# Clustering and Characterization of the Lactation Curves of Dairy Cows Using K-Medoids Clustering Algorithm

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Backgrounds

## 3. Materials and Methods

#### 3.1. Dataset

#### 3.2. Clustering

- $d(A,B)$: Distance between lactation dataset A and B
- ${m}_{A,i}$: Amount of milk yield on ${i}^{th}$ day in lactation dataset A
- ${m}_{B,i}$: Amount of milk yield on ${i}^{th}$ day in lactation dataset B
- N: The total number of milking days

- ${\mu}_{A}$: A median dataset of a cluster A
- $d(\mathit{X},{\mu}_{A})$: Euclidean distance between lactation datasets $\mathit{X}$ and ${\mu}_{A}$

#### 3.3. Characterization and Comparison of the Clusters

#### 3.4. Statistical Analysis

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Conventional | Automatic | Total | ||||
---|---|---|---|---|---|---|

Milking System | Milking System | |||||

Mean | SE | Mean | SE | Mean | SE | |

Total milking days (day) | 103,086 | 203,444 | 306,530 | |||

Milking days (day/lactation) | 207.00 | 5.99 | 243.94 | 5.20 | 230.13 | 3.98 |

Number of animals | 287 | 437 | 724 | |||

Number of total lactations | 498 | 834 | 1332 | |||

- Number of primiparous cows | 191 | 269 | 460 | |||

- Number of multiparous cows | 307 | 565 | 872 | |||

Parity | 2.24 | 0.06 | 2.58 | 0.06 | 2.45 | 0.04 |

Daily milk yield (L/day/lactation) | 30.75 | 0.52 | 32.82 | 0.39 | 32.06 | 0.31 |

Total milk yield (L/lactation) | 7056 | 224 | 8349 | 189 | 7876 | 146 |

Parity | Number of Lactations | Milking Days | Daily MY | Total MY | |||
---|---|---|---|---|---|---|---|

(Cows) | (Day) | (L/Day/Cow) | (L/Cow/Lactation) | ||||

Mean | SD | Mean | SD | Mean | SD | ||

1 | 111 | 358.95 | 73.55 | 31.65 | 4.86 | 11,303 | 2649 |

2 | 101 | 377.30 | 78.56 | 36.00 | 5.41 | 13,545 | 3403 |

3 | 61 | 375.21 | 69.06 | 36.89 | 6.12 | 13,786 | 3310 |

≥4 | 57 | 406.58 | 81.59 | 37.09 | 6.60 | 14,981 | 3601 |

Model | Lactation Curve | |
---|---|---|

Wood (1967) | $y=a\xb7{t}^{b}\xb7{e}^{-c\xb7t}$ | [5] |

Wilmink (1987) | $y=a+b\xb7{e}^{-k\xb7t}+c\xb7t$ | [6] |

Dijkstra (1997) | $y=a\xb7{e}^{b\xb7\frac{1-{e}^{-c\xb7t}}{c}-d\xb7t}$ | [7] |

Cluster | Total | p-Value | ||||||
---|---|---|---|---|---|---|---|---|

(a) | (b) | (c) | (d) | (e) | (f) | |||

No. of lactations | 119 | 64 | 50 | 47 | 38 | 12 | 330 | |

Primiparous | 17 | 28 | 18 | 10 | 28 | 10 | 111 | |

Multiparous | 102 | 36 | 32 | 37 | 10 | 2 | 219 | |

Parity | 2.61 ^{a} | 2.16 ^{c} | 2.30 ^{b} | 2.57 ^{a} | 1.63 ^{d} | 1.25 ^{e} | 2.31 | <0.001 |

(0.12) | (0.17) | (0.19) | (0.20) | (0.20) | (0.18) | (0.07) | (0.16) | |

70 DIM | Aug. 1. | Sep. 29. | Oct. 10. | Aug. 20. | Sep. 30. | Jun. 30. | Aug. 31. | 0.272 |

in a lactation (days) | (21.79) | (24.92) | (26.28) | (37.73) | (32.88) | (90.25) | (12.47) | (28.93) |

