The stochastic quantization method and its application to ... · The stochastic quantization method and its application to the numerical simulation of volcanic conduit dynamics under

Solid Earth, 1, 49–59, 2010www.solid-earth.net/1/49/2010/doi:10.5194/se-1-49-2010© Author(s) 2010. CC Attribution 3.0 License.

Solid Earth

The stochastic quantization method and its application to thenumerical simulation of volcanic conduit dynamics under randomconditions

E. Peruzzo1,*, M. Barsanti2,3, F. Flandoli2, and P. Papale3

1Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy2Dipartimento di Matematica Applicata, Universita di Pisa, Via Buonarroti 1c, 56127, Pisa, Italy3Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Via della Faggiola 32, 56126, Pisa, Italy* Invited contribution by E. Peruzzo, one of the EGU, Young Scientists Outstanding Poster Paper Award winners 2009

Received: 16 February 2010 – Published in Solid Earth Discuss.: 4 March 2010Revised: 11 May 2010 – Accepted: 16 June 2010 – Published: 22 June 2010

Abstract. Stochastic Quantization (SQ) is a method for theapproximation of a continuous probability distribution witha discrete one. The proposal made in this paper is to ap-ply this technique to reduce the number of numerical simula-tions for systems with uncertain inputs, when estimates of theoutput distribution are needed. This question is relevant involcanology, where realistic simulations are very expensiveand uncertainty is always present. We show the results of abenchmark test based on a one-dimensional steady model ofmagma flow in a volcanic conduit.

1 Introduction

Since the demand for eruption scenario forecast in the worldis pressing, there is a strong need for using complex physicalmodels and numerical codes in order to get information aboutthe possible eruptive conditions at many hazardous volca-noes (e.g.,Sparks, 2003; Neri et al., 2007). Such models candescribe volcanic processes thoroughly, but this ability oftenresults in high computational costs: a single simulation canrequire a time of the order of days to weeks to be completed.

Correspondence to:E. Peruzzo([email protected])

On the other hand, since volcanic systems are largely in-accessible to direct observation, the models which describethem often involve intrinsically uncertain quantities. As aconsequence, some of the input data required by the numer-ical codes should be considered as random variables ratherthan as fixed parameters. Therefore, the most direct way ofobtaining information about the probability of possible erup-tive scenarios would be to implement a Monte Carlo (MC)method. As a drawback, this would require a large numberof numerical simulations (e.g. 104), while often a number ofthe order of 10 simulations cannot be exceeded.

It is thus fundamental to be able to choose the input datavalues in such a way that the simulations provide the maxi-mum amount of information possible about the output quan-tities. Furthermore, the selection of the “best” sets of valuesof input data should be guided by fairly general principles,which could be applied to a large class of models and nu-merical codes and are not devised for a particular situation.The strategy presented in this paper is indeed general, sincethe choice of the optimal sets of input data does not involvethe numerical code at all. In this respect, a different (andsomewhat complementary) approach to the problem wouldbe that of using the simulations to construct a function whichis a reasonably good approximation to the complex code and,at the same time, can be evaluated with a low computationaleffort (Sacks et al., 1989; Currin et al., 1991). The simplifiedfunction produced could then be used in a Monte Carlo sim-ulation. In this case, the choice of the sets of input data used

Published by Copernicus Publications on behalf of the European Geosciences Union.

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50 E. Peruzzo et al.: The SQ method for the numerical simulation of volcanic conduit dynamics under random conditions

to find the approximating function is less critical, while thefocus is on the approximation of the numerical code. A pos-sibility that has been explored in the mathematical literatureis that of employing Bayesian inference to select a functionwith the required properties (Kennedy and O’Hagan, 2001).Our approach, on the other hand, aims at defining an optimalset of input data, that sufficiently describes the output distri-bution. Future developments may involve a mixed approach,whereby some gross properties of the numerical code are ex-ploited to guide the choice of the optimal sets of input dataand corresponding output distribution.

