The probability of truth or falseness of a null hypothesis that assumes variables have a random spatial pattern is tested by computing the z-scores and two-tailed p-values (Figure 3.1) (Mitchell, 2005; McKillup and Dyar, 2010). Positive z values (> +1.96) indicate significant pos- itive autocorrelation, whereas negative z values (< -1.96) indicate significant dissimilarity among neighboring observations.
The z-score is a measure of the distance, in standard deviation units, of an observed point from the mean of a normal (random) distribution of a population (assumed by the null hypothe- sis). The z-score of -1, 0, or +1, indicates one standard deviation below the mean, the mean itself, or one standard deviation above the mean, respectively. For a normal distribution, there is about 68% chance that an observation will fall between -1 and +1 z-scores (i.e., within one standard deviation). The chance for randomness increases to 95% for an observation to fall between -1.96 and +1.96 z-scores (Table 3.1). This means that there is a 5% chance that an observation will be
outside the random range, i.e., 2.5% for a z-score less than -1.96 and 2.5% for a z-score greater than +1.96 (Table 3.1).
High z-scores (-1.96 > z-score > +1.96) can be used to reject the null hypothesis of ran- domness (Mitchell, 2005; McKillup and Dyar, 2010). While the z-score gives a measure for the standard deviation, the p-value (probability value) provides the probability for the observed spa- tial pattern to have been generated by a random process (assumed by the null hypothesis). Both z-scores and p-values relate to the tail of the bell-shaped standard normal distribution. A small p-value falsifies the null hypothesis (of randomness) at a specific confidence level (90%, 95%, or 99%) based on the z-score value (Table 3.1). The significance level at which the null hypothesis is rejected is an indication of the strength of the evidence provided by the sample data against the null hypothesis, in favor of an alternative hypothesis (e.g., clustering or dispersion). The p-value is the smallest significance level (highest confidence level) at which the null hypothesis can be falsified.
The commonly stated confidence levels of 90%, 95%, and 99% correspond with the 0.10, 0.05, and 0.01 significance levels, respectively (Table 3.1). Thus, the z-score associated with a 95% confidence level is between -1.96 and +1.96 standard deviations, while the p-value at the same level is 5%. Therefore, if the test statistic is beyond +1.96, the null hypothesis (of spatial randomness) would be rejected by assuming a statistically significant difference at least at the 95% confidence level. Moreover, when the z-score associated with a 99% confidence level is between -2.58 and +2.58 standard deviations, the p-value would be very small (<0.01) (Table 3.1), and the null hypothesis can be rejected with high confidence (Ebdon, 1985; Goodchild, 1986; Griffith, 1987; Mitchell, 2005; Langlois, 2013)
Table 3.1. The p-values and z-scores for common confidence levels.
Figure 3.1. The Test Statistic for normal frequency distribution. Null Hypothesis (H_0) can be rejected if
-1.96 < z test statistic > 1.96 (see Table 3.1).
3.2.3 Polyline feature class
The data classification scheme for each of the four fault groups, i.e., Basin and Range normal fault system, cross normal fault system, and the regional N-S and E-W striking sets, was described in Chapter 2.
3.2.3.1 Line density map
Line density maps for the population of the BR, CF, E-W, and N-S faults were made to display their spatial distribution and concentration in different domains applying the Spatial Ana- lyst in ArcGIS 10. The Line Density tool measures fault trace lengths per unit area by drawing a
z-score (Standard Deviations) p-value (Probability) Confidence level
< -1.65 or > +1.65 < 0.10 90%
< -1.96 or > +1.96 < 0.05 95%
circle around each raster cell center by a “search radius” (i.e., optional distance to calculate den-
sity based on the linear unit), and multiplying the length of the portion of each fault trace that falls within the circle by its “population field value” (Figure 3.2).
Fault traces that fall within the search area are summed, and this number is divided by the circle’s area to calculate the line density (Silverman, 1986; Bornmann and Waltman, 2011). For instance, consider a raster cell with its circular neighborhood (Figure 3.3), in which L1 and L2 are
portions of the lengths of two fault traces which fall within the circle, and V1 and V2 represent
the corresponding population field values for the fault traces. In this case, the line density is cal- culated using the formula: Density = {(L1 * V1) + (L2 * V2)}/ (circle area).
3.2.3.2 Linear Directional/Orientational Mean (LDM)
The mean orientation for each set of fault traces was calculated using the Linear Direc- tional/Orientational Mean (LDM) tool in ArcGIS’s Spatial Statistics. A directional mean of a set
of fault traces is the direction angle of a resultant average line constructed by connecting the starting point of the first trace to the end point of the last fault trace in the set. In this case, the y- axis and x-axis components are the sine and cosine functions of the direction angle (i.e., azimuth) of individual fault traces, respectively.
Figure 3.2. Circular raster cell partially covering two polyline objects. See text for explanation.
The tangent of the direction angle of the resultant fault traces (tan θR) is given by the ra-
tio of the sum of the sine of all angles that represent the y-component of the traces to the sum of the cosine of all angles that represent the x-component extent of the fault set. The inverse of tan θR gives the linear directional/orientational mean (LDM) for the resultant line for a sub-parallel
set of fault traces, measured clockwise from North as is given by the following equation (Wong and Lee, 2005; Mitchell, 2005):
(Eqn. 3.1)
Since the orientation of a line is independent of its length (Krivoruchko, 2011), fault trac- es are considered to have a unit length during the calculation of the LDM. Because fault traces on maps are horizontal and have no meaningful starting and ending points (i.e., are not vectors), their LDM is calculated based on the orientation, rather than direction of the lines (i.e., the two ends are equivalent). This means that if a fault trace which is oriented 010o (or equally 190o) was arbitrarily digitized, for the USGS database, by someone clicking on the starting point in the NE and an ending point in the SW (i.e., 190o), instead of the other way around, the calculation of the LDM first involves converting all trends to range between 000o and 180o. In this method, the
orientation of the LDM, for a sub-parallel set of fault traces, is given by the angle (Table 3.4), measured clockwise from North (Wong and Lee, 2005; Mitchell, 2005).
The Circular Variance (CV) of each fault set represents the deviation of the orientation of the fault traces from their mean. The Circular Variance (CV) spans from 0, where all traces of a fault set have the same or very similar orientation, to 1, where all fault traces are in opposite di- rections.
The Circular Variance is calculated from the following equation wherei is the same as that defined above for the equation of the LDM:
(Eqn. 3.2)