arXiv subjects

Andrea Saltelli

Publications and source records attributed to Andrea Saltelli.

At least 19 recordsLinked to original sources

arXiv2026

Improving Discrepancy Measures for Global Sensitivity Analysis

Sensitivity analysis methods based on Sobol' total-order indices ($T_i$) are well-founded but computationally demanding. A recently proposed ersatz discrepancy measure offers a cheaper alternative by quantifying deviations from uniformity in input--output scatterplots, yet lacks theoretical grounding and has not been benchmarked against other data-given estimators. We introduce an adjusted ersatz discrepancy that rank-transforms the output before gridding and imputes isolated empty cells via a Moore-neighbourhood rule, substantially improving agreement with $T_i$. We prove, via a copula-theoretic argument, that the adjustment is a consistent screening statistic with a zero condition, an explicit full-support ceiling bounding its use as a magnitude estimator, and a documented failure mode for purely interaction-mediated dependencies. We benchmark the adjusted ersatz against three zero-extra-cost comparators -- polynomial chaos expansion (PCE), PCE-derived Shapley effects, and a PAWN-type maximum Kolmogorov--Smirnov index -- across seven benchmark functions and a real-world hydrological model. The adjusted ersatz is the only estimator achieving perfect rank agreement on a non-smooth hydrological output where PCE is misspecified. A joint sensitivity analysis of five algorithmic parameters shows grid resolution, not the imputation threshold or sampling method, drives performance variability.

stat.ME↗

arXiv2024

Enhancing Variable Importance in Random Forests: A Novel Application of Global Sensitivity Analysis

The present work provides an application of Global Sensitivity Analysis to supervised machine learning methods such as Random Forests. These methods act as black boxes, selecting features in high--dimensional data sets as to provide accurate classifiers in terms of prediction when new data are fed into the system. In supervised machine learning, predictors are generally ranked by importance based on their contribution to the final prediction. Global Sensitivity Analysis is primarily used in mathematical modelling to investigate the effect of the uncertainties of the input variables on the output. We apply it here as a novel way to rank the input features by their importance to the explainability of the data generating process, shedding light on how the response is determined by the dependence structure of its predictors. A simulation study shows that our proposal can be used to explore what advances can be achieved either in terms of efficiency, explanatory ability, or simply by way of confirming existing results.

stat.ML↗

arXiv2023

Discrepancy measures for sensitivity analysis

While sensitivity analysis improves the transparency and reliability of mathematical models, its uptake by modelers is still scarce. This is partially explained by its technical requirements, which may be hard to understand and implement by the non-specialist. Here we propose a sensitivity analysis approach based on the concept of discrepancy that is as easy to understand as the visual inspection of input-output scatterplots. Firstly, we show that some discrepancy measures are able to rank the most influential parameters of a model almost as accurately as the variance-based total sensitivity index. We then introduce an ersatz-discrepancy whose performance as a sensitivity measure matches that of the best-performing discrepancy algorithms, is simple to implement, easier to interpret and orders of magnitude faster.

stat.AP↗

arXiv2022

Mind the hubris: complexity can misfire

Here we briefly reflect on the philosophical foundations that ground the quest towards ever-detailed models and identify four practical dangers derived from this pursuit: explosion of the model's uncertainty space, model black-boxing, computational exhaustion and model attachment. We argue that the growth of a mathematical model should be carefully and continuously pondered lest models become extraneous constructs chasing the Cartesian dream.

cs.GL↗

arXiv2022

Variance-based sensitivity analysis: The quest for better estimators and designs between explorativity and economy

Variance-based sensitivity indices have established themselves as a reference among practitioners of sensitivity analysis of model outputs. A variance-based sensitivity analysis typically produces the first-order sensitivity indices $S_j$ and the so-called total-effect sensitivity indices $T_j$ for the uncertain factors of the mathematical model under analysis. The cost of the analysis depends upon the number of model evaluations needed to obtain stable and accurate values of the estimates. While efficient estimation procedures are available for $S_j$, this availability is less the case for $T_j$. When estimating these indices, one can either use a sample-based approach whose computational cost depends on the number of factors or use approaches based on meta modelling/emulators. The present work focuses on sample-based estimation procedures for $T_j$ and tests different avenues to achieve an algorithmic improvement over the existing best practices. To improve the exploration of the space of the input factors (design) and the formula to compute the indices (estimator), we propose strategies based on the concepts of economy and explorativity. We then discuss how several existing estimators perform along these characteristics. We conclude that: a) sample-based approaches based on the use of multiple matrices to enhance the economy are outperformed by designs using fewer matrices but with better explorativity; b) among the latter, asymmetric designs perform the best and outperform symmetric designs having corrective terms for spurious correlations; c) improving on the existing best practices is fraught with difficulties; and d) ameliorating the results comes at the cost of introducing extra design parameters.

