# A Vector Representation of Lactation Curves for Dairy Cows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Conventional Regression Model

#### 2.2. Piecewise Linear Regression Representation

#### 2.3. Data Resources

#### 2.4. Evaluation

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Approximated LCs of the models listed in Table 1.

**Figure 3.**Fitting procedure of the PWLR model. (

**a**) initial anchor points setting, (

**b**) adding a new critical point, (

**c**) final fitting result.

**Figure 5.**BIC test results of the PWLR model for the LC groups in Table 2.

Model | Function for LC | |
---|---|---|

Brody (1924) | $y=a\xb7{e}^{-b\xb7t}-a\xb7{e}^{-c\xb7t}$ | [16] |

Wood (1967) | $y=a\xb7{t}^{b}\xb7{e}^{-c\xb7t}$ | [3] |

Cobby (1978) | $y=a-b\xb7t-a\xb7{e}^{-c\xb7t}$ | [17] |

Wilmink (1987) | $y=a+b\xb7{e}^{-k\xb7t}+c\xb7t$ | [4] |

Rook (1993) | $y=a\xb7\left[\frac{1}{1+\frac{b}{c+t}}\right]\xb7{e}^{-d\xb7t}$ | [6] |

Dijkstra (1997) | $y=a\xb7{e}^{b\xb7\frac{1-{e}^{-c\xb7t}}{c}-d\xb7t}$ | [2] |

Group | A | B | C | D | E | F |
---|---|---|---|---|---|---|

Number of cows | 119 | 64 | 50 | 47 | 38 | 12 |

- distribution | 36.1% | 19.4% | 15.2% | 14.2% | 11.5% | 3.6% |

- primiparous ratio | 14.3% | 43.8% | 36.0% | 21.3% | 73.7% | 83.3% |

Parity | 2.61 | 2.16 | 2.30 | 2.57 | 1.63 | 1.25 |

(0.12) ^{†} | (0.17) | (0.19) | (0.20) | (0.20) | (0.18) | |

Total Milk Yield (liter/cow) | 10,713 | 9930 | 10,351 | 10,504 | 9509 | 9806 |

(165) | (261) | (252) | (275) | (305) | (459) | |

Peak Milk Yield (liter/cow) | 53.08 | 46.39 | 47.14 | 54.25 | 42.70 | 45.82 |

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

Peak Day (day) | 59.92 | 86.25 | 119.68 | 54.94 | 144.00 | 119.92 |

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

^{†}standard error.

**Table 3.**LC vectors for the LC groups in Table 2.

V | Group A | Group B | Group C | ||||||
---|---|---|---|---|---|---|---|---|---|

$d$ | $y$ | $r$ | $d$ | $y$ | $r$ | $d$ | $y$ | $r$ | |

${\mathbf{P}}_{0}$ | 10 | 39.13 | 12.77 | 10 | 27.95 | 27.63 | 10 | 29.74 | 34.71 |

${\mathbf{P}}_{1}$ | 28 | 45.49 | 5.94 | 15 | 29.59 | 2.62 | 37 | 37.10 | 10.19 |

${\mathbf{P}}_{2}$ | 36 | 46.60 | 1.62 | 25 | 35.18 | 3.84 | 54 | 40.67 | 5.92 |

${\mathbf{P}}_{3}$ | 51 | 47.46 | 3.83 | 33 | 37.64 | 1.91 | 67 | 41.90 | 4.58 |

${\mathbf{P}}_{4}$ | 71 | 46.94 | 3.42 | 50 | 40.51 | 3.53 | 90 | 41.38 | 7.59 |

${\mathbf{P}}_{5}$ | 106 | 43.52 | 8.32 | 74 | 42.05 | 6.03 | 100 | 42.07 | 2.90 |

${\mathbf{P}}_{6}$ | 134 | 42.58 | 7.05 | 147 | 37.28 | 27.63 | 186 | 40.15 | 34.71 |

