A decision fusion-based method for global sensitivity analysis of complicated experiments

In real world, the relationship between experimental input and output has become more and more complex, which brings great challenges to the sensitivity analysis. This paper proposes a decision fusion based global sensitivity analysis method for complicated experiments, which not only provides quantitative evluation of the input factor influence on experimental results, but also mines the correlation and form the explicit criteria in IF-THEN fomation for further guidance. The theory of decision information system and continuous attribute discretization is presented first for transforming the experimental input and output into a decision table. In order to calculate the sensitivity of the factors and extract valid correlation criterions between conditional attributes and decision attributes simultaneously, the discrimination matrx is utilized for attribute reduction. Then a sensitivity analysis method based on decision fusion is proposed by organically assembling experiment design, attribute discretization, the discrimization matrix, and attribute reduction. Finally, the effectiveness and practicality of the proposed method were verified by the application of sensitivity analysis in hypersonic vehicle re-entry trajectory experiment.


Introduction
In various fields, simulation and experiments are the primary means for theories test and validation.In response to increasingly complicated real-world problems and richer simulation techniques, study on sensitivity analysis (SA) becomes more necessary, which provides an influence assessment of the critical factors to the experimental results.Moreover, the requirement of sensitivity analysis includes not only the influence assessment, but also the knowledge reveals the correlation between inputs and outputs.SA, an essential tool for modelling and simulation, has long been a hot research topic for many years.SA is divided into local and global categories by the effect of targeting single or multiple parameters on the model output at a time.The classical local SA method is the extended One Factor at a Time (OAT) method named Morris method, which is present by Max D. Morris in 1991 [1].The Morris method provides a single time single factor method, which gives quantitative ranking of the magnitude of the sensitivity of each parameter of the model and is simple and effective in screening out significant factors.In 1990, I.M. Sobol', a famous Russian mathematician, proposed a global sensitivity analysis method based on variance.The basic idea of the method is to study the effect of the variance of the input parameters on the output variance [2].The study and application of SA methods has been further developed.Faniran investigates the global stability of disease-free and epidemic equilibria by means of global Lyapunov functions, using a normalized forward method to derive a sensitivity index for the basic reproduction number [3].Kang proposed an integrated framework for global sensitivity analysis 2 (GSA) and calibration, called GSA-CAL, which was used to identify input parameters that have a small impact on the difference between the simulation output and the observed results.By dropping these less influential input parameters from the calibration process, this study reduces the computational intensity of the calibration [4].With the development of experimental techniques, the focus of SA research is not on how to provide analytical efficiency, but on how to distil the knowledge that guides experimental design.Meanwhile, the Rough Set Theory (RST), as a powerful tool for decision fusion and knowledge mining, provide an innovative solution for SA.RST aims to reduce the attributes of real-world decision systems and mine the minimal set containing key information, which draws a lot of research attention.In terms of experimental data processing, Han proposed a generalized rough set-based information filling technique for failure analysis of thruster experimental data, which extracted the key factors affecting the failure of thruster experiments and formed the guidelines to instruct the design [5].Chen proposed a rough set-based method for updating decision rules on attribute values' coarsening and refining.He developed dynamic maintenance of decision rules for the complexity analysis and extensive experiments on UCI data [6].Fuzzy set theory is receiving more and more attention in practical applications, and these researches involve attribute discretization, interval-valued systems, attribute reduction, information filling, etc [7].Inspired by such approaches, this paper adopts the fuzzy set theory to solve the sensitivity analysis problem of complicated experiments, tries to transform the input-output relationship into a decision information system, and explores the minimum set of rules by attribute discretization and attribute reduction, so as to form effective criterion and knowledge to guide the experiment optimization.In this paper, the decision information system and continuous attribute discretization of the decision fusion theory is presented first for constructing a discrete-attribute decision table.In order to calculate the importance of the experimental factors as well as to extract valid correlation criterions between conditional attributes and decision attributes, a discrimination matrix-based attribute reduction method is demonstrated consequently.As a result, a sensitivity analysis method based on decision fusion is proposed by organically assembling experiment design, attribute discretization, the discrimization matrix, and attribute reduction.Finally, the effectiveness and practicality of the proposed method were verified by the application of sensitivity analysis in hypersonic vehicle re-entry trajectory experiments.

Decision Information System
In real-life applications, an information system is defined by a quadruples ( ) is the non-empty finite objects set, called a theoretical domain.

