Einblick - Computational Complexity - # Sensitivity Analysis Methodology

A New Paradigm for Sensitivity Analysis: Leveraging Factorial Experiments to Assess Input Importance

Q: How can the proposed paradigm be extended to handle non-scalar outputs, such as functional outputs or multivariate outputs?

The proposed paradigm for global sensitivity analysis can be extended to handle non-scalar outputs by redefining the sensitivity maps and divergence functions to accommodate the structure of functional or multivariate outputs. For functional outputs, one can consider sensitivity maps that evaluate the variability of the output as a function of the input variations over a defined domain. This involves using divergence functions that measure differences between functions, such as the L2 norm or other integral-based metrics, to quantify the sensitivity of the output function to changes in the input parameters. For multivariate outputs, the sensitivity analysis can be approached by treating each component of the output vector separately or by employing a joint sensitivity map that captures the interactions between the different output dimensions. This can be achieved by defining a multi-dimensional divergence function that assesses the variability across the output components simultaneously. The factorial experiment framework can be adapted to include multiple outputs, allowing for the computation of main and interaction effects across the different output dimensions. By leveraging the flexibility of the proposed paradigm, one can systematically analyze the sensitivity of complex models with non-scalar outputs while maintaining interpretability and coherence in the results.

Q: What are the computational and statistical challenges in estimating the weighted factorial effects in high-dimensional settings, and how can they be addressed?

Estimating weighted factorial effects in high-dimensional settings presents several computational and statistical challenges. One major challenge is the curse of dimensionality, where the number of possible input combinations grows exponentially with the number of inputs, making it computationally infeasible to evaluate all combinations in a factorial design. This can lead to a significant increase in the number of model evaluations required, which is often prohibitive in practice. To address this challenge, one can employ techniques such as dimensionality reduction, where methods like principal component analysis (PCA) or feature selection algorithms are used to identify and retain only the most influential inputs. Additionally, adaptive sampling methods, such as Latin Hypercube Sampling (LHS) or surrogate modeling techniques, can be utilized to efficiently explore the input space without exhaustively evaluating all combinations. Statistically, estimating the weighted factorial effects accurately in high dimensions can be complicated by the presence of interactions and dependencies among inputs. To mitigate this, one can use regularization techniques or Bayesian approaches that incorporate prior knowledge about the input-output relationships, allowing for more robust estimates of the factorial effects. Furthermore, employing cross-validation methods can help assess the reliability of the estimates and ensure that the model generalizes well to unseen data.

Q: What insights can be gained by exploring the connections between sensitivity analysis and cooperative game theory more deeply?

Exploring the connections between sensitivity analysis and cooperative game theory can yield valuable insights into the interpretation and application of sensitivity indices. In cooperative game theory, the Shapley value provides a fair distribution of payoffs among players based on their contributions to the overall outcome. Similarly, in sensitivity analysis, indices like the Shapley effect can be interpreted as measures of input importance, reflecting how much each input contributes to the variability of the output. By delving deeper into these connections, one can enhance the understanding of how to allocate importance among inputs in a more equitable manner, particularly in scenarios where inputs may have synergistic or antagonistic interactions. This perspective can lead to the development of new sensitivity indices that incorporate notions of fairness and cooperation, potentially improving the interpretability of results in complex models. Moreover, the framework of cooperative game theory can provide a structured approach to address issues of input dependence and interaction effects, allowing for a more nuanced analysis of how inputs work together to influence the output. This can facilitate the identification of key input combinations that drive model behavior, leading to more informed decision-making in model calibration and uncertainty quantification. Overall, the interplay between sensitivity analysis and cooperative game theory opens avenues for innovative methodologies that enhance the robustness and applicability of sensitivity analysis in various fields, including environmental modeling, finance, and engineering.

Kernkonzepte

A new paradigm for global sensitivity analysis is proposed, which leverages the concepts of factorial experiments to define main and interaction effects without relying on the Sobol-Hoeﬀding functional decomposition. This approach allows for arbitrary input distributions, output variability measures, and provides well-defined and interpretable sensitivity indices.

Zusammenfassung

The article presents a new paradigm for global sensitivity analysis that departs from the traditional Sobol-Hoeﬀding functional decomposition approach. The key ideas are:

Sensitivity maps are defined as functions that quantify the sensitivity of the output to different subsets of the inputs. These maps satisfy three axioms that capture the intuitive notion of sensitivity.
Sensitivity maps can be identified with factorial experiments, where the presence or absence of inputs defines the factors. This allows for a natural definition of main and interaction effects, without relying on any functional decomposition.
Weighted factorial effects are introduced, which generalize the standard factorial effects. This allows for the recovery of well-known sensitivity indices like Sobol indices and Shapley effects as special cases.
The total output variability can be decomposed in different ways using appropriate weight functions, leading to Sobol-like or Shapley-like decompositions.
The concept of a dual sensitivity map is introduced, which provides an alternative perspective on the sensitivity analysis.

