Isotonic Regression As a Function of Comments
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Suppose you have data giving the height, shirt size, and weight of some people, and you want to generate a regression function to predict the weight of a person given their height and shirt size. Further, you only want to assume that for a fixed height, weight is a nondecreasing function of shirt size, and for a fixed shirt size, weight is a nondecreasing function of height. The optimal regression function under these weak assumptions is known as isotonic regression. Isotonic regression is being increasingly used as researchers find it difficult to justify parametric assumptions such as linearity; or when the independent variables have an ordering but no natural metric, such as S less than M less than L less than XL shirt sizes; or when the independent variables form a tree instead of a multidimensional grid. I'll give some results motivated by reviewer comments, one asinine and one helpful. When error is measured with the L_\infty metric the regression is not unique and results concerning the "best best" isotonic regression will be presented. Also, algorithms for L_1 will be used in algorithms for exact solutions for L_2. I'll also give some open questions on nonparametric regression. The algorithms are serial, so parallel-phobic individuals need not worry.