Regression models using the LINEX loss to predict lower bounds for the number of points for approximating planar contour shapes

Abstract

Researchers in statistical shape analysis often analyze outlines of objects. Even though these contours are infinite-dimensional in theory, they must be discretized in practice. When discretizing, it is important to reduce the number of sampling points considerably to reduce computational costs, but to not use too few points so as to result in too much approximation error. Unfortunately, determining the minimum number of points needed to achieve sufficiently approximate the contours is computationally expensive. In this paper, we fit regression models to predict these lower bounds using characteristics of the contours that are computationally cheap as predictor variables. However, least squares regression is inadequate for this task because it treats overestimation and underestimation equally, but underestimation of lower bounds is far more serious. Instead, to fit the models, we use the LINEX loss function, which allows us to penalize underestimation at an exponential rate while penalizing overestimation only linearly. We present a novel approach to select the shape parameter of the loss function and tools for analyzing how well the model fits the data. Through validation methods, we show that the LINEX models work well for reducing the underestimation for the lower bounds.

Keywords: 62H35; 62J05; 62J20; LINEX loss function; Linear regression; lower bound prediction; planar contours; shape analysis.

Figure 1.

Change of shape due to approximation as k increases; approximated polygons for $k=4,10,25,40$, and 200 in blue on original contours in red.

Figure 2.

Choosing 50 points along the contour using the three different parameterizations: (left) arc-length, (middle) equal curvature, and (right) Latecki parametrization.

Figure 3.

Plots of (a) a vs cost function and (b) the RMSE power ratio as a function of the tuning parameter a for the $k_{LA}$ model for selecting the value of a to use.

Figure 4.

Histograms of residuals for $k_{DC}$ model using the specified loss function. (a) Least squares and (b) LINEX.

Figure 5.

Analysis of the relative costs and residuals of the pears under $k_{LC}$. (a) Relative cost. (b) Shape 153 and (c) Predicted vs. observed k.

Figure 6.

(a) The distribution of average cost for testing sets consisting of 10 randomly selected shapes. The average cost for the group of dogs is shown as a red asterisk. (b) A scatterplot of the maximum value of relative cost of testing sets of size 10 for $k_{LA}$ versus the sample index. Each testing set is color-coded by the total cost of that set of observations. (c) The same scatterplot re-coded to indicate the presence of shapes 62 and 233.

Figure 7.

The distribution of average cost for testing sets consisting of (a) 20, (b) 5, and (c) 88 randomly selected shapes. The average costs for the various groups of shapes are shown as red symbols.

Figure 8.

Plot of K vs.error under $k_{DA}$.

Figure 9.

The originally observed contour for shape 233 and the approximation resulting from the lower bound predicted by the $k_{LA}$ model.