corrcoef
Correlation coefficients
Syntax
Description
returns
the matrix of correlation
coefficients for R
= corrcoef(A
)A
, where the columns of A
represent
random variables and the rows represent observations.
[
returns the matrix of correlation
coefficients and the matrix of p-values for testing the hypothesis
that there is no relationship between the observed phenomena (null
hypothesis). Use this syntax with any of the arguments from the previous
syntaxes. If an off-diagonal element of R
,P
] =
corrcoef(___)P
is smaller
than the significance level (default is 0.05
),
then the corresponding correlation in R
is considered
significant. This syntax is invalid if R
contains
complex elements.
___ = corrcoef(___,
returns any of the output arguments from the previous syntaxes with additional
options specified by one or more Name,Value
)Name,Value
pair arguments.
For example, corrcoef(A,'Alpha',0.1)
specifies a 90%
confidence interval, and corrcoef(A,'Rows','complete')
omits
all rows of A
containing one or more NaN
values.
Examples
Random Columns of Matrix
Compute the correlation coefficients for a matrix with two normally distributed, random columns and one column that is defined in terms of another. Since the third column of A
is a multiple of the second, these two variables are directly correlated, thus the correlation coefficient in the (2,3)
and (3,2)
entries of R
is 1
.
x = randn(6,1); y = randn(6,1); A = [x y 2*y+3]; R = corrcoef(A)
R = 3×3
1.0000 -0.6237 -0.6237
-0.6237 1.0000 1.0000
-0.6237 1.0000 1.0000
Two Random Variables
Compute the correlation coefficient matrix between two normally distributed, random vectors of 10 observations each.
A = randn(10,1); B = randn(10,1); R = corrcoef(A,B)
R = 2×2
1.0000 0.4518
0.4518 1.0000
P-Values of Matrix
Compute the correlation coefficients and p-values of a normally distributed, random matrix, with an added fourth column equal to the sum of the other three columns. Since the last column of A
is a linear combination of the others, a correlation is introduced between the fourth variable and each of the other three variables. Therefore, the fourth row and fourth column of P
contain very small p-values, identifying them as significant correlations.
A = randn(50,3); A(:,4) = sum(A,2); [R,P] = corrcoef(A)
R = 4×4
1.0000 0.1135 0.0879 0.7314
0.1135 1.0000 -0.1451 0.5082
0.0879 -0.1451 1.0000 0.5199
0.7314 0.5082 0.5199 1.0000
P = 4×4
1.0000 0.4325 0.5438 0.0000
0.4325 1.0000 0.3146 0.0002
0.5438 0.3146 1.0000 0.0001
0.0000 0.0002 0.0001 1.0000
Correlation Bounds
Create a normally distributed, random matrix, with an added fourth column equal to the sum of the other three columns, and compute the correlation coefficients, p-values, and lower and upper bounds on the coefficients.
A = randn(50,3); A(:,4) = sum(A,2); [R,P,RL,RU] = corrcoef(A)
R = 4×4
1.0000 0.1135 0.0879 0.7314
0.1135 1.0000 -0.1451 0.5082
0.0879 -0.1451 1.0000 0.5199
0.7314 0.5082 0.5199 1.0000
P = 4×4
1.0000 0.4325 0.5438 0.0000
0.4325 1.0000 0.3146 0.0002
0.5438 0.3146 1.0000 0.0001
0.0000 0.0002 0.0001 1.0000
RL = 4×4
1.0000 -0.1702 -0.1952 0.5688
-0.1702 1.0000 -0.4070 0.2677
-0.1952 -0.4070 1.0000 0.2825
0.5688 0.2677 0.2825 1.0000
RU = 4×4
1.0000 0.3799 0.3575 0.8389
0.3799 1.0000 0.1388 0.6890
0.3575 0.1388 1.0000 0.6974
0.8389 0.6890 0.6974 1.0000
The matrices RL
and RU
give lower and upper bounds, respectively, on each correlation coefficient according to a 95% confidence interval by default. You can change the confidence level by specifying the value of Alpha
, which defines the percent confidence, 100*(1-Alpha)
%. For example, use an Alpha
value equal to 0.01 to compute a 99% confidence interval, which is reflected in the bounds RL
and RU
. The intervals defined by the coefficient bounds in RL
and RU
are bigger for 99% confidence compared to 95%, since higher confidence requires a more inclusive range of potential correlation values.
[R,P,RL,RU] = corrcoef(A,'Alpha',0.01)
R = 4×4
1.0000 0.1135 0.0879 0.7314
0.1135 1.0000 -0.1451 0.5082
0.0879 -0.1451 1.0000 0.5199
0.7314 0.5082 0.5199 1.0000
P = 4×4
1.0000 0.4325 0.5438 0.0000
0.4325 1.0000 0.3146 0.0002
0.5438 0.3146 1.0000 0.0001
0.0000 0.0002 0.0001 1.0000
RL = 4×4
1.0000 -0.2559 -0.2799 0.5049
-0.2559 1.0000 -0.4792 0.1825
-0.2799 -0.4792 1.0000 0.1979
0.5049 0.1825 0.1979 1.0000
RU = 4×4
1.0000 0.4540 0.4332 0.8636
0.4540 1.0000 0.2256 0.7334
0.4332 0.2256 1.0000 0.7407
0.8636 0.7334 0.7407 1.0000
NaN
Values
Create a normally distributed matrix involving NaN
values, and compute the correlation coefficient matrix, excluding any rows that contain NaN
.
