Showing papers by "Jerome H. Friedman published in 1983"

Graph-Theoretic Measures of Multivariate Association and Prediction

Abstract: Interpoint-distance-based graphs can be used to define measures of association that extend Kendall's notion of a generalized correlation coefficient. We present particular statistics that provide distribution-free tests of independence sensitive to alternatives involving non-monotonic relationships. Moreover, since ordering plays no essential role, the ideas are fully applicable in a multivariate setting. We also define an asymmetric coefficient measuring the extent to which (a vector) $X$ can be used to make single-valued predictions of (a vector) $Y$. We discuss various techniques for proving that such statistics are asymptotically normal. As an example of the effectiveness of our approach, we present an application to the examination of residuals from multiple regression.

Multidimensional Additive Spline Approximation

Abstract: We describe an adaptive procedure that approximates a function of many variables by a sum of (univariate) spline functions $s_m $ of selected linear combinations $a_m \cdot x$ of the coordinates \[ \phi (x) = \sum_{1 \leqq m \leqq M} {s_m ( a_m \cdot x)}. \] The procedure is nonlinear in that not only the spline coefficients but also the linear combinations are optimized for the particular problem. The sample need not lie on a regular grid, and the approximation is affine invariant, smooth, and lends itself to graphical interpretation. Function values, derivatives, and integrals are inexpensive to evaluate.

M and n plots

Abstract: Publisher Summary This chapter discusses M and N plots and describes a method for scatterplotting four-dimensional data. Conventional scatterplots use dots on a rectangular coordinate system to graphically represent two-dimensional vectors. An M and N plot represents (M+N)-dimensional vectors by plotting M coordinates in one coordinate system and N coordinates in a second coordinate system. The two dots representing a point are connected by a straight line segment. Thus, scatterplots are 2 and 0 plots. A 1 and 0 plot—the dot plot—is sometimes used to plot one-dimensional data, plotting each point as a dot. There are two ways to draw M and N plots of two-dimensional data: the conventional scatterplot (2 and 0 plot) and the 1and 1 plot. A 1 and 1 plot represents a two-dimensional point (x,y) by 2 dots and a line segment on a pair of parallel coordinate axis. The basic ingredients of a 2 and 2 plot are a pair of coordinate axes and a segment. With a PRIM 9-1ike graphical device, it is possible to draw 3 and 3 plots of six-dimensional data.