${\mathbf{P}}_{7}$ | 177 | 38.41 | 9.05 | 186 | 36.16 | 9.63 | 220 | 38.33 | 10.18 |

${\mathbf{P}}_{8}$ | 212 | 33.01 | 11.31 | 226 | 33.04 | 10.15 | 254 | 32.30 | 12.07 |

${\mathbf{P}}_{9}$ | 262 | 30.66 | 12.77 | 264 | 32.44 | 9.72 | 266 | 31.63 | 4.86 |

${\mathbf{P}}_{10}$ | 280 | 28.81 | 1.99 | 280 | 31.10 | 1.81 | 280 | 29.09 | 1.60 |

Group D | Group E | Group F | |||||||

$\mathit{d}$ | $\mathit{y}$ | $\mathit{r}$ | $\mathit{d}$ | $\mathit{y}$ | $\mathbf{r}$ | $\mathit{d}$ | $\mathit{y}$ | $\mathit{r}$ | |

${\mathbf{P}}_{0}$ | 10 | 39.53 | 22.03 | 10 | 23.93 | 22.40 | 10 | 30.77 | 47.19 |

${\mathbf{P}}_{1}$ | 15 | 40.77 | 3.54 | 39 | 31.78 | 10.24 | 32 | 37.67 | 13.38 |

${\mathbf{P}}_{2}$ | 30 | 47.21 | 8.63 | 54 | 33.50 | 4.61 | 40 | 38.00 | 7.74 |

${\mathbf{P}}_{3}$ | 37 | 48.10 | 2.91 | 69 | 34.01 | 5.05 | 50 | 36.24 | 7.64 |

${\mathbf{P}}_{4}$ | 65 | 46.87 | 14.92 | 86 | 35.91 | 6.02 | 88 | 36.12 | 32.69 |

${\mathbf{P}}_{5}$ | 95 | 42.94 | 16.07 | 99 | 35.72 | 5.53 | 104 | 31.73 | 12.87 |

${\mathbf{P}}_{6}$ | 125 | 36.38 | 15.30 | 111 | 37.24 | 4.81 | 117 | 34.19 | 11.03 |

${\mathbf{P}}_{7}$ | 152 | 33.97 | 9.34 | 154 | 36.76 | 16.66 | 132 | 33.03 | 13.41 |

${\mathbf{P}}_{8}$ | 197 | 34.92 | 18.58 | 187 | 37.61 | 10.16 | 178 | 37.52 | 33.21 |

${\mathbf{P}}_{9}$ | 213 | 36.14 | 6.41 | 209 | 36.38 | 9.08 | 251 | 38.97 | 47.19 |

${\mathbf{P}}_{10}$ | 280 | 33.27 | 22.03 | 280 | 34.81 | 22.40 | 280 | 38.32 | 10.07 |

LC | Group | Whole | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | A | B | C | D | E | F | Mean | Set | |

${e}_{m}$ | Brody | 2.553 | 0.788 | 2.073 | 2.682 | 1.261 | 2.019 | 1.896 | 0.742 |

Wood | 0.668 | 0.953 | 1.147 | 2.675 | 0.507 | 1.978 | 1.321 | 0.609 | |

Cobby | 1.009 | 0.767 | 1.999 | 2.547 | 1.261 | 2.019 | 1.600 | 0.663 | |

Wilmink | 0.668 | 0.524 | 1.006 | 2.546 | 0.551 | 1.881 | 1.196 | 0.244 * | |

Rook | 0.635 | 0.600 | 1.053 | 2.563 | 0.524 | 1.881 | 1.209 | 0.313 | |

Dijkstra | 0.693 | 0.464 | 1.121 | 2.368 | 0.578 | 2.018 | 1.207 | 0.248 | |

PWLR | 0.338 * | 0.388 * | 0.470 * | 0.563 * | 0.459 * | 0.973 * | 0.532 * | 0.310 | |

${e}_{g}$ | Brody | 4.400 | 3.559 | 4.214 | 5.010 | 3.395 | 4.302 | 4.147 | 4.176 |

