# Age-Length Key by the Hoening et al. (1993, 1994) method

### Description

Generation of Age-Length Keys (ALK) using incomplete data,
by the method proposed by Hoenig *et al.* (1993,
1994).

### Usage

1 2 3 |

### Arguments

`Ak` |
A list of |

`fik` |
A list of |

`fiz` |
A list of vectors of equal length ( |

`age_classes` |
A vector with the name of each age class. |

`length_classes` |
A vector with the name of each age class. |

`threshold` |
The value at which convergence is considered to be achieved: see ‘details’. |

`maxiter` |
The maximum number of iterations of the EM algorithm: see ‘details’. |

`name` |
A string with the name of the ALK. |

`description` |
A string describing the ALK. |

### Details

Calculates an ALK using the generalized method proposed by
Hoenig *et al.* (1993, 1994), which uses an undefined
number of data sets with known and unknown age information.

The returned `ALKr`

object contains information on the
convergence threshold that was used, the number of
iterations ran, and if convergence was reached.

#### Convergence

The method proposed by Hoenig *et al.* (1993, 1994) is
based on the EM algorithm as defined by Dempster *et
al.* (1997), and it generates the ALK by a series of
iterations which are repeated until convergence is
acheived.

Let `Nz`

be a list of matrices containing the number
of fish in each length and age class for each of the
`z`

populations with unknown age information and with
length distribution specified by `fiz`

. Convergence is
tested by evaluating the greatest of the absolute
differences between all pairs of `Nz`

matrices
generated on the current and previous iterations:
`max(mapply("-", Nz, Nz.old))`

.

### Value

A list of `ALKr`

objects, one for each item in the
`fiz`

list, each containing a matrix with the
probability of an individual of age `j`

having length
`i`

, i.e. *P(i|j)*, the vectors of age and length
classes, and information about the method used to generate
the key.

### References

Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum
Likelihood from Incomplete Data via the EM Algorithm.
*Journal of the Royal Statistical Society. Series B
(Methodological)*, **39**/1, 1-38. DOI:
`10.2307/2984875`

Hoenig, J.M., Heisey, D.M., Hanumara, R.C. (1993). Using
Prior and Current Information to Estimate Age Composition:
a new kind of age-length key. *ICES CM Documents
1993*, 10.

Hoenig, J.M., Heisey, D.M., Hanumara, R.C. (1994). A
computationally simple approach to using current and past
data in age-length key. *ICES CM Documents 1994*, 5.

### See Also

inverse_ALK kimura_chikuni hoenig_heisey gascuel

### Examples

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