Description Usage Arguments Author(s) Source Examples

This function provides the values of the second derivative of the beta profile. The second derivative represents the deceleration of the profile for any given value of "beta". For "beta" close to -1, high values of the second derivative denote high dominance because the profile tends to be extremely curved when some prevalent species are present in a community. For the interval [0,1] the second derivative tends always towards zero.

1 | ```
betasecond(x,n)
``` |

`x` |
A data matrix with the abundance of the species (the rows are the communities and the columns indicate the species living in a community). The user can freely choose to use the absolute or relative frequencies. |

`n` |
The number of points of the domain [-1,1] that the user wants to consider in computing the functional tools. It is suggested to choose a number in the interval [3,10000] because a value of less than 3 has little sense; indeed, the major indices of biodiversity are the richness index, the index of Shannon and the index of Simpson (that we get for beta equal to -1, 0, +1, respectively). On the other hand, a value greater than 10000 is not very interesting because with a value of approximately 100 we already have a good approximation. The examples provided in this package always divide the domain into 20 intervals. The default value is n=20. |

Fabrizio Maturo, Francesca Fortuna, Tonio Di Battista

ENVIRONMENTAL MONITORING THROUGH FUNCTIONAL BIODIVERSITY TOOLS - T. Di Battista, F. Fortuna, F. Maturo - Ecological Indicators (2016) 60, 237-247 - online (2015). DOI: 10.1016/j.ecolind.2015.05.056.

PARAMETRIC FUNCTIONAL ANALYSIS OF VARIANCE FOR FISH BIODIVERSITY ASSESSMENT - T. Di Battista, F. Fortuna, F. Maturo - Journal of Environmental Informatics. Special issue of Marine and Freshwater Environmental Engineering and Management, Journal of Environmental Informatics. In press.

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