Clustering analyses to assess HVAC noise in real-world conditions

Domenico De Salvio 1 , Dario D’Orazio, Massimo Garai Department of Industrial Engineering, University of Bologna, Italy Viale del Risorgimento 2, 40136 Bologna, Italy

ABSTRACT The assessment of systems’ noise during measurements is usually carried out in an empty state of the spaces under study. Long-term monitoring in real-world conditions can provide insights into the performance of systems during the day. Thus, it becomes useful to find methods able to separate coexisting sound sources in the same space. Clustering techniques can supply the lack of robust methods to perform the segregation. Previous works have shown reliable results about distinguishing human and mechanical noises through sound level meter long-term measurements. Sound pressure levels (SPLs) were post-processed via Gaussian Mixture Model and K-means clustering with a three-step algorithm. The first step is clustering validation and concerns the assessment of the optimal number of clusters among the candidate models. Then, SPLs were divided into the number of clusters obtained by the validation. Finally, statistical and metrical features were used to label the sound sources whether mechanical or human, depending on the algorithm that performed the clustering. Carrying out these algorithms during real-time conditions makes it possible to monitor the actual HVAC noise providing more accurate analyses. Further studies will focus on broadening the ability to analyze the mechanical noise into its components, such as continuous and discontinuous sources.

1. INTRODUCTION

The lack of ability to separate sound sources is one of the main issues concerning sound level meter measurements [1]. The background noise in spaces is usually assessed in empty state. Thus, these analyses are not carried out in real conditions. Further, statistical analyses conducted through percentile levels do not rely on robust assumptions, especially for temporal factors [2]. Long-term monitoring can provide big amount of data, besides the recording of useful insights about the real mode of operation of systems. Thus, machine learning can be exploited to investigate large acquisition of sound pressure levels. Clustering techniques can provide di ff erent algorithms to explore data finding valuable relationships [3]. Previous works have used Gaussian Mixture Model and K-means clustering to analyze di ff erent sound environments separating the sound sources. These algorithms have been used to identify sound sources in school and o ffi ces through sound level measurements [2,4]. Basing on the procedure proposed in previous works, the present study deepens the analysis of a long-term monitoring of two working days in three o ffi ces. Previous findings pointed out the ability

1 domenico.desalvio2@unibo.it

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

to separate the mechanical noise due to the di ff erent working devices inside the o ffi ce from the speech of employees. The present work proposes a cluster analysis of the resulting cluster associated to the mechanical sources. The goal is to explore useful insights about the mechanical sources in real-world conditions.

2. THEORY AND CONTEXT

In this work, three o ffi ces represent the case studies. The activities carried out within are di ff erent in each of them. Following, a brief description of the spaces is presented:

O ffi ce A - Open plan with 8-12 workers. The amount of employees can vary during the day. This is a sales o ffi ce. Hence, the activity is mainly focused on outside of the room communication.

O ffi ce B - Open plan with 10-12 workers. The amount of employees can vary during the day. This is a design o ffi ce. Hence, the activity can be more or less collaborative.

O ffi ce C - Small o ffi ce with a maximum of 2 workers. The amount of employees can vary during the day. The activity carried out has a small amount of collaborative work.

This study was conducted after COVID19. Thus, besides the occupancy’s variations, all o ffi ces were treated with screens 120 cm height.

2.1. Gaussian Mixture Model The Gaussian Mixture Model (GMM) is an iterative method to describe random distributions as a sum of Gaussian curves. It is frequently used as an unsupervised technique in machine learning. The assumption is that all data points are generated from a mixture of Gaussian distributions [5]. The Gaussian probability density function f ( x i ) of a set of observations x 1 , ..., x n can be expressed as a sum of k Gaussian densities f k ( x i , µ k , σ 2 k ):

K X

k = 1 π k f k ( x i , µ k , σ 2 k ) (1)

f ( x i ) ≃

where π k are the mixing proportions [6], non-negative quantities that sum to one; that is,

0 ≤ π k ≤ 1 ( k = 1 , . . . , K )

and K X

k = 1 π k = 1 .

