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en:div-ind [2017/05/24 21:44]
David Zelený
en:div-ind [2019/03/22 22:03] (current)
David Zelený
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-====== Diversity analysis ======+Section: [[en:​diversity_analysis]]
 ===== Diversity indices ===== ===== Diversity indices =====
 +
  
 [[{|width: 7em; background-color:​ light; color: firebrick}div-ind|**Theory**]] [[{|width: 7em; background-color:​ light; color: firebrick}div-ind|**Theory**]]
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 [[{|width: 7em; background-color:​ white; color: navy}div-ind_examples|Examples]] [[{|width: 7em; background-color:​ white; color: navy}div-ind_examples|Examples]]
 [[{|width: 7em; background-color:​ white; color: navy}div-ind_exercise|Exercise {{::​lock-icon.png?​nolink|}}]] [[{|width: 7em; background-color:​ white; color: navy}div-ind_exercise|Exercise {{::​lock-icon.png?​nolink|}}]]
-==== Theory ==== 
  
 This section will overview commonly used indices measuring diversity of ecological community (species richness, Shannon index, Simpson index). We will also introduce the measures of evenness, concept of effective number of species and general framework of Hill numbers. This section will overview commonly used indices measuring diversity of ecological community (species richness, Shannon index, Simpson index). We will also introduce the measures of evenness, concept of effective number of species and general framework of Hill numbers.
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 === Comparison of species richness, Shannon index and Simpson index === === Comparison of species richness, Shannon index and Simpson index ===
 +In case of perfectly even communities,​ the Shannon and Gini-Simpson index increases non-linearly with number of species in the community; Gini-Simpson index increases faster. This relationship also illustrates that Gini-Simpson index changes very fast in low species richness values (0.5 for //S// = 2, 0.67 for //S// = 3, 0.75 for //S// = 4, ... 0.9 for //S// = 10), and with richness over 10 it changes much slower (0.95 for //S// = 20 and 0.99 for //S// = 100).
 +
 +{{:​obrazky:​shannon-and-simpson-on-sp-richness.png?​direct&​400|}}
 +
 Dependence of the three diversity indices (richness, Shannon and Simpson) on the (un)evenness and diversity of the community is illustrated below. Dependence of the three diversity indices (richness, Shannon and Simpson) on the (un)evenness and diversity of the community is illustrated below.
  
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 (called also //​equitability//​) is calculated from Simpson’s effective number of species divided by observed number of species. Effective number of species (ENS) is the number of equally abundant species which would need to be in a community so as it has the same Simpson’s index as the one really calculated (more about the concept of //effective number of species// below). In case of Simpson’s D, effective number of species is simply 1/D. (called also //​equitability//​) is calculated from Simpson’s effective number of species divided by observed number of species. Effective number of species (ENS) is the number of equally abundant species which would need to be in a community so as it has the same Simpson’s index as the one really calculated (more about the concept of //effective number of species// below). In case of Simpson’s D, effective number of species is simply 1/D.
  
-=== Effective numbers of species ===+=== Effective numbers of species ​(ENS) ===
 <WRAP right box 35%> <WRAP right box 35%>
 **Effective number of species** **Effective number of species**
   * for species richness = S   * for species richness = S
   * for Shannon index = //​e//<​sup>//​H//</​sup> ​ (exponential of Shannon entropy index)   * for Shannon index = //​e//<​sup>//​H//</​sup> ​ (exponential of Shannon entropy index)
-  * for Simpson index = 1/𝐷 (reciprocal of Simpson concentration index)+  * for Simpson index = 1///D// (reciprocal of Simpson concentration index)
 </​WRAP>​ </​WRAP>​
 Lou Jost (2002) argued that to call Shannon and Simpson (or Ginni-Simpson,​ respectively) indices as //​diversity//​ is misleading, since diversity should be measured in intuitive units of //​species//,​ while each of the two indices have different units (Shannon //bits// and Simpson //​probability//​)((Jost (2002) argues: "The radius of a sphere is an index of its volume but is not itself the volume, and using the radius in place of the volume in engineering equations will give dangerously misleading results. This is what biologists have done with diversity indices. The most common diversity measure, the Shannon-Wiener index, is an entropy, giving the uncertainty in the outcome of a sampling process.... Entropies are reasonable indices of diversity, but this is no reason to claim that entropy //is// diversity."​)). This problem can be overcome by introducing concept of ''​effective number of species''​ (ENS, MacArthur 1965), i.e. number of species in equivalent community (i.e. the one which has the same value of diversity index as the community in question) composed of equally-abundant species. In cace of perfectly even community, ENS is equal to species richness; for unevenn communities,​ ENS is always smaller than S. Each of the indices above can be converted into effective number of species following a simple formulas. ​   Lou Jost (2002) argued that to call Shannon and Simpson (or Ginni-Simpson,​ respectively) indices as //​diversity//​ is misleading, since diversity should be measured in intuitive units of //​species//,​ while each of the two indices have different units (Shannon //bits// and Simpson //​probability//​)((Jost (2002) argues: "The radius of a sphere is an index of its volume but is not itself the volume, and using the radius in place of the volume in engineering equations will give dangerously misleading results. This is what biologists have done with diversity indices. The most common diversity measure, the Shannon-Wiener index, is an entropy, giving the uncertainty in the outcome of a sampling process.... Entropies are reasonable indices of diversity, but this is no reason to claim that entropy //is// diversity."​)). This problem can be overcome by introducing concept of ''​effective number of species''​ (ENS, MacArthur 1965), i.e. number of species in equivalent community (i.e. the one which has the same value of diversity index as the community in question) composed of equally-abundant species. In cace of perfectly even community, ENS is equal to species richness; for unevenn communities,​ ENS is always smaller than S. Each of the indices above can be converted into effective number of species following a simple formulas. ​  
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 <​m>​{}^2{D}~=~1/​D</​m>​ <fs x-small>​(reciprocal of Simpson index)</​fs>​\\ <​m>​{}^2{D}~=~1/​D</​m>​ <fs x-small>​(reciprocal of Simpson index)</​fs>​\\
 </​WRAP>​ </​WRAP>​
-Mark Hill (British scientist ​whoamong others, introduced ​DCA, Twinspan, and recallibrated ​Ellenberg species indicator values for Britain) realized that species richness, Shannon entropy and Simpson'​s concentration index are all members of the same family of diversity indices, later called as Hill numbers. Individual Hill numbers differ by the parameter //q//, which quantifies how much the measure discounts rare species when calculating diversity. Hill number for //q// = 0 is simply species richness, for //q// = 1((In fact the Hill's formula is not defined for //q// = 1, but it can be shown that when q approaches 1 from below or above, the index gets equal to exponential Shannon.)) it is effective number of species derived from Shannon entropy, and for //q// = 2 it is ENS for Simpson index. For //q// > 0, indices discount rare species, while for //q// < 0 the indices discount common species and focus on number of rare species (usually not meaningful).+Mark Hill (British scientist, ​known also for introducing Detrended correspondence analysis (DCA), Twinspan, and recallibrating ​Ellenberg species indicator values for Britain) realized that species richness, Shannon entropy and Simpson'​s concentration index are all members of the same family of diversity indices, later called as Hill numbers. Individual Hill numbers differ by the parameter //q//, which quantifies how much the measure discounts rare species when calculating diversity. Hill number for //q// = 0 is simply species richness, for //q// = 1((In fact the Hill's formula is not defined for //q// = 1, but it can be shown that when q approaches 1 from below or above, the index gets equal to exponential Shannon.)) it is **Shannon diversity**,​ i.e. effective number of species derived from Shannon entropy, and for //q// = 2 it is **Simpson diversity**,​ i.e. ENS for Simpson ​concentration ​index. For //q// > 0, indices discount rare species, while for //q// < 0 the indices discount common species and focus on number of rare species (usually not meaningful).
  
