An Adaptive Decomposition–Ensemble Modeling Method for Multi-Category Power Materials Demand Forecasting with Uncertainty Quantification

Abstract

Accurate demand forecasting with uncertainty quantification is critical for materials management in power grid enterprises, yet existing methods struggle to capture multi-scale temporal dynamics across heterogeneous material categories while providing reliable confidence estimates. This paper proposes an Adaptive Decomposition–Ensemble Modeling (ADEM) method that integrates adaptive CEEMDAN (Complete Ensemble Empirical Mode Decomposition with Adaptive Noise) with category-specific depth selection, a heterogeneous ensemble of a GBM (Gradient Boosting Machine), ELM (Extreme Learning Machine), and SVR (Support Vector Regression) with per-component optimized weights, and Bayesian uncertainty quantification with conformal calibration for distribution-free coverage guarantees. Experiments on real-world data spanning 18 material categories over 60 months demonstrate that ADEM consistently outperforms 14 baselines spanning statistical, machine learning, deep learning, and decomposition-based methods in both point prediction accuracy and prediction interval quality. Rolling-origin evaluation across six temporal windows further exhibits the robustness and statistical significance of these improvements.

1. Introduction

The power grid industry faces unprecedented challenges in material management due to the increasing complexity of infrastructure projects, the diversity of material categories, and the critical nature of maintaining continuous electricity supply [1]. Power materials encompass a wide spectrum, including high-voltage equipment, construction supplies, emergency repair materials, and specialized technical components, each exhibiting distinct demand patterns influenced by project schedules, seasonal variations, policy changes, and unforeseen operational requirements [2,3]. Accurate demand forecasting is fundamental to optimizing inventory levels, reducing holding costs, minimizing emergency procurement expenses, and ensuring project execution without material shortages that could compromise grid reliability.

Time series forecasting methodologies have undergone substantial evolution in recent decades, progressing from traditional statistical approaches through machine learning techniques to contemporary deep learning architectures. Traditional statistical methods such as Autoregressive Integrated Moving Average (ARIMA) models and exponential smoothing have been extensively applied in supply chain demand forecasting, offering interpretability and computational efficiency [4,5]. The advancement of machine learning [6] subsequently introduced powerful nonlinear modeling capabilities, with Support Vector Machines, Artificial Neural Networks, and ensemble methods such as Random Forests and Gradient Boosting Machines demonstrating superior performance in capturing complex demand patterns [7,8]. More recently, deep learning [9] architectures, including Long Short-Term Memory (LSTM) networks [10,11], have achieved notable results by leveraging their capacity to model long-range temporal dependencies through gating mechanisms. Transformer architectures, originally developed for natural language processing, have also been adapted for time series prediction with the promise of capturing long-range dependencies through self-attention mechanisms [12,13].

Despite this methodological progression, each paradigm exhibits inherent limitations that constrain its effectiveness in forecasting the demand for power materials. Traditional statistical methods operate under restrictive linear assumptions and stationarity requirements [14] that conflict with time-varying structures driven by infrastructure investment cycles while facing difficulties incorporating exogenous variables and requiring separate model identification for each material category. Machine learning approaches, though capable of nonlinear modeling, typically treat time series as static feature vectors, potentially losing important temporal dependencies inherent in sequential data [15]. LSTM-based methods suffer from accumulated error propagation in multi-step forecasting scenarios, diminishing temporal correlation over extended prediction horizons, and limited interpretability that compromises user confidence. Transformer architectures exhibit quadratic computational complexity with respect to sequence length, restricting the historical horizon that can be effectively processed, while their attention mechanisms frequently fail to discriminate between relevant and irrelevant temporal patterns and may degenerate into simple multilayer perceptrons in time series applications. Perhaps most critically, these methods—whether statistical, machine learning, or deep learning—typically provide only point predictions without quantifying forecast uncertainty, limiting their utility for risk-aware decision making in supply chain operations where prediction confidence directly influences inventory buffer sizing and procurement timing decisions [16].

To address these limitations, this paper proposes a novel Adaptive Decomposition–Ensemble Modeling (ADEM) method for multi-category power materials demand forecasting with uncertainty quantification. The fundamental mechanism underlying ADEM is that complex demand signals—characterized by multi-scale temporal dynamics, nonlinear dependencies, and the superposition of trend, seasonal, and irregular fluctuations driven by diverse factors such as project schedules, policy changes, and operational contingencies—can be more effectively predicted by first decomposing them into simpler components with distinct temporal characteristics, then leveraging specialized models that exploit their comparative advantages for different component types, and finally providing reliable confidence estimates to support risk-aware decision making. This decompose–ensemble-quantify paradigm enables the model to systematically address the multi-scale nature of power materials demand while accommodating the heterogeneous characteristics across different material categories. Specifically, the proposed model employs a Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN)-based algorithm [17] to adaptively extract intrinsic mode functions (IMFs) reflecting different time scales of demand fluctuations—ranging from long-term trends to short-term irregular variations—with the optimal number of such functions automatically determined for each material category through energy contribution analysis and predictive validation, thereby separating demand signals into trend, seasonal, and irregular components tailored to category-specific characteristics. The decomposed components are subsequently processed by a heterogeneous ensemble architecture where a Gradient Boosting Machine (GBM), Extreme Learning Machine (ELM) and Support Vector Regression (SVR) are assigned optimized weights through constrained optimization, allowing each type of model to handle components according to its comparative advantages. Furthermore, to quantify prediction confidence, a Bayesian module constructs prediction intervals through posterior predictive distributions with conformal calibration, providing grounded coverage guarantees for risk-aware inventory optimization and procurement planning. The main contributions of this paper are as follows:

• An adaptive CEEMDAN-based decomposition–ensemble framework specifically designed for multi-category power materials demand forecasting is proposed. The framework employs an adaptive CEEMDAN strategy that automatically determines category-specific decomposition depth, a heterogeneous ensemble architecture that assigns the GBM, ELM, and SVR to different signal components based on their comparative advantages, and a Multi-Dimensional Feature Engineering Module that captures domain-specific driving factors, including project schedules and cross-category correlations. These mechanisms collectively enable robust forecasting across diverse material categories with varying demand characteristics.
• A Bayesian uncertainty quantification module that provides grounded prediction intervals is introduced for risk-aware decision making. This module constructs prediction intervals through posterior predictive distributions, augmented with conformal calibration to ensure finite-sample coverage guarantees. Unlike existing models that focus solely on point predictions, the introduced approach addresses the critical need for uncertainty estimates in inventory optimization and procurement planning, enabling power grid enterprises to make informed decisions under demand uncertainty conditions.
• Extensive experiments on real-world power materials demand data spanning 18 specialized categories over 60 months demonstrate that ADEM consistently achieves the lowest prediction error and the highest prediction interval coverage among all 14 compared baselines spanning statistical, machine learning, deep learning, and decomposition-based methods. Rolling-origin evaluation across six temporal windows further confirms the robustness and statistical significance of these improvements.

The remainder of this paper is organized as follows. Section 2 reviews related work on demand forecasting and CEEMDAN. Section 3 presents the detailed methodology of the proposed ADEM framework, including the adaptive CEEMDAN, heterogeneous ensemble architecture, Bayesian uncertainty quantification module, and feature engineering framework. Section 4 describes the experimental setup, including dataset characteristics, baseline methods, evaluation metrics, and implementation details. Section 5 reports comprehensive experimental results under both fixed-partition and rolling-origin evaluation protocols, including ablation studies, category-level analysis with feature importance, uncertainty quantification analysis, sensitivity analysis of key hyperparameters, computational cost analysis, and managerial implications. Section 6 concludes the paper and outlines future research directions.

2. Related Work

This section provides a comprehensive review of related research, organized into two main categories: time series forecasting methods for demand prediction and CEEMDAN.

2.1. Time Series Forecasting Methods for Demand Prediction

Time series forecasting has been extensively studied across various domains, with methodologies evolving from classical statistical approaches to advanced machine learning and deep learning methods.

(1) Statistical Methods: Classical statistical methods remain foundational in demand forecasting applications. The autoregressive integrated moving average (ARIMA) model, formalized by Box and Jenkins [4], has been widely adopted for its interpretability and theoretical grounding. Hyndman and Athanasopoulos [5] provided comprehensive treatments of exponential smoothing methods, demonstrating their effectiveness in capturing trend and seasonal patterns. Taylor and Letham [18] introduced Prophet, a decomposable time series model designed for business forecasting that handles missing data and outliers robustly. However, these methods inherently assume linear relationships within the data, limiting their capacity to model complex nonlinear dynamics prevalent in industrial demand patterns.

(2) Machine Learning Methods: The advent of machine learning has substantially expanded the forecasting toolkit. SVR, introduced by Drucker et al. [19], leverages kernel functions to capture nonlinear relationships while maintaining computational tractability. Random Forests, proposed by Breiman [20], aggregate predictions from multiple decision trees to reduce variance and improve generalization. The GBM, formalized by Friedman [8], constructs additive models through sequential optimization of differentiable loss functions. Chen and Guestrin [21] developed XGBoost, an efficient and scalable implementation that has achieved state-of-the-art performance across numerous forecasting benchmarks [22]. Extreme Learning Machines (ELMs), proposed by Huang et al. [23], offer rapid training through random initialization of hidden layer weights, making them attractive for real-time applications.

(3) Deep Learning Methods: Deep learning architectures have demonstrated remarkable capabilities in modeling sequential dependencies. Long short-term memory (LSTM) networks, introduced by Hochreiter and Schmidhuber [10], address the vanishing gradient problem through gating mechanisms, enabling learning of long-range temporal dependencies. Cho et al. [24] proposed gated recurrent units (GRU) as a simplified variant with comparable performance. The transformer architecture, proposed by Vaswani et al. [25], employs self-attention mechanisms to model dependencies without recurrence constraints. Lim et al. [13] developed the Temporal Fusion Transformer (TFT), incorporating interpretable attention mechanisms specifically designed for multi-horizon forecasting. Zhou et al. [26] proposed Informer, addressing the quadratic complexity limitation of vanilla transformers through ProbSparse self-attention. Zeng et al. [27] challenged the necessity of complex architectures by showing that simple linear models (DLinear) can outperform sophisticated transformers on many benchmarks, sparking considerable debate in the community. Das et al. [28] introduced a decoder-only foundation model for time series forecasting, achieving zero-shot performance competitive with supervised approaches. Despite their impressive performance, these deep learning methods predominantly provide point predictions without quantifying prediction uncertainty, limiting their applicability in risk-sensitive decision-making contexts.

2.2. CEEMDAN

Signal decomposition techniques have gained substantial attention for their ability to extract multi-scale temporal patterns from complex time series. Among various decomposition methods, CEEMDAN, proposed by Torres et al. [17], has emerged as a particularly effective approach for handling nonlinear and non-stationary signals prevalent in industrial demand data.

