Data-driven models for atmospheric air temperature forecasting at a continental climate region

Which is subjected to: X r i¼1 ai a� i � ¼ 0 ð5Þ ai; a� i 2 0; r ½ � ð6Þ Where ρ are the cost factor, ai; a� i � 0 are the Lagrange multiplier factors. The linear SVR can be written as follows f x ð Þ ¼ X r i¼1 ai a� i � xi; x ð Þ þ b ð7Þ This equation may be considered inappropriate for solving many engineering problems because of its linear characteristic, while engineering problems often need a non-linear regres- sion analysis. Therefore, in order to switch the input data to a much higher-dimensional space, nonlinear kernel functions are utilized. In this regard, the radial bases kernel function (RBF) is used in this study and can be expressed mathematically in the equation below. K xi; x ð Þ ¼ exp b k x xi k 2 ð Þ ð8Þ Where K(xi, x) represents the kernel function and β is the bandwidth of K(xi, x). 2.3 Regression tree and quantile regression tree A decision tree (DT) is a supervised machine learning-based technique that uses labeled data (data with known target attributes) to carry out simulations with the help of classification and regression algorithms [47]. In general, DT’s consist of three types of nodes: decision Root nodes, internal nodes, and leaf nodes, where each node or leaf denotes a class label while the branches denote the outcome of the test performed [48]. The technique splits the input dataset on the basis of the most significant splitter or differentiator in the input variables. This process of data division and selection of the most significant attribute in the dataset is governed by the

classification and regression algorithms. The technique follows a top-down approach as the top portion holds all the observations at one spot, which splits into two or more branches that further split. This approach is also referred to as the greedy approach, as it only incorporates the current nodes without focusing on the future nodes [49]. The decision tree algorithm con- tinues to run until a stop criterion such as the minimum number of observations etc., is attained. Once this criterion is achieved and a decision tree is developed, many nodes are detected as outliers which may be addressed through the tree pruning method. This, in turn, improves the forecasting accuracy of the DT-based model. In the same method that regression minimizes cost function (i.e., squared-error loss) when forecasting a single point estimate, the quantile regression tree (QRT) minimizes the loss func- tion in forecasting a particular quantile. The median, or 50th percentile, is the most commonly used quantile, and the quantile loss is just the sum of absolute errors in this case. Additionally, quantiles can be used as endpoints of prediction intervals; for instance, the 10th and 90th per- centiles define an 80-percentile range in the middle. It appears that the quantile loss differs according to the evaluated quantile, with higher quantiles penalizing more for negative errors and lower quantiles penalizing more for positive errors. Accordingly, in this study, we used the median (the 50th percentile), which is the most well-known quantile.

In the fields of artificial intelligence and search algorithms, pruning is a data compression method used to minimize the size of decision trees by deleting parts of the tree that are deemed PLOS ONE Air temperature forecasting: Mutli-time scale PLOS ONE | https://doi.org/10.1371/journal.pone.0277079 November 3, 2022 6 / 31