Test for a Normal Distribution and Data Continuity

                The application of geostatistical interpolation methods (Kriging) is dependant upon the assumption of a normal distribution of the data, and a constant mean and constant variance (second order stationarity, as discussed in the next section).  In addition to interpolation methods, the Geostatistical Wizard also has exploratory spatial data analysis options (ESDA) that allow for examination of data distribution characteristics (ESRI 2003).  This type of analysis can predict  problems that may arise in interpolation and suggest modeling modifications to correct these problems.  Results of this data analysis are presented in Table 2 and Figures 3-5.

    When the mean and median are approximately equal and the skewness approaches zero, a normal (symmetric) data distribution is supported (Kleinbaum et al. 1998).  Both cases are true for the 1 hour fuel data (Table 2).  The Kurtosis is a measure of the “peakedness” of the data (flat vs. high peak around the mean or median).  A high kurtosis indicates that the variance is strongly affected by extreme values in the data.  The 1 hour fuel data show a fairly low degree of kurtosis, suggesting that the variance is controlled by frequent and moderate sized deviations from the mean (Wikipedia Kurtosis 2009), resulting in a lower variance.  In contrast, the 10 hour fuels show a marked divergence in mean vs. median, greater skewness and greater kurtosis.  This suggests that the data distribution may not be normal.  The high degree of kurtosis presents the greatest problem because it indicates that the variance is strongly affected by rare extreme values (Neter and Wasserman 1974). The higher kurtosis results in an expected higher variance.  The lack of a normal distribution in the 10 hour fuel data predicts that there may be some problems encountered when Kriging, and more data analysis is warranted.

Figure 3A. 1-Hour Fuels

Figure 3B. 10-Hour Fuels

    Normal QQ-Plots are constructed by plotting data quantiles vs. the standard normal value (ESRI 2003).  The straight line shown on these plots represents the General Normal QQ Plot.   When the Normal QQ Plot (points) approximate the General QQ Plot (straight line) then the data distribution is normal (ESRI 2003).  Figure 4A demonstrates the close agreement expected when the data distribution is in fact normal, as with the 1 hour dataset.  Figure 4B shows the expected deviations from normality in the 10 hour dataset.   However, as shown in the histogram for the 10 hour data, the extreme values are on the high (volume) end of the distribution, even though most of the10 hour data do approximate a normal distribution according to the Quantiles (Figure 4B).  Therefore, the data will be model as a normal distribution, with the knowledge that some parameter adjustments may be required in the Kriging process, and that the accuracy of the predictions could be reduced.

Figure 4A. Normal QQ Plot for 1-Hour Fuels

Figure 4B. Normal QQ Plot for 19-Hour Fuels

Trend Analysis performed using the Geostatistical Wizard ESDA permits the examination of trends in the data.  This is especially important when setting Kriging parameters such as whether or not to use a directional variogram, and in the case of Universal Kriging, determining the need for a de-trending polynomial function.  The 1 Hour fuels appear to have three clusters of data that approximate the mesa top regions (Figure 5A), and showing some degree of volume increase from south to north.  The 10 hour fuels show a very gradual trend to higher fuel volumes that corresponds with the change in elevation across the region  from southeast to northwest (Figure 5B).  This suggests that polynomial de-trending and/or a directional variogram may be required when Kriging interpolation methods. Compare to the elevation trends in Figure 2.

Figure 5A. 1-Hour Fuels Trend Analysis

Figure 5B. 10-Hour Fuels Trend Analysis

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