sampling based on sobol sequences for monte carlo techniques applied to building ... · ·...

SAMPLING BASED ON SOBOL′ SEQUENCES FOR MONTE CARLO TECHNIQUESAPPLIED TO BUILDING SIMULATIONS

Sebastian Burhenne1,∗, Dirk Jacob1, and Gregor P. Henze2

1Fraunhofer Institute for Solar Energy Systems, Freiburg, Germany2University of Colorado, Boulder, USA

∗Corresponding author. E-mail address: [email protected]

ABSTRACTMonte Carlo (MC) techniques are commonly used toperform uncertainty and sensitivity analyses. A key el-ement of MC methods is the sampling of input param-eters for the simulation, where the goal is to explorethe entire input space with a reasonable sample size(N ). The sample size determines the computationalcost of the analysis since N is equal to the requirednumber of simulation runs. Quasi-random (QR) se-quences such as the Sobol′ sequences are designed togenerate a sample that is uniformly distributed overthe unit hypercube. In this paper, sampling basedon Sobol′ sequences is compared with other standardsampling procedures with respect to typical buildingsimulation applications. The work revealed that forthe most of the analyzed aspects the sampling basedon Sobol′ sequences performs better than the other in-vestigated sampling techniques.

INTRODUCTIONDue to the substantial influence of uncertain param-eters on building performance, uncertainty and sensi-tivity analyses will become an important part of thebuilding performance simulation and the design pro-cess of low energy buildings. In an uncertainty analy-sis the modeler quantifies the uncertainty in the modeloutput given the uncertainty in the model input. Thisgoes often hand in hand with a sensitivity analysiswhere the aim is to apportion the uncertainty in themodel output to the uncertainty in the model input(Saltelli et al., 2008, pg. 1). Both analyses give in-sights to the driving parameters or variables of themodel and the model structure.Several examples of Monte Carlo based uncertaintyand sensitivity analyses applied to building sim-ulations exist (Lomas and Eppel, 1992; de Wit,2003; Mara and Tarantola, 2008; Macdonald, 2009;Burhenne et al., 2010a,b). Compared to samplingmethods already applied in various building perfor-mance simulation applications (e.g., random sam-pling, stratified sampling and Latin Hypercube sam-pling) the sampling based on Solbol′ quasi-random se-quences is expected to be more effective in exploringthe input parameter space. This space is a unit hy-percube (Ω) with k dimensions. Exploring the unithypercube with a sufficient sample density becomes

numerically expensive when the number of analyzedparameters (k) is large as its volume increases dramat-ically with k.In this paper different sampling techniques are ana-lyzed with respect to the estimator of the mean of theresult and how quick this estimator converges to thetrue mean (i.e., expected value) with respect to thesample size. Another analyzed measure of the perfor-mance of the sampling strategy is its robustness. Ro-bustness can be measured via the standard error of theestimated mean. This is done using multiple MC sim-ulations and analyzing their results. A way to visualizethe robustness is to compare the empirical cumulateddensity functions (CDFs) of several repetitions of theMC simulation (Helton and Davis, 2003).Macdonald analyzed the performance of random sam-pling, stratified sampling and Latin hypercube sam-pling applied to the evaluation of a building model(Macdonald, 2009). This paper extends his work byapplying a sampling technique based on Sobol′ se-quences to a building simulation model. Furthermore,models with different properties than the model usedby Macdonald are analyzed. As a test case, a simplemathematical model and a typical building simulationmodel are used.

SAMPLING TECHNIQUESSampling is the process of exploring the domain of in-terest (e.g., x1). That can be done randomly, wherethe random numbers are independent realizations of arandom variable (Sobol’ and Levitan, 1999). In com-puter experiments, pseudo-random numbers or quasi-random numbers are used. These numbers are gener-ated using an algorithm or a sequence of numbers thatfulfill requirements as if they were true random num-bers. The properties of the samples can be analyzed bythe use of statistical tests (Sobol’ and Levitan, 1999).In the context of this paper the language and environ-ment R for statistical computing is used to generate thesamples (R Development Core Team, 2010).

