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to calculate the proportion of space in compliance that month. The current estima¬ 
tion procedure for obtaining predicted values is Inverse Distance Weighting (IDW), 
a non-statistical spatial interpolator that uses the observed data to calculate a 
weighted average as a predicted value for each location on the prediction grid. The 
method calculates the weight associated with a given observation as the inverse of 
the square of the distance between the prediction location and the observation. 
The panel considered several interpolation methods in addition to IDW. Of these, 
kriging methods emerged as a principal alternative approach for populating the grid 
of prediction locations. Non-parametric methods were also considered. These 
include Loess regression or cubic spline methods. These approaches could be advan¬ 
tageous in that they are statistical methods that provide levels of error, but panel 
analyses and deliberations have been insufficient to provide definitive statements on 
this class of methods. Table 3.2 which appears in Section 3.3 summarizes our deter¬ 
minations. 
3.1 KRIGING OVERVIEW 
Kriging is a spatial interpolation technique that arose out of the field of geostatistics, 
a subfield of statistics that deals with the analysis of spatial data. Kriging and the 
field of geostatistics has been employed in a wide variety of environmental applica¬ 
tions and is generally accepted as a method for performing statistically optimal 
spatial interpolations (Cressie 1991, Schabenberger and Gotway 2004, Diggle and 
Ribeiro 2006). Applications of kriging in water related research can be found in 
(Kitanidis 1997, Wang and Liu 2005,0uyang et al. 2006). References on kriging 
methodology, geostatistics, and their related statistical development can be found in 
(Cressie 1991, Diggle et al. 1998, Schabenberger and Gotway 2004, Diggle and 
Ribeiro 2006). 
Kriging can equivalently be formulated in terms of a general linear regression model 
Y (s) =/3 0 + £, X|(s) • • • +/3 p X p (s) + e(s) (1) 
with s representing a generic spatial location vector (usually 2-D) assumed to vary 
continuously over some domain of interest, Y(s) the outcome of interest measured at 
s, X](s).X p (s,) potential covariates indexed by location s, and their associated 
regression effects /3j, . . . , /3 p . Note that covariates must be known at every predic¬ 
tion location. The elements of the spatial vector s can be used as covariates for 
modeling spatial trends. On the other hand water quality measures such as salinity 
which may have a strong association with the outcome of interest, is of limited value 
as a covariate because it is not known at all prediction locations. The uncertainty in 
this regression relationship is modeled with the random error term e{s ) assumed to 
have zero mean and constant variance. Spatial data like the type sampled in the 
Chesapeake Bay water-quality criteria assessments often exhibit a property known 
as (positive) spatial dependence, observations closer together are more similar than 
those further away. This property is accounted for in model (1) by allowing e(s) to 
have a spatial correlation structure. 
Some further specifics on e(s) are warranted. Common distributional assumptions on 
e(s ) include normality or log-normality, although kriging can be performed based on 
appendix a 
The Cumulative Frequency Diagram Method for Determining Water Quality Attainment 
