Function calculates the heterogeneity matrix for the one-dimensional series.
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F |
the series to be checked for structural changes |
... |
further arguments passed to |
B |
integer, length of base series |
T |
integer, length of tested series |
L |
integer, window length for the decomposition of the base series |
neig |
integer, number of eigentriples to consider for calculating projections |
x |
'hmatr' object |
col |
color palette to use |
main |
plot title |
xlab,ylab |
labels for 'x' and 'y' axis |
The heterogeneity matrix (H-matrix) provides a consistent view on the structural discrepancy between different parts of the series. Denote by F_{i,j} the subseries of F of the form: F_{i,j} = ≤ft(f_{i},…,f_{j}\right). Fix two integers B > L and T ≥q L. Let these integers denote the lengths of base and test subseries, respectively. Introduce the H-matrix G_{B,T} with the elements g_{ij} as follows:
g_{ij} = g(F_{i,i+B}, F_{j,j+T}),
for i=1,…,N-B+1 and j=1,…,N-T+1, that is we split the series F into subseries of lengths B and T and calculate the heterogeneity index between all possible pairs of the subseries.
The heterogeneity index g(F^{(1)}, F^{(2)}) between the series F^{(1)} and F^{(2)} can be calculated as follows: let U_{j}^{(1)}, j=1,…,L denote the eigenvectors of the SVD of the trajectory matrix of the series F^{(1)}. Fix I to be a subset of ≤ft\{1,…,L\right\} and denote \mathcal{L}^{(1)} = \mathrm{span}\,≤ft(U_{i},\, i \in I\right). Denote by X^{(2)}_{1},…,X^{(2)}_{K_{2}} (K_{2} = N_{2} - L + 1) the L-lagged vectors of the series F^{(2)}. Now define
g(F^{(1)},F^{(2)}) = \frac{∑_{j=1}^{K_{2}}{\mathrm{dist}\,^{2}≤ft(X^{(2)}_{j}, \mathcal{L}^{(1)}\right)}} {∑_{j=1}^{K_{2}}{≤ft\|X^{(2)}_{j}\right\|^{2}}},
where \mathrm{dist}\,(X,\mathcal{L}) denotes the Euclidean distance between the vector X and the subspace \mathcal{L}. One can easily see that 0 ≤q g ≤q 1.
object of type 'hmatr'
Golyandina, N., Nekrutkin, V. and Zhigljavsky, A. (2001): Analysis of Time Series Structure: SSA and related techniques. Chapman and Hall/CRC. ISBN 1584881941
ssa
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