[GRASS-SVN] r46156 - in grass-addons/grass7/raster/r.fuzzy:
r.fuzzy.logic r.fuzzy.set r.fuzzy.system
svn_grass at osgeo.org
svn_grass at osgeo.org
Sun May 1 19:59:50 EDT 2011
Author: neteler
Date: 2011-05-01 16:59:50 -0700 (Sun, 01 May 2011)
New Revision: 46156
Modified:
grass-addons/grass7/raster/r.fuzzy/r.fuzzy.logic/r.fuzzy.logic.html
grass-addons/grass7/raster/r.fuzzy/r.fuzzy.set/r.fuzzy.set.html
grass-addons/grass7/raster/r.fuzzy/r.fuzzy.system/r.fuzzy.system.html
Log:
html slightly prettified
Modified: grass-addons/grass7/raster/r.fuzzy/r.fuzzy.logic/r.fuzzy.logic.html
===================================================================
--- grass-addons/grass7/raster/r.fuzzy/r.fuzzy.logic/r.fuzzy.logic.html 2011-05-01 23:56:53 UTC (rev 46155)
+++ grass-addons/grass7/raster/r.fuzzy/r.fuzzy.logic/r.fuzzy.logic.html 2011-05-01 23:59:50 UTC (rev 46156)
@@ -1,13 +1,21 @@
<h2>OPTIONS</h2>
<DL>
<DT><b>xmap</b></DT>
-<DD>Name of input x membership operand. This map must be of type FCELL with range 0 :1 and may require null values. Otherwise program will print error message and stops.
+<DD>Name of input x membership operand. This map must be of type FCELL with
+range 0 :1 and may require null values. Otherwise program will print error
+message and stops.
</DD>
<DT><b>xmap</b></DT>
-<DD>Name of input y membership operand. This map must be of type FCELL with range 0 :1 and may require null values. Otherwise program will print error message and stops. This map is optional bit is required for all operation except NOT
+<DD>Name of input y membership operand. This map must be of type FCELL with
+range 0 :1 and may require null values. Otherwise program will print error
+message and stops. This map is optional bit is required for all operation except
+NOT
</DD>
<DT><b>operator</b></DT>
-<DD>A fuzzy set operators are generalization of crisp operators. There is more than one possible generalization of every opeartor. There are three operations: fuzzy complements, fuzzy intersections, and fuzzy unions. Addational implication operator is also provided.
+<DD>A fuzzy set operators are generalization of crisp operators. There is more
+than one possible generalization of every opeartor. There are three operations:
+fuzzy complements, fuzzy intersections, and fuzzy unions. Addational implication
+operator is also provided.
<ul>
<li>fuzzy intersection (<B>AND</B>) use T-norm of given family for calculation;
<li>fuzzy union (<B>OR</B>) use T-conorm of given family for calculation;
@@ -16,23 +24,32 @@
</ul>
</DD>
<DT><b>family</b></DT>
-<DD>T-norms, T-conorms and residuals are a generalization of the two-valued logical conjunction, disjunction and implication used by boolean logic, for fuzzy logics. Because there is more than one possible generalisation of logial operations, r.fuzzy.logic provides 6 most popular families for fuzzy operations:
+<DD>T-norms, T-conorms and residuals are a generalization of the two-valued
+logical conjunction, disjunction and implication used by boolean logic, for
+fuzzy logics. Because there is more than one possible generalisation of logial
+operations, r.fuzzy.logic provides 6 most popular families for fuzzy operations:
<ul>
<li><b>Zadeh</b> with minimum (Godel) t-norm and maximum T-conorm;
<li><b>product</b> with product T-norm and probabilistic sum as T-conorm;
<li><b>drastic</b> with drastic T-norm and drastic T-conorm;
<li><b>Łukasiewicz</b> with Łukasiewicz T-norm and bounded sum as a T-conorm;
-<li><b>Fodor</b> with nilpotent minimum as T-norm and nilpotent maximum as T-conorm;
-<li><b>Hamacher</b> (simplified) with Hamacher product as T-norm and Einstein sum as T-conorm;
+<li><b>Fodor</b> with nilpotent minimum as T-norm and nilpotent maximum as
+T-conorm;
+<li><b>Hamacher</b> (simplified) with Hamacher product as T-norm and Einstein
+sum as T-conorm;
</ul>
There is no residuum for drastic and Hamacher families.
