DCCA (version 0.1.1)

Km: Matrix K

Description

This is an auxiliary function which computes expression (18) in Prass and Pumi (2019). It creates an \(m+1\) by \(m+1\) matrix defined by \(K = J'QJ\) where \(J\) is a \(m+1\) by \(m+1\) lower triangular matrix with all non-zero entries equal to one and \(Q\) is a \(m+1\) by \(m+1\) given by \(Q = I - P\) where \(P\) is the projection matrix into the subspace generated by degree \(nu+1\) polynomials and \(I\) is the \(m+1\) by \(m+1\) identity matrix.

Usage

Km(m = 3, nu = 0, J = NULL, Q = NULL)

Arguments

m

a positive integer greater or equal than \(nu\) indicating the size of the window for the polinomial fit.

nu

a non-negative integer denoting the degree of the polinomial fit applied on the integrated series.

J, Q

optional: the matrices such that \(K = J'QJ\). If both matrices are provided, \(m\) and \(nu\) are ignored.

Value

an \(m+1\) by \(m+1\) matrix corresponding to expression (18) in Prass and Pumi (2019).

References

Prass, T.S. and Pumi, G. (2019). On the behavior of the DFA and DCCA in trend-stationary processes <arXiv:1910.10589>.

See Also

Jn which creates the matrix \(J\), Qm which creates \(Q\) and Pm which creates \(P\).

Examples

# NOT RUN {
K = Km(m = 3, nu = 0)
K
# same as
m = 3
J = Jn(n = m+1)
Q = Qm(m = m, nu = 0)
K = Km(J = J, Q = Q)
K

# }