Vektorrelation

Vektorrelation is a term used in mathematics, computer science, and related fields to describe a measure that quantifies the relationship between two vectors. Depending on the context, it may refer to different but related concepts of similarity or linear association between vector components.

Two common interpretations are:

- Cosine-based vector correlation (cosine similarity): This measures the alignment of two vectors regardless of their magnitudes. For vectors u and v, it is defined as (u · v) / (||u|| ||v||), where · is the dot product and ||·|| is the Euclidean norm. The value ranges from -1 to 1, with 1 indicating identical direction, -1 opposite direction, and 0 orthogonality.

- Pearson-style vector correlation across components: Here u and v are treated as sequences of paired components (u1, v1), (u2, v2), …, (un, vn). The correlation is r = cov(u, v) / (std(u) std(v)), where cov is the sample covariance and std is the standard deviation computed over i = 1..n. This measures linear dependence between corresponding components and is sensitive to centering and scaling.

Relationships and notes:

- If the vectors are centered (means removed) and standardized to unit variance, cosine similarity and Pearson-like correlation can coincide in certain settings.

- The two measures answer different questions: orientation similarity (cosine) versus linear dependency of components (Pearson-style).

Applications include pattern recognition, document and image similarity, sensor fusion, and multivariate data analysis. Limitations include dependence on vector length, sensitivity to noise, and the choice of definition.