The likelihood function for a mixture model with K univariate Normal components is:

e − ( xi − µ k ) 2

n Y

K X

n Y

K X

k = 1 π k 1 q

2 σ 2 k . (2)

L ( x ) =

k = 1 π k f k ( x i ) =

2 πσ 2 k

i = 1

i = 1

In the present work, the GMM implements the Expectation-Maximization algorithm to fit the original distribution [7].

2.2. K-means clustering The K-means clustering algorithm is an iterative method which aims to minimize the distance among data and clusters. The set of observations x 1 , . . . , x n can be clustered into a set of K clusters, C = { c k ; k = 1 , . . . , K } , where µ k is the mean of cluster c k . In the present work, the distance to minimize is represented by the Euclidean distance. The squared Euclidean distances between µ k and the points in cluster c k is defined as:

J ( c k ) = X

x i ∈ c k || x i − µ k || 2 . (3)

Thus, the aim is to minimize it over all K clusters:

K X

X

x i ∈ c k || x i − µ k || 2 . (4)

J ( C ) =

k = 1

The process ends when the convergence is reached. This is outlined in two steps: first, the optimal partition for a given set of µ k is found; then, the cluster centroids are computed once C is fixed [8].

3. EXPERIMENTAL PROCEDURE

The present study follows the same procedure proposed in a previous work, here briefly outlined [4]. A sound level meter monitored the entire working day in three di ff erent o ffi ces. Octave band filtered (from 125 up to 4000 Hz) short-term sound pressure levels (SPLs) have been recorded every 0.1 seconds. A short interval of acquisition allows to record a high resolution of the time history of the activity within the spaces. Indeed, the sound pressure levels recorded during the breaks of the speech refer to other active sound sources, like HVAC and mechanical and electronic devices. The occurrences of the SPLs obtained are processed via two clustering algorithms, GMM and KM. First, the optimal number of clusters is found through the Akaike Information Criterion for GMM and the silhouette coe ffi cient for KM [9,10]. Results found for both algorithms a number of sources equal to 2. Thus, carrying out a clustering analysis in each octave band, it is possible to reconstruct the spectra of the sound sources. The tendencies of the spectra obtained refer to the two main active sound sources in the o ffi ces: the speech and the mechanical noise. Table 1 shows the results obtained by the first clustering analysis. The SPLs of each sound source of each algorithm are shown per each o ffi ce and day. The standard deviations SD and the average intra-cluster distances AICD of data points are shown in brackets. These latter represent the feature used to label the sources as mechanical or speech. A standard deviation of 5 dB is set as threshold for the tag [11]. A value lower than the threshold is deemed as mechanical, otherwise as speech. A correlation between SD and AICD were investigated to find the correspondent threshold for AICD. These two parameters measure the size of the clusters. Previous findings showed a tight correlation for large clusters (R 2 = 0 . 96). Thus, the ones associated to the speech. More spread data were found in the tight clusters associated to the mechanical sources instead (R 2 = 0 . 47). Hence, a cluster analysis was carried out following the same procedure already described only on the mechanical clusters.

Table 1: Results of the clustering carried out over long-term monitoring of the three o ffi ces. The o ffi ce, the correspondent measurement day, the algorithm and the kind of source are shown. Measured SPLs are shown for each octave band from 125 to 4000 Hz, besides the A-weighted values. Moreover, the correspondent A-weighted continuous-equivalent level L Aeq , T measured through the sound level meter is shown.