CEEMDAN originated from the empirical mode decomposition (EMD) framework proposed by Huang et al. [29], which decomposes a signal $\mathbf{y}={[y_1,y_2,\dots ,y_T]}^T$ into a finite set of IMFs and a residual component through an iterative sifting procedure. The original signal can be reconstructed as

$$\mathbf{y}\left(t\right)=\sum _{k=1}^K{\mathbf{IMF}}_k\left(t\right)+\mathbf{r}\left(t\right)$$

(1)

where ${\mathbf{IMF}}_k\left(t\right)$ denotes the k-th IMF capturing oscillatory components at different time scales, and $\mathbf{r}\left(t\right)$ represents the final residue reflecting the underlying trend. Let $E_k(·)$ denote the operator that extracts the k-th IMF through EMD, and let $w^{\left(i\right)}\left(t\right)$ represent the i-th realization of unit-variance white Gaussian noise. The CEEMDAN algorithm computes the first mode by averaging over I noise-added realizations:

$${\overline{\mathbf{IMF}}}_1\left(t\right)={\displaystyle \frac{1}{I}}\sum _{i=1}^I{E}_1\left(\mathbf{y}\left(t\right)+{\epsilon }_0{w}^{\left(i\right)}\left(t\right)\right)$$

(2)

where ${\epsilon }_0$ controls the initial noise amplitude. The first residue is then calculated as $r_1\left(t\right)=\mathbf{y}\left(t\right)-{\overline{\mathbf{IMF}}}_1\left(t\right)$. For subsequent modes $k=2,\dots ,K$, CEEMDAN adaptively injects noise that has been filtered through the decomposition operator:

$${\overline{\mathbf{IMF}}}_k\left(t\right)={\displaystyle \frac{1}{I}}\sum _{i=1}^I{E}_1\left(r_{k-1}\left(t\right)+{\epsilon }_{k-1}{E}_{k-1}\left(w^{\left(i\right)}\left(t\right)\right)\right)$$

(3)

with residues updated as $r_k\left(t\right)=r_{k-1}\left(t\right)-{\overline{\mathbf{IMF}}}_k\left(t\right)$. The noise coefficients ${\epsilon }_k$ are typically set as ${\epsilon }_k={\epsilon }_0·\mathrm{std}\left(r_k\right)$ to adapt to the signal energy at each stage. This procedure continues until the residue $\mathbf{r}\left(t\right)=r_K\left(t\right)$ becomes monotonic or until a predefined maximum number of IMFs has been extracted. The complete signal reconstruction is given by

$$\mathbf{y}\left(t\right)=\sum _{k=1}^K{\overline{\mathbf{IMF}}}_k\left(t\right)+\mathbf{r}\left(t\right).$$

(4)

CEEMDAN achieves exact reconstruction with negligible error and produces IMFs with high decomposition quality [30]. These properties make CEEMDAN particularly suitable for demand forecasting applications where accurate extraction of trend, seasonal, and irregular components is essential.

CEEMDAN has been successfully applied across diverse forecasting domains [31,32,33]. Li et al. [34] combined CEEMDAN with LSTM networks for short-term load forecasting, demonstrating significant accuracy improvements. Niu et al. [35] developed a CEEMDAN-based hybrid model for wind power prediction that achieved superior performance in capturing multi-scale fluctuations. Ribeiro et al. [36] applied CEEMDAN to electricity price forecasting, showing that adaptive decomposition substantially enhances prediction accuracy. However, most existing studies employ fixed decomposition depths across all series, failing to account for heterogeneous signal characteristics. This limitation motivates our adaptive strategy that automatically determines optimal decomposition depth based on category-specific signal properties through energy contribution analysis and predictive validation.

To contextualize the choice of CEEMDAN within the broader landscape of signal decomposition methods, it is instructive to compare its properties with the principal alternatives. The original EMD [29] suffers from mode mixing, whereby signals of disparate scales are erroneously merged into a single IMF, degrading the interpretability and predictability of the extracted components. Ensemble EMD (EEMD) [37] mitigates mode mixing by averaging over multiple noise-added trials but introduces residual noise that cannot be fully eliminated and does not guarantee exact signal reconstruction. Variational Mode Decomposition (VMD) [38] reformulates decomposition as a constrained optimization problem, yielding compact frequency-band modes; however, VMD requires a priori specification of the number of modes K, which is precisely the parameter that varies across material categories and that our adaptive strategy aims to determine automatically. Wavelet-based methods [39] offer multi-resolution analysis with well-characterized frequency localization but require the selection of a mother wavelet and decomposition level, introducing additional design choices whose optimality is data-dependent and difficult to validate without domain-specific prior knowledge. By contrast, CEEMDAN [17,30] achieves exact reconstruction with negligible residual noise, produces IMFs free from mode mixing through adaptive noise injection, and—critically for the present application—does not require a priori specification of the number of modes, thereby providing the flexible foundation upon which the proposed adaptive depth selection strategy can operate.

It is important to note that while CEEMDAN, ensemble learning, and Bayesian uncertainty quantification are individually well-established techniques, existing hybrid forecasting approaches combine them without adequately addressing three fundamental limitations. First, prior CEEMDAN-based methods [34,35,36] uniformly apply a fixed decomposition depth across all series, disregarding the heterogeneous signal characteristics among different material categories. Second, most decomposition–ensemble strategies assign a single model or uniform weights to all decomposed components, thereby failing to exploit the distinct predictability characteristics inherent at different frequency scales. Third, existing decomposition–ensemble methods rarely incorporate uncertainty estimation, and those that do typically lack formal coverage guarantees that account for the additive reconstruction structure inherent in signal decomposition frameworks. The proposed ADEM advances beyond these approaches through three tightly coupled methodological innovations: (a) an adaptive three-criteria voting mechanism that automatically determines category-specific decomposition depth $K^{\ast \left(c\right)}$ by jointly considering energy contribution, statistical significance, and predictive validation; (b) a component-type-aware heterogeneous ensemble architecture with per-component optimized weights ${\mathbf{w}}_k^{\left(c\right)}$ that enables model specialization across distinct frequency bands; and (c) a decomposition-compatible Bayesian uncertainty quantification module augmented with conformal calibration to provide distribution-free finite-sample coverage guarantees. It is this systematic integration of adaptive decomposition, component-specialized ensemble forecasting, and calibrated uncertainty quantification—rather than any individual component—that constitutes the principal methodological contribution of this work.

3. Proposed ADEM for Multi-Category Power Materials Demand Forecasting with Uncertainty Quantification

3.1. Problem Formulation

We formulate the multi-category power materials demand forecasting problem with uncertainty quantification as follows. Consider a dataset $\mathcal{D}=\left\{({\mathbf{x}}_i,y_i,c_i)\right|i=1,2,\dots ,N\}$ containing N historical demand records across C material categories over T time periods. Each sample ${\mathbf{x}}_i={[x_{i,1},x_{i,2},\dots ,x_{i,M}]}^T\in {\mathbb{R}}^{M\times 1}$ represents a feature vector with dimensionality M, where $y_i\in {\mathbb{R}}^{+}$ denotes the corresponding demand quantity, and $c_i\in \{1,2,\dots ,C\}$ indicates the material category index. Based on the category labels $c_i$, the dataset $\mathcal{D}$ is partitioned into C category-specific subsets, namely, ${\mathcal{D}}_1,{\mathcal{D}}_2,\dots ,{\mathcal{D}}_C$, where $\mathcal{D}={\mathcal{D}}_1\cup {\mathcal{D}}_2\cup \dots \cup {\mathcal{D}}_C$, and ${\mathcal{D}}_c=\left\{({\mathbf{x}}_i,y_i,c_i)\right|c_i=c;i=1,2,\dots ,N_c\}$ for $c=1,2,\dots ,C$. Here, $N_c$ represents the cardinality of the c-th material category subset.

The primary objective is to construct a forecasting function $\mathcal{F}:{\mathbb{R}}^M\to {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ that maps the input feature vector $\mathbf{x}$ to a triplet $(\widehat{y},{\widehat{y}}_{lower},{\widehat{y}}_{upper})$, where $\widehat{y}$ denotes the point prediction, and $[{\widehat{y}}_{lower},{\widehat{y}}_{upper}]$ represents the prediction interval with a specified confidence level $1-\alpha$. Formally, the forecasting task can be expressed as

$${\widehat{y}}_{t+h}^{\left(c\right)}=\mathcal{F}({\mathbf{x}}_t^{\left(c\right)};{\mathrm{\Theta }}_{ADEM})$$

(5)

where ${\widehat{y}}_{t+h}^{\left(c\right)}$ represents the predicted demand for material category c at forecast horizon h, ${\mathbf{x}}_t^{\left(c\right)}$ denotes the feature vector constructed from historical observations up to time t, and ${\mathrm{\Theta }}_{ADEM}$ encompasses all learnable parameters within the ADEM. The uncertainty quantification objective requires that the constructed prediction intervals satisfy the coverage guarantee:

$$P(y_{t+h}^{\left(c\right)}\in [{\widehat{y}}_{lower},{\widehat{y}}_{upper}])\ge 1-\alpha$$

(6)

where $y_{t+h}^{\left(c\right)}$ denotes the actual demand value.

3.2. Framework Overview

The architecture of the proposed ADEM, illustrated in Figure 1, comprises five interconnected modules organized in a sequential pipeline. Table 1 provides a concise summary of the input, output, and key technique of each module. Detailed descriptions of the individual modules are presented in the following subsections.

Figure 1. The architecture of the proposed ADEM.

Table 1. Summary of the five modules in the proposed ADEM framework.

No.	Module	Input	Output	Key Technique
1	Adaptive CEEMDAN	Raw demand series ${\mathbf{y}}^{\left(c\right)}$	IMFs $\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}$ and residual ${\mathbf{r}}^{\left(c\right)}$ with optimal depth $K^{\ast \left(c\right)}$	Three-criteria voting for adaptive depth selection
2	Multi-Dimensional Feature Engineering	Decomposed IMFs, residual, and auxiliary data	Feature matrices ${\mathbf{X}}_k^{\left(c\right)}\in {\mathbb{R}}^{(T-W)\times M_k}$	Statistical, project, temporal, and cross-category feature extraction
3	Heterogeneous Ensemble Forecasting	Feature matrices and target IMF values	Component-level predictions ${\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)$	GBM + ELM + SVR with per-component optimized weights
4	Bayesian Uncertainty Quantification	Ensemble predictions and validation residuals	Variance estimates ${\sigma }_k^{2,\left(c\right)}$ and calibrated quantiles $q_{1-\alpha }$	Bootstrap variance estimation with conformal calibration
5	Intelligent Aggregation	Component predictions and variance estimates	Prediction triplet $({\widehat{y}}_{t+h}^{\left(c\right)},{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)})$	Additive reconstruction with variance propagation

Module 1: Adaptive CEEMDAN Module. This module receives the raw demand time series ${\mathbf{y}}^{\left(c\right)}={[y_1^{\left(c\right)},y_2^{\left(c\right)},\dots ,y_T^{\left(c\right)}]}^T\in {\mathbb{R}}^{T\times 1}$ for each material category c as input. Through adaptive signal decomposition, this module produces a set of IMFs $\{{\overline{\mathbf{IMF}}}_1^{\left(c\right)},{\overline{\mathbf{IMF}}}_2^{\left(c\right)},\dots ,{\overline{\mathbf{IMF}}}_{K^{\ast \left(c\right)}}^{\left(c\right)}\}$ and a residual trend component ${\mathbf{r}}^{\left(c\right)}$ as output, where $K^{\ast \left(c\right)}$ denotes the automatically determined optimal decomposition depth for category c. The decomposition satisfies the reconstruction property ${\mathbf{y}}^{\left(c\right)}={\sum }_{k=1}^{K^{\ast \left(c\right)}}{\overline{\mathbf{IMF}}}_k^{\left(c\right)}+{\mathbf{r}}^{\left(c\right)}$.

Module 2: Multi-Dimensional Feature Engineering Module. This module takes the decomposed components ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ from the first module, together with auxiliary information including project attributes, calendar variables, and cross-category demand series, as input. It transforms these raw inputs into comprehensive feature matrices ${\mathbf{X}}_k^{\left(c\right)}\in {\mathbb{R}}^{(T-W)\times M_k}$ for each decomposed component, where W denotes the sliding window length and $M_k$ represents the number of engineered features for component k. These feature matrices serve as the training data for downstream forecasting models.

