# Towards Age Determination of Southern King Crab (Lithodes santolla) Off Southern Chile Using Flexible Mixture Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Finite Mixtures of Flexible Distributions

- (a)
- when $U\sim Gamma(\nu /2,\nu /2)$ in (3), $\nu >0$, we obtain the skew-t distribution (ST) [24] denoted by $Z\sim ST(\xi ,{\sigma}^{2},\eta ,\nu )$ and with the pdf given by$$\begin{array}{c}\hfill f\left(z\right)=\frac{2}{\sigma}t({z}_{0};\nu )T\left(\right)open="("\; close=")">\eta {z}_{0}\sqrt{\frac{\nu +1}{\nu +{z}_{0}^{2}}};\nu +1,\end{array}$$
- (b)
- The stochastic representation (2) of SMSN allows to determine the exact density of conditional distributions necessary for the ECME algorithm. That is, using Lemma 2 of Basso et al. [23], variable Y in (2) is represented conveniently by a hierarchical representation and, in this form, makes it possible to obtain the conditional maximization (CM) steps.

#### 2.2. Growth Modeling

#### 2.3. Implementation

**Modal decomposition.**The FMST model was carried out for southern king crab LFD with $\nu $ degrees of freedom and m components by zone, year, and sex. For example, degrees of freedom $\nu =5$ indicate a high presence of heavy-tails in LFD [15,16]. The order of m depends on the reported BIC for each combination, where the ’best’ model for each m is selected through the smallest BIC.**Age-class assignment.**From Step 1, take into account that m modes provide m modes used for age-class determination, which is at least m. Considering the classified carapace lengths, a bar chart of means is built where gray and white colors alternate and represent classified age classes. The cluster means are ordered and grouped into cohorts, so that no year is repeated within each group. “Premise II” of Reference [11] is considered as a criterion to determine the cohort point between age classes; textually, this is: strong assumption that each year no more than one cohort (means that it is not possible that two mean carapace lengths with the same year index fall into the same age class) and no less than one cohort (means that it is not possible that two consecutive mean carapace lengths with the same year index fall into two nonconsecutive age classes) enters the population. Given that age classes 0 and 1 include individuals with molt frequency decreasing with age (six to seven molts in the first and four to five in the second year), we opted to label the first group with Year ’2’ and then estimate the vBGF parameters.**vBGF model.**Given the estimated year in Step 2, the formed age-length data serve to evaluate the vBGF (5) growth function.

## 3. Computational Implementation

#### 3.1. Data

#### 3.2. Computational Aspects

`R`[27]. All computational estimations was made under Linux v. 4.15 and MacOS v. 10.13 operating systems. Particularly, to estimate the mixture of distributions, we used the mixsmsn package, developed by Prates et al. [28]. The mixsmsn package considers the Expectation-Maximization algorithm [29] for FMST modal decomposition. For the vBGF estimates and initial values, the

`nls`and

`FSA`packages were used, respectively. More details appear in Reference [30].

## 4. Results

#### 4.1. Males

#### 4.2. Females

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIC | Akaike’s information criterion |

BIC | Bayesian information criterion |

CM | Conditional maximization |

ECME | Expectation/conditional maximization either |

FMN | Finite mixture of normal |

FMST | Finite mixture of skew-t |

FM-SMSN | Finite mixture of scale mixtures of skew-normal |

LFD | Length-frequency data |

LQ | Lower quantile |

SMSN | Scale mixtures of skew-normal |

SE | Standard error |

SN | Skew-normal |

ST | Skew-t |

UQ | Upper quantile |

vBGF | von Bertalanffy growth function |

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**Figure 1.**Study area restricted to Chile (${50}^{\circ}$06${}^{\prime}$–${55}^{\circ}$59${}^{\prime}$ S, ${76}^{\circ}$36${}^{\prime}$–${66}^{\circ}$41${}^{\prime}$ W). Red and green points correspond to northern (${50}^{\circ}$06${}^{\prime}$–${53}^{\circ}$15${}^{\prime}$ S, ${76}^{\circ}$36${}^{\prime}$–${72}^{\circ}$18${}^{\prime}$ W) and southern (${54}^{\circ}$13${}^{\prime}$–${55}^{\circ}$59${}^{\prime}$ S, ${69}^{\circ}$40${}^{\prime}$–${66}^{\circ}$41${}^{\prime}$ W) individuals, respectively.

**Figure 3.**(

**a**) Barplot of assigned age from FMST modal decomposition, (

**b**) vBGF model fits for estimated age-length, and (

**c**) its respective absolute residuals for male southern king crabs.

**Figure 5.**(

**a**) Barplot of assigned age from FMST modal decomposition, (

**b**) vBGF model fits for estimated age-length, and (

**c**) its respective absolute residuals for female southern king crabs.

**Table 1.**Descriptive statistics for the southern king crab grouped by sex and year. LQ and UQ stand for Lower Quantile and Upper Quantile, respectively.

