WeB12.3
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
Computationally Efficient Uncertainty Propagation and Reduction using the Stochastic Response Surface Method Sastry S. Isukapalli, Suhrid Balakrishnan and Panos G. Georgopoulos
Abstract— A computationally efficient means for propagation of uncertainty in computational models is provided by the Stochastic Response Surface Method (SRSM), which facilitates uncertainty analysis through the determination of statistically equivalent reduced models. SRSM expresses random outputs in terms of a “polynomial chaos expansion” of Hermite polynomials, and uses an efficient collocation scheme with regression to determine the coefficients of the expansion. This polynomial form then allows straightforward determination of statistics such as the mean and variance, and of first and second order sensitivity information. Further improvements in computational efficiency are achieved by coupling SRSM with automated source code differentiation tools, which produce code for partial derivatives with respect to model inputs and parameters. One such combination is the SRSM-ADIFOR, a combination of SRSM with the Automatic DIfferentiation of FORtran (ADIFOR). ADIFOR provides estimates of partial derivatives from a single model run, and this partial derivative information is used in the determination of the coefficients of the polynomial chaos expansions. Furthermore, SRSM can be used in conjunction with Bayesian methods such as Markov Chain Monte Carlo (MCMC) methods to reduce uncertainties by incorporating observational information in estimates of model parameters. An overview of the application of SRSM, SRSM-ADIFOR, and the combined SRSM and MCMC methods to complex mechanistic models describing environmental systems is presented here, and the advantages over traditional techniques are discussed.
I. INTRODUCTION Comprehensive uncertainty analyses of computational mechanistic models are essential, especially when these models are used in decision making. This is especially true for environmental applications, where complex computer programs are often used to model and analyze real phenomena. The systematic accounting of parametric uncertainty in such models is valuable, as this aids in the quantification of the degree of confidence in model predictions. Uncertainty almost always exists in such systems in both the observed data and in the parameters themselves, and its disregard may easily lead to ill-fitted or poorly calibrated models. A traditional “uncertainty analysis or error analysis” typically focuses on “data uncertainty”, and consists of (a) the characterization (“encoding”) of uncertainty in model parameters/inputs via their probability density functions (pdf s), and (b) the propagation of these pdf s through model equations to obtain the pdf s of selected output metrics. In S. Isukapalli (
[email protected]) and P. Georgopoulos (
[email protected]) are affiliated with the Environmental and Occupational Health Sciences Institute, a Joint Institute of UMDNJRW Johnson Medical School and Rutgers University, New Jersey. S. Balakrishnan (
[email protected]) is affiliated with the Department of Computer Science, Rutgers University, New Jersey
0-7803-8682-5/04/$20.00 ©2004 IEEE
this study, the focus is on computationally efficient methods for propagation of uncertainty in input data and parameters. Uncertainty analysis using traditional Monte Carlo methods, e.g. involving standard or Latin Hypercube sampling, for propagating uncertainty and developing probability densities of model outputs, may require performing a large number of model simulations. This, however, can be prohibitive in the case of 3D finite element or finite difference models, which typically reduce to the numerical solution of millions of simultaneous ordinary differential equations. Therefore, there is a need to study computationally efficient alternative techniques for uncertainty propagation. II. STOCHASTIC RESPONSE SURFACE METHOD (SRSM) The SRSM [1] can be viewed as an extension to the classical deterministic Response Surface Method (RSM) [2], [3]. The main difference between the RSM and the SRSM is that in the former the inputs are deterministic variables, whereas in the latter the inputs are random variables. Conceptually, the propagation of input uncertainty through a model using SRSM is accomplished as follows: (1) input uncertainties are expressed in terms of a set of random variables (input transformation), (2) a functional form is assumed for selected outputs or output metrics (approximation of output form), and (3) the parameters of the functional approximation are determined (parameter estimation). The approximation can then be used in deriving the statistics of model outputs. A. Transforming model inputs In the SRSM, the approach for transforming model inputs is based on the principle that random variables with wellbehaved (square-integrable) pdf s can be represented as functions of a set of independent random variables [4], [5], [6], [7]. The above integrability requirement is usually satisfied in physical systems of concern, so these transformations are generally applicable to a wide variety of physical models. Normal random variables have been extensively studied and their functions are typically well behaved and well tractable [4], [5], [6], [8]. Hence, they are used in the SRSM. In the SRSM, the inputs are represented as functions of independent identically distributed (iid) normal random variables {ξi }ni=1 , where n is the number of independent inputs, and each ξi has zero mean and unit variance. These random variables are referred to as “Standard Random Variables” (srvs). Once the inputs are expressed as functions
