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Statistics « Travis V. Anderson

Engineer by day, Superhero by night — Amazing all the time

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Publication in the Journal of Mechanical DesignLast September, ASME’s Journal of Mechanical Design published a technical paper I wrote as part of my mechanical engineering master’s research.

This peer-reviewed paper is about error propagation through design models. Essentially, a Taylor series is often used to propagate error through closed-form engineering models. However, designers typically assume outputs are Gaussian, and neglect any higher-order statistics (such as skewness and kurtosis). This paper shows that skewness and kurtosis can also be propagated using a Taylor series with little additional computational cost, but tremendous improvements in overall accuracy.

Title Propagating Skewness and Kurtosis Through Engineering Models for Low-Cost, Meaningful, Non-Deterministic Design
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.
Journal Journal of Mechanical Design
Publication Date October 2012
Volume 134
Issue 10
Publication URL
Written by Travis Anderson on January 4th, 2013 , Mechanical Engineering, Patents & Publications, Statistics

Publication in the Journal of Mechanical DesignThis month, ASME’s Journal of Mechanical Design published a technical paper I wrote as part of my mechanical engineering master’s research.

This peer-reviewed paper is about error propagation through design models. The formula typically used to analytically propagate error is based on a first-order Taylor series expansion, and consequently, it can be wrong by one or more orders of magnitude for nonlinear systems. Using a higher-order Taylor series does improve the accuracy of the approximation, but this comes at higher and higher computational cost. This paper presents a technique for error propagation that achieves higher-order accuracy but without the additional higher-order cost. This is accomplished by predicting the Taylor series truncation error and applying a “correction factor” to the lower-order model.

Title Efficient Propagation of Error Through System Models for Functions Common in Engineering
Abstract System modeling can help designers make and verify design decisions early in the design process if the model’s accuracy can be determined. The formula typically more
Journal Journal of Mechanical Design
Publication Date January 4, 2012
Volume 134
Issue 1
Publication URL

Thesis Defense PresentationLast Tuesday, I defended my thesis entitled Efficient, Accurate, and Non-Gaussian Statistical Error Propagation Through Nonlinear, Closed-Form, Analytical System Models.

Basically, my thesis uses a Taylor series expansion to estimate the distribution of system model outputs resultant from propagating error in system inputs through a system model. I show how variance can be propagated more efficiently and accurately, producing fourth-order accuracy with only second-order computational cost. I also show that output skewness and kurtosis can be estimated.

All of my committee members thought I did really well, and were very impressed by my presentation, the quality and clarity of my writing, and the research work I have done. One of my committee members, Dr. Chase, was particularly excited about my work. He’s in his seventies (he retired last year, but still continues teaching as an adjunct faculty) and has spent his entire career in the field of assembly uncertainty and tolerance analysis. He’s really excited about the contributions I’ve made to uncertainty analysis and its applicability to tolerance analysis. He said mine was one of the best defenses he’s ever been to ("And I’ve been here a lot of years!"), and he asked for a copy of my defense presentation to use as an example for all his current and future grad students.

Since my work was so math-intensive (not very advanced math, but a lot of it), I’ve had Dr. Fullwood, who has his Ph.D. in math, review my papers before I submitted them for publication. He said I’m an exceptional writer. My advisor, Dr. Mattson, said there’s not a single thing that could’ve gone better in my presentation.

Written by Travis Anderson on July 31st, 2011 , Downloads, Mechanical Engineering, Patents & Publications, Statistics Tags: , ,