A metric for quantifying nonlinearity in k-dimensional complex-valued functions

Larry C Llewellyn, Michael R Grimaila, Douglas D Hodson and Scott Graham

The Journal of Defense Modeling and Simulation, 2024, vol. 21, issue 1, 5-15

Abstract: Modeling and simulation is a proven cost-efficient means for studying the behavioral dynamics of modern systems of systems. Our research is focused on evaluating the ability of neural networks to approximate multivariate, nonlinear, complex-valued functions. In order to evaluate the accuracy and performance of neural network approximations as a function of nonlinearity (NL), it is required to quantify the amount of NL present in the complex-valued function. In this paper, we introduce a metric for quantifying NL in multi-dimensional complex-valued functions. The metric is an extension of a real-valued NL metric into the k -dimensional complex domain. The metric is flexible as it uses discrete input–output data pairs instead of requiring closed-form continuous representations for calculating the NL of a function. The metric is calculated by generating a best-fit, least-squares solution (LSS) linear k -dimensional hyperplane for the function; calculating the L2 norm of the difference between the hyperplane and the function being evaluated; and scaling the result to yield a value between zero and one. The metric is easy to understand, generalizable to multiple dimensions, and has the added benefit that it does not require a closed-form continuous representation of the function being evaluated.

Keywords: Nonlinearity metric; complex-valued functions; modeling; simulation (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:sae:joudef:v:21:y:2024:i:1:p:5-15

DOI: 10.1177/15485129221080399

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