Canonical analysis
Encyclopedia
In 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 r², 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 correlation
R, 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 correlation
s ρ1, ρ2, ..., and by the sets of canonical weights C and D.
for 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 analysis
and factor analysis
, each with its characteristic set of commonalities, eigenvalues and eigenvectors.
One can also construct relationships which are made to agree with constraint restrictions arising from theory or to agree with common sense/intuition. These are called maximum correlation models.(Tofallis, 1999)
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, canonical analysis (from Gk.κανων bar, measuring rod
Measuring rod
A measuring rod is a tool used to physically measure lengths and survey areas of various sizes. Most measuring rods are round or square sectioned, however they can be flat boards. Some have markings at regular intervals...
, 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
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
r, coefficient of determination r², 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 correlation
Multiple correlation
In statistics, multiple correlation is a linear relationship among more than two variables. It is measured by the coefficient of multiple determination, denoted as R2, which is a measure of the fit of a linear regression...
R, 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 correlation
Canonical correlation
In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross-covariance matrices. If we have two sets of variables, x_1, \dots, x_n and y_1, \dots, y_m, and there are correlations among the variables, then canonical correlation analysis will...
s ρ1, ρ2, ..., and by the sets of canonical weights C and D.
Canonical analysis
Canonical analysis belongs to a group of methods which involve solving the characteristic equationCharacteristic equation
Characteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...
for 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
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
’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 analysis
Principal components analysis
Principal component analysis is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. The number of principal components is less than or equal to...
and factor analysis
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, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...
, each with its characteristic set of commonalities, eigenvalues and 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, let's 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, i.e. can some form of X represent Y. It works by finding the linear combination of X variables, i.e. X1, X2 etc., and linear combination of Y variables, i.e. 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.One can also construct relationships which are made to agree with constraint restrictions arising from theory or to agree with common sense/intuition. These are called maximum correlation models.(Tofallis, 1999)