# Using saturation rational function models to calculate yield adjustment factors across varied milking frequencies

## TL;DR

This paper compares two mathematical models to calculate how often milking cows affects milk yield, finding that one model is more accurate while another works better with limited data.

## Contribution

The study introduces a polynomial rational function model for calculating yield adjustment factors and compares it with an exponential model.

## Key findings

- The polynomial rational function model showed higher accuracy with RMSE = 0.004 and R2 = 0.999.
- The exponential model was more robust with limited data.
- Using 2× milking data to derive adjustment factors is possible but requires caution in extrapolation.

## Abstract

Summary: Milking frequency significantly affects milk yield in dairy cows, with higher frequency generally resulting in increased lactation milk yields. The increase follows a nonlinear pattern, with diminishing returns as milking frequency rises, and eventually reaches a saturation point at higher milking frequencies. This study compared 2 saturation rational functions (RF) to derive yield-adjustment factors across different milking frequencies.

Summary: Milking frequency significantly affects milk yield in dairy cows, with higher frequency generally resulting in increased lactation milk yields. The increase follows a nonlinear pattern, with diminishing returns as milking frequency rises, and eventually reaches a saturation point at higher milking frequencies. This study compared 2 saturation rational functions (RF) to derive yield-adjustment factors across different milking frequencies.

•The single-parameter exponential RF model shows greater robustness with limited data.•The 3-parameter polynomial RF model offers enhanced model flexibility.•Both models provide good model fitting and accurate predictions with data support.•Deriving adjustment factors by leveraging 2× milking records is evaluated.

The single-parameter exponential RF model shows greater robustness with limited data.

The 3-parameter polynomial RF model offers enhanced model flexibility.

Both models provide good model fitting and accurate predictions with data support.

Deriving adjustment factors by leveraging 2× milking records is evaluated.

Milking frequency significantly affects milk yield in dairy cows, with higher frequency generally leading to greater lactation yields. The increase follows a nonlinear pattern, showing diminishing returns and eventually reaching saturation as milking frequency rises. This study introduces a polynomial rational function model to derive yield adjustment factors across different milking frequencies. Formulated as a ratio of 2 polynomials, this model has 3 parameters to capture the initial increase in yield and the saturation rate, offering enhanced flexibility across various milking frequencies. We compared its performance to a recently proposed exponential rational function model. Both models demonstrated a good fit to varying milking frequency data up to 10× and satisfactorily predicted yield adjustment factors for milking frequencies where data were absent. The polynomial rational function model exhibited a higher accuracy (root mean square error [RMSE] = 0.004; R2 = 0.999), achieving greater accuracy across a broader range of varied milking frequencies, compared with the exponential rational function model (RMSE = 0.011; R2 = 0.994). Nevertheless, the latter model proved more robust to limited data coverage of milking frequencies. This study also evaluated the strategy of leveraging 2× milking data to derive yield adjustment factors across different frequencies. However, caution is advised when extrapolating far beyond the data-supported frequency range.