Global Sensitivity Analysis with Small Sample Sizes: Ordinary Least

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Global Sensitivity Analysis with Small Sample Sizes: Ordinary Least Squares Approach Michael J. Davis, Wei Liu, and Raghu Sivaramakrishnan J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b09310 • Publication Date (Web): 21 Dec 2016 Downloaded from http://pubs.acs.org on December 27, 2016

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The Journal of Physical Chemistry

Global Sensitivity Analysis with Small Sample Sizes: Ordinary Least Squares Approach Michael J. Davis*, Wei Liu, and Raghu Sivaramakrishnan Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, IL 60439.

ABSTRACT. A new version of global sensitivity analysis is developed in this paper. This new version coupled with tools from statistics, machine learning, and optimization can devise small sample sizes that allow for the accurate ordering of sensitivity coefficients for the first 10-30 most sensitive chemical reactions in complex chemical-kinetic mechanisms, and is particularly useful for studying the chemistry in realistic devices. A key part of the paper is calibration of these small samples.

Because these small sample sizes are developed for use in realistic

combustion devices, the calibration is done over the ranges of conditions in such devices, with a test case being the operating conditions of a compression ignition engine studied earlier. Compression-ignition engines operate under low-temperature combustion conditions with quite complicated chemistry making this calibration difficult, leading to the possibility of false positives and false negatives in the ordering of the reactions. So an important aspect of the paper is showing how to handle the trade-off between false positives and false negatives using ideas from the multi-objective optimization literature. The combination of the new global sensitivity method and the calibration are sample sizes a factor of approximately 10 times smaller than were available with our previous algorithm.

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1. INTRODUCTION The sensitivity of chemical kinetic modeling to the uncertainties of the rate coefficients in the model has been a focus of considerable research over the last several decades.1-16 The sensitivities are often measured locally, that is close to the nominal values of the rate coefficients and this is referred to as local sensitivity analysis.2,17 The sensitivity can also be calculated over the full range of the estimated uncertainties of the reaction rate coefficients and this is referred to as global sensitivity analysis.1 Global sensitivity analysis (GSA) has been an increasing area of research for chemical kinetic models over the last 20 years,1,13 although it has a longer history.18 Global sensitivity analysis can be used to explore the full probability density of the uncertainty14,19 something that cannot be accomplished within a local framework. In a recent publication, we along with several co-authors studied the effect of individual rate coefficients on the ignition properties in simulations of a compression-ignition engine (e.g., a diesel engine).20 It was shown in Ref. 20 that ignition delay times for the compression-ignition engine depended strongly on some of the reaction rate coefficients. In earlier applications of global sensitivity analysis we typically used at least 40 simulations per reaction to generate accurate global sensitivity analysis.13,14 Because the reaction mechanism developed in Ref. 21 and used in Ref. 20 for engine simulations contains 914 reactions this would mean approximately 36,000 simulations would be required, making it difficult to perform GSA for this type of detailed engine simulation, as each engine simulation takes on the order of several processor weeks to complete.20 In order to make


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GSA of engine simulations for realistic chemical mechanisms relatively routine, we have developed new ways of doing small-sample GSA as reported here. The reduction in sample size for GSA was accomplished in three stages. The first stage is the development of a re-designed algorithm for global sensitivity analysis that reduces the number of simulations significantly. The second stage in the reduction relies on model and reaction selection. Model selection is often used in machine learning22-25 and involves procedures to pick the best model out of a group of models. The model-selection problem is expanded somewhat in this study to the selection of a model that is most appropriate for a given sample size. “Reaction selection” is a shorthand description for describing the process of choosing a limited number of reactions for which the target (e.g., ignition delay time) is most sensitive. The number of reactions selected will once again depend on the sample size. Another way of viewing the model and reaction selection procedure is as a calibration: what sample size and what model will accurately find, for example, the 10 most sensitive chemical reactions in an engine simulation? This calibration is done over the range of chemical conditions that are deemed important for the ignition process in the engine simulations. The first two stages of the reduction described in this paper reduce sample sizes by approximately a factor of ten. The third and final stage of the development of small-sample size GSA reduces the sample size by another factor of 3-5 using sparse regression techniques26,27 in conjunction with the calibration techniques developed in this paper. Although we study the same chemical model in this paper as in the engine simulations,20 this model is embedded in a much simpler physical system, a constant pressure, “zerodimensional” system, a well-stirred reactor.28 We will study this simpler model as a way of calibrating the accuracy of small-sample size GSA. We accomplish the calibration by


