全局灵敏度分析：建模与模拟的基石

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"全球敏感性分析" 全局灵敏度分析（Global Sensitivity Analysis，GSA）是一种统计方法，用于评估模型中的输入参数变化如何影响输出结果的不确定性。在模型建模和数值模拟领域，全局敏感性分析是至关重要的工具，它帮助识别哪些输入参数对模型预测最敏感，从而提供关于模型行为的深入理解。描述中提到，模型建模在不同学术领域中往往各自独立发展，这可能引发了一个问题：模型建模是否更像一门手艺而非科学？换句话说，是否存在一套通用规则来构建描述自然或人为系统的数学模型？全局敏感性分析正试图解答这个问题，通过量化参数敏感性，它有助于确定哪些规则和假设对于模型的准确性和可靠性最为关键。《全球敏感性分析：基础指南》是由Andrea Saltelli、Marco Ratto等作者合著的一本专著，其中涵盖了全球敏感性分析的理论、方法和实践应用。这本书的出版机构是John Wiley & Sons Ltd，旨在为科研人员、工程师和政策制定者提供一个理解和应用全局敏感性分析的综合资源。全局敏感性分析的应用广泛，包括但不限于环境科学、工程设计、社会科学和经济学等领域。它可以帮助研究人员识别关键参数，优化实验设计，减少不必要的计算成本，并提高模型的预测能力。常见的全局敏感性分析方法有 Sobol' 方法、Morris方法、FAST方法等，这些方法能够量化单个参数以及参数之间的交互效应对模型输出的影响。在实际操作中，全局敏感性分析通常包括以下几个步骤： 1. 定义模型：明确模型的目标、输入参数和输出结果。 2. 参数化：为输入参数设定可能的取值范围，通常通过实验数据或专家知识确定。 3. 执行模型：使用不同的参数组合运行模型，生成输出结果的样本集。 4. 敏感性分析：应用适当的统计技术（如方差分解或响应面方法）来量化参数的敏感度。 5. 结果解释：识别并解读对模型最敏感的参数，以及参数之间的相互作用。 6. 验证和确认：通过额外的实验或数据验证敏感性分析的结果。全局敏感性分析的重要性在于它能够帮助模型开发者理解哪些参数的变动可能导致模型预测的重大变化，从而改进模型结构，提高模型的稳健性和实用性。此外，这种方法也适用于识别模型中的不确定性和局限性，为决策者提供更有依据的建议。在模型开发和应用过程中，全局敏感性分析是一个不可或缺的工具，它促进了不同学科间的交流和合作，推动了建模科学的发展。

MODELS AND SENSITIVITY ANALYSIS 3

modellers can generate different nonequivalent descriptions of it, that is,

models whose outputs are compatible with the same set of observations but

whose structures are not reconcilable with one another.

While this may be disturbing to a student accustomed to the beauty and

apparent self-evidence of physical laws, practitioners of modelling have

come to live with the rather unpleasant reality that more than one model

may be compatible with the same set of data or evidence. Some have gone

so far as to coin a word for this paradox: equifinality – Beven (1993,

2001), see also Saltelli et al. (2004, pp. 173–178) – meaning that different

models can lead to the same end. Others refer to the phenomenon as model

indeterminacy.

Since Galileo’s time scientists have had to deal with the limited capacity

of the human mind to create useful maps of ‘world’ into ‘model’. The

emergence of ‘laws’ can be seen in this context as the painful process

of simplification, separation and identification which leads to a model of

uncharacteristic simplicity and beauty.

1.1.3 Models and Uncertainty

What makes modelling and scientific inquiry in general so painful is uncer-

tainty. Uncertainty is not an accident of the scientific method, but its

substance.

Modellers and philosophers of science have debated the issue of model

indeterminacy at length (Oreskes et al., 1994). Most modellers today would

probably agree that a model cannot be validated, in the sense of ‘be proven

true’. Rather, it is more defensible and correct to say that a model has

been extensively corroborated, meaning by this that the model has survived

a series of tests – be they formal, of internal consistency, or relative to

the model’s capacity to explain or predict the ‘world’ in a convincing and

parsimonious way.

When models fail publicly, the ensuing controversy can be devastating

for the scientific parties involved.