 Dependence of species richness, Shannon diversity (effective number of species based on Shannon entropy index) and Simpson'​s diversity (effective number of species based on Simpson'​s index) on (un)evenness and diversity is illustrated below. Dependence of species richness, Shannon diversity (effective number of species based on Shannon entropy index) and Simpson'​s diversity (effective number of species based on Simpson'​s index) on (un)evenness and diversity is illustrated below.
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 === Diversity profiles === === Diversity profiles ===
-{{ :​obrazky:​diversity-profile-even-mod-high.png?​direct&​400|}} +It is possible to draw the effective number of species as a function of coefficient //q// - increasing //q// decreases the impact of rare species on the measure of diversity. The value for //q// = 0 equals to species richness (in the diagram displayed by squares), for //q// = 1 equals to Shannon diversity (circles) and for //q// = 2 Simpson diversity (triangles). The shape of the diversity profile considers the differences in evenness between the three communities;​ the more is the community species abundance uneven, the faster the curve declines with increasing coefficient //q//. The future will see what exactly can this form of diversity visualization bring new.
-It is possible to draw the effective number of species as a function of coefficient //q// - increasing //q// decreases the impact of rare species on the measure of diversity. The value for //q// = 0 equals to species richness (in the diagram displayed by squares), for //q// = 1 equals to Shannon diversity (circles) and for //q// = 2 Simpson diversity (triangles).+
  
 +{{:​obrazky:​diversity-profile-even-mod-high.png?​direct|}}
  
-The summary ​of various ​diversity ​expressions is in the table below. ​ +=== Summary ​of values for diversity ​measures discussed ​in this chapter === 
-                  ​^ Community A\\ (perfectly even)  ^ Community B\\ (moderately uneven) ​ ^ Community C\\ (highly uneven) ​ ^ +                                               ^ Community A\\ (perfectly even)  ^ Community B\\ (moderately uneven) ​ ^ Community C\\ (highly uneven) ​ ^ 
-Species richness ​ |  12                             ​| ​ 12                                |  12                            | +Species richness ​                              ​|  12                             ​| ​ 12                                |  12                            | 
-Shannon entropy ​  ​|  2.48                            |  1.81                                   ​|  0.87                              +Shannon entropy ​                               |  2.48                           ​|  1.81                              |  0.87                          
-Simpson index     ​|  0.92                               ​|  0.79                                  |  0.46                              +Simpson index                                  |  0.92                           ​|  0.79                              |  0.46                          
-Shannon evenness ​ |  1                               ​|  0.73                                  |  0.35                              +Shannon evenness ​                              ​|  1                              |  0.73                              |  0.35                          
-Simpson evenness ​ |  1                               ​|  0.38                                  |  0.15                              |+Simpson evenness ​                              ​|  1                              |  0.38                              |  0.15                          | 
 +| Shannon diversity (<​sup>//​1//</​sup>//​D//,​ N1)  |  12                              |  6.14                               ​| ​ 2.39                           | 
 +| Simpson diversity (<​sup>//​2//</​sup>//​D//,​ N2)  |  12                              |  4.66                               ​| ​ 1.86                           |
  
en/div-ind.1495633483.txt.gz · Last modified: 2017/10/11 20:36 (external edit)