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--- determine_weights_of_evidence_coefficients [2015/10/11 23:29]
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+++ determine_weights_of_evidence_coefficients [2026/08/28 03:17] (current)
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 ====== Determine Weights Of Evidence Coefficients ======
 ===== 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 the Weights of Evidence coefficients for selected spatial variables with respect to the transitions occurring between an initial and a final map. Weights of Evidence is a Bayesian method, originally used in geology to point out favorable areas for phenomena such as mineralization, adapted here to measure the empirical relationship between spatial variables (represented by categorical or continuous maps) and land-use and land-cover change. For each transition and each variable, this container evaluates how much more (or less) likely the transition is within each of the variable's ranges than would be expected by chance, expressing that relationship as a weight and a contrast.
+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  ^
-| Initial Landscape  | [[ Categorical Map Type]]  | Initial map of land use and cover classes.  |
+| Initial Landscape  | [[Categorical Map Type]]  | Initial map of land use and cover classes.  |
 | Final Landscape  | [[Categorical Map Type]]  | Final map of land use and cover classes.  |
-| Ranges  | [[Weights Type]]  | Pre-defined intervals for continuous gray-tone variable.  |
+| Ranges  | [[Weights Type]]  | Pre-defined intervals for each continuous or categorical spatial variable, for each transition, for which the Weights of Evidence coefficients will be calculated.  |
 ===== Optional Inputs =====
 ^ Name  ^ Type  ^ Description  ^ Default Value  ^
-| Fix Abnormal Weights  | [[Boolean Value Type]]  | 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. This is an advanced port.  | No  |
+| Potential Transition Criteria  | [[Boolean Value Type]]  | Affects how a possible but unexecuted transition from class C to class P is identified. If true, any cell that transitions from C to some other class X (different from P) is considered a case where the C to P transition was possible but did not happen. If false, only a cell that stays in class C is considered such a case. This affects the analysis of concurrent transitions, and is an advanced port.  | Yes  |
 ===== Outputs =====
-^ Name ^ Type ^ Description ^
+^ Name  ^ Type  ^ Description  ^
-| Weights  | [[ Weights Type]]  | 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 transition or transitions.  |
-| Report  | [[Table Type]]  | Table containing a full report of Weights of Evidence coefficient calculation. These are essentially the same results showed in the message log, but in a table format.  |
+| Report  | [[Table Type]]  | Table with the same information reported to the message log during the calculation: for each transition, variable and range, the columns "Transition_From", "Transition_To", "Variable", "Range_Lower_Limit", "Range_Upper_Limit", "Possible_Transitions", "Executed_Transitions", "Weight_Coefficient", "Contrast", and "Significant".  |
 ===== Group =====
 [[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.
+For each range of a variable, this container compares how often the transition actually occurred within that range against how often it could have occurred, and expresses the difference as a weight; the difference between the weight for the presence of the range and the weight for its absence is the contrast, which reflects the overall strength of the association between that range and the transition. A contrast is considered statistically significant, with 95% probability, when its magnitude is larger than 1.96 times its standard deviation.
-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)
+This method was adapted from Agterberg & Bonham-Carter (1990), Goodacre et al. (1993), and Bonham-Carter (1994).
-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//).
+**References**
-<m>O{delim{lbrace}{D|B}{rbrace}}={P{delim{lbrace}{D|B}{rbrace}}}/{P{delim{lbrace}{overline{D}|B}{rbrace}}}</m> (2)
-<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 =====
 DetermineWeightsOfEvidenceCoefficients
+===== Usage examples =====
+See practical examples of this functor in [[lesson_18|Lesson 18: Building a land-use and land-cover change simulation model]]