exact propagation of uncertainties in multiplicative models

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Exact propagation of uncertainties inmultiplicative modelsPhongtape Wiwatanadate a b & H. Gregg Claycamp aa Department of Environmental and Occupational Health ,University of Pittsburgh , 260 Kappa Drive, Pittsburgh, PA, 15238Phone: (412) 967–6524 Fax: (412) 967–6524b Department of Community Medicine, Faculty of Medicine ,Chiang Mai University , Chiang Mai, 50200, Thailand Phone:66–53–945472 to 4 Fax: 66–53–945472 to 4Published online: 02 Dec 2008.

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Human and Ecological Risk Assessment: Vol. 6, No. 2, pp. 355-368 (2000)

Exact Propagation of Uncertainties in MultiplicativeModels

Phongtape Wiwatanadate1,* and H. Gregg ClaycampDepartment of Environmental and Occupational Health, Pittsburgh, PA

ABSTRACT

Propagation of uncertainties has been widely accepted as an integral compo-nent in health risk assessments. Historically, the approximations based onTaylor series expansions have been applied, but they are valid only when themodel is linear or the coefficients of variation are small. Another approximationapproach is Monte Carlo analysis. It is simpler and more practical for riskmanagement but may become cumbersome if the model involves many variablescontaining large uncertainties. Recently, the exact log transform technique hasbeen proposed for multiplicative models with lognormal uncertainties. Thepurpose of this paper is to present a novel exact analytical method as a comple-mentary approach to simulation for general multiplicative models with indepen-dent variables, without a constraint on the types of uncertainty distributions. Italso provides good estimates of the mean and variance of highly skewed distri-butions, determination of their existence, and verification of sufficient simula-tion sample size for simulation method. Two case examples are given to dem-onstrate the applications.

KEY WORDS: Propagation of uncertainties; uncertainty analysis; multiplicativemodel; probability; Monte Carlo analysis.

INTRODUCTION

Propagation of uncertainties or uncertainty analysis is an integral componentof a variety of scientific endeavors such as health risk assessment, physics, chem-istry, engineering, and economics. Depending on the type of calculation leading

1Department of Environmental and Occupational Health, University of Pittsburgh, 260Kappa Drive, Pittsburgh, PA 15238. Tel: (412) 967-6524. Fax: (412) 624-1020.

*A11 correspondence should be addressed at the Department of Community Medicine,Faculty of Medicine, Chiang Mai University, Chiang Mai, 50200, Thailand. Tel: 66-53-945472 to 4. Fax: 66-53-945476.

1080-7039/00/$.50© 2000 by ASP

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to the final results, numerical or analytical methods are employed to determinethe propagation of uncertainties from inputs to results (e.g., Seiler, 1987). Thereare two general approaches for the propagation of uncertainties: approximateand exact. The two most common approximate techniques are method of mo-ment (Taylor series expansion) and Monte Carlo simulation. The simplest methodof moment is based on the first-order or Gaussian approximation (Parratt, 1987;Bevington, 1969; Morin, 1988; Lyons, 1991) that works well for the uncertaintydistributions with small coefficients of variation (CVs)—the ratios of standarddeviation to the mean. If more exact results are required, especially in largeuncertainties and complex multivariate models, the higher-order derivative termscannot be ignored. Unfortunately, this introduces a significant degree of tediousalgebra.

In contrast to the method of moments, the Monte Carlo analysis is far simpler.Once the probability density function (PDF) of each input variable is established,the results can be easily obtained by using a computer to simulate and compute thevalues. The only limitations to the technique are that the resulting output isobviously an estimate of the exact result, for which the accuracy of the estimatedepends on the number of random samples and the quality of random numbergenerators. While the output will converge when the number of simulationsapproaches infinity, diis is often impractical due to significant computational time.This is true when the model involves many inputs with large CVs. In certainsituations, several million simulations are required in order to achieve stability(Willard and Critchfield, 1986).

