77:
251:, the entities can be characters and the types of the different letters of the alphabet. The most commonly used diversity indices are simple transformations of the effective number of types (also known as 'true diversity'), but each diversity index can also be interpreted in its own right as a measure corresponding to some real phenomenon (but a different one for each diversity index).
1235:) in strings of text. The idea is that the more letters there are, and the closer their proportional abundances in the string of interest, the more difficult it is to correctly predict which letter will be the next one in the string. The Shannon entropy quantifies the uncertainty (entropy or degree of surprise) associated with this prediction. It is most often calculated as follows:
36:
2037:
2860:. The original Simpson index λ equals the probability that two entities taken at random from the dataset of interest (with replacement) represent the same type. Its transformation 1 − λ, therefore, equals the probability that the two entities represent different types. This measure is also known in ecology as the probability of interspecific encounter (
124:
2723:
Since the mean proportional abundance of the types increases with decreasing number of types and increasing abundance of the most abundant type, λ obtains small values in datasets of high diversity and large values in datasets of low diversity. This is counterintuitive behavior for a diversity index,
258:
True diversity, or the effective number of types, refers to the number of equally abundant types needed for the average proportional abundance of the types to equal that observed in the dataset of interest (where all types may not be equally abundant). The true diversity in a dataset is calculated by
2588:
The interpretation of λ as the probability that two entities taken at random from the dataset of interest represent the same type assumes that the first entity is replaced to the dataset before taking the second entity. If the dataset is very large, sampling without replacement gives approximately
1728:
2139:
values, and the smaller the corresponding
Shannon entropy. If practically all abundance is concentrated to one type, and the other types are very rare (even if there are many of them), Shannon entropy approaches zero. When there is only one type in the dataset, Shannon entropy exactly equals zero
3417:
Morris, E. Kathryn; Caruso, Tancredi; Buscot, François; Fischer, Markus; Hancock, Christine; Maier, Tanja S.; Meiners, Torsten; Müller, Caroline; Obermaier, Elisabeth; Prati, Daniel; Socher, Stephanie A.; Sonnemann, Ilja; Wäschke, Nicole; Wubet, Tesfaye; Wurst, Susanne (September 2014).
490:
254:
Many indices only account for categorical diversity between subjects or entities. Such indices, however do not account for the total variation (diversity) that can be held between subjects or entities which occurs only when both categorical and qualitative diversity are calculated.
1363:, and these have since become the most popular bases in applications that use the Shannon entropy. Each log base corresponds to a different measurement unit, which has been called binary digits (bits), decimal digits (decits), and natural digits (nats) for the bases 2, 10 and
765:
915:
is often referred to as the order of the diversity. It defines the sensitivity of the true diversity to rare vs. abundant species by modifying how the weighted mean of the species' proportional abundances is calculated. With some values of the parameter
2425:
1717:
1547:
2032:{\displaystyle H'=-\ln p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}=\ln \left({1 \over p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}}\right)=\ln \left({1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}\right)}
1168:) is simply the number of species, e.g. at a particular site. Richness is a simple measure, so it has been a popular diversity index in ecology, where abundance data are often not available. If true diversity is calculated with
2554:
of the types of interest, with the proportional abundances themselves being used as the weights. Proportional abundances are by definition constrained to values between zero and one, but it is a weighted arithmetic mean, hence
2269:
2827:
2832:
This simply equals true diversity of order 2, i.e. the effective number of types that is obtained when the weighted arithmetic mean is used to quantify average proportional abundance of types in the dataset of interest.
2966:
890:
2728:–Simpson index (1 − λ). Both of these have also been called the Simpson index in the ecological literature, so care is needed to avoid accidentally comparing the different indices as if they were the same.
2589:
the same result, but in small datasets, the difference can be substantial. If the dataset is small, and sampling without replacement is assumed, the probability of obtaining the same type with both random draws is:
284:
2692:
1356:
Although the equation is here written with natural logarithms, the base of the logarithm used when calculating the
Shannon entropy can be chosen freely. Shannon himself discussed logarithm bases 2, 10 and
1314:
2532:
3482:
Spellerberg, Ian F., and Peter J. Fedor. (2003) A tribute to Claude
Shannon (1916–2001) and a plea for more rigorous use of species richness, species diversity and the ‘Shannon–Wiener’Index.
610:
1353:
th species in the dataset of interest. Then the
Shannon entropy quantifies the uncertainty in predicting the species identity of an individual that is taken at random from the dataset.
2724:
so often, such transformations of λ that increase with increasing diversity have been used instead. The most popular of such indices have been the inverse
Simpson index (1/λ) and the
243:. The entities of interest are usually individual organisms (e.g. plants or animals), and the measure of abundance can be, for example, number of individuals, biomass or coverage. In
3124:
Tucker, Caroline M.; Cadotte, Marc W.; Carvalho, Silvia B.; Davies, T. Jonathan; Ferrier, Simon; Fritz, Susanne A.; Grenyer, Rich; Helmus, Matthew R.; Jin, Lanna S. (May 2017).
2280:
1369:, respectively. Comparing Shannon entropy values that were originally calculated with different log bases requires converting them to the same log base: change from the base
142:
1558:
3284:
Tuomisto, H (2010). "A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity".
3930:
571:
2971:
The Gibbs–Martin index of sociology, psychology, and management studies, which is also known as the Blau index, is the same measure as the Gini–Simpson index.
895:
and the term inside the parentheses is called the basic sum. Some popular diversity indices correspond to the basic sum as calculated with different values of
2171:
1419:
1060:
values is used, and each species is exactly weighted by its proportional abundance (in the weighted geometric mean, the weights are the exponents). When
2742:
2543:
is richness (the total number of types in the dataset). This equation is also equal to the weighted arithmetic mean of the proportional abundances
2870:
789:
485:{\displaystyle {}^{q}\!D={1 \over M_{q-1}}={1 \over {\sqrt{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}
3499:
3900:
3383:
Chao, Anne; Chiu, Chun-Huo; Jost, Lou (2016), "Phylogenetic
Diversity Measures and Their Decomposition: A Framework Based on Hill Numbers",
3920:
2595:
2720:
is the total number of entities in the dataset. This form of the
Simpson index is also known as the Hunter–Gaston index in microbiology.
2469:
The measure equals the probability that two entities taken at random from the dataset of interest represent the same type. It equals:
2079:
values themselves being used as the weights (exponents in the equation). The term within the parentheses hence equals true diversity
3836:
3813:
3402:
178:
160:
63:
3126:"A guide to phylogenetic metrics for conservation, community ecology and macroecology: A guide to phylogenetic metrics for ecology"
49:
3483:
3557:
Herfindahl, O. C. (1950) Concentration in the U.S. Steel
Industry. Unpublished doctoral dissertation, Columbia University.