In the first section of this paper we present our approach tothe problem, showing that the stochastic quantization (SQ)method provides a possible solution. In Sects. 2 and 3 webriefly sketch the basic principles of one-dimensional andmulti-dimensional quantization. In Sect. 4 we show the ap-plication of our strategy to an ideal case, while in Sect. 5we turn to a more realistic situation, involving the one-dimensional steady model of magma flow in a volcanic con-duit described in (Papale, 2001). We show that, given a max-imum numberN of simulations, the results obtained usingthe SQ method to chooseN sets of values of input data arebetter than those produced byN MC simulations.

The origins of the theory of quantization of probability dis-tributions date back to the three fundamental articles (Oliveret al., 1948; Bennett, 1948; Panter and Dite, 1951); thefield of application was that of signal processing, in partic-ular modulation and analog-to-digital conversion. See (Grayand Neuhoff, 1998) for a survey of the method and of itsengineering applications, besides a huge list of references(mainly in the field of engineering); the book (Graf andLuschgy, 2000) gives a mathematically rigorous treatmentof the subject.

2 Outline of the strategy

In order to illustrate our strategy, consider a practical situ-ation: suppose that a numerical code for the simulation ofsome volcanic processes has the random variableX amongits input data. Likewise,X can be a collection of input ran-dom variables(X1,...,Xd), i.e. ad-dimensional random vec-tor. Let Y be one relevant output quantity of the numericalcode. We denote byf (x1,...,xd) andg(y) the probabilitydensity functions ofX andY , respectively;f is assumed tobe known, while nothing is known aboutg. We also supposethat the numerical code has such a high degree of complex-ity that the maximum numberN of affordable simulations isvery small, of the order of 10. It is thus not possible to collectinformation aboutg using a MC method.

Our strategy (see Fig.1) consists in three main steps andan optional fourth step.

1. FindN values of the random vectorX,(x

(1)1 ,...,x

(1)d

),...,

(x

(N)1 ,...,x

(N)d

)

andN corresponding weights(w(1),...,w(N)

),

with∑N

i=1w(i)=1, such that the discrete probability

distributionf which assigns the weightw(i) to the point(x

(i)1 ,...,x

(i)d

)(for i=1,...,N) is the optimal approxi-

mation off , among all the discrete probability distri-butions concentrated inN points. The meaning of opti-mality is to be precised later.

2. PerformN numerical simulations to compute theN cor-responding valuesy(1),...,y(N) of the random variableY . We represent the action of the numerical code on theinput data through a functionϕ, so that

y(i)=ϕ

(x

(i)1 ,...,x

(i)d

)for i=1,...,N.

3. Build a discrete approximationg of g by assigning theweightw(i) to the pointy(i) for i = 1,...,N .

4. Useg to build a continuous probability distribution.

The core of the problem is thus to find the “best” dis-cretization of a fixed continuous probability distributionf(step 1). Stochastic quantization is a mathematical theorywhich allows to accomplish this task by giving a definitionof the optimal discretization and providing an algorithm tofind it. Once this is done, a discrete approximation of the un-known output probability distribution is automatically pro-duced (steps 2 and 3). Concerning step 3, it is a rigorous factthat the transformation of a discrete probability distributionby a functionϕ is the discrete probability distribution hav-ing the same weigths on the image points. Thus step 3 is themost natural choice. If we would know the output densityg,a better choice would be the weigths given by the SQ algo-rithm applied tog and the given output points. But we donot knowg. It is an open problem to improve step 3 in thisdirection.

Though the discrete probability distributiong can be usedto estimate the values of some parameters ofg (e.g. its mean,its variance, some quantiles), its graphical representationgives poor insight into the main qualitative features ofg. Thisis one of the motivations of step 4, which can be carried outthrough a kernel smoothing algorithm (e.g.,Wand and Jones,1995). Moreover, it may be useful for other purposes, likerandom number generation, to have a continuous distributionin output. The main idea of kernel smoothing algorithmsis to smear out each weightw(i) around the correspondingpoint y(i) according to a fixed rule and then to sum up allthe contributions. This leads to the continuous probabilitydistribution

gKS(y) =1

Nh

N∑i=1

K

(y − y(i)

h

), (1)

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E. Peruzzo et al.: The SQ method for the numerical simulation of volcanic conduit dynamics under random conditions 51

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(c) Input discretization, with N = 10. (c) Output discretization, with N = 10.

