# Addressing the Non-Stationarity and Complexity of Time Series Data for Long-Term Forecasts
## 1 Introduction  
Time series forecasting (TSF) plays a crucial role in various real-life applications, including transportation [1], health care [2, 3], and energy management [4]. In such TSF-dependent fields, accurate forecasts over extended horizons contribute to long-term planning and resilience to future challenges. For instance, in sales forecasting, historical sales data can inform inventory management to anticipate future demands. Interest in accurate TSF methods has been established. Traditional approaches such as Kalman filters [5], hidden Markov models [6], and statistical models such as auto regressive integrated moving average (ARIMA) [7] have been commonly used. However, these models often require external input and struggle with long-sequence time series forecasting (LSTF). Similarly, a variety of deep learning architectures have emerged for LSTF. Temporal convolutional networks (TCNs) [8, 9] and their variants have demonstrated success in LSTF [10, 11, 12, 13], achieving a wide receptive field with reduced computational complexity through dilated convolutions. However, TCNs are limited in their ability to capture temporal relationships. Recently, transformer architectures have garnered attention for their capacity to model one-to-one relationships in sequential data [14], proving effective in LSTF [4, 15]. Nonetheless, despite their ability to model one-to-one relationships, transformers require massive computational resources and they also appear to struggle in capturing the inter- and intra-series patterns in time series data. Despite the abundance of architectures designed to address TSF and LSTF, many of these models overlook key challenges inherent in real-life time series data. First, a significant portion of time series data are non-stationary, with statistical properties such as mean and variance changing over time [16]. Second, complex patterns exist within inter- and intra-series relationships of these time series data, posing difficulties for modeling [17]. Overlooking these issues can lead to inaccurate forecasts [17; 18]. Recent efforts using modern deep learning techniques have begun to address these challenges. For instance, an adaptive recurrent neural network model was suggested to handle time series non-stationarity [19], while another work suggested a simple normalization technique to mitigate distribution shifts in time series. This approach involved normalization followed by the restoration of the distribution to recover withdrawn information [18]. Additionally, techniques such as the disentanglement of time series into trends and seasonality have been beneficial for improving forecasts [20]. Moreover, learning features of time series across multiple temporal scales has shown promise [21]. In our approach, we draw inspiration from traditional methods to tackle non-stationary time series and unravel complex patterns. Initially, we consider the widely used statistical ARIMA model, which uses weighted averages of past data point differences to predict future values [7]. We suggest using a separate block to compute these differences, ensuring only past data points with similar phase positions corresponding to each input data point are used. Additionally, prior to forecasting, we restore non-stationarity information to enhance forecast accuracy. Next, we perform spectral decomposition of the time series data to extract suitable features before learning the features for forecasts. Spectral decomposition, commonly employed to decompose input into its frequency components for detailed analysis, is leveraged to simplify intricate time series patterns into spectral components, enabling the learning of hidden features for long-term forecasts. Finally, inspired by the success of mixer architectures such as multilayer perceptron mixer [22] and ConvMixer [23] for mixing spatial and channel-wise data points, we deploy ConvMixer to capture the intra- and inter-series dependencies from the spectral representation of the multivariate time series signals.