O ffi ce Day Algorithm Source Frequency octave band A-weighted L Aeq , T 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz

56.5 Human 53.2 (4.3) 52.3 (5.3) 52.6 (8.5) 42.5 (8.6) 37.6 (8.5) 32.6 (8.4) 51.9 (7.7)

55.4 Human 52.9 (4.3) 53.0 (5.6) 51.8 (8.0) 43.4 (8.0) 38.2 (7.7) 32.8 (7.4) 51.8 (7.1)

55.0 Human 49.6 (5.1) 49.3 (6.4) 49.3 (8.3) 44.3 (7.4) 40.2 (7.0) 35.1 (6.8) 50.6 (7.1)

54.5 Human 49.1 (5.0) 48.2 (6.5) 47.9 (8.8) 43.2 (7.7) 39.3 (7.4) 33.6 (6.8) 49.5 (7.5)

53.2 Human 46.1 (7.4) 46.2 (7.5) 45.4 (8.8) 39.6 (8.1) 35.9 (8.3) 29.5 (8.5) 46.3 (7.9)

51.7 Human 43.8 (7.1) 44.1 (7.2) 45.5 (8.8) 38.6 (8.4) 34.7 (8.7) 29.6 (8.2) 45.7 (8.1)

GMM Mechanical 47.7 (1.7) 43.8 (1.4) 37.0 (1.2) 30.3 (1.3) 24.8 (2.3) 19.8 (3.7) 39.8 (1.0)

KM Mechanical 47.8 (1.8) 44.3 (1.9) 38.3 (3.0) 32.5 (3.5) 27.6 (4.2) 22.3 (4.7) 41.3 (2.9)

GMM Mechanical 47.2 (1.7) 43.8 (1.7) 37.7 (1.5) 31.7 (1.8) 26.6 (2.5) 22.3 (3.6) 40.2 (1.4)

KM Mechanical 47.4 (1.8) 44.3 (2.1) 38.9 (3.0) 33.3 (3.3) 28.4 (3.7) 23.3 (3.9) 41.5 (2.8)

GMM Mechanical 40.9 (2.2) 38.7 (2.4) 34.4 (2.2) 33.7 (2.1) 29.8 (2.3) 25.2 (3.1) 38.7 (2.1)

KM Mechanical 41.9 (2.7) 40.1 (3.3) 36.4 (3.9) 35.2 (3.2) 31.7 (3.5) 26.5 (3.6) 40.7 (3.5)

GMM Mechanical 41.7 (1.9) 38.2 (1.9) 33.3 (1.7) 32.7 (1.6) 28.7 (1.9) 23.8 (2.6) 37.9 (1.7)

KM Mechanical 42.3 (2.3) 39.4 (2.9) 35.0 (1.1) 34.0 (3.0) 30.5 (3.2) 25.2 (3.4) 39.5 (3.1)

GMM Mechanical 35.4 (3.2) 35.7 (3.2) 33.8 (4.8) 29.0 (4.2) 22.4 (3.7) 14.2 (1.1) 34.7 (3.5)

KM Mechanical 36.2 (3.6) 36.5 (3.6) 34.2 (4.5) 30.3 (4.4) 25.9 (4.9) 20.6 (4.7) 36.6 (4.3)

GMM Mechanical 34.7 (2.4) 33.5 (2.2) 31.4 (3.8) 27.7 (4.3) 22.0 (4.5) 16.9 (3.0) 33.5 (3.5)

KM Mechanical 35.3 (3.0) 34.5 (3.0) 32.3 (4.2) 28.2 (4.2) 23.6 (4.8) 19.2 (4.3) 34.7 (4.1)

Human 55.5 (3.1) 55.1 (3.9) 57.2 (5.9) 49.6 (6.4) 43.6 (6.4) 37.8 (6.0) 57.0 (5.4)

Human 55.3 (3.1) 55.8 (4.1) 56.4 (5.6) 49.2 (6.0) 43.3 (5.7) 36.9 (5.4) 56.3 (5.1)

Human 52.3 (3.6) 53.0 (4.5) 54.1 (5.8) 49.6 (5.7) 45.2 (5.2) 39.1 (5.0) 55.0 (5.1)

Human 51.8 (3.6) 52.3 (4.8) 53.6 (6.1) 49.3 (6.1) 44.9 (5.7) 37.9 (5.1) 54.6 (5.5)