1162:
simply quantifies how many different types the dataset of interest contains. For example, species richness (usually noted
203:
relatedness among the types. Diversity indices are statistical representations of different aspects of biodiversity (e.g.
3706:
2475:
1406:
of the proportional abundances of the types. Specifically, it equals the logarithm of true diversity as calculated with
1224:
3420:"Choosing and using diversity indices: insights for ecological applications from the German Biodiversity Exploratories"
2568:
By comparing the equation used to calculate λ with the equations used to calculate true diversity, it can be seen that
1241:
89:
3084:
3058:
2837:
2454:
to measure the degree of concentration when individuals are classified into types. The same index was rediscovered by
1232:
760:{\displaystyle {}^{1}\!D={1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}=\exp \left(-\sum _{i=1}^{R}p_{i}\ln(p_{i})\right)}
3741:
Berger, Wolfgang H.; Parker, Frances L. (June 1970). "Diversity of
Planktonic Foraminifera in Deep-Sea Sediments".
3094:
3005:
value in the dataset, i.e., the proportional abundance of the most abundant type. This corresponds to the weighted
3578:"Numerical index of the discriminatory ability of typing systems: an application of Simpson's index of diversity"
3069:
1081:, the species weights exactly cancel out the species proportional abundances, such that the weighted mean of the
3387:, Topics in Biodiversity and Conservation, vol. 14, Springer International Publishing, pp. 141–172,
2991:
1403:
3925:
2849:
2420:{\displaystyle {}^{q}H=\ln \left({1 \over {\sqrt{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}\right)=\ln({}^{q}\!D)}
2144:
496:
275:
3089:
2864:) and the Gini–Simpson index. It can be expressed as a transformation of the true diversity of order 2:
604:
approaches 1 is well defined and the corresponding diversity is calculated with the following equation:
236:
3330:
Tuomisto, H (2010). "A consistent terminology for quantifying species diversity? Yes, it does exist".
774:
calculated with natural logarithms (see above). In other domains, this statistic is also known as the
510:
equals the average proportional abundance of the types in the dataset as calculated with the weighted
3752:
3654:
3643:
Hurlbert, S.H. (1971). "The nonconcept of species diversity: A critique and alternative parameters".
3530:
3431:
3341:
3295:
3249:
3193:
1712:{\displaystyle H'=-(\ln p_{1}^{p_{1}}+\ln p_{2}^{p_{2}}+\ln p_{3}^{p_{3}}+\cdots +\ln p_{R}^{p_{R}})}
98:
55:
1015:
increases the effective weight given to the most abundant species. This leads to obtaining a larger
3897:
3625:
2459:
2455:
595:
248:
247:, the entities of interest can be people, and the types of interest various demographic groups. In
212:
3852:"Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample"
199:) there are in a dataset (e.g. a community). Some more sophisticated indices also account for the
3874:
3776:
3723:
3686:
3670:
3365:
3209:
3053:
2987:
3704:
Gibbs, Jack P.; William T. Martin (1962). "Urbanization, technology and the division of labor".
3832:
3809:
3768:
3743:
3678:
3645:
3607:
3465:
3447:
3398:
3357:
3184:
3155:
3099:
2451:
2128:. The more unequal the abundances of the types, the larger the weighted geometric mean of the
527:
is richness (the total number of types in the dataset), and the proportional abundance of the
2462:. As a result, the same measure is usually known as the Simpson index in ecology, and as the
3866:
3760:
3715:
3662:
3597:
3589:
3538:
3455:
3439:
3388:
3349:
3303:
3257:
3201:
3145:
3137:
3104:
3074:
3006:
2857:
2463:
2156:
1151:
511:
260:
232:
227:, the types of interest are usually species, but they can also be other categories, such as
208:
204:
3851:
3026:
approaches infinity, and hence equals the inverse of the true diversity of order infinity (
2458:
in 1950. The square root of the index had already been introduced in 1945 by the economist
1199:
has been a popular diversity index in the ecological literature, where it is also known as
547:
3904:
3079:
3064:
3043:
971:
771:
3756:
3658:
3534:
3435:
3345:
3299:
3253:
3197:
3182:
Hill, M. O. (1973). "Diversity and evenness: a unifying notation and its consequences".
76:
3825:
3797:
3460:
3419:
3150:
3125:
3048:
1220:
1206:
1005:
value, which is the proportional abundance of the most abundant species in the dataset.
958:
17:
3602:
3577:
544:. The proportional abundances themselves are used as the nominal weights. The numbers
3914:
3802:
3593:
3307:
3261:
2140:(there is no uncertainty in predicting the type of the next randomly chosen entity).
1214:
945:
200:
3878:
3780:
3690:
3566:
Hirschman, A. O. (1945) National power and the structure of foreign trade. Berkeley.
3369:
2725:
2264:{\displaystyle {}^{q}H={\frac {1}{1-q}}\;\ln \left(\sum _{i=1}^{R}p_{i}^{q}\right)}
1542:{\displaystyle H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}=-\sum _{i=1}^{R}\ln p_{i}^{p_{i}}}
3764:
215:), which are useful simplifications for comparing different communities or sites.
3843:
See chapter 5 for an elaboration of coding procedures described informally above.
3892:
3393:
2822:{\displaystyle {\frac {1}{\lambda }}={1 \over \sum _{i=1}^{R}p_{i}^{2}}={}^{2}D}
2054:
1124:
is generally limited to non-negative values. This is because negative values of
274:
of the proportional abundances of the types in the dataset, and then taking the
3870:
3353:
776:
244:
3451:
2430:
This means that taking the logarithm of true diversity based on any value of
3847:
3332:
3286:
2961:{\displaystyle 1-\lambda =1-\sum _{i=1}^{R}p_{i}^{2}=1-{\frac {1}{{}^{2}D}}}
240:
3772:
3682:
3469:
3361:
3159:
934:
assumes familiar kinds of weighted means as special cases. In particular,
885:{\displaystyle {}^{q}\!D=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}
3611:
2975:
2585:. The original Simpson's index hence equals the corresponding basic sum.
984:
3907:
gives some examples of estimates of
Simpson's index for real ecosystems.
3727:
3674:
3213:
224:
196:
3443:
3141:
3543:
3518:
2687:{\displaystyle \ell ={\frac {\sum _{i=1}^{R}n_{i}(n_{i}-1)}{N(N-1)}}}
1181:) equals the actual number of types, which is identical to Richness (
3719:
3666:
3205:
1130:
would give rare species so much more weight than abundant ones that
2102:
When all types in the dataset of interest are equally common, all
228:
1067:, the weight given to abundant species is exaggerated, and when
905:
Sensitivity of the diversity value to rare vs. abundant species
783:
The general equation of diversity is often written in the form
3502:. The Bell System Technical Journal, 27, 379–423 and 623–656.