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TABLE I
als (3) with unknown coefficients, as follows:
R EPRESENTATION OF COMMON UNIVARIATE DISTRIBUTIONS AS FUNCTIONALS OF NORMAL RANDOM VARIABLES
yj = aj,0 + Distribution Type Uniform (a, b) Normal (µ,σ)
Transformation „ « √ 1 1 + erf(ξ/ 2) a + (b − a) 2 2 µ + σξ
Weibull (a)
exp(µ + σξ) !3 r 1 1 ab ξ +1− 9a 9a „ « √ 1 1 1 + erf(ξ/ 2) − log λ 2 2 y 1/a , where y is exponential(1)
Extreme Value
− log(y), where y is exponential(1)
Lognormal (µ,σ) Gamma (a,b) Exponential (λ)
+
n
n
aj,i1 Γ1 (ξi1 ) +
i1 =1 n n
n n
aj,i1 i2 Γ2 (ξi1 , ξi2 )
i1 =1 i2 =1
aj,i1 i2 i3 Γ3 (ξi1 , ξi2 , ξi3 ) + . . . ,
(2)
i1 =1 i2 =1 i3 =1
where Γp (ξi1 , . . . , ξip ) = (−1)p e 2 ξ ξ 1
of these srvs, the output metrics can also be represented as functions of the same set of srvs. The minimum number of srvs needed to represent the inputs is defined as the “number of degrees of freedom” in input uncertainty. Since model outputs are deterministic functions of model inputs, they have at most the same number of degrees of freedom in uncertainty. In simple cases, the uncertainty in the ith model input Xi , is expressed as a function of the ith srv, ξi . Some examples are shown in Table I. In more complex cases, such as when input distributions are provided in terms of empirical data, or when inputs are correlated, more elaborate transformations are used, as follows: 1) Transforming inputs represented by empirical cdfs: When the uncertainty in a model input x is specified by an empirical cumulative distribution function (cdf ), in the form Fx (x) = g(x), where g(x) can be an algebraic function, a look-up table, or a numerical subroutine. In such cases, a transformation can be used as follows: x = g −1 (z), where g(x) is the cdf of x, and √ 1 1 z = (1) + erf(ξ/ 2) 2 2 2) Transforming correlated inputs: When the random inputs are correlated in a simple manner, the interdependence of a set of n correlated random variables is often described by a covariance matrix, Σ. For a set of correlated variables with mean of the ith element µi and variance σi , the following steps based on [5] are used: ∗ ∗ • create new matrix Σ such that Σi,j = Σi,j /(σi σj ) • construct new vector Y , where Yi = (xi − µi )/σi T ∗ • construct vector Z = HY , where HH = Σ , and • express the model inputs as xi = µi + σi z i . Here, the problem reduces to finding a nonsingular matrix H such that HH T = Σ∗ . Since covariance matrices are positive semidefinite, one can always find a nonsingular H. B. Approximating model outputs Each model output is expressed as a series expansion in terms of the srvs as multidimensional Hermite polynomi-
T
p
1 T ∂ e− 2 ξ ξ (3) ∂ξi1 . . . ∂ξip
The symmetric version of (2), where Γp (ξi1 , . . . , ξip ) is assumed to be symmetric, is known as polynomial chaos expansion [6], where asymmetric terms of the same order with different permutations of parameters are lumped together. The polynomial chaos expansion is commonly used for approximating functions of normal random variables because the multidimensional Hermite polynomials form an orthogonal basis for iid normal random variables [6]. In the SRSM, symmetric terms in (2) are aggregated, and deterministic coefficients are assigned to each unique term. These coefficients are then estimated using model calculations at a set of sample points. C. Estimating coefficients of output approximation In the SRSM, a regression based technique is used to obtain estimates of the probability density of the variables under consideration. In this approach, a set of points is selected from regions of high probability; the model outputs at the points in this set is used for calculating the unknown coefficients. These points are selected using a simple heuristic technique. For each term of the series expansion, a “corresponding” basis point is selected. For example, the point corresponding to the constant is the origin; i.e., all the standard normal variables (ξi s) are set to value zero. For terms involving only one variable, the regression basis points are selected by setting all other ξi s to zero value, and by setting the corresponding variable equal to the roots of the next order Hermite polynomial. For terms involving two or more random variables, the values of the corresponding variables are set to the values of the roots of the higher order polynomial, and so on. If more points “corresponding” to a set of terms are available than needed, the points which are closer to the origin are preferred, as they typically fall in regions of higher probability. Further, when there is still an unresolved choice, the regression basis points are chosen so that the overall distribution of these points is more symmetric with respect to the origin. The advantage of this method is that the behavior of the model is captured reasonably well at points corresponding to regions of high probability. Further, the selection of a higher number of points than the minimum required leads to additional stability in the estimation of parameters and corresponding probability densities.
