Correlation coefficients

r = corrcoef(X) r = corrcoef(X,Y)

| Matrix where each row is an observation and each column is a variable. |

| Matrix where each row is an observation and each column is a variable. |

`corrcoef`

for financial time series objects
is based on the MATLAB^{®} `corrcoef`

function.
See `corrcoef`

in the MATLAB documentation.

`r=corrcoef(X)`

calculates a matrix `r`

of
correlation coefficients for an array `X`

, in which
each row is an observation and each column is a variable.

`r=corrcoef(X,Y)`

, where `X`

and `Y`

are
column vectors, is the same as `r=corrcoef([X Y])`

.
`corrcoef`

converts `X`

and `Y`

to
column vectors if they are not; that is, `r = corrcoef(X,Y)`

is
equivalent to `r=corrcoef([X(:) Y(:)])`

in that case.

If `c`

is the covariance matrix, ```
c=
cov(X)
```

, then `corrcoef(X)`

is the matrix
whose (* i,j*) 'th element is

`c`

`i,j`

`sqrt`

(`c`

`i,i`

`c`

(`j,j`

`[r,p]=corrcoef(...)`

also returns `p`

,
a matrix of `p`

-values for testing the hypothesis
of no correlation. Each `p`

-value is the probability
of getting a correlation as large as the observed value by random
chance, when the true correlation is zero. If `p`

(* i,j*)
is less than 0.05, then the correlation

`r`

(`i,j`

`[r,p,rlo,rup]=corrcoef(...)`

also returns
matrices `rlo`

and `rup`

, of the
same size as `r`

, containing lower and upper bounds
for a 95% confidence interval for each coefficient.

`[...]=corrcoef(...,'PARAM1',VAL1,'PARAM2',VAL2,...)`

specifies
additional parameters and their values. Valid parameters are:

`'alpha'`

— A number between 0 and 1 to specify a confidence level of 100*(1-ALPHA)%. Default is 0.05 for 95% confidence intervals.`'rows'`

— Either`'all'`

(default) to use all rows,`'complete'`

to use rows with no`NaN`

values, or`'pairwise'`

to compute`r`

() using rows with no`i,j`

`NaN`

values in columnor`i`

.`j`

The `p`

-value is computed by transforming the
correlation to create a t-statistic having `N`

–
2 degrees of freedom, where `N`

is the number of
rows of `X`

. The confidence bounds are based on an
asymptotic normal distribution of 0.5*log((1 + r)/(1 – r)),
with an approximate variance equal to 1/(N – 3). These bounds
are accurate for large samples when `X`

has a multivariate
normal distribution. The `'pairwise'`

option can
produce an `r`

matrix that is not positive definite.

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