MY (L/day) | 39.53 ^{a} | 36.64 ^{d} | 38.20 ^{c} | 38.76 ^{b} | 35.09 ^{e} | 36.18 ^{d} | 38.03 | 0.006 |

(0.61) | (0.96) | (0.93) | (1.02) | (1.13) | (1.69) | (0.39) | (0.88) | |

MY (L) | 10,713 ^{a} | 9930 ^{d} | 10,351 ^{c} | 10,504 ^{b} | 9509 ^{e} | 9806 ^{d} | 10,305 | 0.006 |

(164.73) | (261.38) | (252.30) | (274.96) | (305.46) | (459.28) | (104.65) | (238.49) | |

Peak DIM (days) | 59.92 ^{d} | 86.25 ^{c} | 119.68 ^{b} | 54.94 ^{e} | 144.00 ^{a} | 119.92 ^{b} | 85.24 | <0.001 |

(2.73) | (4.61) | (8.42) | (6.11) | (9.67) | (25.73) | (3.01) | (6.01) | |

Peak MY (L) | 53.08 ^{b} | 46.39 ^{d} | 47.14 ^{c} | 54.25 ^{a} | 42.70 ^{e} | 45.82 ^{d} | 49.59 | <0.001 |

(0.83) | (1.24) | (1.08) | (1.34) | (1.33) | (2.26) | (0.54) | (1.13) |

Model | Parameter | Cluster | Total | |||||
---|---|---|---|---|---|---|---|---|

(a) | (b) | (c) | (d) | (e) | (f) | |||

Wood | a | 24.6645 | 15.9437 | 12.5155 | 44.8198 | 12.2953 | 30.2838 | 22.1761 |

(0.3161) | (0.3522) | (0.3401) | (2.1754) | (0.1697) | (1.3543) | (0.2843) | ||

b | 0.2142 | 0.2738 | 0.3406 | 0.0170 | 0.2743 | 0.0374 | 0.2000 | |

(0.0037) | (0.0063) | (0.0076) | (0.0143) | (0.0038) | (0.0127) | (0.0037) | ||

c | 0.0039 | 0.0033 | 0.0035 | 0.0016 | 0.0018 | ≈0 | 0.0029 | |

(≈0) | (0.0001) | (0.0001) | (0.0001) | (≈0) | (0.0001) | (≈0) | ||

${\u03f5}_{f}$ (L) | 0.6680 | 0.9529 | 1.1474 | 2.6746 | 0.5068 * | 1.9778 | 0.6090 | |

${\u03f5}_{c}$ | 0.6125 | 0.8121 | 0.8298 | 0.8827 | 0.8343 * | 1.2126 | 0.9087 | |

Wilmink | a | 54.2924 | 46.0936 | 65.3834 | 47.6263 | 42.0509 | 6604.6668 | 47.5097 |

(0.1707) | (0.1337) | (2.2023) | (0.3927) | (0.4718) | (0.1981) | (0.0581) | ||

b | −24.9865 | −31.9822 | −38.4509 | −58.5113 | −21.6106 | −6570.6721 | −25.8178 | |

(0.7756) | (0.6105) | (1.9260) | (35.6232) | (0.3697) | (0.1981) | (0.3245) | ||

c | −0.0937 | −0.0549 | −0.1200 | −0.0594 | −0.0247 | ≈0 | −0.0582 | |

(0.0009) | (0.0007) | (0.0072) | (0.0023) | (0.0019) | (9.1156) | (0.0003) | ||

k | 0.0494 | 0.0495 | 0.0124 | 0.1715 | 0.0195 | ≈0 | 0.0543 | |

(0.0020) | (0.0012) | (0.0009) | (0.0496) | (0.0008) | (0.0014) | (0.0008) | ||

${\u03f5}_{f}$ (L) | 0.6676 * | 0.5240 | 1.0063 * | 2.5458 | 0.5509 | 1.8807 * | 0.2436 * | |

${\u03f5}_{c}$ | 0.6119 * | 0.7866 | 0.8230 * | 0.8682 | 0.8359 | 1.1974 * | 0.9006 * | |

