Data depth for simple orthogonal regression with application to crack orientation

This paper studies tangential and simplicial data depth for simple orthogonal regression. Given N points in the plane, simple orthogonal regression means that we wish to determine the line through the origin that has smallest distance to the points measured in the direction orthogonal to the line. For both depth notions, it is proved that two lines which are orthogonal to each other, i.e. two lines forming a cross, have the same depth. Depth-based orthogonal regression can thus merely fit crosses, not lines. We investigated the robustness properties of maximum depth estimators using the notion of exact fit. Another topic the paper covers is the testing of the hypothesis that the data points form a cross-like pattern. After a simple transformation, such a test can be based on the biggest data depth. The paper discusses an application of this test for the investigation of stress fractures in materials.

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