(a) Known input distribution. (b) Unknown output distribution.

Fig. 1. Graphical representation of the first three steps of our strategy, withd=1 andN=10. The lower graphs represent the discreteapproximations of the input and output probability distributions produced by the SQ method. The weightw(i) is the area of the rectangleassociated to thei-th point fori = 1,...,N .

whereK is a smooth probability distribution andh is a mea-sure of the width of the interval over which each weight isspread. h is chosen according to an optimality criterion,based on the minimization of some kind of error resultingfrom the substitution ofg with gKS . K is usually chosen tobe a unimodal probability density symmetric about zero, butits exact expression does not affect very much the result.

3 Quantization of univariate probability distributions(d=1)

As a first step, in order to give a precise meaning to the ex-pression “best approximation”, we introduce a distance be-tween probability distributions. Consequently, the optimaldiscretization off can be defined as the discrete probabilitydistributionf which has the minimum distance fromf .

Since we have to compare discrete and continuous proba-bility distributions, it is easier to rely on the cumulative distri-bution functions, especially in the cased=1, in which thereis only one input random variableX. Let F be the cumula-tive distribution function associated with the densityf andF the one associated withf : namely (see Fig.2),

F(x) =

∫ x

xmin

f (t)dt,

F (x)=∑

x(i) ≤ x

f(x(i)

)=

∑x(i) ≤ x

w(i),

for xmin ≤ x ≤ xmax,

wherexmin andxmax are the minimum and maximum valuesof X, x(i) are the points in whichf is concentrated andw(i)

are the corresponding weights.For instance, we can define a distance betweenf andf as

d(f,f ) =

∫ xmax

xmin

∣∣F(x) − F (x)∣∣dx (2)

(see Fig.3). Therefore, the procedure consists in searchingfor the discrete probability distributionf which minimizesthe quantityd(f,f ), among all the discrete distributions con-centrated inN points.

4 Quantization of multivariate probability distributions(d>1)

A more complicated situation arises when there are severalrandom variablesX1,...,Xd in input, or equivalently a sin-gle random vectorX=(X1,...,Xd). If X1,...,Xd are inde-pendent andf1(x1),...,fd(xd) are their probability densityfunctions, thenf (x1,...,xd) = f1(x1) × ... × fd(xd) is the

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probability density function ofX. If X1,...,Xd are not inde-pendent, the probability density functionf (x1,...,xd) of X

is not related tof1(x1),...,fd(xd) in an obvious way. How-ever, the independency hypothesis is not necessary for theSQ algorithm.

A definition of distance similar to the one in Eq.2 couldbe given, but it is easier and more appropriate to use anotherdefinition of distance, based on the random variables ratherthan on their cumulative distribution functions.

Let X be a discrete random vector with probability distri-bution f , approximating the continuous random vectorX; itseems quite natural to choose, as a measure of the distancebetweenf andf , the mean value of the error

∣∣X−X∣∣ result-

ing from the substitution ofX with X:

d(f,f ) = E[∣∣X − X

∣∣]. (3)

The distance defined by Eq. (3) could be computed numer-ically, but the calculation is easier and faster (especially inhigh dimension) via a MC method; Appendix A shows in de-tail how it can be performed. The MC simulations involvedin the algorithm use only samples ofX and X, which canbe generated easily, and not samples ofY , whose generation

is out of reach. The randomness on the value ofd(f,f )

(and, as a consequence, on the values of the “optimal” pointsx(1),...,x(N)) due to the choice of a stochastic algorithm canbe made negligible, provided that the numberM of MC sim-ulations is large enough. Moreover, the stochastic algorithmeasily provides the weightsw(1),...,w(N) (see Appendix A).

