SMOOTH DENSITY SPATIAL QUANTILE REGRESSION

Abstract

We derive the properties and demonstrate the desirability of a model-based method for estimating the spatially-varying effects of covariates on the quantile function. By modeling the quantile function as a combination of I-spline basis functions and Pareto tail distributions, we allow for flexible parametric modeling of the extremes while preserving non-parametric flexibility in the center of the distribution. We further establish that the model guarantees the desired degree of differentiability in the density function and enables the estimation of non-stationary covariance functions dependent on the predictors. We demonstrate through a simulation study that the proposed method produces more efficient estimates of the effects of predictors than other methods, particularly in distributions with heavy tails. To illustrate the utility of the model we apply it to measurements of benzene collected around an oil refinery to determine the effect of an emission source within the refinery on the distribution of the fence line measurements.

Figure 1:

Example set of normalized B-spline (left) and corresponding I-spline (right) basis functions. Dotted vertical lines indicate knot locations.

Figure 2:

Examples of quantile function (top row) and corresponding density function (bottom row) constructed using different bases.

Figure 3:

Benzene measurements by time and location. Source locations, e₁ and e₂, are shown in black. Points have been jittered slightly to improve visibility.

Figure 4:

Wind roses for different seasons.

Figure 5:

Estimated predictor effect by quantile and location.

Figure 6:

Spatial variation in the effect of the predictors on the 95^th quantile of fenceline benzene measurements.