Random samplingA random sample can be generated by a pseudo-random number generator which is available in manysoftware packages. A sample is randomly distributedin a defined interval according to some distribution

Proceedings of Building Simulation 2011: 12th Conference of International Building Performance Simulation Association, Sydney, 14-16 November.

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(e.g., uniform distribution in the interval [0,1], henceXi ∼ U(0, 1) with i = 1, 2, ..., N ). For small samplesizes (N ), the samples can contain clusters and gaps asshown in Figure 1 on line a. Regions with gaps are nottaken into account in the statistical analyses for anyuncertainty or sensitivity analysis and function valuesin the regions with clusters are overemphasized in thecalculations (Saltelli et al., 2008, pg. 83). The sampleon line b was drawn using the same pseudo-randomnumber generator but shows a better coverage of theinterval.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b

a

Figure 1: Two examples of sampling with a pseudo-random number generator.

The unbiased mean and variance of the model outputcan be calculated by the following equations (Saltelliet al., 2008, pg. 59):

Y =1N

N∑i=1

yi (1)

Var(Y ) =1

N − 1

N∑i=1

(yi − Y )2. (2)

The mean and the variance resulting from the sampleand calculated with these two equations are uncertain.Based on the central limit theorem, the uncertainty inthe estimate of the mean can be quantified with thestandard error

SE(Y ) =

√Var(Y )N

. (3)

This equation shows that the uncertainty decreasesslowly when N increases since it depends on thesquare root of N .In the following example a three-dimensional param-eter space is used to illustrate the properties of thesampling methods. However, the reader should keepin mind that exploring the parameter space becomesharder as the number of analyzed parameters (k) in-creases. The sampling is performed according to a uni-form distribution in the interval [0,1]. Figure 2 showsa three-dimensional plot of the parameter space x1, x2

and x3 with N = 128. The number was chosen be-cause of the properties of the sampling based on Sobol′

sequences which will be explained later. For pseudo-random sampling anyN can be chosen but for the sakeof comparability N = 128 was used.With a three-dimensional plot it is a difficult task tocheck if the parameter space is explored in a proper

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0.00.20.40.60.81.0

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x 2, r

ando

mx 3, r

ando

m

Figure 2: Three-dimensional plot of the pseudo-randomly sampled points in the parameter space x1,x2 and x3. The color of the points varies from redto black depending on the value of x2. That colorvariation allows a better interpretation of the three-dimensional plot.

way. For this reason Figure 3 shows the variablesx1, x2 and x3 plotted against each other in two-dimensional plots. The plots for random samplingshow clusters and gaps.

Stratified samplingFigure 1 showed that a random sample may containclusters and gaps. Using a stratified sampling tech-nique can solve that problem. In a scheme which ap-plies stratified sampling, the domain of xi is dividedinto subintervals. Each of the subintervals contains thesame number of sample points. These points are sam-pled randomly within each subinterval using a pseudo-random number generator. If one compares Figure 1with Figure 4 it is obvious that the stratified samplingtechnique ensures the avoidance of clusters and gapsat a certain resolution.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b

a

Figure 4: Two examples of stratified sampling. Theposition of the points within each subinterval is chosenrandomly.

The mean and variance can be calculated in the sameway as for pseudo-random sampling (see Equations 1and 2).In multivariate stratified sampling the same techniqueis applied. Figure 5 shows a two-dimensional param-eter space with a stratified sampling with 10 strata foreach parameter. That results in 100 cells where onepoint is in each cell. For a given resolution stratifiedsampling results in less uncertain mean and varianceestimates than pseudo-random sampling (Saltelli et al.,


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Figure 3: Three sampled parameters plotted against each other in pairs using different sampling techniques. Leftto right: x1 vs. x2, x1 vs. x3, and x2 vs. x3. Top to bottom: Pseudo-random, stratified sampling, Latin hypercubesampling and sampling based on Sobol ′ sequences.