Module 3: Heterogeneous Ensemble Forecasting Module. This module receives the component-specific feature matrices ${\left\{{\mathbf{X}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and corresponding target vectors ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ as input. It trains three base learners—a GBM, an ELM, and SVR—on each component and determines optimized combination weights. The output consists of component-level point predictions ${\left\{{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and ${\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$ for the specified forecast horizon h.

Module 4: Bayesian Uncertainty Quantification Module. This module takes the component-level predictions from the ensemble module and the associated prediction residuals on validation data as input. Through Bayesian inference and conformal calibration, it produces component-level variance estimates ${\left\{{\sigma }_k^{2,\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and calibrated quantile functions as output. These uncertainty estimates enable the construction of prediction intervals with coverage guarantees.

Module 5: Intelligent Aggregation Module. This module receives the component-level point predictions ${\left\{{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)\right\}}_{k=1}^{K^{\ast \left(c\right)}}$, ${\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$, and the uncertainty estimates ${\left\{{\sigma }_k^{2,\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ as input. It reconstructs the final demand forecast by summing the component predictions and propagates uncertainty through the aggregation process. The final output is the prediction triplet $({\widehat{y}}_{t+h}^{\left(c\right)},{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)})$, consisting of the point prediction and the $(1-\alpha )$ prediction interval bounds.

3.3. Detailed Design of Individual Modules

3.3.1. Adaptive CEEMDAN Module

Introduced in Section 2.2, CEEMDAN decomposes a signal into a set of IMFs and a residual component through adaptive noise-assisted ensemble averaging. In the proposed ADEM method, this decomposition module receives the raw demand time series ${\mathbf{y}}^{\left(c\right)}={[y_1^{\left(c\right)},y_2^{\left(c\right)},\dots ,y_T^{\left(c\right)}]}^T\in {\mathbb{R}}^{T\times 1}$ for each material category c as input. Following Equations (1)–(4), CEEMDAN produces a collection of IMFs $\{{\overline{\mathbf{IMF}}}_1^{\left(c\right)},{\overline{\mathbf{IMF}}}_2^{\left(c\right)},\dots ,{\overline{\mathbf{IMF}}}_{K_{max}}^{\left(c\right)}\}$ and a residual trend ${\mathbf{r}}^{\left(c\right)}$ as output, where $K_{max}$ denotes the maximum number of extractable modes determined by the termination criterion. A critical limitation of existing CEEMDAN-based forecasting methods is their reliance on fixed decomposition depths across all series, neglecting the heterogeneous signal characteristics among different material categories. To address this limitation, an adaptive IMF selection strategy is proposed that automatically determines the optimal decomposition depth $K^{\ast \left(c\right)}$ for each category based on the following three complementary criteria.

The first criterion is the Energy Contribution Threshold, which assesses how much of the original signal’s energy is captured by the retained IMFs. The rationale is that an IMF accounting for a substantial proportion of the total signal energy carries significant predictive information and should be retained, whereas an IMF with negligible energy contribution is likely dominated by noise and can be safely discarded without loss of forecasting accuracy. The energy contribution of each IMF is quantified as its proportion of the total signal energy, including both IMFs and the residual component:

$$E_k^{\left(c\right)}={\displaystyle \frac{{\sum }_{t=1}^T{\left|{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(t\right)\right|}^2}{{\sum }_{j=1}^{K_{max}}{\sum }_{t=1}^T|{\overline{\mathbf{IMF}}}_j^{\left(c\right)}{\left(t\right)|}^2+{\sum }_{t=1}^T{\left|{\mathbf{r}}^{\left(c\right)}\left(t\right)\right|}^2}},$$

(7)

and the cumulative energy contribution up to the k-th IMF is defined as

$$C{E}_k^{\left(c\right)}=\sum _{j=1}^k{E}_j^{\left(c\right)}.$$

(8)

Note that the residual ${\mathbf{r}}^{\left(c\right)}$ is always retained as the trend component. The threshold criterion therefore selects the minimum number of IMFs such that, together with the residual, they capture at least ${\tau }_E$ of the total signal energy: $K_E^{\ast \left(c\right)}=min\{k:C{E}_k^{\left(c\right)}+E_r^{\left(c\right)}\ge {\tau }_E\}$.

The second criterion is Statistical Significance Testing using the Ljung–Box test, which evaluates whether each IMF contains statistically significant autocorrelation structure that warrants explicit modeling. The underlying rationale is that an IMF exhibiting significant temporal dependencies contains learnable temporal patterns amenable to forecasting, whereas an IMF whose autocorrelation structure is indistinguishable from white noise provides no additional predictive value and should be excluded from the modeling pipeline. The Ljung–Box test statistic is computed as

$$Q_{LB}^{(k,c)}=T(T+2)\sum _{l=1}^L{\displaystyle \frac{{\widehat{\rho }}_l^2\left({\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right)}{T-l}}$$

(9)

where ${\widehat{\rho }}_l$ denotes the sample autocorrelation at lag l, and L represents the maximum lag order. Under the null hypothesis that the IMF is white noise, this statistic follows a chi-squared distribution with L degrees of freedom. The number of significant IMFs is determined as $K_S^{\ast \left(c\right)}=max\{k:Q_{LB}^{(k,c)}>{\chi }_{1-{\alpha }_Q,L}^2\}$, where ${\alpha }_Q$ denotes the significance level (typically 0.05), ensuring that only components with meaningful temporal structure are retained.

The third criterion is predictive validation through expanding-window cross-validation, which directly assesses the marginal predictive improvement contributed by each additional IMF while strictly preserving temporal ordering. For each candidate decomposition depth k, a forecasting model is trained using the first k IMFs. Specifically, for each split point $t_s\in \{T_{min},T_{min}+1,\dots ,T-1\}$, the model is trained on observations $\{1,\dots ,t_s\}$ and evaluated on the one-step-ahead prediction at $t_s+1$, ensuring that no future information is used during training. The cross-validated prediction error is computed as

$$C{V}_k^{\left(c\right)}={\displaystyle \frac{1}{T-T_{min}}}\sum _{t_s=T_{min}}^{T-1}{(y_{t_s+1}^{\left(c\right)}-{\widehat{y}}_{t_s+1}^{(c,k)})}^2.$$

(10)

where $T_{min}$ denotes the minimum training length required for stable model fitting (set to 24 months in this study). The optimal number based on predictive performance is $K_P^{\ast \left(c\right)combinedCEEMDANwithLSTMnetworks}=arg{min}_{k}C{V}_k^{\left(c\right)}$. This criterion prevents both over-decomposition, which could introduce noise, and under-decomposition, which could discard valuable predictive information.

The final optimal decomposition depth integrates these three criteria through a voting mechanism:

$$K^{\ast \left(c\right)}=\mathrm{median}\{K_E^{\ast \left(c\right)},K_S^{\ast \left(c\right)},K_P^{\ast \left(c\right)}\}.$$

(11)

This approach ensures robust selection by balancing energy preservation, statistical significance, and predictive ability across different material categories with varying demand characteristics. The complete adaptive decomposition depth selection procedure is summarized in Algorithm 1.

Algorithm 1 Adaptive decomposition depth selection

Require: Demand series ${\mathbf{y}}^{\left(c\right)}$, energy threshold ${\tau }_E$, significance level ${\alpha }_Q$, maximum lag L, minimum training length $T_{min}$

Ensure: Optimal decomposition depth $K^{\ast \left(c\right)}$

1: Apply CEEMDAN to ${\mathbf{y}}^{\left(c\right)}$, extract ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}}_{k=1}^{K_{max}}$ and ${\mathbf{r}}^{\left(c\right)}$   2: Criterion 1 (Energy Contribution):   3:     Compute $E_k^{\left(c\right)}$ via Equation (7) for $k=1,\dots ,K_{max}$   4:     Compute cumulative energy $C{E}_k^{\left(c\right)}$ via Equation (8)   5:     Set $K_E^{\ast \left(c\right)}=min\{k:C{E}_k^{\left(c\right)}+E_r^{\left(c\right)}\ge {\tau }_E\}$   6: Criterion 2 (Statistical Significance):   7:     Compute Ljung–Box statistic $Q_{LB}^{(k,c)}$ via Equation (9) for $k=1,\dots ,K_{max}$   8:     Set $K_S^{\ast \left(c\right)}=max\{k:Q_{LB}^{(k,c)}>{\chi }_{1-{\alpha }_Q,L}^2\}$   9: Criterion 3 (predictive validation): 10:     Compute expanding-window CV error $C{V}_k^{\left(c\right)}$ via Equation (10) for $k=1,\dots ,K_{max}$ 11:     Set $K_P^{\ast \left(c\right)}=arg{min}_{k}C{V}_k^{\left(c\right)}$ 12: Voting: 13:     Collect candidates $\mathcal{K}=\{K_E^{\ast \left(c\right)},K_S^{\ast \left(c\right)},K_P^{\ast \left(c\right)}\}$ 14: if all three values in $\mathcal{K}$ are equal then 15:     $K^{\ast \left(c\right)}\leftarrow$ the unanimous value 16: else if exactly two values in $\mathcal{K}$ agree then 17:     $K^{\ast \left(c\right)}\leftarrow$ the majority value (median equals the agreed value) 18: else 19:     $K^{\ast \left(c\right)}\leftarrow \mathrm{median}\left(\mathcal{K}\right)$, selecting the intermediate estimate 20: end if 21: return  $K^{\ast \left(c\right)}$

The median operator is preferred over majority vote because it always returns a well-defined scalar even when all three criteria disagree, and it naturally selects the intermediate estimate, thereby avoiding both over-decomposition (which introduces noise) and under-decomposition (which discards predictive information). Compared to a weighted combination, the median is parameter-free and does not require additional tuning of combination weights.

Following the adaptive decomposition, the extracted IMFs are further categorized into three semantic groups based on their frequency characteristics to inform subsequent feature engineering and model selection. Specifically, IMFs with mean instantaneous frequencies exceeding a threshold $f_H$ are classified as irregular components that capture short-term fluctuations, noise, and random demand shocks ${\mathcal{H}}^{\left(c\right)}=\{k:{\overline{f}}_k^{\left(c\right)}>f_H\}$, where ${\overline{f}}_k^{\left(c\right)}$ denotes the mean instantaneous frequency of ${\overline{\mathbf{IMF}}}_k^{\left(c\right)}$. IMFs with mean instantaneous frequencies within the range $[f_L,f_H]$ are categorized as seasonal components that reflect periodic patterns such as quarterly procurement cycles and annual budget allocations: ${\mathcal{M}}^{\left(c\right)}=\{k:f_L\le {\overline{f}}_k^{\left(c\right)}\le f_H\}$. IMFs with mean instantaneous frequencies below $f_L$, together with the residual ${\mathbf{r}}^{\left(c\right)}$, constitute the trend components that represent long-term demand trajectories influenced by infrastructure development plans and policy changes: ${\mathcal{L}}^{\left(c\right)}=\{k:{\overline{f}}_k^{\left(c\right)}<f_L\}\cup \left\{{\mathbf{r}}^{\left(c\right)}\right\}$. This categorization enables component-specific modeling strategies that exploit the distinct predictability characteristics at different time scales.

3.3.2. Multi-Dimensional Feature Engineering Module

Once the demand signal has been decomposed into IMFs and residual through the previous module, this module is to construct informative features that enable effective learning by the forecasting models. It transforms the decomposed components into structured feature matrices suitable for machine learning models. The input consists of the IMFs ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and residual ${\mathbf{r}}^{\left(c\right)}$ obtained from the adaptive CEEMDAN module, along with auxiliary project and calendar information. The output is a set of component-specific feature matrices ${\{{\mathbf{X}}_k^{\left(c\right)}\in {\mathbb{R}}^{(T-W)\times M_k}\}}_{k=1}^{K^{\ast \left(c\right)}}$, where W denotes the sliding window length and $M_k$ represents the feature dimensionality for component k.