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D. Evaluating convergence of the approximation
Generate a set of random inputs
Once the coefficients of the polynomial chaos expansion are obtained using one of the methods described above, the convergence of the approximation is determined through comparison with the results from a higher order approximation. The next order polynomial chaos expansion is used, and the process for the estimation of unknown coefficients is repeated. If the estimates of pdf s of output metrics agree closely, the expansion is assumed to have converged, and the higher order approximation is used to calculate the pdf s of output metrics. If the estimates differ significantly, yet another series approximation, of the next order, is used, and the entire process is repeated until convergence is reached.
Input Distributions
Select a set of srvs and transform inputs in terms of srvs
Original Model
Output distributions
Wrap model with srvs as inputs
Express outputs as series in srvs with unknown coefficients
ADIFOR application
Generate a set of regression points
Derivative code (w.r.t. srvs)
Estimate coefficients of output approximation
E. Obtaining statistics of the model outputs The output representations in terms of srvs can be directly used in the determination of the statistical properties of the outputs, such as the individual moments, joint moments, correlations between the outputs as well as correlations between inputs and outputs. Although this can be done analytically, it is computationally inexpensive to perform this step numerically by generating a large number of random samples of (ξ1 , . . . , ξn ). Corresponding values for model inputs at these points are calculated using the input transformations discussed in Sec II-A. Outputs at these points are calculated through the approximation in (2). All statistical properties of model inputs and outputs are then calculated from these samples. III. USING SRSM WITH SUPPLEMENTARY MODEL INFORMATION Typically, running a computational model yields the model outputs for a combination of model inputs. However, there are computational tools that can be used in obtaining the partial derivatives of model outputs with respect to model inputs at any point from a single model run. One such technique is known as automatic differentiation, which involves direct manipulation of a model code to generate a corresponding “derivative calculating code.” The resulting code can then be compiled in a standard manner, and used to calculate derivatives in the model, along with the model outputs. Given the source code, and the information about the dependent and independent variables of interest, the automatic differentiation tools can generate the corresponding derivative information. The main advantage of the automated differentiation method is that all the partial derivatives can be estimated from a single model run, whereas many other methods often require at least one model run per partial derivative that is estimated. Further, the partial derivatives can be calculated with the accuracy of machine precision [9]. Another advantage of this method is that it can be used in conjunction with complex numerical models that can be considered as “black-box” structures, since the method requires just the model source code and not the underlying mathematical equations.