stat.AP↗

arXiv2021

sensobol: an R package to compute variance-based sensitivity indices

The R package "sensobol" provides several functions to conduct variance-based uncertainty and sensitivity analysis, from the estimation of sensitivity indices to the visual representation of the results. It implements several state-of-the-art first and total-order estimators and allows the computation of up to third-order effects, as well as of the approximation error, in a swift and user-friendly way. Its flexibility makes it also appropriate for models with either a scalar or a multivariate output. We illustrate its functionality by conducting a variance-based sensitivity analysis of three classic models: the Sobol' (1998) G function, the logistic population growth model of Verhulst (1845), and the spruce budworm and forest model of Ludwig, Jones and Holling (1976).

stat.CO↗

arXiv2021

Unpacking uncertainty in the modelling process for energy policy making

This paper explores how the modelling of energy systems may lead to undue closure of alternatives by generating an excess of certainty around some of the possible policy options. We exemplify the problem with two cases: first, the International Institute for Applied Systems Analysis (IIASA) global modelling in the 1980s; and second, the modelling activity undertaken in support of the construction of a radioactive waste repository at Yucca Mountain (Nevada, USA). We discuss different methodologies for quality assessment that may help remedy this issue, which include NUSAP (Numeral Unit Spread Assessment Pedigree), diagnostic diagrams, and sensitivity auditing. We demonstrate the potential of these reflexive modelling practices in energy policy making with four additional cases: (i) stakeholders evaluation of the assessment of the external costs of a potential large-scale nuclear accident in Belgium in the context of the ExternE (External Costs of Energy) project; (ii) the case of the ESME (Energy System Modelling Environment) for the creation of UK energy policy; (iii) the NETs (Negative Emission Technologies) uptake in Integrated Assessment Models (IAMs); and (iv) the Ecological Footprint (EF) indicator. We encourage modellers to widely adopt these approaches to achieve more robust and inclusive modelling activities in the field of energy modelling.

stat.AP↗

arXiv2021

A comprehensive comparison of total-order estimators for global sensitivity analysis

Sensitivity analysis helps identify which model inputs convey the most uncertainty to the model output. One of the most authoritative measures in global sensitivity analysis is the Sobol' total-order index, which can be computed with several different estimators. Although previous comparisons exist, it is hard to know which estimator performs best since the results are contingent on the benchmark setting defined by the analyst (the sampling method, the distribution of the model inputs, the number of model runs, the test function or model and its dimensionality, the weight of higher order effects or the performance measure selected). Here we compare several total-order estimators in an eight-dimension hypercube where these benchmark parameters are treated as random parameters. This arrangement significantly relaxes the dependency of the results on the benchmark design. We observe that the most accurate estimators are Razavi and Gupta's, Jansen's or Janon/Monod's for factor prioritization, and Jansen's, Janon/Monod's or Azzini and Rosati's for approaching the "true" total-order indices. The rest lag considerably behind. Our work helps analysts navigate the myriad of total-order formulae by reducing the uncertainty in the selection of the most appropriate estimator.

stat.AP↗

arXiv2021

Physics to the rescue?

A vast body of literature addresses the complex nature of science's reproducibility crisis. In contrast with this perceived complexity, some recent papers from the discipline of physics suggests that irreproducibility does not point to a systemic crisis, but is, on the contrary, a sign that the science system works properly. These works, while acknowledging the difference between physics and other disciplines mired in the reproducibility crisis, hint that all disciplines could learn from Physics. The present work suggests that this optimistic message, when addressed to struggling disciplines, may invite complacency over other relevant dimensions of crisis, delay its solution, and get into the way of a truly joint effort from all disciplines to tackle the important social and environmental predicaments of the present age.

physics.soc-ph↗

arXiv2020

Is VARS more intuitive and efficient than Sobol' indices?

