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Distance Measures for Quantifying the Differences in Microstructures
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
P and Q are the normalized histograms representing the distributions being compared, and N is the number of bins in each histogram [91]. The Hellinger distance is a true metric, whereas the Bhattacharyya coefficient is not, as it does not satisfy the triangle inequality. The Hellinger distance is convenient as it is straightforward to compute and interpret. Since it is a bin-by-bin similarity measure, we expect that d(P1, P3), and d(P1, P4) should be 1.0, indicating that they are completely different distributions and do not overlap. This is indeed the case as indicated in Table 17.1. This also illustrates one limitation of bin-by-bin measures; P1 and P3 are identical except for a lateral shift, while P4 is not identical to P1. Even so, the Hellinger distance, and any other bin-by-bin distances, do not give any indication that P3 is more like P1 than P4.
Document Clustering: The Next Frontier
Published in Charu C. Aggarwal, Chandan K. Reddy, Data Clustering, 2018
David C. Anastasiu, Andrea Tagarelli, George Karypis
The Hellinger distance is a metric directly derived from the Bhattacharyya coefficient [57], which offers an important geometric interpretation in that it represents the cosine between any two vectors that are composed by the square root of the probabilities of their mixtures. Formally, the Hellinger distance is defined as HL(p,q)=1−BC(p,q) where BC(p,q)=∑i=1Rp(xi)q(xi) is the Bhattacharyya coefficient for the two PMFs p and q.
Moving Fast and Slow: Analysis of Representations and Post-Processing in Speech-Driven Automatic Gesture Generation
Published in International Journal of Human–Computer Interaction, 2021
Taras Kucherenko, Dai Hasegawa, Naoshi Kaneko, Gustav Eje Henter, Hedvig Kjellström
Figure 5a presents speed histograms across all joints. We observe no difference between using different features on this scale. Since the results in Figure 5a are averaged over all joints, they do not indicate whether all the joints move naturally. To address this we also analyze the speed distribution for the elbows (Figure 5b) and wrists (Figure 5c). Hands convey the most important gesture information, suggesting that these plots are more informative. To aid comparison we complement the visuals by computing a numerical distance measure between the different histograms and the ground truth. We use the Hellinger distance , which is a distance metric between two probability distributions. For two normalized histograms and , it is defined by:
Estimation of urban arterial travel time distribution considering link correlations
Published in Transportmetrica A: Transport Science, 2020
Wenwen Qin, Xiaofeng Ji, Feiwen Liang
In order to confirm the fitting performance of the non-parametric approach, the Hellinger distance (HD) are employed to assess the similarity between the estimated distribution and the empirical (actual) one. For discrete distributions, the HD between the estimated distribution and the empirical one is defined as (Spiegelhalter et al. 2002): where, and are the estimated and observed probabilities for an observation . Besides, the HD satisfies two requirements in Equation (21).
Data-driven distributionally robust risk parity portfolio optimization
Published in Optimization Methods and Software, 2022
Next, we present the square of the Hellinger distance The Hellinger distance is a proper distance metric and, by its definition, is bounded between zero and one. We define as the squared Hellinger distance to improve computational tractability in practice.