2159:
is a generalization of the Shannon entropy to other values of
117:
70:
29:
2436:
gives the Rényi entropy corresponding to the same value of
1336:
th type of letter in the string of interest. In ecology,
2565:, which is reached when all types are equally abundant.
2143:
In machine learning the Shannon index is also called as
1347:
is often the proportion of individuals belonging to the
195:
is a method of measuring how many different types (e.g.
138:
94:
3385:
Biodiversity Conservation and Phylogenetic Systematics
2466:
or the Herfindahl–Hirschman index (HHI) in economics.
2873:
2745:
2598:
2478:
2283:
2174:
1731:
1561:
1422:
1244:
792:
613:
550:
287:
1099:even when all species are not equally abundant. At
133:
may be too technical for most readers to understand
3824:
3801:
2960:
2821:
2686:
2527:{\displaystyle \lambda =\sum _{i=1}^{R}p_{i}^{2},}
2526:
2419:
2263:
2031:
1711:
1541:
1308:
884:
759:
565:
484:
2410:
1330:is the proportion of characters belonging to the
1309:{\displaystyle H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}}
802:
623:
594:, the above equation is undefined. However, the
297:
3061:, a diversity index applied to political parties
3325:
3323:
3321:
3319:
3317:
3279:
3277:
3275:
3273:
3271:
3512:
3510:
3508:
2120:, and the Shannon index hence takes the value
987:, the weighted generalized mean with exponent
3494:
3492:
8:
2450:The Simpson index was introduced in 1949 by
1112:, hence equals the actual number of species
2836:The index is also used as a measure of the
2708:is the number of entities belonging to the
219:Effective number of species or Hill numbers
64:Learn how and when to remove these messages
3235:
3233:
3231:
3229:
3227:
3225:
3223:
3177:
3175:
3173:
3171:
3169:
2578:, i.e., true diversity as calculated with
2208:
2057:equals the weighted geometric mean of the
1074:, the weight given to rare species is. At
3823:Cover, Thomas M.; Thomas, Joy A. (1991).
3601:
3542:
3459:
3392:
3240:Jost, L (2006). "Entropy and diversity".
3149:
2946:
2944:
2937:
2922:
2917:
2907:
2896:
2872:
2810:
2808:
2795:
2790:
2780:
2769:
2759:
2746:
2744:
2646:
2633:
2623:
2612:
2605:
2597:
2515:
2510:
2500:
2489:
2477:
2404:
2402:
2370:
2357:
2352:
2342:
2332:
2321:
2314:
2309:
2287:
2285:
2282:
2250:
2245:
2235:
2224:
2190:
2178:
2176:
2173:
2014:
2009:
2004:
1994:
1983:
1978:
1973:
1945:
1940:
1935:
1920:
1915:
1910:
1898:
1893:
1888:
1876:
1871:
1866:
1856:
1835:
1830:
1825:
1810:
1805:
1800:
1788:
1783:
1778:
1766:
1761:
1756:
1730:
1698:
1693:
1688:
1661:
1656:
1651:
1630:
1625:
1620:
1599:
1594:
1589:
1560:
1531:
1526:
1521:
1505:
1494:
1478:
1462:
1452:
1441:
1421:
1300:
1284:
1274:
1263:
1243:
1219:. The measure was originally proposed by
860:
856:
845:
840:
830:
819:
814:
796:
794:
791:
743:
724:
714:
703:
671:
666:
661:
651:
640:
635:
630:
617:
615:
612:
554:
549:
460:
456:
445:
440:
430:
419:
414:
391:
378:
373:
363:
353:
342:
335:
330:
313:
304:
291:
289:
286:
179:Learn how and when to remove this message
161:Learn how and when to remove this message
145:, without removing the technical details.
27:How many different types are in a dataset
3931:Summary statistics for categorical data
3859:Environmental and Ecological Statistics
3116:
3500:A mathematical theory of communication
2848:The Gini-Simpson Index is also called
2986:The Berger–Parker index, named after
1049:, the weighted geometric mean of the
143:make it understandable to non-experts
7:
1027:value and a smaller true diversity (
922:, the value of the generalized mean
2053:values equals 1 by definition, the
1381:is obtained with multiplication by
1106:, the effective number of species,
1009:Generally, increasing the value of
223:When diversity indices are used in
2974:The quantity is also known as the
2736:The inverse Simpson index equals:
25:
1175:, the effective number of types (
88:to comply with Knowledge (XXG)'s
45:This article has multiple issues.
3594:10.1128/JCM.26.11.2465-2466.1988
3308:10.1111/j.1600-0587.2009.05880.x
3262:10.1111/j.2006.0030-1299.14714.x
770:which is the exponential of the
122:
75:
34:
3576:Hunter, PR; Gaston, MA (1988).
3484:Global Ecology and Biogeography
1118:. In the context of diversity,
53:or discuss these issues on the
3827:Elements of Information Theory
2678:
2666:
2658:
2639:
2414:
2398:
1706:
1576:
877:
865:
749:
736:
477:
465:
1:
3765:10.1126/science.168.3937.1345
2165:than 1. It can be expressed:
3707:American Sociological Review
944:corresponds to the weighted
3921:Measurement of biodiversity
3394:10.1007/978-3-319-22461-9_8
3085:Measurement of biodiversity
3059:Effective number of parties
2838:effective number of parties
1233:Shannon information content
582:effective number of species
3947:
3519:"Measurement of diversity"
1149:
278:of this. The equation is:
259:first taking the weighted
3893:Simpson's Diversity index
3354:10.1007/s00442-010-1812-0
3070:Generalized entropy index
1552:This can also be written
1201:Shannon's diversity index
3626:"Growing Decision Trees"
2978:in population genetics.
1223:in 1948 to quantify the
1033:) value with increasing
101:may contain suggestions.
86:may need to be rewritten
3871:10.1023/A:1026096204727
3804:Introduction to Ecology
3517:Simpson, E. H. (1949).
2992:Frances Lawrence Parker
2976:expected heterozygosity
1404:weighted geometric mean
994:approaches the maximum
18:Simpson diversity index
3498:Shannon, C. E. (1948)
2962:
2912:
2854:Gini's diversity index
2823:
2785:
2688:
2628:
2528:
2505:
2421:
2337:
2265:
2240:
2033:
1999:
1713:
1543:
1510:
1457:
1310:
1279:
886:
835:
761:
719:
656:
567:
486:
435:
358:
3850:; Shen, T-J. (2003).
