analysis of uncertainty in building acoustic predictions using monte-carlo methods

4aAAc17: Analysis of uncertainty in

building acoustic predictions using Monte-

Carlo methods

Ralph T. Muehleisen

Civil, Architectural, and Environmental Engineering

Director of the Miller Acoustics Lab

Illinois Institute of Technology

[email protected]

Overview

• Quick Review of Analytic Method and Applicability

• Quick Review of Monte Carlo Numerical Method

• Discussion of Assignment of Probability

Distribution Functions (PDF) to input variables

• Application to Building Acoustics

– Calculation of STC/Rw

4aAAc17 Uncertainty in building acoustics using Monte Carlo Methods, Muehleisen

Methods of Prediction Uncertainty Analysis

The uncertainty analysis of computer predictions

is usually done in one of two ways:

• Analytic Methods

– Well described in the ISO Guide to the Expression of

Uncertainty in Models (GUM)

• Numerical (statistical) Methods

– One popular method described in Supplement 1 to

GUM


Analytical Uncertainty Analysis Methods

For a function y=f (x1,x2,…,xn), the GUM says the

uncertainty in y, u2(y), given the uncertainties in x,

u2(xi), and u(xi, xj) will be:

This equation is called the law of propagation of

uncertainty, is based on a 1st order Taylor

expansion of f(x). This is essentially a

linearization of f(x) for error propagation


21

2 2

1 1 1

( ) ( ) 2 ( , )N N N

i i j

i i j ii i i

f f fu y u x u x x

x x x

Applicability of Analytic Equation

• The partial derivatives can be obtained

– Difficult if y is determined by numerical solution such

as FEM, BEM, or CFD analysis

• The uncertainties in xi are not too large

• The uncertainties in xi are of similar value

• Uncertainties in xi are uncorrelated

• The probability distributions (PDF) of the xi are

symmetric (not necessarily Gaussian though)

• The PDF of y will be close to Gaussian or a t-

distribution


Simple Example where Analytic Method Fails

• Suppose y=x12

• The PDF of y is not close to Gaussian or a t-

distribution so analytical method does not apply

– Cannot use standard analytic equation to estimate

uncertainty bounds on y given known uncertainty in x


Numerical Analysis Methods

If the analytical requirements do not hold, or if

finding the partial derivatives is difficult, numerical

methods can be used

In these cases, the full probability distribution can

be propagated through the model using

numerical methods. One such numerical method

is the Monte Carlo Method (MCM) and its

implementation is described in Supplement 1 to

the GUM.


Propagation of Distributions

• Probability distribution functions are obtained (or

assigned) for each input and numerical methods

are used to find the probability distribution of the

model output


Input Probability Distribution Functions (PDF)

• Selecting the proper PDF can be important– Selecting something with correct range may be more

important than getting the exact shape right

• Equipment and product manufacturers usually give accuracy of some sensor or measurement as x x with no indication of the uncertainty PDF.– Users often incorrectly take this to mean the quantity must

be bounded between x x and therefore assign a uniform PDF between x- x and x+ x

– The meaning of the x usually the 95% confidence intervals, not the bounds and values in the middle are more likely than at the ends. Use a Gaussian with a standard deviation equal to ½ of the x values


Gaussian (Normal) Distribution

• This is the most important distribution and is the distribution expected when the error of many sources gets added together– Most sensor uncertainties should be

modeled as Gaussian

• Since many times the limits are physically bounded, a truncated Gaussian distribution might be used

• Generate a Gaussian with a standard deviation equal to ½ the 95% confidence interval (since 95% is a 2 result)– e.g. if the temp sensor is accurate to 1.5 K, generate a Gaussian

distribution with the mean at the input temp with =0.75


Beta Distribution

The Beta Distribution is always bounded between 0 and 1 but can take on many different shapes depending upon its two input parameters, one of which is the uniform distribution

• Absorption coefficient measurements which can

have large, but bounded error can be effectively

modeled by a beta distribution with the

parameters selected to give a narrow peak at the

measured absorption value


Rayleigh and Log-Normal Distributions

In some situations the probability should be asymmetric without a true bound on one side– In this case an asymmetric

distribution like a Rayleigh or Log-Normal Distribution is good

• Things like construction defects will always reduce the TL so modeling a construction defect with such a Rayleigh or Log-Normal distribution can be very effective


Application to Building Acoustics

• Estimation of Reverberation Time (RT or T60)– Sabine and Eyring equations can be easily differentiated by

hand, but the uncertainty in the inputs are not necessarily small and the PDFs of the inputs are not necessarily symmetric

• Estimation of STC or Rw– Assigning of STC/Rw is a very nonlinear procedure and an

analytic derivative cannot be computed

– Uncertainty in TL is asymmetric if the possibility of construction defects which drop the TL are to be included in the probability

– If defects are to be measured though reductions in TL will happen at various frequencies so some correlation between the inputs is a good idea


Application to STC/Rw

• Computation of STC or Rw for a construction first

entails measurement of TL and then a very non-

linear procedure to get STC/Rw.

