluwer Acadeic, The Netherlands, 1991, pp G. L. Bretthorst, Bayesian Spectru Analysis and Paraeter Estiation Springer-Verlag, New York, L., Creer, H. A. Müller, and T. J. Schultz, Principles and Applications of Roo Acoustics Applied Science, London, S. F. Gull, Bayesian Inductive Inference and Maxiu Entropy, in Maxiu-Entropy and Bayesian Methods in Science and Engineering, Vol. 1, edited by G. J. Erickson and C. R. Sith Kluwer Acadeic, The Netherlands, 1988, pp W. H. Jefferys and J. O. Berger, Ockha s Razor and Bayesian Analysis, A. Sci. 80, J. Acoust. Soc. A., Vol. 113, No. 5, May 003 N. Xiang and P. M. Goggans: Decay ties in coupled spaces 697