Cryptocurrency markets are characterized by ex-treme volatility, making accurate forecasts essential for effective risk management and informed trading strategies. Traditional deterministic (point) forecasting methods are inadequate for capturing the full spectrum of potential volatility outcomes, underscoring the importance of probabilistic approaches. To address this limitation, this paper introduces probabilistic fore-casting methods that leverage point forecasts from a wide range of base models, including statistical (HAR, GARCH, ARFIMA) and machine learning (e.g. LASSO, SVR, MLP, Random Forest, LSTM) algorithms, to estimate conditional quantiles of cryp-tocurrency realized variance. To the best of our knowledge, this is the first study in the literature to propose and systematically evaluate probabilistic forecasts of variance in cryptocurrency markets based on predictions derived from multiple base models. Our empirical results for Bitcoin demonstrate that the Quantile Estimation through Residual Simulation (QRS) method, partic-ularly when applied to linear base models operating on log-transformed realized volatility data, consistently outperforms more sophisticated alternatives. Additionally, we highlight the robustness of the probabilistic stacking framework, providing comprehensive insights into uncertainty and risk inherent in cryptocurrency volatility forecasting. This research fills a sig-nificant gap in the literature, contributing practical probabilistic forecasting methodologies tailored specifically to cryptocurrency markets.

In this paper, we propose a new short-term load forecasting (STLF) model based on contextually enhanced hybrid and hierarchical architecture combining exponential smoothing (ES) and a recurrent neural network (RNN). The model is composed of two simultaneously trained tracks: the context track and the main track. The context track introduces additional information to the main track. It is extracted from representative series and dynamically modulated to adjust to the individual series forecasted by the main track. The RNN architecture consists of multiple recurrent layers stacked with hierarchical dilations and equipped with recently proposed attentive dilated recurrent cells. These cells enable the model to capture short-term, long-term and seasonal dependencies across time series as well as to weight dynamically the input information. The model produces both point forecasts and predictive intervals. The experimental part of the work performed on 35 forecasting problems shows that the proposed model outperforms in terms of accuracy its predecessor as well as standard statistical models and state-of-the-art machine learning models.

Short-term load forecasting (STLF) is challenging due to complex time series (TS) which express three seasonal patterns and a nonlinear trend. This article proposes a novel hybrid hierarchical deep-learning (DL) model that deals with multiple seasonality and produces both point forecasts and predictive intervals (PIs). It combines exponential smoothing (ES) and a recurrent neural network (RNN). ES extracts dynamically the main components of each individual TS and enables on-the-fly deseasonalization, which is particularly useful when operating on a relatively small dataset. A multilayer RNN is equipped with a new type of dilated recurrent cell designed to efficiently model both short and long-term dependencies in TS. To improve the internal TS representation and thus the model's performance, RNN learns simultaneously both the ES parameters and the main mapping function transforming inputs into forecasts. We compare our approach against several baseline methods, including classical statistical methods and machine learning (ML) approaches, on STLF problems for 35 European countries. The empirical study clearly shows that the proposed model has high expressive power to solve nonlinear stochastic forecasting problems with TS including multiple seasonality and significant random fluctuations. In fact, it outperforms both statistical and state-of-the-art ML models in terms of accuracy.

In this paper, we investigate meta-learning for combining forecasts generated by models of different types. While typical approaches for combining forecasts involve simple averaging, machine learning techniques enable more sophisticated methods of combining through meta-learning, leading to improved forecasting accuracy. We use linear regression, knearest neighbors, multilayer perceptron, random forest, and long short-term memory as meta-learners. We define global and local meta-learning variants for time series with complex seasonality and compare meta-learners on multiple forecasting problems, demonstrating their superior performance compared to simple averaging.

The decomposition of a time series is an essential task that helps to understand its very nature. It facilitates the analysis and forecasting of complex time series expressing various hidden components such as the trend, seasonal components, cyclic components and irregular fluctuations. Therefore, it is crucial in many fields for forecasting and decision-making processes. In recent years, many methods of time series decomposition have been developed, which extract and reveal different time series properties. Unfortunately, they neglect a very important property, i.e., time series variance. To deal with heteroscedasticity in time series, the method proposed in this work – a seasonal-trend-dispersion decomposition (STD) – extracts the trend, seasonal component and component related to the dispersion of the time series. We define STD decomposition in two ways: with and without an irregular component. We show how STD can be used for time series analysis and forecasting.

Short-term load forecasting (STLF) is a challenging problem due to the complex nature of the time series expressing multiple seasonality and varying variance. This paper proposes an extension of a hybrid forecasting model combining exponential smoothing and dilated recurrent neural network (ES-dRNN) with a mechanism for dynamic attention. We propose a new gated recurrent cell - attentive dilated recurrent cell, which implements an attention mechanism for dynamic weighting of input vector components. The most relevant components are assigned greater weights, which are subsequently dynamically fine-tuned. This attention mechanism helps the model to select input information and, along with other mechanisms implemented in ES-dRNN, such as adaptive time series processing, cross-learning, and multiple dilation, leads to a significant improvement in accuracy when compared to well-established statistical and state-of-the-art machine learning forecasting models. This was confirmed in the extensive experimental study concerning STLF for 35 European countries.

