tailieunhanh - Data Analysis Machine Learning and Applications Episode 3 Part 7

Tham khảo tài liệu 'data analysis machine learning and applications episode 3 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A New Interval Data Distance Based on the Wasserstein Metric 707 dTD A B f f aA x b - a - u v y v - u 2dxdy -1 2-1 2 1 a b u v 2 I 1 b-a 2 I v-u 2 2 2 3 2 2 In practice they consider the expected value of the distance between all the points belonging to interval A and all those points belonging to interval B. In their paper they ensure that it is a distance but it is easy to observe that the distance does not satisfy the first properties mentioned above. Indeed the distance of an interval by itself is equal to zero only if the interval is thin 17 a b a b M 2 _1_ 1 r Í b-a 2 Í b-a 21 2 Í b-a 2 dTD A A Í- 3HV 2 0 2 Hausdorff-based distances. The most common distance used for the comparison of two sets is the Hausdorff distance 2. Considering two sets A and B of points of Rn and a distance d x y where x G A and y G B the Hausdorff distance is defined as follows dH A B max sup inf d x y sup inf d x y 3 xeAyGB yGBxGA J If d x y is the L1 City block distance then Chavent et al. 2002 proved that dH A B max a - u b - v I a b - u r I I I- - -1 4 An analytical formulation of this metric using the Euclidean distance has been devised Book 2005 . Lq distances between the bounds of intervals. A family of distances between intervals has been proposed by De Carvalho et al. 2006 . Considering a set of interval data described into a space Rp the metric of norm q is defined as _ A1 q dLq A B I 12 a - u q b - v q 5 1 1 J They also showed that if the norm is L then d dH in L1 norm . The same measure was extended De Carvalho 2007 to an adaptive one in order to take into account the variability of the different clusters in a dynamical clustering process. 3 Our proposal Wasserstein distance If we suppose a uniform distribution of points an interval of reals A t a b can be expressed as the following type of function 2 The name is related to Felix Hausdorff who is well-known for the separability theorem on topological spaces at the end of the 19th century. 708 Rosanna Verde and Antonio .