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quantitative comparisons of small-sample size GSA with more fully converged large-sample sized versions. In order for the calibration to be useful for the engine simulations we study the global sensitivities for a range of these systems that are meant to cover the range of chemical conditions in the engine simulations most relevant for ignition. The reason we undertook this project was the desire to extend the results of Ref. 20, where the sensitivity of ignition delay times in a compression-ignition engine model was studied for a few reactions. Global sensitivity analysis allows the study of the sensitivities of all the chemical reactions in a systematic fashion. Because the compression-ignition engine simulations are computationally intensive, it led us to devise methods to limit the number of simulations, i.e. the “sample size”, which would be necessary to obtain an accurate ordering of sensitivity coefficients. The goal of an accurate ordering of the sensitivity coefficients is more limited than many applications of sensitivity analysis (see for, example Ref. 15 and the papers cited there) and is one of the reasons that the sample sizes can be small. In our earlier applications of global sensitivity analysis,13-16 we used the highdimensional model representation (HDMR, for example, Ref. 5-9), but in this paper are presenting a new algorithm. The motivation for using this new algorithm was that it was straightforward to develop a sparse version of the algorithm using the well-studied techniques described, for example in Refs.. 23, 26, and 27, that have the potential to significantly reduce the sample size further. The change in the algorithm also allows us to use many techniques developed in the regression literature, including diagnostics, as described below. Section 2 will discuss background material, including the chemical model and mechanism and introduce global sensitivity analysis. Section 3 systematically introduces and


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develops a set of methods to evaluate and calibrate small-sample size global sensitivity analysis and discusses how the space of chemical conditions will be studied. Section 4 presents results and develops and addresses the important role that trade-offs between false positives and false negatives play in choosing small sample sizes to use in device simulations. Section 5 presents additional discussion and makes several conclusions.

2. BACKGROUND 2.1 Biodiesel Ignition and Kinetics Model. In the US, traditional combustion sources and processes have been historically used to meet a dominant fraction of the energy requirements. This scenario is unlikely to change in the next few decades as is evident in the annual reports published by the US Energy Information Administration.29 An even more revealing prediction30 indicates that in 2035 the vehicle fleet will still be predominantly powered by internal combustion (IC) engines. With the recognition that transportation is a significant source of greenhouse gas (GHG) emissions, increasingly stringent emissions compliance legislations31,32 have been put forth and this has led to a rapid growth in the usage of alternative biofuels in IC engines. Biodiesel is an alternative fuel that either when used as is or in blends with conventional petro-derived diesel has been shown33,34 to offer substantial emissions advantages in CI engines. In a recent study,21 we have developed detailed and reduced kinetic models for a biodiesel surrogate (a blend of n-heptane and methylbutanoate) with a view to applying the reduced model in multi-dimensional CI engine simulations.20


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The detailed model developed for the biodiesel surrogate in Ref. 21 included 661 species and 3019 reactions and this was reduced using the directed relation graph (DRG) based method. Specifically, DRG-X21 followed by DRG-aided sensitivity analysis was utilized to generate a reduced version of the model21 with 166 species and 914 reactions. This reduced model is used in all the simulations reported here. The reduced model is used in a systematic GSA study of auto-ignition delays for a homogeneous, adiabatic system at constant pressure as implemented in SENKIN.28 Autoignition delays are defined as the time delay to which the temperature rises by 400 K from the initial temperature. The results from this study serve to be calibration benchmarks for comparison against small sample size GSA for the multi-dimensional CI engine simulations.

