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

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`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. |

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.

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))`

.

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.

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.

inverse_ALK kimura_chikuni hoenig_heisey gascuel

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