-For more details see <a href="http://www.jstatsoft.org/v31/i02">Meyer D, Hornik K (2009)</a>; <a href="http://en.wikipedia.org/wiki/T-norm">T-norms</a>;
+For more details see <a href="http://www.jstatsoft.org/v31/i02">Meyer D, Hornik
+K (2009)</a>; <a href="http://en.wikipedia.org/wiki/T-norm">T-norms</a>;
</DD>
</DL>
<h2>OUTPUTS</h2>
<DL>
<DT><b>output</b></DT>
-<DD>Map containing result of two-values operations. Multivalued operations will be avaialbel in the future. Map is always of type FCELLS and contains values from 0 (no membership) to 1 (full membership). Values between 0 and 1 indicate partial membership</DD>
+<DD>Map containing result of two-values operations. Multivalued operations will
+be avaialbel in the future. Map is always of type FCELLS and contains values
+from 0 (no membership) to 1 (full membership). Values between 0 and 1 indicate
+partial membership</DD>
</DL>
<h2>SEE ALSO</h2>
<em>
@@ -41,22 +58,29 @@
</em>
-
<h2>REFERENCES</h2>
-Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.<P>
+Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353.
+doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.<P>
-Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger. ISBN 0-85274-583-4.<P>
+Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger.
+ISBN 0-85274-583-4.<P>
-Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.<P>
+Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and
+applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.<P>
-Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory: foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN 0133410587.<P>
+Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory:
+foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN
+0133410587.<P>
-Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal of Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.<P>
+Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal of
+Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.<P>
-Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets. R~package version~1.0, URL http://CRAN.R-project.org/package=sets.<P>
+Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets.
+R~package version~1.0, URL http://CRAN.R-project.org/package=sets.<P>
<h2>AUTHOR</h2>
Jarek Jasiewicz
+<p><i>Last changed: $Date$</i>
Modified: grass-addons/grass7/raster/r.fuzzy/r.fuzzy.set/r.fuzzy.set.html
===================================================================
--- grass-addons/grass7/raster/r.fuzzy/r.fuzzy.set/r.fuzzy.set.html 2011-05-01 23:56:53 UTC (rev 46155)
+++ grass-addons/grass7/raster/r.fuzzy/r.fuzzy.set/r.fuzzy.set.html 2011-05-01 23:59:50 UTC (rev 46156)
@@ -1,17 +1,23 @@
<h2>OPTIONS</h2>
<DL>
<DT><b>input</b></DT>
-<DD>Name of input raster map to be fuzified. This map may be of any type and may require null values.
+<DD>Name of input raster map to be fuzified. This map may be of any type and may
+require null values.
</DD>
<DT><b>points</b></DT>
-<DD>A list containing 4 (A,B,C,D) or 2 A,B) points defining set boundaries. Points must not to be in map range, but it may lead to only 0 o 1 membership for the whole map. For "both" side parameters range between A and D defines base, but range between B and C core of the fuzzy set. Between A and B and C and D are set's boundaries. If side is "both" it require 4 points, else 2 points.
+<DD>A list containing 4 (A,B,C,D) or 2 A,B) points defining set boundaries.
+Points must not to be in map range, but it may lead to only 0 o 1 membership for
+the whole map. For "both" side parameters range between A and D defines base,
+but range between B and C core of the fuzzy set. Between A and B and C and D are
+set's boundaries. If side is "both" it require 4 points, else 2 points.
<center>
<h3>Fuzzy set definition:</h3>
<img src="set.png" border=1>
</center>
</DD>
<DT><b>side</b></DT>
-<DD>Option indicate if set is fuzzified of both sides (both), left or right side. See description for details.
+<DD>Option indicate if set is fuzzified of both sides (both), left or right
+side. See description for details.