Human 50.3 (5.3) 50.5 (5.4) 49.8 (6.4) 40.5 (5.7) 41.2 (5.7) 35.7 (5.9) 51.0 (5.7)

Human 48.3 (4.9) 48.8 (5.3) 50.0 (6.4) 43.0 (6.1) 39.7 (6.2) 34.8 (6.0) 50.4 (5.9)

B 1

B 2

C 1

C 2

A 1

A 2

4. RESULTS AND DISCUSSIONS

Results of the cluster analysis carried out on the mechanical sources are shown in Figure 1. The mechanical clusters have been separated into two di ff erent classes and the spectra have been reconstructed from 125 to 4000 Hz. In all the o ffi ces, the lowest sources indicated as Mech1 show the same results for both algorithms GMM and KM. Di ff erences are noticed for the highest sources indicated as Mech2. This is expected because Mech1 has the lowest SPLs and the lowest variances too. Thus, for these sources, homoscedasticity is almost fulfilled [12]. Tendencies of Mech1 diverge only in the highest frequency bands, 2 and 4 kHz, during the day 1 of the o ffi ce A and the 4 kHz octave band for both days in o ffi ce C. However, the SPLs referring to the 4 kHz band are very small and they can be influenced by the electric noise of the sound level meter [4]. The sound source Mech2 shows higher di ff erences between the algorithms. Nevertheless, in the lowest frequencies, the 125 and 250 Hz octave bands, variances are very low in each case. This is the expected behaviour of a mechanical sources. It is worth noting that the spectra remain almost the same for each day in each o ffi ce, especially in o ffi ce B. The latter is a design o ffi ce. Thus, can be deduced that the mechanical noise is more constant and less a ff ected by more random sound sources like phones. In o ffi ce B, the GMM seems to be less sensitive to shape a proper cluster for Mech2. In this case KM seems to be more reliable. Higher di ff erences between the days have been noticed in o ffi ce A, the sales o ffi ce. Here, external calls and dynamic activity are more usual. Figure 2 shows the correlation between SD and AICD after the cluster analysis has been performed on the mechanical sources. Grey data points represent the speech cluster, while blue and orange points represent respectively Mech1 and Mech2. The correlation between the two metrics confirms to be very strong in the human cluster. However, the R-squared coe ffi cient obtained by the separation of the mechanical sources decreased per each cluster but summing up their values the R-squared value of the global mechanical cluster is obtained. The separation of the mechanical sources does not follow linearly the correlation between SD and AICD. The clusters shape a second threshold around the value of SD equal 1 dB. This value could represent a new feature to separate and label mechanical sources.

5. CONCLUSIONS

The present work deepened the analysis of the mechanical noise in real-world conditions. A sound level meter measurement monitored the activity in three di ff erent o ffi ces for two days each. Then, a cluster analysis has been performed via Gaussian Mixture Model and K-means clustering to separate the human speech from the mechanical noise. Further clusters have been found in the mechanical source to investigate the ability of these algorithms to split di ff erent mechanical contributions. Results show how both algorithms are capable to find non-trivial spectra of two di ff erent mechanical sources. Moreover, the correlation between the standard deviation and the average intra-cluster distance shows useful insights about a possible threshold between the mechanical sources. A standard deviation equal 1 dB seems to be a significant esteem to separate di ff erent mechanical contributions. Further steps of this preliminary analysis will involve other o ffi ces to increase the database and find more robust insights about the dynamical behaviour of the mechanical sources in o ffi ces.

REFERENCES

[1] Marc Green and Damian Murphy. Environmental sound monitoring using machine learning on mobile devices. Applied Acoustics , 159:107041, 2020.

