eventstar | R Documentation |

The function `eventstar`

finds the minimum (`q^*`

) of the
evenness profile based on the Tsallis entropy. This scale factor
of the entropy represents a specific weighting of species
relative frequencies that leads to minimum evenness of the
community (Mendes et al. 2008).

```
eventstar(x, qmax = 5)
```

`x` |
A community matrix or a numeric vector. |

`qmax` |
Maximum scale parameter of the Tsallis entropy to be used in
finding the minimum of Tsallis based evenness
in the range |

The function `eventstar`

finds a characteristic value of the scale
parameter `q`

of the Tsallis entropy corresponding to
minimum of the evenness (equitability) profile based on Tsallis entropy.
This value was proposed by Mendes et al. (2008) as `q^*`

.

The `q^\ast`

index represents the scale parameter of
the one parameter Tsallis diversity family that leads to
the greatest deviation from the maximum equitability given the relative
abundance vector of a community.

The value of `q^\ast`

is found by identifying the minimum
of the evenness profile over scaling factor `q`

by
one-dimensional minimization. Because evenness profile is
known to be a convex function, it is guaranteed that underlying
`optimize`

function will find a unique solution
if it is in the range `c(0, qmax)`

.

The scale parameter value `q^\ast`

is used to
find corresponding values of diversity (`H_{q^\ast}`

),
evenness (`H_{q^\ast}(\max)`

),
and numbers equivalent (`D_{q^\ast}`

). For calculation
details, see `tsallis`

and Examples below.

Mendes et al. (2008) advocated the use of `q^\ast`

and corresponding diversity, evenness, and Hill numbers, because
it is a unique value representing the diversity profile, and is
is positively associated with rare species in the community,
thus it is a potentially useful indicator of certain
relative abundance distributions of the communities.

A data frame with columns:

`qstar` |
scale parameter value |

`Estar` |
Value of evenness based on normalized Tsallis
entropy at |

`Hstar` |
Value of Tsallis entropy at |

`Dstar` |
Value of Tsallis entropy at |

See `tsallis`

for calculation details.

Values for `q^\ast`

found by Mendes et al. (2008) ranged
from 0.56 and 1.12 presenting low variability, so an
interval between 0 and 5 should safely encompass
the possibly expected `q^\ast`

values in practice,
but profiling the evenness and changing the value of
the `qmax`

argument is advised if output values
near the range limits are found.

Eduardo Ribeiro Cunha edurcunha@gmail.com and Heloisa Beatriz Antoniazi Evangelista helobeatriz@gmail.com, with technical input of Péter Sólymos.

Mendes, R.S., Evangelista, L.R., Thomaz, S.M.,
Agostinho, A.A. and Gomes, L.C. (2008) A unified
index to measure ecological diversity and species
rarity. *Ecography* **31**, 450–456.

Jost, L. (2007) Partitioning diversity into independent alpha and beta components.
*Ecology* **88**, 2427–2439.

Tsallis, C. (1988) Possible generalization of Boltzmann-Gibbs statistics.
*J. Stat. Phis.* **52**, 479–487.

Tsallis entropy: `tsallis`

```
data(BCI)
(x <- eventstar(BCI[1:5,]))
## profiling
y <- as.numeric(BCI[10,])
(z <- eventstar(y))
q <- seq(0, 2, 0.05)
Eprof <- tsallis(y, scales=q, norm=TRUE)
Hprof <- tsallis(y, scales=q)
Dprof <- tsallis(y, scales=q, hill=TRUE)
opar <- par(mfrow=c(3,1))
plot(q, Eprof, type="l", main="Evenness")
abline(v=z$qstar, h=tsallis(y, scales=z$qstar, norm=TRUE), col=2)
plot(q, Hprof, type="l", main="Diversity")
abline(v=z$qstar, h=tsallis(y, scales=z$qstar), col=2)
plot(q, Dprof, type="l", main="Effective number of species")
abline(v=z$qstar, h=tsallis(y, scales=z$qstar, hill=TRUE), col=2)
par(opar)
```

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