statistics, canonical analysis (from Gk."κανων" bar, measuring rod, ruler) belongs to the family of regression methods for data analysis. Regression analysis quantifies a relationship between a predictor variable and a criterion variable by the coefficient of correlation"r", coefficient of determination , and the standard regression coefficient β. Multiple regression analysis expresses a relationship between a set of predictor variables and a single criterion variable by the multiple correlationR, multiple coefficient of determination R², and a set of standard partial regression weights β1, β2, etc. Canonical variate analysis captures a relationship between a set of predictor variables and a set of criterion variables by the canonical correlations ρ1, ρ2, ..., and by the sets of canonical weights C and D.
Canonical analysis belongs to a group of methods which involve solving the
characteristic equationfor its latent roots and vectors. It describes formal structures in hyperspace invariant with respect to the rotation of their coordinates. In this type of solution, rotation leaves many optimizing properties preserved, provided it takes place in certain ways and in a subspace of its corresponding hyperspace. This rotation from the maximum intervariate correlation structure into a different, simpler and more meaningful structure increases the interpretability of the canonical weights C and D. In this the canonical analysis differs from Harold Hotelling’s (1936) canonical variate analysis (also called the canonical correlation analysis), designed to obtain maximum (canonical) correlations between the predictor and criterion canonical variates. The difference between the canonical variate analysis and canonical analysis is analogous to the difference between the principal components analysisand factor analysis, each with its characteristic set of communalities, eigenvaluesand eigenvectors.
Canonical analysis (simple)
Canonical analysis is a multivariate technique which is concerned with determining the relationships between groups of variables in a data set. The data set is split into two groups, lets call these groups X and Y, based on some common characteristics. The purpose of Canonical analysis is then to find the relationship between X and Y, IE can some form of X represent Y. It works by finding the linear combination of X variables, IE X1, X2 etc and linear combination of Y variables, IE Y1, Y2 etc which are most highly correlated. This combination is known as the "first canonical variates" which are usually denoted U1 and V1, with the pair of U1 and V1 being called a "canonical function". The next canonical function, U2 and V2 are then restricted so that they are uncorrelated with U1 and V1. Everything is scaled so that the variance equals 1 .
* Cliff, N. and Krus, D. J. (1976) Interpretation of canonical variate analysis: Rotated vs. unrotated solutions. "Psychometrika," 41, 1, 35-42. [http://www.visualstatistics.net/Statistics/New%20Cliff%20CA/newCliff%20CA.asp (Request reprint).]
* Hotelling, H. (1936) Relations between two sets of variates. "Biometrika," 28, 321-377
* Krus, D.J., et al. (1976) Rotation in canonical analysis. "Educational and Psychological Measurement," 36, 725-730. [http://www.visualstatistics.net/Statistics/Rotation%20in%20CA/Rotation%20in%20CA.asp (Request reprint).]
* Liang, K.H., Krus, D.J., & Webb, J.M. (1995) K-fold crossvalidation in canonical analysis. "Multivariate Behavioral Research", 30, 539-545. [http://www.visualstatistics.net/Statistics/K-fold%20CA/k-fold%20CA.asp (Request reprint).]
Wikimedia Foundation. 2010.
Look at other dictionaries:
canonical analysis — See multivariate analysis … Dictionary of sociology
Canonical provision — is a term of the canon law of the Roman Catholic Church, signifying regular induction into a benefice.AnalysisIt comprises three distinct acts the designation of the person, canonical institution, and installation. In various ways a person may be … Wikipedia
Canonical correlation — In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross covariance matrices.DefinitionGiven two column vectors X = (x 1, dots, x n) and Y = (y 1, dots, y m) of random variables with finite… … Wikipedia
Canonical form — Generally, in mathematics, a canonical form (often called normal form or standard form) of an object is a standard way of presenting that object. Canonical form can also mean a differential form that is defined in a natural (canonical) way; see… … Wikipedia
Canonical form (Boolean algebra) — In Boolean algebra, any Boolean function can be expressed in a canonical form using the dual concepts of minterms and maxterms. Minterms are called products because they are the logical AND of a set of variables, and maxterms are called sums… … Wikipedia
Canonical coordinates — In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of… … Wikipedia
Canonical Criticism — This term, though not welcomed by its exponents, has been applied to a kind of biblical scholarship developed by a small group of university teachers in North America. It began with the breakdown in the 1960s of the attempt by Biblical Theology… … Dictionary of the Bible
Canonical units — In astrodynamics, A canonical unit is a unit of measurement defined in terms of an object s reference orbit. In this system, a reference mass, for example the sun, is assumed to be 1 mass unit and the mean distance from the orbiting object to the … Wikipedia
multivariate analysis — Univariate analysis consists in describing and explaining the variation in a single variable. Bivariate analysis does the same for two variables taken together (covariation). Multivariate analysis (MVA) considers the simultaneous effects of many… … Dictionary of sociology
Linear discriminant analysis — (LDA) and the related Fisher s linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate two or more classes of objects or events. The… … Wikipedia