3424:Ecology and Evolution
3090:Qualitative variation
2994:, equals the maximum
2963:
2892:
2824:
2765:
2732:Inverse Simpson index
2689:
2608:
2529:
2485:
2422:
2317:
2266:
2220:
2042:Since the sum of the
2034:
1979:
1714:
1544:
1490:
1437:
1311:
1259:
887:
815:
762:
699:
636:
575:Hill numbers of order
568:
566:{\displaystyle ^{q}D}
487:
415:
338:
2871:
2743:
2596:
2476:
2281:
2172:
1729:
1559:
1420:
1402:) is related to the
1242:
1211:, and (erroneously)
790:
611:
548:
285:
3757:1970Sci...168.1345B
3751:(3937): 1345–1347.
3659:1971Ecol...52..577H
3535:1949Natur.163..688S
3436:2014EcoEv...4.3514M
3346:2010Oecol.164..853T
3300:2010Ecogr..33....2T
3254:2006Oikos.113..363J
3198:1973Ecol...54..427H
2982:Berger–Parker index
2927:
2800:
2520:
2460:Albert O. Hirschman
2456:Orris C. Herfindahl
2368:
2255:
2021:
1952:
1927:
1905:
1883:
1842:
1817:
1795:
1773:
1705:
1668:
1637:
1606:
1538:
1396:The Shannon index (
850:
678:
521:. In the equation,
450:
389:
249:information science
3903:2005-12-19 at the
3798:Colinvaux, Paul A.
3130:Biological Reviews
3095:Relative abundance
3054:Cultural diversity
2988:Wolfgang H. Berger
2958:
2913:
2844:Gini–Simpson index
2819:
2786:
2684:
2524:
2506:
2417:
2348:
2261:
2241:
2029:
2000:
1931:
1906:
1884:
1862:
1821:
1796:
1774:
1752:
1709:
1684:
1647:
1616:
1585:
1539:
1517:
1306:
882:
836:
757:
657:
596:mathematical limit
563:
482:
436:
369:
3898:Diversity indices
3588:(11): 2465–2466.
3444:10.1002/ece3.1155
3430:(18): 3514–3524.
3142:10.1111/brv.12252
3100:Species diversity
2956:
2802:
2754:
2682:
2452:Edward H. Simpson
2383:
2381:
2206:
2068:values, with the
2023:
1954:
680:
404:
402:
325:
189:
188:
181:
171:
170:
163:
116:
115:
90:quality standards
68:
16:(Redirected from
3938:
3882:
3856:
3842:
3830:
3819:
3807:
3785:
3784:
3738:
3732:
3731:
3701:
3695:
3694:
3640:
3634:
3633:
3622:
3616:
3615:
3605:
3582:J Clin Microbiol
3573:
3567:
3564:
3558:
3555:
3549:
3548:
3546:
3544:10.1038/163688a0
3514:
3503:
3496:
3487:
3480:
3474:
3473:
3463:
3414:
3408:
3407:
3396:
3380:
3374:
3373:
3327:
3312:
3311:
3281:
3266:
3265:
3237:
3218:
3217:
3179:
3164:
3163:
3153:
3121:
3105:Species richness
3075:Gini coefficient
3032:
3025:
3019:
3007:generalized mean
3004:
2967:
2965:
2964:
2959:
2957:
2955:
2951:
2950:
2945:
2938:
2926:
2921:
2911:
2906:
2858:Machine Learning
2856:in the field of
2828:
2826:
2825:
2820:
2815:
2814:
2809:
2803:
2801:
2799:
2794:
2784:
2779:
2760:
2755:
2747:
2719:
2713:
2707:
2693:
2691:
2690:
2685:
2683:
2681:
2661:
2651:
2650:
2638:
2637:
2627:
2622:
2606:
2584:
2577:
2571:
2564:
2553:
2542:
2533:
2531:
2530:
2525:
2519:
2514:
2504:
2499:
2464:Herfindahl index
2441:
2435:
2426:
2424:
2423:
2418:
2409:
2408:
2403:
2388:
2384:
2382:
2380:
2369:
2367:
2356:
2347:
2346:
2336:
2331:
2315:
2310:
2292:
2291:
2286:
2270:
2268:
2267:
2262:
2260:
2256:
2254:
2249:
2239:
2234:
2207:
2205:
2191:
2183:
2182:
2177:
2164:
2145:Information gain
2138:
2127:
2119:
2112:
2098:
2090:
2084:
2078:
2067:
2052:
2038:
2036:
2035:
2030:
2028:
2024:
2022:
2020:
2019:
2018:
2008:
1998:
1993:
1974:
1959:
1955:
1953:
1951:
1950:
1949:
1939:
1926:
1925:
1924:
1914:
1904:
1903:
1902:
1892:
1882:
1881:
1880:
1870:
1857:
1841:
1840:
1839:
1829:
1816:
1815:
1814:
1804:
1794:
1793:
1792:
1782:
1772:
1771:
1770:
1760:
1739:
1718:
1716:
1715:
1710:
1704:
1703:
1702:
1692:
1667:
1666:
1665:
1655:
1636:
1635:
1634:
1624:
1605:
1604:
1603:
1593:
1569:
1548:
1546:
1545:
1540:
1537:
1536:
1535:
1525:
1509:
1504:
1483:
1482:
1467:
1466:
1456:
1451:
1430:
1412:
1401:
1392:
1380:
1374:
1368:
1362:
1352:
1346:
1335:
1329:
1315:
1313:
1312:
1307:
1305:
1304:
1289:
1288:
1278:
1273:
1252:
1186:
1180:
1174:
1167:
1161:
1152:Species richness
1141:
1135:
1129:
1123:
1117:
1111:
1105:
1098:
1091:
1080:
1073:
1066:
1059:
1048:
1038:
1032:
1026:
1014:
1004:
993:
982:
970:to the weighted
969:
957:to the weighted
956:
943:
933:
921:
914:
900:
891:
889:
888:
883:
881:
880:
864:
855:
851:
849:
844:
834:
829:
801:
800:
795:
766:
764:
763:
758:
756:
752:
748:
747:
729:
728:
718:
713:
681:
679:
677:
676:
675:
665:
655:
650:
631:
622:
621:
616:
603:
593:
572:
570:
569:
564:
559:
558:
543:
532:
526:
520:
512:generalized mean
509:
491:
489:
488:
483:
481:
480:
464:
455:
451:
449:
444:
434:
429:
405:
403:
401:
390:
388:
377:
368:
367:
357:
352:
336:
331:
326:
324:
323:
305:
296:
295:
290:
273:
261:generalized mean
237:functional types
184:
177:
166:
159:
155:
152:
146:
126:
125:
118:
111:
108:
102:
79:
71:
60:
38:
37:
30:
21:
3946:
3945:
3941:
3940:
3939:
3937:
3936:
3935:
3911:
3910:
3905:Wayback Machine
3889:
3854:
3846:
3839:
3822:
3816:
3796:
3793:
3791:Further reading
3788:
3740:
3739:
3735:
3720:10.2307/2089624
3703:
3702:
3698:
3667:10.2307/1934145
3642:
3641:
3637:
3624:
3623:
3619:
3575:
3574:
3570:
3565:
3561:
3556:
3552:
3516:
3515:
3506:
3497:
3490:
3481:
3477:
3416:
3415:
3411:
3405:
3382:
3381:
3377:
3329:
3328:
3315:
3283:
3282:
3269:
3239:
3238:
3221:
3206:10.2307/1934352
3181:
3180:
3167:
3123:
3122:
3118:
3114:
3109:
3080:Isolation index
3065:Gamma diversity
3044:Alpha diversity
3039:
3027:
3021:
3018:
3010:
3003:
2995:
2984:
2943:
2942:
2869:
2868:
2846:
2807:
2764:
2741:
2740:
2734:
2715:
2709:
2706:
2698:
2662:
2642:
2629:
2607:
2594:
2593:
2579:
2573:
2569:
2556:
2552:
2544:
2538:
2474:
2473:
2448:
2437:
2431:
2401:
2338:
2316:
2305:
2284:
2279:
2278:
2219:
2215:
2195:
2175:
2170:
2169:
2160:
2153:
2137:
2129:
2121:
2114:
2111:
2103:
2092:
2086:
2080:
2077:
2069:
2066:
2058:
2051:
2043:
2010:
1969:
1941:
1916:
1894:
1872:
1861:
1852:
1831:
1806:
1784:
1762:
1732:
1727:
1726:
1694:
1657:
1626:
1595:
1562:
1557:
1556:
1527:
1474:
1458:
1423:
1418:
1417:
1407:
1397:
1388:
1382:
1376:
1370:
1364:
1358:
1348:
1345:
1337:
1331:
1328:
1320:
1296:
1280:
1245:
1240:
1239:
1229:Shannon entropy
1193:
1182:
1176:
1169:
1163:
1157:
1154:
1148:
1137:
1131:
1125:
1119:
1113:
1107:
1100:
1093:
1090:
1082:
1075:
1068:
1061:
1058:
1050:
1043:
1034:
1028:
1025:
1016:
1010:
1003:
995:
988:
978:
972:arithmetic mean
964:
951:
938:
932:
923:
917:
910:
907:
896:
810:
809:
793:
788:
787:
772:Shannon entropy
739:
720:
695:
691:
667:
614:
609:
608:
599:
588:
551:
546:
545:
542:
534:
528:
522:
515:
508:
499:
410:
409:
359:
337:
309:
288:
283:
282:
272:
263:
221:
193:diversity index
185:
174:
173:
172:
167:
156:
150:
147:
139:help improve it
136:
127:
123:
112:
106:
103:
93:
80:
39:
35:
28:
23:
22:
15:
12:
11:
5:
3944:
3942:
3934:
3933:
3928:
3923:
3913:
3912:
3909:
3908:
3895:
3888:
3887:External links
3885:
3884:
3883:
3865:(4): 429–443.
3844:
3837:
3820:
3814:
3792:
3789:
3787:
3786:
3733:
3714:(5): 667–677.
3696:
3653:(4): 577–586.
3635:
3617:
3568:
3559:
3550:
3504:
3488:
3486:12.3, 177-179.
3475:
3409:
3403:
3375:
3340:(4): 853–860.
3313:
3267:
3248:(2): 363–375.
3219:
3192:(2): 427–432.
3165:
3136:(2): 698–715.
3115:
3113:
3110:
3108:
3107:
3102:
3097:
3092:
3087:
3082:
3077:
3072:
3067:
3062:
3056:
3051:
3049:Beta diversity
3046:
3040:
3038:
3035:
3014:
2999:
2983:
2980:
2969:
2968:
2954:
2949:
2941:
2936:
2933:
2930:
2925:
2920:
2916:
2910:
2905:
2902:
2899:
2895:
2891:
2888:
2885:
2882:
2879:
2876:
2845:
2842:
2830:
2829:
2818:
2813:
2806:
2798:
2793:
2789:
2783:
2778:
2775:
2772:
2768:
2763:
2758:
2753:
2750:
2733:
2730:
2702:
2695:
2694:
2680:
2677:
2674:
2671:
2668:
2665:
2660:
2657:
2654:
2649:
2645:
2641:
2636:
2632:
2626:
2621:
2618:
2615:
2611:
2604:
2601:
2548:
2535:
2534:
2523:
2518:
2513:
2509:
2503:
2498:
2495:
2492:
2488:
2484:
2481:
2447:
2444:
2428:
2427:
2416:
2413:
2407:
2400:
2397:
2394:
2391:
2387:
2379:
2376:
2373:
2366:
2363:
2360:
2355:
2351:
2345:
2341:
2335:
2330:
2327:
2324:
2320:
2313:
2308:
2304:
2301:
2298:
2295:
2290:
2272:
2271:
2259:
2253:
2248:
2244:
2238:
2233:
2230:
2227:
2223:
2218:
2214:
2211:
2204:
2201:
2198:
2194:
2189:
2186:
2181:
2152:
2149:
2133:
2107:
2073:
2062:
2047:
2040:
2039:
2027:
2017:
2013:
2007:
2003:
1997:
1992:
1989:
1986:
1982:
1977:
1972:
1968:
1965:
1962:
1958:
1948:
1944:
1938:
1934:
1930:
1923:
1919:
1913:
1909:
1901:
1897:
1891:
1887:
1879:
1875:
1869:
1865:
1860:
1855:
1851:
1848:
1845:
1838:
1834:
1828:
1824:
1820:
1813:
1809:
1803:
1799:
1791:
1787:
1781:
1777:
1769:
1765:
1759:
1755:
1751:
1748:
1745:
1742:
1738:
1735:
1720:
1719:
1708:
1701:
1697:
1691:
1687:
1683:
1680:
1677:
1674:
1671:
1664:
1660:
1654:
1650:
1646:
1643:
1640:
1633:
1629:
1623:
1619:
1615:
1612:
1609:
1602:
1598:
1592:
1588:
1584:
1581:
1578:
1575:
1572:
1568:
1565:
1550:
1549:
1534:
1530:
1524:
1520:
1516:
1513:
1508:
1503:
1500:
1497:
1493:
1489:
1486:
1481:
1477:
1473:
1470:
1465:
1461:
1455:
1450:
1447:
1444:
1440:
1436:
1433:
1429:
1426:
1384:
1341:
1324:
1317:
1316:
1303:
1299:
1295:
1292:
1287:
1283:
1277:
1272:
1269:
1266:
1262:
1258:
1255:
1251:
1248:
1221:Claude Shannon
1192:
1189:
1150:Main article:
1147:
1144:
1092:values equals
1086:
1054:
1020:
1007:
1006:
999:
992: − 1
975:
962:
959:geometric mean
949:
927:
906:
903:
893:
892:
879:
876:
873:
870:
867:
863:
859:
854:
848:
843:
839:
833:
828:
825:
822:
818:
813:
808:
805:
799:
768:
767:
755:
751:
746:
742:
738:
735:
732:
727:
723:
717:
712:
709:
706:
702:
698:
694:
690:
687:
684:
674:
670:
664:
660:
654:
649:
646:
643:
639:
634:
629:
626:
620:
562:
557:
553:
538:
519: − 1
514:with exponent
503:
493:
492:
479:
476:
473:
470:
467:
463:
459:
454:
448:
443:
439:
433:
428:
425:
422:
418:
413:
408:
400:
397:
394:
387:
384:
381:
376:
372:
366:
362:
356:
351:
348:
345:
341:
334:
329:
322:
319:
316:
312:
308:
303:
300:
294:
267:
220:
217:
187:
186:
169:
168:
130:
128:
121:
114:
113:
83:
81:
74:
69:
43:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3943:
3932:
3929:
3927:
3926:Index numbers
3924:
3922:
3919:
3918:
3916:
3906:
3902:
3899:
3896:
3894:
3891:
3890:
3886:
3880:
3876:
3872:
3868:
3864:
3860:
3853:
3849:
3845:
3840:
3838:0-471-06259-6
3834:
3829:
3828:
3821:
3817:
3815:0-471-16498-4
3811:
3806:
3805:
3799:
3795:
3794:
3790:
3782:
3778:
3774:
3770:
3766:
3762:
3758:
3754:
3750:
3746:
3745:
3737:
3734:
3729:
3725:
3721:
3717:
3713:
3709:
3708:
3700:
3697:
3692:
3688:
3684:
3680:
3676:
3672:
3668:
3664:
3660:
3656:
3652:
3648:
3647:
3639:
3636:
3631:
3627:
3621:
3618:
3613:
3609:
3604:
3599:
3595:
3591:
3587:
3583:
3579:
3572:
3569:
3563:
3560:
3554:
3551:
3545:
3540:
3536:
3532:
3529:(4148): 688.
