dissimilarities

Dissimilarity is a quantitative measure of how different two objects are. Formally, a dissimilarity (or distance) is a function d that assigns a nonnegative real number to a pair of objects from a given domain, with d(x, x) = 0. If a dissimilarity also satisfies symmetry d(x, y) = d(y, x) for all x and y, and, in the stricter case, the triangle inequality d(x, z) ≤ d(x, y) + d(y, z), it is called a metric. Not all dissimilarities satisfy all metric properties; in data analysis the term distance is often used loosely.

Different data types necessitate different measures. For binary or categorical data, the Hamming distance counts mismatches, while Jaccard and Sörensen-Dice distances compare set-based features. For real-valued vectors, common choices include L1 (Manhattan) and L2 (Euclidean) distances; Mahalanobis distance accounts for feature correlations; Bray-Curtis and Canberra distances are sensitive to magnitude in certain domains. In high-dimensional, sparse spaces such as text data, cosine distance (one minus cosine similarity) is frequently employed. There are also specialized measures like Levenshtein (edit) distance for strings.

Properties and considerations are important when selecting a dissimilarity. The choice influences clustering, classification, and visualization outcomes. Some algorithms require metric properties to guarantee certain guarantees, but non-metric dissimilarities can still be useful with appropriate methods. Normalization and scaling can affect results, so data preparation matters.

Applications of dissimilarities include clustering, nearest-neighbor methods, anomaly detection, and information retrieval. Related concepts include similarity, distance, and metric spaces; see also distance measures, metrics, and multidimensional scaling.