Unlike the approximate methods, the exact analytical approaches have beenrarely developed and reported in the literature. Seiler (1987) has explored theexact analytical approaches extensively, but the formulas are limited because ofsome assumptions and considerable numerical computations that tend to makethem inapplicable. In risk calculations that can be expressed as multiplicativemodels using lognormal distributions, Slob (1994) proposed an exact techniqueusing a log transform to convert the multiplicative model into an additive modelthat is easily solved with the first-order Taylor series expansion. The approachhas been extended to a systematic analytical method to distinguish betweenuncertainty and variability (Rai and Krewski, 1998). However, multiple lognor-mal distributions are not always the case in health risk assessments. For ex-ample, when uncertainty relates to lack of knowledge, triangular or uniformdistributions have been suggested as suitable conservative estimates of actualdistributions (Slob, 1994; FinleyeZ al, 1994). Furthermore, in risk assessmentsthat rely on distributions other than the lognormal, using lognormal distribu-tions as approximations unnecessarily contributes to uncertainty in the overallprocess.

In this paper, a novel method based on probability theories is presented to showhow the exact formulas for propagation of uncertainties of multiplicative modelscan be derived without restrictions on the type of uncertainty distributions. Twoexamples of multiplicative models chosen from the literature are given to demon-strate application of the method.

356 Hum. Ecol. Risk Assess. Vol. 6, No. 2, 2000

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Exact Propagation of Uncertainties in Multiplicative Models

THEORY

In this paper a multiplicative model is defined as the entity Y obtained from anumber of operations of input variable JQ using multiplication, division, and/orexponentiation only, in the form of

X, (1)

The power a will be considered as a known constant, whereas 5$ is the variableincorporated with uncertainty, which, in this paper is defined as both imprecisionin an estimate of an exposure factor and actual differences among members of apopulation (variability). Although the uncertainty is usually described as a lognor-mal distribution, no such constraint is necessary in the exact method presentedhere. In other words, any probability distribution is valid as long as its mean andstandard deviation can be defined.

For the sake of simplicity, die multiplicative model will be considered as threedistinct models separately: products of distributions, quotients of distributions, andproducts of powers of distributions. The analytical results of each model aresummarized in Table 1. In order to conserve space, the definitions of mean andvariance are reviewed in Appendix A, and the formal proofs are shown only for theproducts of distributions (see Appendix B).

Table 1. General Analytical Results of Separate Multiplicative Models.

Model

Products of

Distributions

Quotients of

Distributions

Products of

Powers of

Distributions

General Form Mean of Z

Z =aXlX1...Xm

I-I

Z

Hz - a \ [Hjr,I.I

fj[\fkl)dx\

Hz

Variance of Z

L '-i ' /-i ' J

Hum. Ecol. Risk Assess. Vol. 6, No. 2, 2000 357

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From Table 1, it is noteworthy that, unlike the products of the distributions, themean of Z in the model of quotients of distributions will depend on the integralterms that do not always exist. For example, Khuri (1993) has shown diat for any

normally distributed random variable X, its £(X~J)or I '" 'does not exist,

because of an improper integral with singularity at x = 0 if/(0)>0, when E{Xrl) doesnot exist. This can also be proven by using convergence tests for improper integralsof the third kind (Spiegel, 1963). Similarly, depending on the existence of the meanand the integral terms, the variance may not always exist. In other words, if themean does not exist the variance does not either, but even if the mean does exist,the variance might not unless all integral terms exist. For example, the first negativemoment of a Weibull distribution with a PDF of f(x) = 2xe~'2 converges, but its secondnegative moment diverges.

Regarding die model of products of powers of distributions, it is, in fact, the generalmultiplicative model. The existence of Ujand o\ again will depend on the integral termsthat in turn depend on types of uncertainty distributions. For example, using conver-gence tests for improper integrals of the first and second kinds (Spiegel, 1963) showsdiat if J[x) is normally distributed, die integral will converge if and only if the poweris greater than - 1 , whereas if J{x) is lognormally distributed the integral will alwaysconverge regardless of the value of die power. Also, (Xzand a\ need not be existing ornonexistent at the same time. In die special case, if all of diea, are positive integers, diemean and variance can be expressed in die moment generating function forms as:

where die moment generating function of a random variableXis defined as, provided diat die expectation exists, and a( is the of order of derivatives of

iit]evaluated at t= 0 (Kinney, 1997).

