# dcor.xy

##### Distance Correlation Statistic and t-Test

Distance correlation t-test of multivariate and functional independence (wrapper functions of energy package).

- Keywords
- multivariate, htest, nonparametric

##### Usage

`dcor.xy(x, y, test = TRUE, metric.x, metric.y, par.metric.x, par.metric.y, n)`dcor.dist(D1, D2)

bcdcor.dist(D1, D2, n)

dcor.test(D1, D2, n)

##### Arguments

- x
data (fdata, matrix or data.frame class) of first sample.

- y
data (fdata, matrix or data.frame class) of second sample.

- test
if TRUE, compute bias corrected distance correlation statistic and the corresponding t-test, else compute distance correlation statistic.

- metric.x, metric.y
Name of metric or semi-metric function used for compute the distances of

`x`

and`y`

object respectively. By default,`metric.lp`

for functional data and`metric.dist`

for multivariate data.- par.metric.x, par.metric.y
List of parameters for the corresponding metric function.

- n
The sample size used in bias corrected version of distance correlation, by default is the number of rows of

`x`

.- D1
Distances of first sample (x data).

- D2
Distances of second sample (y data).

##### Details

These wrapper functions extend the functions of the `energy`

package
for multivariate data to functional data. Distance correlation is a measure
of dependence between random vectors introduced by Szekely, Rizzo, and
Bakirov (2007).
`dcor.xy`

performs a nonparametric t-test of multivariate or functional
independence in high dimension. The distribution of the test statistic is
approximately Student t with \(n(n-3)/2-1\) degrees of freedom and for
\(n \geq 10\) the statistic is approximately distributed as standard
normal. Wrapper function of `energy:::dcor.ttest`

. The t statistic is
a transformation of a bias corrected version of distance correlation (see SR
2013 for details). Large values (upper tail) of the t statistic are
significant.
`dcor.test`

similar to `dcor.xy`

but only for distance matrix.
`dcor.dist`

compute distance correlation statistic. Wrapper function
of `energy::dcor`

but only for distance matrix
`bcdcor.dist`

compute bias corrected distance correlation statistic.
Wrapper function of `energy:::bcdcor`

but only for distance matrix.

##### Value

`dcor.test`

returns a list with class `htest`

containing

`method`

description of test`statistic`

observed value of the test statistic`parameter`

degrees of freedom`estimate`

bias corrected distance correlation`bcdcor(x,y)`

`p.value`

p-value of the t-test`data.name`

description of data

`dcor.xy`

returns the previous list with class `htest`

and

`D1`

the distance matrix of`x`

`D2`

the distance matrix of`y`

`dcor.dist`

returns the distance correlation statistic.

`bcdcor.dist`

returns the bias corrected distance correlation
statistic.

##### References

Szekely, G.J. and Rizzo, M.L. (2013). The distance correlation
t-test of independence in high dimension. *Journal of Multivariate
Analysis*, Volume 117, pp. 193-213.

Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007), Measuring and Testing
Dependence by Correlation of Distances, *Annals of Statistics*, Vol. 35
No. 6, pp. 2769-2794.

##### See Also

`metric.lp`

amd `metric.dist`

.

##### Examples

```
# NOT RUN {
x<-rproc2fdata(100,1:50)
y<-rproc2fdata(100,1:50)
dcor.xy(x, y,test=TRUE)
dx <- metric.lp(x)
dy <- metric.lp(y)
dcor.test(dx, dy)
bcdcor.dist(dx, dy)
dcor.xy(x, y,test=FALSE)
dcor.dist(dx, dy)
# }
# NOT RUN {
# }
```

*Documentation reproduced from package fda.usc, version 2.0.1, License: GPL-2*