Classification of Damage Functions Sample Clauses

Classification of Damage Functions. As mentioned above, Fig. 7 shows all 140 damage functions (normalised by their maximum damage value at a 10 m flood level) composing our proposed library. As can be seen at first glance, the functions exhibit a wide range of shapes. While some curves indicate a convex increase over the whole range of sea levels, others show a saturation effect at high flood levels. The functions vary considerably in regard to the surge value for which damages start to pick-up. For example, a hypothetical surge of 8 metres results in some urban clusters in the loss of about 20% of the maximum damages. This means that 80% of the damages are expected to occur between an 8 and 10 metre surge. In contrast, some curves start with very high damages at moderate sea levels and exhibit only a low damage increase above 6 m flood height. In summary, the curves exhibit different characteristics within certain ranges of flood levels. It is the goal of this section to characterise these behaviours and to categorise the functions into groups of similar characteristics. By doing so, the huge variety of curves shall be disentangled. In order to differentiate the fundamental shapes of the curves, we will use a k-means clustering. The approach requires a distance measure between curves, which we define as the sum of the squared differences at all data points (x = 0 m, 0.5 m, 1 m, .. ., 10 m). I.e. we define the distance d between two functions f and g as d(f, g) := Σ · − · (f (0.5 k) g(0.5 k))2 (13) = (f (0) − g(0))2 + (f (0.5) − g(0.5))2 + ... + (f (9.5) − g(9.5))2 + (f (10) − g(10))2 . As a visual inspection suggests, we assume the existence of two large groups of functions and accordingly we set k = 2 in order to separate the bunch of curves. The resulting groups of functions are shown in Fig. 8. In total, 92 curves are assigned to group 1 and 48 to group 2. The two group means exhibit different characteristics. While the mean of group 1 is a convex curve, the mean of group 2 shows a slight saturation effect for high flood levels and suggests a sigmoidal shape. Accordingly, we will refer to the two groups of functions as following as convex and sigmoidal, respectively. The difference between the two groups can interpreted as follows. While the curves in both cases follow an increase of a certain exponent, the sigmoidal functions experience a saturating effect for sea levels below 10 m. The convex functions are also expected to reach such a saturation, this cannot be detected below 10 m sur...