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## Directional analysis of point datasets |

A somewhat different form of directional analysis is used within many packages to summarize the distribution of point sets around a mean center. The average variation in the distance of points around the mean can be viewed as a circle or set of circles at a set of standard distances (like standard deviations in univariate statistics). In addition, separate variations in x- and y-coordinate values may be used to generate a standard distance ellipse, with major and minor axes reflecting the directional variation of the point pattern.

Figure 4‑79 illustrates this for the location of churches on Romney Marsh in SE England. The two ellipses are set at one and two standard deviations respectively. Determination of the standard deviations in x and y are not made using the original data, but using transformed coordinates selected by minimizing the squared deviations from the point set to transformed (rotated) x and y axes. This is similar to the process of computing a linear regression on the (x,y) coordinate pairs and using the slope of the best fit line in angular measure to calculate the adjusted standard deviations.

The formula for the angle of (clockwise) rotation of the y-axis is somewhat lengthy:

Using this value the two standard deviations can then be computed in much the same way as for conventional standard deviations, but with the degrees of freedom in this case being n‑2, where n is the number of points in the sample (for n large the divisor will be close to n, as per the common formulation of standard distance).

The formulas cited are those used within Crimestat, which have been adjusted to ensure the ellipse axes are the correct length and the formula is consistent if the standard deviations in x- and y- are equal:

Software packages may provide information on the rotational angle and other attributes of the standard deviational ellipse. For example, Crimestat includes the angle of rotation and the individual standard deviation values, together with the area of the standard deviational ellipse (which is simply A=SDxSDyπ; the lengths of the two ellipse axes are 2SDx and 2SDy). For the data illustrated in Figure 4‑79 the clockwise rotation of the y-axis is 53.2 degrees and the ratio of major to minor axis length, or coefficient of eccentricity, is 1.76.

Figure 4‑79 Standard distance circle and ellipses

Crimestat includes directional data in another technique, Correlated Walk Analysis (CWA). CWA is an analytical procedure based on examining a sequence of events (e.g. crimes, animal movements) in terms of location, bearing (direction from a given point) and time lapse. Its objective is to predict where subsequent events, e.g. burglaries, may occur. Random walks, of various types (see further below), may be used as part of such analysis. For more details of these methods and their applications see the Crimestat manual, Chapters 4 and 9, and the more recent paper by Demšar et al. (2021).

An essential part of such analyses is the examination of patterns in the distribution of point data as the distance and direction of examination is varied. A simple Correlated Random Walk (CRW) simulation facility wass provided with Hawth’s Tools/GME for ArcGIS that enabled model paths to be generated and saved as features (Figure 4‑80). This facility highlights the impact of model parameters on the resulting path form (in this example using CRW as the basic model, but other models such as pure random/Brownian motion and so-called persistent random walks are also supported).

Graphs that plot the average difference in values for pairs of points in a region, over given blocks of distance, optionally grouped into different blocks of directions, are known as correlograms or variograms. If the region is isotropic there will be no significant differences between the variograms drawn for one direction (e.g. 0‑30 degrees) or from another (e.g. 60‑90 degrees), whereas with significant anisotropy the shapes of these variograms will differ. Hence if modeling is to utilize such information, different models or model parameters may be required to accommodate these directional differences (these issues are covered in greater detail in Section 6.7.1, Core concepts in Geostatistics).

Figure 4‑80 Correlated Random Walk simulation

A. 500 step CRW, variable (random uniform) step length, directional model N(0,1) degrees |
B. 500 step CRW, variable (random uniform) step length, directional model N(30,15) degrees |