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There are numerous ASTM standard test methods where force and displacement are recorded and the data analysis requires that the slope of the force-displacement record be determined. Demands for greater accuracy and the availability of computers have led to the widespread use of simple linear regression. However, computers are not good at determining what data to include in the regression, so the analyst must manually select the upper and lower limits of the regression region, thereby introducing subjectivity into the analysis. Fixed fit ranges that are often used for linear regression can lead to slope bias for data sets that exhibit curvature within the fixed range. This is particularly true for data sets that have an initial curvature or that have a small linear region. Two approaches that provide a powerful tool for examining a data set to determine the linear region are reduced displacement and analysis of residuals. The latter was incorporated into a fully automated algorithm for slope determination by analysis of residuals. This study looked at how noise, digital resolution, and sampling rate influence the determination of slope using this algorithm. Twelve synthetically generated data sets were analyzed to provide insight into how each of these data sets’ characteristics affected the resulting slope. It was determined that slope error from linear regression is a complex interaction between the shape of the data in the nonlinear regions and the data set characteristics. Noise has more effect on slope error than digital resolution over the ranges considered. The algorithm proved robust in that, even with typical noise and digital resolution, slope error in data sets with small linear regions was less than about 2 %.
linear regression, mechanical testing, analysis of residuals
Graham, Stephen M.
United States Naval Academy, Mechanical Engineering Dept., Annapolis, MD