Abstract
Global sensitivity analysis seeks to detect influential input factors contributing to a black-box model’s specific decisions. This aligns with a key objective of AI explainability: Clarifying and interpreting the behavior of machine learning algorithms by identifying the features that influence their decisions-a significant approach for mitigating the computational burden associated with processing high-dimensional data. Various techniques are proposed for sensitivity analysis; however, each of these methods focuses on different mathematical aspects, which can lead to varying conclusions about the impact or importance of each feature. Therefore, it remains unclear which of these algorithms are most suitable for machine learning models and, in particular, deep learning models. Our goal is to examine the influential features identified by each sensitivity analysis algorithm and evaluate their role in helping deep learning models make accurate decisions. In this article, first, we present the mathematical foundations underlying Global Sensitivity algorithms and explain the rationale behind selecting the important features identified by each method. We then provide a comparative case study on global sensitivity analysis methods and propose a methodology to evaluate the efficacy of these methods by conducting a case study on MNIST digit dataset classification. Our study highlights the most effective global sensitivity analysis methods for detecting the key factors influencing the digit data classification.
Keywords: Global sensitivity analysis, Influential features, Black-box models, Feature selection, Explainable AI, Digit classification, Deep neural networks
Introduction
In the era of deep learning and the rapid advancement of powerful and complicated Artificial Intelligence (AI) models consisting of numerous layers and millions of parameters, the demand for understanding the decision-making process of black-box models is on the rise [1]. Through computational analysis, XAI seeks to demystify the decision-making processes of these models, offering insights into how they arrive at specific conclusions and predictions [2, 3].In the era of machine learning, the importance of explainability has become even more pronounced, as AI systems are deployed across an ever-expanding range of applications. XAI has been applied to various types of data, including video [4], text [5], AIS [6], causal [7], genomic data [8], and various applications such as art [9], medicine [10], finance [11] and education [12]. Explainability methods can be broadly divided into model-agnostic or model-free and model-specific approaches. Model-agnostic methods can be applied to any trained machine learning model regardless of the learning mechanism and model architecture. Statistical analysis and sensitivity analysis are two common approaches from this category. Statistical methods such as Spearman rank correlation coefficient (SPEA), Standard Regression Coefficient (SRC), Partial Correlation Coefficient (PCC), Pearson correlation coefficient [13] can be referred to as a few approaches which are employed for finding the association between input parameters as well as input parameters and model output. Sensitivity analysis methods, on the other hand, refer to a broad approach for examining relationships between a system’s inputs and outputs. According to Saltelli et al., sensitivity analysis investigates the process of how the uncertainty in a model’s output can be attributed to different sources of uncertainty within the model’s inputs [14].
In contrast to model-agnostic techniques, model-specific methods are tailored for particular machine learning models, such as those developed to explain the inner workings of deep neural networks. Consequently, a substantial body of research is dedicated to developing explanations of the predictions made by deep learning models through network dissection [15], gradient optimization and visualization [16, 17], saliency maps and visualization [18, 19], loss landscapes [20, 21], and semantic analysis [22–24].
From another perspective, explainability methods can take a global or local processing approach. Basically, global explainability methods can be applied for analyzing the effect of parameters on the overall output of the model, whereas local techniques are applicable to study this effect for individual cases. Global explainability methods focus on explaining the overall behavior of a machine learning model by varying the entire range of input factors and examining the joint effect and interaction between them. Therefore, global explainability methods can be applied for analyzing the effect of parameters on the overall model output, whereas local techniques are applicable to study this effect for individual cases. Examples of global approaches include Partial Dependence Plot (PDP) [25], (Individual Conditional Expectation) ICE [26] and global sensitivity analysis algorithms. In global approach, all input parameters are allowed to change simultaneously across all possible ranges of values. In contrast, local explianibilty methods study the effect of a parameter by exploring its local vicinity while keeping all other parameters fixed at their baseline values [27, 28]. Local interpretability methods typically make the underlying assumption that the machine learning model exhibits non-linear relationships and independence between the input parameters. Consequently, if input factors exhibit significant interactions, local interpretability methods can produce misleading or inaccurate results, as they rely on the assumption of parameter independence [29]. Many statistical methods assume that data points are independent and cannot be applied effectively without strong evidence supporting this assumption. Shapley values [30] and LIME (Local Interpretable Model-Agnostic Explanations) [31] are two of the most widely used local explainablity approaches, which try to provide an explanation for individual instances and are based primarily on the independence assumption [32]. In contrast, global explainablity methods can be applied without the need to make assumptions about the relationships between the input parameters. Nevertheless, there is no universal consensus on how to measure the impact of parameters using global explaniability methods. Different global techniques are grounded in varying mathematical foundations. As a result, they can produce divergent rankings or measures of parameter importance when applied to the same machine learning model.
