Similarity

Specific definitions

Different fields provide differing definitions of similarity:

In mathematics

In computer science

In other fields

See also

Similarity (network science)

Similarity in network analysis occurs when two nodes (or other more elaborate structures) fall in the same equivalence class.

There are three fundamental approaches to constructing measures of network similarity: structural equivalence, automorphic equivalence, and regular equivalence. There is a hierarchy of the three equivalence concepts: any set of structural equivalences are also automorphic and regular equivalences. Any set of automorphic equivalences are also regular equivalences. Not all regular equivalences are necessarily automorphic or structural; and not all automorphic equivalences are necessarily structural.

Visualizing similarity and distance

Clustering tools

Agglomerative Hierarchical clustering of nodes on the basis of the similarity of their profiles of ties to other nodes provides a joining tree or Dendrogram that visualizes the degree of similarity among cases - and can be used to find approximate equivalence classes.

Multi-dimensional scaling tools

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level.

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.

Similarity relation (music)

In music, a similarity relation or pitch-class similarity is a comparison between sets of the same cardinality (two sets containing the same number of pitch classes), based upon shared pitch class and/or interval class content.

Allen Forte originally designated four types: R_p (maximal similarity with respect to pitch class), R₀ (minimal similarity), R₁ (first order maximal similarity), and _R2 (second order maximal similarity). In R_p one pitch class is different, in R₀ all are different, and in R₁ and R₂ four interval classes are the same.

Rp is defined for sets S₁ and S₂ of cardinal number n and S₃ of cardinal number n-1 as:

Meaning that S₁ and S₂ each have all the pitch-classes of S₃ (transposed or inverted), plus one.

Sources

Further reading

Similarity (psychology)

Similarity refers to the psychological nearness or proximity of two mental representations. It is one of the principles of grouping in Gestalt psychology According to Daniel Schachter "Regions that are similar in color, lightness, shape, or texture are perceived as belonging to the same object" Research in cognitive psychology has taken a number of approaches to the concept of similarity. Each of them is related to a particular set of assumptions about knowledge representation.

Cognitive psychological approaches

Mental distance approaches

Mental distance approaches (Shepard, 1962) assume that mental representations can be conceptualized as some kind of mental space. Concepts are represented as points within the space. Similarity between concepts is a function of the distance between the concepts in space. Concepts represented by points that are near to each other are more psychologically similar than are points that are conceptually distant. A strength of this approach is there are many mathematical techniques for deriving spaces from data such as multidimensional scaling (Shepard, 1962) and latent semantic analysis (Laudauer & Dumais, 1997).