2.2 Chemical reaction rate coefficient uncertainties. Chemical reaction rates are generally imprecise, with an uncertainty range between 10% to several orders of magnitude.35 The evaluation of the effect of this uncertainty is embodied in a set of tools in the field of uncertainty and sensitivity analysis, which has a long history in the study of chemical kinetic modeling. The uncertainty is often examined locally in the form of local sensitivity analysis.5 Local sensitivity evaluates how changes in the rate coefficients effect the outcome of a chemical kinetics simulation when the rate coefficients are changed in the vicinity of the nominal rate coefficients. The sensitivity is generally written in terms of a differential, as the changes are made in the near vicinity of the rate coefficients


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Global sensitivity analysis calculates the effect of the changes in rate coefficients over their whole range of uncertainty.1,13,18 The method of global sensitivity analysis used in this paper follows a long tradition of uncertainty analysis in the chemical kinetics literature and beyond. The uncertainty of the target is represented by the variance in the target (e.g., ignition delay times or species concentrations) induced by varying all rate coefficients within their ranges of uncertainties. To obtain the global sensitivity of a particular rate coefficient the variance is decomposed in a set of partial variances for each reaction leading to a set of sensitivity coefficients. The mathematical details of the global sensitivity analysis developed and used in this paper are presented in Sec. 3.1. The analysis is based on the Arrhenius form of the rate coefficient, as discussed in several places. 13 The range of uncertainty for a rate coefficient is often represented as a multiplicative factor on the A-factor in the Arrhenius form of the rate coefficient,13-16 with the range of uncertainty of A written as:

∆log(A) = 2logf

.

(2.1a)

To perform global sensitivity analysis, a set of rate coefficients are generated by sampling the uncertainty ranges of all the A-factors:

A c = fc A 0 ,

(2.1b)

where A0 is the A-factor for the nominal rate constant and fc lies within the uncertainty range and is sampled randomly for the range in Eq. (2.1a):

log(fc ) = (2u-1)logf ,

(2.1c)


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where u is a random number between 0 and 1. Section 3.1 shows how the results from a series of simulations with rate coefficients sampled in this manner are used to find global sensitivity coefficients. 3. METHODS This section describes the main tools that lead to the selection of small sample sizes that accurately order the first several sensitivity coefficients. Section 3.1 presents the new algorithm and shows how the sensitivity coefficients are extracted. The algorithm described in Sec. 3.1 requires expansion coefficients and Sec. 3.2 describes how these are calculated. The full use of the algorithm requires diagnostic tools related to convergence of the fit and Sec. 3.2 also describes how these are generated. One set of models in Sec. 3.1 requires interaction terms, which are cross-terms in the expansion. Because we include all the reactions in our models, there are a very large number of potential interaction terms. Section 3.3 describes how a subset of these cross-terms are chosen based on the sensitivity coefficients derived from models without interaction terms, coupled with an analysis of subsets of interaction terms. Section 3.4 introduces the global sensitivity analysis of the system studied here and demonstrates how complex the sensitivity spectrum can be. The chemical system studied here is a high-dimensional system and it is difficult to adequately sample such a large dimensional space, which is a well-known problem,36 making the diagnosis of the accuracy important. Sections 3.5 and 3.6 are detailed, because it was our experience that there were many pitfalls in trying to get an accurate ordering of the sensitivities. Section 3.5 examines how the in-sample and out-of sample errors vary with sample size. Section 3.5 also demonstrates how to choose among the models introduced in Sec. 3.1. Because the sample sizes are small for the