The selection of feature categories is guided by the distinct driving factors identified in power materials demand analysis. Unlike generic time series forecasting where only historical values serve as predictors, power materials demand is jointly driven by signal-intrinsic temporal patterns, infrastructure project schedules, calendar-induced procurement cycles, and supply-chain interdependencies across material categories. Accordingly, four categories of features are constructed to comprehensively characterize demand dynamics. The first category comprises decomposed signal features that capture the statistical properties and temporal dependencies of each IMF. Operating on individual IMFs rather than the raw demand series ensures that the extracted statistics characterize dynamics within a single frequency band, avoiding the confounding of trend, seasonal, and irregular variations that would occur if statistics were computed on the original broadband signal. Within a sliding window of length W, the rolling mean is computed as

$${\mu }_k^{\left(c\right)}\left(t\right)={\displaystyle \frac{1}{W}}\sum _{j=t-W+1}^t{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(j\right)$$

(12)

The rolling standard deviation ${\sigma }_{roll,k}^{\left(c\right)}\left(t\right)$, skewness, and kurtosis are similarly extracted to characterize local distributional properties. Additionally, lag features are constructed as ${\mathbf{x}}_{lag}^{(k,c)}\left(t\right)={[{\overline{\mathbf{IMF}}}_k^{\left(c\right)}(t-1),{\overline{\mathbf{IMF}}}_k^{\left(c\right)}(t-2),\dots ,{\overline{\mathbf{IMF}}}_k^{\left(c\right)}(t-p)]}^T$, where the maximum lag order p is determined by the partial autocorrelation function.

The second category consists of project attribute features that encode infrastructure project information. In the power grid industry, large-scale infrastructure projects (e.g., transmission line construction, substation upgrades) are the primary demand drivers for high-value material categories such as transformers and high-voltage cables. Project phase transitions—from planning through execution to completion—produce systematic shifts in procurement volume that cannot be captured by historical demand patterns alone, necessitating explicit encoding of project-level information. Binary phase indicators ${\mathbf{x}}_{phase}\left(t\right)={[I_{planning}\left(t\right),I_{execution}\left(t\right),I_{completion}\left(t\right)]}^T$ represent the current project stage, where $I_{phase}\left(t\right)=1$ if the project is in the specified phase at time t. The normalized budget allocation ratio is defined as

$$x_{budget}^{\left(c\right)}\left(t\right)={\displaystyle \frac{B_t^{\left(c\right)}}{{\sum }_{c^{\prime }=1}^C{B}_t^{\left(c^{\prime }\right)}}}$$

(13)

where $B_t^{\left(c\right)}$ denotes the budget allocated to category c at time t.

The third category includes temporal pattern features that capture cyclical variations. Power materials procurement follows strong calendar-driven periodicity: quarterly budget allocation cycles produce regular demand peaks, while fiscal year boundaries trigger year-end procurement surges across most material categories. Trigonometric encoding is adopted over dummy variables to represent these cyclical patterns continuously, preserving the proximity relationship between adjacent months (e.g., December and January) that binary indicators would fail to capture. Specifically, the monthly cyclical features are computed as

$$x_{month,sin}\left(t\right)=sin\left({\displaystyle \frac{2\pi ·month\left(t\right)}{12}}\right)$$

(14)

$$x_{month,cos}\left(t\right)=cos\left({\displaystyle \frac{2\pi ·month\left(t\right)}{12}}\right)$$

(15)

Quarterly patterns and fiscal year boundary indicators are constructed analogously to capture administrative cycles.

The fourth category comprises cross-category correlation features that exploit inter-category dependencies. Material categories within the same infrastructure project are often procured in coordinated sequences—for example, cable procurement typically precedes transformer installation by one to two months in substation construction projects. These lead–lag relationships create predictive signals: an observed surge in one category can foreshadow demand increases in downstream categories. Unlike the preceding three feature categories that operate on individual IMF components, cross-category correlations are computed on the raw demand series to capture aggregate inter-category dependencies that may not be preserved within any single frequency band after decomposition. The rolling cross-correlation at lag l between categories c and $c^{\prime }$ is computed as

$${\rho }^{(c,c^{\prime })}(t,l)={\displaystyle \frac{{\sum }_{j=t-W+1}^t(y_j^{\left(c\right)}-{\overline{y}}^{\left(c\right)})(y_{j-l}^{\left(c^{\prime }\right)}-{\overline{y}}^{\left(c^{\prime }\right)})}{\sqrt{{\sum }_{j=t-W+1}^t{(y_j^{\left(c\right)}-{\overline{y}}^{\left(c\right)})}^2}\sqrt{{\sum }_{j=t-W+1}^t{(y_{j-l}^{\left(c^{\prime }\right)}-{\overline{y}}^{\left(c^{\prime }\right)})}^2}}}$$

(16)

Finally, the complete feature vector for component k at time t is formed by concatenating all four categories:

$${\mathbf{x}}_{k,t}^{\left(c\right)}={[{\mathbf{x}}_{decomp}^{(k,c)}\left(t\right);{\mathbf{x}}_{project}\left(t\right);{\mathbf{x}}_{temporal}\left(t\right);{\mathbf{x}}_{correlation}^{\left(c\right)}\left(t\right)]}^T\in {\mathbb{R}}^{M_k}$$

(17)

The four feature categories are designed to be complementary: decomposed signal features provide the autoregressive backbone, project attributes inject exogenous demand drivers, temporal features encode calendar-induced periodicity, and cross-category correlations capture supply-chain interdependencies. Together, these four groups ensure that the forecasting models have access to both endogenous temporal dynamics and exogenous contextual information, enabling robust prediction across material categories with diverse demand characteristics.

3.3.3. Heterogeneous Ensemble Forecasting Module

With the feature matrices prepared, this module employs multiple base learners to generate robust component-level predictions. The rationale for using a heterogeneous ensemble is that different learners exhibit complementary strengths: some excel at capturing nonlinear patterns while others provide robustness or computational efficiency. The input consists of the feature matrices ${\left\{{\mathbf{X}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and corresponding target IMF values from the previous feature engineering module. The output consists of component-level predictions ${\left\{{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and ${\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$ for forecast horizon h, along with optimized ensemble weights.

The choice of base learners is guided by three complementary requirements arising from the decomposed component structure. Trend components, characterized by smooth nonlinear shifts driven by project schedules and policy changes, demand models that excel at capturing feature interactions. Seasonal components, exhibiting quasi-periodic patterns, benefit from rapid model fitting with implicit regularization to avoid overfitting on limited training samples. Irregular components, dominated by sporadic demand spikes and outliers, require robustness to large deviations. Based on these requirements, three learners spanning distinct architectural paradigms are selected: Gradient Boosting Machines (boosted decision trees) for trend components, Extreme Learning Machines (single-hidden-layer feedforward networks with analytical weight determination) for seasonal components, and Support Vector Regression (kernel-based regression with $\epsilon$-insensitive loss) for irregular components. This architectural diversity maximizes ensemble complementarity, which is widely recognized as a key driver of ensemble forecasting accuracy [20]. Alternative gradient boosting implementations such as XGBoost and LightGBM offer similar theoretical properties; the standard GBM formulation is adopted for methodological clarity and ease of reproducibility.

The mathematical formulations of the three base learners are detailed below. The first is the Gradient Boosting Machine (GBM), which constructs an additive model through sequential fitting:

$$f_{GBM}(\mathbf{x};{\theta }_{GBM})=\sum _{b=1}^{N_B}\eta ·h_b(\mathbf{x};{\theta }_b)$$

(18)

where $N_B$ denotes the number of boosting iterations, $\eta$ represents the learning rate, and $h_b(·)$ is the b-th weak learner. The GBM effectively captures nonlinear relationships and feature interactions, making it particularly suitable for trend and seasonal components.

The second is the Extreme Learning Machine (ELM), which provides computational efficiency through random weight initialization:

$$f_{ELM}(\mathbf{x};{\theta }_{ELM})={\mathit{\beta }}^T·g({\mathbf{W}}_{in}·\mathbf{x}+{\mathbf{b}}_h)$$

(19)

where ${\mathbf{W}}_{in}$ and ${\mathbf{b}}_h$ are randomly initialized, $g(·)$ is the activation function, and $\mathit{\beta }$ is determined analytically as

$$\mathit{\beta }={({\mathbf{H}}^T\mathbf{H}+\lambda \mathbf{I})}^{-1}{\mathbf{H}}^T\mathbf{y}$$

(20)

with hidden layer output matrix $\mathbf{H}\in {\mathbb{R}}^{T\times N_h}$, where $N_h$ denotes the number of hidden neurons, and regularization parameter $\lambda$. The ELM is well-suited for high-frequency components that require rapid model updates.

The third is Support Vector Regression (SVR), which offers robustness to outliers through kernel-based mapping:

$$f_{SVR}(\mathbf{x};{\theta }_{SVR})=\sum _{i\in SV}({\mu }_i-{\mu }_i^{\ast })K({\mathbf{x}}_i,\mathbf{x})+b$$

(21)

where ${\mu }_i,{\mu }_i^{\ast }$ are Lagrange multipliers, $K(·,·)$ is the radial basis function kernel, and $SV$ denotes the support vector set. SVR is particularly valuable for components affected by occasional demand spikes.

To leverage the complementary strengths of these learners, the ensemble weights ${\mathbf{w}}_k^{\left(c\right)}={[w_{k,1}^{\left(c\right)},w_{k,2}^{\left(c\right)},w_{k,3}^{\left(c\right)}]}^T$ are optimized separately for each component. The optimization objective seeks the convex combination of base learner predictions that minimizes the squared prediction error on the validation set, thereby ensuring that each learner contributes in proportion to its demonstrated predictive accuracy for the specific component under consideration. Formally, the weights are determined by solving

$$\underset{{\mathbf{w}}_k^{\left(c\right)}}{min}\sum _{t=1}^{T_{val}}{\left({\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(t\right)-\sum _{m=1}^3{w}_{k,m}^{\left(c\right)}{\widehat{\overline{\mathbf{IMF}}}}_{k,m}^{\left(c\right)}\left(t\right)\right)}^2+{\lambda }_w{\parallel {\mathbf{w}}_k^{\left(c\right)}\parallel }_2^2$$

(22)

subject to

$$\sum _{m=1}^3{w}_{k,m}^{\left(c\right)}=1,w_{k,m}^{\left(c\right)}\ge 0,m=1,2,3$$

(23)

where ${\widehat{\overline{\mathbf{IMF}}}}_{k,m}^{\left(c\right)}\left(t\right)$ denotes the prediction from base learner m and ${\lambda }_w$ is the regularization parameter. This constrained quadratic programming problem is solved using active-set methods.

Once the weights are determined, the ensemble prediction for each component is computed as

$${\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)=\sum _{m=1}^3{w}_{k,m}^{\left(c\right)}·f_m({\mathbf{x}}_{k,t}^{\left(c\right)};{\theta }_m)$$

(24)

3.3.4. Bayesian Uncertainty Quantification Module

While the ensemble module provides point predictions, practical materials management requires understanding the confidence associated with these forecasts. This module constructs prediction intervals with theoretical coverage guarantees to support risk-aware decision making. The input consists of the component-level predictions from the previous ensemble forecasting module and prediction residuals on validation data. The output consists of component-level variance estimates ${\left\{{\sigma }_k^{2,\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and calibrated prediction interval bounds.

