Solar activity directly influences the heliospheric environment and lives on the Earth. Sunspot number (SN) is one of the most crucial and commonly predicted solar activity indices. The prediction of SN time series is a challenging problem owing to its non-stationary, non-Gaussian and nonlinear nature. Therefore, improving the forecasting accuracy of SN time series is an important and challenging task. Motivated from this, in this paper, we have proposed a hybridization of the autoregressive integrated moving average (ARIMA); exponential smoothing with error, trend and seasonality (ETS); and support vector machine (SVM) to predict monthly and yearly SN time series. In this method, first ARIMA, ETS and SVM with linear kernel function are applied to the SN time series and the maximum of forecasts are determined to obtain the forecasts on linear component. Then the residual series is obtained by subtracting the forecasts on linear component from SN time series. The residual series is considered as nonlinear and modeled using SVM with Gaussian kernel function. Then the forecasts on linear component are added with the forecasts on nonlinear component to obtain the final forecasts. To evaluate the efficiency of the proposed method, three constituent models, one of the most popular deep learning models long short-term memory (LSTM), four hybrid methods, four ensemble methods are considered. Furthermore, two horizons, monthly and yearly sunspot time series are considered to evaluate the robustness of the proposed method. Results indicate the statistical superiority of the proposed methods over different horizons considering different accuracy measures.

中文翻译：

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使用 ARIMA、ETS 和 SVM 方法的混合预测太阳黑子时间序列

太阳活动直接影响日光层环境并生活在地球上。太阳黑子数（SN）是最重要和最常预测的太阳活动指数之一。SN 时间序列的预测由于其非平稳、非高斯和非线性特性而成为一个具有挑战性的问题。因此，提高SN时间序列的预测精度是一项重要且具有挑战性的任务。受此启发，在本文中，我们提出了自回归综合移动平均线（ARIMA）的混合；具有误差、趋势和季节性（ETS）的指数平滑；和支持向量机 (SVM) 来预测每月和每年的 SN 时间序列。在这种方法中，首先 ARIMA，将具有线性核函数的 ETS 和 SVM 应用于 SN 时间序列，并确定预测的最大值以获得对线性分量的预测。然后通过从SN时间序列中减去对线性分量的预测得到残差序列。残差序列被认为是非线性的，并使用具有高斯核函数的 SVM 进行建模。然后将线性分量的预测值与非线性分量的预测值相加得到最终的预测值。为了评估所提出方法的效率，考虑了三个组成模型、最流行的深度学习模型之一长短期记忆 (LSTM)、四种混合方法、四种集成方法。此外，考虑到每月和每年的太阳黑子时间序列这两个视界来评估所提出方法的稳健性。