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. relative concentration.

Illustrate with trees for SEW 2019.

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

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 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 of specialisation is \[\ln{S} - \sum_s{p_s \ln {\frac{1}{p_s}}}\]

Specialisation is opposed to diversity.

Parametric entropy (Tsallis 1988; Brülhart 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.

Clarify the links between relative and absolute diversity/ubiquity.

Deal with the MAUP.

In search for biodiversity in public gardens:

• Choose the arrondissement where public gardens are more diverse on average.

• Choose the one where they are more different from each other.

Absolute or relative diversity \(\leftrightarrow\) \(\alpha\) or \(\beta\) diversity.

Rules:

• Entropy(arrondissement) = entropy(within parks) + entropy(between parks)

• Diversity(arrondissement) = Diversity(within parks) X Diversity(between parks)

• Dimensions of diversity: # Species = # Species/park X # Parks

and, similarly:

• Spatial concentration(all trees) = concentration(within species) X concentration(between species)

Diversity/Ubiquity increases when data are aggregated:

• Scale effect of the MAUP.

• Actually increases by between diversity: not a problem but a feature.

The biodiversity of Paris trees is comparable to that of the most diverse tropical forests.

Many introduced species.

High variability:

• Streets are not diverse. Sanitary issues.

• Some parks imitate nature, some are collections.

• Cemeteries are more diverse than parks (at low \(q\)).

Diversity \(\leftrightarrow\) Specialisation.

Ubiquity \(\leftrightarrow\) Spatial concentration.

Hill numbers unify many metrics.

Key importance of the order of diversity/ubiquity:

• Theil\(\leftrightarrow q=1\);

• Ellison and Glaeser\(\leftrightarrow q=2\).

Partitioning solves the scale effect of the MAUP.