Sex | Year | Min. | LQ | Median | Mean | UQ | Max. | SD | n |
---|---|---|---|---|---|---|---|---|---|

Males | 2007 | 34 | 111 | 122 | 120.1 | 132 | 176 | 17.86 | 1734 |

2008 | 39 | 112 | 122 | 121.340 | 132 | 193 | 18.567 | 4983 | |

2009 | 31 | 113 | 125 | 123.604 | 136 | 177 | 18.158 | 5759 | |

2010 | 39 | 104 | 116 | 115.230 | 127 | 171 | 18.261 | 3424 | |

2011 | 14 | 106 | 121 | 118.791 | 134 | 175 | 21.725 | 6282 | |

2012 | 60 | 108 | 119 | 116.589 | 128 | 170 | 16.636 | 5453 | |

2013 | 34 | 113 | 124 | 121.719 | 134 | 179 | 18.437 | 4256 | |

2014 | 39 | 115 | 126 | 125.210 | 136 | 182 | 17.499 | 80,612 | |

2015 | 58 | 107 | 116 | 115.510 | 126 | 171 | 16.132 | 13,683 | |

Females | 2007 | 35 | 100 | 112 | 110.730 | 122 | 162 | 16.426 | 1586 |

2008 | 35 | 104 | 113 | 112.751 | 122 | 165 | 15.365 | 5883 | |

2009 | 33 | 105 | 113 | 113.319 | 121 | 168 | 14.105 | 7717 | |

2010 | 46 | 98 | 108 | 107.192 | 116 | 165 | 15.137 | 3273 | |

2011 | 30 | 94 | 105 | 105.844 | 117 | 165 | 17.602 | 7366 | |

2012 | 61 | 93 | 101 | 100.306 | 108 | 152 | 11.554 | 5530 | |

2013 | 43 | 102 | 110 | 109.949 | 118 | 167 | 14.113 | 3508 | |

2014 | 38 | 104 | 112 | 111.552 | 120 | 212 | 14.364 | 59,090 | |

2015 | 58 | 97 | 104 | 103.903 | 112 | 144 | 11.156 | 22,852 |

**Table 2.**Bayesian information criterion (BIC) values of the finite mixture of normal (FMN) and finite mixture of skew (FMST) fitted models ($m=2,\dots ,9$) for each specification (sex and year). The smallest values for each model and year are marked in bold.

Number of Modes (m) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Sex | Model | Year | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Males | FMN | 2007 | 21,660.32 | 25,110.58 | 28,584.18 | 32,027.82 | 35,511.00 | 38,942.26 | 42,404.44 | 45,867.89 |

2008 | 62,812.42 | 72,732.02 | 82,687.67 | 92,642.98 | 102,607.25 | 112,582.18 | 122,534.11 | 132,498.96 | ||

2009 | 72,131.15 | 83,620.96 | 95,128.46 | 106,629.39 | 118,143.52 | 129,656.02 | 141,144.20 | 152,665.18 | ||

2010 | 43,261.16 | 50,089.41 | 56,927.93 | 63,774.23 | 70,585.84 | 77,434.39 | 84,261.89 | 91,108.33 | ||

2011 | 81,316.49 | 93,847.13 | 106,403.04 | 118,966.49 | 131,524.34 | 144,081.13 | 156,638.06 | 169,188.31 | ||

2012 | 67,406.53 | 78,299.57 | 89,191.86 | 100,106.07 | 111,008.67 | 121,889.19 | 132,817.15 | 143,720.88 | ||

2013 | 53,475.91 | 61,958.09 | 70,463.77 | 78,974.94 | 87,477.20 | 95,984.53 | 104,494.43 | 113,008.14 | ||

2014 | 1,010,474.45 | 1,171,030.24 | 1,331,999.31 | 1,493,074.55 | 1,654,157.23 | 1,815,515.11 | 1,976,135.29 | 2,137,650.73 | ||

2015 | 169,223.51 | 196,429.18 | 223,791.31 | 251,162.38 | 278,521.06 | 305,856.42 | 332,855.51 | 360,116.56 | ||

FMST | 2007 | 13,364.51 | 13,371.75 | 13,375.64 | 14,843.43 | 14,878.02 | 14,892.64 | 14,910.46 | 14,932.68 | |

2008 | 48,624.99 | 48,585.88 | 48,595.47 | 42,986.84 | 43,009.54 | 43,059.00 | 43,061.88 | 43,096.80 | ||