Compute the inputs corresponding to the regression points
Legend:
(SRSM),
Original Model
(SRSM−ADIFOR),
(Monte Carlo)
Fig. 1. A schematic depiction of the steps involved in the application of the SRSM and SRSM with automated differentiation method ADIFOR
In the context of the SRSM, this means that the partial derivatives with respect to the srvs can also be calculated by using wrapper code around the computational model. When partial derivative information from the model is available in terms of the srvs, the following equation can be used in conjunction with (2): n n n ∂yr ∂Γ1 (ξi1 ) ∂Γ2 (ξi1 , ξi2 ) = ar,i1 + ar,i1 i2 ∂ξj i =1 ∂ξj ∂ξj i =1 i =1 1
+
n n n
i1 =1 i2 =1 i3
1
2
∂Γ3 (ξi1 , ξi2 , ξi3 ) ar,i1 i2 i3 + ... ∂ξj =1
(4)
The SRSM-ADIFOR technique used the automatic differentiation tool ADIFOR (Automatic DIfferentiation of FORtran) in conjunction with the SRSM to study uncertainty propagation. Fig 1 outlines the steps involved in the application of the SRSM and SRSM-ADIFOR for computational models. IV. USING THE REDUCED SRSM MODEL IN CONJUNCTION WITH BAYESIAN METHODS FOR UNCERTAINTY REDUCTION For many environmental problems, prior information on physically reasonable values of model parameters is often available from expert analysis/opinion/experience, as is calibration information in the form of actual field measurements
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of the model predicted output quantities (observational data). This leads to an interesting and challenging parameter estimation problem, namely, the adjustment of the prior uncertain information in view of the comprehensive environmental model and the observed field data. When the data generating mechanism is viewed probabilistically this leads naturally to a Bayesian inference problem. The typical approach to Bayesian inference for complex models (where closed form or analytic solutions for the posterior distribution of parameters given the data are unattainable) is to employ numerical techniques for Bayesian inference, e.g. Markov Chain Monte Carlo (MCMC) methods. In the case of computationally intensive models, the time and resources required by these methods can prove to be prohibitively expensive when applied directly to the complex environmental models themselves (which may have substantial individual simulation run times). Therefore, techniques such as the SRSM can be used to provide a statistically equivalent reduced model which serves as a computationally efficient but sufficiently accurate surrogate for the original model in the numerical Bayesian inference step. Once the Bayesian inference procedure is complete, and model parametric input uncertainty is estimated in a manner consistent with observed data, the resulting joint posterior distribution of the inputs can be utilized to estimate uncertainty in the outputs of the full model. This can once again be done accurately and efficiently via the use of the SRSM. A typical application of the above methodology could be as follows: given a deterministic model f (θ), that predicts the outputs of a complex environmental process but requires substantial run-time, prior probability distributions on uncertain inputs P (θ), and field observations/data d, the aim is to determine data and model consistent input distributions P (θ|d), and further based on these posterior input distributions determine how uncertainty propagates through the environmental model and obtain output distributions. The joint posterior distribution of the model is given by Bayes rule as P (θ|d) ∝ P (d|θ)P (θ), where P (θ) is the prior probability and P (d|θ) is the likelihood function, which can be evaluated given a probability (statistical) model for the data, an instantiation of the parameters and the observed data. In general, P (θ|d) cannot be estimated directly, because it is usually not possible to sample from the likelihood function. However, it is possible to calculate the value of the likelihood function for a given realization of the model parameters, and MCMC exploits this property to generate samples from P (θ|d) when the chain has converged. A sufficiently large number of these samples can be used as a good numerical approximation of P (θ|d). Here, a standard MCMC algorithm, the Metropolis algorithm [10], [11] is employed. The Metropolis algorithm requires a symmetric proposal distribution (Jt (θ∗ |θt−1 ) = Jt (θt−1 |θ∗ )) that defines the transition of the Markov Chain, and an initial realization of the parameter vector θ0 . This involves obtaining a re-
