19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 9 th INERNAIONAL CONGRESS ON ACOUSICS MADRID, -7 SEPEMBER 007 BAYESIAN ANALYSIS IN CHARACERIZING SOUND ENERGY DECAY: A QUANIAIVE HEORY OF INFERENCE IN ROOM ACOUSICS PACS: B; Mc Xiang, Ning ; Goggans, Paul M. ; Jasa, oislav. Gaduate Poga in Achitectue Acoustics, and Dept. of Electical, Copute, and Systes Eng., Rensselae Polytechnic Institute, oy, NY 80, USA. xiangn@pi.edu. Depatent of Electical Eng., Univesity of Mississippi, Univesity, MS 38677, USA ABSRAC Concet hall designs that incopoate coupled had chabes have been dawing inceasing attention in the oo-acoustics counity. One ipotant task in analyzing the acoustics of coupled spaces is the evaluation of decay paaetes associated with the single- o doublesloped decay chaacteistics of Schoede decay functions deived fo expeientally deteined oo ipulse esponses. aditionally, howeve, chaacteization of double-sloped enegy decay fo oo ipulse esponses has been vey challenging. his wok applies Bayesian pobabilistic infeence to cope with the deands of estiating divese decay paaetes fo Schoede decay functions. Recent effot in oo-acoustic enegy decay analysis using Bayesian pobability theoy deonstates that Bayesian pobability theoy, along with the Ockha s Razo inheently ebedded within Bayesian faewok can help acousticians to quantify single-slope o ulti-slope enegy decay chaacteistics. his pape discusses decay ode deteination, estiation of decay paaetes and thei uncetainties within Bayesian faewok. INRODUCION Revebeation ties ae the ost coon quantity to chaacteize sound enegy decays in enclosues, assuing that the sound enegy decays oe o less exponentially. Yet soeties, the sound enegy can also decay non-exponentially as found aleady by Eying [], sound enegy decay functions on a logaithic scale ae not geneally linea fo coupled spaces having diffeent natual evebeation ties o even fo a single-space oo with nonunifoly distibuted absoption and no diffusing schee. In effect, the sound enegy decay in a singlespace oo can be of ulti-ate chaacte. Sound enegy decays ae coonly obtained via oo ipulse esponses though acoustical easueents o copute-siulation techniques followed by Schoede s backwad integation []. Additional analysis pocedues ae needed to infe elevant paaetes fo the pupose of enegy decay chaacteization. Recent woks have established a paaetic odel [3-4] fo the Schoede decay functions accoding to the natue of Schoede s backwad integation. When analyzing sound enegy decays given the Schoede decay functions, we should fist ask the question how any decay ates (slopes) ae thee in the data? befoe any effot in estiating evebeation / decay ties and elated paaetes. ISO 38 [5] states in cases whee the (logaithic) decay cuve is not a staight line a unique evebeation tie cannot be said to exist. If thee would be a double-slope decay in the data, one would still ask the data to povide the paaetes associated with a single-slope decay o with a tiple-slope decay, one could not get ight answe. Moe subtly, if diffeences between a double-slope and a tiple-slope epesentation ae sall, what is a ustifying eason fo choosing the double-slope epesentation? aking this oo-acoustic application as an exaple, this pape discusses how Bayesian pobability theoy as a quantitative theoy of infeence, is applied in oo-acoustics applications. Using the established Schoede decay odels [3-4], the oo-acoustics enegy decay analysis equies two levels of odel-based infeence [6]; one level of the infeence is the task of odel copaison and selection (MCS) aong a set of copeting odels. Anothe level of infeence assues that a paticula odel is tue and then deteines the posteio pobability function (PPDF) fo the odel paaetes. hese PPDF ae used to estiate the odel paaetes, so-called odel-based paaete estiation (PE). his pape will biefly epot ou ecent effot in Bayesian PE [4,7,8] and ou on-going effot in Bayesian MCS [9-0] fo the oo-acoustic enegy decay analysis.

