May, 24th 2019

Recent developments in biodiversity measurement (Marcon et al. 2014).

Possible interfertilisation with measures of specialisation and spatial concentration in economics (Marcon 2019).

Applied to Paris trees data: explore the biodiversity of trees.

Urban trees provide many services (Taylor and Hochuli 2015). More diversity: more amenities and more resistance.

Show similar approaches in different disciplines.

Discuss economic issues:

MAUP,

absolute vs.Â relative concentration.

Illustrate with trees for SEW 2019.

Paris open data: https://opendata.paris.fr

Dataset: 200,000 trees in Paris located in 20 arrondissements.

Streets, Beltway,

Public gardens, Cemeteries,

Schools, Nurseries, Sports facilities.

Reproducible research: on GitHub.

Made with R, knitr, sp, entropart.

Hybrid plane tree, horse chestnut, scholar tree, lime tree, Norway maple, sycamore mapleâ€¦

Species | Number of Trees |
---|---|

Platanus x hispanica | 33297 |

Aesculus hippocastanum | 15507 |

Sophora japonica | 10714 |

Tilia tomentosa | 6987 |

Acer platanoides | 4423 |

Acer pseudoplatanus | 4417 |

We want to measure the diversity of trees in Paris arrondissements.

Equivalent in economics: measure the specialisation of Paris arrondissements.

Information theory is appropriate.

An experiment with several outcomes \(\{r_1, \dots, r_s, \dots, r_S \}\):

- The probability to obtain \(r_s\) is \(p_s\).

Information function: \(I(p_s)>0, p_s\in [0, 1[\), decreasing to \(I(1)=0\).

Definition: rarity is \(1/p_s\).

The logarithm of rarity is Shannonâ€™s information function.

The expectation of the information carried by an individual is Shannonâ€™s entropy: \[\sum_s{p_s \ln {\frac{1}{p_s}}}\]

Theilâ€™s index of specialisation is \[\ln{S} - \sum_s{p_s \ln {\frac{1}{p_s}}}\]

Specialisation is opposed to diversity.

Parametric entropy (Tsallis 1988; BrĂĽlhart and Traeger 2005)

Deformed logarithm: \(\ln_q x = \frac{x^{1-q} -1}{1-q}\)

Tsallis entropy \(\sum_s{p_s \ln_q {\frac{1}{p_s}}}\) is the average (deformed, of order \(q\)) logarithm of rarity.

The order \(q\) stresses small or high probabilities.

Entropy of order 0: the number of possible outcomes (-1), called

*richness*.Entropy of order 1: Shannon (\(\ln{S}-\) Theil)

Entropy of order 2: Simpson (1-Herfindahl).

The number of equiprobable outcomes that have the same entropy as the observed system (Hill 1973). Exponential of entropy (Marcon et al. 2014).

We want to measure the spatial ubiquity of species.

Consider a species.

The event \(r_i\) *A tree of arrondissement \(i\) belongs to the species* has probability \(p_i\).

Ubiquity is the effective number of arrondissements the species is present in.

Spatial concentration is opposed to ubiquity.