alization at time step t from a proposal distribution Jt (θ∗ |θt−1 ). The new realization is accepted with probability min(1, P (d|θ∗ )P (θ∗ )/P (d|θt−1 )P (θt−1 )), or the parameter vector stays at the old state. The Markov chain generated in this process will eventually converge to the distribution used in the calculation of the acceptance criteria (which is the joint posterior distribution of model parameters, in this case) for any form of the proposal distribution given that the Markov chain is ergodic [12]. As a large number of iterations of the steps of the algorithm are usually required for convergence, the SRSM polynomials can instead be used as a computationally efficient surrogate model of the process. More precisely, instead of running the full model to compute the likelihood P (d|θ), the fully specified SRSM polynomials of the form shown in (2) to compute an approximate likelihood. It must be noted that the probabilistic setup remains unaffected; except that the outputs produced by SRSM polynomial expansions are used instead of the complex model outputs f (θ) [13]. V. APPLICATIONS TO ENVIRONMENTAL MODELING A. Application of SRSM and SRSM-ADIFOR to study uncertainties in atmospheric modeling The Reactive Plume Model, version IV (RPM-IV), is a standard regulatory model used for estimating pollutant concentrations in the atmosphere, resulting from the emissions from point sources such as industrial stacks [14], [15]. It uses either point source emission estimates or initial plume concentrations as inputs, and calculates downwind concentrations, as the plume expands. This model simulates complex photochemistry and dispersion processes in the atmosphere. RPM-IV adopts a Lagrangian framework to simulate the evolution of the plume [16]. In this model, the pollutant mass is initially divided into cells containing equal amounts of pollutants. As the plume expands, the individual cells expand in volume and pollutant mass is transferred across cell boundaries in two phases: (a) an “entrainment” phase, where the expanding cell boundaries entrain the pollutants from other cells, and (b) a “detrainment” phase, where the pollutants diffuse across cell boundaries, due to concentration gradients. Further, the pollutants in each cell undergo chemical transformation governed by the Carbon Bond-IV mechanism, which describes the complex non-linear gasphase atmospheric photochemistry through a set of 95 reactions among 35 surrogate chemical species corresponding to organic bonds/functional groups [17]. This application focuses on the effect that uncertainty in emission estimates has on predicted downwind ozone levels. Ozone is formed in polluted air through a complex mechanism involving reactions between oxides of nitrogen (NOx ) and volatile organic compounds (VOCs) in the presence of sunlight. However, there is a significant level of uncertainty in developing estimates of the emissions of VOCs and NOx , as well as their compositions (or speciations, i.e.,
2240
160
Monte Carlo −− 10000 runs SRSM −− 600 runs
Probability density (1/ppm)
140
SRSM−ADIFOR −− 80 runs 120 100 80 60 150 40 20
100
0 0.06
0.08
50 0.1
0.12
0.14
0.16
0.18
0
Plume Evolution Time (minutes)
Ozone concentration at the plume centerline Fig. 2. Evaluation of the SRSM and SRSM-ADIFOR. Estimates of uncertainty in the ozone concentrations calculated by the photochemical plume model RPM-IV. The probability densities of the average plume concentration (ppm) for different plume evolution times as estimated by the SRSM, SRSM-ADIFOR, and standard Monte Carlo method are shown.
the fractions of various chemicals within these groups of compounds). The uncertainties arise due to a variety of reasons. For example, these estimates are typically derived from hourly averages projected from annual or seasonal averages of emissions [18]. Since there is a significant variation in the load and operating conditions, emission estimates for specific situations are likely to include significant uncertainties. The following distributions for the emissions of VOCs and NOx were used in this example: (a) the amounts of VOCs and NOx released were assumed to have normal distributions with a standard deviation of 20% of the mean value, and (b) the chemical compositions of VOCs and NOx were assumed to follow a Dirichlet distribution [19]. The Dirichlet distribution satisfies the condition that the sum of mole fractions is unity. According to this distribution the molefraction of the ith compound, yi , is given by yi = n xi /( i=1 xi ), where xi is an independent random variable, representing the the amount (in moles) of the ith compound; here truncated normal distributions were assumed for all xi s, with a nominal standard deviation of 40% of mean value. The output metrics considered are the average ozone concentration in the plume for selected distances downwind from the source. In this case study, the VOC group consists of five subgroups (paraffins, ethylene, olefins, toluene, and xylene), and the NOx group consists of NO and NO2 . Thus, the total number of uncertain parameters for the model
