# Factor Analysis Algorithm

A Factor Analysis Algorithm is an analysis algorithm that can be applied by a factor analysis system (to solve a factor analysis task).

**AKA:**FA, Independent Factor Analysis.**Context:**- It can range from being a Linear Factor Analysis Algorithm to being a Non-Linear Factor Analysis Algorithm.

**Counter-Example(s):****See:**Dimensionality Compression Algorithm, Component Analysis, Fisher Discriminant Analysis.

## References

### 2013

- http://en.wikipedia.org/wiki/Factor_analysis
**Factor analysis**is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. In other words, it is possible, for example, that variations in three or four observed variables mainly reflect the variations in fewer unobserved variables. Factor analysis searches for such joint variations in response to unobserved latent variables. The observed variables are modeled as linear combinations of the potential factors, plus “error” terms. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset. Computationally this technique is equivalent to low rank approximation of the matrix of observed variables. Factor analysis originated in psychometrics, and is used in behavioral sciences, social sciences, marketing, product management, operations research, and other applied sciences that deal with large quantities of data.Factor analysis is related to principal component analysis (PCA), but the two are not identical. Latent variable models, including factor analysis, use regression modelling techniques to test hypotheses producing error terms, while PCA is a descriptive statistical technique.

^{[1]}There has been significant controversy in the field over the equivalence or otherwise of the two techniques (see exploratory factor analysis versus principal components analysis).^{[citation needed]}

- ↑ Bartholomew, D.J.; Steele, F.; Galbraith, J.; Moustaki, I. (2008).
*Analysis of Multivariate Social Science Data*. Statistics in the Social and Behavioral Sciences Series (2nd ed.). Taylor & Francis. ISBN 1584889608.

### 2002

- (Fodor, 2002) ⇒ Imola K. Fodor. (2002). “A Survey of Dimension Reduction Techniques." LLNL technical report, UCRL ID-148494
- QUITE: This section follows [41]. Like PCA,
**factor analysis (FA)**is also a linear method, based on the second-order data summaries. First suggested by psychologists, FA assumes that the measured variables depend on some unknown, and often unmeasurable, common factors. Typical examples include variables de¯ned as various test scores of individuals, as such scores are thought to be related to a common "intelligence" factor. The goal of**FA**is to uncover such relations, and thus can be used to reduce the dimension of datasets following the factor model.

- QUITE: This section follows [41]. Like PCA,

### 2000

- (Valpola, 2000) ⇒ Harri Valpola. (2000). “Bayesian Ensemble Learning for Nonlinear Factor Analysis." PhD Dissertation, Helsinki University of Technology.
- QUOTE: factor analysis: A technique for finding a generative model which can represent some of the statistical structure of the observations. Usually refers to linear factor analysis where the generative model is linear.

### 1998

- (Johnson & Wichern, 1998) ⇒ Richard A. Johnson, and Dean W. Wichern. (1998). “Applied Multivariate Statistical Analysis, 4th ed." Prentice hall, 1992. ISBN:013834194X
- QUOTE: Factor analysis can be considered an extension of principal components analysis. Both can be viewed as attempts to approximate the covariance matrix [math]\Sigma[/math]. however, the approximation based on the factor analysis model is more elaborate. They primary question in factor analysis is whether the data are consistent with a prescribed structure.