In this paper, we explore global sensitivity analysis (GSA) methods in depth and compare their effectiveness through a case study using deep learning modeling and a large image dataset. GSA methods have a wide range of applications and their scope extends beyond machine learning. The scope of these methods covers different applications, such as engineering [33, 34], finance [34], and biology [35]. Although sensitivity analysis (SA) has been around for several decades, there has been no comprehensive, systematic comparison of the functionality and performance of Global Sensitivity Analysis (GSA) algorithms or a clear understanding of their fundamental differences that assist in method selection. This often creates challenges for developers, decision makers, and end users in selecting the right tool for their needs. In this study, we discuss that the choice of GSA method greatly influences the conclusions drawn about input feature importance. To this end, it is critical to understand the strengths and weaknesses of each method and carefully consider the nature of the model and the specific goals of the analysis before selecting an appropriate method. Herman and Usher developed a Python implementation of the most widely used sensitivity analysis methods in the Sensitivity Analysis Library (SALib) package [36]. However, they did not offer a comparable study to examine the differences and effectiveness of each algorithm. Stein et al., provided a review of several GSA methods [8]. However, they compared the robustness and accuracies of these algorithms using two conventional ML algorithms, namely linear regression and random forests, and tested their performance using the Black-Box Optimization Benchmarking (BBOB) test suite consisting of single-objective standard test functions [37]. Loose and Lemiatre provided a review article and categorized several of these methods based on their computational cost and complexity on a flood example [38]. Borgonovo and Plischke disscussed some of the local and global SA methods using a classical example [22]. Given the widespread use and applicability of advanced deep learning systems in the visual domain, as well as the common practice of utilizing visual images for research purposes, this study focuses on applying a deep learning technique to a visual dataset. Our goal is to address the mentioned issues in an effective comparison, by applying deep learning modeling on a standard large-scale visual dataset to provide a practical understanding about the performance of each of these algorithms. Our contributions are as follows: We offer a comprehensive overview of the mechanism behind global sensitivity analysis algorithms, propose a methodology for systematic measurement of their performance through a visual digit classification task on large-scale image dataset using deep learning modeling, and present a comparative analysis of the feature importance identified by each algorithm. This paper is structured as follows. We first review the global sensitivity analysis methods; then, we propose a methodology for comparing the performance of the discussed GSA methods. We present the results and conclude the paper in the last section.
Global sensitivity analysis methods
Global Sensitivity Analysis explains the global effect of input parameters on the whole system performance. They consider the full range of variation in the input parameters, not just local changes. This makes these methods particularly useful for complex models. The general paradigm of GSA methods consists of two phases of sampling and analysis. Initially, during sampling, values for the input parameters 
of a model are selected to explore how these values influence the output. Sampling techniques for a variety of SA methods are discussed in [13]. The output vector Y is then produced based on the trained model f for each of the generated samples in the analysis phase:
 |
1 |
Finally, the impact of each input parameter or variable is analyzed and evaluated. GSA methods can be roughly categorized into four different groups, i.e., variance-based methods, derivative-based methods, density-based methods, and feature additive methods. In the following subsections, we explore the common GSA methods from each of these categories, present their mathematical underlying, and explain their mechanism.
Variance-based methods
Variance-based methods are based on the assumption that variance is sufficient to describe the output uncertainty, an assumption made by Saltelli et al. [39–41]. The rationale behind this approach is based on the variance of the expected output value conditioned on the input parameters. A higher variance suggests a lower level of importance for a parameter.
Sobol
The Sobol method is one of the most popular and well-established approaches to GSA. It relies on the decomposition of the variance of the model output under the assumption that the inputs are independent and uncorrelated. In this method, the total variance of the output Y (V(Y)) is decomposed into variances from individual parameters and their combinations. Equations (1) to (4) illustrate the variance decomposition of the variable Y in the Sobol‘s approach.
 |
2 |
 |
3 |
 |
4 |
 |
5 |
Sobol sensitivity analysis assesses the impact of each input parameter, both in isolation and in conjunction with other parameters, resulting in the derivation of first- and higher-order sensitivity indices. First-order (
), second-order (
), total-order (
) and higher order sensitivity indices are calculated to accurately reflect the influence of individual inputs and the interaction between them. Equations (2) and (3) measure the first and second order variations, while Eqs. (6) and (7) show the first and second order sensitivity indices, which indicate the fraction of variance of Y caused by factors 
in isolation and the variance derived from interaction between factors 
and 
, respectively.
 |
6 |
 |
7 |
Furthermore, the total-order sensitivity index is explained by Eq. (8) and is the summation over all the variances induced by factor 
both individually and collaboratively with other factors (denoted by 
).
 |
8 |
Sobol indices are relatively easy to interpret. It works well for both linear and non-linear models and has a strong statistical foundation. However, it can be computationally expensive, especially when applied to high-dimensional models. As the number of input parameters increases, estimating higher-order interactions (e.g., third or fourth order) becomes more challenging.
