# Towards a Vectorial Approach to Predict Beef Farm Performance

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Aim and Scope

#### 2.2. The Dataset

#### 2.3. Standard vs. Vectorial Approaches: Genetic Programming

#### 2.4. Standard vs. Vectorial Approaches: Experimental Settings

`{plus; minus; times; mydivide}`, where

`plus, minus, and times`indicate the usual operators of binary addition, subtraction, and multiplication, respectively, while

`mydivide`represents the protected division, which returns the numerator when the denominator is equal to zero. Likewise, we chose proper functions for VE-GP. Differently from ST-GP, suitable functions are indeed provided to manage scalar and vectors [18,19]. For the considered problem, we used

`{VSUMW; V_W; VprW; VdivW; V_mean; V_min; V_meanpq; V_minpq}`. The first four operators represent the elementwise sum, difference, product, and the protected division between two vectors or between a scalar and a vector, respectively, e.g.,

`VSUMW([2,3.5,4,1],[1,0,1,2.5] = [3,3.5,5,3.5])`. The mean and minimum of a vector return the corresponding value for the whole vector (standard aggregate functions

`V_mean`and

`V_min`) or for a selected range $[p,q]$ inside the vector, where p and q are positive integers with $0<p\le q$ (parametric aggregate functions

`V_meanpq`and

`V_minpq`), e.g.,

`V_mean([2,3.5,4,1]) = 2.6`, whereas

`V_mean`${}_{3,4}$

`([2,3.5,4,1]) = 2.5`. The fact that standard and parametric aggregate functions collapse the vectorial variable into a single value allows one to handle all the information contained in the vector or part of it. In addition to crossover and mutation, the algorithm is provided with an operator reserved for the mutation of the aggregate function parameters. It allows p and q to evolve in order to detect the most informative window in which to apply thereafter the aggregate function. The set of terminals was composed of the predictors in Table 2 for both ST- and VE-GP.

## 3. Results

#### 3.1. ST-GP vs. VE-GP

#### 3.2. Comparisons of ST-GP and VE-GP with Other ML Methods

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PLF | Precision Livestock Farming |

ML | Machine Learning |

ANABORAPI | National Association of Piemontese Cattle Breeders |

GP | Genetic Programming |

ST-GP | Standard Genetic Programming |

VE-GP | Vectorial Genetic Programming |

EA | Evolutionary Algorithm |

KNN | k-Nearest Neighbors |

NN | Neural Network |

LM | Linear Model |

GLMNET | Generalized Linear Model with Elastic Net Regularization |

RNN | Recurrent Neural Network |

LSTM | Long Short-Term Memory |

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**Figure 1.**ST-GP (

**up**) and VE-GP (

**down**) fitness evolution plots. For each generation, the graph plots the median of the 10 median fitness achieved by the best 7 models on the validation sets and correspondingly the performance achieved on the learning and test sets.

**Figure 2.**RMSEs on both the learning and test sets for the different algorithms. Learning results are plotted in yellow (

**left**) and test results are plotted in blue (

**right**) for each technique.

**Table 1.**Standard Data Panel. Structure of the dataset. The farms are listed horizontally, as well as the reference year, the variables vertically.

FARM | YEAR | PRIMIPAROUS | PLURIPAROUS | HEIFERS | INTERPARTO |
---|---|---|---|---|---|

Farm 1 | 2014 | 22 | 36 | 7 | 365 |

Farm 1 | 2015 | 10 | 46 | 13 | 375 |

Farm 1 | 2016 | 16 | 47 | 12 | 381 |

Farm 1 | 2017 | 14 | 46 | 11 | 375 |

Farm 1 | 2018 | 16 | 47 | 12 | 374 |

Farm 2 | 2014 | 11 | 90 | 9 | 396 |

Farm 2 | 2015 | 10 | 93 | 9 | 391 |

Farm 2 | 2016 | 9 | 95 | 7 | 380 |

Farm 2 | 2017 | 7 | 97 | 10 | 387 |

Farm 2 | 2018 | 9 | 92 | 11 | 385 |

Farm 3 | 2014 | 7 | 42 | 3 | 414 |

Farm 3 | 2015 | 4 | 43 | 4 | 439 |

Farm 3 | 2016 | 4 | 44 | 10 | 452 |

Farm 3 | 2017 | 10 | 44 | 11 | 425 |

Farm 3 | 2018 | 9 | 60 | 4 | 473 |

**Table 2.**Final set of variables used for the benchmarked problem. The bottom line represents the dependent variable Y, i.e., the target for the predicted models.