is nine (including two parameters representing the total amounts of VOCs and NOx ). Fig 2 presents the pdf s for the calculated ozone concentrations at downwind distance of 20km are presented in. Here, 600 model runs were used for a third order SRSM, and 80 model runs were used for SRSM-ADIFOR. These results were compared with the results from 10,000 Monte Carlo simulations. The results indicate that the output pdf s estimated by SRSM and SRSM-ADIFOR agree with the results from the Monte Carlo simulations, while requiring much fewer model runs. Furthermore, the polynomial form approximation of model outputs in terms of srvs, which in turn can be represented as functions of model inputs, enables the calculation of other metrics of importance (e.g. statistics of joint probability distributions). B. Application of SRSM and Bayesian MCMC for uncertainty reductions in ground water modeling The Subsurface Flow and Contaminant Transport (FACT) code is a transient three-dimensional, finite element code designed to simulate isothermal groundwater flow, moisture movement, and solute transport in variably saturated and fully saturated subsurface porous media [20]. The code is designed specifically to handle complex multi-layer and/or heterogeneous aquifer systems in an efficient manner and accommodates a wide range of boundary conditions. The code uses simple rectangular (plane or brick) elements that
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can be deformed in the vertical dimension to accommodate stratigraphic variations. The simulation of saturated groundwater flow using FACT requires model inputs such as three-dimensional conductivity fields and a two-dimensional recharge rate field, which are inherently uncertainty. However, this uncertainty can be fairly well characterized through expert opinion based on years of collected data and experience gained through traditional trial and error model calibration. Thus, substantial prior knowledge exists for the inputs of interest. FACT is computationally quite demanding, averaging about half an hour per FACT code simulation (when run on a Sun Fire 280R with two 750 MHz UltraSPARC III CPUs and 4GB memory). Quantities of interest in the uncertainty analysis tasks are model outputs such as hydraulic head values (for which actual data are available from field measurements) and stream baseflow rates. In order to make analysis tractable, the problem was simplified by assigning an independent global multiplier to each of the three-dimensional conductivity field variables as well as one to the two-dimensional recharge rate field. The global multiplier ensures that relative spatial variations dictated by characterization and subsequent model calibration are preserved, while the mean value of the field is perturbed [21]. The main output variables monitored were the simulated hydraulic head values in the various aquifers (which are direct model outputs) and the stream baseflows in the main discharge regions (Fourmile, Crouch and McQueen Branches and the Upper Three Runs) which are model post processed results. Expert opinion based priors on the global multipliers were used as the input distributions for the application of the SRSM (Table II). Collocation points were generated based on these input distributions, the full model was run at these specified points and results were collated in order to obtain the coefficients of the SRSM polynomials for each of the outputs under consideration (all 667 well hydraulic head values and stream baseflow rates). Both second and third order SRSM expansions were fit to the full model outputs requiring only 51 and 191 simulations, respectively. The convergence of the resulting output distributions obtained by SRSM polynomials of second and third order revealed that second order SRSM polynomials were sufficient for the representation of this system. Thus, a second order SRSM approximate model as simulated by FACT was obtained. MCMC simulation using the Metropolis-Hastings algorithm was subsequently implemented on the SRSM reduced model. The priors were chosen as described above, and proposal distributions each chosen to be Gaussian with current parameter realization values as their mean values and variance set at 20% variance of their corresponding prior. This satisfies the symmetric proposal distribution requirement of Metropolis algorithm. In addition, in order to determine the likelihood function values, three independent error models were utilized for each of the three output groups corresponding each aquifer. The model chosen was