The posterior predictive distribution is approximated through residual-based bootstrap aggregation independently for each decomposed component. It is important to distinguish this procedure from standard i.i.d. bootstrap applied to raw observations, which would destroy the temporal structure of the demand series and lead to underestimated uncertainty. Instead, the residual bootstrap operates in three stages: (i) the ensemble model is fitted to the original training data to obtain a set of fitted values; (ii) the prediction residuals ${\widehat{e}}_k^{\left(c\right)}\left(t\right)={\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(t\right)-{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}\left(t\right)$ are computed; and (iii) bootstrap replicates are generated by resampling these residuals with replacement and adding them back to the fitted values, producing perturbed training sets that preserve the underlying signal structure while introducing controlled variability. The i.i.d. resampling of residuals—rather than block bootstrap or other dependency-aware alternatives—is justified at the component level by the frequency-band separation achieved through CEEMDAN decomposition: because each IMF captures oscillations within a narrow frequency range, the prediction residuals within a single component exhibit substantially reduced temporal autocorrelation compared to the original broadband demand series [17]. This property renders the component-level residuals approximately exchangeable, satisfying the theoretical requirement for valid i.i.d. resampling [40].

For the k-th component, let ${\left\{{\widehat{\overline{\mathbf{IMF}}}}_k^{(c,s)}\right\}}_{s=1}^S$ denote predictions from S ensemble model instances trained on residual-based bootstrap resamples, and let ${\overline{\widehat{\overline{\mathbf{IMF}}}}}_k^{\left(c\right)}={\displaystyle \frac{1}{S}}{\sum }_{s=1}^S{\widehat{\overline{\mathbf{IMF}}}}_k^{(c,s)}$ denote their mean. The epistemic uncertainty, arising from limited training data, is estimated as the variance across bootstrap predictions. The aleatoric uncertainty, representing inherent randomness in demand, is derived from component-level validation residuals. Conceptually, the epistemic term quantifies the degree of disagreement among model instances trained on different data subsets, reflecting reducible uncertainty due to finite sample size, while the aleatoric term captures the irreducible residual variation that persists even when the model is well-specified. Their summation provides a comprehensive characterization of total prediction uncertainty at the component level. The combined variance estimate for component k is

$${\sigma }_k^{2,\left(c\right)}=\underset{\mathrm{epistemic}\mathrm{uncertainty}}{\underbrace{{\displaystyle \frac{1}{S}}\sum _{s=1}^S{\left({\widehat{\overline{\mathbf{IMF}}}}_k^{(c,s)}-{\overline{\widehat{\overline{\mathbf{IMF}}}}}_k^{\left(c\right)}\right)}^2}}+\underset{\mathrm{aleatoric}\mathrm{uncertainty}}{\underbrace{{\displaystyle \frac{1}{T_{val}}}\sum _{t=1}^{T_{val}}{\left({\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(t\right)-{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}\left(t\right)\right)}^2}}$$

(25)

The variance estimate for the residual component is computed analogously:

$${\sigma }_r^{2,\left(c\right)}={\displaystyle \frac{1}{S}}\sum _{s=1}^S{\left({\widehat{\mathbf{r}}}^{(c,s)}-{\overline{\widehat{\mathbf{r}}}}^{\left(c\right)}\right)}^2+{\displaystyle \frac{1}{T_{val}}}\sum _{t=1}^{T_{val}}{\left({\mathbf{r}}^{\left(c\right)}\left(t\right)-{\widehat{\mathbf{r}}}^{\left(c\right)}\left(t\right)\right)}^2$$

(26)

Based on this component-level variance estimate, the $(1-\alpha )$ prediction interval for each component under Gaussian assumptions is constructed as

$$[{\widehat{\overline{\mathbf{IMF}}}}_{k,lower}^{\left(c\right)},{\widehat{\overline{\mathbf{IMF}}}}_{k,upper}^{\left(c\right)}]=[{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}-z_{1-\alpha /2}·{\sigma }_k^{\left(c\right)},{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}+z_{1-\alpha /2}·{\sigma }_k^{\left(c\right)}]$$

(27)

where $z_{1-\alpha /2}$ denotes the $(1-\alpha /2)$-th quantile of the standard normal distribution and ${\sigma }_k^{\left(c\right)}=\sqrt{{\sigma }_k^{2,\left(c\right)}}$.

However, the Gaussian assumption may not always hold in practice. To ensure finite-sample coverage guarantees without distributional assumptions, conformal calibration is applied as a post-processing step. The nonconformity score quantifies how many aggregated standard deviations each calibration observation deviates from its corresponding forecast. The empirical quantile of these scores then serves as a data-driven scaling factor that replaces the Gaussian critical value $z_{1-\alpha /2}$ in Equation (27), thereby ensuring valid coverage probability regardless of the true error distribution. To ensure compatibility with component-level uncertainty propagation, the nonconformity score is normalized by the aggregated standard deviation ${\sigma }_{agg}^{\left(c\right)}=\sqrt{{\sum }_{k=1}^{K^{\ast \left(c\right)}}{\sigma }_k^{2,\left(c\right)}+{\sigma }_r^{2,\left(c\right)}}$, which pools variance contributions across all decomposed components:

$$R_i={\displaystyle \frac{|y_i^{\left(c\right)}-{\widehat{y}}_i^{\left(c\right)}|}{{\sigma }_{agg}^{\left(c\right)}}}$$

(28)

The calibrated prediction interval is then constructed as

$$[{\widehat{y}}_{lower}^{cal},{\widehat{y}}_{upper}^{cal}]=[{\widehat{y}}^{\left(c\right)}-q_{1-\alpha }·{\sigma }_{agg}^{\left(c\right)},{\widehat{y}}^{\left(c\right)}+q_{1-\alpha }·{\sigma }_{agg}^{\left(c\right)}]$$

(29)

where $q_{1-\alpha }$ denotes the $(1-\alpha )$-th quantile of the normalized nonconformity scores $\left\{R_i\right\}$ on the calibration set. Since $q_{1-\alpha }$ is now a dimensionless calibration factor, it can be applied with different variance estimates across components and their aggregation. This procedure guarantees that prediction intervals achieve the target coverage rate on exchangeable test data.

3.3.5. Intelligent Aggregation Module

The final module integrates the component-level predictions and uncertainty estimates to produce the complete forecast output. This module reconstructs the final demand forecast by reversing the decomposition process. The input consists of the predicted IMFs ${\left\{{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and predicted residual ${\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$ from the previous ensemble forecasting module, together with uncertainty estimates ${\left\{{\sigma }_k^{2,\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ from the uncertainty quantification module. The output is the prediction triplet $({\widehat{y}}_{t+h}^{\left(c\right)},{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)})$ that serves as the final framework output.

The point prediction is obtained by summing all component-level forecasts, mirroring the original signal decomposition:

$${\widehat{y}}_{t+h}^{\left(c\right)}=\sum _{k=1}^{K^{\ast \left(c\right)}}{\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)+{\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$$

(30)

For uncertainty propagation, since the decomposed components capture dynamics at different frequency bands, their prediction errors can be reasonably assumed to be approximately independent. Under this assumption, the aggregated variance is computed as:

$${\sigma }_{agg}^{2,\left(c\right)}=\sum _{k=1}^{K^{\ast \left(c\right)}}{\sigma }_k^{2,\left(c\right)}+{\sigma }_r^{2,\left(c\right)}$$

(31)

where ${\sigma }_r^{2,\left(c\right)}$ denotes the variance estimate for the residual component. The additive variance aggregation assumes that prediction errors across decomposed components are approximately independent. This assumption is grounded in the signal-theoretic properties of CEEMDAN: by construction, the sifting procedure extracts components occupying distinct frequency bands with minimal spectral overlap [17,30], and the residual at each extraction stage is orthogonal to the mode just removed. Consequently, the prediction errors at different frequency scales are driven by largely non-overlapping sources of variation—trend errors reflect misspecification of long-term dynamics, seasonal errors arise from periodicity estimation inaccuracies, and irregular errors stem from unpredictable demand shocks—reducing the likelihood of systematic cross-component error correlation. Nevertheless, since all components originate from the same underlying signal, residual correlations cannot be entirely excluded in practice. To safeguard against this possibility, the conformal calibration step (Equations (28) and (29)) is applied at the aggregate level rather than at the component level: the nonconformity scores are computed on the reconstructed total prediction error $|y_i^{\left(c\right)}-{\widehat{y}}_i^{\left(c\right)}|$, which inherently accounts for any cross-component error dependence. This design ensures that the resulting prediction intervals achieve the target coverage probability regardless of whether the component-level independence assumption holds exactly.

The final calibrated prediction interval is then constructed by combining the aggregated point prediction with the propagated uncertainty:

$$[{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)}]=[{\widehat{y}}_{t+h}^{\left(c\right)}-q_{1-\alpha }·{\sigma }_{agg}^{\left(c\right)},{\widehat{y}}_{t+h}^{\left(c\right)}+q_{1-\alpha }·{\sigma }_{agg}^{\left(c\right)}]$$

(32)

where $q_{1-\alpha }$ is the calibration quantile obtained from the previous uncertainty quantification module. This aggregation approach ensures that the final prediction intervals appropriately reflect cumulative uncertainty across all decomposed components, providing reliable confidence bounds for downstream inventory optimization and procurement planning.

3.4. Training Procedure

The training process of the proposed ADEM method proceeds through a sequential pipeline where each module is configured based on the outputs of preceding modules. The overall input to the training procedure is the historical demand dataset $\mathcal{D}=\left\{({\mathbf{x}}_i,y_i,c_i)\right|i=1,2,\dots ,N\}$ partitioned into training, validation, and calibration sets. The overall output is the complete set of trained model parameters ${\mathrm{\Theta }}_{ADEM}$, including CEEMDAN configurations, feature engineering specifications, base learner parameters, ensemble weights, and calibration quantiles.

The training procedure consists of six sequential steps, as summarized in Algorithm 2. The first step is data preprocessing, where the raw demand data for each material category undergo min-max normalization

$${\tilde{y}}_t^{\left(c\right)}={\displaystyle \frac{y_t^{\left(c\right)}-min\left({\mathbf{y}}^{\left(c\right)}\right)}{max\left({\mathbf{y}}^{\left(c\right)}\right)-min\left({\mathbf{y}}^{\left(c\right)}\right)}}$$

(33)

This normalization ensures numerical stability during model training and enables fair comparison across material categories with different demand magnitudes.

Algorithm 2 ADEM training procedure

Require: Historical demand dataset $\mathcal{D}$, confidence level $1-\alpha$

Ensure: Trained model parameters ${\mathrm{\Theta }}_{ADEM}$

1: Step 1: Data Preprocessing   2: for each material category $c=1,2,\dots ,C$ do   3:     Apply min-max normalization to obtain ${\tilde{\mathbf{y}}}^{\left(c\right)}$   4: end for   5: Step 2: Adaptive Decomposition   6: for each material category $c=1,2,\dots ,C$ do   7:     Apply CEEMDAN to ${\tilde{\mathbf{y}}}^{\left(c\right)}$   8:     Determine optimal depth $K^{\ast \left(c\right)}$ using adaptive selection strategy   9:     Extract ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and ${\mathbf{r}}^{\left(c\right)}$ 10: end for 11: Step 3: Feature Engineering 12: for each category c and component $k=1,2,\dots ,K^{\ast \left(c\right)}$ do 13:     Construct feature matrix ${\mathbf{X}}_k^{\left(c\right)}$ using Equations (12)–(17) 14: end for 15: Step 4: Base Learner Training 16: for each category c and component k do 17:     Train GBM, ELM, and SVR on $({\mathbf{X}}_k^{\left(c\right)},{\overline{\mathbf{IMF}}}_k^{\left(c\right)})$ 18:     Optimize hyperparameters via cross-validation 19: end for 20: Step 5: Ensemble Weight Optimization 21: for each category c and component k do 22:     Solve Equations (22) and (23) to obtain ${\mathbf{w}}_k^{\left(c\right)}$ 23: end for 24: Step 6: Uncertainty Calibration 25: for each material category c do 26:     Compute variance estimates ${\sigma }_k^{2,\left(c\right)}$ using Equation (25) 27:     Compute nonconformity scores on calibration set 28:     Determine calibration quantile $q_{1-\alpha }$ 29: end for 30: return  ${\mathrm{\Theta }}_{ADEM}=\{K^{\ast \left(c\right)},{\theta }_{GBM},{\theta }_{ELM},{\theta }_{SVR},{\mathbf{w}}_k^{\left(c\right)},q_{1-\alpha }\}$

The second step is adaptive decomposition, where the CEEMDAN algorithm is applied to each material category’s normalized demand series ${\tilde{\mathbf{y}}}^{\left(c\right)}$. The adaptive IMF selection strategy described in Section 3.3.1 determines the category-specific optimal decomposition depth $K^{\ast \left(c\right)}$ and extracts the corresponding IMFs ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and residual ${\mathbf{r}}^{\left(c\right)}$.

