# Kendall's tau correlation coefficient for bivariate functional data

### Description

This function computes the Kendall's tau correlation coefficient for a bivariate functional dataset, with either a max or area-under-curve order order relation between univariate functional elements (components).

### Usage

1 | ```
cor_kendall(mfD, ordering = "max")
``` |

### Arguments

`mfD` |
a bivariate functional dataset whose Kendall's tau
coefficient must be computed, in form of bivariate |

`ordering` |
the ordering relation to use on functional observations,
either |

### Details

Given a bivariate functional dataset, with first components *X_1(t),
X_2(t), …, X_N(t)* and second components *Y_1(t), Y_2(t), …,
Y_N(t)*, the function exploits either the order relation based on the maxima
or the area-under-curve relation to compare data and produce concordances and
discordances, that are then used to compute the tau coefficient.

See the references for more details.

### Value

The function returns the Kendall's tau correlation coefficient for
the bivariate dataset provided with `mfData`

.

### References

Valencia, D., Romo, J. and Lillo, R. (2015). A Kendall correlation
coefficient for functional dependence,
*Universidad Carlos III de Madrid technical report*,
`http://EconPapers.repec.org/RePEc:cte:wsrepe:ws133228`

.

### See Also

`mfData`

, `area_ordered`

,
`max_ordered`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | ```
#### TOTALLY INDEPENDENT COMPONENTS
N = 2e2
P = 1e3
grid = seq( 0, 1, length.out = P )
# Creating an exponential covariance function to simulate guassian data
Cov = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )
# Simulating (independent) gaussian functional data with given center and
# covariance function
Data_1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )
Data_2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )
# Using the simulated data as (independent) components of a bivariate functional
# dataset
mfD = mfData( grid, list( Data_1, Data_2 ) )
# Correlation approx. zero (components were created independently)
cor_kendall( mfD, ordering = 'max' )
# Correlation approx. zero (components were created independently)
cor_kendall( mfD, ordering = 'area' )
#### TOTALLY DEPENDENT COMPONENTS
# Nonlinear transform of first component
Data_3 = t( apply( Data_1, 1, exp ) )
# Creating bivariate dataset starting from nonlinearly-dependent components
mfD = mfData( grid, list( Data_1, Data_3 ) )
# Correlation very high (components are nonlinearly dependent)
cor_kendall( mfD, ordering = 'max' )
# Correlation very high (components are nonlinearly dependent)
cor_kendall( mfD, ordering = 'area' )
``` |