CASE STUDIES

To appreciate die applications of diis proposed technique, two multiplicativemodels from die risk assessment literature are presented. The first one involvesmultiplication and division, whereas the second one is die product of powers ofdistributions. The chosen examples are intended to illustrate the use of diis exact


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method on commonly encountered risk assessment models. The input variables inboth models are assumed to be independent and die models are assumed to be validaccording to the audiors' discussions.

Incremental Lifetime Cancer Risk from Ingestion of Contaminated Soil

Thompson, Burmaster, and Crouch (1992) presented a hypothetical exposuremodel to estimate the incremental lifetime cancer risk (ILCR) from incidentalingestion of soil contaminated with benzene using the expression:

ILCR = ADD(life) • CPF (4)

where ADD (life) is the average daily dose of benzene, averaged over lifetime duringwhich exposure occurs and CPF is the cancer potency factor of benzene. TheADD (life) can be estimated from

(5)WDinYYinL

Table 2 shows the variables and constants used in this model. Applying the formulasin Table 1 to Eqs. 4 and 5 yields

dwl-u.2 ( 7 )

Note that the integration is evaluated over the positive range since the integrandat negative values is undefined and zero is more than five standard deviations belowthe mean; thus, the error in considering only positive values is completely negligible(Chambless, Dubose, and Sensintaffar, 1994). It can be proven that both integralterms diverge (see earlier section); however, because most of the data are confinedaround their means, each integral can be approximated and converted into conver-gence by integrating over the range of mean ± 5SD limits as long as the lower limitis nonzero and nonnegative. Strictly speaking, if the CV is less than 0.2, thedivergence problem can be safely ignored, as the probability of obtaining observa-


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Whvatandate and Claycamp

Table 2. Variables and Constants Used in Benzene Exposure Model."

Name, symbol Units Distribution" Meanc Variance"7

Cancer potency factor, CPF

Soil concentration, Cs

Soil ingcstion rate, SIngR

Relative bioavailability, RBA

Exposure days per week, DpW d/wk Constant (1)

Exposure weeks per year, WpY wk/yr Constant (20)

Exposure years per life, YpL yr/life Constant (10)

Average body weight, W kg Normal (47,83)

Days in year, DinY d/yr Constant (364)

Years in lifetime, YinL yr/life Constant (70)

(kgd)/mg Lognormal (-4.33,0.67) 0.016S 1.539E-04

mg/kg Lognormal (0.84,0.77) 3.1157 7.8556

mg/d Lognormal (3.44,0.80) 42.9484 1653.6193

Constant (1)

47 68.89

"Source: modified from Table 1 of Thompson, Burmaster, and Crouch (1992).

*For a lognormal, the mean and standard deviation in parentheses are the underlying normaldistribution parameters in log-scale.

•Mean and variance are the population mean and variance (see Appendix D for conversionformulas).

tions beyond five standard deviations away from the mean is quite small (Morganand Henrion, 1990). There are no closed forms for both integrals, but they can beassessed by using numerical integration algorithms that are available in most math-ematical software packages. In this example, the Mathcad® version 7 was used toperform the calculations. It follows that the integral term in Eq. 6 approximatelyequals to 0.022 and that in Eq. 7 approximately equals to 5.037E04 (the range ofintegration is [5.5, 88.5] with the tolerance of 0.001 and the number of significantdigits of 4). For the numerical integration algorithms used in the Mathcad®,readers are referred to Appendix C. Thus,nILCR= 3.81E — 10 and OlLCir* 8.16E — 10.The Monte Carlo analysis performed by Thompson, Burmaster, and Crouch (1992)resulted in the mean value of 3.74E - 10. (Note thatO",LCR was not provided.)