<center>
<h3>Boundary definition:</h3>
<img src=boundary.png border=1><br>
@@ -22,33 +28,52 @@
<h2>OUTPUTS</h2>
<DL>
<DT><b>output</b></DT>
-<DD>Map containing membership value of original map. Map is alvays of type FCELLS and contains values from 0 (no membership) to 1 (full membership). Values between 0 and 1 indicate partial membership</DD>
+<DD>Map containing membership value of original map. Map is alvays of type
+FCELLS and contains values from 0 (no membership) to 1 (full membership). Values
+between 0 and 1 indicate partial membership</DD>
</DL>
<h2>FUZZY SET PARAMETERS</h2>
<DL>
<DT><b>boundary</b></DT>
-<DD>Parameter definied the shape of the fuzzy boundary. The default and most popular is S-shaped, linear, J-shaped and G-shaped boundaries are also available. The same boundaires are appled to the both sides.</DD>
+<DD>Parameter definied the shape of the fuzzy boundary. The default and most
+popular is S-shaped, linear, J-shaped and G-shaped boundaries are also
+available. The same boundaires are appled to the both sides.</DD>
<DT><b>shape</b></DT>
-<DD>Optional shape modifier. Range from -1 to 1. The default value is 0 and shoud not be changed in most of the time. The negative values indicate more dilatant set, the positive values more concentrate set. See description for details.
+<DD>Optional shape modifier. Range from -1 to 1. The default value is 0 and
+shoud not be changed in most of the time. The negative values indicate more
+dilatant set, the positive values more concentrate set. See description for
+details.
<center>
<h3>Impact of shape parameter on shape boundary:</h3>
<img src=shape.png border=1>
</center>
</DD>
<DT><b>height</b></DT>
-<DD>Optional height modifier. Range from 0 to 1. The default value is 1 and indicate full membership beteen points B and C. If height is lesser than one the maximum memebrship is equal to height. See image: Fuzzy set definition.</DD>
+<DD>Optional height modifier. Range from 0 to 1. The default value is 1 and
+indicate full membership beteen points B and C. If height is lesser than one the
+maximum memebrship is equal to height. See image: Fuzzy set definition.</DD>
</DL>
<h2>DESCRIPTION</h2>
<H4>Definition of fuzzy set</H4>
-Fuzzy sets are sets whose elements have degrees of membership. Zadeh (1965) introduced Fuzzy sets as an extension of the classical notion of set. Classical membership of elements in a set are binary terms: an element either belongs or does not belong to the set. Fuzzy set theory use the gradual assessment of the membership of elements in a set. A membership function valued in the real unit interval [0, 1]. Classical sets, are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Classical sets are in fuzzy set theory usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is imprecise, such as most of the GIS operations.
+Fuzzy sets are sets whose elements have degrees of membership. Zadeh (1965)
+introduced Fuzzy sets as an extension of the classical notion of set. Classical
+membership of elements in a set are binary terms: an element either belongs or
+does not belong to the set. Fuzzy set theory use the gradual assessment of the
+membership of elements in a set. A membership function valued in the real unit
+interval [0, 1]. Classical sets, are special cases of the membership functions
+of fuzzy sets, if the latter only take values 0 or 1. Classical sets are in
+fuzzy set theory usually called crisp sets. The fuzzy set theory can be used in
+a wide range of domains in which information is imprecise, such as most of the
+GIS operations.
<h2>NOTES</H2>
<H4>Calculation of boundary shape</H4>
-Depending on type of the boundary different equation are used to determine its shape:
+Depending on type of the boundary different equation are used to determine its
+shape:
<P>
<B>Linear:</B> the membership is calculated according following equation:<br>
<pre><code>
@@ -72,7 +97,8 @@
where m: shape parameter.
</code></pre>
-For default shape parameter = 0 m is = 2 whcich is most common parameter for that equation.
+For default shape parameter = 0 m is = 2 whcich is most common parameter for
+that equation.
<P>
<B>G-shaped and J shaped:</B> it use following equations:
@@ -95,22 +121,28 @@
</em>
-
<h2>REFERENCES</h2>
-<P>Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.</P>
+<P>Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353.
+doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.</P>
-<P>Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger. ISBN 0-85274-583-4.</P>
+<P>Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger.
+ISBN 0-85274-583-4.</P>
-<P>Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.</P>
+<P>Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and
+applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.</P>
-<P>Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory: foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN 0133410587.</P>
+<P>Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory:
+foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN
+0133410587.</P>
-<P>Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal of Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.</P>
+<P>Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal
+of Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.</P>
-Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets. R~package version~1.0, URL http://CRAN.R-project.org/package=sets.<P>
+Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets.