 
12 , , , m AT a a a = is the non-empty finite attribute set, is the value field, where a V denotes the value field attribute a , :  .Moreover, an information system is called decision information system if its attribute set consists of a conditional attribute set C and a decision attribute set D , and meets the condition CD =.Generally, a decision information system is denoted as R be a general binary relationship, B AT  be a subset of the attribute set, and within its limits, define the set ( ) Bi Sx , called the information granule: According to the definition of information granules and the equivalence relations, the division of the conditional object set of the decision information system can be obtained with the qualification of conditional attribute combination： Similarly, the division of the decision object set can be obtained with the qualification of the set of decision attributes: Therefore, the core of decision fusion lies in how to describe the decision object equivalence classes by the conditional equivalence classes, and to obtain the classification criterion or decision criterion.If the upper and lower approximate sets of all equivalence classes are combined together under the decision attribute qualification, we can get the upper and lower approximate sets of the whole division D described by division C , which are denoted as

Continuous attribute discretization
In practical decision fusion applications, there are various types of data, mostly containing both continuous and discrete attributes; therefore, when dealing with these data with rough set methods, it is necessary to discretize them first.For a decision information system with continuous attributes, effective discretization requires partitioning the space of continuous attribute values using the smallest possible set of breakpoints, while maintaining the original indistinguishable relationships in the decision table.
Given a decision information system ( ) ( ) ( ) Where: The decision table is compatible when ( ) The premise of effective discretization of continuous attributes is to ensure that the indistinguishable relationship of the decision table remains unchanged, that is, the overall information of the decision table does not incur loss.It is obvious from the definition of decision table compatibility that ensuring the indistinguishability of decision tables is to keep its compatibility unchanged.In this paper, we use the compatibility as feedback information to identify whether the discretization needs to be adjusted.Calculating the compatibility of the decision table can ensure that the indistinguishable relationships in the decision table as a whole do not change and no loss of information occurs.The process of the compatibility-based discretization for continuous attributes is as follows: • Step 1: Calculate the initial compatibility of the original information system ( ) • Step 2: Determine the number of discretization categories k according to the size of the information system, and construct a decision table by discretizing each conditional attribute and decision attribute using the equidistant method [8].
• Step 3: Calculate the compatibility of the decision table ( ) P dQ by equation ( 6) and compare it with the initial compatibility ( ) , then the algorithm terminates and goes to Step 5.
• Step 4: Take 1 kk =+, and redisperse each conditional attribute, and calculate the compatibility of each conditional attribute.When all conditional attributes are discretized, go to Step 3.
• Step 5: The output result is the decision table after discretization.

Discrimination matrix-based attribute reduction for decision systems
Attribute reduction is one of the important functions of rough set theory.Since the importance of each conditional attribute is different for the decision attributes, the purpose of attribute reduction is to eliminate the redundant conditional attributes and reserve the more important conditional attributes for the decision fusion while keeping the classification ability of the decision table intact, so as to finally reduce the decision table.In a decision information system, the element values in the discrimination matrix represent the set of conditional attributes that can distinguish between two objects [9].In the discrimination matrix, an attribute with a high occurrence frequency means that it can distinguish more objects, while an attribute with a low occurrence frequency means that the attribute can distinguish fewer objects.In the extreme case, an attribute that does not occur in the discrimination matrix actually implies that it should be deleted directly.Therefore, the number of occurrences in the discrimination matrix can be used as the criterion for assessing the importance of an attribute.If an attribute has more occurrences in the discrimination matrix, that represents a greater distinguishing capability and a higher importance.Conversely, if an attribute has fewer occurrences, that indicates a lower distinguishing capability and a lower importance.Given a decision information system Where , 1, 2,3,..., i j U = .We use discrimination function  to indicate the discrimination relationship of all objects in the decision information system S.
Hence, the discrimination function  can be given: According to the properties of Boolean operations, the set of all the minimal analytic formulation of the discrimination matrix is the reduction of the decision table.In other words, the discrimination matrix is consisted of all the core attributes.Therefore, the attribute importance is calculated in terms of the occurrence frequency of conditional attributes in the discrimination matrix as follows: The discrimination matrix provides a complete description of the similarities and differences among all objects in a decision information system, and the obtained reduction results are extremely comprehensive.At the same time, this method based on Boolean operations can directly obtain the kernel properties of the information system and find all the approximate reduction by logical operations, eliminating the process of repeated trial and error and improving the overall computational efficiency.

Decision fusion-based sensitivity analysis procedure
In this paper, the sensitivity analysis method based on decision fusion aims to explore the influence and the internal correlation criterion of the experimental factors towards the output results, thus involving experimental design, attribute discretization, and attribute reduction based on the discrimination matrix.The decision fusion-based sensitivity analysis procedure is as follows: • Step 1 (Experimental design): For getting the desired distribution of experimental results, the experimental design needs to be conducted before running the experiment.Here we utilize the orthogonal design for allowing correlation between input variables； • Step 2 (Experiment running): After the input variables are determined, the experiment can be executed for obtaining the output data.Here we treat the experimental execution process as a black box, only focusing on the relationships between its inputs and outputs, and regard it as an information system as a whole.• Step 3 (Attribute discretization): the inputs and outputs of the experiment constitute an information system, the method in Section 2.2 is needed here for continuous attribute discretization in order to achieve a decision table composed of discrete variables.