Total Body Surface AreaA model to estimate the distribution of total body surface area (SA) from height

(H) and weight (W) in male subpopulation was used by Slob (1994) to demonstratethe log transform method as


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SA =aHbWc (8)

where H is assumed to be normally distributed with mean 175.7 and SD 7.30 (cm)and W is assumed to be lognormally distributed with mean 4.35 and SD 0.17 (In ofkg), whereas a, b, c are constants(a =0.0239, b= 0.417, and c = 0.517). By applyingthe relevant formulas shown in Table 1 to Eq. 8, we obtain the mean and varianceof SAas

exp(9)

wcdw\

exp

2<w

w2cdw[-ii2SA

(10)

Unlike the previous case example, the convergence tests show that all integralterms converge (see earlier section). Again, there are no closed forms found for allof the integrals, but they can be assessed by the numerical integration techniques.However, some mathematical software packages will show "overflow" errors if onetries to perform the computations. For example, the Mathcad® version 7 used inthis case showed the error message when evaluating the expression -^i-175.7)2 as:"found a number with a magnitude greater than 10 307 while trying to evaluate thisexpression", as the maximum digits that the Mathcad® can handle is 10307. (SeeAppendix C for numerical integration algorithms used in the Mathcad®.) Never-theless, for the same rationale mentioned in the previous example, the approxima-tions can be validly constrained over the integration ranges of p. ± 5a for a normaldistribution and exp(|l ± 5a) for a lognormal distribution. The approximations ofthe integral terms in Eq. 9 are 8.6294 and 9.5145, and of those in Eq. 10 are 74.4882and 91.2268, respectively. Therefore, it follows that ^A = 1.962 and G^- 0.176. Forthe sake of comparison, if we assume that SA is lognormally distributed, then wehave HinsA~ 0.670 andalnSA~ 0.090. (See Appendix D for conversion formulas.) ThealnSA can be compared favorably to that of Slob (1994), who reported qnSA« 0.090.


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Whvatandate and Claycamp

It should be noted that this analytical method does not require the assumption oflognormal distributions of inputs as the log transform technique does.

DISCUSSION

There is no doubt that the log transform technique for uncertainty analysis ofmultiplicative models is easy and useful when all the uncertainties in the model arelognormally distributed—a common occurrence in risk assessment. Even if a giveninput variable is not truly lognormal, Slob (1994) has shown that this method stillworks rather well by approximating a normal distribution with a lognormal distribu-tion as long as the CV is small. However, if the models are the combination ofadditive and multiplicative models (e.g., the exposure from different routes suggest-ing the addition of quantities), and/or the uncertainty distributions are other thanlognormal distribution, diis technique will no longer be valid.

The exact analytical method presented here overcomes the constraints on thetypes of input distributions. Furthermore, in certain models, it can be combined tothe additive model, which is readily analyzed with the first-order Taylor seriesexpansion. The only limitation is one shared with the other techniques: the inputvariables are assumed to be independent. Yet, as in other methods, the dependen-cies among die variables can be derived, but are beyond the scope of diis paper.

As already discussed in die second section, die madiematical expressions haveshown that in certain situations die mean and/or the variance of die outputdistribution may not exist for all values of the parameters. For example, diis is truewhen the range of integration of a normally distributed input variable widi a power< -1 includes zero. Moreover, diey enable us to predict diat sometimes die meanof the output distribution exists whereas die variance does not. However, eventhough the mean and/or die variance might not exist for all parameters, tiiislimitation is readily overcome if we integrate over die range diat contains nonzeroand represents almost all of die observations (e.g., u± 5a in a normal distributionwidi CV roughly less dian 0.2). This will explain die instability in die Monte Carloanalysis in certain circumstances no matter how large the number of die simulationruns.