2008, pg. 80). Note that the required sample size fordoing this design is

N = mk. (4)

Hence, for a stratified sampling with 10 strata (m) and5 parameters (k) a sample size (N ) of 105 = 100, 000is required.In Figure 3 the stratified sampling shows gaps andclusters like the random sampling. The reason is thata sample size of 125 (128 like for the other techniquescould not be used because of the property described byEquation 4) leads to 5 strata which is not sufficient toavoid visible clusters and gaps.

Latin hypercube samplingLatin hypercube sampling (LHS) is a particular kindof stratified sampling. One feature is that each param-eter is stratified over s > 2 intervals (levels) wherethe same number of points are located in each interval(Saltelli et al., 2008, pg. 76). An example of a Latin

hypercube sampling with two parameters, 10 intervalsand a sample size (N ) of 10 is shown in Figure 6. InFigure 6 the unique property of the sampled points of aLHS is easily visible: each sampled point is associatedwith one of the 10 rows and one of the 10 columns.In Figure 3 the Latin hypercube sampling shows gapsand clusters like the random sampling because thesample size is too small to generate a sample with thesame density across the parameter space.The mean and variance can be calculated in the sameway as for pseudo-random sampling (see Equations 1and 2). The LH sampling was implemented using theR package lhs (Carnell, 2009).

Sampling based on Sobol′ sequencesThe investigated method is based on Sobol′ se-quences. Sobol′ sequences belong to the family ofquasi-random sequences which are designed to gen-erate samples of multiple parameters as uniformly aspossible over the multi-dimensional parameter space(Saltelli et al., 2010). The biggest difference to


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Figure 5: Scatterplot of a two-dimensional stratifiedsampling. The position of the points within each cellis chosen randomly.

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x 2, L

HS

Figure 6: Scatterplot of a LHS: x1 vs. x2. The col-ored dotted lines indicate to which intervals the sam-pled point belongs. For illustration reasons this is justplotted for two points. However, each point has thatproperty.

pseudo-random numbers is that the sample values arechosen under consideration of the previously sampledpoints and thus avoiding the occurrence of clusters andgaps.One criterion for assessing the performance of a goodsampling method is the discrepancy in the explorationof the multi-dimensional parameter space. The dis-crepancy metric was defined by Ilya M. Sobol′ and isthe maximum deviation between the theoretical den-sity dt = 1/N and the point density di in an arbitraryhyper-parallelepiped (Pi) within the parameter space(hypercube) (Saltelli et al., 2010). The sampling basedon Sobol′ sequences is designed to generate sampleswith low discrepancy.Figure 3 shows that the points produced by a sam-pling based on Sobol′ sequences are more evenly dis-

tributed than the points produced with the other sam-pling techniques. As a result, the discrepancy in theexploration of the multi-dimensional parameter spaceis lower compared to the other sampling techniques.The mean and variance can be calculated in the sameway as for the other sampling techniques (see Equa-tions 1 and 2).One result of the low discrepancy is that the estimatedmean of a function Y(x1, x2, ..., xk) evaluated at thepoints of a sampled input matrix

Min =

x(1)1 x

(1)2 · · · x

(1)k

x(2)1 x

(2)2 · · · x

(2)k

......

. . ....

x(N−1)1 x

(N−1)2 · · · x

(N−1)k

x(N)1 x

(N)2 · · · x

(N)k

will converge quicker to the true mean than in the caseof pseudo-random sampling (Saltelli et al., 2008, pg.83). How quick depends on the model structure andwill be analyzed later. The properties of the Sobol′

sequences require a sample size of

N = 2j (5)

where j ∈ N+. The sampling based on Sobol′ se-quences was implemented using the R package rand-toolbox. For the repetitions of the MC simulation thesampling was done using the Owen type of scramblingwith a random seed (Dutang, 2009).

TEST MODELSSimple mathematical modelThe first model used in this paper is a simple math-ematical model. It has the advantage that analyticalsolutions are available which leads to a straight for-ward analysis. The model was introduced by StefanoTarantola (Tarantola, 2010).The 2-dimensional model equation is

f(x1, x2) = 4x21 + 3x2. (6)

with the input distributions

x1, x2 ∼ U(−12,

12

).