This work presents a hybrid and hierarchical deep learning model for midterm load forecasting. The model combines exponential smoothing (ETS), advanced long short-term memory (LSTM), and ensembling. ETS extracts dynamically the main components of each individual time series and enables the model to learn their representation. Multilayer LSTM is equipped with dilated recurrent skip connections and a spatial shortcut path from lower layers to allow the model to better capture long-term seasonal relationships and ensure more efficient training. A common learning procedure for LSTM and ETS, with a penalized pinball loss, leads to simultaneous optimization of data representation and forecasting performance. In addition, ensembling at three levels ensures a powerful regularization. A simulation study performed on the monthly electricity demand time series for 35 European countries confirmed the high performance of the proposed model and its competitiveness with classical models such as ARIMA and ETS as well as state-of-the-art models based on machine learning.

Feedforward neural networks are widely used as universal predictive models to fit data distribution. Common gradient-based learning, however, suffers from many drawbacks making the training process ineffective and time-consuming. Alternative randomized learning does not use gradients but selects hidden node parameters randomly. This makes the training process extremely fast. However, the problem in randomized learning is how to determine the random parameters. A recently proposed method uses autoencoders for unsupervised parameter learning. This method showed superior performance on classification tasks. In this work, we apply this method to regression problems, and, finding that it has some drawbacks, we show how to improve it. We propose a learning method of autoencoders that controls the produced random weights. We also propose how to determine the biases of hidden nodes. We empirically compare autoencoder based learning with other randomized learning methods proposed recently for regression and find that despite the proposed improvement of the autoencoder based learning, it does not outperform its competitors in fitting accuracy. Moreover, the method is much more complex than its competitors.

Randomized methods of neural network learning suffer from a problem with the generation of random parameters as they are difficult to set optimally to obtain a good projection space. The standard method draws the parameters from a fixed interval which is independent of the data scope and activation function type. This does not lead to good results in the approximation of the strongly nonlinear functions. In this work, a method which adjusts the random parameters, representing the slopes and positions of the sigmoids, to the target function features is proposed. The method randomly selects the input space regions, places the sigmoids in these regions and then adjusts the sigmoid slopes to the local fluctuations of the target function. A bias-variance tradeoff is controlled by the region size. This brings very good results in the approximation of the complex target functions when compared to the standard fixed interval method and other methods recently proposed in the literature.

This work presents a Long Short-Term Memory (LSTM) network for forecasting a monthly electricity demand time series with a one-year horizon. The novelty of this work is the use of pattern representation of the seasonal time series as an alternative to decomposition. Pattern representation simplifies the complex nonlinear and nonstationary time series, filtering out the trend and equalizing variance. Two types of patterns are defined: x-pattern and y-pattern. The former requires additional forecasting for the coding variables. The latter determines the coding variables from the process history. A hybrid approach based on x-patterns turned out to be more accurate than the standard LSTM approach based on a raw time series. In this combined approach an x-pattern is forecasted using a sequence-to-sequence LSTM network and the coding variables are forecasted using exponential smoothing. A simulation study performed on the monthly electricity demand time series for 35 European countries confirmed the high performance of the proposed model and its competitiveness to classical models such as ARIMA and exponential smoothing as well as the MLP neural network model.

Many forecasting models are built on neural networks. The key issues in these models, which strongly translate into the accuracy of forecasts, are data representation and the decomposition of the forecasting problem. In this work, we consider both of these problems using short-term electricity load demand forecasting as an example. A load time series expresses both the trend and multiple seasonal cycles. To deal with multi-seasonality, we consider four methods of the problem decomposition. Depending on the decomposition degree, the problem is split into local subproblems which are modeled using neural networks. We move from the global model, which is competent for all forecasting tasks, through the local models competent for the subproblems, to the models built individually for each forecasting task. Additionally, we consider different ways of the input data encoding and analyze the impact of the data representation on the results. The forecasting models are examined on the real power system data from four European countries. Results indicate that the local approaches can significantly improve the accuracy of load forecasting, compared to the global approach. A greater degree of decomposition leads to the greater reduction in forecast errors.

In this paper, a new forecasting model based on artificial immune system (AIS) is proposed. The model is used for short-term electrical load forecasting as an example of forecasting time series with multiple seasonal cycles. Artificial immune system learns to recognize antigens (AGs) representing two fragments of the time series: 1) fragment preceding the forecast (input vector) and 2) forecasted fragment (output vector). Antibodies as recognition units recognize AGs by selected features of input vectors and learn output vectors. In the test procedure, new AG with only input vector is recognized by some antibodies (ABs). Its output vector is reconstructed from activated ABs. The unique feature of the proposed AIS is the embedded property of local feature selection. Each AB learns in the clonal selection process its optimal subset of features (a paratope) to improve its recognition and prediction abilities. In the simulation studies the proposed model was tested on real power system data and compared with other AIS-based forecasting models as well as neural networks, autoregressive integrated moving average, and exponential smoothing. The obtained results confirm good performance of the proposed model.

A new multiclass classifier based on immune system principles is proposed. The unique feature of this classifier is the embedded property of local feature selection. This method of feature selection was inspired by the binding of an antibody to an antigen, which occurs between amino acid residues forming an epitope and a paratope. Only certain selected residues (so-called energetic residues) take part in the binding. Antibody receptors are formed during the clonal selection process. Antibodies binding (recognizing) with most antigens (instances) create an immune memory set. This set can be reduced during an optional apoptosis process. Local feature selection and apoptosis result in data-reduction capabilities. The amount of data required for classification was reduced by up to 99%. The classifier has only two user-settable parameters controlling the global-local properties of the feature space searching. The performance of the classifier was tested on several benchmark problems. The comparative tests were performed usingk-NN, support vector machines, and random forest classifiers. The obtained results indicate good performance of the proposed classifier in comparison with both other immune inspired classifiers and other classifiers in general.