KernReg

Introduction

KernReg provides a pure-Python routine for local polynomial kernel regression based on Wand & Jones (1995) and their accompanying R package KernSmooth. In addition, KernReg comes with an automatic bandwidth selection procedure that minimizes the residual squares criterion proposed by Fan & Gijbels (1996).

KernReg allows for the estimation of a regression function as well as their vth derivatives using the Gaussian density function as kernel weights. The degree of the polynomial p must be equal to v + 1, v + 3, v + 5, or v + 7.

Local polynomial fitting provides a simple way of finding a functional relationship between two variables (where X typically denotes the predictor, and Y the response variable) without the imposition of a parametric model. It is a natural extension of local mean smoothing, as described by Nadaraya (1964) and Watson (1964). Instead of fitting a local mean, local polynomial smooting involves fitting a local pth-order polynomial via locally weighted least-squares. The Nadaraya–Watson estimator is thus equivalent to fitting a local polynomial of degree zero. Local polynomials of higher order have better bias properties and, in general, do not require bias adjustment at the boundaries of the regression space.

Installation

Install KernReg via PyPI.

$ pip install kernreg

Quick-Start

import kernreg as kr

motorcycle = kr.get_example_data()
x, y = motorcycle["time"], motorcycle["accel"]

# By default, only x and y need to be provided.
# Derivative = 0 is chosen by default
# and hence the polynomial degree = 0 + 1.
rslt_default = kr.locpoly(x, y)
kr.plot(x, y, rslt_default, "motorcycle_default_fit.png")

# We can improve on the default specification by
# choosing a higher order polynomial
rslt_user = kr.locpoly(x, y, degree=3)
kr.plot(x, y, rslt_user, "motorcycle_user_fit.png")

References

Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability, 66. Chapman & Hall.

Wand, M.P. & Jones, M.C. (1995). Kernel Smoothing. Monographs on Statistics and Applied Probability, 60. Chapman & Hall.

Wand, M.P. and Ripley, B. D. (2015). KernSmooth: Functions for Kernel Smoothing for Wand and Jones (1995). R package version 2.23-18.

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* The image is taken from futurist illustrator Arthur Radebaugh's (1906–1974) Sunday comic strip Closer Than We Think!, which was published by the Chicago Tribune - New York News Syndicate from 1958 to 1963.