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dimensionality of the space, the ordering of the reactions will vary with different samples of the same size and Sec. 3.6 introduces methods for examining this feature of the calculations. Based on our previous work,20 we were able to discern a range of chemical conditions that contributed most to the ignition in the engine simulations. Because we wish to have samples that will accurately order reactions in these engine simulations, all small samples need to be calibrated over this set of conditions, which is what is developed in Sec. 3.7. 3.1 Modified Algorithm for Global Sensitivity Analysis. The results of the simulations described in Sec. 2.2 lead to a series of ignition delay times. These ignition delay times are a function of the uncertainties described in Eqs. (2.1a) – (2.1c). The ignition delay times are written as a function of the uncertainties in one of the following polynomial expansions:

τ ({u i }) = (1)

nr

r

∑∑a

ik

u ik (3.1a)

i=1 k=1

τ ({u i }) = (2)

nr

r

∑∑a i=1 k=1

ik

u + k i

nt

ms

∑ ∑ {b

qm

}(s) Q (s) (u i, u j )

q=1 m=1

(3.1b)

“r” refers to the order of the polynomial and “Q” refers to an interaction term. The subscripts inside the “Q” terms refer to pairs of reactions chosen from the list of possible interaction terms which is quite large for the chemical model described in Sec. (2.1). The interaction terms include pairs of reactions and “s” refers to either 2, 3 or 4 indicating


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quadratic, cubic, or quartic interaction terms. There are 914 reactions and thus over 417,000 possible interaction terms (0.5 x 914 x 913). The quadratic, cubic and quartic interaction terms are:

Q(2) (u i , u j ) = u i u j (m s = 1)

(3.2a)

Q(3) (u i , u j ) = u i u j, u 2i u j, u i u 2j (m s = 3) (3.2b)

Q(4) (u i , u j ) = u i u j, u 2ii u j, u i u 2j , u 3i u j, u i2u 2j , u i u 3j (m s = 6)

(3.2c)

The choice of the specific set of the “nt” interactions in the summation for Eq. (3.1b) will be discussed in Sec. (3.3). The notation in the second summation of Eq. (3.1b) refers to an enumerated list of nt interaction terms with a specific number of terms listed in Equations (3.2a) – (3.2c). The interaction model is defined by the enumerated list of interaction terms and the value of “s” which leads to ms terms for each of the nt interaction terms. Equations (3.1a) and (3.1b) are written as if there is no intercept in the expansion, because the data can be centered in such a way that the intercept is 0. However, some of the calculations carried out here include an intercept term. This difference does not affect the sensitivity coefficients in any way and we do not distinguish the two types of calculations in the paper.


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The coefficients in Eqs. (3.1a) and (3.1b) are estimated with linear regression,37 which results from minimizing the least squares error.37 In previous publications linear regression was also used to find the sensitivity coefficients.13 However, in those papers a series of fits were done for each reaction and each pair of reactions, rather than the single fits of Eqs. (3.1a) or (3.1b). With the earlier algorithm, the chemical model studied here would require 914 fits, one for each reaction, rather than the single, much larger fits in Eqs. (3.1a) or (3.1b). The “divide and conquer” strategy of the earlier work generally requires less computer time and much less computer memory. However, we have found that a smaller number of simulations are needed with the current method to calculate accurate sensitivity coefficients. For example, in a previous paper14 it was indicated that accurate sensitivity coefficients require approximately 40 computer simulations per reaction with the older algorithm, which for the present chemical model would be over 36,000 simulations. As demonstrated in the rest of the paper, accurate sensitivity coefficients can be obtained with considerably less simulations than that. There is another advantage of using the expansions in Eqs. (3.1a) and (3.1b), compared to our earlier method. Sparse regression26,27 can be used to calculate the coefficients in Eqs. (3.1a) and (3.1b). Accurate sensitivity coefficients can be calculated using sparse regression with even less simulations than what is used in this paper. The calculated regression coefficients in Eq. (3.1a) or (3.1b) are used to calculate sensitivity coefficients. To obtain the ith sensitivity coefficient, integration of all u’s, except ui is performed. For example, a quadratic expansion for Eq. (3.1b) gives the following:


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1 ni τ (u i ) = q i + a i1u i + a i2 u + ∑ b iju i 2 j=1 (2)

2 i

(3.3a)

where qi is an integration constant, and as discussed in sec. 2.2, the u’s are between 0 and 1 as indicated in Eq. (2.1c). The summation refers to all ni interaction terms which include ui [Sec. (3.3)]. Collecting terms further leads to:

1 ni τ (u i ) = q i + c i1u i + a i2 u , c i1 = a i1 + ∑ b ij 2 j=1 (2)

2 i

(3.3b)

The variance of this curve leads to a sensitivity coefficient and because the constant, qi, does not contribute to the variance, it will be dropped in the rest of the development. For the function in Eq. (3.3b), the average of the ignition delay time and the average of the square of the ignition delay time are:

1 1 < τ > i = c i1 + a i2 , 2 3 1 1 1 < τ 2 > i = c 2i1 + c i1 a i2 + a 2i2 3 2 5 .

(3.4a)

(3.4b)

The variance along the curve, referred to as the “partial variance”, is:


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Vi = < τ > i − ( < τ > i ) 2

Vi =

2

1 1 1 1  1 = c 2i1 + c i1a i2 + a 2i2 −  c i1 + a i2  2 3 2 5 3 

1 2 1 4 2 c i1 + c i1a i2 + a i2 12 6 45

2

,

(3.4c)

leading to the sensitivity coefficient, which is defined as:

Si ≡

Vi V,

(3.4d)

“V” is the numerical variance of the ignition delay time. Note that the superscript over τ used in previous equations was dropped in Eqs. (3.4a) – (3.4d). Equations (3.4a) – (3.4d) define a first-order sensitivity coefficient.1,2 Secondorder1,2 sensitivities can also be calculated. These describe the interaction of two reactions. The second-order sensitivities are calculated for two-reaction surfaces formed by integrating out all terms that do not include the two reactions. An example of such a surface is presented here for a quadratic expansion:

1 2

1 2

τ (2) (u i , u j ) = q ij + (c i1 − b ij ) u i + a i2 u 2i + (c j1 − b ij ) u j + a j2 u 2j + b iju i u j

(3.5a) The following quantity can be calculated from Eq. (3.5a):


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Vij = τ 2 (u i, u j ) − τ (u i , u j )

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2

,

(3.5b)

leading to a second-order sensitivity coefficient1,2:

Sij =

Vij V

=

Vij − (Vi + Vj) V

(3.5c)

In the development leading to Eq. (3.5c), the integration constant, qij, is once again dropped, as it does not contribute to the partial variance. In previous work13 difficulties arose in calculating Sij accurately, because the quantities in Eq. (3.5c), Vi, Vj and Vij, were calculated from three different fits. In fact, it was common to have negative values of Sij unless the sample sizes were very large. In Ref. 16 one million simulations were used to converge a significant number of second-order coefficients in a study of butanol ignition and speciation for a chemical mechanism with 1443 reactions. There are no such difficulties for the present method, as all variables in Eq. (3.5a) and (3.5b) come from the same fit. Starting from these equations straightforward, but tedious, algebra can be used to simplify the interaction-term sensitivities for the quadratic fit: 2 bsw V = 144 int sw

(3.6a)


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and cubic fit:

int Vsw =

1 11 1 21 2 2 21 2 (bsw + b12 [(b12 sw + b sw ) + sw ) + (b sw ) ] 144 2160

(3.6b)

Because these terms are obviously positive, they demonstrate that second-order sensitivity coefficients calculated for the relevant expansions are positive even if the fits are poor (i.e., the b’s are not accurately converged). Note that Eq. (3.6b) refers to Eq. (3.1b). The sum of first-order and second-order sensitivity coefficients satisfy the following relationship: nr

nr

∑S + ∑ ∑S i

i=1

j

Global Sensitivity Analysis with Small Sample Sizes: Ordinary Least

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