Figure 7 shows a contour plot of the model.The expected value (E(f(x1, x2))) is

E(f(x1, x2)) =∫ 0.5

−0.5

∫ 0.5

−0.5

(4x21 + 3x2)dx1dx2

=13. (7)

The availability of the analytical solution makes it pos-sible to compare the mean of the function which is anestimator of the expected value to the true expectedvalue.


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-0.4 -0.2 0.0 0.2 0.4

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x 2

Figure 7: Contour plot of the simple mathematicalmodel.

Building simulation modelThe building simulated is a typical German buildingwith a net floor area of 436 m2. The model wasintroduced in (Burhenne and Jacob, 2008; Burhenneet al., 2010b). There is no air-conditioning availablein the building and the heat is emitted by radiatorsequipped with thermostatic control valves. The build-ing is equipped with sensors (outside temperature, heatmeter, room temperatures etc.) to allow for a valida-tion of the simulation. The main building parametersare shown in Table 1 and Figure 8 is a 3D-plan of thebuilding.

Table 1: Building parameters.

parameter value unitAV (area to volume ratio) 0.38 m2

m3

U -value (mean U-value) 0.53 Wm2K

Awin (total window area) 106 m2

Figure 8: 3D-plan of the building.

Monte Carlo (MC) simulations require many simula-tion runs and are therefore computationally expensive.Especially the tests with repetitions of the analysis re-quire many simulations but are necessary to evaluatethe performance of the different sampling techniques.In this paper more than 100,000 one-year simulations

were executed to perform the analysis (100 repetitionsfor 4 techniques with a sample size of 256 for eachMC simulation). The simulations were performed inparallel on a Linux-based computer cluster with 96processor cores. However, in order to reduce the com-puting time, it is desirable to use an appropriate sim-ple model for thermal building simulation. As a zonemodel the simple hourly method (SHM) according tothe ISO 13790 standard is used (ISO 13790, 2008).This zone model is based on five resistances and onecapacity. The model was calibrated for this building;it showed a good agreement with the measured roomtemperature and the heating demand of the building(Burhenne and Jacob, 2008). The object-oriented andequation-based modeling language Modelica is usedto describe the system (Elmqvist, 1997) and the simu-lations are conducted using the software Dymola 2012(Dassault Systemes AB, 2011).In actuality, the building is an office building heated bya gas boiler. For this analysis, however, it is assumedthat it is a residential building with 12 occupants. Asolar thermal collector with 25 m2 area and a 2,000liter storage tank are modeled. The collector modelis implemented in Modelica using a plug flow modeldescription (Isakson and Eriksson, 1994). The collec-tor flow rate is controlled by an on/off controller; thestorage tank is modeled as a simple one resistor / onecapacitor (R1-C1) network. The radiation processoris implemented according to an equation-based model(written in the modeling language Neutral Model For-mat; (Sahlin, 1996)) from the simulation software IDAICE (Sahlin et al., 2004). The solar thermal systemis designed for domestic hot water and space heating.When the heat from this solar thermal system is notsufficient, a gas boiler meets the remainder of the loadof the building. Further details on the model can befound in (Burhenne et al., 2010b). The primary modelresult is the annual solar fraction of the solar thermalsystem.It is assumed that the mass flow rate of the domes-tic hot water (m) and the air change rate (ACH) areuncertain. Furthermore, the number of people (occ)present at a particular time and the set point for theroom temperature (set) cannot be determined exactly.Therefore, these four values (m, ACH , occ, set) arevaried in the MC simulation. The schedule for the do-mestic hot water flow rates is generated with a programwhich was developed in the Solar Heating and Cool-ing Program of the International Energy Agency (IEA-SHC), Task 26: Solar Combisystems (Jordan and Va-jen, 2003). The air change rates, the number of oc-cupants and the room temperature set point are alsoimplemented using a schedule. The variation of theschedule values is implemented by multiplying a sam-pled scaling factor or adding a scaling summand. Thescaling parameters are listed in Table 2. Note that thevalues for the scaling summands for the number of oc-cupants and the set point are rounded (integers for the


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occupancy and values with one fractional digit for theset point).

