曼哈顿距离、欧几里得距离与契比雪夫距离 ——葛振兴曼哈顿距离（Taxicab Geometry）

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曼哈顿距离、欧几里得距离与契比雪夫距离 ——葛振兴曼哈顿距离（TAXICAB GEOMETRY） • Taxicab geometry, considered by Hermann Minkowski in 19th-century Germany, is a form of geometry in which the usual distance function of metric or Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. FORMAL DEFINITION APPLICATION • 1、Compressed sensing • 2、Differences of frequency distributions Next COMPRESSED SENSING • Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. • Shannon-Nyquist sampling theorem Back FREQUENCY DISTRIBUTIONS • For example, in RNA splicing positional distributions of hexamers, which plot the probability of each hexamer appearing at each given nucleotide near a splice site, can be compared with L1-distance. Each position distribution can be represented as a vector where each entry represents the likelihood of the hexamer starting at a certain nucleotide. A large L1-distance between the two vectors indicates a significant difference in the nature of the distributions while a small distance denotes similarly shaped distributions. This is equivalent to measuring the area between the two distribution curves because the area of each segment is the absolute difference between the two curves' likelihoods at that point. When summed together for all segments, it provides the same measure as L1-distance. EUCLIDEAN DISTANCE • In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" (i.e. straight-line) distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric. A generalized term for the Euclidean norm is the L2 norm or L2 distance. DEFINITION *AKRITEAN DISTANCE akrdis(x1, y1, x2, y2, a) = euclidean(x1, y1, x2, y2)*(1-a) + manhattan(x1, y1, x2, y2)*a • In computer science, the Akritean distance or akrdis is a combination of the Euclidean and the Manhattan distance, designed for use in path finding algorithms as the heuristic distance function. • A* search algorithm CHEBYSHEV DISTANCE • In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. DEFINITION • L∞ metric OTHER DISTANCE • 1、Mahalanobis distance • 2、Minkowski distance • 3、Haversine formula .

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