3528:
3524:
3520:
3513:
3511:
3509:
3505:
3501:
3495:
3493:
3489:
3485:
3479:
3476:
3471:
3467:
3462:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3429:
3425:
3421:
3413:
3410:
3406:
3404:9783319224602
3400:
3395:
3390:
3386:
3379:
3376:
3371:
3367:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3335:
3334:
3326:
3324:
3322:
3320:
3318:
3314:
3309:
3305:
3301:
3297:
3293:
3289:
3288:
3280:
3278:
3276:
3274:
3272:
3268:
3263:
3259:
3255:
3251:
3247:
3243:
3236:
3234:
3232:
3230:
3228:
3226:
3224:
3220:
3215:
3211:
3207:
3203:
3199:
3195:
3191:
3187:
3186:
3178:
3176:
3174:
3172:
3170:
3166:
3161:
3157:
3152:
3147:
3143:
3139:
3135:
3131:
3127:
3120:
3117:
3111:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3041:
3036:
3034:
3031:
3024:
3017:
3013:
3008:
3002:
2998:
2993:
2989:
2981:
2979:
2977:
2972:
2952:
2947:
2939:
2934:
2931:
2928:
2923:
2918:
2914:
2908:
2903:
2900:
2897:
2893:
2889:
2886:
2883:
2880:
2877:
2874:
2867:
2866:
2865:
2863:
2859:
2855:
2851:
2850:Gini impurity
2843:
2841:
2839:
2834:
2816:
2811:
2804:
2796:
2791:
2787:
2781:
2776:
2773:
2770:
2766:
2761:
2756:
2751:
2748:
2739:
2738:
2737:
2731:
2729:
2727:
2721:
2718:
2712:
2705:
2701:
2675:
2672:
2669:
2663:
2655:
2652:
2647:
2643:
2634:
2630:
2624:
2619:
2616:
2613:
2609:
2602:
2599:
2592:
2591:
2590:
2586:
2582:
2576:
2566:
2563:
2559:
2551:
2547:
2541:
2521:
2516:
2511:
2507:
2501:
2496:
2493:
2490:
2486:
2482:
2479:
2472:
2471:
2470:
2467:
2465:
2461:
2457:
2453:
2446:Simpson index
2445:
2443:
2440:
2434:
2411:
2405:
2395:
2392:
2389:
2385:
2377:
2374:
2371:
2364:
2361:
2358:
2353:
2349:
2343:
2339:
2333:
2328:
2325:
2322:
2318:
2311:
2306:
2302:
2299:
2296:
2293:
2288:
2277:
2276:
2275:
2274:which equals
2257:
2251:
2246:
2242:
2236:
2231:
2228:
2225:
2221:
2216:
2212:
2209:
2202:
2199:
2196:
2192:
2187:
2184:
2179:
2168:
2167:
2166:
2163:
2158:
2157:Rényi entropy
2151:Rényi entropy
2150:
2148:
2146:
2141:
2136:
2132:
2125:
2118:
2113:values equal
2110:
2106:
2100:
2096:
2089:
2083:
2076:
2072:
2065:
2061:
2056:
2050:
2046:
2025:
2015:
2011:
2005:
2001:
1995:
1990:
1987:
1984:
1980:
1975:
1970:
1966:
1963:
1960:
1956:
1946:
1942:
1936:
1932:
1928:
1921:
1917:
1911:
1907:
1899:
1895:
1889:
1885:
1877:
1873:
1867:
1863:
1858:
1853:
1849:
1846:
1843:
1836:
1832:
1826:
1822:
1818:
1811:
1807:
1801:
1797:
1789:
1785:
1779:
1775:
1767:
1763:
1757:
1753:
1749:
1746:
1743:
1740:
1736:
1733:
1725:
1724:
1723:
1722:which equals
1699:
1695:
1689:
1685:
1681:
1678:
1675:
1672:
1669:
1662:
1658:
1652:
1648:
1644:
1641:
1638:
1631:
1627:
1621:
1617:
1613:
1610:
1607:
1600:
1596:
1590:
1586:
1582:
1579:
1573:
1570:
1566:
1563:
1555:
1554:
1553:
1532:
1528:
1522:
1518:
1514:
1511:
1506:
1501:
1498:
1495:
1491:
1487:
1484:
1479:
1475:
1471:
1468:
1463:
1459:
1453:
1448:
1445:
1442:
1438:
1434:
1431:
1427:
1424:
1416:
1415:
1414:
1410:
1405:
1400:
1394:
1391:
1387:
1379:
1373:
1367:
1361:
1354:
1351:
1344:
1340:
1334:
1327:
1323:
1301:
1297:
1293:
1290:
1285:
1281:
1275:
1270:
1267:
1264:
1260:
1256:
1253:
1249:
1246:
1238:
1237:
1236:
1234:
1231:, related to
1230:
1226:
1222:
1218:
1216:
1210:
1208:
1202:
1198:
1197:Shannon index
1191:Shannon index
1190:
1188:
1185:
1179:
1172:
1166:
1160:
1153:
1145:
1143:
1140:
1136:would exceed
1134:
1128:
1122:
1116:
1110:
1103:
1097:
1089:
1085:
1078:
1071:
1064:
1057:
1053:
1046:
1040:
1037:
1031:
1023:
1019:
1013:
1002:
998:
991:
986:
981:
976:
973:
967:
963:
960:
954:
950:
947:
946:harmonic mean
941:
937:
936:
935:
930:
926:
920:
913:
909:The value of
904:
902:
899:
874:
871:
868:
861:
857:
852:
846:
841:
837:
831:
826:
823:
820:
816:
811:
806:
803:
797:
786:
785:
784:
781:
779:
778:
773:
753:
744:
740:
733:
730:
725:
721:
715:
710:
707:
704:
700:
696:
692:
688:
685:
682:
672:
668:
662:
658:
652:
647:
644:
641:
637:
632:
627:
624:
618:
607:
606:
605:
602:
597:
591:
585:
583:
579:
576:
560:
555:
552:
541:
537:
531:
525:
518:
513:
506:
502:
498:
474:
471:
468:
461:
457:
452:
446:
441:
437:
431:
426:
423:
420:
416:
411:
406:
398:
395:
392:
385:
382:
379:
374:
370:
364:
360:
354:
349:
346:
343:
339:
332:
327:
320:
317:
314:
310:
306:
301:
298:
292:
281:
280:
279:
277:
270:
266:
262:
256:
252:
250:
246:
242:
238:
234:
230:
226:
218:
216:
214:
210:
206:
202:
198:
194:
183:
180:
165:
162:
154:
144:
140:
134:
131:This article
129:
120:
119:
110:
100:
96:
91:
87:
84:This article
82:
78:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
3862:
3858:
3826:
3803:
3748:
3742:
3736:
3711:
3705:
3699:
3650:
3644:
3638:
3629:
3620:
3585:
3581:
3571:
3562:
3553:
3526:
3522:
3478:
3427:
3423:
3412:
3384:
3378:
3337:
3331:
3291:
3285:
3245:
3241:
3189:
3183:
3133:
3129:
3119:
3029:
3022:
3020:values when
3015:
3011:
3000:
2996:
2985:
2973:
2970:
2861:
2853:
2847:
2835:
2831:
2735:
2722:
2716:
2714:th type and
2710:
2703:
2699:
2696:
2587:
2580:
2574:
2567:
2561:
2557:
2549:
2545:
2539:
2536:
2468:
2449:
2438:
2432:
2429:
2273:
2161:
2154:
2142:
2134:
2130:
2123:
2116:
2108:
2104:
2101:
2094:
2087:
2081:
2074:
2070:
2063:
2059:
2048:
2044:
2041:
1721:
1551:
1408:
1398:
1395:
1389:
1385:
1377:
1371:
1365:
1359:
1355:
1349:
1342:
1338:
1332:
1325:
1321:
1318:
1228:
1212:
1204:
1200:
1196:
1194:
1183:
1177:
1170:
1164:
1158:
1155:
1138:
1132:
1126:
1120:
1114:
1108:
1101:
1095:
1087:
1083:
1076:
1069:
1062:
1055:
1051:
1044:
1041:
1035:
1029:
1021:
1017:
1011:
1008:
1000:
996:
989:
979:
965:
952:
939:
928:
924:
918:
911:
908:
897:
894:
782:
775:
769:
600:
589:
586:
581:
577:
574:
539:
535:
529:
523:
516:
504:
500:
494:
268:
264:
257:
253:
222:
201:phylogenetic
192:
190:
175:
157:
148:
132:
104:
95:You can help
85:
61:
54:
48:
47:Please help
44:
3294:(1): 2–22.
2055:denominator
983:approaches
573:are called
533:th type is
497:denominator
3915:Categories
3112:References
777:perplexity
276:reciprocal
245:demography
241:haplotypes
151:April 2020
107:April 2020
50:improve it
3831:. Wiley.
3808:. Wiley.
3630:MathWorks
3452:2045-7758
3333:Oecologia
3287:Ecography
2935:−
2894:∑
2890:−
2881:λ
2878:−
2767:∑
2752:λ
2673:−
2653:−
2610:∑
2600:ℓ
2487:∑
2480:λ
2396:
2375:−
2362:−
2319:∑
2303:
2222:∑
2213:
2200:−
1981:∏
1967:
1929:⋯
1850:
1819:⋯
1750:
1744:−
1682:
1673:⋯
1645:
1614:
1583:
1574:−
1515:
1492:∑
1488:−
1472:
1439:∑
1435:−
1294:
1261:∑
1257:−
1156:Richness
872:−
817:∑
734:
701:∑
697:−
689:
638:∏
472:−
417:∑
396:−
383:−
340:∑
318:−
213:dominance
99:talk page
56:talk page
3901:Archived
3879:20389926
3848:Chao, A.
3800:(1973).
3781:29553922
3773:17731043
3691:25837001
3683:28973811
3470:25478144
3370:19902787
3362:20978798
3160:26785932
3037:See also
1737:′
1567:′
1428:′
1375:to base
1250:′
1213:Shannon–
1205:Shannon–
1146:Richness
985:infinity
233:families
209:evenness
205:richness
3753:Bibcode
3744:Science
3728:2089624
3675:1934145
3655:Bibcode
3646:Ecology
3612:3069867
3531:Bibcode
3461:4224527
3432:Bibcode
3342:Bibcode
3296:Bibcode
3250:Bibcode
3214:1934352
3194:Bibcode
3185:Ecology
3151:5096690
3009:of the
2572:equals
2091:equals
1227:(hence
1225:entropy
225:ecology
197:species
137:Please
3877:
3835:
3812:
3779:
3771:
3726:
3689:
3681:
3673:
3610:
3603:266921
3600:
3523:Nature
3468:
3458:
3450:
3401:
3368:
3360:
3212:
3158:
3148:
2697:where
2537:where
2085:, and
1319:where
1215:Weaver
1207:Wiener
1072:< 1
1065:> 1
229:genera
211:, and
97:. The
3875:S2CID
3855:(PDF)
3777:S2CID
3724:JSTOR
3687:S2CID
3671:JSTOR
3366:S2CID
3242:Oikos
3210:JSTOR
2852:, or
1217:index
1209:index
1042:When
961:, and
587:When
239:, or
3833:ISBN
3810:ISBN
3769:PMID
3679:PMID
3608:PMID
3466:PMID
3448:ISSN
3399:ISBN
3358:PMID
3156:PMID
2990:and
2726:Gini
2560:≥ 1/
2155:The
2115:1 /
1195:The
1094:1 /
495:The
3867:doi
3761:doi
3749:168
3716:doi
3663:doi
3598:PMC
3590:doi
3539:doi
3527:163
3456:PMC
3440:doi
3389:doi
3350:doi
3338:164
3304:doi
3258:doi
3246:113
3202:doi
3146:PMC
3138:doi
3033:).
