In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, a distance matrix is a matrix An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied. These operations have many of the properties of (two-dimensional array) containing the distances Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific, taken pairwise, of a set of points. It is therefore a symmetric Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according N×N matrix containing non-negative reals as elements, given N points in Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity. The number of pairs of points N×(N-1)/2 is the number of independent elements in the distance matrix.
Distance matrices are related to adjacency matrices In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix, with the differences that (a) the latter only provides the information which vertices are connected but does not tell about costs or distances between the vertices and (b) an entry of a distance matrix is smaller if two elements are closer, while "close" (connected) vertices yield larger entries in an adjacency matrix.
Examples
For example, suppose these data are to be analyzed, where pixel In digital imaging, a pixel is a single point in a raster image. The pixel is the smallest addressable screen element, it is the smallest unit of picture which can be controlled. Each pixel has its own address. The address of a pixel corresponds to its coordinates. Pixels are normally arranged in a 2-dimensional grid, and are often represented euclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the is the distance metric In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When a topological space has a topology that can be described by a metric, we say that the.
Raw dataThe distance matrix would be:
| a | b | c | d | e | f | |
|---|---|---|---|---|---|---|
| a | 0 | 184 | 222 | 177 | 216 | 231 |
| b | 184 | 0 | 45 | 123 | 128 | 200 |
| c | 222 | 45 | 0 | 129 | 121 | 203 |
| d | 177 | 123 | 129 | 0 | 46 | 83 |
| e | 216 | 128 | 121 | 46 | 0 | 83 |
| f | 231 | 200 | 203 | 83 | 83 | 0 |
These data can then be viewed in graphic form as a heat map A heat map is a graphical representation of data where the values taken by a variable in a two-dimensional map are represented as colors. A very similar presentation form is a tree map. The term is also used to mean its thematic application as a choropleth map. In this image, black denotes a distance of 0 and white is maximal distance.
Graphical ViewIn bioinformatics Bioinformatics is the application of information technology and computer science to the field of molecular biology. The term bioinformatics was coined by Paulien Hogeweg in 1979 for the study of informatic processes in biotic systems. Its primary use since at least the late 1980s has been in genomics and genetics, particularly in those areas of, distance matrices are used to represent protein Proteins are organic compounds made of amino acids arranged in a linear chain and folded into a globular form. The amino acids in a polymer are joined together by the peptide bonds between the carboxyl and amino groups of adjacent amino acid residues. The sequence of amino acids in a protein is defined by the sequence of a gene, which is encoded structures in a coordinate-independent manner, as well as the pairwise distances between two sequences in sequence space. They are used in structural Structural alignment attempts to establish equivalences between two or more polymer structures based on their shape and three-dimensional conformation. This process is usually applied to protein tertiary structures but can also be used for large RNA molecules. In contrast to simple structural superposition, where at least some equivalent residues and sequential In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or Protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Aligned sequences of nucleotide or amino acid residues are typically represented as rows within a matrix alignment, and for the determination of protein structures from NMR Nuclear magnetic resonance is a property that magnetic nuclei have in a magnetic field and applied electromagnetic (EM) pulse or pulses, which cause the nuclei to absorb energy from the EM pulse and radiate this energy back out. The energy radiated back out is at a specific resonance frequency which depends on the strength of the magnetic field or X-ray crystallography X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and diffracts into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal.
Sometimes it is more convenient to express data as a similarity matrix A similarity matrix is a matrix of scores which express the similarity between two data points. Similarity matrices are strongly related to their counterparts, distance matrices and substitution matrices.
See also
- Data clustering Cluster analysis or clustering is the assignment of a set of observations into subsets so that observations in the same cluster are similar in some sense. Clustering is a method of unsupervised learning, and a common technique for statistical data analysis used in many fields, including machine learning, data mining, pattern recognition, image
- Computer Vision Computer vision is the science and technology of machines that see. As a scientific discipline, computer vision is concerned with the theory behind artificial systems that extract information from images. The image data can take many forms, such as video sequences, views from multiple cameras, or multi-dimensional data from a medical scanner
Categories: Metric geometry Metric geometry is a branch of geometry with metric spaces as the main object of study. It is applied mostly to Riemannian geometry and group theory | Bioinformatics Bioinformatics and Computational biology are interdisciplinary fields of research, development and application of algorithms, computational and statistical methods for management and analysis of biological data, and for solving basic biological problems | Matrices |