2 DECAY MODELS AND MODEL SELECION Schoede decay odels Steady-state sound enegy decays afte sound souces deceased ae deteined by Schoede integation [] fo oo ipulse esponses expeientally easued o nueically siulated fo enclosues unde test. D = [ d, d, K, d K ] epesents Schoede decay function data, ( ) stands fo atix tanspose, K is the total nube of data points in D. Schoede decay function odels have been established based on the natue of Schoede's integation [4,8]: D = G A + e, (Eq.) which appoxiates the data D with an eo vecto e. A is a colun vecto of ( +) aplitude coefficients, teed linea paaete vecto. G is a atix of K ( +). is the nube of decay ates o odel ode. he th colun of g k whee tk t K = ex 3.8 t k / ) ex 3.8 t K / ) is th decay tie to be deteined fo fo G = 0 fo =,, is given by its atix eleent K, (Eq.),, =. he quantity t epesents the uppe liit of Schoede's integation, 0 k < K. Recent woks [3,4,7,8] K have expeientally poven the validity of this odel. Figue shows one exaple of Schoede decay functions evaluated fo scale-odel coupled spaces and its odel function with popely estiated odel paaetes. G A 0 Figue.-Copaison between Schoede decay function (D) easued in eal halls and thei odel functions GA. Sound enegy decay (Schoede) function easued in scale-odel coupled spaces. Model copaison and Ockha s azo Once the Schoede decay function data D ae available, we conside -ode Schoede decay odel: F = G Α. (Eq.3) he odel copaison in the sound enegy decay analysis is to deteine which odel the data ae in favou aong two copeting odels. Howeve, the coect appoach is NO siply to choose the odel that fits the data best: oe coplex odels can always fit the data bette 9 th INERNAIONAL CONGRESS ON ACOUSICS ICA007MADRID

3 [6]. Bayes theoe foulates the poble in tes of posteio pobability density function Figue.-Ockha s azo: aong copeting hypotheses, favou the siplest one. Bayesian infeence autoatically ebodies Ockha s azo, which quantitatively penalizes ovepaaeteized odels. (PPDF) of odel F : F p ( D F, F D, =. (Eq.4) D I ) Backgound infoation I includes the infoation that two (e.g. =, ) odels in eq.() though eq.(3) ae copeting to each othe to epesent the data easonably well in a sense that the eo vecto is of finite value. hese odels can be copaed on the basis of the following atio [6]: F D, F p ( D F, =. (Eq.5) F D, F p ( D F, able I.- Calculated odel paaetes including odds fo a data set easued in Student Union (Univesity of Mississippi), showing that the data pefe the nd odel ode []. Decay odel ode st ode nd ode 3 d ode Aplitude A0 (db) Aplitude A (db) Decay tie (s) Aplitude A (db) Decay tie (s) Aplitude A3 (db) Decay tie (s) Log odds log0 O o see how Bayesian odel copaison incopoates Ochha s azo, conside the case whee and can both fit the data equally well. he fist (pio) atio on the ight-hand side F F in Eq. 5 indicates how uch ou initial beliefs favous odel ove, the second (likelihood) atio easues how well the available data wee odelled by F. Assigning equal pio F F ) ( I F F F in copaison with p = would iply no subective pefeence to eithe of odels, leaving only the likelihood atio on the ight-hand side. If F is highe ode (oe 9 th INERNAIONAL CONGRESS ON ACOUSICS ICA007MADRID 3

4 coplex) odel than, it ust spead its pedictive pobability oe thinly ove the data space than F p ( D F, F as shown in Fig.. In the ovelapped egion of possible data set space whee the data ae copatible with both odels, the siple F will becoe oe pobable than. he atio in Eq. 5 allows us to evaluate the plausibility of two altenative odels in the light of data D [6]. he Bayesian foulation associated with Eq.5 inheently ipleents Ockha s azo: it favous siple odels with geate pedictive powe povided they give a good fit of the data. Bayesian odel copaison autoatically contains a logic echanis fo penalizing ove-paaeteized odels [9]. Howeve, if the data pefe a highe ode odel, its pedictive powe ust have aleady ovecoe the Ockha s penalty. Ou ecent effot in applying quantitative ethods of Bayesian MCS in oo-acoustic analysis is docuented