2009 | 62,001.87 | 61,869.83 | 61,871.81 | 49,233.68 | 49,267.38 | 49,274.14 | 49,321.24 | 49,333.27 | ||

2010 | 27,032.18 | 27,020.63 | 27,026.80 | 29,699.00 | 29,729.53 | 29,749.19 | 29,785.83 | 29,779.53 | ||

2011 | 63,056.19 | 63,046.12 | 62,964.81 | 56,325.63 | 56,358.11 | 56,391.05 | 56,417.41 | 56,439.62 | ||

2012 | 42,731.45 | 42,730.32 | 42,738.57 | 45,750.64 | 45,783.98 | 45,817.96 | 45,845.73 | 45,874.98 | ||

2013 | 28,366.28 | 28,334.04 | 28,340.50 | 36,570.12 | 36,607.13 | 36,640.77 | 36,676.77 | 36,698.06 | ||

2014 | 687,830.5 | 687,524.6 | 687,307.9 | 687,305.38 | 687,289.56 | 687,347.91 | 687,457.90 | 687,328.14 | ||

2015 | 114,496.6 | 114,461.6 | 114,494.8 | 114,453.80 | 114,554.55 | 114,353.79 | 114,252.30 | 11,4437.46 | ||

Females | FMN | 2007 | 19,713.96 | 22,852.88 | 26,023.90 | 29,192.11 | 32,359.81 | 35,532.03 | 38,696.17 | 41,875.65 |

2008 | 72,139.24 | 83,843.33 | 95,605.08 | 107,358.69 | 119,124.74 | 130,859.57 | 142,656.47 | 154,376.82 | ||

2009 | 92,895.77 | 108,137.59 | 123,550.34 | 138,974.63 | 154,408.95 | 169,848.59 | 185,267.58 | 200,701.04 | ||

2010 | 40,113.74 | 46,619.98 | 53,153.97 | 59,699.58 | 66,235.46 | 72,768.60 | 79,308.71 | 85,864.49 | ||

2011 | 92,517.32 | 107,138.55 | 121,826.14 | 136,551.47 | 151,272.19 | 166,021.39 | 180,718.50 | 195,463.84 | ||

2012 | 64,868.23 | 75,887.82 | 86,941.12 | 97,998.73 | 109,057.37 | 120,093.74 | 131,144.66 | 142,200.95 | ||

2013 | 42,350.73 | 49,353.13 | 56,369.57 | 63,378.44 | 70,397.06 | 77,409.93 | 844,25.82 | 91,438.30 | ||

2014 | 716,320.29 | 834,395.94 | 951,934.44 | 1,070,071.78 | 1,188,213.60 | 1,306,378.32 | 1,424,541.82 | 1,542,624.05 | ||

2015 | 266,204.82 | 311,746.05 | 357,317.35 | 403,020.59 | 448,584.81 | 494,297.28 | 539,896.96 | 585,635.92 | ||

FMST | 2007 | 13,402.09 | 13,430.81 | 13,456.18 | 13,478.17 | 13,504.66 | 13,532.48 | 13,553.48 | 13,570.40 | |

2008 | 48,671.75 | 48,659.36 | 48,695.67 | 48,725.50 | 48,763.35 | 48,781.16 | 48,803.79 | 48,834.31 | ||

2009 | 62,050.53 | 61,946.29 | 61,976.07 | 62,004.46 | 62,041.64 | 62,066.87 | 62,104.48 | 61,535.19 | ||

2010 | 27,074.84 | 27,087.65 | 27,118.21 | 27,143.20 | 27,167.77 | 27,189.12 | 27,213.60 | 27,237.89 | ||

2011 | 63,104.52 | 63,122.07 | 63,068.38 | 63,074.54 | 63,102.86 | 63,149.36 | 63,181.02 | 63,210.95 | ||

2012 | 42,777.77 | 42,803.12 | 42,837.84 | 42,876.89 | 42,906.37 | 42,925.71 | 42,971.24 | 42,985.32 | ||

2013 | 28,409.42 | 28,401.83 | 28,432.94 | 28,465.27 | 28,492.60 | 28,527.44 | 28,556.07 | 28,583.88 | ||

2014 | 480,353.4 | 480,016.9 | 480,040.3 | 479,967.99 | 479,778.60 | 479,668.53 | 479,583.39 | 479,593.39 | ||

2015 | 174,927.2 | 174,939.8 | 174,752.3 | 174,863.17 | 174,746.23 | 174,709.63 | 174,691.60 | 174,687.12 |