FAST
Like other variance-based methods, the Fourier amplitude sensitivity test (FAST) decomposes the total variance of the model output into contributions from each input parameter and their interactions. FAST is based on periodic search sampling using a period search function and applies a decomposition of variance based on the Fourier Transform [42, 43]. For a comparison of sampling approaches for this method, refer to [42]. The FAST method works by analyzing how changes in inputs affect the frequency components of the model output [43]. It describes the decomposition of variance using transfer functions as defined by Eq. (9):
 |
9 |
where 
are integer frequencies. The model output can be obtained by:
 |
10 |
The variance of Y is then defined according to Eq. (11) by estimating the expected value of Y and can be approximated by Eq. (12) by incorporating Parseval’s theorem:
 |
11 |
 |
12 |
where 
and 
are Fourier coefficients:
 |
13 |
 |
14 |
The first-order sensitivity indexes of FAST are then computed by employing Eq. (15).
 |
15 |
The total order sensitivity index or total effect of FAST can be obtained by considering all first order effects as well as all higher order effects that exclude the index i.
 |
16 |
FAST achieves a better estimate in terms of robustness and speed of convergences than Sobol and can be applied to non-linear and non-monotonic models. [44, 45]. FAST relies on frequency-based sampling, which might not be intuitive and interpretable for non-expert users. FAST’s implementation can be complex and may require specialized knowledge. Although FAST can be applied to stochastic models, in certain cases where there is much noise or large variability in the model output, the sensitivity indices might be less accurate.
RBD and FAST_RBD
When the number of inputs increases, FAST encounters numerous sources of error, and it results in poor estimation considering the computational cost of deriving all the higher-order terms. The RBD and hybrid FAST_RBD (HFR) methods have been developed to overcome this computational burden of FAST.
In contrast to FAST, which explores the space using different frequencies for each parameter (
), RBD takes a single frequency (
) for all parameters which can be determined randomly and can be set to 1 for the sake of simplicity [46]. This reduces the computational complexity of the algorithm; however, the search curve cannot cover the whole space completely. In order to avoid this problem, random permutation of the coordinate of sample points (i.e., design points) is used to increase the stochastic nature of the algorithm [47].
 |
17 |
In hybrid FAST_RBD, the k parameters are grouped into partitions of equal size while a particular frequency is assigned to each partition. Hence, HFR strikes a balance between the accuracy of FAST and the computational efficiency of RBD [46].
Derivative based methods
This approach is based on the determination of the sensitivity indices by computing the first-order partial derivative of the model output with respect to the input variables 
. A higher derivative value suggests a higher level of sensitivity. Two significant methods falling under this category are Morris and DGSM. A major limitation of DGSM is that it requires the model to be differentiable. For non-smooth or discontinuous models, calculating accurate derivatives may be computationally costly or impossible.
Morris
The basic idea of the Morris method is based on the calculation of elementary effects (EE) for each input factor by dividing the range of each factor into p levels and considering 
as a predetermined multiple of 
, where 
for exploring the grid space. The EE (or the elementary effect) is then calculated by perturbing each parameter by a small amount 
and assessing the effect on the output while keepin the other parameters fixed. This is done for all input parameters to assess their individual sensitivities:
 |
18 |
After sampling 
and computing the elementary effects (
) for the r trajectories (or repetitions), the average and standard deviation of the elementary effects are calculated. These are referred to as the 
(mean) and 
(standard deviation) of the elementary effects for the parameter 
:
 |
19 |
 |
20 |
Campolongo et al. proposed a revised version of 
called 
by considering the absolute value of 
in order to mitigate the issue of cancellation of opposite signs in non-monotonic models [48]. Higher values of 
indicate a greater influence of 
on the output, while higher values of 
suggest an increased interaction between 
and the other variables or a non-linear effect [49]. We categorized Morris method as a derivative-based method since it involves computing changes in the model output based on changes in input parameters, and in that sense, it shares some conceptual similarity with derivative calculation. The Morris method is computationally less expensive than other methods like Sobol, especially when applied to high-dimensional models. It uses a relatively small number of model evaluations to estimate the sensitivity of the model, making it faster than variance-based methods that require a large number of simulations. Morris can be applied to a wide variety of models and domains. It is a flexible and easy to implement tool that works in different modeling contexts. However, it does not explicitly capture interaction effects between parameters.