It is apparent from die growdi of Monte Carlo-based methods in the riskassessment literature that computer-intensive mediods, once limited by hardwareand software, are becoming a standard practice. It is also evident from theliterature that Monte Carlo mediods are replacing other approximate methodsincluding method of moments. However, as aforementioned, the accuracy of dieestimate depends on the number of random samples. This can be best demon-strated by Figure 1, where die CVs ofZ from the model Z = X2Y3(Xand Farenormally distributed with means 1,2 and standard deviations 0.1, 0.2, respectively)are plotted against the number of simulations in log-scale. It is self-evident thatthe CV of Z approaches the exact value of 0.362 as the number of simulationsincreases. Additionally, we have shown in die present study that classical exactmethods for die propagation of uncertainties can be extended to contemporaryrisk assessment problems to improve accuracy and to identify unstable means andvariances diat might be undetected in Monte Carlo analyses. Conclusively, it

362 Hum. Ecol.-Risk Assess. Vol. 6, No. 2, 2000

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0.375

o

riati

50

•+*

Q>

O

£oo

0.370 -

0.365 -

0.360 -

0.355 -

0.350 -

0.345 -

0.340

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Number of samples

Figure 1. The CVs of Zfrom the model Z = A2P(Xand Fare normally distributed withmeans 1,2 and standard deviations 0.1, 0.2, respectively) are plotted against thenumber of simulations in log-scale ranging from 100 to 1,000,000. The CV ofZ approaches the exact value of 0.362 (dashed line) as the number of simula-tions increases.

allows good estimates of the mean and variance of highly skewed distributions,determination of convergence and divergence (i.e., existence of estimated distri-butions), and verification of sufficient simulation sample size for simulationsperformed to estimate mean and variance.

It should be borne in mind, however, that the analytical method presented here isintended to be a complementary tool to simulation method, because it does not provideas much information (i.e., the percentiles of the distribution, its higher moments, or thenature of the distribution) as the simulation method does. Furthermore, it requires theindependence of variables, whereas the simulation method does not

APPENDIX

A. Mean and VarianceIn order to provide a simple description of a distribution, we need to measure the

values at which the distribution is centered, and how wide the distribution is. Themean \i and the variance O2 are suitable for this. Thus, a is the deviation from themean, and is also known as the "standard deviation" of the distribution.

For a continuous distribution the mean and variance are defined as (Lyons, 1991;Kinney, 1997; Hogg and Tanis, 1993):


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(Al)

Var(X) = a2 = E(X - \i)2 = j /(*)(* - [if dx (A2)

where J[x) is the PDF of X It can be shown that the following relationships hold

G2=E(X2)-[E(X)f (A3)

(A4)

B. Products of Distributions

Using the definitions reviewed in Appendix A, let us begin with a simple two-variable model: Z= XY. If fix) is a PDF of Xand g(y) is a PDF of Y, then the meanof Z can be determined as:

\iz=jjf(x)(xy)g(y)dxdy

The variance of Z is given by


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Replacing (J.z with n x |iY [from Eq. Bl ] :

)\j= j 8(y)\j f(x)(x2y2 -

(B2)

= [ J f(x)x2dxjg(y)y2dy] - ^

Substituting the integral terms of Eq. B2 with Eq. A4 yields

o2z = ( 4 + a2 )(n2

r + a2Y)- \i\\i\ (B3)

Note that the result of this equation, uOy + \iyGl + o^Oy, differs from that of thecommonly used approximate solution (Parratt, 1987; Bevington, 1969; Morin, 1988;Lyons, 1991) by die term of o*.

Likewise, for general products of independent variables in the form ofn

7- Y Y Y - FT Y** ~ "Ai A2 • • • A

n ~ \_ \_ i, where a is a constant, it can be shown that the mean of

/=i

Zwill be

and the variance of Z is


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(B5)

where u ,is the mean and Gx is the standard deviation of corresponding variable .Xj

Numerical Integration Algorithms Used by the Mathcad®

Mathcad® computes definite integrals numerically using a Romberg algo-rithm. It makes successive estimates of the value of the integral and returns avalue when the two most recent estimates differ by less than the value of thebuilt-in variable TOL (tolerance). To compute the integral off(x) from a to b(if b is an infinity, Mathcad® will set it to lO™7), Mathcad® follows these steps(MathSoft, Inc., 1997):

• Calculate an estimate for the integral using the trapezoidal rule.