in [0,]. able I lists esults of odds values along with odel paaetes estiated fo a oo ipulse esponse easued in Student Union at the Univesity of Mississippi. DECAY PARAMEER ESIMAION Afte selecting a decay odel pefeed by data, Bayesian theoy foulates the PPDF of odel paaetes though the pio pobability density and likelihood function via Bayes' theoe: A, D A,, A, D, =, (Eq.6) D whee D I ) acts in the context of paaete estiation as a noalization constant. is a vecto atix of coefficients. A, I ) is the pio distibution function of A and. Bayes' theoe in Eq. 6 epesents how ou pio knowledge A, I ) is odified in the light F Figue 3.-Posteio pobability density function (PPDF) of decay ties evaluated fo a Schoede decay function easued in scale-odel coupled spaces. (a) PPDF in -D epesentation ove, within {0.35, 0.9} s. (b) PPDF in 3-D epesentation ove a subspace evaluated with a gid of 08 x 40. (c) A-zooed PPDF ove within {0.37, 0.47} s, within {0.5, 0.9} s. (d), (e) poection of the line acoss the peak PPDF onto individual decay tie axis. 9 th INERNAIONAL CONGRESS ON ACOUSICS ICA007MADRID 4

5 of data though the likelihood function D A,,. Backgound infoation I hee includes that the Schoede decay odel in Eqs. - descibes the data D easonably well so that all eos in e ae bounded by a finite value. Given finite eos and a easonable odel as only available infoation, application of the pinciple of the axiu entopy [4] assigns a Gaussian distibution to the likelihood function D A,, and an independence to eos e i fo each othe, so that ( D A,, σ, K e e ( ) πσ exp σ p, (Eq.7) = with a finite, but unspecified eo vaiance σ. Pobability theoy povides us with ules of elating and anipulating the quantities, leading to an analytically tactable PPDF in fo of the student- distibution: ( D D q q) ( K ) / D,, (Eq.8) Q = GEΔ with q = Q D and [8]. Figue 3 illustates a PPDF (Eq. 8) ove paaete space {, } using Schoede decay odel ode evaluated fo a oo ipulse esponse easued in scale-odel coupled spaces. Localizing axiu values of the PPDFs, so-called axiu a posteio (MAP) appoaches [4,8] o pobabilistic appoaches based on Makov chain Monte Calo (MCMC) ethods [7,0,] have been used in oo-acoustic enegy decay analysis, whee the uncetainties of infeed decay paaetes can also be quantitatively estiated [7] though a covaiance atix [ ] with its expected atix eleent: with C i R = ( i i R )( = u( ) C i ) u( ), (Eq.9) D, I ) u( ) =, (Eq.0) g( ) g( ) is the ando pocess used to daw R saples fo D, I ) in Eq. 8, and expected ean value of decay tie i i is. In this way the standad deivations of individual estiated paaetes, and dependences between paaetes can also be estiated fo the covaiance atix [ ] via Eqs.9-0 [7]. C i REPARAMEERIZED MODELS FOR AURALLY-SIGNIFICAN ACOUSIC COUPLING Acoustic designes have incopoated coupled had chabes in concet halls [] to achieve, in addition to vaiable acoustics, nonexponential ultiple-sloped decay chaacteistics of sound enegy decay which eet the following conditions: A > A >, K, and < <, K. (Eq. ) he otivation of achieving such decay chaacteistics is that two copleting, yet desiable featues of peceived claity and evebeance ae believed to be siultaneously satisfied []. he coesponding acoustic coupling in achieving such decay chaacteistics will be auallysignificant. If the conditions expessed in Eq. cannot be et, e.g. double-slope sound enegy decays will becoe single-slope natue. Ou ecent effot in applying Bayesian infeential ethods also includes incopoating these conditions (Eq.) as pio infoation [,3] into a e-paaeteized Schoede decay odel. he challenges in assigning pios of odel paaetes and thei estiations have been addessed in Ref. [,3]. Note that the esults given in able I incopoate the conditions given by Eq.. EVALUAIONS OF EXPERIMENAL DAA Bayesian pobabilistic infeence can povide us with odel paaetes, quantifying the sound decay pocess eliably. able II lists a set of evaluation esults fo a oo ipulse esponse, easued in the Student Union at the Univesity of Mississippi, acoss the octave fequency bands fo 5 Hz to khz. Bayesian odel copaison eveals that the data in khz band is 9 th INERNAIONAL CONGRESS ON ACOUSICS ICA007MADRID 5

