Recent advances in deep learning have significantly improved signal denoising by enabling the modeling of complex and nonlinear temporal relationships. However, their performance often declines in real-world settings where data are disrupted by volatile, high-dimensional noise and irregular patterns. To address these challenges, this study presents a novel enhancement of the deep multilayer perceptron through the integration of a hierarchical Kalman filtering framework. This framework introduces a recursively filtering mechanism that combines state-space modeling, spectral decomposition, and multiscale noise isolation. It begins with a Kalman filter that separates raw inputs into trend components and residual noise. A spectral analysis stage then identifies dominant noise frequencies for targeted suppression. Following this, the model iteratively updates its internal state-space parameters based on the evolving statistical properties of the residuals, particularly their power spectral density and autocorrelation decay. To prevent overfitting, convergence thresholds are dynamically tuned based on signal complexity rather than fixed values. By refining noisy inputs into feature-rich signals, the system overcomes key limitations of standard deep multilayer perceptron such as their sensitivity to noise and difficulty modeling fine-grained temporal structures. Empirical evaluations on crude oil price dataset demonstrated substantial improvements. Compared to the deep multilayer perceptron, the proposed model achieved up to 58.2% lower mean absolute error, 82.8% reduction in mean squared error, and 57.0% lower root mean squared error, on the test set after 11 adaptive iterations. Forecasting accuracy improved progressively with each filtering step, and results showed a mean absolute percentage error reduction from 2.79% to 1.21%.

Modeling and forecasting of real-world systems have become one of the most critical needs in different science kinds. Among various factors considered in selecting an appropriate forecasting tool, accuracy is known as the most important criterion. Therefore, the most critical issue in recent forecasting studies is related to improving forecasting accuracy. These studies can be generally categorized into two categories of proposing new single models and developing hybrid models. It has been proven from both theoretical and empirical points of view that combining different models can generate superior results and improve single models' predictive performance. However, despite the popularity and the widespread use of hybrid models, some influential factors, such as the structure of hybridization, number of components, type of components, etc., affect hybrid models' performance that must be appropriately chosen by designers. It is the most challenging subject in the literature of time series forecasting in two recent decades. Several researchers examine different combinations of these factors in order to conclude which one is better. In this way, several different papers have been published in hybridization literature; however, none can prove that its proposed structure is universally better than others. Thus, this paper's primary purpose is to propose an optimal hybrid structure for time series forecasting. The proposed structure's main idea is to simultaneously use remarkable features of series and parallel structures and lift their limitations by hybridization of these two methodologies. In this way, in some parts of the modeling process, parallel structure is used, and in some other parts, the series structure is used. In this paper, a bi-component hybrid model of statistical classic and artificial intelligence models is presented as the initial implementation of the proposed methodology. Its performance is theoretically and empirically evaluated. The optimality of the proposed structure is mathematically demonstrated from the theoretical point of view. It is universally proven that the constructed hybrid model based on the proposed structure will achieve the best performance among all other hybrid models constructed based on series and parallel structures by the same conditions, e.g., number and type of components. In addition, the empirical results of ten benchmark data sets with different characteristics indicate that the profitability of the proposed structure is statistically significant.

Cluster analysis or clustering is one of the most important and widely used techniques for data exploration and knowledge discovery that concerned with partitioning a set of objects in such a way that objects in the same groups, called clusters, are