Environmental monitoring through biodiversity functional measures

Environmental monitoring through biodiversity functional measures T. Di Battista, F. Fortuna and F. Maturo Department of Philosophical, Pedagogical and Quantitative-Economics Sciences - University G. d Annunzio Chieti - Pescara Conference of European Statistics Stakeholders Methodologists, Producers and Users of European Statistics Rome, November 24-25, 2014

Biodiversity assessment Biodiversity and environmental quality Biodiversity is one of the most important indicator of environmental health (Burger et al., 2013) since it decreases in relation to ecosystem stressors. Its measurement is essential for predicting future biological impacts of environmental damages. In this context, the arising problem is to provide a suitable measure of biodiversity in order to quantify its decreased in relation to increasing human stressors. Although many indices have been proposed (Gove et al., 1994), nowadays there is not yet a universally accepted biodiversity measure. Indeed, different community ranking can be obtained according to the diversity index under consideration (Patil and Taillie, 1982). For this reason, the use of a single indicator greatly reduces the complexity of the ecological systems and hides the multidimensional aspect of biodiversity.

Biodiversity assessment Diversity profiles Fort his reason, it is desirable the use of parametric families of diversity indices (Hill, 1973; Patil and Taillie, 1982), which are usually referred to as diversity profiles. A diversity profile is a non-negative, monotone decreasing and convex curve which expresses diversity as a function of the relative abundance vector. The profile portrays the simultaneous values of a large collection of diversity indices in a single diversity spectrum (Patil and Taillie, 1982). Diversity profile plays a fundamental role in comparing different community using the graphical analysis. Indeed, communities with intersecting profiles are not comparable. Problem: the profiles, generally, intersect one or more times in real cases. In this context, it is suitable the introduction of additional tools to improve the analysis of biodiversity profile.

Analysis of derivatives Curvature and Radius of curvature Arc length Functional tools for biodiversity assessment We focus on the use of diversity profiles to assess biodiversity using functional data analysis approach (Di Battista and Gattone, 2009; De Sanctis and Di Battista, 2012; Di Battista and Fortuna, 2013). FDA refers to the analysis of functions in a fixed domain and allows to consider the functional datum as a single entity, rather than a sequences of observations (Ramsay and Silverma, 2005; Ferraty and View, 2006). In particular, we propose three functional tools to inspect diversity profile characteristics in addition to its graphical properties: 1 the derivatives; 2 the radius of curvature; 3 the length of the diversity profile. The combined use of these indicators allows to consider the multidimensional aspect of diversity and to establishes an index-free community ranking to identify critical areas.

Analysis of derivatives Curvature and Radius of curvature Arc length β diversity profiles In particular we refer to the β diversity profile proposed by Patil and Taillie (1982): β = s i=1 (1 p β i ) β p i = 1 s i=1 pβ+1 i β β 1 (1) where: p = (p 1,..., p s) = the relative abundance vector of the i-th species (i = 1, 2,.., s); the value of β denotes the relative importance of richness and evenness Some of the most frequently used indices of diversity are special cases of equation (1); in fact for β = 1 we get the richness index, for lim β 0 the Shannon diversity index and for β = 1 the Simpson index.

Analysis of derivatives Curvature and Radius of curvature Arc length Graphical characteristics of β diversity profiles From an ecological point of view, β diversity profiles presents some graphical characteristics: it is an high curve in case of maximum diversity while it corresponds to the β axis in the opposite case; it tends to a straight line decreasing from 1 to 1 in case of maximum equitability with few species; it is more curve in case of prevalence of few species on the others. According to these aspects, we propose to study the shape of the β profile by analyzing its derivatives, its radius of curvature and its length providing additional information about biodiversity and allowing comparison of different communities.

Analysis of derivatives Curvature and Radius of curvature Arc length Analysis of β profile derivatives The graphical analysis of β profile derivatives highlights the behavior of the slope and of the deceleration of the profile. The first derivatives indicates the presence of dominance or evenness in a community. For β close to 1 great absolute values of it correspond to a strong decrease of the profile and, thus, to great dominance and vice versa. The second derivative represents the deceleration of the profile for any given value of β. For β close to 1 high values of it denote high dominance because β tends to be very convex if there are some prevalent species in a community. In the second part of β domain (0 β 1) the first and the second derivatives tends to zero because the diversity profile tends to be constant.

Analysis of derivatives Curvature and Radius of curvature Arc length Curvature and Radius of curvature Taking advantage of the FDA approach, we propose the use of an index of curvature for the analysis of different communities. Intuitively, the curvature is the amount by which a geometric object deviates from being flat. Given a curve C, the curvature at a given point P, is defined as the curvature of the circumference, said osculating circle, that best approximates the curve near P. For a given function, f (x), the curvature k is given by: k = f (x) (1 + (f (x) 2 )) 3/2 (2) and the radius of curvature R is defined as the reciprocal of the curvature.

