# Contrails and Their Dependence on Meteorological Situations
## 3 Results
### Advanced Statistical Methods for the Evaluation of Data  
Advanced statistical methods were used in the context of the evaluation of measured and obtained contrail data. Linear regression and five classification methods were used to investigate the dependence of contrails on selected variables x(i) from datasets 1 and 2. The results of the observed frequencies of contrails according to their duration confirmed that for most experiments, the classification of contrails into short-lived with a duration of up to 1 min and long-lived contrails with a duration of more than 1 min is suitable.  
The learning and subsequent classification processes were performed for the used classification models. The data used from dataset 1 and dataset 2 were randomly distributed, such that 80% of the selected data was used for training and the remaining 20% for testing. This ratio was maintained each time, but the selection of data for training was randomized to maintain the proportion of data selected relative to the final category. The remaining data were used to test the model for accuracy, precision, sensitivity, and F1 score. The result of the ability to classify the final variable was run against the resulting classifier. Validation was performed on the remaining data that were not used for training. The process of training and testing ensures that the objectivity of the particular classification model is evaluated, and the correct functioning of the model is determined. The random selection should render the validation results more substantiated.  
To evaluate the trained classifier models and to select the best model, a confusion matrix was calculated for all models, in which the values of TP, FN, FP, and TN are given. From these values, the average accuracy of classification (S) is obtained, which summarizes the performance of the classifier models as the number of correct predictions divided by the total number of predictions. Furthermore, the precision (P), sensitivity (R), and F1 score (F1) of all models were obtained by computing the below equations [31]:  
\[S=\frac{TP+TN}{TP+FP+FN+TN}, \tag{4}\]  
\[P=\frac{TP}{TP+FP}, \tag{5}\]  
\[R=\frac{TP}{TP+FN}, \tag{6}\]  
Figure 5: Arithmetic means of relative humidity with respect to water and ice for contrail lifetime.  
\[F1=\frac{TP}{TP+\frac{1}{2}(FP+FN)}, \tag{7}\]  
where TP represents the number of true-positive classifications, FN is the number of false-negative classifications, FP is the number of false-positive classifications, and TN is the number of true-negative classifications. Precision indicates the proportion of positive predictions that are correct. Sensitivity indicates how many true scores were classified correctly. Precision and sensitivity allow for the evaluation of trained classification models and the selection of the best model. From the confusion matrix, the cumulative frequencies of the predicted values against the actual values can be observed. The F1 score is an evaluation metric of machine learning; it expresses a measure of the harmonic mean of accuracy and sensitivity. It ranges from 0-100%, with higher F1 scores indicating a better-quality classifier. It is sensitive to unbalanced datasets.