The third step is feature engineering, where the Multi-Dimensional Feature Engineering Module constructs the component-specific feature matrices ${\left\{{\mathbf{X}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ by extracting statistical, project, temporal, and cross-correlation features from the decomposed components and auxiliary data.

The fourth step is base learner training, where each of the three base learners (GBM, ELM, and SVR) is trained independently on each component’s feature matrix. Hyperparameter optimization is performed using grid search with 5-fold cross-validation to select optimal configurations for tree depth, number of hidden neurons, kernel parameters, and regularization strengths.

The fifth step is ensemble weight optimization, where the component-specific ensemble weights ${\left\{{\mathbf{w}}_k^{\left(c\right)}\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ are determined by solving the constrained quadratic programming problem defined in Equation (22) on the validation set.

The sixth step is uncertainty calibration, where the Bayesian variance estimates are computed from bootstrap model ensembles and validation residuals using Equation (25), followed by conformal calibration on the calibration set to obtain the quantile $q_{1-\alpha }$ that ensures coverage guarantees.

The complete training procedure terminates when all modules have been configured and validated. The resulting trained framework is then ready for deployment in the prediction phase.

3.5. Prediction Procedure

Upon completion of ADEM framework training, the prediction pipeline generates forecasts for new time points in a computationally efficient manner. The input to the prediction procedure consists of the most recent demand observations and auxiliary information up to time t, together with the specified forecast horizon h and material category c. The output is the prediction triplet $({\widehat{y}}_{t+h}^{\left(c\right)},{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)})$ consisting of the point prediction and calibrated prediction interval bounds.

The prediction procedure consists of five sequential steps, as summarized in Algorithm 3. The first step applies the trained CEEMDAN model to decompose the most recent demand observations, using the category-specific optimal decomposition depth $K^{\ast \left(c\right)}$ determined during training. This produces the current IMF values ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(t\right)\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and residual ${\mathbf{r}}^{\left(c\right)}\left(t\right)$.

The second step constructs the feature vector ${\mathbf{x}}_{k,t}^{\left(c\right)}$ for each component using the feature engineering specifications established during training, incorporating the latest decomposed component values, current project status, calendar information, and cross-category observations.

The third step generates component-level predictions by passing each feature vector through the trained base learner ensemble with optimized weights according to Equation (24). The residual component ${\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$ is predicted analogously.

The fourth step reconstructs the final point prediction by summing all component-level forecasts according to Equation (30).

The fifth step constructs the calibrated prediction interval by propagating uncertainty across components using Equation (31) and applying the calibration quantile according to Equation (32).

Algorithm 3 ADEM prediction procedure

Require: Recent observations up to time t, forecast horizon h, category c, trained parameters ${\mathrm{\Theta }}_{ADEM}$

Ensure: Prediction triplet $({\widehat{y}}_{t+h}^{\left(c\right)},{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)})$

1: Step 1: Signal Decomposition   2: Normalize recent observations and apply trained CEEMDAN with depth $K^{\ast \left(c\right)}$   3: Obtain ${\left\{{\overline{\mathbf{IMF}}}_k^{\left(c\right)}\left(t\right)\right\}}_{k=1}^{K^{\ast \left(c\right)}}$ and ${\mathbf{r}}^{\left(c\right)}\left(t\right)$   4: Step 2: Feature Construction   5: for each component $k=1,2,\dots ,K^{\ast \left(c\right)}$ do   6:     Construct feature vector ${\mathbf{x}}_{k,t}^{\left(c\right)}$ using trained specifications   7: end for   8: Step 3: Component-Level Prediction   9: for each component $k=1,2,\dots ,K^{\ast \left(c\right)}$ do 10:     Compute ${\widehat{\overline{\mathbf{IMF}}}}_k^{\left(c\right)}(t+h)$ using Equation (24) 11: end for 12: Predict residual ${\widehat{\mathbf{r}}}^{\left(c\right)}(t+h)$ analogously 13: Step 4: Point Prediction Reconstruction 14: Compute ${\widehat{y}}_{t+h}^{\left(c\right)}$ using Equation (30) 15: Step 5: Prediction Interval Construction 16: Compute ${\sigma }_{agg}^{\left(c\right)}$ using Equation (31) 17: Compute $[{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)}]$ using Equation (32) 18: return $({\widehat{y}}_{t+h}^{\left(c\right)},{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)})$

The complete prediction procedure outputs the forecast triplet for use in downstream inventory optimization and procurement planning applications, enabling risk-aware decision making that accounts for both expected demand and forecast uncertainty.

4. Experimental Setup

4.1. Dataset Description

Experiments were conducted on a real-world power materials demand dataset collected from a provincial power grid company in China, spanning January 2019 to December 2023 (60 months). The dataset encompasses 18 specialized material categories organized into 8 major groups, as summarized in Table 2. These categories exhibit highly heterogeneous demand patterns: infrastructure construction materials are driven by large-scale project schedules with long lead times, whereas emergency repair materials display sporadic, event-driven demand spikes. This diversity provides a rigorous testbed for evaluating the adaptability of the proposed ADEM across distinct temporal dynamics.

Table 2. Power materials category taxonomy.

No.	Major Group	Specific Categories
1	Infrastructure Construction	High-voltage cables, transformers, circuit breakers
2	Technical Renovation	Equipment modernization, grid upgrade materials
3	Marketing	Smart meters, customer service equipment
4	Information and Communication	IT hardware, communication devices, network equipment
5	Operation and Maintenance	Inspection tools, routine maintenance supplies
6	Emergency Repair	Rapid response materials, fault restoration equipment
7	Office Administration	Office supplies, administrative materials
8	Production Tools	Specialized instruments, technical equipment

Figure 2 presents the monthly demand time series for all 18 material categories, organized by the eight major groups defined in Table 2. Several distinguishing patterns are visually apparent. Infrastructure categories (HV cables, transformers, circuit breakers) exhibit pronounced upward trends driven by sustained grid expansion investment, superimposed with regular seasonal fluctuations reflecting quarterly procurement cycles. Marketing categories (smart meters, customer service equipment) display high seasonality with occasional sharp spikes coinciding with large-scale deployment campaigns. In contrast, emergency categories (rapid response, fault restoration) are characterized by highly irregular, event-driven demand patterns with elevated volatility, as evidenced by their high coefficient of variation (CV > 0.30). Office and administrative categories exhibit the most stable demand profiles with low variability (CV < 0.20).

Figure 2. Monthly demand time series for all 18 material categories, organized by the eight major groups. Each panel corresponds to one major group as defined in Table 2, illustrating the heterogeneous demand patterns across categories.

To further characterize the demand heterogeneity, Figure 3 shows enlarged views of six representative categories spanning diverse demand profiles: transformers (stable trend with moderate seasonality, CV = 0.17), smart meters (strong seasonality with occasional spikes, CV = 0.25), rapid response materials (highly irregular with sporadic demand shocks, CV = 0.31), grid upgrade materials (project-driven with gradual growth, CV = 0.19), office supplies (low-amplitude stable pattern, CV = 0.15), and technical equipment (moderate trend with intermittent spikes, CV = 0.22). This diversity validates the necessity of category-specific adaptive decomposition as a fixed modeling strategy cannot simultaneously accommodate the smooth trend-dominated patterns of infrastructure categories and the spike-dominated patterns of emergency categories.

Figure 3. Monthly demand patterns for six representative material categories. Each panel displays the demand trajectory with shaded area fill, annotated with descriptive statistics (mean $\mu$, standard deviation $\sigma$, and coefficient of variation CV).

Table 3 summarizes the descriptive statistics across all 18 categories. The mean demand ranges from 172 (admin materials) to 3427 (transformers), spanning two orders of magnitude in scale. The coefficient of variation ranges from 0.14 (admin materials) to 0.41 (fault restoration), quantifying the substantial heterogeneity in demand volatility. Categories with positive skewness (e.g., inspection tools: 2.14, equip. modernization: 2.14) are characterized by occasional large demand surges, whereas categories with near-zero or negative skewness (e.g., transformers: 0.04, office supplies: −0.32) exhibit more symmetric demand distributions. These statistical characteristics inform the adaptive decomposition strategy in the proposed ADEM method: categories with high CV and positive skewness require deeper decomposition to separate sporadic spikes from underlying patterns, while categories with low CV can be effectively modeled with fewer IMFs.

Table 3. Descriptive statistics of monthly demand across 18 material categories.

No.	Category	Mean	Std	CV	Min	Max	Skewness
1	HV Cables	1755	425	0.24	976	4331	2.01
2	Transformers	3427	567	0.17	2355	4610	0.04
3	Circuit Breakers	1164	269	0.23	773	2508	1.80
4	Equip. Modernization	921	268	0.29	530	2115	2.14
5	Grid Upgrade	1332	250	0.19	835	2032	0.02
6	Smart Meters	2238	565	0.25	1372	4764	1.49
7	Customer Svc. Equip.	608	170	0.28	315	1241	1.80
8	IT Hardware	580	157	0.27	326	1124	0.91
9	Comm. Devices	436	121	0.28	249	900	1.40
10	Network Equip.	758	212	0.28	404	1453	1.25
11	Inspection Tools	456	125	0.27	299	1001	2.14
12	Maintenance Supplies	730	124	0.17	498	1057	0.33
13	Rapid Response	245	76	0.31	62	425	−0.17
14	Fault Restoration	358	147	0.41	107	802	1.67
15	Office Supplies	220	34	0.15	131	278	−0.32
16	Admin Materials	172	24	0.14	121	223	−0.29
17	Specialized Instr.	570	164	0.29	328	1098	1.59
18	Technical Equip.	613	135	0.22	332	1070	0.56

Prior to conducting the forecasting experiments, two evaluation protocols were established to ensure both rigorous benchmarking and robust assessment of temporal consistency. The first is a fixed chronological partition that provides the primary experimental setting with clearly delineated functional roles for each data subset. The second is a complementary rolling-origin (expanding-window) protocol that systematically varies the temporal boundary between training and test data, yielding multiple independent evaluation windows to quantify metric variability and enable non-parametric statistical significance testing.

Under the fixed chronological partition, the dataset was divided into training (January 2019–December 2021, 36 months), validation (January 2022–December 2022, 12 months), calibration (January 2023–June 2023, 6 months), and test (July 2023–December 2023, 6 months) sets. The validation set was used for ensemble weight optimization (Equation (22)) and hyperparameter selection, while the calibration set was reserved exclusively for computing the conformal nonconformity scores (Equation (28)) and determining the calibration quantile $q_{1-\alpha }$. This separation ensures that the coverage guarantee provided by conformal calibration is not compromised by data reuse. This temporal split avoids information leakage and reflects realistic deployment conditions where models must generalize to future periods.