Analysis of derivatives Curvature and Radius of curvature Arc length Curvature and Radius of curvature From an ecological point of view, the graphical analysis of the radius of curvature (or of the curvature) reflects the community composition since it is a good indicator of the lack of balance among species. In the first part of the domain ( 1 β 0), β tends to be more convex in case of high dominance; so the radius of curvature is very small.

Analysis of derivatives Curvature and Radius of curvature Arc length Length of a curve The arc length represents the length of a curve if it were rectified. Let f be a continuosly differentiable function on the closed interval [a, b]; then the arc length of f from x = a to x = b is the integral: b ( 2 L = 1 + f (x)) dx (3) a In an ecological framework, the arc length allows to rank communities given the number of species by means of a scalar measure; in fact it is very high in case of maximum dominance and vice versa. The main advantage in using profile length is that the ordering among communities can be investigated without the graphical analysis. Moreover, it provides a scalar measure of diversity preserving its multidimensional aspect.

Lichen biodiversity Biodiversity of epiphytic lichens in the province of Genoa The functional tools proposed have been applied to a real data set concerning epiphytic lichen biodiversity of the province of Genoa in Liguria region, northwestern Italy. Lichen biodiversity provides useful information about the global conditions affecting the environment over a given area (Giordani et al., 2002). These organisms, in fact, are particularly sensitive to environmental stresses, especially with regard to pollution and climate change. Data on lichen abundance have been collected following the standards suggested by Asta et al. (2002). The survey lasted from 2002 to 2003 and involves a total of 196 epiphytic lichen species and 47 plots.

Lichen biodiversity Biodiversity of epiphytic lichens in the province of Genoa: β profile

Lichen biodiversity β profiles for the sites with s = 8 in the province of Genoa Following we focus on comparing habitats with the same number of species. The β profile provides a clear ranking among sites with the same richness except for s = 8 and s = 19.

Lichen biodiversity First derivative of β for the sites with s = 8 For β close to 1, the first derivatives highlight greater evenness for the site number 40 compared to the number 11 (low absolute value of its first derivative).

Lichen biodiversity Second derivative of β for the sites with s = 8 in the province of Genoa The second derivative emphasizes that the site number 11 has greater dominance respect to the number 40 because its deceleration is higher.

Lichen biodiversity Radius of curvature of β for the sites with s = 8 in the province of Genoa The radius of curvature shows that in the communities number 11 and 40 there is greater lack of balance among species. This aspect is particularly evident for sites number 40 which reaches the lowest value of R for β = 0.2.

Lichen biodiversity Profile length for the sites with s = 8 in the province of Genoa Although functional tools have been used, it is clear that the graphical analysis often involves problems of interpretation due especially to the complexity of the phenomenon under study. In this context, a useful functional tool to rank sites with the same richness is the profile length. The profiles, then, can be ranked according to their length even if they intersect each other. Indeed, a profile with high diversity presents a lower length and vice versa. For the sites with s = 8, the profile lengths allows to establish a unique ranking among them from the higher to the less diverse: 1 site number 21: L=6.8384; 2 site number 11: L=6.9509; 3 site number 40: L=6.9942.

Lichen biodiversity Ranking of the sites according to richness and profile length in the province of Genoa Rank Plot code Richness Profile length Rank Plot code Richness Profile length 1 27 40 38.4157 26 8 17 15.6684 2 26 36 34.4174 27 37 14 12.5495 3 42 34 32.4487 28 39 14 12.5846 4 47 33 31.4504 29 9 13 11.6432 5 10 32 30.4621 30 19 13 11.8387 6 28 29 27.4367 31 5 10 8.6606 7 29 29 27.4512 32 20 10 8.7244 8 32 28 26.4499 33 34 10 8.8810 9 35 28 26.5409 34 21 8 6.8384 10 17 27 25.4591 35 11 8 6.9509 11 7 25 23.4555 36 40 8 6.9942 12 23 24 22.4775 37 12 7 5.8335 13 6 23 21.4639 38 22 7 5.9286 14 44 23 21.4944 39 18 6 4.8915 15 45 23 21.5292 40 2 6 5.0736 16 13 22 20.4904 41 3 6 5.0956 17 16 20 18.5266 42 33 6 5.1609 18 15 19 17.4987 43 38 5 4.0491 19 41 19 17.5214 44 4 3 2.5035 20 30 19 17.5259 45 14 3 3.0939 21 31 19 17.5284 46 1 1 2.0100 22 25 19 17.5445 46 36 1 2.0100 23 46 18 16.5558 24 43 18 16.5564 25 24 18 16.5736

Lichen biodiversity Conclusions Biodiversity assessment is a key component of environmental sustainability. One of the most common biodiversity measure is the diversity profile; however it presents two critical aspects: 1 it is not always easily interpretable about the communities composition; 2 it is based on the graphical analysis and do not lead to a ranking if the profiles intersect each other. Our aim is to provide additional models for biodiversity profile evaluation by mean of functional data analysis and, in particular, using three tools: derivatives, radius of curvature and arc length. The first two looks for solving the first drawback; while the arc length focuses on solving the second criticism.

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