# Calculate the Linear Recurrence Relation

### Description

Calculates the Linear Recurrence Relation given the one-dimensional 'ssa' object.

### Usage

1 2 3 4 5 6 7 8 9 10 11 | ```
## S3 method for class '1d.ssa'
lrr(x, groups, reverse = FALSE, ..., drop = TRUE)
## S3 method for class 'toeplitz.ssa'
lrr(x, groups, reverse = FALSE, ..., drop = TRUE)
## Default S3 method:
lrr(x, eps = sqrt(.Machine$double.eps),
reverse = FALSE, ..., orthonormalize = TRUE)
## S3 method for class 'lrr'
roots(x, ..., method = c("companion", "polyroot"))
## S3 method for class 'lrr'
plot(x, ..., raw = FALSE)
``` |

### Arguments

`x` |
SSA object holding the decomposition or matrix containing the basis vectors in columns
for |

`groups` |
list, the grouping of eigentriples used to derive the LRR |

`reverse` |
logical, if 'TRUE', then LRR is assumed to go back |

`...` |
further arguments to be passed to |

`drop` |
logical, if 'TRUE' then the result is coerced to lrr object itself, when possible (length of 'groups' is one) |

`eps` |
Tolerance for verticality checking |

`method` |
methods used for calculation of the polynomial roots: via eigenvalues
of companion matrix or R's standard |

`raw` |
logical, if 'TRUE' then |

`orthonormalize` |
logical, if 'FALSE' then the basis is assumed orthonormal. Otherwise, orthonormalization is performed |

### Details

Produces the linear recurrence relation from the series. The default implementation works as follows.

Denote by *U_i* the columns of matrix *x*. Denote by
*\tilde{U}_{i}* the same vector *U_i* but without the
last coordinate. Denote the last coordinate of *U_i* by
*π_i*. The returned value is

*
\mathcal{R} = \frac{1}{1-ν^2}∑_{i=1}^{d}{π_i \tilde{U}_{i}},
*

where

*
ν^2 = π_1^2 + … + π_d^2.
*

For `lrr.ssa`

case the matrix *U* used is the matrix of basis
vector corresponding to the selected elementary series.

For `reverse = 'TRUE'`

everything is the same, besides the
last coordinate substituted for the first coordinate.

### Value

Named list of object of class 'lrr' for `lrr`

function call,
where elements have the same names as elements of `groups`

(if group is unnamed, corresponding component gets name ‘Fn’,
where ‘n’ is its index in `groups`

list).
Or the object itself if 'drop = TRUE' and groups has length one.

Vector with the roots of the of the characteristic
polynomial of the LRR for `roots`

function call. Roots are
ordered by moduli decreasing.

### See Also

`Rssa`

for an overview of the package, as well as,
`ssa`

,
`parestimate`

,

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 | ```
# Decompose 'co2' series with default parameters
s <- ssa(co2, L = 24)
# Calculate the LRR out of first 3 eigentriples
l <- lrr(s, groups = list(1:3))
# Calculate the roots of the LRR
r <- roots(l)
# Moduli of the roots
Mod(r)
# Periods of three roots with maximal moduli
2*pi/Arg(r)[1:3]
# Plot the roots
plot(l)
``` |