(0.166) ^{†} | (0.184) | (0.185) | (0.244) | (0.213) | (0.527) | (0.253) | (0.093) | ||

Wood | 3.887 | 3.431 | 3.896 | 5.040 | 3.166 | 4.245 | 3.944 | 3.894 | |

(0.145) | (0.16) | (0.177) | (0.241) | (0.191) | (0.495) | (0.235) | (0.085) | ||

Cobby | 4.033 | 3.390 | 4.178 | 5.052 | 3.351 | 4.303 | 4.051 | 4.007 | |

(0.153) | (0.163) | (0.185) | (0.257) | (0.193) | (0.527) | (0.246) | (0.089) | ||

Wilmink | 3.898 | 3.296 | 3.686 | 5.017 | 3.063 | 4.167 | 3.854 | 3.822 | |

(0.154) | (0.163) | (0.168) | (0.255) | (0.188) | (0.499) | (0.238) | (0.089) | ||

Rook | 3.812 | 3.328 | 3.809 | 4.958 | 3.095 | 4.178 | 3.863 | 3.812 | |

(0.145) | (0.159) | (0.176) | (0.239) | (0.19) | (0.492) | (0.234) | (0.085) | ||

Dijkstra | 3.760 | 3.253 | 3.717 | 4.846 | 3.045 | 4.148 | 3.795 | 3.741 | |

(0.146) | (0.159) | (0.167) | (0.238) | (0.187) | (0.504) | (0.234) | (0.085) | ||

PWLR | 2.721 * | 2.524 * | 2.898 * | 3.049 * | 2.479 * | 2.984 * | 2.776 * | 2.738 * | |

(0.097) | (0.115) | (0.135) | (0.169) | (0.171) | (0.379) | (0.178) | (0.058) |

^{†}standard deviation; * min in each column; e

_{m}and e

_{g}represent mean fitting error and intra-fitting error each; Mean indicates the error average of group A–F, whereas Whole set means the error average of entire LCs.

LC | Group | Whole | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | A | B | C | D | E | F | Mean | Set | |

${e}_{c}$ | Brody | 4.436 | 3.956 | 4.068 | 4.605 | 3.709 | 7.460 | 4.289 | 4.524 |

Wood | 4.036 | 3.821 | 3.763 * | 4.682 | 3.486 * | 7.442 | 4.095 | 4.445 | |

Cobby | 4.115 | 3.931 | 4.030 | 4.598 | 3.709 | 7.460 | 4.211 | 4.508 | |

Wilmink | 4.035 | 3.787 | 3.780 | 4.601 | 3.513 | 7.210 | 4.075 | 4.429 | |

Rook | 4.021 | 3.796 | 3.773 | 4.618 | 3.505 | 7.205 | 4.073 | 4.431 | |

Dijkstra | 4.024 | 3.782 | 3.824 | 4.540 | 3.519 | 7.434 | 4.078 | 4.424 * | |

PWLR | 3.987 * | 3.776 * | 3.765 | 4.141 * | 3.542 | 6.619 * | 3.952 * | 4.450 |

_{c}represents cross-fitting error; Mean indicates the error average of group A–F, whereas Whole set means the error average of entire LCs.

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**MDPI and ACS Style**

Lee, S.; Park, J.
A Vector Representation of Lactation Curves for Dairy Cows. *Agriculture* **2022**, *12*, 395.
https://doi.org/10.3390/agriculture12030395

**AMA Style**

Lee S, Park J.
A Vector Representation of Lactation Curves for Dairy Cows. *Agriculture*. 2022; 12(3):395.
https://doi.org/10.3390/agriculture12030395

**Chicago/Turabian Style**

Lee, Seonghun, and Jaehwa Park.
2022. "A Vector Representation of Lactation Curves for Dairy Cows" *Agriculture* 12, no. 3: 395.
https://doi.org/10.3390/agriculture12030395