|
||
i |
X |
Y
|
1 |
Latitude (deg) |
Y = X/90 |
2 |
Longitude (deg) |
Y = X/150 |
3 |
Elevation (ft) |
Y = X/10000 |
4 |
Mean annual precipitation (in.) |
Y = X/100 |
5 |
Ratio of minimum average two-month total precipitation to maximum average precipitation |
Y = X |
6 |
Beginning month j = 1, , 12 of minimum average two-month precipitation |
Y = sin(2 j/12) |
7 |
Beginning month j = 1, , 12 of maximum average two-month precipitation |
Y = sin(2 j/12) |
Remembering that the purpose of the clustering was to produce a set of stations for which each station in the set responds to the same physical controlling processes, that is, all stations in the set exhibit the same precipitation climate (as defined by the variables upon which the clustering is based), and also for which the set is homogeneous solely with respect to annual precipitation amounts (a requirement for the follow-on precipitation probability analyses), it was known a priori that the areal extent of a region would be relatively small. For convenience of computation, the contiguous United States was split into four overlapping quadrants. The sites in each quadrant were then clustered.
The output from both the average linkage and Ward's methods was very similar. For each quadrant, 2 through 14 clusters were subjectively reviewed to insure spatial continuity and physical reasonableness. The overlap areas between adjacent quadrants were also examined to ensure consistency of results from the separate quadrant cluster analyses. Cluster members were occasionally moved to other clusters to meet the spatial continuity requirements. The review resulted in an initial regionalization consisting of between 7 and 11 clusters per quadrant; most of the clusters were large in areal extent. The clusters were subjectively determined to be reasonable in the sense that they depicted areas that could easily be justified on the basis of controlling physical processes.
Homogeneity of annual precipitation amounts within a region was evaluated by using L-moment techniques. Scatterplots of L-CV and L-skew versus L-kurtosis show compact groupings when the data are homogeneous. Based on this idea, Hosking and Wallis (1993) defined measures of discordancy and homogeneity for regional data. Given a region, a discordancy measure based on the individual site L-CV, L-skew, and L-kurtosis vector difference from the region centroid identifies those sites that are grossly different from the region as a whole. It is a guideline rather than a formal statistical test because the data are not assumed to come from identical multivariate distributions. A homogeneity measure estimates the degree of heterogeneity within a group of sites. This measure assumes that in a homogeneous region, all sites will have the same population L-moments, but that sample L-moments will differ because of sampling variability. It compares the dispersion of the observed L-CVs at the sites to the dispersion that would be expected in a homogeneous region; the expected dispersion is obtained through simulation.
The discordancy and homogeneity measures were computed for each of the initial regions defined by the cluster analyses. If the homogeneity test showed a cluster to be heterogeneous, the stations in the cluster were separated by the clustering algorithms into smaller groupings. This iterative procedure ended when the smaller groupings either were homogeneous or appeared to display random geographical patterns that could not be justified on physical grounds. At this point, the homogeneity and discordancy measures for the stations within a cluster were generally acceptable.
The final result of the clustering process was a division of the HCN sites into 111 regions, of which 108 were accepted as homogeneous for annual precipitation by the homogeneity measure of Hosking and Wallis (1993). Only in three areas did it prove impossible to define homogeneous regions: Nevada, central Colorado, and the Olympic peninsula of Washington State.
Regional average L-moments were computed and used to fit the three-parameter generalized extreme value, Pearson type III (gamma), generalized logistic (as defined by Hosking 1990, Table 1), and lognormal distributions. Two-parameter distributions were not considered because the regions are typically large enough so that the third parameter can be estimated with sufficient accuracy.
A measure constructed by Hosking and Wallis (1993) was used to evaluate the goodness of fit. This measure is based on the difference between L-kurtosis of the fitted distribution and the regional average L-kurtosis of the sample data. Assessment of goodness of fit is based on L-kurtosis, the fourth L-moment, because the first three L-moments are used to estimate the three parameters of the distribution.
Counts were made by duration, region, and starting month of the number of times a distribution was acceptable. The gamma was found to be acceptable most often for precipitation totals over all durations. The lognormal and generalized extreme value distributions were acceptable almost as often as the gamma for durations longer than six months. The generalized logistic was acceptable least often.
In many regions more than one distribution passed the goodness-of-fit test. This means that the amount of data in the region was not sufficient to enable discrimination between the distributions. This is not surprising since some of the distributions closely resemble each other over certain ranges of skewness. At the low skewness values typical of 12-month precipitation, for example, the lognormal and gamma distributions are very similar; they both reduce to the normal distribution when the skewness is zero. When more than one distribution is accepted by the goodness-of-fit test, the estimated quantiles may be expected to be very similar except in the extreme tails of the distributions. In our study we were concerned not with extreme tail quantiles, but only with quantiles in the range 0.02 to 0.98, and the differences between estimated quantiles were generally small compared with the root-mean-square errors of the quantile estimates. We therefore deemed it adequate to use any of the distributions that passed the goodness-of-fit test.