Under the rolling-origin protocol, six forecast origins were placed at months 30, 33, 36, 39, 42, and 45 of the 60-month observation period, each producing a six-month-ahead forecast horizon, as illustrated in Figure 4. For each origin $t_o$, the training set comprises months 1 through $t_o$, and the test set comprises months $t_o+1$ through $t_o+H$ ($H=6$). A validation subset (last 20% of the training period) and a calibration subset (subsequent 15% before the test boundary) were carved out internally for each window, following the same functional roles as in the fixed partition. This protocol ensures that temporal ordering is strictly preserved within every window and that no future information leaks into the training data.

Figure 4. Rolling-origin evaluation scheme. Six expanding-window origins (months 30–45) each produce a six-month test forecast, yielding six independent evaluation windows.

4.2. Baseline Methods

The proposed ADEM was compared against a comprehensive set of 14 baseline methods spanning four methodological families:

1.

Statistical methods: ARIMA [4], Exponential Smoothing [5], and Prophet [18].

2.

Machine learning methods: Random Forest [20], Gradient Boosting Machine (GBM) [8], and Support Vector Regression (SVR) [19].

3.

Deep learning methods: LSTM [10], GRU [24], Temporal Convolutional Network (TCN) [41], transformer [25], and TFT-Transformer [13].

4.

Decomposition-based methods: EMD-ELM [23,29], VMD-LSTM [42], and Wavelet-SVM [43].

All methods used identical data partitions and the same feature engineering pipeline (where applicable) to ensure fair comparison. Specifically, under the fixed chronological partition, all methods shared the same training/validation/calibration/test splits. Under the rolling-origin protocol, all methods were evaluated on the same six expanding-window partitions with identical forecast origins and horizons. For methods that do not natively produce prediction intervals (i.e., all baselines except ADEM), intervals were constructed using the residual standard deviation on the training set scaled by the Gaussian critical value $z_{1-\alpha /2}$, providing a consistent uncertainty estimation baseline for PICP and MPIW comparison.

4.3. Evaluation Metrics

Model performance was assessed using nine complementary metrics organized into three categories.

Point prediction accuracy was measured by Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and symmetric MAPE (sMAPE). Additionally, overall prediction accuracy (Accuracy) is reported as the percentage of forecasts falling within $\pm 15\%$ of actual demand, a threshold commonly adopted in power materials planning practice.

Prediction interval quality was evaluated by Prediction Interval Coverage Probability (PICP), defined as the fraction of actual values falling within the predicted interval, and Mean Prediction Interval Width (MPIW), which measures interval sharpness. An ideal prediction interval achieves PICP close to the nominal confidence level $1-\alpha$ with the smallest possible MPIW.

Comprehensive probabilistic forecast quality was assessed by two additional metrics that jointly evaluate calibration and sharpness in a single score. The Continuous Ranked Probability Score (CRPS) measures the integrated squared difference between the predictive cumulative distribution function and the empirical distribution of the observation:

$$\mathrm{CRPS}={\displaystyle \frac{1}{N_{test}}}\sum _{i=1}^{N_{test}}{\int }_{-\infty }^{\infty }{\left(F_i\left(z\right)-\mathbf{1}(z\ge y_i)\right)}^{2}dz$$

(34)

where $F_i\left(z\right)$ denotes the predictive cumulative distribution function for the i-th test sample, approximated as a Gaussian with mean ${\widehat{y}}_i^{\left(c\right)}$ and variance ${\sigma }_{agg}^{2,\left(c\right)}$, and $\mathbf{1}(·)$ is the indicator function. The Winkler Score [44] simultaneously penalizes interval width and coverage violations:

$$W_i=\left\{\begin{array}{cc}{\delta }_i\hfill & \mathrm{if}{y}_i\in [{\widehat{y}}_{lower}^{\left(c\right)},{\widehat{y}}_{upper}^{\left(c\right)}]\hfill \\ {\delta }_i+{\displaystyle \frac{2}{\alpha }}({\widehat{y}}_{lower}^{\left(c\right)}-y_i)\hfill & \mathrm{if}{y}_i<{\widehat{y}}_{lower}^{\left(c\right)}\hfill \\ {\delta }_i+{\displaystyle \frac{2}{\alpha }}(y_i-{\widehat{y}}_{upper}^{\left(c\right)})\hfill & \mathrm{if}{y}_i>{\widehat{y}}_{upper}^{\left(c\right)}\hfill \end{array}\right.$$

(35)

where ${\delta }_i={\widehat{y}}_{upper}^{\left(c\right)}-{\widehat{y}}_{lower}^{\left(c\right)}$ denotes the interval width and $\alpha$ is the significance level. Lower values of both CRPS and Winkler Score indicate better probabilistic forecast quality. Together, these two metrics complement PICP and MPIW by penalizing both miscalibration and excessive interval width within a unified scoring framework, providing a more nuanced assessment of uncertainty quantification performance.

4.4. Implementation Details

The implementation details for each module of the proposed ADEM framework are described below, and a complete summary of all hyperparameters is provided in Table 4 to facilitate independent reproduction.

Table 4. Complete hyperparameter configuration. Values marked with ^† are selected via grid search; all others are predefined.

Module/Parameter	Symbol	Value
CEEMDAN
Noise realizations	I	500
Initial noise amplitude	${\epsilon }_0$	$0.2·\mathrm{std}\left({\mathbf{y}}^{\left(c\right)}\right)$
Energy threshold	${\tau }_E$	0.95
Ljung–Box significance level	${\alpha }_Q$	0.05
Maximum lag	L	12
Frequency threshold (low)	$f_L$	0.06 cycles/month
Frequency threshold (high)	$f_H$	0.20 cycles/month
Min. training length (CV)	$T_{min}$	24 months
Feature Engineering
Sliding window length	W	12 months
GBM
Boosting iterations	$N_B$	500
Maximum tree depth	–	6
Learning rate	$\eta$	0.05
ELM
Hidden neurons	$N_h$	200
Activation function	$g(·)$	Sigmoid
Regularization	$\lambda$	${10}^{-4}$
SVR
Kernel	–	RBF
Regularization †	$C_{svr}$	10
Kernel width †	$\gamma$	0.1
Ensemble and Uncertainty
Weight regularization	${\lambda }_w$	${10}^{-3}$
Bootstrap resamples	S	100
Confidence level	$1-\alpha$	0.95
Deep Learning Baselines
Early stopping patience	–	10 epochs
Learning rate (LSTM/GRU)	–	${10}^{-3}$
Hidden units (LSTM/GRU)	–	64
Attention heads (transformer)	–	4
Dropout rate	–	0.1

The CEEMDAN module was configured with $I=500$ noise realizations and initial noise amplitude ${\epsilon }_0=0.2·\mathrm{std}\left({\mathbf{y}}^{\left(c\right)}\right)$. The energy threshold was set to ${\tau }_E=0.95$ and the Ljung–Box significance level to ${\alpha }_Q=0.05$, with maximum lag $L=12$. The frequency thresholds for IMF classification (Section 3.3.1) were set to $f_L=0.06$ cycles/month and $f_H=0.20$ cycles/month, corresponding to semi-annual and quarterly periodicity boundaries respectively. These values were derived from the physical periodicity of the monthly demand data: an annual cycle corresponds to $1/12\approx 0.083$ cycles/month and a quarterly cycle to $1/3\approx 0.333$ cycles/month, with the thresholds positioned to separate trend (<semi-annual), seasonal (semi-annual to quarterly), and irregular (>quarterly) frequency bands. The minimum training length for the expanding-window cross-validation in the adaptive decomposition depth selection (Equation (10)) was set to $T_{min}=24$ months, ensuring sufficient data for stable model fitting within each cross-validation fold.

For the feature engineering module, the sliding window length was $W=12$ months, corresponding to one full annual cycle to capture complete seasonal patterns within each rolling window.

For the ensemble forecasting module, the GBM used $N_B=500$ boosting iterations with maximum tree depth 6 and learning rate $\eta =0.05$; the ELM employed $N_h=200$ hidden neurons with sigmoid activation and regularization $\lambda ={10}^{-4}$; and SVR used a radial basis function kernel with parameters selected via grid search over $C_{svr}\in \{1,10,100\}$ and $\gamma \in \{0.01,0.1,1\}$, yielding optimal values $C_{svr}=10$ and $\gamma =0.1$ across the majority of material categories. The ensemble weight regularization parameter was set to ${\lambda }_w={10}^{-3}$.

For the uncertainty quantification module, $S=100$ bootstrap resamples were employed, and the confidence level was set to $1-\alpha =0.95$. All deep learning baselines were trained with early stopping (patience = 10 epochs) on the validation set (January 2022–December 2022 under the fixed chronological partition; the corresponding internal validation subset in each rolling-origin window).

All experiments were implemented in Python 3.12 using scikit-learn 0.24 for the GBM, SVR, and Random Forest, with an ELM implemented as a custom module following the formulation in Equations (19) and (20). CEEMDAN was performed using the PyEMD library with version 0.4.0. Experiments were executed on a standard workstation equipped with an Intel i7-12700 processor and 32 GB RAM, without GPU acceleration.

5. Results and Discussion

5.1. Overall Performance Comparison

5.1.1. Fixed-Partition Results

Table 5 presents the performance comparison across all 18 material categories on the test set under the fixed chronological partition. The proposed ADEM achieves the best performance across all evaluation metrics, attaining 6.0% MAPE and 88.0% prediction accuracy with a PICP of 93.5%.

Table 5. Overall performance comparison under the fixed chronological partition, aggregated across 18 material categories.

Method	MAPE (%)	RMSE	MAE	sMAPE (%)	PICP (%)	MPIW	Accuracy (%)	CRPS	Winkler
ARIMA	30.2	689.8	349.3	31.4	70.4	631.9	31.5	296.8	7445.1
Exp. Smoothing	30.9	607.4	321.8	32.6	73.1	693.3	38.9	272.7	6441.8
Prophet	24.5	433.4	258.4	29.7	74.1	620.8	42.6	211.7	4348.9
Random Forest	15.7	385.8	189.6	16.1	76.9	397.8	64.8	165.3	4276.5
GBM	14.8	300.6	169.6	15.4	82.4	617.8	64.8	141.6	2582.0
SVR	15.7	306.1	166.9	15.5	81.5	479.4	58.3	134.8	2581.1
LSTM	23.7	383.2	243.1	23.2	77.8	624.7	40.7	196.8	3958.1
GRU	21.2	408.3	248.9	21.3	79.6	838.1	50.0	208.8	4006.9
TCN	23.3	448.0	251.7	22.6	80.6	681.2	46.3	206.8	4360.4
Transformer	19.4	378.6	199.4	19.0	84.3	728.5	55.6	162.9	2991.9
TFT-Transformer	9.1	219.0	110.2	9.4	89.8	522.8	81.5	94.6	1807.3
EMD-ELM	17.7	420.9	219.2	18.2	83.3	776.2	56.5	187.7	3962.7
VMD-LSTM	17.0	348.9	193.3	18.5	87.0	858.7	57.4	162.1	2801.3
Wavelet-SVM	18.3	400.8	192.1	19.2	83.3	618.8	63.0	158.9	3224.4
ADEM (Ours)	6.0	121.8	69.4	5.9	93.5	592.2	88.0	64.2	800.4

ADEM achieves a 24.2 percentage point (pp) MAPE reduction over ARIMA, a 17.7 pp reduction over LSTM, a 13.4 pp reduction over the vanilla transformer, and a 3.1 pp reduction over TFT-Transformer.