Sex | Year | ${\mathit{\mu}}_{1}$ | ${\mathit{\mu}}_{2}$ | ${\mathit{\mu}}_{3}$ | ${\mathit{\mu}}_{4}$ | ${\mathit{\mu}}_{5}$ | ${\mathit{\mu}}_{6}$ | ${\mathit{\mu}}_{7}$ | ${\mathit{\mu}}_{8}$ | ${\mathit{\mu}}_{9}$ |
---|---|---|---|---|---|---|---|---|---|---|

Males | 2007 | 117.098 | 120.580 | - | - | - | - | - | - | - |

2008 | 79.255 | 103.574 | 121.668 | 126.899 | 142.613 | - | - | - | - | |

2009 | 58.301 | 102.911 | 115.216 | 122.973 | 138.269 | - | - | - | - | |

2010 | 101.505 | 119.793 | 125.396 | - | - | - | - | - | - | |

2011 | 81.542 | 100.494 | 116.096 | 123.987 | 140.691 | - | - | - | - | |

2012 | 91.457 | 121.495 | 125.046 | - | - | - | - | - | - | |

2013 | 94.073 | 126.019 | 128.364 | - | - | - | - | - | - | |

2014 | 95.159 | 115.189 | 126.915 | 131.681 | 142.104 | 152.129 | - | - | - | |

2015 | 76.686 | 82.263 | 98.657 | 112.971 | 119.165 | 130.680 | 141.562 | 149.571 | - | |

Females | 2007 | 106.905 | 115.245 | - | - | - | - | - | - | - |

2008 | 103.470 | 116.723 | 123.758 | - | - | - | - | - | - | |

2009 | 60.367 | 63.205 | 81.713 | 93.794 | 100.643 | 107.161 | 111.463 | 121.416 | 132.965 | |

2010 | 98.687 | 106.437 | 125.761 | - | - | - | - | - | - | |

2011 | 83.598 | 101.347 | 110.181 | 127.319 | - | - | - | - | - | |

2012 | 95.546 | 99.692 | - | - | - | - | - | - | - | |

2013 | 104.431 | 106.143 | 121.669 | - | - | - | - | - | - | |

2014 | 83.145 | 94.126 | 101.085 | 107.903 | 113.760 | 116.014 | 125.594 | 137.902 | - | |

2015 | 75.346 | 86.977 | 97.093 | 103.001 | 107.625 | 109.844 | 112.652 | 115.984 | 119.786 |

**Table 4.**Von Bertalanffy growth function (vBGF) estimates, standard error (SE), Student (t) value, and p-value, Pr($>\left|t\right|$), for the southern king crab for each sex. Estimated correlations between vBGF parameters are in the last three columns.

Sex | Parameter | Estimate | SE | t | Pr($>\left|\mathit{t}\right|$) | ${\mathit{L}}_{\mathbf{\infty}}$ | K | ${\mathit{t}}_{0}$ |
---|---|---|---|---|---|---|---|---|

Males | ${L}_{\infty}$ | 176.756 | 12.865 | 13.739 | <0.001 | 1.000 | - | - |

K | 0.151 | 0.032 | 4.773 | <0.001 | −0.985 | 1.000 | - | |

${t}_{0}$ | −1.678 | 0.519 | −3.231 | 0.002 | −0.897 | 0.956 | 1.000 | |

Females | ${L}_{\infty}$ | 134.799 | 4.065 | 33.162 | <0.001 | 1.000 | - | - |

K | 0.220 | 0.031 | 7.029 | <0.001 | −0.950 | 1.000 | - | |

${t}_{0}$ | −1.302 | 0.442 | −2.946 | 0.005 | −0.830 | 0.952 | 1.000 |

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## Share and Cite

**MDPI and ACS Style**

Contreras-Reyes, J.E.; López Quintero, F.O.; Yáñez, A.A.
Towards Age Determination of Southern King Crab (*Lithodes santolla*) Off Southern Chile Using Flexible Mixture Modeling. *J. Mar. Sci. Eng.* **2018**, *6*, 157.
https://doi.org/10.3390/jmse6040157

**AMA Style**

Contreras-Reyes JE, López Quintero FO, Yáñez AA.
Towards Age Determination of Southern King Crab (*Lithodes santolla*) Off Southern Chile Using Flexible Mixture Modeling. *Journal of Marine Science and Engineering*. 2018; 6(4):157.
https://doi.org/10.3390/jmse6040157

**Chicago/Turabian Style**

Contreras-Reyes, Javier E., Freddy O. López Quintero, and Alejandro A. Yáñez.
2018. "Towards Age Determination of Southern King Crab (*Lithodes santolla*) Off Southern Chile Using Flexible Mixture Modeling" *Journal of Marine Science and Engineering* 6, no. 4: 157.
https://doi.org/10.3390/jmse6040157