that the hydraulic head value at the well locations was taken to be equal to that predicted by the FACT model (or its SRSM based approximation) but slightly corrupted by some additive zero mean Gaussian noise. The variance of the distributions were left as additional parameters to be estimated by the MCMC simulation (non-informative/flat priors were used for the variances of the error model). Sets of highly overdispersed values were chosen as initial points of the MCMC simulation and 4 chains were successfully run for 10,000 iterations. After confirming convergence, the final joint posterior distribution of the input parameters was obtained after discarding the initial burn-in (the first half of the converged chain) for the chain with the least variance in the most variables. The four MCMC chains using the SRSM second order reduced model of the system, converged to practically identical joint posterior distributions. Convergence characteristics of the chains were confirmed via two well known MCMC multiple chain convergence criteria, Gelman and Rubin [22] and Brooks and Gelman √ [23]. The maximum value of the Gelman and Rubins R metric was found to be 1.007 over all the parameters. The maximum Potential Shrink Rate Factor (PRSF) value (used by the Brooks and Gelman criteria) over all the parameters was found to be approximately 1.01 (in both cases, values close to 1 were deemed convergent). Subsequent analysis of the joint posterior distribution was based on the final 5000 realizations (discarding the initial half of the chain as burn-in) of the converged chain whose variance in the most input parameters was found to be minimal, namely chain 4. The uncertainty in the posteriors, when compared to the priors, as evaluated by Shannon’s E[ln(p)] metric [24], was found to decrease for all the variables, as indicated by the positive value of information gain, which is defined as IG = −E[ln(p0 )]−(−E[ln(p∗ )]) where p0 and p∗ are the prior and posterior probability density functions obtained for each uncertain variable. Five model inputs were considered in this study: the vertical hydraulic conductivity field values for four physical locations (GCU, LAZ, TCZ, and UAZ), and the recharge rate global multiplier (RECH). The computed values of IG were 2.18, 1.62, 3.37, 0.8 and 6.93 for the five model inputs considered: GCU Kv , LAZ Kh , TCZ Kv , UAZ Kh and RECH distributions, respectively. The decrease in the uncertainty can also be inferred by comparing summary statistics of the prior and posterior distributions II. Thus an uncertainty reduction has indeed been achieved by the Bayesian “fusion” of the field observations and the model predictions as approximated by the SRSM expansions. The analysis also provided some information on input parameter correlations. Since the Bayesian analysis results in a full joint posterior distribution of the parameters, the linear coefficient of correlation between parameter values was easily calculable, Table III. These correlations provide insight into how further modeling efforts may be improved. Finally, the joint posterior distribution was further used for the
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TABLE II
this work are solely the responsibility of the authors and do not necessarily reflect the views of the USEPA, USDOE, or their contractors.
C OMPARISON OF STATISTICS OF THE PRIOR AND POSTERIOR DISTRIBUTIONS OF THE UNCERTAIN INPUTS . Variable (Units)
Prior (µ, σ)
Posterior (µ, σ)
Log10 normal
-5, 0.75
-4.94, 0.73
Distribution
R EFERENCES
Type GCU Kv (ft/d) LAZ Kh (ft/d)
Log10 normal
0.9685, 0.15
1.00, 0.03
TCCZ Kv (ft/d)
Log10 normal
-2.2, 0.5
-2.65, 0.27
UAZ Kh (ft/d)
Log10 normal
0.9823, 0.15
1.15, 0.05
RECH (in/yr)
Normal
18, 3.1
21.04, 1.13
TABLE III M ATRIX OF AVERAGE C ORRELATION C OEFFICIENTS O BTAINED ACROSS A LL MCMC C HAINS GCU
LAZ
TCZ
UAZ
RECH
GCU
1.00
-0.05
0.00
-0.15
-0.12
LAZ
-0.05
1.00
0.45
0.47
0.86
TCZ
0.00
0.45
1.00
0.22
0.42
UAZ
-0.15
0.47
0.22
1.00
0.83
RECH
-0.12
0.86
0.42
0.83
1.00
final uncertainty analysis of the stream baseflows and well location hydraulic head values [13]. VI. DISCUSSION An overview of the Stochastic Response Surface Method and its applications in conjunction with automated differentiation methods and Bayesian inference methods is presented here in the context of computationally efficient uncertainty propagation and reduction. The SRSM provides a computationally effective means for studying uncertainty propagation, and is easily extensible. Further, the SRSM can be coupled with other sensitivity analysis methods, as shown by the application of the SRSM-ADIFOR. The SRSM can also be used in a Bayesian context to address inverse solution problems, as shown in the application of the SRSM in conjunction with Bayesian MCMC method. The SRSM can thus be used in addressing problems related to characterization and reduction of uncertainty. While the application of the SRSM to multiple computational models has yielded useful results, there is clearly a need for evaluating the theoretical aspects of the approximation, specifically with respect to convergence when dealing with higher order approximations. Ongoing efforts focus on these issues along with approaches for more robust approximation of coefficient of the approximation. VII. ACKNOWLEDGMENTS This work has been funded in part by the USEPA University Partnership Agreement (UPA) for EOHSI’s Center for Exposure and Risk Modeling (CERM: EPA CR-827033), and by a grant to the IRM, CRESP from USDOE, Instrument DE-FG2600NT 40938. The viewpoints expressed in
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