• In the ANSI E-90 TL measurement standard, the

maximum 95% confidence interval, TL, for a TL

testing lab must be less than


1/3 Octave Band Freq Confidence Interval TL

80 6 dB

100 4 dB

125, 160 3 dB

200, 250 2 dB

315-4000 1 dB

STC/Rw Cont

• There is limited information about the interlab

repeatability and confidence intervals for TL

measurements, so use the intralab requirements

• Assign a Normal PDF with = TL/2, truncated

at 3 to add to the TL at each frequency to

represent measurement error in TL

• Include a second PDF to model the uncertainty in

typical construction. This PDF should always

reduce the TL since the in-field TL is always less

than the lab TL and construction defects always

reduce TL


Using a Rayleigh Distribution for Defects

• Rayleigh distribution is

bounded on one side and with

an extending tail on the other.

– This can be used to model the

reduction in TL from construction

defects

• It is named after Rayleigh so it

must be the right one to use for

acoustics!

– A Log-Normal PDF might work

well too.


Example: Constructed STC of a steel stud wall

• As an example consider estimating the STC/Rw

value of a common steel stud wall with STC 39

• Look up TL from a manufacturer or text

• Assign a PDF at each frequency of the input with

a long tail toward lower TL to simulate

construction defects. In this case I chose a

Rayleigh distribution that has a mean value of 3

dB. This leads to a most likely value of

constructed STC to be about 36


MATLAB Code


% dTL is the 95% confidence intervals for the measurements for E90 meas In dB

dTL=[6 4 3 3 2 2 1 1 1 1 1 1 1 1 1 1];

sigma = dTL/2; % the 95% confidence interval is 2 standard deviations

for I=1:Nmcm

% model construction defects with Rayleigh distribution with beta=3

% this will give an average field TL about 3-4 dB below the Lab Value

Defect=random('rayleigh',3);

% find the monte carlo value of TL by adding measurement variance at

% each frequency, subtracting the common TL defect , and rounding

TLin=round(tnormrand(TL,sigma,3*sigma)-Defect);

STCMCM(I)=stc_calc(TLin); % compute STC from TL

RwMCM(I)=Rw_calc(TLin); % compute Rw from TL

end


22 24 26 28 30 32 34 36 38 400

0.5

1

1.5

2x 10

4

1 1 5 20 63 194 5091300

2889

5443

9508

13858

17777

19470

16799

9843

2259

61

mode=36 mean=35.195 variance=4.2297 N=100000

STC Histogram using Normal + Rayleigh Input PDFN

um

ber

of O

ccura

nces

STC

22 24 26 28 30 32 34 36 38 400

0.5

1

1.5

2x 10

4

1 1 3 14 57 180 4771246

2778

5309

9418

13805

17857

19620

16985

9926

2262

61

mode=36 mean=35.221 variance=4.1595 N=100000

Rw Histogram using Normal + Rayleigh Input PDF

Num

ber

of O

ccura

nces

Rw

What can we learn from this example?

Assuming the PDF assignments are correct:

• The most likely value of in-field STC or Rw is 36

(The lab values are both 39)

• The variance for both STC and Rw exceeds 4

• The STC and Rw distributions are asymmetric

• More than 1% of the time we could expect an

STC/RW as low as 30

– This is an example of an important thing to know and

shows the utility of MCM for architectural acoustics

calculations


Conclusions about MCM

• When errors are large or asymmetric in the inputs to a model, Monte Carlo Methods are one way of finding the range and statistics of expected values in the output

• For nonlinear models such as STC and Rw the MCM can be used with Normal and Rayleigh distributions on inputs to simulate measurement error and construction defects at the same time.

• More research should be done to be able to characterize the input PDFs better, especially if we want to include construction defects


analysis of uncertainty in building acoustic predictions using monte-carlo methods

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