Histogram Ordering

Mateu Sbert

doi:10.1109/ACCESS.2021.3058577

Histogram Ordering

2021-01-01

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DOI：10.1109/ACCESS.2021.3058577

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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.

Keywords:

Histograms

Harmonic analysis

Probability distribution

Licenses

Image processing

Monte Carlo methods

Distribution functions

Histograms

ordering

quasi-arithmetic mean

traffic flow

histogram equalization

income distribution

Histogram Ordering

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