Because of user friendliness considerations about the atlas, it was decided to compute quantile values, if possible, from only one distribution function for all regions and durations. Based on the counts of acceptable fits, the gamma was chosen. However, the gamma was not acceptable for all time periods and regions. Two conditions preclude the use of the gamma: first, the goodness-of-fit measure finds it unacceptable; second, the region is heterogeneous. For this second condition there is no reason to assume that a single distribution will give a good fit to every site's data within the heterogeneous region. The Wakeby five-parameter distribution was chosen as the single fitted distribution for a heterogeneous region. The Wakeby was also chosen as the distribution for a region where the gamma was unacceptable. The generalized extreme value, generalized logistic, and lognormal were not chosen because they were rarely acceptable when the gamma was unacceptable.
Once a distribution was chosen, quantile values were calculated from the regional average L-moments. In dry areas for the shorter durations, some of the values were negative. Because precipitation amounts are calculated by multiplying a site or regional mean precipitation amount by a quantile value, negative values violate the physical lower bound of zero precipitation totals. Consequently, a mixed model was used, of the form
F(x) = p + (1-p) G(x) , |
where F is the cumulative distribution function (cdf) of precipitation amounts, p is the probability that the precipitation amount is zero as estimated by the proportion of zero values in the data for the region, and G is the cdf of the distribution of non-zero precipitation amounts as estimated from the regional average L-moments of the non-zero data values. For distribution fitting, the L-moments were computed from only the non-zero data; L-moments from both the non-zero and zero data were used for defining regions.
As stated previously, the distribution G was initially chosen to be gamma in homogeneous regions for which the gamma distribution was accepted by the goodness-of-fit criterion, and Wakeby otherwise. However, G was constrained to have a lower bound of zero when this was necessary to obtain non-negative quantiles for all the probabilities of interest (the lowest of these is 0.02). When constrained estimation was necessary, the four-parameter Wakeby with fixed lower bound of zero was fit. A gamma distribution with zero lower bound was not used because it has only two free parameters, and it rarely gave a good fit to the data.
In spite of its wide range of distributional shapes, the Wakeby distribution cannot be fit to all data sets because there are some L-moment values that no Wakeby distribution attains. The Wakeby distributions were fit using Hosking's (1991) implementation of the algorithm of Landwehr et al. (1979). In this implementation, when the full five-parameter Wakeby distribution cannot be fit, successive attempts are made to fit four-parameter Wakeby and three-parameter Wakeby (three-parameter Pareto) distributions until a successful fit is achieved.
Quantile values were assessed by their bias and root-mean-squared error (rmse). These quantities cannot be calculated analytically. Instead, a Monte Carlo simulation procedure was used. Simulated data were generated for a hypothetical region with the same number of sites and the same record lengths as the actual region, drawn from the distribution that was fit to the actual regional data. Quantile estimates were calculated for the sites in this simulated region. For bias and rmse estimates, the simulation was repeated 500 times. The 500 sets of errors in the simulated quantile estimates were accumulated and averaged to yield approximations to the bias and rmse of the quantile estimates calculated from the actual data.
For all durations, for quantiles between 0.02 and 0.50, the bias is negligible, and the rmse is less than 0.10. For the shorter durations, for quantiles greater than 0.50, the bias and rmse are only slightly larger. This bias and standard error generally decrease as the duration increases.
An exception is the Pareto distribution. It was used to fit a few of the shorter duration samples, but at the higher quantiles, especially 0.98, confidence is minimal. However, since the Pareto consistently underestimates the precipitation at these higher quantiles, the values are conservative in terms of drought planning.
The only confidence measures used are the bias and rmse determined from the simulation. Confidence intervals were not computed. Following usual practice, intervals could be constructed by adding and subtracting the product of the rmse of the estimated quantile value and the appropriate percentage point from the standardized normal distribution to the quantile estimate. This construction assumes that quantile estimates are normally distributed. The validity of this assumption is, however, dubious for extreme quantiles, for arid areas, and for quantiles close to zero. It may also be questionable for other quantiles and other areas. Unless the assumption of normality of quantile estimates is verified, the usual practice of constructing confidence intervals is strongly discouraged.
Preliminary estimates made in the early stages of atlas preparation indicated that drought frequencies displayed orderly patterns. The final frequency patterns are even more orderly than anticipated. Over large areas of the United States, the estimates of the once in 50-year low precipitation, the 0.02 quantile, vary little, and vary in the directions a climatologist or hydrologist would expect. While this is especially true for durations of 12 months and longer, it is also true for the shorter durations, although for the shorter durations the seasonal distribution of precipitation is a prominent feature. What might not be so obvious is that there are substantial seasonal differences in the frequency distributions at the shorter time periods.
There are a few places that display marked differences in frequency distributions from the surrounding or nearby territory. The largest difference in the eastern states is found in Key West, Florida (which resulted in the Key West precipitation station being the only station in a cluster). In the western states, the largest differences are in the California and Nevada deserts and in western Washington state.
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revised 1 Aug 2006
