A technique which measures the dimension of a point cloud in the vicinity of each observation is discussed. The basic idea is as follows. First, each observation is taken in turn as the center of a ball. A formula inspired by a stochastic model is applied to the observations contained in the ball. The result is a ‘raw’ local dimensions for the observation at the center. The raw dimensions are then smoothed, either by kernel smoothing or recursive partitioning. By measuring dimension locally, low-dimensional structure can be recognized even if it is nonlinear. Three examples of the use of the technique are given.