Derivative based global sensitivity measure
Morris method for importance measurement is based on the approximation of 
, where 
is the sample space. The derivative-based global sensitivity measure (DGSM) is a generalization of Morris and is measured as:
 |
21 |
A low value of 
signifies non-important factors of low influence [50]. It is also shown that there is a link between the DGSM values and Sobol’s total index [39].
Density based methods
The density-based (or moment-independent) approach investigates the entire distribution of a model measured by its Probability Density Function and finds the sensitivity of the model based on the variation to its density function without relying on any moments of the output.
DELTA
Delta is a density-based sensitivity method. It assumes that all input variables are independent. However, this method can be used in the presence of correlation between variables. The idea behind this method is to find the shift s between the unconditional density (or cumulative distribution) of Y (i.e., 
) and the conditional density of Y given that one variable 
is fixed with a constant value x (i.e., 
). The Delta sensitivity index considers the expected value of s:
 |
22 |
where 
is the marginal density of 
. The sensitivity index of factor 
is then measured by 
:
 |
23 |
This Delta sensitivity index remains unaffected by monotonic transformations and is normalized (
) [51]. This method can provide a visualization of results about the probability density function; however, it relies on knowing or estimating the probability distributions of the input variables.
Method
In this section, we propose a method for making a comparison between the performance of the sensitive analysis methods discussed in the previous section. In our study, we concentrate on feature selection of each methods and their corresponding classification accuracy as a basis for evaluation. To this end, we first train a simple two-layer Convolutional Neural Network (CNN) on MNIST data. The MNIST dataset contains handwritten digits, with 60,000 train and 10,000 test examples. Each image consists of 784 pixels and is labeled into 10 categories, corresponding to the numbers 0 through 9. For the deep learning model, we employ 2D convolutional operators followed by ReLU and Maxpooling. For linear classification, we used DropOut with probability of 0.2 and added nonlinearity with a ReLU layer. The complete architecture of this network model is shown in Fig. 1. We trained this network on MNIST training data using the train image set and evaluated the performance by measuring the percentage of correct classification on the test set using stochastic gradient descent with a learning rate of 0.01 and achieved 99% classification accuracy. Then we applied each of the SA methods using this trained CNN on the MNIST test data and obtained importance feature ranking for each image. The ranking identifies the influential pixels to make a correct prediction about the category of digits.
Fig. 1.
Architecture of the deep learning model trained on the MNIST dataset. The network consists of two sequential blocks dedicated to representation learning and classification
To achieve this objective, we propose a methodology to evaluate the performance of SA indices offered by each algorithm on the digit classification task. For this purpose, the average SA indices of all image pixels are sorted both descendingly and ascendingly to study the effect of influential and non-influential factors correspondingly. The sorted vectors are then segmented into seven blocks. The last block comprises all 784 pixels of an image from the MNIST dataset, while subsequent blocks gradually decrease by 100 pixels until reaching the first block, which contains 84 pixels. To evaluate the influential factors, the SA indices are sorted in descending order, with the first block containing the most important pixels. Conversely, to assess the non-influential factors, the SA indices are sorted in ascending order, with the first block containing the least important pixels, as determined by each SA algorithm. We subsequently assessed the classification performance of the test data solely based on the basis of the pixels segmented in each block. Through this approach, we can measure the impact of both the influential and non-influential factors of SA algorithms on the digit classification task. Figure 2 illustrates the methodology procedure. For the implementation of SA algorithms, we applied the Sensitivity Analysis Library in Python (SALIB1).2
Fig. 2.
