# corrcoef

Correlation coefficients

## Syntax

``R = corrcoef(A)``
``R = corrcoef(A,B)``
``````[R,P] = corrcoef(___)``````
``````[R,P,RL,RU] = corrcoef(___)``````
``___ = corrcoef(___,Name,Value)``

## Description

example

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

example

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

example

``````[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.```

example

``````[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.```

example

````___ = 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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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 ```

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 ```

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 ```

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 ```

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

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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

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 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)`

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`

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 semi-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 semi-definite.

Data Types: `char`

## Output Arguments

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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-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.

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.

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,

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

 Fisher, R.A. Statistical Methods for Research Workers, 13th Ed., Hafner, 1958.

 Kendall, M.G. The Advanced Theory of Statistics, 4th Ed., Macmillan, 1979.

 Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. Numerical Recipes in C, 2nd Ed., Cambridge University Press, 1992.