Models are often used in highly polar-

ized contexts and uncertainty may be used instrumentally. ‘All parties deal

with environmental information in a selective way, or even manipulate it’,

observed a Dutch environmental scientist (In ’t Veld, 2000). Fabricated

‘That is what we meant by science. That both question and answer are tied up with uncer-

tainty, and that they are painful. But that there is no way around them. And that you hide

nothing; instead, everything is brought out into the open’ (Høeg, 1995).

For the modelling credibility crisis in the Netherlands’ RIVM Laboratories see Van der Sluijs

(2002). See also Mac Lane (1988) for another example.

MODELS AND SENSITIVITY ANALYSIS 5

associated with ‘fitness for purpose’ – that is, with a specific purpose – than

with the model’s intrinsic fabric.

A post-normal view of the modes of scientific production in relation to

policy is given in Funtowicz (2004). Models as metaphors are discussed in

Ravetz (2006).

1.1.4 How to Set Up Uncertainty and Sensitivity Analyses

As mentioned at the beginning of the chapter, our definition of sensitivity

analysis involves models, model input and model output. We now try to

define model input in relation to the nature and purpose of the model, as

well as to the set-up of the uncertainty and sensitivity analyses. A model

can be:

•

Diagnostic or prognostic. In other words, we try to distinguish between

models used to understand a law and models used to predict the

behaviour of a system given a supposedly understood law. Models can

thus range from wild speculations used to play what-if games (e.g. models

for the existence of extraterrestrial intelligence) to models which can be

considered accurate and trusted predictors of a system (e.g. a control

system for a chemical plant).

•

Data-driven or law-driven. A law-driven model tries to put together

accepted laws which have been attributed to the system, in order to

predict its behaviour. For example, we use Darcy’s and Ficks’ laws to

understand the motion of a solute in water flowing through a porous

medium. A data-driven model tries to treat the solute as a signal and to

derive its properties statistically. Advocates of data-driven models like to

point out that these can be built so as to be parsimonious, i.e. to describe

reality with a minimum of adjustable parameters (Young et al., 1996).

Law-driven models, by contrast, are customarily overparametrized, as

they may include more relevant laws than the amount of available data

would support. For the same reason, law-driven models may have a

greater capacity to describe the system under unobserved circumstances,

while data-driven models tend to adhere to the behaviour associated with

the data used in their estimation. Statistical models (such as hierarchical

or multilevel models) are another example of data-driven models.

Many other categorizations of models are possible,

and the definition of

model input depends on the particular model under study. For the purpose

Bell et al. (1988) distinguish between formal (axiomatic), descriptive and normative models

(rules an agent should follow to reach a target). The examples in this book are descriptive

models.

MODELS AND SENSITIVITY ANALYSIS 7

from N ¯ 



 and likewise for the distribution function of the other

factors.

⎡

⎢

⎣



1



1



1





2



2



2



   



N −1



N −1



N −1





N



N



N



⎤

⎥

⎦

(1.1)

•

We can then compute (‘run’ is the conventional term) the model for all

vectors 

j



j

 thereby producing a set of N values of a model

output Y

⎡

⎢

⎣

1

2



N −1

N

⎤

⎥

⎦

(1.2)

These steps constitute our uncertainty analysis. From these we can

compute the average output, its standard deviation, the quantiles of its

distribution, confidence bounds, plot the distribution itself and so on. It is

clear that in this analysis, sometimes called a ‘parametric bootstrap’,

our

inputs are the model’s parameters. Having performed this uncertainty anal-

ysis we can then move on to a sensitivity analysis, in order to determine

which of the input parameters are more important in influencing the uncer-

tainty in the model output. However, we defer this step in order to continue

our discussion of model input.

Note that for the purpose of the uncertainty analysis just described we

consider as relevant inputs only our estimated parameters. All other types

of information fed into the model, e.g. the observations, physical or math-

ematical constants, internal model variables (e.g. number of grid points if

the model needs a mesh), are disregarded – that is, we do not allow them

to vary and hence they cannot cause variation in the output.

In Figure 1.4 we have played the uncertainty analysis game differently by

sampling the observations rather than the parameters. We have a limited

set of observations, and we are aware that different subsets of these could

Note that this model output Y

may be different from the model output used in the estimation

step.

Bootstrapping is the process of repeatedly sampling ‘with replacements’. For example, if we

want to estimate the average sum of three Bingo chips, we could do this by extracting three

random chips from the Bingo bag, computing their average, putting the chips back into the

bag and extracting again. With a sufficiently large number of extractions we could determine

the average sum being sought, and this strategy would be called a bootstrap of the Bingo chips.

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