The Variogram Analysis of Response Surfaces (VARS) has been proposed by Razavi and Gupta as a new comprehensive framework in sensitivity analysis. According to these authors, VARS provides a more intuitive notion of sensitivity and it is much more computationally efficient than Sobol' indices. Here we review these arguments and critically compare the performance of VARS-TO, for total-order index, against the total-order Jansen estimator. We argue that, unlike classic variance-based methods, VARS lacks a clear definition of what an "important" factor is, and show that the alleged computational superiority of VARS does not withstand scrutiny. We conclude that while VARS enriches the spectrum of existing methods for sensitivity analysis, especially for a diagnostic use of mathematical models, it complements rather than substitutes classic estimators used in variance-based sensitivity analysis.

stat.AP↗

arXiv2020

A sensitivity analysis of the PAWN sensitivity index

The PAWN index is gaining traction among the modelling community as a sensitivity measure. However, the robustness to its design parameters has not yet been scrutinized: the size ($N$) and sampling ($\varepsilon$) of the model output, the number of conditioning intervals ($n$) or the summary statistic ($θ$). Here we fill this gap by running a sensitivity analysis of a PAWN-based sensitivity analysis. We compare the results with the design uncertainties of the Sobol' total-order index ($S_{Ti}^*$). Unlike in $S_{Ti}^*$, the design uncertainties in PAWN create non-negligible chances of producing biased results when ranking or screening inputs. The dependence of PAWN upon ($N,n,\varepsilon, θ$) is difficult to tame, as these parameters interact with one another. Even in an ideal setting in which the optimum choice for ($N,n,\varepsilon, θ$) is known in advance, PAWN might not allow to distinguish an influential, non-additive model input from a truly non-influential model input.

stat.AP↗

arXiv2019

Are the results of the groundwater model robust?

De Graaf et al. (2019) suggest that groundwater pumping will bring 42--79\% of worldwide watersheds close to environmental exhaustion by 2050. We are skeptical of these figures due to several non-unique assumptions behind the calculation of irrigation water demands and the perfunctory exploration of the model's uncertainty space. Their sensitivity analysis reveals a widespread lack of elementary concepts of design of experiments among modellers, and can not be taken as a proof that their conclusions are robust.

physics.ao-ph↗

arXiv2019

Discussion Paper: Should statistics rescue mathematical modelling?

Statistics experiences a storm around the perceived misuse and possible abuse of its methods in the context of the so-called reproducibility crisis. The methods and styles of quantification practiced in mathematical modelling rarely make it to the headlines, though modelling practitioners writing in disciplinary journals flag a host of problems in the field. Technical, cultural and ethical dimensions are simultaneously at play in the current predicaments of both statistics and mathematical modelling. Since mathematical modelling is not a discipline like statistics, its shortcomings risk remaining untreated longer. We suggest that the tools of statistics and its disciplinary organisation might offer a remedial contribution to mathematical modelling, standardising methodologies and disseminating good practices. Statistics could provide scientists and engineers from all disciplines with a point of anchorage for sound modelling work. This is a vast and long-term undertaking. A step in the proposed direction is offered here by focusing on the use of statistical tools for quality assurance of mathematical models. By way of illustration, techniques for uncertainty quantification, sensitivity analysis and sensitivity auditing are suggested for incorporation in statistical syllabuses and practices.

stat.ME↗

arXiv2019

A new sample-based algorithms to compute the total sensitivity index

Variance-based sensitivity indices have established themselves as a reference among practitioners of sensitivity analysis of model output. It is not unusual to consider a variance-based sensitivity analysis as informative if it produces at least the first order sensitivity indices S_j and the so-called total-effect sensitivity indices T_j for all the uncertain factors of the mathematical model under analysis. Computational economy is critical in sensitivity analysis. It depends mostly upon the number of model evaluations needed to obtain stable values of the estimates. While efficient estimation procedures independent from the number of factors under analysis are available for the first order indices, this is less the case for the total sensitivity indices. When estimating T_j, one can either use a sample-based approach, whose computational cost depends fromon the number of factors, or approaches based on meta-modelling/emulators, e.g. based on Gaussian processes. The present work focuses on sample-based estimation procedures for T_j and tries different avenues to achieve an algorithmic improvement over the designs proposed in the existing best practices. We conclude that some proposed sample-based improvements found in the literature do not work as claimed, and that improving on the existing best practice is indeed fraught with difficulties. We motivate our conclusions introducing the concepts of explorativity and efficiency of the design.

