3.4.3. Selection of Training and Test Data

Selecting the test data from a random sub-sample of the complete data makes the training and test data become chronologically intertwined. This is a shortcoming because it does not reflect the practical purpose of a statistical model predicting the EE of an EAF [1]. From a process perspective, a statistical model will predict on heats that are from a future point in time with respect to the heats whose data have been used to adapt the parameters of the model, i.e., training data. To account for this shortcoming, the test data will be selected in chronological order from the training data. The test data will be all heats produced from 1st of February 2020 to the 28th of February 2020 and the training data will the heats produced from the 10th of November 2018 through January 2020. The start date of the training data was selected based on a furnace upgrade that was completed on the 9th of November 2018. This amounted to 4032 training data points and 263 test data points before data treatment. The heats in the training data and test data will be referred to as *training heats* and *test heats*, respectively.

#### 3.4.4. Model Performance Metrics

The performance of the models will be compared using two fundamental metrics. These are the coefficient of determination, *R*2, and the regular error metric.

The adjusted-*R*<sup>2</sup> value should be used instead of the regular *R*<sup>2</sup> value when comparing models that use different number of input variables. The reason is because each additional input variable increases the *R*2-value when the number of data points is fixed [24]. The adjusted-*R*<sup>2</sup> value can be calculated as follows:

$$
\bar{R}^2 = 1 - (1 - R^2) \frac{n - 1}{n - v - 1} \tag{7}
$$

where *n* is the number of data points, is the number of input variables, and *R*<sup>2</sup> is the regular R-square value.

The regular error metric was chosen in favor of the absolute error metric, because in a practical context, an overestimated prediction of EE is vastly different from an underestimated prediction of EE. The regular error metric can be defined as follows:

$$E\_i = y\_i - \hat{y}\_i \tag{8}$$

where *yi* is the true value, *y*ˆ*<sup>i</sup>* as the predicted value, *i* ∈ 1, 2, . . . , *n*, and *n* is the number of data points. Using all the data points under consideration, the standard deviation, mean, minimum, and maximum, error values are defined the ordinary way.

The EE consumption, and therefore the unit of error, will be expressed in kWh/heat rather than in kWh/t tapped steel. The main reason for this choice is that the former varies only by consumed EE while the latter varies both by consumed EE and the yield. This is a more challenging problem as it requires the statistical model to adapt to both the consumed EE as well as the yield, which is defined as the weight of tapped steel divided by weight of charged scrap. Furthermore, the tap weight is dependent on factors such as the slag created by oxidation in the process and on the total amount of dust generated. These two examples are, in turn, affected by process times, amount of oxygen injected, additives, and the charge mix.