• Step 4 (Attribute reduction):
A discrimination matrix is constructed, and the importance of the conditional attributes is calculated according to the Equation (10).Then, keep updating the resolution matrix by removing the attribute items containing the highest frequencies until the resolution matrix is empty.Finally, the attribute reduction consisting of multiple high importance attributes can be acquired, which indicates the correlation between the factors and the experimental results.

Application
To verify the practicality of the proposed sensitivity analysis method based on decision fusion, a dynamics model with seven uncertain physical characteristics parameters is given for optimizing the reentry trajectory of hypersonic vehicle.According to the hypersonic vehicle dynamics model of re-entry trajectory, the seven parameters is shown in Table 1, including three dimensionless initial states, two aerodynamic parameters and two physical parameters.For convenience, the range of each parameter variation in the numerical simulation is also listed in Table 1.According to the methodological flow, the experimental design was performed firstly.Considering the experiment scale and computational cost, we design three level for the seven uncertain parameters.Then the experiment matrix is constructed as shown in Table 2.The range of values between the different levels is presented in the fourth column of Table 1.Here we treated the trajectory generation model as a black box and obtained the maximum range based on the parameter design of the seven factors as input, and the data smax results are also shown in Table 2.
7339113 Obviously, the input and output data constituted a typical information system with discrete values for the conditional attributes and continuous values for the decision attributes.In order to accomplish decision fusion, it is necessary to perform the continuous attribute discretization at first.After the discretization procedure elaborated in Section 2.2, when 3 k = , a discrete attribute decision table with an overall compatibility of 1 is obtained, as shown in Table 3. .Then the discrimination matrix is updated by removing the combination of the conditional attribute terms with the highest frequency, and the conditional attribute with the next highest frequency is calculated as Moreover, the decision criterion is obtained by substituting the reduction result into the decision table: For a hypersonic vehicle, the goal is to optimize its design parameters to achieve maximum range.Under a unified dynamics model, the method in this paper contributes a design optimization criterion that focuses on adjusting the initial altitude and the ejection angle to a higher level, or keeping the initial altitude constant and reducing the ejection angle, which can achieve a larger range.For comparison, an extreme difference analysis method is used based on the simulation results obtained from this experimental design [10].This method is a traditional sensitivity analysis method for accessing the effect of factors upon the experimental results, which is simple and effective.For the uncertain parameter j x , the range of maximum range performance is defined by x values its kth level, and is given by where n is the number of tests for the kth level of parameter j x , and jkm s the result of the m-th test among the n tests.Till now, we have obtained the ranges of the maximum range performance with respect to each uncertain parameter.Then sensitivity of maximum range to the jth influence factor j x could be computed by s s s , respectively.The distribution of the sensitivity coefficients can be calculated by the above equations as shown in Figure 1(a).The distribution of factor importance calculated by the proposed method is shown in Figure 1(b).Notice that since the principles of the two methods are different, the scales of importance results between each other are also different.Obviously, the importance sorting of the two categories stay consistent, with the initial altitude factor corresponding to the conditional attribute 1 c having the highest influence on the maximum range, followed by the track angle factor corresponding to the conditional attribute 3 c .Further, the application of the proposed method also acquired expressive reduction results by decision fusion.By deriving intuitive correlation criterion from reduction, it serve as a guidance to optimize the experiment execution in practice.

Conclusion
To overcome the increasingly complicated correlation between experimental factors and output results, this paper proposes a decision fusion based global sensitivity analysis method for complicated experiments, not only provides quantitative evluation of the input factor influence on experimental results, but also mines the correlation and form the explicit criteria in IF-THEN fomation to provide further guidance.The decision information system and continuous attribute discretization is presented first for transforming the experimental input and output into a decision table.In order to calculate the sensitivity of the factors and extract valid correlation criterions between conditional attributes and decision attributes simultaneously, the discrimination matrx is utilized for attribute reduction.Then a sensitivity analysis method based on decision fusion is proposed by organically assembling experiment design, attribute discretization, the discrimization matrix, and attribute reduction.Finally, the effectiveness and practicality of the proposed method were verified by the application of sensitivity analysis in hypersonic vehicle re-entry trajectory experiments.
the essence of these two sets is the upper and lower approximate set for the whole information system.
D V f = with conditional attribute set C and decision attribute set D , and AT C D = , CD =.The discrimination matrix is defined as a UU  matrix, which is consists of following elements:

3 c . After removing the combination containing 3 c
, the algorithm ends with the empty discrimination matrix.And the final reduction result can be given as 13 Reduction c c =.
mean value of maximum range performance when uncertain parameter j

1 .
The 14th Asia Conference on Mechanical and Aerospace Engineering Journal of Physics: Conference Series 2746 (2024) 012004 IOP Publishing doi:10.1088/1742-6596/2746/1/0120048 (a) the extreme difference analysis result (b) the decision fusion based analysis result Figure Sensitivity of maximum range to uncertain parameters.

Table 2 .
Parameters and results of orthogonal experiment.

Table 3 .
Discrete attribute decision table.