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Special section: Methods for Uncertainty Computing Either Uncertainty Propagation or Optimization Under Uncertainty

# Propagating Skewness and Kurtosis Through Engineering Models for Low-Cost, Meaningful, Nondeterministic Design

[+] Author and Article Information
Travis V. Anderson

Design Exploration Research Group, Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602travisanderson@byu.edu

Christopher A. Mattson1

Design Exploration Research Group, Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602mattson@byu.edu

1

Corresponding author.

J. Mech. Des 134(10), 100911 (Sep 28, 2012) (9 pages) doi:10.1115/1.4007389 History: Received July 29, 2011; Revised March 19, 2012; Published September 21, 2012; Online September 28, 2012

## Abstract

System models help designers predict actual system output. Generally, variation in system inputs creates variation in system outputs. Designers often propagate variance through a system model by taking a derivative-based weighted sum of each input’s variance. This method is based on a Taylor-series expansion. Having an output mean and variance, designers typically assume the outputs are Gaussian. This paper demonstrates that outputs are rarely Gaussian for nonlinear functions, even with Gaussian inputs. This paper also presents a solution for system designers to more meaningfully describe the system output distribution. This solution consists of using equations derived from a second-order Taylor series that propagate skewness and kurtosis through a system model. If a second-order Taylor series is used to propagate variance, these higher-order statistics can also be propagated with minimal additional computational cost. These higher-order statistics allow the system designer to more accurately describe the distribution of possible outputs. The benefits of including higher-order statistics in error propagation are clearly illustrated in the example of a flat-rolling metalworking process used to manufacture metal plates.

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Topics: Errors , Functions

## Figures

Figure 1

Predicted output distributions obtained from propagating (a) mean and variance only, and (b) mean, variance, skewness, and kurtosis. Actual system output distribution is shown in (c).

Figure 2

Examples of negative (left) and positive (right) skewness

Figure 3

The excess kurtosis of various common statistical distributions

Figure 4

The flat-rolling manufacturing process whereby plates or sheets of metal are made. Material is drawn between two rollers, which reduces the material’s thickness.

Figure 5

Distribution of the coefficient of friction in a flat-rolling metalworking process

Figure 6

Predicted output distributions obtained from propagating (a) mean and variance only, and (b) mean, variance, skewness, and kurtosis. Actual system output distribution is shown in (c).

Figure 7

Accuracy (distribution overlap) of different error propagation methods

Figure 8

Computational cost of different error propagation methods, as measured by MATLAB execution time

Figure 9

Sensitivity of the accuracy of the predicted output distribution to error in the derivative approximations

## Errata

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