Spatial Smoothing - Cont.

The loess procedure determines the fitted value z by performing a local regression for each county. The transformed loss costs are fitted to the independent variables of latitude and longitude including an interaction term. The set of fitted values over all counties defines the fitted surface. Each local regression includes only those points that are in the neighborhood of the point being fit, where a neighborhood consists of the nearest k% of points in the sample space. A small value of k results in greater local accuracy, whereas a large value of k results in a smoother surface. For this analysis, 100% of the data points have been included in each neighborhood. However, greater weight is assigned to nearby counties than the more distant counties by the use of the tri-cube formula (1 – d3)3. The distance d between any two counties x0 and xi is defined as d = |x0 – xi| / max(|x0 – xk|), where the denominator is computed over all values k within the neighborhood. Here, each point xi represents the joint latitude and longitude coordinates at the center of a specified county. The coordinates of the county midpoints must be transformed to the Euclidean coordinate system using a distance preserving projection prior to their use in the loess procedure.

The best fit local regression for each county is determined by using a maximum likelihood technique under the assumption that residuals have a normal distribution with constant variance. The residuals are weighted by the actual cumulative liability (i.e., exposure) for each county in order to improve the accuracy of the smoothed results. As in actuarial credibility, the loss costs of counties with larger weight, as measured by cumulative liability, reflect their own experience to a greater degree than counties with smaller weight. The amount of smoothing produced by this process can be evaluated from a comparison of Charts 22 and 23, which show cotton loss costs for Crop Hail insurance in the southeastern states. The perspective on these maps is looking west from the Atlantic Ocean, with Florida shown to the left and North Carolina to the right.