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A high quality distance function that measures the difference between instances is essential in many real-world applications and research fields. For example, in instance-based learning, the distance function plays the most important role. A large number of distance functions have been proposed. For nominal attributes, Value Difference Metric (VDM) is one of the state-of-the-art and widely used distance functions. However, it needs to estimate the conditional probabilities, which drops its efficiency in computing the distance between instances. Besides, a practical issue that arises in estimating the conditional probabilities is that the denominators can be zero or very small. This makes them either undefined or very large. Therefore, an efficient distance function that can measure the difference between two instances but without the practical issue confronting VDM is desirable. In this paper, we propose a novel distance function: Frequency Difference Metric (FDM). FDM is just based on the joint frequencies of class labels and attribute values, instead of the conditional probabilities. Extensive empirical studies show that FDM performs almost as well as VDM in terms of accuracy, but significantly outperforms VDM in terms of efficiency. This work provides a very simple, efficient, and effective distance function that can be widely used in many real-world applications and research fields.

The bisector of two plane curve segments (other than lines and circles) has, in general, no simple — i.e., rational — parameterization, and must therefore be approximated by the interpolation of discrete data. A procedure for computing ordered sequences of point/tangent/curvature data along the bisectors of polynomial or rational plane curves is described, with special emphasis on (i) the identification of singularities (tangent–discontinuities) of the bisector; (ii) capturing the exact rational form of those portions of the bisector with a terminal footpoint on one curve; and (iii) geometrical criteria the characterize extrema of the distance error for interpolants to the discretely–sample data. G¹ piecewise– parabolic and G² piecewise–cubic approximations (with O(h⁴) and O(h⁶) convergence) are described which, used in adaptive schemes governed by the exact error measure, can be made to satisfy any prescribed geometrical tolerance.

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.