Some Derivations and Criticisms

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Simpson Index
$\backslash sum\_i\{p\_i^2\}$
with
$p\_i$ =
$n\_i/\backslash sum\_i\{n\_i\}$
and
$n\_i$ is
the abundance of species
$i$
derivation: the probability to choose a specific species
twice is
$p\_i*p\_i$ and
thus the probability that any two chosen specimens belong both to
any species
$i$ is the sum
over all species
$i$.
criticism: almost none, it is easy to understand and, as a
true probability, bounded between 0 and 1. Of course we shouldn't
interprete the "choosing" of a specimen as catching one (this has
definitely a different probability), but rather say that the
sample is representative for the population, what is still
questionable but difficult to avoid.
Rarefaction$$
$E(S)\; =\; \backslash sum\_i(\{1\backslash begin\{pmatrix\}\; N\; N\_i\; \backslash \backslash \; n\backslash end\{pmatrix\}\; /\; \backslash begin\{pmatrix\}\; N\; \backslash \backslash \; n\; \backslash end\{pmatrix\}\})$
where
$N\_i$ is the
number of specimens for species $i$ in a
population,
$N$ is the total
number of specimens and $n$ the size of
the sample (in specimens)$N$
$$
$$$N$derivation:
the expectation value for the number of species to choose from an
abundance vector is the sum over the probabilities of all species i
in the population, which in turn is one minus the probability to
miss the species i. The probability to miss a species i is equal to
the number of possible combinations in the sample without that
species $\backslash begin\{pmatrix\}\; N\; \; N\_i\; \backslash \backslash \; n\; \backslash end\{pmatrix\}$
divided by the number of possible combinations with that
species $\backslash begin\{pmatrix\}\; N\; \backslash \backslash \; n\; \backslash end\{pmatrix\}$.
criticism: we have to assume both, that the abundance
distribution of our measured sample is representative for the
distribution in the population and that choosing a specimen from the
population is a Laplace experiment (all elemenary probabilities are
equal). Both assumptions don't hold, and so applications of a proper
rarefaction require that we know how close we are to saturation in
advance, and thus are circular. What is specially dangerous, is that
every rarefaction curve shows some sort of convergence, like in this
study. And ceterum censeo, I don't see how Monte Carlo
methods like Jackknifing make sense, if we have an analytical
formula.
Entropy
$entropy\; =\; 1*\backslash sum\_i\{p\_i*\backslash ln\backslash left(\{p\_i\}\backslash right)\}$
derivation: coming soon
criticism: the usually given formula repesents the entropy of
an ideal mixture, like, e.g. noble gases, which behave like an ideal
gas
$pV\; =\; nRT$. It is
just the ideal gas constant $R$ which is
missing. Whith interacting species, like polar molecules, this
formula doesn't make much sense and if we deal with a chemical
reaction, the entropy of mixture has usually a negligible
contribution. And flies do react with each other... (The same
arguments hold if we choose Boltzmann's microstates for a
derivation, or if we rename the term entropy to the thereby defined
"information content".)
Evenness
$entropy\; /\; max(entropy)$ =
$\backslash sum\_i\{p\_i*\backslash ln\backslash left(\{p\_i\}\backslash right)\}\; /\; P*\backslash ln\backslash left(\{P\}\backslash right)$ where
$p\_i$ is again
the probability to choose species $i$ in the
population and
$P$ is the
probability to choose species $i$ if all
species where equally abundant
derivation: none, once entropy is defined
criticism: even if the concept of entropy in ecology is
questionable, the evenness , as a number between 0 and 1, gives a
feeling how close a distribution is to the case where all species
are equally abundant.
Chao1
$S\_\{expected\}\; =\; S\_\{obs\}\; +\; f\_1^2/(2*f\_2)$
where $f\_1$ is the
number of singletons in a sample (the number of species caught only
once) and $f\_2$ the number
of doubletons
derivation: Chao, A. (1984) Nonparametric estimation of the
number of classes in a population. Scandinavian Journal of
Statistics, 11, 265–270.
criticism: not yet