canonvars
Syntax
Description
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 0.2780 -0.0147
-7.1287 -0.7867 0.0678 0.8910
-7.4898 -0.2654 -0.4915 0.2587
-6.8132 -0.6706 -0.7707 -0.2776
-8.1323 0.5145 -0.0240 -0.4797
-7.7019 1.4617 0.3962 -0.4742
-7.2126 0.3558 -1.0245 -0.3291
-7.6053 -0.0116 0.0437 -0.2531
-6.5606 -1.0152 -1.1049 0.1391
-7.3431 -0.9473 0.0965 -0.1066
⋮
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
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
Qe and
Qh in Multivariate Analysis of Variance for Repeated Measures.
Version History
Introduced in R2023b
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