3.6 KB · ⇊ 2 parts · ⤓ Micron 3.6 KB · ⤓ HTML 22.1 KB

top

In mathematics, the indicator vector, characteristic vector, or incidence vector of a subset T of a set S is the vector x T := ( x s ) s ∈ ∈ S {\displaystyle x_{T}:=(x_{s})_{s\in S}} such that x s = 1 {\displaystyle x_{s}=1} if s ∈ ∈ T {\displaystyle s\in T} and x s = 0 {\displaystyle x_{s}=0} if s ∉ ∉ T . {\displaystyle s\notin T.}

If S is countable and its elements are numbered so that S = { s 1 , s 2 , … … , s n } {\displaystyle S=\{s_{1},s_{2},\ldots ,s_{n}\}} , then x T = ( x 1 , x 2 , … … , x n ) {\displaystyle x_{T}=(x_{1},x_{2},\ldots ,x_{n})} where x i = 1 {\displaystyle x_{i}=1} if s i ∈ ∈ T {\displaystyle s_{i}\in T} and x i = 0 {\displaystyle x_{i}=0} if s i ∉ ∉ T . {\displaystyle s_{i}\notin T.}

To put it more simply, the indicator vector of T is a vector with one element for each element in S, with that element being one if the corresponding element of S is in T, and zero if it is not.cite-ref-1[1]cite-ref-2[2]cite-ref-3[3]

An indicator vector is a special (countable) case of an indicator function.

• Example

• Notes

If S is the set of natural numbers N {\displaystyle \mathbb {N} } , and T is some subset of the natural numbers, then the indicator vector is naturally a single point in the Cantor space: that is, an infinite sequence of 1's and 0's, indicating membership, or lack thereof, in T. Such vectors commonly occur in the study of arithmetical hierarchy.

cite-note-11. ↑ citerefmirkin1996Mirkin, Boris Grigorʹevich (1996). Mathematical Classification and Clustering. Springer. p. 112. ISBN 0-7923-4159-7. Retrieved 10 February 2014.

cite-note-22. ↑ citerefvon-luxburg2007von Luxburg, Ulrike (2007). "A Tutorial on Spectral Clustering" (PDF). Statistics and Computing. 17 (4): 2. Archived from the original (PDF) on 6 February 2011. Retrieved 10 February 2014.

cite-note-33. ↑ citereftaghavi2008Taghavi, Mohammad H. (2008). Decoding Linear Codes Via Optimization and Graph-based Techniques. p. 21. ISBN 9780549809043. Retrieved 10 February 2014.