Page 62 - Spring 2015

objective bayesian Analysis in Acoustics
example above, a predictive model for sound energy decay functions consists of a number of exponential decay terms and one noise term. Models with one, two, or three rates of decay can be denoted as H1,H2,H3, in which the multiple de- cay times, T1,T2, or T3, via decay constants are of practical concern.
This room-acoustic problem is readily soluble using the two levels of Bayesian inference (Xiang, 2015). In Figure 4, the data analysis begins with a set of data points (in the time domain) and a set of models with unspecified number of ex- ponential terms. Given the experimental (sound energy de- cay) data, architectural acousticians need the answer to the higher level question of how many exponential terms there are in the energy decay data before estimating detailed decay parameters as indicated in Figure 2.
sparse sensors for Direction
of Arrival estimation
Sound source localization using a two-microphone ar- ray in, e.g., communication acoustics is a cost-efficient yet challenging technique because just two omnidirectional microphones set a known distance apart from each other (but not binaural configurations) provide incomplete infor- mation. Escolano et al. (2014) reported an application us- ing generalized cross-correlation (GCC) between the two microphone signals with a dedicated phase transformation (GCC-PHAT) as a prediction model for the direction of ar- rivals. In this application, a complete solution embodies two levels of inference, model selection to estimate the number of concurrent sound sources in a noisy, reverberant environ- ment and parameter estimation to determine the direction of arrivals. The GCC-PHAT model-based processing of two microphone signals involves the predictive models H1,H2,H3 ..., with one, two, or three sources and so on. Yet there is no need to impose the number of concurrent sources a priori because it is sufficient to apply Bayesian model selection so as to estimate the number of concurrent sources. Once the model is selected, given the two microphone signals, to con- tain the selected number of concurrent sources, it is possible to estimate their angular parameters (direction of arrivals).
bayes’ theorem represents how our initial belief, or our prior knowledge, is updated in the light of the data.
Figure 6. Geoacoustic transdimensional inversion for seabed struc- ture. hi, Layer thickness; ci, sound speed; ρi, density; αi, attenuation. From Dettmer et al. (2010) and Dosso et al. (2014).
Geoacoustic Transdimensional
Inversion of seabed Properties
Acoustics is widely used to study the seabed. Figure 6 shows an at-sea experiment (Dettmer et al., 2010; Dosso et al., 2014) in which a ship tows an impulsive acoustic source past a moored receiver that records reflections off the seafloor and subbottom layers; the data are processed as reflection coefficients as a function of incident angle and frequency. The aim is to estimate a geoacoustic model of the seabed in- cluding an unknown number of layers, with unknown layer thickness hi , sound speed ci , density ρi , and attenuation αi in each layer. The two levels of inference are carried out simultaneously using transdimensional inversion to sample probabilistically over models with differing numbers of lay- ers. The result is a set of depth-dependent probability pro- files for the geoacoustic parameters (c,ρ,α) averaged over the number of layers, with the advantage that the uncer- tainty in the number of layers is included in the parameter uncertainty estimates.
Concluding Remarks
Bayesian probabilistic analysis has recently been applied to an increasing extent in acoustic science and engineer- ing. Objective Bayesian analysis ensures that the output of Bayes’ theorem is a posterior probability distribution based precisely on the information put into it, not more and not less, and that the information has been utilized systemati- cally and objectively. Many data analysis tasks in acoustics embody two levels of inference, namely, model selection and, within the chosen model, parameter estimation. Objec- tive Bayesian analysis provides solutions of both levels using Bayes’ theorem within a unified framework.
60 | Acoustics Today | Spring 2015