A = randn(5,3); A(1,3) = NaN; A(3,2) = NaN; A
A = 5×3
0.5377 -1.3077 NaN
1.8339 -0.4336 3.0349
-2.2588 NaN 0.7254
0.8622 3.5784 -0.0631
0.3188 2.7694 0.7147
R = corrcoef(A,'Rows','complete')
R = 3×3
1.0000 -0.8506 0.8222
-0.8506 1.0000 -0.9987
0.8222 -0.9987 1.0000
Use 'all'
to include all NaN
values in the calculation.
R = corrcoef(A,'Rows','all')
R = 3×3
1 NaN NaN
NaN NaN NaN
NaN NaN NaN
Use 'pairwise'
to compute each two-column correlation coefficient on a pairwise basis. If one of the two columns contains a NaN
, that row is omitted.
R = corrcoef(A,'Rows','pairwise')
R = 3×3
1.0000 -0.3388 0.4649
-0.3388 1.0000 -0.9987
0.4649 -0.9987 1.0000
Input Arguments
A
— Input array
matrix
Input array, specified as a matrix.
If
A
is a scalar,corrcoef(A)
returnsNaN
.If
A
is a vector,corrcoef(A)
returns1
.
Data Types: single
| double
Complex Number Support: Yes
B
— Additional input array
vector | matrix | multidimensional array
Additional input array, specified as a vector, matrix, or multidimensional array.
A
andB
must be the same size.If
A
andB
are scalars, thencorrcoef(A,B)
returns1
. IfA
andB
are equal, however,corrcoef(A,B)
returnsNaN
.If
A
andB
are matrices or multidimensional arrays, thencorrcoef(A,B)
converts each input into its vector representation and is equivalent tocorrcoef(A(:),B(:))
orcorrcoef([A(:) B(:)])
.If
A
andB
are 0-by-0 empty arrays,corrcoef(A,B)
returns a 2-by-2 matrix ofNaN
values.
Data Types: single
| double
Complex Number Support: Yes
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: R = corrcoef(A,'Alpha',0.03)
Alpha
— Significance level
0.05 (default) | number between 0 and 1
Significance level, specified as a number between 0 and 1. The value
of the 'Alpha'
parameter defines the percent
confidence level, 100*(1-Alpha
)%, for the correlation
coefficients, which determines the bounds in RL
and
RU
.
Data Types: single
| double
Rows
— Use of NaN
option
'all'
(default) | 'complete'
| 'pairwise'
Use of NaN
option, specified as one of these values:
'all'
— Include allNaN
values in the input before computing the correlation coefficients.'complete'
— Omit any rows of the input containingNaN
values before computing the correlation coefficients. This option always returns a positive semi-definite matrix.'pairwise'
— Omit any rows containingNaN
only on a pairwise basis for each two-column correlation coefficient calculation. This option can return a matrix that is not positive semi-definite.
Data Types: char
Output Arguments
R
— Correlation coefficients
matrix
Correlation coefficients, returned as a matrix.
For one matrix input,
R
has size[size(A,2) size(A,2)]
based on the number of random variables (columns) represented byA
. The diagonal entries are set to one by convention, while the off-diagonal entries are correlation coefficients of variable pairs. The values of the coefficients can range from -1 to 1, with -1 representing a direct, negative correlation, 0 representing no correlation, and 1 representing a direct, positive correlation.R
is symmetric.For two input arguments,
R
is a 2-by-2 matrix with ones along the diagonal and the correlation coefficients along the off-diagonal.If any random variable is constant, its correlation with all other variables is undefined, and the respective row and column value is
NaN
.
P
— P-values
matrix
P-values, returned as a matrix. P
is symmetric
and is the same size as R
. The diagonal entries
are all ones and the off-diagonal entries are the p-values for each
variable pair. P-values range from 0 to 1, where values close to 0
correspond to a significant correlation in R
and
a low probability of observing the null hypothesis.
RL
— Lower bound for correlation coefficient
matrix
Lower bound for correlation coefficient, returned as a matrix. RL
is
symmetric and is the same size as R
. The diagonal
entries are all ones and the off-diagonal entries are the 95% confidence
interval lower bound for the corresponding coefficient in R
.
The syntax returning RL
is invalid if R
contains
complex values.
RU
— Upper bound for correlation coefficient
matrix
Upper bound for correlation coefficient, returned as a matrix. RU
is
symmetric and is the same size as R
. The diagonal
entries are all ones and the off-diagonal entries are the 95% confidence
interval upper bound for the corresponding coefficient in R
.
The syntax returning RL
is invalid if R
contains
complex values.
More About
Correlation Coefficient
The correlation coefficient of two random variables is a measure of their linear dependence. If each variable has N scalar observations, then the Pearson correlation coefficient is defined as
where and are the mean and standard deviation of A, respectively, and and are the mean and standard deviation of B. Alternatively, you can define the correlation coefficient in terms of the covariance of A and B:
The correlation coefficient matrix of two random variables is the matrix of correlation coefficients for each pairwise variable combination,
Since A and B are always directly correlated to themselves, the diagonal entries are just 1, that is,
References
[1] Fisher, R.A. Statistical Methods for Research Workers, 13th Ed., Hafner, 1958.
[2] Kendall, M.G. The Advanced Theory of Statistics, 4th Ed., Macmillan, 1979.
[3] Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. Numerical Recipes in C, 2nd Ed., Cambridge University Press, 1992.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The
corrcoef
function supports tall arrays with the following usage
notes and limitations:
A
andB
must be tall arrays of the same size, even if both are vectors.Inputs
A
andB
cannot be scalars forcorrcoef(A,B)
.The second input
B
must be 2-D.The
'pairwise'
option is not supported.
For more information, see Tall Arrays.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
Row-vector input is only supported when the first two inputs are vectors and nonscalar.
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™.
The corrcoef
function
fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray
(Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
See Also
plotmatrix
| cov
| mean
| std
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