Methodology for comparing SA methods. CNN model is trained using the train set. This model is then analyzed on the test data by each of the SA algorithms. SA algorithms identify the important factors. Finally, the important factors are sorted and segmented and the impact of each segment is evaluated
Results and conclusions
We evaluated the effect of the influential factors identified by each of the SA algorithms discussed in Sect. 2 based on the methodology described in Sect. 3. The number of samples applied is listed in Table 1 and is determined using a grid search in the interval of 50 to 10,000 and step size of 100, which strikes a balance between optimizing performance and minimizing the number of samples required. In order to understand the efficiency of important features selected by each SA algorithm, we demonstrate the transition of accuracy of classification for each segmented block of influential and non-inflectional factors in Fig. 3. A clear incremental trend is observable in both graphs from the first block containing 84 pixels in the left to the final block which holds the whole pixels of image. This trend verifies that when more important features are accumulated, the accuracy of the classification improves. For all SA methods, the accuracy attained by the first block, which contains 84 most important features falls below 80%. Similarly, the accuracy achieved by the first block of non-influential factors drops below 60% because it employs the least important features. This confirms the efficacy of the SA indexes in accurately identifying the critical regions that significantly influence correct classification. Nevertheless, a notable distinction arises in the first block of pixels chosen by each of the SA methods.
Table 1.
The sampling space size for each of the SA methods
| SA method | Number of samples |
|---|---|
| Morris | 50 in 4 levels |
| Sobol | 300 |
| FAST | 100 |
| RBD | 400 |
| Delta | 1000 |
| DGSM | 1000 |
Fig. 3.
Accuracy of MNIST classification on test set data achieved by blocks of a: influential factors, b non-influential factors. The factors are sorted and segmented into pixel blocks. Each block begins with 84 pixels, with the number of pixels in each subsequent block increasing by 100. The final block, located on the right, contains all 784 pixels
Comparing the results in Fig. 3a, it can be concluded that the 
Sobol index, as well as 
and 
indices of Morris method present superior classification precisions. Furthermore, using the top 684 most important pixels segmented in block 6 produces results that closely resemble those of block 7. This indicates that the additional 100 pixels are redundant and contribute no significant information for classification. The results presented in Fig. 3b also reveal that the least influential factors identified by the 
Sobol index along with Morris 
and 
indices derive the lowest accuracy. This puts forward that these indices efficiently identify influential and non-influential features for CNN network. Morris mu and FAST S1 are the next indices that demonstrate notable performance. These methods have previously demonstrated successful results for a variety of applications. Morris has been chosen by many authors due to its efficiency, lower computational cost and interpretablity [52, 53]. Sobol and FAST have also been chosen for complex black-box systems and shown to provide effective results [54–58]. A similar conclusion can be derived from Fig. 4 which compares the impact of the most influential and non-influential factors. The optimal SA methods are characterized by a substantial accuracy with influential factors and a comparatively lower accuracy with non-influential factors, highlighting a significant disparity in the achieved accuracies between the two factors. It is evident that 
and 
from Morris and 
from Sobol demonstrate a significant disparity in accuracy of classification. The obtained sensitivity indices are also visualized in Fig. 5 to reveal the pixel locations and their impact on classification results. Since MNIST digits are centered in the images, we expect higher saliency in the central area and lower saliency in the surrounding regions. The results are inline with our previous observation and suggest that the saliency maps returned by Morris, Sobol and FAST specify important pixels more effectively compared to the rest of the discussed methods in this paper. As expected, our case study demonstrated that different global explainablity methods yield different rankings of feature importance when applied to the same machine learning model. This supports our hypothesis that not all sensitivity analysis methods are equally efficient for every problem, as they utilize different data properties to analyze models. Therefore, it is essential to conduct rigorous analysis for each specific problem and carefully select the most appropriate global explainablity approach that is suitable for a particular task. We proposed methodology for identification of the most effective sensitivity analysis method for a given task. Additionally, we provided a detail review about each GSA method along with their pros and cons that would help end users to pick a suitable method for each particular task. Essentially, we conducted our experiments using deep neural networks on a standard large dataset and provided a comparison on the performance of GSA method. Further studies are needed to determine the generalizability of sensitivity analysis methods across different tasks and different data types. One issue about applying GSA methods is the computational cost of sampling and analyzing all the generated samples. In future works, we will investigate efficient sampling methods for enhancing in the quality of SA indices.
Fig. 4.
Comparison between the most influential and non-influential factors. The red bars represent the classification accuracy achieved using the most influential features identified by each algorithm, while the blue bars show the accuracy for non-influential features
Fig. 5.
Visualization of the sensitivity indices of each method trained on MNIST classification task. The higher values which are highlighted in yellow indicate more influential features as determined by each algorithm
Funding
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.
Data availability
The dataset analyzed during the current study is available at https://www.tensorflow.org/datasets/catalog/mnist
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Consent to participate and consent to publish were obtained from all participants.
Competing interests
On behalf of all authors, the corresponding author states that there is no competing interest.
Footnotes
Publisher’s Note
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The dataset analyzed during the current study is available at https://www.tensorflow.org/datasets/catalog/mnist