corrcoef

Correlation coefficients

Syntax

• ___ = corrcoef(___,Name,Value)
example

Description

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R = corrcoef(A) returns the matrix of correlation coefficients for A, where the columns of A represent random variables and the rows represent observations.

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R = corrcoef(A,B) returns coefficients between two random variables A and B.

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[R,P] = corrcoef(___) 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 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.

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[R,P,RL,RU] = corrcoef(___) includes matrices containing lower and upper bounds for a 95% confidence interval for each coefficient. This syntax is invalid if R contains complex elements.

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___ = corrcoef(___,Name,Value) returns any of the output arguments from the previous syntaxes with additional options specified by one or more 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

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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 =

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 =

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 =

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 =

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 coeffcients.

A = randn(50,3);
A(:,4) = sum(A,2);
[R,P,RL,RU] = corrcoef(A)
R =

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 =

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 =

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 =

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 alpha equal to 0.01 to compute a 99% confidence interval, which is reflected in the bounds RL and RU. The intervals defined by the coefficent 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 =

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 =

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 =

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 =

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 =

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 =

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 =

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 omited.

R = corrcoef(A,'rows','pairwise')
R =

1.0000   -0.3388    0.4649
-0.3388    1.0000   -0.9987
0.4649   -0.9987    1.0000

Input Arguments

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A — Input arraymatrix

Input array, specified as a matrix.

• If A is a scalar, corrcoef(A) returns NaN.

• If A is a vector, corrcoef(A) returns 1.

Data Types: single | double
Complex Number Support: Yes

B — Additional input arrayvector | matrix | multidimensional array

Additional input array, specified as a vector, matrix, or multidimensional array.

• A and B must be the same size.

• If A and B are scalars, then corrcoef(A,B) returns 1. If A and B are equal, however, corrcoef(A,B) returns NaN.

• If A and B are matrices or multidimensional arrays, then corrcoef(A,B) converts each input into its vector representation and is equivalent to corrcoef(A(:),B(:)) or corrcoef([A(:) B(:)]).

• If A and B are 0-by-0 empty arrays, corrcoef(A,B) returns a 2-by-2 matrix of NaN values.

Data Types: single | double
Complex Number Support: Yes

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: R = corrcoef(A,'alpha',0.03)

'alpha' — Significance level0.05 (default) | number between 0 and 1

Significance level, specified as a number between 0 and 1. alpha 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 all NaN values in the input before computing the correlation coefficients.

• 'complete' — Omit any rows of the input containing NaN values before computing the correlation coefficients. This option always returns a positive definite matrix.

• 'pairwise' — Omit any rows containing NaN only on a pairwise basis for each two-column correlation coefficient calculation. This option can return a matrix that is not positive definite.

Data Types: char

Output Arguments

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R — Correlation coefficientsmatrix

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 by A. 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-valuesmatrix

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 coefficientmatrix

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 coefficientmatrix

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.

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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

$\rho \left(A,B\right)=\frac{1}{N-1}\sum _{i=1}^{N}\left(\frac{\overline{{A}_{i}-{\mu }_{A}}}{{\sigma }_{A}}\right)\left(\frac{{B}_{i}-{\mu }_{B}}{{\sigma }_{B}}\right),$

where ${\mu }_{A}$ and ${\sigma }_{A}$ are the mean and standard deviation of A, respectively, and ${\mu }_{B}$ and ${\sigma }_{B}$ are the mean and standard deviation of B. Alternatively, you can define the correlation coefficient in terms of the covariance of A and B:

$\rho \left(A,B\right)=\frac{\mathrm{cov}\left(A,B\right)}{{\sigma }_{A}{\sigma }_{B}}.$

The correlation coefficient matrix of two random variables is the matrix of correlation coefficients for each pairwise variable combination,

$R=\left(\begin{array}{cc}\rho \left(A,A\right)& \rho \left(A,B\right)\\ \rho \left(B,A\right)& \rho \left(B,B\right)\end{array}\right).$

Since A and B are always directly correlated to themselves, the diagonal entries are just 1, that is,

$R=\left(\begin{array}{cc}1& \rho \left(A,B\right)\\ \rho \left(B,A\right)& 1\end{array}\right).$

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.