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Introduction

The sixteen measurses that make up the HMI set come from the areas of economics, ecology, and information theory. Collectively, we refer to these sixteen measures as the HMI set (i.e., heuristic measures of interestingness).

The measures can be used for more than just ranking the interestingness of generalized relations using domain generaliztion graph. For example, alternative methods could be used to guide the generation of summaries, such as Galois lattices, conceptual graphs, or formal concept analysis. Also, summarties could more generally include views generated from databases or summary tables generated from data cubes.

Sixteen Characteristics of the HMI Set

Each measure shares three important properties

Variables used in describing of the HMI set

IVariance
Based on the sample variance from classical statistics, IVariance measures the weighted average of the squared deviations of the probabilities pi from the mean probability q-bar, where the weight assigned to each squared deviation is 1/(m - 1).

ivariance


ivariance sample

ISimpson
A variance-like measure based on the Simpson index, ISimpson measures the extent to which the counts are distributed over the tuples in a summary, rather than being concentrated in any single one of them.

simpson


simpson sample

IShannon
Based on a relative entropy measure from information theory (known as the Shannon index), IShannon measures the average information content in the tuples of a summary.

shannon

ITotal
Based on the Shannon index from information theory, ITotal measures the total information content in a summary.

total

IMax
Based on the Shannon index from information theory, IMax measures the maximum possible information content in a summary.

max

IMcIntosh
Based on a heterongeneity index from ecology, IMcIntosh views the counts in a summary as the coordinates of a point in a multidimensional space and measures the modified Euclidean distance from this point to the origin.

macintosh

ILorenz
Based on the Lorenz curve from statistics, economics, and social science, ILorenz measures the average value of the Lorenz curve derived from the probabilities pi associated with the tuples in a summary. The Lorenz curve is a series of straight lines in a square of unit length, starting from the origin and going successively to points (p1,q1),(p1 + p2, q1 + q2), . . .. When the pi's are all equal, the Lorenz curve coincides with the diagonal that cuts the unit square into equal halves. When the pi's are not all equal, the Lorenz curve is below the diagonal.

lorenz

IGini
Based on the Gini coefficient which is defined in terms of the Lorenz curve, IGini measures the ratio of the area between the diagonal (i.e., the line of equality) and Lorenz curve, and the total area below the diagonal.

gini

IBerger
Based on a dominance index from ecology, IBerger measures the proportional dominance of the tuple in a summary with the highest probability pi

berger

ISchutz
Based on an inequality measure from the economics and social science, ISchutz measures the relative mean deviation of the observed distribution of the counts in a summary from a uniform distribution of the counts.

schutz

IBray
Based on a community similarity index from the ecology, IBray measures the percentage of similarity between the observed distribution of the counts in a summary and a uniform distribution of the counts.

bray

IWhittaker
Based on a community similarity index from ecology, IWhittaker measures the fraction of similarity between the observed distribution of the counts in a summary and a uniform distribution of the counts.

whittaker

IKullback
Based on a distance measure from information theory, IKullback measures the distance between the observed distribution of the counts in a summary and a uniform distribution of the counts.

kullback

IMacArthur
Based on the Shannon index from information theory, IMacArthur combines two summaries, and then measures the difference between the amount of information contained in the combined distribution and the amount of contained in the average of the two original distributions.

macarthur

ITheil
Based on a distance measure from the information theory, ITheil measures the distance between the observed distribution of the counts in a summary and a uniform distribution of the counts.

theil

IAtkinson
Based on a measure of inequality from economics, IAtkinson measures the percentage to which the population in a summary would have to be increased to achieve the same level of interestingness as if the counts in the summary were uniformly distributed.

atkinson