**7.1 Partial autocorrelation function: for average daily milk production**

In Box–Jenkins models, the partial autocorrelation plot or partial correlogram is also often employed for model identification. In the same way, there is one lag number. As a result, we must test the MA model to ensure that it is adequate. That is, under the partial autocorrelations graph, the average daily milk output shows that there is a point outside the bottom boundaries, which is MA [1]. In other words, the distributions of average daily milk output are neither balanced nor equal, as observed in **Figure 7**.

*Predicting Trends, Seasonal Effects, and Future Yields in Cow's Milk through Time… DOI: http://dx.doi.org/10.5772/intechopen.105704*

**Figure 6.**

*Autocorrelation plot.*

**Figure 7.** *Partial autocorrelation plot.*

#### **7.2 ARMA process**

By combining the autoregressive and moving average processes, we obtain a very general time series model, ARMA (1,1).

#### **7.3 ARIMA process**

ARIMA (p, d, q) stands for autoregressive integrated moving average process, where d denotes the number of times the data is differenced before it is an ARMA (p, q). As a result, the ARIMA model is ARIMA (p, d, q) = ARIMA (1,2,1).

### **7.4 ARIMA model: for average daily milk production**

Final estimates of parameters.



As we have seen from the MINITAB output, the ARIMA model (1, 2, 1) equation is described as follows.

$$\mathbf{Y}\_t = -0.2066 \mathbf{Y}\_{t-1} + 0.997 \mathbf{\tilde{e}}\_{t-1} + \mathbf{e}\_t$$

#### *7.4.1 Testing of parameters*

The final estimates are those that reduce the sum of squared errors to the point where no other estimates yield lower sums of squared errors. As shown in the MINITAB output, a model should include significant parameters. The p-value of ARIMA (1, 2, 1) is less than the significance level (=0.05). This means the parameters are significantly different from zero and have the smallest sum-squared error possible. Then it has parameters that are statistically significant. As a result, the model is adequate.

#### *7.4.2 Forecasting*

The process of obtaining the forecast point and the final model in its original form is as follows.

$$Y\_t = -0.2066Y\_{t-1} + 0.9977e\_{t-1} + e\_t$$


*We can use the 95% confidence interval (CI) defined above to assess the accuracy of the anticipated number. We can state that the forecasted value is accurate since the entire forecast values are found between the lower and upper intervals.*

### **8. Conclusions**

The average amount of milk produced at Andassa dairy farm is dropping. The data for 179 days reveal a high degree of variability in daily milk production

*Predicting Trends, Seasonal Effects, and Future Yields in Cow's Milk through Time… DOI: http://dx.doi.org/10.5772/intechopen.105704*

compared to other days, implying that the amount of milk produced varies greatly from day to day. Because the slope of the trends over the 179 days is 16.066, the amount of milk is falling. The daily milk production graph in the autocorrelations and partial autocorrelations graphs reveals that the top and lower boundaries do not encompass the entire observation. For ARIMA (1, 2, 1), a parameter with a p-value less than the level of significance (0.05) is a parameter. This indicates that the parameter is significantly distinct from zero and has the smallest value.