Variable Name | Variable Description | |
---|---|---|

1 | $COWS$ | Consistency for cows, i.e., number of cows |

2 | $HEIFERS$ | Consistency for heifers, i.e., number of heifers |

3 | $INTP$ | Calving interval in days, based on currently pregnant cows |

4 | ${C}_{PAR}$ | Average parity |

5 | $ETA$_$PART$_1 | Age at first calving |

6 | ${C}_{EASE}$ | N. of cows that delivered with easy calving |

7 | ${H}_{EASE}$ | N. of primiparous that delivered with easy calving |

8 | ${C}_{PART}$_${}_{IND}$ | Calving ease (EBV for cows) |

9 | ${H}_{PART}$_${}_{IND}$ | Birth ease (EBV for heifers) |

10 | $TF{A}_{BIRTH}$ | Birth ease (EBV for A.I. bulls) |

11 | $TF{A}_{PAR}$ | Calving ease (EBV for A.I. bulls) |

12 | $UBA06$ | UBA referred to bovines 6 months–2 years old |

13 | $UBA04$ | UBA referred to bovines 4–6 months old |

14 | ${N}_{ELIM}$ | N. of dead calves in the first 60 days after birth |

15 | ${N}_{TOT}$ | Total number of calves born |

16 | ${N}_{BALIVE}$ | Total number of calves born alive |

17 | $CORRECT$ | Percentage of calves born without defects (e.g., Macroglossia, Arthrogryposis) |

18 | $CONSANG$_$NEW$ | Consanguinity calculated on future calves |

19 | Y | N. of weaned calves per cow per year (2) |

**Table 3.**Dataset configuration from 2017–2018. On the left side, the input scalar variables ${X}_{17,1},{X}_{17,2},\dots ,{X}_{17,m}$. On the right side, the scalar target ${Y}_{18}$.

2017 | 2018 | ||||||
---|---|---|---|---|---|---|---|

${X}_{17,j,1}$ COWS | ${X}_{17,j,2}$ COW_AGE | ${X}_{17,j,3}$ CALVING_INT | ${X}_{17,j,4}$ N_CALVING | ${Y}_{18,j}$ | |||

FARM 1- | 104 | 3020 | 387 | 60 | 0.95 | ||

FARM 2- | 54 | 3112 | 425 | 54 | 0.9 | ||

FARM 3- | 63 | 2824 | 515 | 48 | 0.69 | ||

… | 49 | 3131 | 466 | 49 | 0.67 | ||

108 | 2766 | 407 | 50 | 0.85 | |||

74 | 3448 | 459 | 62 | 0.84 |

**Table 4.**Vectorial panel dataset configuration for 2014–2018. On the left side, the input vectorial variables ${\mathbf{X}}_{t,j,i}=[{X}_{14,j,i},{X}_{15,j,i},{X}_{16,j,i},{X}_{17,j,i}]$, with $t\in \{14,\dots ,17\}$, $i=1,\dots ,m$, and $j=1,\dots ,n$. On the right side, the scalar target variable ${Y}_{18}$.

2014–2017 | 2018 | |||||
---|---|---|---|---|---|---|

${X}_{t,1,j}$ COWS | ${X}_{t,2,j}$ COW_AGE | ${X}_{t,3,j}$ CALVING_INT | ${Y}_{18,j}$ | |||

FARM 1- | [98, 101, 107, 104] | [2999, 3001, 2998, 3020] | [391, 391, 380, 387] | 0.95 | ||

FARM 2- | [61, 49, 53, 54] | [3076, 3002, 3056, 3112] | [408, 376, 402, 425] | 0.9 | ||

FARM 3- | [53, 55, 64, 63] | [2799, 2813, 2802, 2824] | [367, 376, 406, 515] | 0.69 | ||

… | [31, 36, 47, 49] | [3102, 3075, 3009, 3131] | [434, 480, 461, 466] | 0.67 | ||

[102, 99, 105, 108] | [2704, 2795, 2789, 2766] | [404, 371, 395, 407] | 0.85 | |||

[69, 71, 75, 74] | [3401, 3388, 3406, 3448] | [387, 367, 373, 459] | 0.84 |

Parameter | Description |
---|---|

ST-GP | |

Maximum number of generations | 40 |

Population size | 250 |

Selection Method | Lexicographic Parsimony Pressure |

Elitism | Keepbest |

Initialization Method | Ramped half and half |

Tournament Size | 2 |

Subtree Crossover Rate | 0.7 |

Subtree Mutation Rate | 0.1 |

Subtree Shrinkmutation Rate | 0.1 |

Subtree Swapmutation Rate | 0.1 |

Maxtreedepth | 17 |

VE-GP | |

Maximum number of generations | 40 |

Population size | 250 |

Selection Method | Lexicographic Parsimony Pressure |

Elitism | Keepbest |

Initialization Method | Ramped half and half |

Tournament Size | 2 |

Subtree Crossover Rate | 0.7 |

Subtree Mutation Rate | 0.3 |

Mutation of aggregate function parameters | 0.2 |

Maxtreedepth | 17 |

**Table 6.**Parameters used to perform ML techniques with caret package in R and the Deep Learning Toolbox in MATLAB.

ML Technique | Parameters |
---|---|

knn | k = 15 |

nnet | size = 7; decay = 0.2 |

glmnet | $\alpha $ = 0.8, $\lambda $ = 0.85 |

LSTM | hidden units = 200; epochs = 50;batchsize = 1; learning algorithm = adam. |

**Table 7.**Frequency of use of each variable among the best 10 individuals found by ST-GP (left column) and VE-GP (right column).