The proposed paradigm offers several advantages over the traditional approach: it can handle arbitrary input distributions, it is not restricted to variance-based analysis, and the interpretation of main and interaction effects is more straightforward. The connection to factorial experiments also opens up the possibility of leveraging techniques from experimental design to tackle high-dimensional problems.

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A new paradigm for global sensitivity analysis

by Gildas Mazo ... um arxiv.org 09-11-2024

https://arxiv.org/pdf/2409.06271.pdf

A new paradigm for global sensitivity analysis

Tiefere Fragen

How can the proposed paradigm be extended to handle non-scalar outputs, such as functional outputs or multivariate outputs?

The proposed paradigm for global sensitivity analysis can be extended to handle non-scalar outputs by redefining the sensitivity maps and divergence functions to accommodate the structure of functional or multivariate outputs. For functional outputs, one can consider sensitivity maps that evaluate the variability of the output as a function of the input variations over a defined domain. This involves using divergence functions that measure differences between functions, such as the L2 norm or other integral-based metrics, to quantify the sensitivity of the output function to changes in the input parameters.
For multivariate outputs, the sensitivity analysis can be approached by treating each component of the output vector separately or by employing a joint sensitivity map that captures the interactions between the different output dimensions. This can be achieved by defining a multi-dimensional divergence function that assesses the variability across the output components simultaneously. The factorial experiment framework can be adapted to include multiple outputs, allowing for the computation of main and interaction effects across the different output dimensions. By leveraging the flexibility of the proposed paradigm, one can systematically analyze the sensitivity of complex models with non-scalar outputs while maintaining interpretability and coherence in the results.

What are the computational and statistical challenges in estimating the weighted factorial effects in high-dimensional settings, and how can they be addressed?

Estimating weighted factorial effects in high-dimensional settings presents several computational and statistical challenges. One major challenge is the curse of dimensionality, where the number of possible input combinations grows exponentially with the number of inputs, making it computationally infeasible to evaluate all combinations in a factorial design. This can lead to a significant increase in the number of model evaluations required, which is often prohibitive in practice.
To address this challenge, one can employ techniques such as dimensionality reduction, where methods like principal component analysis (PCA) or feature selection algorithms are used to identify and retain only the most influential inputs. Additionally, adaptive sampling methods, such as Latin Hypercube Sampling (LHS) or surrogate modeling techniques, can be utilized to efficiently explore the input space without exhaustively evaluating all combinations.
Statistically, estimating the weighted factorial effects accurately in high dimensions can be complicated by the presence of interactions and dependencies among inputs. To mitigate this, one can use regularization techniques or Bayesian approaches that incorporate prior knowledge about the input-output relationships, allowing for more robust estimates of the factorial effects. Furthermore, employing cross-validation methods can help assess the reliability of the estimates and ensure that the model generalizes well to unseen data.

What insights can be gained by exploring the connections between sensitivity analysis and cooperative game theory more deeply?

Exploring the connections between sensitivity analysis and cooperative game theory can yield valuable insights into the interpretation and application of sensitivity indices. In cooperative game theory, the Shapley value provides a fair distribution of payoffs among players based on their contributions to the overall outcome. Similarly, in sensitivity analysis, indices like the Shapley effect can be interpreted as measures of input importance, reflecting how much each input contributes to the variability of the output.
By delving deeper into these connections, one can enhance the understanding of how to allocate importance among inputs in a more equitable manner, particularly in scenarios where inputs may have synergistic or antagonistic interactions. This perspective can lead to the development of new sensitivity indices that incorporate notions of fairness and cooperation, potentially improving the interpretability of results in complex models.
Moreover, the framework of cooperative game theory can provide a structured approach to address issues of input dependence and interaction effects, allowing for a more nuanced analysis of how inputs work together to influence the output. This can facilitate the identification of key input combinations that drive model behavior, leading to more informed decision-making in model calibration and uncertainty quantification.
Overall, the interplay between sensitivity analysis and cooperative game theory opens avenues for innovative methodologies that enhance the robustness and applicability of sensitivity analysis in various fields, including environmental modeling, finance, and engineering.