In appendix A we also show that, for univariate distribu-tions, the distances in Eqs. (2) and (3) lead to the same opti-mal discretization: therefore the definition given by Eq. (3),besides having an intuitive motivation, is a natural general-ization of that given by Eq. (2).

5 Testing SQ in a simple case

In order to assess the quality of the approximation of the out-put probability distribution given by the SQ, we first considera simple test case.

– There is a 2-dimensional random vectorX=(X1,X2) ininput; X1 andX2 are independent and their probabil-ity distributions are both gaussians, truncated at 0 andat 1; the distribution ofX1 has mean 0.5 and standard



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00

0

011

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N = 5 N = 10 N = 15

N = 20 N = 25 N = 30

N = 40 N = 50

percentiles percentiles percentiles


percentiles percentiles

Fig. 4. Quantile-quantile plots. On the x-axis there are the true values of the quantiles ofg, on the y-axis their estimates given by the SQ.The orange daggers represent the percentiles, from 1 to 99. If a dagger is close to the dashed liney=x, the SQ gives a good estimate of thecorresponding percentile. Different graphs refer to different numerosities of the SQ.

deviation 0.2, that ofX2 has mean 0.6 and standard de-viation 0.1.

– The relation betweenX and the output random variableY is known and has a simple analytical expression:

Y = ϕ(X1,X2)

=1

8(X2

1 + X1)(X22 + X2) +

1

4(X1 + X2). (4)

In such a simple case it is possible to implement a MCmethod with very high numerosity (e.g. 105), producing anestimate ofg which can be considered exact for practical pur-poses. This version ofg can be directly compared to the re-sults produced by the SQ method and by MC methods withvariable numerosity, in order to establish which one gives thebest approximation ofg: for instance, the SQ method can becompared to a MC with the same numerosity, or to other lownumerosity MC simulations. For each numerosity, the MCcan be repeated several times, thanks to the simplicity of the

functionϕ; consequently, it is possible to estimate the prob-ability that the performances of the SQ are better than thoseof the MC (see Fig.5).

Figure4 compares some quantiles ofg, whose values arevery well known thanks to the high numerosity MC, to theirestimates produced by the SQ method. These estimates areobtained via a linear interpolation of the cumulative distribu-tion associated tog. Figure4 shows that, as the numerosityN of the SQ grows, the estimates of the percentiles ofg glob-ally improve. N=15 is a first good compromise; then theimprovement is not so strong, untilN=40 or 50, where theresult is almost perfect. On the other hand, the central quan-tiles do not become better and better but fluctuate around thetrue values in an unpredictable manner. This is the reasonwhy the estimate of a parameter sometimes gets worse evenif N increases, as is shown in Fig.5: for instance, the esti-mate of the median ofg (red curve) gets worse asN passesfrom 5 to 10. This makes it possible for MC simulations torapidly achieve better estimates of the median than 10 pointsSQ as their numerosity increases.



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C)

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mean

st.dev.q5

q25

q50

q75

q95

N = 5 N = 10

N = 20 N = 25 N = 30

N = 40 N = 50

N = 15

Fig. 5. Comparison between the SQ and MC simulations. On thex axis there is the numerosity of the MC, in logarithmic scale; on the

y axis there is the probability that errSQ<errMC, where errSQ=

∣∣∣p−pSQ

∣∣∣ and errMC=

∣∣∣p−pMC

∣∣∣, p is the real value of a parameter of the

output probability distributiong, pSQ andpMC are its estimates given by SQ and MC, respectively. The curves represent the results forvarious parameters: mean, standard deviation and the 5%, 25%, 50%, 75%, 95% quantiles. The horizontal dashed line marks the value 0.5of probability. Different graphs refer to different values of the numerosityN of the SQ. In each graph, a vertical dashed line is drawn incorrespondance of the MC numerosity equal toN .

However, Fig.5 also shows that there is a general tendencyto improvement asN grows. The whole bundle of curves, infact, moves gradually towards the upper right corner of thegraph: this means that the probability of the estimates givenby the SQ being better than those given by the MC is gener-ally increasing asN grows. Furthermore, it is evident that,if N is sufficiently high (N>10 in this case), the SQ methodgives better results than MC simulations with the same nu-merosity with probability greater than 0.5.

