May, 24th 2019

Motivation

Methodological exercise

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.

Aims

Show similar approaches in different disciplines.

Discuss economic issues:

  • MAUP,

  • absolute vs. relative concentration.

Illustrate with trees for SEW 2019.

Data

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.

Most abundant species

Hybrid plane tree, horse chestnut, scholar tree, lime tree, Norway maple, sycamore maple…

Species Number of Trees
Platanus x hispanica 33297
Aesculus hippocastanum 15507
Sophora japonica 10714
Tilia tomentosa 6987
Acer platanoides 4423
Acer pseudoplatanus 4417

Species distribution

Measures of diversity

Aims

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

Equivalent in economics: measure the specialisation of Paris arrondissements.

Information theory is appropriate.

Measuring Uncertainty

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 Shannon’s information function.

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

Theil’s index

Theil’s index of specialisation is \[\ln{S} - \sum_s{p_s \ln {\frac{1}{p_s}}}\]

Specialisation is opposed to diversity.

Generalized Entropy

Parametric entropy (Tsallis 1988; Brülhart and Traeger 2005)

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

Formalism

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).

Hill Numbers

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

Diversity of Parisian green spaces

Richness of arrondissements

Shannon’s Diversity

Simpson’s Diversity of arrondissements

Measures of spatial concentration

Similar approach

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.

Ubiquity of abundant species