DCCA (version 0.1.1)

Kkronm: The product of Kronecker Product of some Arrays

Description

This is an auxiliary function and requires some context to be used adequadely. It computes equation (19) in Prass and Pumi (2019), returning a square matrix defined by $$K* = (Jm \%x\% J*)'(Q \%x\% Q)(Jm \%x\% J*)$$ where:

• \(J\) is an \((m+1)*(h+1) - m*h*s\) by \((m+1)*(h+1) - m*h*s\) lower triangular matrix with all non-zero entries equal to one, with \(s = 1\) if overlap = TRUE and \(s = 0\), otherwise;

• \(Jm\) corresponds to the first \(m+1\) rows and columns of \(J\);

• \(J*\) corresponds to the last \(m+1\) rows of \(J\);

• \(Q = I-P\), where \(P\) is the \(m+1\) by \(m+1\) projection matrix into the subspace generated by degree \(nu+1\) polynomials.

Usage

Kkronm(m = 3, nu = 0, h = 0, overlap = TRUE, K = NULL)

Arguments

m

a positive integer 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.

h

an integer indicating the lag.

overlap

logical: if true (the default), overlapping boxes are used for calculations. Otherwise, non-overlapping boxes are applied.

K

optional: the matrix defined by \(K = J'QJ\). This is used to calculate \(K* = (Jm \%x\% J*)'(Q \%x\% Q)(Jm \%x\% J*)\). For details see (19) in Prass and Pumi (2019). If this matrix is provided \(mu\) is ignored.

Value

an \((m+1)[(m+1)*(h+1) - m*h*s]\) by \((m+1)[(m+1)*(h+1) - m*h*s]\) matrix, where \(s = 1\) if overlap = TRUE and \(s = 0\), otherwise. This matrix corresponds to equation (19) 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 Km which creates \(K\).

Examples

# NOT RUN {
m = 3
h = 1
J = Jn(n = m+1+h)
Q = Qm(m = m, nu = 0)

# using K
K = Km(J = J[1:(m+1),1:(m+1)], Q = Q)
Kkron0 = Kkronm(K = K, h = h)

# using m and nu
Kkron = Kkronm(m = m, nu = 0, h = h)

# using  kronecker product from R
K = Km(J = J[1:(m+1),1:(m+1)], Q = Q)
Kh = rbind(matrix(0, nrow = h, ncol = m+1+h),
           cbind(matrix(0, nrow = m+1, ncol = h), K))
KkronR = K %x% Kh

# using  the definition K* = (Jm %x%  J)'(Q %x%  Q)(Jm %x%  J)
J_m = J[1:(m+1),1:(m+1)]
J_h = J[(h+1):(m+1+h),1:(m+1+h)]
KkronD = t(J_m %x%  J_h)%*%(Q %x%  Q)%*%(J_m %x%  J_h)


# comparing the results
sum(abs(Kkron0 - Kkron))
sum(abs(Kkron0 - KkronR))
sum(abs(Kkron0 - KkronD))  # difference due to rounding error

# }
# NOT RUN {
# Function Kkronm is computationaly faster than a pure implementation in R:

m = 100
h = 1
J = Jn(n = m+1)
Q = Qm(m = m, nu = 0)

# using Kkronm
t1 = proc.time()
Kkron = Kkronm(m = m, nu = 0, h = 1)
t2 = proc.time()
# elapsed time:
t2-t1

# Pure R implementation:
K = Km(J = J, Q = Q)
Kh = rbind(matrix(0, nrow = h, ncol = m+1+h),
           cbind(matrix(0, nrow = m+1, ncol = h), K))
t3 = proc.time()
KkronR = K %x% Kh
t4 = proc.time()
# elapsed time
t4-t3

# }