+R~package version~1.0, URL http://CRAN.R-project.org/package=sets.<P>
<h2>AUTHOR</h2>
Jarek Jasiewicz
-
+<p><i>Last changed: $Date$</i>
Modified: grass-addons/grass7/raster/r.fuzzy/r.fuzzy.system/r.fuzzy.system.html
===================================================================
--- grass-addons/grass7/raster/r.fuzzy/r.fuzzy.system/r.fuzzy.system.html 2011-05-01 23:56:53 UTC (rev 46155)
+++ grass-addons/grass7/raster/r.fuzzy/r.fuzzy.system/r.fuzzy.system.html 2011-05-01 23:59:50 UTC (rev 46156)
@@ -2,27 +2,53 @@
<DL>
<DT><b>maps</b></DT>
-<DD>A text file containing maps name and fuzzy sets connected with map definition. Maps in text file must exist on a search path, except the output map. The name of output map name is the output name parameter. In map file outmput map is marked by special name <B>_OUTPUT_</B> If maps are in different mapsets the name require @. Map names in database cannot contain following symbols: <B> %,$ and #</B>. Every map name must start with map name identifier:<B> %</B>. Every set definition connected with cetrain map must follow the map name and must start with set identifier :<B> $</B>. The set definition must be in braces { } and requires parameters separated by semicolon. Any whitespaces like spaces, tabs, empty lines are allowed and may used to visual format of rule file.
+<DD>A text file containing maps name and fuzzy sets connected with map
+definition. Maps in text file must exist on a search path, except the output
+map. The name of output map name is the output name parameter. In map file
+outmput map is marked by special name <B>_OUTPUT_</B> If maps are in different
+mapsets the name require @. Map names in database cannot contain following
+symbols: <B> %,$ and #</B>. Every map name must start with map name
+identifier:<B> %</B>. Every set definition connected with cetrain map must
+follow the map name and must start with set identifier :<B> $</B>. The set
+definition must be in braces { } and requires parameters separated by semicolon.
+Any whitespaces like spaces, tabs, empty lines are allowed and may used to
+visual format of rule file.
<PRE>$ set_name {side; points; boundary_shape; hedge; height }</PRE>
<UL>
-<li><B>set_name</B>: Any name of the fuzzy set. Must not contain symbols: <i> %,$ and #</i>
-<li><B>side</B>: Option indicate if set is fuzzified of both sides (both), left or right side. Available: <i>both, left, right</i>.
-<li><B>points</B>: A list containing 4 (A,B,C,D) or 2 A,B) points separated by comma. Points definine location of sets of boundaries. Points may not to be in map range, but it may lead to only 0 o 1 membership for the whole map. For "both" side parameters range between A and D defines base, but range between B and C core of the fuzzy set. Between A and B and C and D are set's boundaries. If side is "both" it require 4 points, else 2 points. Points values must be not-decreasing.
-<li><B>shape</B>: Parameter definied the shape of the fuzzy boundary. Available: <i>sshaped, linear, jshaped, gshaped</i>. The same boundaires are appled to both sides of fuzzy set.
-<li><B>hedge</B>: Shape modifier the positive number means dilatation (power the fuzzy set by 2) the negative means concetration (square root of fuzzy set). The number means number of dilatation/concetration applied on fuzzy set.
-<li><B>height</B>: Height modifier. Range from 0 to 1. The value 1 and indicate full membership beteen points B and C. If height is lesser than one the maximum memebrship is equal to height.
+<li><B>set_name</B>: Any name of the fuzzy set. Must not contain symbols: <i>
+%,$ and #</i>
+<li><B>side</B>: Option indicate if set is fuzzified of both sides (both), left
+or right side. Available: <i>both, left, right</i>.
+<li><B>points</B>: A list containing 4 (A,B,C,D) or 2 A,B) points separated by
+comma. Points definine location of sets of boundaries. Points may not to be in
+map range, but it may lead to only 0 o 1 membership for the whole map. For
+"both" side parameters range between A and D defines base, but range between B
+and C core of the fuzzy set. Between A and B and C and D are set's boundaries.
+If side is "both" it require 4 points, else 2 points. Points values must be
+not-decreasing.
