centroid

Noun

centroid (plural centroids)

1. The point at which gravitational force (or other universally and uniformly acting force) may be supposed to act on a given rigid, uniformly dense body; the centre of gravity or centre of mass.
   • The center of gravity of any body or geometrical magnitude is by definition the same as the centroid of a certain system of parallel forces. It will be convenient, therefore, to use the word centroid in most cases instead of center of gravity.[…]The centroid of any area may be found by the...1892, Leander Miller Hoskins, The Elements of Graphic Statics, MacMillan and Co., pages 151–152
   • For example, if a postal code area (PCA) has 1000 distinct residences, we might suppose all 1000 residences are at the centroid of the PCA. Centroids are commonly used, for example, with geographic information systems and CD-ROM phone books (Francis, Lowe, Rushton and Rayco 1999).2004, Richard L. Francis, Timothy J. Lowe, Arie Tamir, “7: Demand Point Aggregation for Local Models”, in Zvi Drezner, Horst W. Hamacher, editors, Facility Location: Applications and Theory...
2. (specifically) The point of intersection of the three medians of a given triangle; the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of the three vertices.
3. the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of a given finite set of points.
4. An analogue of the centre of gravity of a nonuniform body in which local density is replaced by a specified function (which can take negative values) and the place of the body's shape is taken by the function's domain.
   • The centroid of an arbitrary function f is given by #92;frac#123;#92;intxf(x)dx#125;#123;#92;intf(x)dx#125;, where the integrals are calculated over the domain of f.

   Synonyms:

   barycentre, centre of area, centre of gravity, centre of mass, centre of volume, geometric centre, geometric centroid, function centroid

5. (cluster analysis) the arithmetic mean (alternatively, median) position of a cluster of points in a coordinate system based on some application-dependent measure of distance.
   • The k-means procedure classifies a given data set by using a user defined number of clusters, k, a priori. The centroids can be placed randomly, or algorithmically, but it should be noted that the initial placement will affect the result. The next step is to analyze each point within the data set...2011, Ross Maciejewski, Data Representations, Transformations, and Statistics for Visual Reasoning, Morgan & Claypool Publishers, page 34
   • The k-means clustering algorithm (MacQueen, 1967) is a widely used clustering method that applies relocation of points to find a locally optimal partitioning of a dataset. In k-means, the total distance between each data point and a representative point (centroid) of the cluster to which it is...2012, Biswanath Panda, Joshua S. Herbach, Sugato Basu, Roberto J. Bayardo, “2: MapReduce and its Application to Massively Parallel Learning of Decision Tree Ensembles”, in Ron Bekkerman, Mikhail...
6. Given a tree of n nodes, either (1) a unique node whose removal would split the tree into subtrees of fewer than n/2 nodes, or (2) either of a pair of adjacent nodes such that removal of the edge connecting them would split the tree into two subtrees of exactly n/2 nodes.
   • 1974 [Prentice-Hall], Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, 2017, Dover, page 248,
     Just as in the case of centers of a tree (Section 3-4), it can be shown that every tree has either one centroid or two centroids. It can also be shown that if a tree has...
   • A node x in a tree T is called a centroid of T if the removal of x will make the size of each remaining connected component no greater than |T|/2. A tree may have at most two centroids, and if there are two then one must be a neighbor of the other [6, 5]. Throughout this paper, we specify the...2009, Hao Yuan, Patrick Eugster, An Efficient Algorithm for Solving the Dyck-CFL Reachability Problem on Trees, Giuseppe Castagna (editor), Programming Languages and Systems: 18th European Symposium...

   Synonyms:

   tree centroid