6 Application of SQ to volcanic conduit dynamics

As a test application of the SQ approach to a volcanologicallyrelevant case, we consider a one dimensional steady model ofmagma flow in a cilindrical conduit with fixed diameter anduniform temperature (Papale, 2001). This model is ideal for

testing SQ, since it provides a set of volcanologically rele-vant, strongly non-linear equations relating input and outputdistributions in a complex, unpredictable way, despite keep-ing the computational time small enough (order of minutesfor each simulation) to allow a MC simulation withN=103.The output distribution given by this MC is reasonably closeto the exact one and can be used for comparison with SQ.Hence, this first application of SQ to a volcanologically rele-vant case is also a further test of the method.

Among the several input quantities that are intrinsicallyuncertain we choose two of them, namely, the diameterD ofthe conduit and the total mass fraction of waterwH20. Thesetwo quantities are known to largely control the conduit flowdynamics and the associated mass flow-rate (e.g.,Wilson etal., 1980; Papale et al., 1998). D and wH20 are thereforeconsidered as random variables. We assign to each of them aprobability distribution (truncated gaussian) and we study the



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Fig. 6. Graph(a) represents the probability distribution of the input random vector(D,wH2O

), while in graph(b) there is its discretization

produced by the SQ withN=20. Graph(c) shows a comparison between the approximation ofg resulting from 103 MC simulations andgKS,which is obtained through the application of a kernel smoothing algorithm to the discrete distributiong given by the SQ withN=10,15,20.Graph(d) represents a comparison between the cumulative distribution function resulting from 103 MC simulations and those resulting fromthe SQ withN=10,15,20, without application of kernel smoothing.

corresponding probability distribution of the logarithm of themass-flow ratem. The latter is a volcanologically relevantquantity, as it defines the intensity of an eruption and as itlargely affects the impact on the surroundings (Valentine andWohletz, 1989; Todesco et al., 2002).

Figure 6a shows the assumed continuous distribution ofthe two random variableswH20 andD, while Fig. 6b illus-trates the result of the application of the SQ method to dis-cretize the distribution in 20wH20-D pairs. In order to testthe SQ method, the discretization has been done also with

15 and 10wH20-D pairs. The results have been compared, interms of output probability density (Fig.6c) and cumulativeprobability distribution (Fig.6d).

The three SQ cases withN=10, 15 and 20 reproduce wellthe mode of the distribution at a mass flow-rate of about108 kg/s, but fail in predicting correctly the shape of the dis-tribution, resulting in a larger and less skewed curve withrespect to the MC case. The larger values of density pre-dicted by SQ at the high mass flow-rate tail largely explainthe lower values of the mode with respect to the MC case.



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Fig. 7. Quantile-quantile plots, analogous to Fig.4. On the x-axis there are the estimates of the quantiles ofg given by a MC simulation withN = 103, which are close to the true values.

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rrS

Q<

err M

C) MC numerosity

mean

st.dev.q5

q25

q50

q75

q95

N = 10

N = 20

N = 1510 20 50 100 200

10 20 50 100 200

Fig. 8. Comparison between the SQ and MC simulations, analogousto Fig.5.

On the contrary, the left tail of the distribution, correspond-ing to the minimum mass flow-rates, is predicted accuratelyby the SQ. The improvement due to increased numerosity ofSQ from 10 to 20 is clearly visible from the cumulative plotsin Fig. 6d.