+<li><B>shape</B>: Parameter definied the shape of the fuzzy boundary. Available:
+<i>sshaped, linear, jshaped, gshaped</i>. The same boundaires are appled to
+both sides of fuzzy set.
+<li><B>hedge</B>: Shape modifier the positive number means dilatation (power the
+fuzzy set by 2) the negative means concetration (square root of fuzzy set). The
+number means number of dilatation/concetration applied on fuzzy set.
+<li><B>height</B>: Height modifier. Range from 0 to 1. The value 1 and indicate
+full membership beteen points B and C. If height is lesser than one the maximum
+memebrship is equal to height.
</UL>
<P>An example of fuzzy sets definiton:
<PRE>$ moderate {both; 90,100,120,130; sshaped; 0; 1}</PRE>
<B>Special notes about sets definition for output map:</B><BR>
-These sets shall be created as triangular (both sides) sets with linear boundaries, without hedge and height modifiers:
+These sets shall be created as triangular (both sides) sets with linear
+boundaries, without hedge and height modifiers:
<PRE>$ moderate {both; 0,20,20,40; linear; 0; 1}</PRE>
</DD>
<DT><b>rules</b></DT>
-<DD>A text file containing rules for classification.Th typical fuzzy rule consists of one or more antecedents and one consequent:
+<DD>A text file containing rules for classification.Th typical fuzzy rule
+consists of one or more antecedents and one consequent:
<PRE>IF elev IS high AND distance IS low THEN probability IS small
@@ -31,7 +57,12 @@
consequent: probability IS small
</PRE>
-The rule file has his own syntax. Because module creates only one result map, the map name is ommited. Every rule starts with $ and consist of consequent name and antecedents in braces { }. All maps and sets used in atecednets must be included in the maps file. At the begining of the calculation program checks if all names and sets are included in maps file. Names of the rules must be same as sets names of the output map. The rules file use following symbols:
+The rule file has his own syntax. Because module creates only one result map,
+the map name is ommited. Every rule starts with $ and consist of consequent name
+and antecedents in braces { }. All maps and sets used in atecednets must be
+included in the maps file. At the begining of the calculation program checks if
+all names and sets are included in maps file. Names of the rules must be same as
+sets names of the output map. The rules file use following symbols:
<UL>
<li>IS is symbolised by <B>=</B>
<li>IS NOT is symbolised by <B>~</B>
@@ -52,35 +83,49 @@
In most cases default options shoud not be changed.
<DL>
<DT><b>family</b></DT>
-<DD>AND and OR operations in fuzzy logic are made with T-norms, T-conorms. T-norms, T-conorms are a generalization of the two-valued logical conjunction and disjunction used by boolean logic, for fuzzy logics. Because there is more than one possible generalisation of logial operations, r.fuzzy.system provides 6 most popular families for fuzzy operations:
+<DD>AND and OR operations in fuzzy logic are made with T-norms, T-conorms.
+T-norms, T-conorms are a generalization of the two-valued logical conjunction
+and disjunction used by boolean logic, for fuzzy logics. Because there is more
+than one possible generalisation of logial operations, r.fuzzy.system provides 6
+most popular families for fuzzy operations:
<ul>
<li><b>Zadeh</b> with minimum (Godel) t-norm and maximum T-conorm;
<li><b>product</b> with product T-norm and probabilistic sum as T-conorm;
<li><b>drastic</b> with drastic T-norm and drastic T-conorm;
<li><b>Lukasiewicz</b> with Lukasiewicz T-norm and bounded sum as a T-conorm;
-<li><b>Fodor</b> with nilpotent minimum as T-norm and nilpotent maximum as T-conorm;
-<li><b>Hamacher</b> (simplified) with Hamacher product as T-norm and Einstein sum as T-conorm;
