Where does similarity collide?

In geometry, two figures are similar when they have the same shape but may differ in size: corresponding angles are equal and corresponding lengths stand in a fixed ratio, so one figure can be carried onto the other by uniform scaling together with rotation, translation, and reflection. Euclid's Elements (Book VI, circa 300 BCE) defined similar rectilinear figures by equal angles and proportional sides and built the theory of proportion that supports them. Here similarity is a yes-or-no relation between shapes, settled by exact proportion, and it is not a graded score of resemblance between data objects or a judgment made by a person.[1]

Unit

A pair of figures that either are or are not similar, with a single scale ratio when they are

Recovery means

Not applicable

Measured by

Established by proving that corresponding angles are equal and corresponding lengths hold a constant ratio, or by exhibiting a similarity transformation between the figures

In data analysis, a similarity measure is a number that says how alike two objects are given their features, usually scaled so that identical objects score at the top and unrelated ones near the bottom. Jaccard's 1912 The distribution of the flora in the alpine zone introduced the first such coefficient, the ratio of shared attributes to total attributes between two samples, and Salton, Wong, and Yang's 1975 A vector space model for automatic indexing made the cosine of the angle between feature vectors the standard measure for comparing documents. Here similarity is a graded score computed from data and symmetric in its two inputs, which sets it apart from the exact shape relation of geometry and from the feature-weighted judgment a person reports.[2][3]

Unit

A score between two objects, often on a fixed range such as 0 to 1

Recovery means

Not applicable

Measured by

Computed from feature overlap or from the angle or distance between vectors, using coefficients such as Jaccard's ratio or the cosine measure

In cognitive psychology, similarity is the resemblance a person perceives between two things, treated as a mental quantity that predicts how people generalize, sort, and remember. Tversky's 1977 Features of similarity modeled it as a contrast of features: the judged similarity of two items rises with the features they share and falls with the features that set them apart, weighted by context and by which item is taken as the subject of the comparison. Because those weights can differ, this similarity is often asymmetric and can break the distance rules of a geometric space, so it is neither the exact shape relation of geometry nor the symmetric score of a data similarity measure.[4]

Unit

A person's rating of how alike two items are, in a given context

Recovery means

Not applicable

Measured by

Elicited by similarity ratings, sorting, and generalization or confusion data, and modeled by feature contrast or by multidimensional scaling