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- :
**cov***(*`x`) - :
**cov***(*`x`,`opt`) - :
**cov***(*`x`,`y`) - :
**cov***(*`x`,`y`,`opt`) Compute the covariance matrix.

If each row of

`x`and`y`is an observation, and each column is a variable, then the (`i`,`j`)-th entry of`cov (`

is the covariance between the`x`,`y`)`i`-th variable in`x`and the`j`-th variable in`y`.cov (

`x`) = 1/(N-1) * SUM_i (`x`(i) - mean(`x`)) * (`y`(i) - mean(`y`))where

*N*is the length of the`x`and`y`vectors.If called with one argument, compute

`cov (`

, the covariance between the columns of`x`,`x`)`x`.The argument

`opt`determines the type of normalization to use. Valid values are- 0:
normalize with

*N-1*, provides the best unbiased estimator of the covariance [default]- 1:
normalize with

*N*, this provides the second moment around the mean

Compatibility Note:: Octave always treats rows of

`x`and`y`as multivariate random variables. For two inputs, however, MATLAB treats`x`and`y`as two univariate distributions regardless of their shapes, and will calculate`cov ([`

whenever the number of elements in`x`(:),`y`(:)])`x`and`y`are equal. This will result in a 2x2 matrix. Code relying on MATLAB’s definition will need to be changed when running in Octave.**See also:**corr.

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**corr***(*`x`) - :
**corr***(*`x`,`y`) Compute matrix of correlation coefficients.

If each row of

`x`and`y`is an observation and each column is a variable, then the (`i`,`j`)-th entry of`corr (`

is the correlation between the`x`,`y`)`i`-th variable in`x`and the`j`-th variable in`y`.corr (

`x`,`y`) = cov (`x`,`y`) / (std (`x`) * std (`y`))If called with one argument, compute

`corr (`

, the correlation between the columns of`x`,`x`)`x`.**See also:**cov.

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`r`=**corrcoef***(*`x`) - :
`r`=**corrcoef***(*`x`,`y`) - :
`r`=**corrcoef***(…,*`param`,`value`, …) - :
*[*`r`,`p`] =**corrcoef***(…)* - :
*[*`r`,`p`,`lci`,`hci`] =**corrcoef***(…)* Compute a matrix of correlation coefficients.

`x`is an array where each column contains a variable and each row is an observation.If a second input

`y`(of the same size as`x`) is given then calculate the correlation coefficients between`x`and`y`.`param`,`value`are optional pairs of parameters and values which modify the calculation. Valid options are:`"alpha"`

Confidence level used for the bounds of the confidence interval,

`lci`and`hci`. Default is 0.05, i.e., 95% confidence interval.`"rows"`

Determine processing of NaN values. Acceptable values are

`"all"`

,`"complete"`

, and`"pairwise"`

. Default is`"all"`

. With`"complete"`

, only the rows without NaN values are considered. With`"pairwise"`

, the selection of NaN-free rows is made for each pair of variables.

Output

`r`is a matrix of Pearson’s product moment correlation coefficients for each pair of variables.Output

`p`is a matrix of pair-wise p-values testing for the null hypothesis of a correlation coefficient of zero.Outputs

`lci`and`hci`are matrices containing, respectively, the lower and higher bounds of the 95% confidence interval of each correlation coefficient.

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**spearman***(*`x`) - :
**spearman***(*`x`,`y`) -
Compute Spearman’s rank correlation coefficient

`rho`.For two data vectors

`x`and`y`, Spearman’s`rho`is the correlation coefficient of the ranks of`x`and`y`.If

`x`and`y`are drawn from independent distributions,`rho`has zero mean and variance`1 / (N - 1)`

, where*N*is the length of the`x`and`y`vectors, and is asymptotically normally distributed.`spearman (`

is equivalent to`x`)`spearman (`

.`x`,`x`)

- :
**kendall***(*`x`) - :
**kendall***(*`x`,`y`) -
Compute Kendall’s

`tau`.For two data vectors

`x`,`y`of common length*N*, Kendall’s`tau`is the correlation of the signs of all rank differences of`x`and`y`; i.e., if both`x`and`y`have distinct entries, then1

`tau`= ------- SUM sign (`q`(i) -`q`(j)) * sign (`r`(i) -`r`(j)) N (N-1) i,jin which the

`q`(i) and`r`(i) are the ranks of`x`and`y`, respectively.If

`x`and`y`are drawn from independent distributions, Kendall’s`tau`is asymptotically normal with mean 0 and variance`(2 * (2N+5)) / (9 * N * (N-1))`

.`kendall (`

is equivalent to`x`)`kendall (`

.`x`,`x`)

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