Proposed BSIE model

It is desirable to find a parameterizable model forO from which statistically realisticBSIEs can be gen- erated. AGMMis used to model the distribution [118], selected for its flexibility in finding the best fit to the observed distribution through the use of multiple Gaussians. ForNGnumber of Gaussians, the joint

probability distribution ofO can be approximated as

P (O) =

NG

∑

i₌₁

π_{iN (O ∣ ¯µi}, ¯Σ_{i) ,} (3.34)

where each N (O ∣ ¯_µi, ¯Σi) is a component of theGMMwith its own mean ¯µiand covariance ¯Σi, andπi

is the mixing coefficient, with ∑NG

i₌₁πi = 1 and 0 ≤ πi ≤ 1. The method of expectation maximization is a common approach for finding a best-fitGMM[118], and is used in this work to fit a givenO. A more

3.5 Proposed BSIE model 67

risk of overfitting to the specific data considered.

In this work, the trade-off between model complexity and goodness-of-fit is evaluated using two model selection methods. The first model selection method used isC-fold cross-validation [118], where the training data is segmented intoC sets, out of which C − 1 sets are used for training and the remaining set is used for validation. The procedure is then repeated for all possible choices of the validation set and the evaluation metrics from theC repetitions are averaged. In the training stage, the best-fitGMM

parameters are found using theN_Gnumber of Gaussians considered, and in the validation stage, the log- likelihood of the validation data given the trainedGMMis computed. A drawback of the cross-validation method, besides being computationally expensive, is that training is performed on separate datasets and could therefore lead to different optimal parameters. The second model selection method used is the Bayesian information criterion (BIC), which provides a single measure of performance over the whole training data. It is defined as [119]

BIC = NGln(Nobs) − 2 ln L(θGMM), (3.35)

whereN_obsis the number of observations inO and L(_θGMM) is the maximized likelihood function of the best-fitGMMwith parameters {πi, ¯µi, ¯Σi}_{∀i∈{1 ... N}G}. A smallerBICis desirable as it minimizes the

negative log-likelihood of theGMMparameters, with a penalty term for the number of Gaussians. In the remainder of this section, theGMMparameters and their corresponding cross-validation log- likelihoods andBICs are computed to describeBSIEmodels for various acoustic scenarios using both simulated and realAIRs.

The simulatedAIRs were obtained using the image method [13,14] with the following acoustic setup. A 5-channel endfire uniform linear array (ULA) with an inter-microphone spacing of 4 cm was placed in the middle of a simulated room. A source was stepped around the centre of theULAat a 1 m radius and at five equally spaced angles from 0○to 180○on the horizontal plane. At each angle, theAIRs were simulated with sampling frequencyf_{s= 8 kHz and a total of 30}AIRs were obtained for each angle, each corresponding to a randomly generated room between 3 × 3 × 2 m and 4 × 5 × 3 m. Two sets ofAIRs were simulated as above withT60∈ {250, 400} ms and their lengths were truncated to L = T60fscoefficients.

Additionally, real measuredAIRs were taken from the MARDY database [120], which comprises 8- channel impulse responses for 3 source-to-microphone distances at {1, 2, 3} m and 3 different source locations, giving a total of 9 different acoustic setups. To maintain consistency with the image method experiments, the number of channels was limited toM = 5 by generating 4 subsets from each 8-channel acoustic setup, where thei-th subset is formed as {mi, . . . ,mi₊₄}. Therefore, in total, 36 experiments

were formed from 4 subsets × 9 acoustic setups.

The microphone signals were then generated from theAIRs as (3.8) usingWGNforv_{m(n) with two}

SNRs, {15, 25} dB, andRNMCFLMSwas used to estimate ˆh. AsBSIalgorithms are known to misconverge in the presence of noise, in this work, the ˆh that gives the minimumNPMover 3600 s ofxm was used.

In practice, algorithm control techniques such as [121] could potentially be used to stop the adaptation process before misconvergence occurs.

For each of the image methodAIRsets and the collective MARDYAIRs, Ξ was computed using

(3.32) and aggregated over allAIRs within the set. Outliers were avoided by removingBSIEratios with absolute values that are within the top 5 percentile, along with their correspondingh coefficients. The

best-fitGMMs were then found for the remaining joint observationsO forNG∈ {1, 2, . . . , 20}.

TheBICand cross-validation results for simulated and realAIRs are given in Fig.3.3and Fig.3.4

respectively. It can be seen that the performance of both metrics improve exponentially as the number ofGMMcomponents increases and does not degrade at anyNGwithin the range investigated. As such,

NG = 20 was arbitrarily chosen for the remainder of this work as a good compromise between model complexity and goodness-of-fit. Examples of the resulting best-fitGMMwithN_{G= 20 are given for both} simulated and realAIRs in Fig.3.5.

In this work, the best-fitGMMparameters were found separately for each acoustic scenario described above. A generalizedGMMcould be found instead by using the joint observations aggregated over all acoustic scenarios. Such a generalized model would be more convenient for practical use, however, it is expected that its accuracy will be lower compared to the individually trained models. In the experiments conducted here, the ranges of theBSIEratios changed withT60andSNR, where a higherT60and lower

SNRresulted in a largerBSIEratio range. As an illustrative example, the acoustic scenario withT60 = 400 ms and SNR = 15 dB has a range of ≈ −21 to ≈ 21 dB, while the scenario with T60 = 250 ms and SNR = 25 dB has a range of ≈ −8.5 to ≈ 8.5 dB.