Anisotropic local constant smoothing for change-point regression function estimation

Abstract

Understanding forest fire spread in any region of Canada is critical to promoting forest health, and protecting human life and infrastructure. Quantifying fire spread from noisy images, where regions of a fire are separated by change-point boundaries, is critical to faithfully estimating fire spread rates. In this research, we develop a statistically consistent smooth estimator that allows us to denoise fire spread imagery from micro-fire experiments. We develop an anisotropic smoothing method for change-point data that uses estimates of the underlying data generating process to inform smoothing. We show that the anisotropic local constant regression estimator is consistent with convergence rate $O\left(n^{-1/{(q+2)}}\right)$. We demonstrate its effectiveness on simulated one- and two-dimensional change-point data and fire spread imagery from micro-fire experiments.

Figures (12)

• Figure 1: An example of one-dimensional change-point data, where there is an abrupt change in the regression function at $X=1.5.$
• Figure 2: An example of two-dimensional change-point data, where there is an abrupt change in the regression function at $X^2+Y^2=(20)^2$
• Figure 3: The three one-dimensional data generating processes used for simulations$.~$The left most figure is the piecewise constant regression function$.~$The middle figure is the continuous regression function$.~$The right most figure is the same continuous regression function with a jump at $X=1.5.$
• Figure 4: Fitting the isotropic and anisotropic local constant estimators to the piecewise constant regression function data and comparing to the underlying data generating process (truth)$.~$This plot uses simulation parameters $n=400$ and $\sigma=0.5$, the uniform kernel function for smoothing, and least-squares cross-validation bandwidth selection method$.$
• Figure 5: The MESEs for the three kernel estimators on the piecewise constant function$.~$Note that $.Z$ means $n=Z00$, e.g$.~$LC.4 is the isotropic local constant estimator with $n=400$ and ALC.16 is the anisotropic estimator with $n=1600.$