Representing the concept of adequacy in mathematical terms can be di cult. The de ni- tion of exact thresholds of the su cient availability of an element can be challenging, due to vagueness and uncertainties associated with such linguistic concepts. Fuzzy reasoning provides a means to express the degree of membership to linguistic concepts, thus trans- lating qualitative elements into quanti able units (for details see e.g. Kropp et al., 2006; Lissner et al., 2012; Zadeh, 1965) and allowing for the consideration of inherent vagueness. By calculating the degree of membership of each variable to a common linguistic category, namely the adequacy of conditions, the diverse range of elements become comparable with regard to their contribution to ful lled AHEAD conditions.
The rst step of the analysis is the fuzzi cation of the base variables with respect to a de ned linguistic category. A function to calculate the degree of membership to the linguistic category is de ned for each variable. In the case of our analysis, the degree of membership µ of each variable to the linguistic category conditions are adequate is determined. Fuzzi ed data sets take continuous values from 0 (adequacy is very low) and 1 (adequacy is very high). For the purpose of determining the ful lment of AHEAD, fuzzy values near 0 re ect a basic level of resource availability, below which development would be compromised. Fuzzy values near 1 indicate a level of su ciency, where basic needs are fully met and conditions are adequate.
Thresholds for membership (ι1, ι2) are de ned to calculate continuous degrees of mem-
bership µziof variable ι through Eq. 3.1 (linear increase), Eq. 3.2 (linear decrease), Eq. 3.3
(exponential increase) and Eq. 3.4 (exponential decrease). For Eq. 3.3 and 3.4, the value of ϵ determines the curvature of the function. For all Equations 3.1 through 3.4 ι1 < ι2
must be true. As the values for ι1 and ι2 critically determine the membership values
for each element and thus the overall result, thresholds have to be context-speci c and re ect the properties of the available data. Threshold values and membership functions for the analysis and are discussed in detail in the following Sec. 2.3 and are summarized in Table 3.1.
Chapter 3: Climate impacts on human livelihoods µzi(ι) = 0, ι≤ ι1 ι−ι1 ι2−ι1, ι1 < ι < ι2 1, ι2 ≤ ι (3.1) µzi(ι) = 1, ι≤ ι1 ι2−ι ι2−ι1, ι1 < ι < ι2 0, ι2 ≤ ι (3.2) µzi(ι) = 0, ι≤ ι1 1 1−exp(−ϵ)× 1− exp −ϵι−ι1 ι2−ι1 , ι1 < ι < ι2 1, ι2 ≤ ι (3.3) µzi(ι) = 1, ι≤ ι1 1 1−exp(−ϵ)× 1− exp−ϵι2−ι ι2−ι1 , ι1 < ι < ι2 0, ι2 ≤ ι (3.4)
Subsequent to their fuzzi cation, variables are aggregated using context-speci c ag- gregation rules in a de ned order (Fig. 3.1).
The choice of aggregation rules should re ect the context of the analysis and be mo- tivated by the properties of the indicators. Fuzzy decision rules thus allow incorporating the content-related properties of and relationships between variables. Operators for the aggregation are de ned analogue to crisp set theory and additional fuzzy operators are available (Mayer et al., 1993). Unlike the strict application of boolean MIN or MAX operators, which result in a strict intersection or union of sets, fuzzy operators allow for compensation through a γ-value, which can take values between 0 and 1 (Equation 3.5 for fuzzy MIN; analogue quanti cation for fuzzy MAX) (Kropp et al., 2001). The intro- duction of γ results in the consideration of the arithmetic mean of all input values to some extent, thus diluting the strict application of the operator to the extent of γ, with values near to 1 resulting in a rather strict application of the operator and values near 0 introducing signi cant compensation. At γ=0 the arithmetic mean of the input values is calculated. µ(z1∧ z2∧ . . . ∧ zn) = γ× min(µz1, µz2, . . . , µzn) + (1− γ) × 1 N N i=1 µzi (3.5) 37
Chapter 3: Climate impacts on human livelihoods
Figure 3.1: Overview of the fuzzy aggregation tree to calculate AHEAD. Detailed expla- nations of each variable as well as the aggregation procedures are given in Sections 2.2 and 2.3.
To assess the ful lment of AHEAD, the characteristics of the contributing elements as well as their relationships determine the rules and order of aggregation, as outlined in Figure 3.1. Initially, the three dimensions of Subsistence, Infrastructure and Societal Structure are aggregated individually. An essential property of the elements of the Subsis- tence dimension is that they are non-substitutable: if one of the elements water, food or clean air is not available, it poses a direct threat to human health and well-being. Indica- tors within this dimension are therefore aggregated using a strict MIN operator with γ=1 (left column of Fig. 3.1). Elements relevant for the Societal Structure dimension, how- ever, may to some extent be substitutable. Low availability of one resource may to some extent be compensated with the high availability of another, which is re ected in using the arithmetic mean (γ=0) (right column of Fig. 3.1). While those elements included in the Infrastructure dimension are not substitutable in a physical sense, high values in one of these domains imply high levels of technological advancement, which motivates the use of the arithmetic mean here (middle column of Figure 3.1). The nal aggregation of the three dimensions to the full index of AHEAD re ects the fact that all three components are required to attain adequate conditions. We aggregate the dimensions Infrastructure and Societal Structure using a fuzzy MIN operator with γ = 0.6. This use of γ accounts for the fact that levels of adequacy in both dimensions are required for ful lled livelihoods, but fully adequate conditions in one area may compensate other de ciencies to the extent of γ. While the order of magnitude and likely ranges of γ can be motivated by the context,
Chapter 3: Climate impacts on human livelihoods
the exact value is to some extent arbitrary within the in the global implementation of the approach. The subsequent aggregation of all dimensions to a measure of AHEAD is performed using a strict MIN operator (γ = 1), again re ecting the non-substitutability of the Subsistence domain.