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--- determine_weights_of_evidence_continuous_occurrences [2012/02/10 02:16]
admin
+++ determine_weights_of_evidence_continuous_occurrences [2026/08/28 03:17] (current)
hermann Sync from local documentation review
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 ====== Determine Weights Of Evidence Continuous Occurrences ======
 ===== Description =====
-This container determines the Weights of Evidence coefficients for selected spatial variables with respect to a transition or set of transitions.
+This container determines Weights of Evidence coefficients for selected spatial variables with respect to a map of occurrences, using the same Weights of Evidence method as [[Determine Weights Of Evidence Coefficients]]. Unlike that container, which works from a transition between an initial and a final map, this one works from a map where each cell holds a count of how many times some event occurred at that location, so the map of occurrences does not need to be binary.
-This is a particular case, in which the map of occurrences does not need to be binary.
+This container accepts one or more [[Name Map]] hooks nested inside it, each assigning a name to a map. These named maps supply the values for the spatial variables referenced by Ranges.
 ===== Inputs =====
-^ Name ^ Type ^ Description ^
+^ Name  ^ Type  ^ Description  ^
-| Occurrences  | [[Map Type|Map]]  | Map with occurrences of some event.  |
+| Occurrences  | [[Map Type]]  | Map of occurrences. Each cell holds the number of times the modeled event occurred at that location.  |
-| Mask  | [[Map Type|Map]]  | Map defining the area to be analyzed. |
+| Mask  | [[Map Type]]  | Map defining, through cells other than its null value, which cells should be analyzed. Its cells must be representable as a 32 Bit Integer.  |
-| Ranges  | [[Weights Type|Weights]]  | Pre-defined intervals for continuous gray-tone variable.  |
+| Ranges  | [[Weights Type]]  | Pre-defined ranges of each spatial variable for which the Weights of Evidence coefficients will be calculated. Only the ranges defined for the first transition are used; ranges defined for any other transition are ignored.  |
 ===== Optional Inputs =====
 ^ Name  ^ Type  ^ Description  ^ Default Value  ^
-| Fix Abnormal Weights  | [[Bool Type|Bool]]  | If true, recalculate abnormal weights. Otherwise, assume abnormal values are zero.  | False  |
+| Fix Abnormal Weights  | [[Boolean Value Type]]  | If true, recalculates weights that would otherwise result in abnormal values, such as those arising from a division by zero. If false, those abnormal values are assumed to be zero.  | No  |
 ===== Outputs =====
-^ Name ^ Type ^ Description ^
+^ Name  ^ Type  ^ Description  ^
-| Weights  | [[ Weights Type|Weights]]  | Obtained coefficients for selected spatial variables with respect to a transition or set of transitions.  |
+| Weights  | [[Weights Type]]  | Weights of Evidence coefficients obtained for the selected spatial variables, with respect to the given occurrences.  |
 ===== Group =====
@@ Line 28: / Line 28: @@
 [[Functor List#Calibration | Calibration]]
 ===== Notes =====
-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.
+Cells with zero occurrences are also used in the calculation: one is always added to the number of occurrences found at a cell before it is used. Only cells located outside the area defined by Mask are excluded from the calculation.
-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, <m>P{delim{lbrace}{D inter B}{rbrace}}=(D inter B)/D</m>, and the fraction of area occupied by pattern (B) with respect to the entire study area (//A//); <m>P{delim{lbrace}{B}{rbrace}}=B/A</m>
+Reports an error if a spatial variable named in Ranges is not provided by a nested Name Map hook.
-<m>P{delim{lbrace}{D|B}{rbrace}}={P{delim{lbrace}{D inter B}{rbrace}}}/{P{delim{lbrace}{B}{rbrace}}}</m> (1)
+Reports an error if Mask has cells that cannot be represented as a 32 Bit Integer.
-Algebraic manipulation allows us to represent the conditional probability in terms of its odds ratio em <m>P{delim{lbrace}{D|B}{rbrace}}/{1-P{delim{lbrace}{D|B}{rbrace}}}</m>, where (<m>overline{D}</m>) 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//).
+This method was adapted from Agterberg & Bonham-Carter (1990), Goodacre et al. (1993), and Bonham-Carter (1994).
-<m>O{delim{lbrace}{D|B}{rbrace}}={P{delim{lbrace}{D|B}{rbrace}}}/{P{delim{lbrace}{overline{D}|B}{rbrace}}}</m> (2)
+**References**
-<m>O{delim{lbrace}{D}{rbrace}}={P{delim{lbrace}{D}{rbrace}}}/{P{delim{lbrace}{overline{D}}{rbrace}}}</m> (4)
-Equation (5) is obtained by rewriting equation (5) in a logit form, where //W<sup>+</sup>// is the positive weight of evidence for occurrence of (//D//) given (//B//). By analogy, //W<sup>–</sup>// is obtained - the corresponding negative weights of evidence -, where (<m>overline{B}</m>) is the absence of (//B//) in equation (6).
-<m>ln{delim{lbrace}{D|B}{rbrace}}=ln{delim{lbrace}{D}{rbrace}}+W^+</m> (5)
-<m>W^- = ln({P{delim{lbrace}{overline{B}|D}{rbrace}}}/{P{delim{lbrace}{overline{B}|overline{D}}{rbrace}}})</m> (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<sup>+</sup>// will be positive and //W<sup>-</sup>// will be negative. The magnitude of the Contrast (//C = W<sup>+</sup> - W<sup>-</sup>//) 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:
-<m>S^2=1/{area(B inter D)}+1/{area(B inter overline{D})}+1/{area(overline{B} inter D)}+1/{area(overline{B} inter overline{D})}</m> (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:
-<m>P{delim{lbrace}{D | B inter C inter D cdots inter N}{rbrace}}=ln{D}+{{W_B}^+}+{{W_C}^+}+{{W_D}^+}+ cdots + {{W_N}^+}</m> (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 <m>P{delim{lbrace}{D|B}{rbrace}}/{1-P{delim{lbrace}{D|B}{rbrace}}}</m>, 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:
-<m>P{delim{lbrace}{i doubleright j | B inter C inter D cdots inter N}{rbrace}}=e^{sum{}{}{W^+}}/{1+e^{sum{}{}{W^+}}}</m> (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.
-==== References ====
 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.
+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.
+===== Internal Name =====
-===== Internal Name =====
+DetermineWeightsOfEvidenceContinuousOccurrences
-DetermineWeightsOfEvidenceContinuousOccurences