6 of single-slope natue, theefoe, only one single decay (evebeation) tie is listed. In othe 4 fequency bands, the odel copaison confis that the data pefe double slope decays. In addition to two decay ties, o the decay tie atio, we ae also inteested in the decay level diffeence fist used in Ref. [4], athe than thei individual values A A ΔL = 0log( A / A ) of,, espectively. he decay level diffeence quantifies how low the second decaying pocess chaacteized by is elative to the fist one of. able II also lists the standad deivations (Std) of i. Successful application of Bayesian infeential analysis in sound enegy decay analysis unabiguously deonstates that othe attepts including linea fit of diffeent potions of decay functions fo evaluating nonexponential decays ae questionable, since a staightfowad choice of pedefined sall potions intoduces abitainess. hose scientifically unsounded attepts, theefoe, should not be pacticed any oe. able II.- Calculated decay paaetes acoss the octave bands fo a oo ipulse esponse easued in the Student Union at Univesity of Mississippi. Band (Hz) (s) Std (s) (s) Std (s) Decay ie Ratio L (db) E E E E E E E E E SUMMARY his wok applies Bayesian pobabilistic infeence to cope with the deanding tasks of estiating divese decay paaetes fo Schoede decay functions. In this application, two levels of infeence ae inevitably involved, decay odel copaison and odel-paaete estiation. Bayesian pobability theoy, elying on extensive uses of Bayes theoe, includes all the ules of elating and odifying pobabilistic quantities, it autoatically ebodies Ockha s Razo to evaluate whethe single-slope o double-slope enegy decay chaacteistics ae favoued by the data. Once the decay odel is selected, Bayesian decay paaete estiation will povide devise decay paaetes, including decay ties, decay level diffeence, and thei uncetainties. Evaluations of expeiental esults using Bayesian appoaches developed in ou ecent effot have deonstated thei effectiveness in the oo-acoustics application. Refeences: [] F. Eying: Revebeation tie easueents in coupled oos. Jounal of the Acoustical Society of Aeica 3 (93) 8-06 [] M. R. Schoede: New ethod of easuing evebeation tie. Jounal of the Acoustical Society of Aeica 37 (965) [3] N. Xiang: Evaluation of evebeation ties using a non-linea egession appoach. Jounal of the Acoustical Society of Aeica 98 (995) - [4] N. Xiang, P. Goggans: Evaluation of decay ties in coupled spaces: Bayesian paaete estiation. Jounal of the Acoustical Society of Aeica 0 (00) [5] ISO 338: Acoustics Measueent of the evebeation tie of oos with efeence to othe acoustical paaetes, , pp. 9 [6] D. MacKay: Infoation heoy, Infeence, and Leaning Algoiths, Cabidge Univesity Pess. 003 [7] N. Xiang, P. Goggans.. Jasa, M. Kleine: Evaluation of decay ties in coupled spaces: Reliability analysis of Bayesian decay tie estiation. Jounal of the Acoustical Society of Aeica 7 (005) [8] N. Xiang,. Jasa: Evaluation of decay ties in coupled spaces: An efficient seach algoith within Bayesian faewok. Jounal of the Acoustical Society of Aeica 0 (006) [9] W. H. Jeffeys, J. O. Bege, Ockha s Razo and Bayesian Analysis, Aeican Scientist 80 (99) 64-7 [0]. Jasa, N. Xiang: Nested sapling in oo-acoustic applications. In Bayesian Infeence and Maxiu Entopy Methods in Science and Engineeing, Ed. K. Knuth et al. (007) in pint [] P. M. Goggans, N. Xiang, and C-Y Chan: Model copaison in oo-acoustical enegy decay analysis. In Bayesian Infeence and Maxiu Entopy Methods in Science and Engineeing, Ed. K. Knuth et al. (007) in pint [] R. Johnson, E. Hahle, and R. Esset: Vaiable coupled cubage fo usic pefoance, Poc. MCHA 95, 995, Kiishia, Japan [3] P. M. Goggans, N. Xiang, C-Y Chan, and Y. Chi: Sound decay analysis in acoustically coupled spaces using a e-paaeteized decay odel. In Bayesian Infeence and Maxiu Entopy Methods in Science and Engineeing, Ed. R. Fishe et. al. (004) th INERNAIONAL CONGRESS ON ACOUSICS ICA007MADRID 6