TY - JOUR

T1 - Histogram Ordering

AU - Sbert, Mateu

AU - Ancuti, Cosmin

AU - Ancuti, Codruta O.

AU - Poch, Jordi

AU - Chen, Shuning

AU - Vila, Marius

N1 - Funding Information:
This work was supported in part by the Romanian Ministry of Education and Research, CNCS—UEFISCDI (PNCDI III), under Project PN-III-P1-1.1-TE-2019-1111 and Project PN-III-P2-2.1-PED-2019-2805. The work of Mateu Sbert and Jordi Poch was supported in part by the Spanish Ministry for Science and Innovation under Grant PID2019-106426RB-C31.
Publisher Copyright:
© 2013 IEEE.

PY - 2021

Y1 - 2021

N2 - Frequency histograms are ubiquitous, being practically used in any field of science. In this paper, we present a partial order for frequency histograms and, to our knowledge, no order of this kind has been yet defined. This order is based on the stochastic order of discrete probability distributions and it has invariance properties that make it unique. First, we model a frequency histogram as a sequence of bins associated with a discrete probability (or relative frequency) distribution. Then, we consider that two histograms are ordered if they are defined on the same sequence of bins and their respective frequency distributions are stochastically ordered. The ordering can be easily spotted because the respective cumulative distribution functions of the frequencies of two ordered histograms do not cross each other. Finally, with each bin we can associate a representative value of the bin, and for two ordered histograms it holds that all quasi-arithmetic means (such as arithmetic, harmonic, and geometric mean) of the representative values weighted by the frequencies are ordered in the same direction than the histograms are. Our theoretical study is supported by three experiments in the fields of image processing, traffic flow, and income distribution.

AB - Frequency histograms are ubiquitous, being practically used in any field of science. In this paper, we present a partial order for frequency histograms and, to our knowledge, no order of this kind has been yet defined. This order is based on the stochastic order of discrete probability distributions and it has invariance properties that make it unique. First, we model a frequency histogram as a sequence of bins associated with a discrete probability (or relative frequency) distribution. Then, we consider that two histograms are ordered if they are defined on the same sequence of bins and their respective frequency distributions are stochastically ordered. The ordering can be easily spotted because the respective cumulative distribution functions of the frequencies of two ordered histograms do not cross each other. Finally, with each bin we can associate a representative value of the bin, and for two ordered histograms it holds that all quasi-arithmetic means (such as arithmetic, harmonic, and geometric mean) of the representative values weighted by the frequencies are ordered in the same direction than the histograms are. Our theoretical study is supported by three experiments in the fields of image processing, traffic flow, and income distribution.

UR - http://www.scopus.com/inward/record.url?scp=85101434243&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85101434243&partnerID=8YFLogxK

U2 - 10.1109/ACCESS.2021.3058577

DO - 10.1109/ACCESS.2021.3058577

M3 - Article

AN - SCOPUS:85101434243

SN - 2169-3536

VL - 9

SP - 28785

EP - 28796

JO - IEEE Access

JF - IEEE Access

M1 - 9352004

ER -