Figure7 compares the estimates of the quantiles of the out-put distribution of mass flow-rates given by MC and SQ withN=10, 15 and 20. The quantiles are obtained by means of alinear interpolation of the cumulative distributions in Fig.6d.As for the analogous Fig.4 (referring to the polynomial mapat Eq.4), the bulk of the distribution is predicted well by theSQ method, but the tails of the distribution are not. Whilea significant improvement clearly emerges fromN=10 toN=15, there is no significant gain in accuracy when moving

Table 1. Comparison between the estimates of some parameters ofthe output distribution given by the SQ withN=10,15,20 pointsand by a MC with 103 simulations. The considered parameters arethe mean, the standard deviation and the 5%, 25%, 50%, 75%, 95%quantiles.

mean st. dev. q5 q25 q50 q75 q95

MC 7.74 0.50 6.71 7.45 7.86 8.12 8.38SQ,N=10 7.75 0.41 7.09 7.32 7.86 8.07 8.32SQ,N=15 7.74 0.44 6.94 7.46 7.82 8.10 8.33SQ,N=20 7.73 0.46 6.85 7.49 7.79 8.09 8.35

from N=15 to N=20. Table1 shows the same results nu-merically for a few selected quantiles.

As for the ideal case discussed above, it is possible to com-pare the performances of the SQ with those of some low nu-merosity MC simulations (Fig.8). This time, the discretedistributions generated by the SQ and by the low numeros-ity MC simulations are both compared to the one obtainedfrom the MC simulation withN=103. In all cases, and forany quantity investigated, the SQ provides a better approxi-mation of the output distribution than the MC with equal nu-merosity. For many quantities a MC with at least hundreds ofsimulations is required in order to exceed the accuracy givenby SQ simulations with numerosity up to 20.

7 Conclusions

The performance of the SQ method has been analyzed bothfor an artificial polynomial map with random input and fora more complex set of non-linear equations. The latter casealso represents an application of the SQ method to the vol-canologically relevant case of steady multiphase magma flowalong a volcanic conduit. Our analysis includes both a com-parison between SQ and pure MC (Monte Carlo) method,and the accuracy of SQ in itself. In both cases the SQ method



provides substantially better estimates of output distributionsthan the MC method with the same number of simulations.This property is already clear with values ofN around 15–20, and it becomes stronger with the increase ofN . The re-sults are less definite for very smallN , like N=5 and some-times 10, where further ideas and research are needed. Ingeneral, MC gives results comparable to SQ only when thenumber of MC simulations is much higher, sometimes byone or more orders of magnitude, than that of SQ. Therefore,the SQ method results in a considerable computational sav-ing for the same degree of accuracy of the estimates. Withvalues ofN of 15 or 20, in general the estimates obtained bySQ are very close to the true ones (or to our better estimatesof the true ones).

In conclusion, the SQ method allows the introduction ofuncertainties in the deterministic approach without requiringexceeding CPU time. This result is promising for the capa-bility of estimating future volcanic scenarios and volcanichazards by means of a merged deterministic-probabilisticapproach, whereby complex deterministic models are em-ployed by taking into account the intrinsic uncertainties in-volved in the definition of the conditions characterizing thevolcanic systems.

Appendix A

The numerical algorithm

This appendix describes in detail how the distanced(f,f ),defined in Eq. (3), is approximately computed via astochastic algorithm. Morevorer, it shows that the op-timal discretization off is found by moving the pointsx(1),...,x(N) in such a way thatd(f,f ) is minimized, i.e. byminimizing a function ofN d-dimensional vectors (functionh in Eq.A3 below).

The starting point is represented by a fundamental result ofthe SQ theory (Graf and Luschgy, 2000, Lemma 3.1), whichstates that, if theN possible values(x

(1)1 ,...,x

(1)d

),...,

(x

(N)1 ,...,x

(N)d

)of the random vectorX, i.e. theN points in whichf is con-centrated, are fixed, then the corresponding optimal weightsw(1),...,w(N) are uniquely determined.

More precisely, the weights are defined as follows.

– Fori=1,...,N , letVi be the region of thed-dimensionalspace such that

x ∈ Vi⇐⇒

∣∣∣x−x(i)∣∣∣=mink=1,...,N

∣∣∣x−x(k)∣∣∣,

where x(k)=

(x

(k)1 ,...,x

(k)d

)for k=1,...,N . Vi is

called the Voronoi region ofx(i) with respect to theset

{x(1),...,x(N)

}; it contains the points which are

closer to x(i) than to any other element of the set{x(1),...,x(N)

}(see figureA1).

