canonvars

Canonical variables

Since R2023b

Syntax

canon = canonvars(maov)

canon = canonvars(maov,factor)

[canon,eigenvec,eigenval] = canonvars(___)

Description

canon = canonvars(maov) returns the values of the canonical variables for the response data in the manova object maov. This syntax is supported for one-way manova objects only.

canon = canonvars(maov,factor) specifies factor canonvars uses to group the response data. This syntax is supported for one-, two-, and N-way manova objects.

[canon,eigenvec,eigenval] = canonvars(___) additionally returns the eigenvectors and eigenvalues canonvars uses to calculate the canonical variables, using any of the input argument combinations in the previous syntaxes.

Examples

Calculate Canonical Response Data for One-Way MANOVA

Load the fisheriris data set.

load fisheriris

The column vector species contains iris flowers of three different species: setosa, versicolor, and virginica. The matrix meas contains four types of measurements for the flower: the length and width of sepals and petals in centimeters.

Perform a one-way MANOVA with species as the factor and the measurements in meas as the response variables.

maov = manova(species,meas);

maov is a one-way manova object that contains the results of the one-way MANOVA.

Calculate the canonical response data for maov.

canon = canonvars(maov)

canon = 150×4

   -8.0618    0.3004    0.2784    0.0062
   -7.1287   -0.7867    0.0008    0.8935
   -7.4898   -0.2654   -0.5095    0.2211
   -6.8132   -0.6706   -0.7477   -0.3347
   -8.1323    0.5145    0.0120   -0.4802
   -7.7019    1.4617    0.4307   -0.4431
   -7.2126    0.3558   -0.9969   -0.4050
   -7.6053   -0.0116    0.0625   -0.2491
   -6.5606   -1.0152   -1.1122    0.0559
   -7.3431   -0.9473    0.1042   -0.0990
      ⋮

The output shows the canonical response data for the first ten observations. Each column of the output corresponds to a different canonical variable.

Create a scatter plot using the first and second canonical variables.

gscatter(canon(:,1),canon(:,2),species)
xlabel("canon1")
ylabel("canon2")

The function calculates the canonical variables by finding the lowest dimensional representation of the response variables that maximizes the correlation between the response variables and the factor values. The plot shows that the response data for the first two canonical variables is mostly separate for the different factor values. In particular, observations with the first canonical variable less than 0 correspond to the setosa group. Observations with the first canonical response variable greater than 0 and less than 5 correspond to the versicolor group. Finally, observations with the first canonical response variable greater than 5 correspond to the virginica group.

Return Canonical Coefficients

Load the carsmall data set.

load carsmall

The variable Model_Year contains data for the year a car was manufactured, and the variable Cylinders contains data for the number of engine cylinders in the car. The Acceleration and Displacement variables contain data for car acceleration and displacement.

Use the table function to create a table from the data in Model_Year and Cylinders.

tbl = table(Model_Year,Cylinders,VariableNames=["Year" "Cylinders"]);

Create a matrix of response variables from Acceleration and Displacement.

y = [Acceleration Displacement];

Perform a two-way MANOVA using the factor values in tbl and the response variables in y.

maov = manova(tbl,y);

maov is a two-way manova object that contains the results of the two-way MANOVA.

Return the canonical response data, canonical coefficients, and eigenvalues for the response data in maov, grouped by the Cylinders factor.

[canon,eigenvec,eigenval] = canonvars(maov,"Cylinders")

canon = 100×2

    2.9558   -0.5358
    4.2381   -0.4096
    3.2798   -0.8889
    2.8661   -0.5600
    2.7996   -1.2391
    6.5913   -0.4348
    7.3336   -0.6749
    6.9131   -1.0089
    7.3680   -0.2249
    5.4195   -1.4126
      ⋮

eigenvec = 2×2

    0.0045    0.4419
    0.0299    0.0081

eigenval = 2×1

    6.5170
    0.0808

The output shows the canonical response data for each canonical variable, and the vectors of canonical coefficients for each canonical variable with their corresponding eigenvalues.

You can use the coefficients in eigenvec to calculate canonical response data manually. Normalize the training data in maov.Y by using the mean function.

normres = maov.Y - mean(maov.Y)

normres = 100×2

   -3.0280   99.4000
   -3.5280  142.4000
   -4.0280  110.4000
   -3.0280   96.4000
   -4.5280   94.4000
   -5.0280  221.4000
   -6.0280  246.4000
   -6.5280  232.4000
   -5.0280  247.4000
   -6.5280  182.4000
      ⋮

Calculate the product of the matrix of normalized response data and matrix of canonical coefficients.

mcanon = normres*eigenvec

mcanon = 100×2

    2.9558   -0.5358
    4.2381   -0.4096
    3.2798   -0.8889
    2.8661   -0.5600
    2.7996   -1.2391
    6.5913   -0.4348
    7.3336   -0.6749
    6.9131   -1.0089
    7.3680   -0.2249
    5.4195   -1.4126
      ⋮

The first ten rows of mcanon are identical to the first ten rows of data in canon.

Check that mcanon is identical to canon by using the max and sum functions.

max(abs(canon-mcanon))

ans = 1×2

     0     0

The zero output confirms that the two methods of calculating the canonical response data are equivalent.

Input Arguments

maov — MANOVA results

manova object

MANOVA results, specified as a manova object. The properties of maov contain the factor values and response data used by canonvars to calculate the canonical response data.

factor — Factor used to group response data

string scalar | character array

Factor used to group the response data, specified as a string scalar or character array. factor must be a name in maov.FactorNames.

Example: "Factor2"

Data Types: char | string

Output Arguments

canon — Canonical response data

n-by-r numeric matrix

Canonical response data, returned as an n-by-r numeric matrix. n is the number of observations in maov, and r is the number of response variables. To get the canonical response data, canonvars normalizes the data in maov.Y and then calculates linear combinations of the normalized data using the canonical coefficients. For more information, see eigenvec.

Data Types: single | double

eigenvec — Canonical coefficients

r-by-r numeric matrix

Canonical coefficients used to calculate the canonical response data, returned as an r-by-r numeric matrix. r is the number of response variables in maov.Y. Each column of eigenvec corresponds to a different canonical variable. The leftmost column of eigenvec corresponds to the canonical variable that is the most correlated to the factor values, and the rightmost column corresponds to the variable that is the least correlated. The canonical variables are uncorrelated to each other. For more information, see Canonical Coefficients.

Data Types: single | double

eigenval — Eigenvalues

r-by-1 numeric vector

Eigenvalues for the characteristic equation canonvars uses to calculate the canonical coefficients, returned as an r-by-1 numeric vector. For more information, see Canonical Coefficients.

Data Types: single | double

More About

Canonical Coefficients

The canonical coefficients are the r eigenvectors of the characteristic equation

$$\{\begin{array}{l}Mv=\lambda v\\ M=H{E}^{-1},\end{array}$$

where H is the hypothesis matrix for maov, E is the error matrix, and r is the number of response variables.

The canonical variables correspond to projections of the response variables into linear spaces with dimensions equal to or smaller than the number of response variables. The first canonical variable is the projection of the response variables into the one-dimensional Euclidean space that has the maximum correlation with the factor values. For 0 < n ≤ r, the nth canonical variable is the projection into the one-dimensional Euclidean space that has the maximum correlation with the factor values, subject to the constraint that the canonical variable is uncorrelated with the previous n – 1 canonical variables. For more information, see Q_e and Q_h in Multivariate Analysis of Variance for Repeated Measures.

Version History

Introduced in R2023b