+<li><b>Fodor</b> with nilpotent minimum as T-norm and nilpotent maximum as
+T-conorm;
+<li><b>Hamacher</b> (simplified) with Hamacher product as T-norm and Einstein
+sum as T-conorm;
</ul>
<P>
<TABLE cellspacing=4>
<TR><TH>Family</TH><TH> T-NORM (AND) </TH><TH>T CONORM (OR) </TH></TR>
<TR><TD>ZADEH</TD><TD> MIN(x,y)</TD><TD>MAX(x,y)</TD></TR>
<TR><TD>PRODUCT</TD><TD> x*y </TD><TD>x + y -x * y</TD></TR>
-<TR><TD>DRASTIC </TD><TD>IF MAX(x, y) == 1 THEN MIN(x, y) ELSE 0 </TD><TD>IF (MIN(x, y) == 0) THEN MAX(x, y) ELSE 1</TD></TR>
+<TR><TD>DRASTIC </TD><TD>IF MAX(x, y) == 1 THEN MIN(x, y) ELSE 0
+</TD><TD>IF (MIN(x, y) == 0) THEN MAX(x, y) ELSE 1</TD></TR>
<TR><TD>LUKASIEWICZ</TD><TD> MAX((x+y-1),0)</TD><TD> MIN((x+y),1)</TD></TR>
-<TR><TD>FODOR </TD><TD>IF (x+y)>1 THEN MIN(x,y) ELSE 0 </TD><TD>IF (x+y<1) THEN MAX(x,y) ELSE 1</TD></TR>
-<TR><TD>HAMACHER</TD><TD> IF (x==y==0) THEN 0 ELSE (x*y)/((x+y)-x*y)</TD><TD> (x+y)/(1+x*y)</TD></TR>
+<TR><TD>FODOR </TD><TD>IF (x+y)>1 THEN MIN(x,y) ELSE 0 </TD><TD>IF
+(x+y<1) THEN MAX(x,y) ELSE 1</TD></TR>
+<TR><TD>HAMACHER</TD><TD> IF (x==y==0) THEN 0 ELSE
+(x*y)/((x+y)-x*y)</TD><TD> (x+y)/(1+x*y)</TD></TR>
</TABLE>
</DD>
<DT><b>imp: implication </b></DT>
-<DD>Imlication determines the method of reshapening of consequents (fuzzy set) by antecedents (single value) :
+<DD>Imlication determines the method of reshapening of consequents (fuzzy set)
+by antecedents (single value) :
<UL>
-<li><B>minimum</B> means the lowest value of the antecedtents and output set definition. It ussually creates trapezoidal consequent set definition.
-<li><B>product</B> means the multiplication of the antecedtents and output set definition. It ussually creates triangular consequent set definition.
+<li><B>minimum</B> means the lowest value of the antecedtents and output set
+definition. It ussually creates trapezoidal consequent set definition.
+<li><B>product</B> means the multiplication of the antecedtents and output set
+definition. It ussually creates triangular consequent set definition.
</UL>
</DD>
<DT><b>defuzz: deffuzyfication method</b></DT>
-<DD>Before deffuzification all consequents are agregated into one fuzzy set. Defuzzification is the process of conversion of aggregated fuzzy set into one crisp value. The r.fuzzy.system provides 5 methods of deffuzyfication:
+<DD>Before deffuzification all consequents are agregated into one fuzzy set.
+Defuzzification is the process of conversion of aggregated fuzzy set into one
+crisp value. The r.fuzzy.system provides 5 methods of deffuzyfication:
<UL>
<li><B>centroid</B> center of mass of the fuzzy set (in practise weighted mean);
<li><B>bisector</B> a value wchich divide fuzzy set on two parts of equal area;
@@ -90,29 +135,47 @@
</UL>
</DD>
<DT><b>res: universe resolution</b></DT>
-<DD>The universe is an interval between the lowest and highest values of consequent and agregated fuzzy sets. The resolution provides number of elements of these fuzzy sets. The minimum and maximum for univese is taken from the minimal and maximal values of fuzzy set definition of output map Because it has strong impact on computation time and precision of deffuzification, values lower than 30 may impact on precision of final result, but values above 200 may slow down computation time.
+<DD>The universe is an interval between the lowest and highest values of
+consequent and agregated fuzzy sets. The resolution provides number of elements
+of these fuzzy sets. The minimum and maximum for univese is taken from the
+minimal and maximal values of fuzzy set definition of output map Because it has
+strong impact on computation time and precision of deffuzification, values lower
+than 30 may impact on precision of final result, but values above 200 may slow
+down computation time.
