Determine Weights Of Evidence Continuous Occurrences

This container determines the Weights of Evidence coefficients for selected spatial variables with respect to a transition or set of transitions.

This is a particular case, in which the map of occurrences does not need to be binary.

Calibration

We have introduced the Weights of Evidence method to spatially model land-use change. Weights of Evidence is a Bayesian method traditionally used by geologists to point out favorable areas for geological phenomena such as mineralization and seismicity (Agterberg & Bonham-Carter, 1990; Goodacre et al. 1993; Bonham-Carter 1994). The Weights of Evidence method was adapted from these authors to calculate empirical relationships of spatial variables, represented by either categorical or gray-tone (continuous variable) maps, with respect to land-use and cove change.

The favorability for the occurrence of an event (D), such as a land-cover change, given a binary map defining the presence or absence of a geographical pattern (B), such as a type of soil, can be expressed by the conditional or posterior probability (equation 1). This is determined by measuring the number of occurrences of (D) - usually, the number of cells (D) in a raster map -, its overlap with the binary pattern, , and the fraction of area occupied by pattern (B) with respect to the entire study area (A);

(1)

Algebraic manipulation allows us to represent the conditional probability in terms of its odds ratio em , where () stands for the absence of (D). Equation (2) can be transformed into equation (3), in which O{D} represents the prior odds ratio of event (D) - equation (4). Prior probability of (D) is then calculated by dividing the total number of cells (D) by the number of cells of the entire study area (A).

(2)

(4)

Equation (5) is obtained by rewriting equation (5) in a logit form, where W⁺ is the positive weight of evidence for occurrence of (D) given (B). By analogy, W^– is obtained - the corresponding negative weights of evidence -, where () is the absence of (B) in equation (6).

(5)

(6)

For cases in which the occurrences of (D) on the binary pattern (B) are found more often than would be expected due to chance, W⁺ will be positive and W^- will be negative. The magnitude of the Contrast (C = W⁺ - W^-) reflects the overall spatial association of the event (D) with the spatial pattern (B). The Contrast, indicating whether there is a relationship of (B) with (D), is considered statically significant with 95% probability if |C| > 1.96 S(C), with the variance of the Contrast determined by:

(7)

This method can be extended to handle multiple predictive maps, so that each weight of evidence represents the degree of association of a spatial pattern (B, C, D, …N) with the occurrence of (D) as follows:

(8)

For modeling transition phenomena, in which (D) stands for a change from class i to j, such as deforestation, is necessary to introduce some modifications to this calculation. First, instead of the entire study area that occupied by the class (i) before changes from i to j take place is used, for example, the former area of forest, as deforestation can only occur in a forested landscape. Second, as we focus on determining the influences of a set of spatial patterns on a modeled transition, we can assume that O{D} is equal to 1. Note that the prior probability of a transition is equivalent to its transition rate, in other words, using the example of deforestation, the net deforestation rate calculated by dividing the number of deforestation cells by the number of forest cells prior to deforestation. In this manner, algebraic manipulation of equation (8), replacing the odds ratio by , leads to the post-probability of a transition i to j, given a particular combination of spatial patterns in a location (x,y), as follows:

(9)

This equation makes the use of GIS overlay analysis very convenient to derive favorability maps for a transition i to j. Indeed, the Weights of Evidence method is easily implemented by cross-tabulating maps, considering that each location (x,y) represents a unique set of overlapping input map classes.

Agterberg, F.P. and Bonham-Carter, G.F., 1990: Deriving weights of evidence from geoscience contour maps for the prediction of discrete events. XXII Int. Symposium AP-COM, 381-395.

Bonham-Carter, G., 1994: Geographic information systems for geoscientists: modelling with GIS. Pergamon, 398 pp.

Goodacre C. M., Bonham-Carter G. F., Agterberg, F. P., Wright D. F., 1993: A statistical analysis of spatial association of seismicity with drainage patterns and magnetic anomalies in western Quebec. Tectonophysics, 217, 205-305.

DetermineWeightsOfEvidenceContinuousOccurences