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This paper compares the most applicable models for the forecasting of time series. Models based on artificial neural networks nonlinear autoregressive neural network with external input (NARX), Elman's neural network, and vector autoregression model were implemented. A special algorithm based on the analysis and prediction of the model residuals, which significantly improves the accuracy of the forecast, was proposed. For the prediction, we used the data of the main greenhouse gases methane (CH4) and water vapor (H2O) concentrations measured in summer 2016 in the surface layer of the atmospheric air on the Arctic island Belyy, Russia. The time interval of 192 h was chosen. The time interval was characterized by the significant daily variations in the CH4 concentration; the H2O concentrations did not have a pronounced trend. Values corresponding to the first 168 h of the interval were used for ANN training, and then, concentrations were predicted for the next 24 h. The accuracy of the prediction was determined by the set of errors and indices. The NARX model was more accurate than models based on Elman network and the VAR. The accuracy gain was from 16.5 to 40% for the models, which predicted the CH4 concentration, and from 21 to 58% for the models, which predicted the H2O concentration. The application of the proposed combined approach made it possible to increase the accuracy of the base model from 1.5 to 20% for the CH4 and from 4 to 20% for the H2O (depending on the corresponding errors and indices). The presented Taylor diagram was also showed the advantage of the proposed approach. [ABSTRACT FROM AUTHOR]