• Fit a polynomial to the sequence of trapezoidal estimates computed so far andthe corresponding subinterval widths. Extrapolate this polynomial to widthzero to produce a Romberg estimate for the integral.

• Compare the two most recent Romberg estimates according to the followingtest:

lestimaten - estimate,,.]! < reltol

where reltol is the larger of TOL and .

• If the two most recent estimates agree according to this test, check alsowhether estimate n-1 and estimate n-2 also agreed. If they did, then returnto the most recent Romberg estimate as the value of the integral. If not, doublethe number of subintervals and return to the first step.

Mathcad® sets a limit on the number of times it will iterate this procedure. If theroutine reaches this limit without converging, or if the integrand is singular at oneor both of the endpoints of the interval of integration, then Mathcad® switches toan open-ended Romberg method. Again, if Madicad® reaches its set limit ofperforming this routine without returning an answer, the integral is marked with anerror indicating that it did not converge.

D. Relationships among Distributional Parameters

Given a lognormal distribution Xfor which fiis the arithmetic mean, a is thearithmetic standard deviation, \iln is the mean value of the distribution of In (X), and


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CTln is the standard deviation of In (A), the following expressions are true (McKone,1994):

(Dl)

(D2)

(D3)

a = ,/[exp(2nln + ofn ][exp(ofn) -1] (D4)

REFERENCESBevington, P.R. 1969. Data Reduction and Error Analysis for the Physical Sciences.McGraw-Hill

Book Company: New York.Chambless, D.A., Dubose, S.S., and Sensintaffar, E.L. 1994. 'Exact' and approximate methods

for analysis of total error in radiation measurements.Health Phys. 66, 313-317.Finley, B., Proctor, D., Scott, P., Harrington, N., Paustenbach, D., and Price, P. 1994.

Recommended distributions for exposure factors frequendy used in health risk assess-ment. Risk Anal 14, 533-553.

Hogg, R.V. and Tanis, E.A. 1993. Probability and Statistical Inference. Macmillan PublishingCompany: New York.

Khuri, A.I. 1993. Advanced Calculus with Applications in Statistis. John Wiley & Sons, New York.Kinney, J. J. 1997. Probability: An Introduction with Statistical Applications. John Wiley & Sons,

New York.Lyons, L. 1991. A Practical Guide to Data Analysis for Physical Science Students Cambridge

University Press: New York.MathSoft, Inc., 1997. Mathcad 7 User's Guide Cambridge, MA.McKone, T.E. 1994. Uncertainty and variability in human exposures to soil contaminants

through home-grown food: a Monte Carlo assessment. Risk Anal 14, 449-463.Morgan, M.G. and Henrion, M. 1990. Uncertainty: A Guide to Dealing with Uncertainty in

Quantitative Risk and Policy Analysis Cambridge University Press, New York.Morin, R.L. 1988. Monte Carlo Simulation in the Radiological Sciences.CRC Press, Florida.Parratt, L.G. 1961. Probability and Experimental Errors in Science: An Elementary Survey. John Wiley

and Sons: New York.Rai, S.N. and Krewski, D. 1998. Uncertainty and variability analysis in multiplicative risk

models. Risk Anal. 18, 37-45.Seiler, F.A. 1987. Error propagation for large errors.Risk Anal. 7, 509-518.Slob, W. 1994. Uncertainty analysis in multiplicative models. Risk Anal. 14, 571-576.


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Spiegel, M.R. 1963. Schaum's Outline of Theory and Problems of Advanced Calculus SchaumPublishing Company. New York.

Thompson, K.M., Burmaster, D.E., and Crouch, E.A.C. 1992. Monte Carlo techniques forquantitative uncertainty analysis in public health risk assessments. Risk Anal. 12, 53-63.

Willard, K.E. and Critchfield, G.C. 1986. Probabilistic analysis of decision trees using symbolicalgebra. Med. Decis. Making 6, 93-100.


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exact propagation of uncertainties in multiplicative models

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