Km {DCCA}

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).

Author(s)

Taiane Schaedler Prass

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

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