Definition of "centroid"

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 following method: Divide the area into parts such that the area and centroid of each part are known. Take the centroids of the partial areas as the points of application of forces proportional respectively to those areas. The centroid of this system of forces is the centroid of the total area, and may be found by the method of Art. 172.

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, Springer-Verlag, page 207

In R n {\displaystyle \mathbb {R} ^{n}} , a centroid is the mean position of all the points in all of the coordinate directions. The centroid of a subset X {\displaystyle {\mathcal {X}}} of R n {\displaystyle \mathbb {R} ^{n}} is computed as follows: Centroid ⁡ ( X ) = ∫ x g ( x ) d x ∫ g ( x ) d x ( 6 ) {\displaystyle \operatorname {Centroid} ({\mathcal {X}})={\frac {\int xg(x)dx}{\int g(x)dx}}\quad \quad \quad \quad \quad \quad (6)} / where the integrals are taken over the whole space R n {\displaystyle \mathbb {R} ^{n}} , and g {\displaystyle g} is the characteristic function of the subset, which is 1 inside X {\displaystyle {\mathcal {X}}} and 0 outside it .

2020, Cheng Zhang, Qiuchi Li, Lingyu Hua, Dawei Song, Assessing the Memory Ability of Recurrent Neural Networks, Giuseppe De Giacomo, et al. (editors), ECAI 2020: 24th European Conference on Artificial Intelligence, IOS Press, page 1660

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 and group it with the nearest centroid according to some distance metric. When all points have been assigned to a group, a new centroid is calculated for each group as a barycenter of the cluster, resulting from the previous step. Once the k new centroids are calculated, the algorithm reiterates through the data set, and each sample is again assigned to a cluster based on its distance to the new centroids. This process is continued until the position[sic] of the centroids no longer change.

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 assigned is minimized. Each iteration of k-means has two steps. In the cluster assignment step, k-means assigns each point to a cluster such that, of all the current cluster centroids, the point is closest to the centroid of that cluster. In the cluster re-estimation step, k-means re-estimates the new cluster centroids based on the reassignments of points to clusters in the previous step. The cluster re-assignment and centroid re-estimation steps proceed in iterations until a specified convergence criterion is reached, such as when the total distance between clusters and centroids does not change substantially from one iteration to another.

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 Bilenko, John Langford, editors, Scaling Up Machine Learning, Cambridge University Press, page 26

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 two centroids, the centroids are adjacent.

1974 [Prentice-Hall], Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, 2017, Dover, page 248

A node x {\displaystyle x} in a tree T {\displaystyle T} is called a centroid of T {\displaystyle T} if the removal of x {\displaystyle x} will make the size of each remaining connected component no greater than | T | / 2 {\displaystyle \vert T\vert /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 centroid to be the one whose numbering is lexicographically smaller (i.e, we number the nodes from 1 to n {\displaystyle n} ). There exists a linear time algorithm to compute the centroid of a tree due to the work of Goldman . We use CT ⁡ ( T ) {\displaystyle \operatorname {CT} (T)} to denote the centroid of T {\displaystyle T} computed by the linear time algorithm.

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, Proceedings, Springer, LNCS 5502, page 186