Review of the role of uncertainties in room acoustics

Transcription

1 Review of the role of uncertainties in room acoustics Ralph T. Muehleisen, Ph.D. PE, FASA, INCE Board Certified Principal Building Scientist and BEDTR Technical Lead Division of Decision and Information Sciences Argonne National Laboratory BEDTR Better Decisions + Better Technology = Better Buildings Building Energy Decision and Technology Research Program

2 Outline What is Uncertainty Types of Uncertainties How to Get Started Steps of Calculating Uncertainty Characterizing Input Uncertainty Recent Applications of Uncertainty in Architectural Acoustics 2

3 What is Uncertainty? Uncertainty = Lack of Knowledge When modeling, uncertainty is reflected in Lack of knowledge of inputs to model Uncertainty in model parameters Lack of knowledge of knowledge of true model uncertainty in model itself When measuring, uncertainty is reflected in Interfering signals Background noises Measurement instrumentation Finite precision, equipment noise, drift, and offsets 3

4 Random and Systematic Uncertainties Random Uncertainties are the result of stochastic fluctuations in the system or in background interference. Not Reproducible Often Unclear Causes Results in Poor Precision Generally Uncorrelated With Each Other and True Value Cannot Always Be Reduced Systematic uncertainties are associated with the nature of the measurement apparatus, choice of model used, assumptions made by modeler or experimenter Reproducible Definite Causes Results in Poor Accuracy Generally Correlated Measurement to Measurement Can Usually Be Reduced 4

5 Understanding Random and Systematic Uncertainty Large Random Uncertainty Large Systematic Uncertainty Large Random and Systematic Uncertainty Small Random and Systematic Uncertainty 5

6 A Few Uncertainties in Room Acoustic Modeling Simplified models make analysis and prediction tractable but create uncertainty Ignore diffraction, refraction Assume pure diffuse or specular reflection Ignore sound structure interaction Use simplified geometry 6

7 Parametric Sweeps Uncertainty Analysis The most common form of uncertainty analysis is parametric sweeps, where inputs are varied over some range and the output range is determined I have only one thing to say JUST DON T DO IT If you are going to go through the effort of varying parameters and doing multiple computations, do a proper uncertainty analysis because then you will know you did it right 7

8 How Can I Get Started in Uncertainty Analysis? Use the GUM Luke! JCGM 100:2008, Guide to the Expression of Uncertainty In Measurements (GUM), is an ISO standard methodology for estimating measurement uncertainties and propagating uncertainties through formulae The supplemental guides are very important! They discuss more advanced topics like strong non-linearities, large parameter uncertainty and mutually dependent (correlated) uncertainty between variables The GUM is the basis of uncertainty estimation for many measurement standards 8

9 How Can I Include Uncertainty in My Analysis? 1. Decide what uncertainty is important and what is not Include uncertainty for parameters with high influence and high uncertainty, perhaps ignore others 2. Characterize the uncertainty of the selected parameters or the model itself 3. Propagate uncertainty through the model Analytic methods if model is very simple and uncertainty is small Numerical methods for other cases 4. Statistically analyze results to make predictions Generate empirical probability and cumulative density functions (PDF and CDF) Get standard statistical measures (mean, median, standard deviation, skew) 9

10 1. Deciding What is Important Sensitivity Analysis should be performed that combines both the size of the uncertainty in parameters with the influence that the parameter has on output For analytic models with little interaction of input parameters we can use partial derivatives to estimate the parameter influence and define the local sensitivity as S local 2 i = σ y xi x i xi =x i The most robust way to decide what is important is to do a global sensitivity analysis that includes the entire parameter space and nonlinear interactions This works for linear and non-linear models with either small or large uncertainties in individual parameters and with strong parameter interaction The first order sensitivity index, S i, and total sensitivity, S ti, are given by S i = V E y x i V y S ti = 1 V E y x i V y where E and V are the mean and variance operators and E[y x i ] is the mean of y given variation in x i and E[y x i ] is the mean of y given variation in all parameters but x i 10

11 3. Propagation of Uncertainty For models with analytic equations and small independent random uncertainty one can use the analytic equation for propagation of uncertainties u(x i ) and covariances u(x i, x j ) (if inputs are correlated) y = f x 1, x 2, x n with known covariances u(x i, x j ) u 2 y = n i=1 n j=1 f f x i x j u 2 x i, x j This method is only accurate if the u(x i, x j ) are small and a simple variance characterizes the uncertainty well The distribution of uncertainty in y is assumed to be Gaussian with a mean of y = f x 1, x 2, x n and variance u(y) 11

12 3. Propagation of Uncertainty For other cases one will usually use numerical methods to propagate uncertainty (Monte Carlo, Important Sampling) x 1 x 2 x n f x 1, x 2, x n y This method requires one to define uncertainty probability distribution functions (PDF) for the model inputs The propagation can be very computationally intensive for many inputs and hard to compute models 12

13 Implementing Monte Carlo: It s All in the Details Careful choice of input uncertainty sampling can make all the difference between fast and slow convergence. Use Latin Hypercube Sampling or Quasirandom numbers whenever practical they almost always converge much faster These methods break CDF into equal probability regions and choose samples from each region rather than completely randomly Example of PDF generated from sampling a Triangle Distribution 300 times 13

14 P(x) F(X) 4. Analyzing the Results Once we have the PDF, P x, we can do a lot of other things including: Calculate the Cumulative Density Function, F X, from the PDF, P x X F x = P X dx x= Get simple statistics (mean, standard deviation, median, mode, etc) x = xp x dx, σ 2 = x x 2 P(x)dx median = F 0.5, mode = max P x Determine confidence intervals for true risk analysis The 5% 95% interval is X 1 to X 2 where F X 1 = 0.05 and F X 2 =

15 2. Characterizing Uncertainty This is often the most difficult part of the whole process. How do we estimate the uncertainty of a parameter? Expert Judgment Most common method but subject to wide variations, individual biases, mistakes Analysis of many independent measurements of the same quantity Results of interlab material testing Uncertainty bounds on measurement standards Check the measurement standard see what info it gives about uncertainty and repeatability Physical Limitations Mother nature has thankfully limited many quantities. Use fundamental physics (conservation of mass and energy) to help you put bounds on some quantities Information Theory Use methods like Maximum Entropy to develop conservative input PDFs from minimal information 15

16 Some Recent Applications Of Uncertainty Vorlander investigated the effect of audience and wall absorption uncertainty on RT, G, and C80 Predictions He ran only 20 Monte Carlo runs so the sampling from the input PDF and resulting output PDFs are fairly ragged Note: Use of Latin Hypercube sampling probably would have improved convergence Vorlander, 2013, JASA 133 (3),

17 More Recent Applications Reynders utilized maximum entropy method to determine PDF for loss factors used in Transmission Loss predictions I think this is a very important new technique for acoustics Input PDF for Block Walls Loss Factor Output PDF for Block Wall TL Reynders, 2014, JASA 135 (4),

18 So Who Do You Wanna Be? Mr. Slick? Midband RT of 0.8 seconds and STI of 0.8. I Guarantee It! Ms. Thoughtful? Midband RT between 0.9 and 0.95 seconds, STI between 0.65 and 0.9 with 95% certainty. BEDTR: Better Decisions + Better Technology = Better Buildings 18

19 Thank you. Questions? Ralph Muehleisen