corrcov
Convert covariance matrix to correlation matrix
Description
Examples
Compare Correlation Matrices Obtained by Two Different Methods
Compare the correlation matrix obtained by applying corrcov
on a covariance matrix with the correlation matrix obtained by direct computation using corrcoef
on an input matrix.
Load the hospital
data set and create a matrix containing the Weight
and BloodPressure
measurements. Note that hospital.BloodPressure
has two columns of data.
load hospital
X = [hospital.Weight hospital.BloodPressure];
Compute the covariance matrix.
C = cov(X)
C = 3×3
706.0404 27.7879 41.0202
27.7879 45.0622 23.8194
41.0202 23.8194 48.0590
Compute the correlation matrix from the covariance matrix by using corrcov
.
R1 = corrcov(C)
R1 = 3×3
1.0000 0.1558 0.2227
0.1558 1.0000 0.5118
0.2227 0.5118 1.0000
Compute the correlation matrix directly by using corrcoef
, and then compare R1
with R2
.
R2 = corrcoef(X)
R2 = 3×3
1.0000 0.1558 0.2227
0.1558 1.0000 0.5118
0.2227 0.5118 1.0000
The correlation matrices R1
and R2
are the same.
Find Standard Deviations from Covariance Matrix
Find the vector of standard deviations from the covariance matrix, and show the relationship between the standard deviations and the covariance matrix.
Load the hospital
data set and create a matrix containing the Weight
, BloodPressure
, and Age
measurements. Note that hospital.BloodPressure
has two columns of data.
load hospital
X = [hospital.Weight hospital.BloodPressure hospital.Age];
Compute the covariance matrix of X
.
C = cov(X)
C = 4×4
706.0404 27.7879 41.0202 17.5152
27.7879 45.0622 23.8194 6.4966
41.0202 23.8194 48.0590 4.0315
17.5152 6.4966 4.0315 52.0622
C
is square, symmetric, and positive semidefinite. The diagonal elements of C
are the variances of the four variables in X
.
Compute the correlation matrix and standard deviations of X
from the covariance matrix C
.
[R,s1] = corrcov(C)
R = 4×4
1.0000 0.1558 0.2227 0.0914
0.1558 1.0000 0.5118 0.1341
0.2227 0.5118 1.0000 0.0806
0.0914 0.1341 0.0806 1.0000
s1 = 4×1
26.5714
6.7128
6.9325
7.2154
Compute the square root of the diagonal elements in C
, and then compare s1
with s2
.
s2 = sqrt(diag(C))
s2 = 4×1
26.5714
6.7128
6.9325
7.2154
s1
and s2
are equal and correspond to the standard deviation of the variables in X
.
Input Arguments
C
— Covariance matrix
matrix
Covariance matrix, specified as a square, symmetric, and positive semidefinite matrix.
For a matrix X that has N
observations (rows) and n random variables (columns),
C
is an n-by-n
matrix. The n diagonal elements of C
are the variances of the n random variables in
X, and a zero diagonal element in
C
indicates a constant variable in
X.
Data Types: single
| double
Output Arguments
R
— Correlation matrix
matrix
Correlation matrix, returned as a matrix that corresponds to the
covariance matrix C
.
Data Types: single
| double
sigma
— Standard deviations
vector
Standard deviations, returned as an n-by-1 vector.
The elements of sigma
are the standard deviations of
the variables in X, the
N-by-n matrix that produces
C
. Row i
in
sigma
corresponds to the standard deviation of column
i
in X.
Data Types: single
| double
More About
Covariance
For two random variable vectors A and B, the covariance is defined as
where N is the length of each column,
μA and
μB are the mean values of
A and B, respectively, and
*
denotes the complex conjugate.
The covariance matrix of two random variables is the matrix of pairwise covariance calculations between each variable,
For a matrix X, in which each column is a random variable composed of observations, the covariance matrix is the pairwise covariance calculation between each column combination. In other words, .
Variance
For a random variable vector A composed of N scalar observations, the variance is defined as
where μ is the mean of A,
Some definitions of variance use a normalization factor of N instead of N–1, but the mean always has the normalization factor N.
Extended Capabilities
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2007b
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