The k-nearest Neighbor Classification of Histogram- and Trapezoid-Valued Data

‎A histogram-valued observation is a specific type of symbolic objects that represents its value by a list of bins (intervals) along with their corresponding relative frequencies or probabilities‎. ‎In the literature‎, ‎the raw data in bins of all histogram-valued data have been assumed to be uniformly distributed‎. ‎A new representation of such observations is proposed in this paper by assuming that the raw data in each bin are linearly distributed‎, ‎which are called trapezoid-valued data‎. ‎Moreover‎, ‎new definitions of union and intersection between trapezoid-valued observations are made‎. This study proposes the k-nearest neighbor technique for classifying histogram-valued data using various dissimilarity measures‎. ‎Further‎, ‎the limiting behavior of the computational complexities based on the performed dissimilarity measures are compared‎. ‎Some simulations are done to study the performance of the proposed procedures‎. ‎Also‎, ‎the results are applied to three various real data sets‎. ‎Eventually‎, ‎some conclusions are stated‎.