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--- determine_weights_of_evidence_coefficients [2011/08/01 21:01]
hermann created
+++ determine_weights_of_evidence_coefficients [2015/10/11 23:29] (current)
admin
@@ Line 8: / Line 8: @@
 ^ Name ^ Type ^ Description ^
-| Initial Landscape  | [[ Categorical Map Type|Categorical Map]]  | 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|Categorical Map]]  | Final map of land use and cover classes.  |
+| Final Landscape  | [[Categorical Map Type]]  | Final map of land use and cover classes.  |
-| Ranges  | [[Weights Type|Weights]]  | Pre-defined intervals for continuous gray-tone variable.  |
+| Ranges  | [[Weights Type]]  | Pre-defined intervals for continuous gray-tone variable.  |
 ===== 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  |
 ===== Outputs =====
 ^ 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]]  | Obtained coefficients for selected spatial variables with respect to a transition or set of 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.  |
 ===== Group =====
@@ Line 37: / Line 40: @@
 <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 (<em>overline{B}</m>) is the absence of (//B//) in equation (6).
+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 vert B}{rbrace}
+<m>ln{delim{lbrace}{D|B}{rbrace}}=ln{delim{lbrace}{D}{rbrace}}+W^+</m> (5)
-<p align="center"><img src="images/WeightsOfEvidence005.gif"> <br>
-(6)
+<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:
-<p>For cases in which the occurrences of (<i>D</i>) on the binary pattern (<i>B</i>) are found more often than would be expected due to chance,
+<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)
-<i>W<sup>+</sup></i> will be positive and <i>W<sup>-</sup></i> will be negative. The magnitude of the Contrast (<i>C = W<sup>+</sup> - W<sup>-</sup></i>) reflects the overall spatial association of the event (<i>D</i>) with the spatial pattern (<i>B</i>). The Contrast, indicating whether there is a relationship of (<i>B</i>) with (<i>D</i>), is considered statically significant with 95% probability if
-<i>|C| > 1.96 S(C)</i>, with the variance of the Contrast determined by:
+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)
-<p align="center"><img src="images/WeightsOfEvidence006.gif"><br>
+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:
-(7)
+<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.
-<p>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 (<i>B, C, D, ...N</i>) with the occurrence of (<i>D</i>) as follows:
+==== References ====
-<p align="center"><img src="images/WeightsOfEvidence007.gif"><br>
-(8)
-<p>For modeling transition phenomena, in which (<i>D</i>) stands for a change from class
-<i>i</i> to <i>j</i>, 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>i</i> to <i>j</i> 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 <i>O{D}</i> 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&nbsp;
-<i>P{D|B}/(1 - P{D|B}</i>, leads to the post-probability of a transition <i>i</i> to <i>j</i>, given a particular combination of spatial patterns in a location <i>(x,y)</i>, as follows:
-<p align="center"><img src="images/WeightsOfEvidence008.gif"><br>
-(9)
-<p>This equation makes the use of GIS overlay analysis very convenient to derive favorability maps for a transition <i>i</i> to <i>j</i>. 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.
-<h2>References</h2>
 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.
-<p>Bonham-Carter, G., 1994: Geographic information systems for geoscientists: modelling with GIS. Pergamon, 398 pp.
-<p>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
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