</DD>
</DL>
<h2>VISUAL OUTPUT</h2>
<DL>
<DT><b>coordinates</b></DT>
-<DD>Coordinates of points for which output: universe, all consequents sets and agregate set. It is useful for visual presentation or detail analysis of fuzzy rules behaviour. In that cases calculations are peroforemd n=only for selected point.</DD>
+<DD>Coordinates of points for which output: universe, all consequents sets and
+agregate set. It is useful for visual presentation or detail analysis of fuzzy
+rules behaviour. In that cases calculations are peroforemd n=only for selected
+point.</DD>
<DT><b>membership only flag</b></DT>
-<DD>Prints for all maps sat of values in map range (map universe) and values of fuzzy sets (linguistic values). Number of values is taken from resolution (default 100). This option is useful for visual control fuzzy set definitions for evrey map.</DD>
+<DD>Prints for all maps sat of values in map range (map universe) and values of
+fuzzy sets (linguistic values). Number of values is taken from resolution
+(default 100). This option is useful for visual control fuzzy set definitions
+for evrey map.</DD>
</DL>
<h2>OUTPUTS</h2>
<DL>
<DT><b>output</b></DT>
-<DD>Map containing deffuzified values. Map is always of type FCELLS and contains values defined in output universe. The output name must be the same as one of maps in maps definition file.
+<DD>Map containing deffuzified values. Map is always of type FCELLS and contains
+values defined in output universe. The output name must be the same as one of
+maps in maps definition file.
</DD>
<DT><b>multipe output flag</b></DT>
-<DD>This flag is used to create fuzzified maps for every rule. The name of the map consist of otput map name, '_' and rule name (for example: output=probs and rule name high, the map name: probs_high). Values of maps ranges from 0 to 1. If map with such name exists will be overwritten without warning.
+<DD>This flag is used to create fuzzified maps for every rule. The name of the
+map consist of otput map name, '_' and rule name (for example: output=probs and
+rule name high, the map name: probs_high). Values of maps ranges from 0 to 1.
+If map with such name exists will be overwritten without warning.
</DD>
</DL>
<h2>NOTES</H2>
<H4>Calculation of boundary shape</H4>
-Depending on type of the boundary different equation are used to determine its shape:
+Depending on type of the boundary different equation are used to determine its
+shape:
<P>
<B>Linear:</B> the membership is calculated according following equation:<br>
@@ -123,7 +186,8 @@
C< value > D -> x = (D-value)/(D-C)
value >= D -> x = 0
</code></pre>
-<B>S-shaped, G-shaped and J shaped:</B> use following equation to sommoth boundary:
+<B>S-shaped, G-shaped and J shaped:</B> use following equation to sommoth
+boundary:
<pre><code>
sin(x * Pi/2)^2 (for S-shaped)
tan(x * Pi/4)^2 (for J-shaped)
@@ -136,32 +200,54 @@
<h2>EXAMPLE</H2>
<P>
-Fuzzy sets are sets whose elements have degrees of membership. Zadeh (1965) introduced Fuzzy sets as an extension of the classical notion of set. Classical membership of elements in a set are binary terms: an element either belongs or does not belong to the set. Fuzzy set theory use the gradual membership of elements in a set. A membership function use values in the real unit interval [0, 1]. Classical sets, are special cases of the membership functions of fuzzy sets and only take values 0 or 1. Classical sets are in fuzzy set theory usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is imprecise, such as most of the GIS operations.
+Fuzzy sets are sets whose elements have degrees of membership. Zadeh (1965)
+introduced Fuzzy sets as an extension of the classical notion of set. Classical
+membership of elements in a set are binary terms: an element either belongs or
+does not belong to the set. Fuzzy set theory use the gradual membership of
+elements in a set. A membership function use values in the real unit interval
+[0, 1]. Classical sets, are special cases of the membership functions of fuzzy
+sets and only take values 0 or 1. Classical sets are in fuzzy set theory usually
+called crisp sets. The fuzzy set theory can be used in a wide range of domains
+in which information is imprecise, such as most of the GIS operations.
<P>
-Suppose we want to determine the flood risk on some area (Spearfish dataset) using two maps: distance to streams and elevation above streams. We can write some common sense rules:
+Suppose we want to determine the flood risk on some area (Spearfish dataset)
+using two maps: distance to streams and elevation above streams. We can write
+some common sense rules:
<PRE>
IF elevation IS low AND distance IS near THEN risk IS veryprobable
IF elevation IS low OR distance IS near THEN risk IS probable
IF elevation IS high AND distance IS far THEN risk IS unprobable
</PRE>
-In clasical boolean sense, we would taken some limits of ideas "near" "far" etc, but what about walues near the limit? The fuzzy set uses partial memberships which abolish these restrictions. In that sense to set "near" belongs all areas with distance no more than 100 m with full membership and from 100 to 200 m with partial membership grater than 0. Over 200 m we can assume that is not near.