2862:PIE
2583:= 2
2570:1/λ
2122:ln(
2093:ln(
1411:= 1
1383:log
1187:).
1173:= 0
1104:= 0
1079:= 0
1047:= 1
977:As
968:= 2
955:= 1
942:= 0
686:exp
598:as
592:= 1
580:or
141:to
3917::
3873:.
3863:10
3861:.
3857:.
3775:.
3767:.
3759:.
3747:.
3722:.
3712:27
3710:.
3685:.
3677:.
3669:.
3661:.
3651:52
3649:.
3628:.
3606:.
3596:.
3586:26
3584:.
3580:.
3537:.
3525:.
3521:.
3507:^
3491:^
3464:.
3454:.
3446:.
3438:.
3426:.
3422:.
3397:,
3364:.
3356:.
3348:.
3336:.
3316:^
3302:.
3292:33
3290:.
3270:^
3256:.
3244:.
3222:^
3208:.
3200:.
3190:54
3188:.
3168:^
3154:.
3144:.
3134:92
3132:.
3128:.
3028:1/
2840:.
2442:.
2393:ln
2300:ln
2210:ln
2147:.
2099:.
2088:H'
1964:ln
1847:ln
1747:ln
1679:ln
1642:ln
1611:ln
1580:ln
1512:ln
1469:ln
1413::
1399:H'
1393:.
1291:ln
1203:,
1142:.
1039:.
1024:−1
931:−1
901:.
780:.
731:ln
584:.
507:−1
271:−1
235:,
231:,
207:,
191:A
59:.
3881:.
3869::
3841:.
3818:.
3783:.
3763::
3755::
3730:.
3718::
3693:.
3665::
3657::
3632:.
3614:.
3592::
3547:.
3541::
3533::
3472:.
3442::
3434::
3428:4
3391::
3372:.
3352::
3344::
3310:.
3306::
3298::
3264:.
3260::
3252::
3216:.
3204::
3196::
3162:.
3140::
3030:D
3023:q
3016:i
3012:p
3001:i
2997:p
2953:D
2948:2
2940:1
2932:1
2929:=
2924:2
2919:i
2915:p
2909:R
2904:1
2901:=
2898:i
2887:1
2884:=
2875:1
2817:D
2812:2
2805:=
2797:2
2792:i
2788:p
2782:R
2777:1
2774:=
2771:i
2762:1
2757:=
2749:1
2717:N
2711:i
2704:i
2700:n
2679:)
2676:1
2670:N
2667:(
2664:N
2659:)
2656:1
2648:i
2644:n
2640:(
2635:i
2631:n
2625:R
2620:1
2617:=
2614:i
2603:=
2581:q
2575:D
2562:R
2558:λ
2550:i
2546:p
2540:R
2522:,
2517:2
2512:i
2508:p
2502:R
2497:1
2494:=
2491:i
2483:=
2439:q
2433:q
2415:)
2412:D
2406:q
2399:(
2390:=
2386:)
2378:1
2372:q
2365:1
2359:q
2354:i
2350:p
2344:i
2340:p
2334:R
2329:1
2326:=
2323:i
2312:1
2307:(
2297:=
2294:H
2289:q
2258:)
2252:q
2247:i
2243:p
2237:R
2232:1
2229:=
2226:i
2217:(
2203:q
2197:1
2193:1
2188:=
2185:H
2180:q
2162:q
2135:i
2131:p
2126:)
2124:R
2117:R
2109:i
2105:p
2097:)
2095:D
2082:D
2075:i
2071:p
2064:i
2060:p
2049:i
2045:p
2026:)
2016:i
2012:p
2006:i
2002:p
1996:R
1991:1
1988:=
1985:i
1976:1
1971:(
1961:=
1957:)
1947:R
1943:p
1937:R
1933:p
1922:3
1918:p
1912:3
1908:p
1900:2
1896:p
1890:2
1886:p
1878:1
1874:p
1868:1
1864:p
1859:1
1854:(
1844:=
1837:R
1833:p
1827:R
1823:p
1812:3
1808:p
1802:3
1798:p
1790:2
1786:p
1780:2
1776:p
1768:1
1764:p
1758:1
1754:p
1741:=
1734:H
1707:)
1700:R
1696:p
1690:R
1686:p
1676:+
1670:+
1663:3
1659:p
1653:3
1649:p
1639:+
1632:2
1628:p
1622:2
1618:p
1608:+
1601:1
1597:p
1591:1
1587:p
1577:(
1571:=
1564:H
1533:i
1529:p
1523:i
1519:p
1507:R
1502:1
1499:=
1496:i
1485:=
1480:i
1476:p
1464:i
1460:p
1454:R
1449:1
1446:=
1443:i
1432:=
1425:H
1409:q
1390:a
1386:b
1378:b
1372:a
1366:e
1360:e
1350:i
1343:i
1339:p
1333:i
1326:i
1322:p
1302:i
1298:p
1286:i
1282:p
1276:R
1271:1
1268:=
1265:i
1254:=
1247:H
1184:R
1178:D
1171:q
1165:S
1159:R
1139:R
1133:D
1127:q
1121:q
1115:R
1109:D
1102:q
1096:R
1088:i
1084:p
1077:q
1070:q
1063:q
1056:i
1052:p
1045:q
1036:q
1030:D
1022:q
1018:M
1012:q
1001:i
997:p
990:q
980:q
974:.
966:q
953:q
948:,
940:q
929:q
925:M
919:q
912:q
898:q
878:)
875:q
869:1
866:(
862:/
858:1
853:)
847:q
842:i
838:p
832:R
827:1
824:=
821:i
812:(
807:=
804:D
798:q
754:)
750:)
745:i
741:p
737:(
726:i
722:p
716:R
711:1
708:=
705:i
693:(
683:=
673:i
669:p
663:i
659:p
653:R
648:1
645:=
642:i
633:1
628:=
625:D
619:1
601:q
590:q
578:q
561:D
556:q
540:i
536:p
530:i
524:R
517:q
505:q
501:M
478:)
475:q
469:1
466:(
462:/
458:1
453:)
447:q
442:i
438:p
432:R
427:1
424:=
421:i
412:(
407:=
399:1
393:q
386:1
380:q
375:i
371:p
365:i
361:p
355:R
350:1
347:=
344:i
333:1
328:=
321:1
315:q
311:M
307:1
302:=
299:D
293:q
269:q
265:M
182:)
176:(
164:)
158:(
153:)
149:(
135:.
109:)
105:(
92:.
66:)
62:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