SPL (dB) &. 8 8 8 Office A Day 1 - Spectra [= = = Mechanical 7 - Gun |—echanical 2 - Gm |- = ~Mechanical 1 = KM [Mechanical 2 -KM 10 125-250-600 +: 1000 ~—-2000 Octave band (Hz)

Office A Day 2 - Spectra 60 50 B40 5 20 [= = = Mechanical? - Gwin 20 }|——echanieal 2 - GMM |- = ~Mechanical1 -KM [Mechanical 2- KM. 10 125 250 600-1000 -~—«2000 Octave band (Hz) 4000

Office B Day 1 - Spectra Mechanical 1-GMM| 20} | Mechanical 2 -GMM Mechanical 1 = KM \——ectanical 2- KM 125-250 «= 800» 1000 2000-4000 Octave band (Hz)

SPL (dB) Office B Day 2 - Spectra 8 & 8 Mechanical 1 GMM ‘Mechanical 2 - GMM| ‘=Mechanical 1 - KM cal 2- KM. 125 250 500 1000-2000 Octave band (Hz) 4000

Figure 1: Spectra obtained from the cluster analysis carried out on the detected mechanical source of both algorithms, GMM and KM. Results are shown per each day and each o ffi ce under study in the octave bands from 125 to 4000 Hz. Blue and red lines represent respectively GMM and KM. Dashed and solid lines represent respectively Mech1 and Mech2.

SPL (dB) Office C Day 1 - Spectra 60 60 40 30 Mechanical 1- GMI ‘Mechanical 2 - GMM| ‘Mechanical 1 = KM [Mechanical 2- KM 5 10 125-250 ©6500 -—«1000-~=«2000 = «4000 Octave band (Hz)

Office C Day 2 - Spectra 60 50 a40 & 30 Mechanical 1 - GN . 20||—ntechanica 2 - Gna Mechanical 1 ~ KM = Mechanical 2 - KM 4 10 125 250-500-1000» 2000 4000 Octave band (Hz)

Figure 2: Correlation between standard deviation SD and average intra-cluster distance AICD. Blue and orange points represent the mechanical sources indicated as Mech1 and Mech2. Grey points represent the speech sources indicated as Human.

[2] Dario D’Orazio, Domenico De Salvio, Laura Anderlucci, and Massimo Garai. Measuring the speech level and the student activity in lecture halls: Visual-vs blind-segmentation methods. Applied Acoustics , 169:107448, 2020. [3] Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning , volume 4. Springer, 2006. [4] Domenico De Salvio, Dario D’Orazio, and Massimo Garai. Unsupervised analysis of background noise sources in active o ffi ces. The Journal of the Acoustical Society of America , 149(6):4049–4060, 2021. [5] Douglas A Reynolds. Gaussian mixture models. Encyclopedia of biometrics , 741(659-663), 2009. [6] Geo ff rey J McLachlan, Sharon X Lee, and Suren I Rathnayake. Finite mixture models. Annual review of statistics and its application , 6:355–378, 2019. [7] Arthur P Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society: Series B (Methodological) , 39(1):1–22, 1977. [8] Stuart Lloyd. Least squares quantization in pcm. IEEE transactions on information theory , 28(2):129–137, 1982. [9] Hirotugu Akaike. A new look at the statistical model identification. IEEE transactions on automatic control , 19(6):716–723, 1974. [10] Peter J Rousseeuw. Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of computational and applied mathematics , 20:53–65, 1987. [11] Pasquale Bottalico and Arianna Astolfi. Investigations into vocal doses and parameters pertaining to primary school teachers in classrooms. The Journal of the Acoustical Society of America , 131(4):2817–2827, 2012. [12] David JC MacKay, David JC Mac Kay, et al. Information theory, inference and learning algorithms . Cambridge university press, 2003.

SD - AICD (Cluster of Cluster) 7 6 y = 0.7071x + 0.1279 R? = 0.9631 5 4 @ = Mechi 3 @ = Mech2 a g y=0.6117x + 1.1821 © Human 2 = 3 R= 0:2396 seseeeees Linear (Mech1) seeteeeee Linear (Mech2) 2 | 9° eG ge GF eee Linear (Human) y = 0.2954x + 1.4627 R? = 0.2386 1 0 0 1 2 3 4 5 6 7 8 9 10 SD (dB)