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TRUE DIVERSITY 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 first taking the weighted generalized mean of the proportional abundances of the types in the dataset, and then taking the inverse of this.

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The equation is:[3][4] The denominator equals average proportional abundance of the types in claude jaccard swiss anti aging dataset as calculated with the weighted generalized mean with exponent q 1. In the equation, R is richness the total number of types in the datasetand the proportional abundance of the ith type is.

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The proportional abundances themselves are used as the nominal weights. It defines the sensitivity of the diversity value to rare vs.

With some values of the parameter q, the generalized mean with exponent q 1 gives familiar kinds of mean as special cases.

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As q approaches infinity, the generalized mean with exponent q 1 approaches the maximum value, which is the proportional abundance of the most abundant species in the dataset. In practice, increasing the value of q hence increases the effective weight given to the most abundant species. This leads to obtaining a larger mean value and a smaller true diversity qD value.

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In the context of diversity, q is generally limited to non-negative values. This is because negative values of q would give rare species so much more weight than abundant ones that would exceed R. The general equation of diversity is often written in the form:[1][2] The term inside the parentheses is called the basic sum.

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Some popular diversity indices correspond to the basic sum as calculated with different values of q. For example, species richness usually notated S of a dataset is the number of different species in the corresponding species list.

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Richness is a simple measure, so it has been a popular diversity index in ecology, where abundance data are often not available for the datasets of interest. Because richness does not take the claude jaccard swiss anti aging of the types into account, it is not the same thing as masti naturale pentru fata si ochi, which does take abundances into account.

The measure was originally proposed by Claude Shannon to quantify the entropy uncertainty or information content in strings of text.

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The Shannon entropy quantifies the uncertainty entropy or degree of surprise associated with this prediction. It is most often calculated as follows: where is the proportion of characters belonging to the ith type of letter in the string of interest. In ecology, is often the proportion of individuals belonging to the ith 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.

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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 e, 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 have been called binary digits bitsdecimal digits decits and natural digits nats for the bases 2, 10 and e, respectively.

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Comparing Shannon entropy values that were originally calculated with different log bases requires converting them to the same log base: change from the base a to base b is obtained with multiplication by logba. The term within the parentheses hence equals true diversity 1D, and H' equals ln 1D.

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The more unequal the abundances of the types, the larger the weighted geometric mean of the 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 themShannon entropy approaches zero. When there is only one type in the dataset, Shannon entropy exactly equals zero there is no uncertainty in predicting the type of the next randomly chosen entity.

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It can be expressed: which equals This means that taking the logarithm of true diversity based on any value of q gives the Rnyi entropy corresponding to the same value of q. Simpson to measure the degree of concentration when individuals are classified into types. Herfindahl in The square root of the index had already been introduced in by the economist Albert O. The original Simpson's index hence equals the corresponding basic sum.

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If the dataset is very large, sampling without replacement gives approximately 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: where is the number of entities belonging to the ith type and N is the total number of entities in the dataset.

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This is counterintuitive behavior for a diversity index, so often such transformations of that increase with increasing diversity have been used instead. Its transformation 1 therefore equals the probability that the two entities represent different types.

This measure is also known in ecology as the probability of interspecific encounter PIE [10] and the GiniSimpson index. This corresponds to the weighted generalized mean of the values when q approaches infinity, and hence equals the inverse of true diversity of order infinity SEE.

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