# (Modified) Band Depth for multivariate functional data

### Description

These functions compute the Band Depth (BD) and Modified Band Depth (MBD) of elements of a multivariate functional dataset.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
multiMBD(Data, weights = "uniform", manage_ties = FALSE)
## S3 method for class 'mfData'
multiMBD(Data, weights = "uniform", manage_ties = FALSE)
## Default S3 method:
multiMBD(Data, weights = "uniform", manage_ties = FALSE)
multiBD(Data, weights = "uniform")
## S3 method for class 'mfData'
multiBD(Data, weights = "uniform")
## Default S3 method:
multiBD(Data, weights = "uniform")
``` |

### Arguments

`Data` |
specifies the the multivariate functional dataset.
It is either an object of class |

`weights` |
either a set of weights (of the same length of |

`manage_ties` |
a logical flag specifying whether the check for ties and
the relative treatment is to be carried out while computing the MBDs in each
dimension. It is directly passed to |

### Details

Given a multivariate functional dataset composed of *N* elements with
*L* components each, *\mathbf{X_1} =( X^1_1(t),* *X^2_1(t),
…, X^L_1(t))*, and a set of *L* non-negative weights,

* w_1, w_2, …, w_L, \qquad ∑_{i=1}^L w_i = 1,*

these functions compute the BD and MBD of each element of the functional dataset, namely:

* BD( \mathbf{X_j} ) = ∑_{i=1}^{L} w_i BD( X^i_j ), \quad \forall
j = 1, … N.*

* MBD( \mathbf{X_j} ) = ∑_{i=1}^{L} w_i MBD( X^i_j ), \quad \forall
j = 1, … N.*

### Value

The function returns a vector containing the depths of each element of the multivariate functional dataset.

### References

Ieva, F. and Paganoni, A. M. (2013). Depth measures for multivariate
functional data, *Communications in Statistics: Theory and Methods*,
41, 1265-1276.

Tarabelloni, N., Ieva, F., Biasi, R. and Paganoni, A. M. (2015). Use of
Depth Measure for Multivariate Functional Data in Disease Prediction: An
Application to Electrocardiograph Signals, *International Journal of
Biostatistics*, 11.2, 189-201.

### See Also

`MBD`

, `BD`

, `toListOfValues`

,
`mfData`

### 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 | ```
N = 20
P = 1e3
grid = seq( 0, 10, length.out = P )
# Generating an exponential covariance function to be used to simulate gaussian
# functional data
Cov = exp_cov_function( grid, alpha = 0.2, beta = 0.8 )
# First component of the multivariate guassian functional dataset
Data_1 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )
# First component of the multivariate guassian functional dataset
Data_2 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )
mfD = mfData( grid, list( Data_1, Data_2 ) )
multiBD( mfD, weights = 'uniform' )
multiMBD( mfD, weights = 'uniform', manage_ties = TRUE )
multiBD( mfD, weights = c( 1/3, 2/3 ))
multiMBD( mfD, weights = c( 1/3, 2/3 ), manage_ties = FALSE )
multiBD( list( Data_1, Data_2 ), weights = 'uniform')
multiMBD( list( Data_1, Data_2 ), weights = 'uniform', manage_ties = TRUE )
multiBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ))
multiMBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ), manage_ties = FALSE )
``` |

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