Covariance function for functional data

Description

S3 method to compute the sample covariance and cross-covariance functions for a set of functional data.

Usage

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cov_fun(X, Y = NULL)

## S3 method for class 'fData'
cov_fun(X, Y = NULL)

## S3 method for class 'mfData'
cov_fun(X, Y = NULL)

Arguments

X

is the (eventually first) functional dataset, i.e. either an object of class fData or an object of class mfData;

Y

is the (optional) second functional dataset to be used to compute the cross-covariance function, either NULL or an fData or mfData object (see the Value section for details).

Details

Given a univariate random function X, defined over the grid I = [a,b], the covariance function is defined as:

C(s,t) = Cov( X(s), X(t) ), \qquad s,t \in I.

Given another random function, Y, defined over the same grid as X, the cross- covariance function of X and Y is:

C^{X,Y}( s,t ) = Cov( X(s), Y(t) ), \qquad s, t \in I.

For a generic L-dimensional random function X, i.e. an L-dimensional multivariate functional datum, the covariance function is defined as the set of blocks:

C_{i,j}(s,t) = Cov( X_i(s), X_j(t)), i,j = 1, ...,L, s,t \in I,

while the cross-covariance function is defined by the blocks:

C^{X,Y}_{i,j}(s,t) = Cov( X_i(s), Y_j(t))

The method cov_fun provides the sample estimator of the covariance or cross-covariance functions for univariate or multivariate functional datasets.

The class of X (fData or mfData) is used to dispatch the correct implementation of the method.

Value

The following cases are given:

  • if X is of class fData and Y is NULL, then the covariance function of X is returned;

  • if X is of class fData and Y is of class fData, the cross-covariance function of the two datasets is returned;

  • if X is of class mfData and Y is NULL, the upper-triangular blocks of the covariance function of X are returned (in form of list and by row, i.e. in the squence 1_1, 1_2, ..., 1_L, 2_2, ... - have a look at the labels of the list with str);

  • if X is of class mfData and Y is of class fData, the cross-covariances of X's components and Y are returned (in form of list);

  • if X is of class mfData and Y is of class mfData, the upper-triangular blocks of the cross-covariance of X's and Y's components are returned (in form of list and by row, i.e. in the squence 1_1, 1_2, ..., 1_L, 2_2, ... - have a look at the labels of the list with str));

In any case, the return type is either an instance of the S3 class Cov or a list of instances of such class (for the case of multivariate functional data).

See Also

fData, mfData, plot.Cov

Examples

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# Generating a univariate functional dataset
N = 1e2

P = 1e2
t0 = 0
t1 = 1

time_grid = seq( t0, t1, length.out = P )

Cov = exp_cov_function( time_grid, alpha = 0.3, beta = 0.4 )

D1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ), Cov = Cov )
D2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ), Cov = Cov )

fD1 = fData( time_grid, D1 )
fD2 = fData( time_grid, D2 )

# Computing the covariance function of fD1

C = cov_fun( fD1 )
str( C )

# Computing the cross-covariance function of fD1 and fD2
CC = cov_fun( fD1, fD2 )
str( CC )

# Generating a multivariate functional dataset
L = 3

C1 = exp_cov_function( time_grid, alpha = 0.1, beta = 0.2 )
C2 = exp_cov_function( time_grid, alpha = 0.2, beta = 0.5 )
C3 = exp_cov_function( time_grid, alpha = 0.3, beta = 1 )

centerline = matrix( c( sin( 2 * pi * time_grid ),
                        sqrt( time_grid ),
                        10 * ( time_grid - 0.5 ) * time_grid ),
                     nrow = 3, byrow = TRUE )

D3 = generate_gauss_mfdata( N, L, centerline,
                       correlations = c( 0.5, 0.5, 0.5 ),
                       listCov = list( C1, C2, C3 ) )

# adding names for better readability of BC3's labels
names( D3 ) = c( 'comp1', 'comp2', 'comp3' )
mfD3 = mfData( time_grid, D3 )

D1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ),
                              Cov = Cov )
fD1 = fData( time_grid, D1 )

# Computing the block covariance function of mfD3
BC3 = cov_fun( mfD3 )
str( BC3 )

# computing cross-covariance between mfData and fData objects
CC = cov_fun( mfD3, fD1 )
str( CC )

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