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

non-critical and repetitive to the regression of instances. By assessing the predictive value of each node in a regression tree, regression tree pruning decreases the danger of overfitting. Nodes that do not increase the anticipated prediction efficiency on new data are substituted with leaves. 2.4 Gradient Boosting Regression (GBR) GBR is an ensemble machine learning approach that enhances the prediction performance of a classical decision tree by incorporating a sequential statistical process called boosting, of which the principle idea is to combine a set of weak predictive models to form a single and high accurate predictive model [50, 51]. The technique applies an iterative procedure, where the estimates of the new tree model (weak learner) are updated with the pseudo residuals (neg- ative gradient of the loss function) of the current learner [52]. This process is repeated until the loss function of the model is reduced to a minimum value, thus improving the forecasting performance of the model. The iterative training process of the GBRT with K decision trees can be briefly explained as follows: For a given training dataset D = {(x1, y1), (x2, y2),. . .. . ., (xn, yn)}, the loss function is com- puted as: L y; f x ð Þ ð Þ ¼ y f x ð Þ ð Þ2 ð9Þ Step 1. Initialize the new tree model (weak learner) with a constant value: f0 x ð Þ ¼ arg minc XN i¼1 L yi; c ð Þ ð10Þ Step 2. Assume the number of iterations m = 1, 2, 3. . .. . .., K (a) For i = 1,2,3. . .. . ..,N.The pseudo residuals of the ith training data is calculated as: rmi ¼ @L y; f xi ð Þ ð Þ @f xi ð Þ � �

f xð Þ¼fm 1 xð Þ ð11Þ (b) Fit a regression tree in terms of rmi, and deduce the leaf node area Rml of the mth tree. Predict the leaf node area of the decision tree to attain an approximate value of the fitting residual. (c) For l = 1,2,3. . .. . .., L. Adopt linear search to attain the value in the leaf node range and minimize the loss function with gradient descent. The best residual fitting value of each blade is as follows: cm ¼ arg minc XN i¼1 L yi; fm 1 xi ð Þ þ c ð Þ ð12Þ (d) Update the regression tree fm x ð Þ ¼ fm 1 x ð Þ þ XL l¼1 cmlI x 2 Rml ð Þ ð13Þ Step 3. Obtain the final model f x ð Þ ¼ fM x ð Þ ¼ XM m¼1 XL l¼1 cmlI x 2 Rml ð Þ ð14Þ PLOS ONE Air temperature forecasting: Mutli-time scale PLOS ONE | https://doi.org/10.1371/journal.pone.0277079 November 3, 2022 7 / 31