Dijkstra | a | 34.0962 | 18.8021 | 28.2765 | 13.6532 | 21.3504 | 12.0427 | 25.1318 |

(0.5020) | (0.3568) | (0.4192) | (8.0760) | (0.2697) | (7.7063) | (0.2315) | ||

b | 0.0197 | 0.0467 | 0.0126 | 0.1950 | 0.0152 | 0.1686 | 0.0346 | |

(0.0011) | (0.0018) | (0.0005) | (0.1342) | (0.0007) | (0.1449) | (0.0009) | ||

c | 0.0357 | 0.0506 | 0.0124 | 0.1525 | 0.0230 | 0.1525 | 0.0520 | |

(0.0015) | (0.0011) | (0.0010) | (0.0356) | (0.0009) | (0.0443) | (0.0008) | ||

d | 0.0026 | 0.0015 | 0.0032 | 0.0016 | 0.0006 | ≈0 | 0.0016 | |

(≈0) | (≈0) | (0.0002) | (0.0001) | (≈0) | (0.0001) | (≈0) | ||

${\u03f5}_{f}$ (L) | 0.6933 | 0.4644 * | 1.1214 | 2.3680 * | 0.5784 | 2.0179 | 0.2479 | |

${\u03f5}_{c}$ | 0.6131 | 0.7852 * | 0.8316 | 0.8510 * | 0.8371 | 1.2012 | 0.9010 |

Cluster | Total | ||||||
---|---|---|---|---|---|---|---|

(a) | (b) | (c) | (d) | (e) | (f) | ||

Wood model ^{1} | |||||||

Peak yield (L) | 46.96 | 40.65 | 42.34 | 45.87 | 37.10 | N/A | 42.34 |

Peak DIM (days) | 54.92 | 82.97 | 97.31 | 10.63 | 152.39 | N/A | 68.97 |

Persistency | −0.0026 | −0.0017 | −0.0016 | −0.0015 | −0.0005 | N/A | −0.0017 |

Dijkstra model ^{2} | |||||||

Peak yield (L) | 47.50 | 41.48 | 42.37 | 46.14 | 37.02 | N/A | 43.13 |

Peak DIM (days) | 56.73 | 67.95 | 110.53 | 31.50 | 140.53 | N/A | 59.11 |

Persistency | −0.0025 | −0.0015 | −0.0020 | −0.0016 | −0.0005 | N/A | −0.0016 |

^{1}The features of LC were calculated by the equations as follows [5,42]: Peak yield (L), $a*{(b/c)}^{b}*{e}^{-b}$; Peak DIM (days), $b/c$; Persistency, $b/{t}_{h}-c$;

^{2}The features of LC were calculated using the following equations [7,42]: Peak yield (L), $a*{(d/b)}^{d/c}*exp[(b-d)/c]$; Peak DIM (days), ${c}^{-1}ln(b/d)$; Persistency, $b*exp(-c*{t}_{h})-d$; * ${t}_{h}=({t}_{m}+{t}_{f})/2$; ${t}_{m}$, Peak DIM; ${t}_{f}$, length of lactation.

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**MDPI and ACS Style**

Lee, M.; Lee, S.; Park, J.; Seo, S.
Clustering and Characterization of the Lactation Curves of Dairy Cows Using *K*-Medoids Clustering Algorithm. *Animals* **2020**, *10*, 1348.
https://doi.org/10.3390/ani10081348

**AMA Style**

Lee M, Lee S, Park J, Seo S.
Clustering and Characterization of the Lactation Curves of Dairy Cows Using *K*-Medoids Clustering Algorithm. *Animals*. 2020; 10(8):1348.
https://doi.org/10.3390/ani10081348

**Chicago/Turabian Style**

Lee, Mingyung, Seonghun Lee, Jaehwa Park, and Seongwon Seo.
2020. "Clustering and Characterization of the Lactation Curves of Dairy Cows Using *K*-Medoids Clustering Algorithm" *Animals* 10, no. 8: 1348.
https://doi.org/10.3390/ani10081348