Table 2: Distribution parameters.parameter distribution µ σ

m scaling factor normal 1 0.1ACH scaling factor normal 1 0.2occ scaling summand normal 0 2set scaling summand normal 0 1

RESULT ANALYSIS AND DISCUSSIONSimple mathematical modelFigure 9 shows a comparison of the different samplingtechniques used to compute the mean of the functionwith different sample sizes. The random samplingshows the worst convergence to the analytical meanof the function. However, the other three techniquesalready converged at a sample size of 64 and the Latinhypercube sampling shows the fastest convergence.

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0.5

0.6

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mean(f(x

1, x2))

Random samplingStratified samplingLatin hypercube samplingSampling based on Sobol' sequences

analytical mean

Figure 9: Comparison of the different sampling tech-niques applied to the evaluation of the simple mathe-matical model.

In this paper, cumulative distribution functions (CDFs)are used to visualize the robustness (i.e., stability)of results obtained by different sampling strategies.The sampling and model evaluation was repeated 100times for each sampling technique and the sample sizeof 256. Figure 10 shows the comparison of the esti-mated CDFs for the simple mathematical model. Onecan see that in this case the robustness of the stratifiedsampling and the sampling based on Sobol′ sequencesis the best followed by the Latin hypercube sampling.The CDFs constructed for the random sampling showthe most variability (i.e., the estimated mean producedby random sampling is most uncertain).

Building simulation modelFigure 11 compares the convergence to the mean (so-lar fraction) for the different sampling strategies forthe building simulation model. The black horizontalline is the mean value after a MC simulation with ran-dom sampling and a sample size of 25,600. Due to thislarge sample size this value can be taken as a refer-ence. It can be seen that the sampling based on Sobol′

sequences and the LH sampling converge very quickand produce reasonable results for very small samplesizes. However, at a sample size of 256 all samplingstrategies converged and produce comparable results.

0 100 200 300 400 500 600

0.18

0.19

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0.23

sample size

mea

n (s

olar

frac

tion)

Random samplingStratified samplingLatin hypercube samplingSampling based on Sobol' sequences

Figure 11: Comparison of the different sampling tech-niques applied to the evaluation of the building simu-lation model.

Figure 12 shows the comparison of the estimatedCDFs for the building simulation model. The modelevaluation was repeated 100 times for each samplingtechnique. The sampling based on Sobol′ sequenceshas the least variation followed by the Latin hypercubeand the stratified sampling. The CDFs constructed forthe random sampling show the most variability.

CONCLUSIONDifferent sampling strategies were analyzed with re-spect to the convergence of the mean estimate to thetrue expected value and their robustness. The perfor-mance of the techniques depends on the number of in-put parameters (k) and the properties of the analyzedmodel (e.g., nonlinearity). The Latin hypercube sam-pling and the sampling based on Sobol′ sequences hadthe fastest convergence of the mean estimates. More-over, the comparisons of the estimated CDFs showedthat the sampling based on Sobol′ sequences has theleast variations in the CDFs. Having less variationsproves that this sampling technique produces the mostrobust results. It is surmised that the higher the numberof analyzed input parameters, the better the samplingbased on Sobol′ sequences performs in comparisonto the other sampling strategies. It is recommended


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Figure 10: Comparison of estimated CDFs for the simple mathematical model with 100 repetitions.

to use either Latin hypercube sampling or samplingbased on Sobol′ sequences when Monte Carlo tech-niques are applied to building performance simula-tions and the sample size is limited because of com-putationally expensive models.

ACKNOWLEDGEMENTThis study was funded by the Reiner LemoineStiftung.

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Figure 12: Comparison of estimated CDFs for the building simulation model with 100 repetitions.

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