Variable | % of Use (ST-GP) | % of Use (VE-GP) |
---|---|---|

X1 $COWS$ | 70% | 100% |

X2 $HEIFERS$ | 10% | 10% |

X3 $INTP$ | 0% | 10% |

X4 ${C}_{PAR}$ | 50% | 0% |

X5 $ETA$_$PART$_1 | 0% | 10% |

X6 ${C}_{EASE}$ | 0% | 10% |

X7 ${H}_{EASE}$ | 0% | 10% |

X8 ${C}_{PART}$_${}_{IND}$ | 0% | 0% |

X9 ${H}_{PART}$_${}_{IND}$ | 0% | 0% |

X10 $TF{A}_{BIRTH}$ | 10% | 0% |

X11 $TF{A}_{PAR}$ | 0% | 0% |

X12 $UBA06$ | 0% | 0% |

X13 $UBA04$ | 20% | 0% |

X14 ${N}_{ELIM}$ | 70% | 40% |

X15 ${N}_{TOT}$ | 0% | 80% |

X16 ${N}_{BALIVE}$ | 60% | 0% |

X17 $CORRECT$ | 30% | 0% |

X18 $CONSANG$_$NEW$ | 20% | 30% |

**Table 8.**Fitness on the test set, number of involved variables, and corresponding percentage for each model evolved by ST-GP (upper table) and VE-GP (lower table) in each of the 10 runs.

Prediction Model | Fitness on Test | N. of Variables | % of Variables |
---|---|---|---|

ST-GP | |||

model 1 | 0.1335 | 9 | 50% |

model 2 | 0.1207 | 6 | 33% |

model 3 | 0.1143 | 11 | 61% |

model 4 | 0.1383 | 8 | 44% |

model 5 | 0.1392 | 7 | 39% |

model 6 | 0.1439 | 7 | 39% |

model 7 | 0.1395 | 8 | 44% |

model 8 | 0.1370 | 6 | 33% |

model 9 | 0.1285 | 15 | 83% |

model 10 | 0.1184 | 7 | 39% |

VE-GP | |||

model 1 | 0.1117 | 5 | 26% |

model 2 | 0.1016 | 3 | 16% |

model 3 | 0.1044 | 9 | 47% |

model 4 | 0.1085 | 8 | 42% |

model 5 | 0.1134 | 3 | 16% |

model 6 | 0.0998 | 8 | 42% |

model 7 | 0.1018 | 4 | 21% |

model 8 | 0.1149 | 4 | 21% |

model 9 | 0.0999 | 8 | 42% |

model 10 | 0.1121 | 3 | 16% |

STGP | KNN | NN | VEGP | GLMNET | LSTM | |
---|---|---|---|---|---|---|

Learning sets | ||||||

Median | 0.1238 | 0.1074 | 0.0967 | 0.1052 | 0.1025 | 0.1011 |

Mean | 0.1220 | 0.1077 | 0.0967 | 0.1054 | 0.1025 | 0.0988 |

Test sets | ||||||

Median | 0.1353 | 0.1151 | 0.1122 | 0.1065 | 0.1057 | 0.1037 |

Mean | 0.1314 | 0.1147 | 0.1128 | 0.1068 | 0.1056 | 0.1034 |

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**MDPI and ACS Style**

Abbona, F.; Vanneschi, L.; Giacobini, M. Towards a Vectorial Approach to Predict Beef Farm Performance. *Appl. Sci.* **2022**, *12*, 1137.
https://doi.org/10.3390/app12031137

**AMA Style**

Abbona F, Vanneschi L, Giacobini M. Towards a Vectorial Approach to Predict Beef Farm Performance. *Applied Sciences*. 2022; 12(3):1137.
https://doi.org/10.3390/app12031137

**Chicago/Turabian Style**

Abbona, Francesca, Leonardo Vanneschi, and Mario Giacobini. 2022. "Towards a Vectorial Approach to Predict Beef Farm Performance" *Applied Sciences* 12, no. 3: 1137.
https://doi.org/10.3390/app12031137