– The optimal approximationX of the random vectorXis defined as follows:X=x(i) if and only if the value ofX belongs toVi . Namely,X is obtained by rounding offX to the nearest vector amongx(1),...,x(N).

– Correspondingly,w(i) (i.e. the probability thatX=x(i))is the weight assigned toVi by the probability distribu-tion f :

w(i)=

∫Vi

f (x)dx.

If X is defined as just described, it can be shown (Graf andLuschgy, 2000, Lemma 3.4) that, in the cased=1,

E[|X−X|

]=

∫ xmax

xmin

∣∣F(x)−F (x)∣∣dx;

moreover, ifX′ is another random variable, with the samepossible valuesx(1),...,x(N) but defined in whatever way,andF ′ is its cumulative distribution function, it can be shownthat

E[|X−X′

|

]≥

∫ xmax

xmin

∣∣F(x) − F ′(x)∣∣dx (A1)

≥

∫ xmax

xmin

∣∣F(x) − F (x)∣∣dx = E

[|X − X|

].

This means that, in the cased=1, minimizing∫ xmaxxmin

∣∣F(x)−F (x)∣∣dx is the same as minimizing

E[|X−X|

], so that the criterion used when there are

several parameters in input is indeed a generalization of thatused when there is only one parameter.

If the points(x(1),...,x(N)

)are fixed, an approximate cal-

culation of the minimum value of the distance in Eq. (3) andof the corresponding optimal weights

(w(1),...,w(N)

)can be

carried out through the following steps (see Fig.A1):

1. generate a large numberM (e.g. M=105) of d-dimensional random vectorsz1,...,zM with probabilitydistributionf ;

2. for each vectorzj , select the indexij such thatzj be-longs toVij , i.e.∣∣∣zj−x(ij )

∣∣∣=mink=1,...,N

∣∣∣zj−x(k)∣∣∣;

3. for k = 1,...,N , assign tox(k) the weight

w(k)=

m(k)

M,



0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

x1 x1

x2

x2

x(1) x(2)

x(3)

x(4)

x(5)

x(6)

x(7)

Fig. A1. Implementation of the stochastic quantization method withd = 2 andN=7. The blue points are a sample of the input randomvectorX=(X1,X2); the orange pointsx(1),...,x(7) are the possible values ofX, which is a discrete approximation ofX. The orange lines

define the Voronoi regions generated by the set{x(1),...,x(7)

}: the region associated tox(i) contains the sample points which are closer to

x(i) than to any other of the orange points.

wherem(k) is the number of vectorszj into the VoronoiregionVk, i.e. the number of indexesj such thatij=k;

4. calculate

d(f,f ) = E[∣∣X − X

∣∣] ≈1

M

M∑j=1

∣∣zj − x(ij )∣∣. (A2)

In our situation, in which the points(x(1),...,x(N)) are notfixed, the function

h(x(1),...,x(N)

)=

1

M

M∑j=1

∣∣zj − x(ij )∣∣ (A3)

must be minimized. The minimization is performed usingPowell’s method (Powell, 1964; Press et al., 2001), whichmoves the pointsx(1),...,x(N), starting from an initial guess;for each new choice ofx(1),...,x(N), the algorithm evaluatesh(x(1),...,x(N)

)going through the steps 1–4 above. The set

of points which produces the minimum value ofh is just theoptimal set of points we are searching for. In order to min-imize the risk of finding local minima, the minimization isrepeated 10 times, varying the initial guesses, and the lowestminimum is taken as the best estimate of the true minimum.

Note that the error in the estimate (Eq.A2) of d(f,f ) isproportional to 1

√M

, so that, for sufficiently high values of

M, it becomes negligible and minimizing

1

M

M∑j=1

∣∣zj−x(ij )∣∣

is the same as minimizingd(f,f ).

Acknowledgements.The authors are grateful to two anonymousreferees whose careful reading of the submitted paper contributedto improve its clarity.

Edited by: R. Moretti

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