stat.AP↗

arXiv2017

Why So Many Published Sensitivity Analyses Are False. A Systematic Review of Sensitivity Analysis Practices

Sensitivity analysis (SA) has much to offer for a very large class of applications, such as model selection, calibration, optimization, quality assurance and many others. Sensitivity analysis offers crucial contextual information regarding a prediction by answering the question "Which uncertain input factors are responsible for the uncertainty in the prediction?" SA is distinct from uncertainty analysis (UA), which instead addresses the question "How uncertain is the prediction?" As we discuss in the present paper much confusion exists in the use of these terms. A proper uncertainty analysis of the output of a mathematical model needs to map what the model does when the input factors are left free to vary over their range of existence. A fortiori, this is true of a sensitivity analysis. Despite this, most UA and SA still explore the input space; moving along mono-dimensional corridors which leave the space of variation of the input factors mostly unscathed. We use results from a bibliometric analysis to show that many published SA fail the elementary requirement to properly explore the space of the input factors. The results, while discipline-dependent, point to a worrying lack of standards and of recognized good practices. The misuse of sensitivity analysis in mathematical modelling is at least as serious as the misuse of the p-test in statistical modelling. Mature methods have existed for about two decades to produce a defensible sensitivity analysis. We end by offering a rough guide for proper use of the methods.

stat.AP↗

arXiv2015

The fallacy of evidence based policy

The use of science for policy is at the core of a perfect storm generated by the insurgence of several concurrent crises: of science, of trust, of sustainability. The modern positivistic model of science for policy, known as evidence based policy, is based on dramatic simplifications and compressions of available perceptions of the state of affairs and possible explanations (hypocognition). This model can result in flawed prescriptions. The flaws become more evident when dealing with complex issues characterized by concomitant uncertainties in the normative, descriptive and ethical domains. In this situation evidence-based policy may concur to the fragility of the social system. Science plays an important role in reducing the feeling of vulnerability of humans by projecting a promise of protection against uncertainties. In many applications quantitative science is used to remove uncertainty by transforming it into probability, so that mathematical modelling can play the ritual role of haruspices. This epistemic governance arrangement is today in crisis. The primacy of science to adjudicate political issues must pass through an assessment of the level of maturity and effectiveness of the various disciplines deployed. The solution implies abandoning dreams of prediction, control and optimization obtained by relying on a limited set of simplified narratives to define the problem and moving instead to an open exploration of a broader set of plausible and relevant stories. Evidence based policy has to be replaced by robust policy, where robustness is tested with respect to feasibility (compatibility with processes outside human control); viability (compatibility with processes under human control, in relation to both the economic and technical dimensions), and desirability domain (compatibility with a plurality of normative considerations relevant to a plurality of actors).

econ.GN↗

arXiv2015

Exploring multi-dimensional spaces: a Comparison of Latin Hypercube and Quasi Monte Carlo Sampling Techniques

Three sampling methods are compared for efficiency on a number of test problems of various complexity for which analytic quadratures are available. The methods compared are Monte Carlo with pseudo-random numbers, Latin Hypercube Sampling, and Quasi Monte Carlo with sampling based on Sobol sequences. Generally results show superior performance of the Quasi Monte Carlo approach based on Sobol sequences in line with theoretical predictions. Latin Hypercube Sampling can be more efficient than both Monte Carlo method and Quasi Monte Carlo method but the latter inequality holds for a reduced set of function typology and at small number of sampled points. In conclusion Quasi Monte Carlo method would appear the safest bet when integrating functions of unknown typology.

stat.AP↗

arXiv2014

Exploring Hoover and Perez's experimental designs using global sensitivity analysis

This paper investigates variable-selection procedures in regression that make use of global sensitivity analysis. The approach is combined with existing algorithms and it is applied to the time series regression designs proposed by Hoover and Perez. A comparison of an algorithm employing global sensitivity analysis and the (optimized) algorithm of Hoover and Perez shows that the former significantly improves the recovery rates of original specifications.

stat.CO↗