Piecewise Linear Models with Guaranteed Closeness to the Data

Abstract

This paper addresses the problem of piecewise linear approximation of point sets without any constraints on the order of data points or the number of model components (line segments). We point out two problems with the maximum likelihood estimate (MLE) that present serious drawbacks in practical applications. One is that the parametric models obtained using a classical MLE framework are not guaranteed to be close to data points. It is typically impossible, in this classical framework, to detect whether a parametric model fits the data well or not. The second problem is related to accurately choosing the optimal number of model components. We first fit a nonparametric density to the data points and use it to define a neighborhood of the data. Observations inside this neighborhood are deemed informative; those outside the neighborhood are deemed uninformative for our purpose. This provides us with a means to recognize when models fail to properly fit the data. We then obtain maximum likelihood estimates by optimizing the Kullback-Leibler Divergence (KLD) between the nonparametric data density restricted to this neighborhood and a mixture of parametric models. We prove that, under the assumption of a reasonably large sample size, the inferred model components are close to their ground-truth model component counterparts. This holds independently of the initial number of assumed model components or their associated parameters. Moreover, in the proposed approach, we are able to estimate the number of significant model components without any additional computation.