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**Factorial \(k\)-means analysis for two-way data.**
*(English)*
Zbl 1051.62056

A discrete clustering model together with a continuous factorial one are fitted simultaneously to two-way data, with the aim of identifying the best partition of the objects, described by the best orthogonal linear combinations of the variables (factors) according to the least-squares criterion. This methodology, named for its features factorial \(k\)-means analysis, has a very wide range of applications since it fulfills a double objective: data reduction and synthesis, simultaneously in the direction of objects and variables; variable selection in cluster analysis, identifying variables that most contribute to determine the classification of the objects.

The least-squares fitting problem proposed here is mathematically formalized as a quadratic constrained minimization problem with mixed variables. An iterative alternating least-squares algorithm based on two main steps is proposed to solve the quadratic constrained problem. Starting from the cluster centroids, the subspace projection is found that leads to the smallest distances between object points and centroids. Updating the centroids, the partition is detected assigning objects to the closest centroids. At each step the algorithm decreases the least-squares criterion, thus converging to an optimal solution. Two data sets are analyzed to show the features of the factorial \(k\)-means model. The proposed technique has a fast algorithm that allows researchers to use it also with large data sets.

The least-squares fitting problem proposed here is mathematically formalized as a quadratic constrained minimization problem with mixed variables. An iterative alternating least-squares algorithm based on two main steps is proposed to solve the quadratic constrained problem. Starting from the cluster centroids, the subspace projection is found that leads to the smallest distances between object points and centroids. Updating the centroids, the partition is detected assigning objects to the closest centroids. At each step the algorithm decreases the least-squares criterion, thus converging to an optimal solution. Two data sets are analyzed to show the features of the factorial \(k\)-means model. The proposed technique has a fast algorithm that allows researchers to use it also with large data sets.

### MSC:

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62H25 | Factor analysis and principal components; correspondence analysis |

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\textit{M. Vichi} and \textit{H. A. L. Kiers}, Comput. Stat. Data Anal. 37, No. 1, 49--64 (2001; Zbl 1051.62056)

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