# Calculate the heterogeneity matrix.

### Description

Function calculates the heterogeneity matrix for the one-dimensional series.

### Usage

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### Arguments

`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 |

### Details

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*.

### Value

object of type 'hmatr'

### References

Golyandina, N., Nekrutkin, V. and Zhigljavsky, A. (2001): *Analysis of
Time Series Structure: SSA and related techniques.* Chapman and Hall/CRC. ISBN 1584881941

### See Also

`ssa`

### Examples

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