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Bhavin Sheth, Jasmine Patel, Quang-Tuan Tran; Do We See a Least Squares World Around Us?. Journal of Vision 2012;12(9):895. doi: 10.1167/12.9.895.
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For years, researchers have fit experimental data using the method of least squares, which minimizes the sum of squared residuals (observed – fitted). There are clear numerical and analytical benefits to this method of data fitting, including the fact that least squares is the maximum likelihood error estimate if the errors follow a gaussian distribution. However, does the computed best fit align with our perceived best fit, and do we perceive the least squares fit to be the best fit? Observers (N=17) viewed a set of points on an x-y plane that were jittered about a straight line, with each point’s noise amplitude drawn from a uniform probability distribution (future experiments will draw from a gaussian distribution). The x-y data were fit with straight lines that yielded either the optimal least squares error, absolute error, or fourth order error. Observers were then asked to choose the line that they perceived as being the best linear fit of the points given. No restrictions on time were placed on response. Observers were significantly less likely to choose the optimal least squares line fit (~25% trials): Statistical tests showed a modest but significant preference across observers for the absolute (39+/-0.5%) and fourth power (34+/- 0.3%) regression fits over the least squares (27+/-0.3%) fit (p<0.05 each). Observers significantly preferred the absolute fit over the fourth order fit if the number of data points was either very low (5 points) or very high (100 points). Preferring the line that optimizes the absolute error argues that we do not weight larger error residuals (non-linearly) more (fourth order fit), or that we trade off between penalizing poor data fits and assigning equal weight (least squares would have been a suitable compromise). Regardless, it is clear that humans do not perceive the least squares fit as optimal.
Meeting abstract presented at VSS 2012
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