+In clasical boolean sense, we would taken some limits of ideas "near" "far" etc,
+but what about walues near the limit? The fuzzy set uses partial memberships
+which abolish these restrictions. In that sense to set "near" belongs all areas
+with distance no more than 100 m with full membership and from 100 to 200 m with
+partial membership grater than 0. Over 200 m we can assume that is not near.
This allow to formulate fuzzy rules for distance map:
<PRE>
near: BELOW 100 = 1; FROM 100 TO 200 = {1 TO 0}; ABOVE 200 = 0;
</PRE>
-To recive final map program calculate partial fuzzy set for all rules and next agregate it into one fuzzy set. These fuzzy sets are created on value sequence called universe. Every set has the number of elements equal to universe resolution. Such set cannot be stored as map so finally is deffuzified with method choosen by user.
+To recive final map program calculate partial fuzzy set for all rules and next
+agregate it into one fuzzy set. These fuzzy sets are created on value sequence
+called universe. Every set has the number of elements equal to universe
+resolution. Such set cannot be stored as map so finally is deffuzified with
+method choosen by user.
<P>
First we need two maps created with r.stream package:
<PRE>
-r.stream.extract elevation=elevation.10m at PERMANENT threshold=2000 stream_rast=streams direction=dirs
+r.stream.extract elevation=elevation.10m at PERMANENT threshold=2000
+stream_rast=streams direction=dirs
r.stream.order stream=streams dir=dirs horton=horton
r.mapcalc "horton3=if(horton>2,horton,null())"
-r.stream.distance stream=streams dir=dirs dem=elevation.10m method=downstream distance=distance elevation=elevation
+r.stream.distance stream=streams dir=dirs dem=elevation.10m method=downstream
+distance=distance elevation=elevation
</PRE>
-Next, to perform analysis we need two files: one with definition of map used in analysis and definition of fuzzy sets for every map, and second with fuzzy rules. For this example:
+Next, to perform analysis we need two files: one with definition of map used in
+analysis and definition of fuzzy sets for every map, and second with fuzzy
+rules. For this example:
<P>MAPS
<PRE>
@@ -192,12 +278,14 @@
r.fuzzy.system maps=flood.map rules=flod.rul output=flood
</PRE>
-Resulting map should look like this below. Yellow color means no risk, red high risk, green, blue end so on moderate risk.
+Resulting map should look like this below. Yellow color means no risk, red high
+risk, green, blue end so on moderate risk.
<CENTER><img src=f_result.png border=1></CENTER><br>
<h2>SEE ALSO</h2>
<em>
+<a href="r.fuzzy.html">r.fuzzy</a>,
<a href="r.fuzzy.logic.html">r.fuzzy.logic</a>,
<a href="r.fuzzy.set.html">r.fuzzy.set</a>,
<a href="r.mapcalc.html">r.mapcalc</a>,
@@ -207,25 +295,26 @@
<h2>REFERENCES</h2>
-Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.<P>
+Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353.
+doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.<P>
-Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger. ISBN 0-85274-583-4.<P>
+Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger.
+ISBN 0-85274-583-4.<P>
-Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.<P>
+Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and
+applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.<P>
-Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory: foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN 0133410587.<P>
+Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory:
+foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN
+0133410587.<P>
-Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal of Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.<P>
+Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal of
+Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.<P>
-Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets. R~package version~1.0, URL http://CRAN.R-project.org/package=sets.<P>
+Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets.
+R~package version~1.0, URL http://CRAN.R-project.org/package=sets.<P>
<h2>AUTHOR</h2>
Jarek Jasiewicz
-
-
-<h2>SEE ALSO</h2>
-<em>
-<a href="r.fuzzy.html">r.fuzzy</a>,
-<a href="r.mapcalc.html">r.mapcalc</a>,
-</em>
+<p><i>Last changed: $Date$</i>
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