Among non-decomposition baselines, the GBM (14.8% MAPE) outperforms all deep learning methods except TFT-Transformer, suggesting that the relatively short time series (60 months) limits the effectiveness of data-hungry architectures such as LSTM and transformers. With only 60 monthly observations per category, deep learning architectures containing thousands of trainable parameters (LSTM: 23.7%, GRU: 21.2%, Transformer: 19.4%) cannot effectively learn their parameter spaces, whereas the proposed ADEM mitigates this limitation by decomposing the forecasting task into simpler component-level problems where compact models with fewer parameters can be trained effectively on limited data. The decomposition-based baselines (EMD-ELM, VMD-LSTM, Wavelet-SVM) consistently outperform their non-decomposition counterparts, validating the benefit of multi-scale signal separation. However, ADEM’s adaptive decomposition depth and heterogeneous ensemble strategy yield substantial additional gains, reducing MAPE by 11.0–12.3 pp relative to these methods.

Figure 5 further illustrates the multi-metric comparison among representative methods. ADEM achieves the most favorable trade-off across all four metrics simultaneously.

Figure 5. Multi-metric performance comparison among representative methods under the fixed chronological partition: (a) MAPE, (b) RMSE, (c) PICP, (d) Accuracy, (e) Winkler Score, and (f) CRPS. ADEM (red bar) achieves the best performance across all six metrics.

Regarding prediction interval quality, ADEM achieves 93.5% PICP with a moderate interval width (MPIW = 592.2), indicating that its intervals are both reliable and informative. Figure 5e,f further present the Winkler Score and CRPS comparisons. ADEM attains the lowest CRPS (64.2) and Winkler Score (800.4), approximately 78% and 89% lower than ARIMA (296.8 and 7445.1) respectively, and 32% and 56% lower than TFT-Transformer (94.6 and 1807.3). These results exhibit that ADEM achieves the most favorable trade-off between prediction interval coverage and width. Although methods with lower PICP (e.g., Random Forest, SVR) produce narrower intervals, their coverage rates fall far below the 95% nominal level, limiting their practical utility for risk-aware decision making. ADEM attains the highest PICP (93.5%) among all methods while maintaining a moderate interval width, yielding the lowest Winkler Score (800.4) that jointly penalizes both coverage violations and excessive width. This superiority is visualized in the PICP–MPIW trade-off plot (Figure 6), where ADEM occupies the topmost position closest to the 95% nominal coverage line.

Figure 6. PICP versus MPIW trade-off across all 15 methods, color-coded by methodological family. ADEM (star) achieves the highest PICP (93.5%), closest to the 95% nominal coverage line, while maintaining a competitive interval width. Methods in the lower-left (e.g., Random Forest, SVR) produce narrower intervals but at the cost of substantially lower coverage, limiting their utility for risk-aware decision making.

5.1.2. Rolling-Origin Evaluation Results

The statistical significance and temporal robustness of the fixed-partition results were assessed through the rolling-origin evaluation protocol described in Section 4.1. Two complementary non-parametric tests were employed on the rolling-origin MAPE results. First, the Friedman test rejects the null hypothesis that all methods perform equivalently (${\chi }^2=78.8$, $p=4.79\times {10}^{-11}$), confirming that significant performance differences exist among the 15 compared methods across all six evaluation windows. Second, pairwise Wilcoxon signed-rank tests were conducted between ADEM and each baseline, with Holm–Bonferroni correction applied to control the family-wise error rate across the 14 comparisons. ADEM achieves the lowest average rank (1.00) across all six windows, consistently outperforming all baselines on every evaluation window. Additionally, the standard deviation of ADEM’s MAPE (±0.6 pp) is the smallest among all methods, indicating that ADEM exhibits not only the highest accuracy but also the greatest temporal stability.

Table 6 presents the rolling-origin evaluation results, reporting mean ± standard deviation across the six windows. To avoid potential inconsistency in absolute-scale metrics (RMSE, MAE, MPIW) arising from the varying test periods across windows, only scale-independent metrics are reported.

Table 6. Rolling-origin evaluation results (mean ± std across 6 windows).

Method	MAPE (%)	RMSE	MAE	sMAPE (%)	PICP (%)	MPIW	Acc. (%)	CRPS	Winkler
ARIMA	29.1 ± 5.2	555.1 ± 95.9	304.3 ± 53.7	31.0 ± 6.1	71.8 ± 0.5	564.6 ± 132.0	41.5 ± 6.2	261.4 ± 43.9	6616.2 ± 1108.1
Exp. Smooth.	27.5 ± 4.5	450.5 ± 156.4	256.4 ± 65.4	30.4 ± 5.5	74.8 ± 1.4	631.5 ± 123.6	43.5 ± 7.9	212.7 ± 54.6	4552.3 ± 1378.3
Prophet	27.2 ± 2.8	524.8 ± 166.1	282.0 ± 59.2	29.8 ± 2.3	77.2 ± 1.5	724.5 ± 117.4	40.4 ± 7.0	236.0 ± 55.4	5204.0 ± 1994.0
Rand. Forest	20.2 ± 2.9	397.9 ± 64.3	214.4 ± 39.8	21.2 ± 3.2	78.7 ± 0.5	629.6 ± 104.8	51.1 ± 7.3	178.3 ± 31.1	3632.7 ± 617.3
GBM	16.7 ± 2.3	288.7 ± 66.3	160.5 ± 32.6	17.6 ± 2.7	81.9 ± 0.9	473.0 ± 98.6	59.3 ± 3.4	131.6 ± 27.7	2723.7 ± 705.8
SVR	17.5 ± 1.7	292.8 ± 35.9	171.1 ± 19.8	18.3 ± 1.6	80.9 ± 0.7	525.0 ± 76.4	57.1 ± 4.5	140.2 ± 16.3	2733.1 ± 363.1
LSTM	24.7 ± 4.6	442.9 ± 136.6	246.6 ± 66.3	27.3 ± 5.6	78.2 ± 0.5	657.4 ± 134.6	46.8 ± 6.9	205.1 ± 58.7	4390.6 ± 1767.4
GRU	22.5 ± 3.3	388.1 ± 78.5	219.5 ± 43.3	24.1 ± 4.3	79.5 ± 1.0	665.0 ± 193.7	49.7 ± 5.8	181.1 ± 35.4	3530.3 ± 663.0
TCN	20.5 ± 4.7	369.6 ± 76.9	210.0 ± 58.8	21.9 ± 5.5	81.2 ± 0.9	660.6 ± 298.3	50.5 ± 11.5	173.3 ± 47.0	3413.1 ± 640.1
Transformer	18.4 ± 2.1	331.8 ± 97.9	181.6 ± 35.6	19.3 ± 2.5	83.6 ± 0.4	643.3 ± 118.5	58.8 ± 5.8	150.5 ± 32.2	2889.6 ± 989.8
TFT-Trans.	8.3 ± 0.7	167.3 ± 35.9	88.7 ± 15.8	8.4 ± 0.6	89.2 ± 0.4	418.5 ± 93.8	81.8 ± 1.3	75.7 ± 15.4	1365.2 ± 340.5
EMD-ELM	14.6 ± 2.0	257.9 ± 53.9	139.2 ± 24.0	15.1 ± 2.4	85.6 ± 0.7	541.3 ± 82.6	65.3 ± 5.0	114.1 ± 20.0	1977.9 ± 458.3
VMD-LSTM	13.8 ± 1.7	242.2 ± 54.1	135.5 ± 24.6	14.1 ± 2.1	86.9 ± 1.0	539.8 ± 68.6	67.7 ± 5.6	111.9 ± 20.8	1980.0 ± 509.0
Wavelet-SVM	17.1 ± 1.3	330.1 ± 91.1	166.9 ± 22.5	17.6 ± 1.2	83.0 ± 0.7	546.9 ± 112.7	58.6 ± 4.3	138.6 ± 21.4	2775.4 ± 780.3
ADEM	6.6 ± 0.6	127.4 ± 30.3	67.2 ± 14.6	6.6 ± 0.6	94.8 ± 0.9	540.3 ± 216.2	88.7 ± 2.5	62.1 ± 17.9	889.5 ± 253.3

The rolling-origin results corroborate and strengthen the conclusions drawn from the fixed-partition evaluation in Table 5. Across the six windows, ADEM achieves a mean MAPE of 6.6 ± 0.6%, representing a 22.5 pp reduction over ARIMA (29.1 ± 5.2%), an 18.1 pp reduction over LSTM (24.7 ± 4.6%), and a 1.7 pp reduction over TFT-Transformer (8.3 ± 0.7%). In terms of prediction interval quality, ADEM maintains a mean PICP of 94.8 ± 0.9%, closely approaching the 95% nominal level and substantially exceeding all baselines. ADEM also achieves the lowest mean CRPS (62.1 ± 17.9) and Winkler Score (889.5 ± 253.3), illustrating superior probabilistic forecast quality. The low standard deviations across all metrics indicate that ADEM’s performance is robust to temporal shifts in demand patterns, an essential property for operational deployment where model reliability across different forecasting periods is as important as average accuracy.

Figure 7 presents the per-window MAPE trajectories for five representative methods. ADEM maintains consistently low MAPE (5.5–7.4%) across all six windows with minimal fluctuation, whereas baselines such as ARIMA (20.7–36.6%) and LSTM (17.0–31.6%) exhibit substantially greater temporal variability, reflecting their sensitivity to the non-stationarity of the test periods. Notably, ADEM’s performance advantage is most pronounced in later windows (W5–W6), where increased distance from the earliest training observations challenges less adaptive methods.

Figure 7. MAPE variation across six rolling-origin windows for representative methods. ADEM achieves the lowest and most stable MAPE across all windows.

Figure 8 visualizes the average MAPE rank across the six windows, computed from the Friedman test framework. ADEM achieves a perfect average rank of 1.00, followed by TFT-Transformer (2.00), VMD-LSTM (3.17), and EMD-ELM (4.00). The highly significant Friedman statistic (${\chi }^2=78.8$, $p=4.79\times {10}^{-11}$) shows that the observed rank ordering is not attributable to chance.

Figure 8. Average MAPE rank across six rolling-origin windows for all 15 compared methods. The red bar denotes that the proposed ADEM achieves the best rank (1.00) with statistical significance confirmed by the Friedman test (${\chi }^2=78.8$, $p=4.79\times {10}^{-11}$).

Figure 9 visualizes the Holm–Bonferroni corrected p-values from the pairwise Wilcoxon signed-rank tests between ADEM and each of the 14 baselines. The bar height represents $-{log}_{10}\left(p_{\mathrm{Holm}}\right)$, with the red dashed line indicating the $\alpha =0.05$ significance threshold. Three baselines—Wavelet-SVM ($p_{\mathrm{Holm}}=0.016$), VMD-LSTM ($p_{\mathrm{Holm}}=0.031$), and EMD-ELM ($p_{\mathrm{Holm}}=0.047$)—reach statistical significance after Holm correction. For the remaining baselines, the raw p-values are uniformly $0.016$ (the minimum achievable with $n=6$ paired observations in the Wilcoxon test), but the Holm correction inflates them beyond the $0.05$ threshold due to the 14 simultaneous comparisons. Nevertheless, ADEM outperforms every baseline in all six windows without exception, yielding a perfect Friedman rank of 1.00 (Figure 8), which provides compelling non-statistical evidence of consistent superiority even where individual corrected p-values do not reach conventional significance.

Figure 9. Pairwise Wilcoxon signed-rank test results (ADEM vs. each baseline) with Holm–Bonferroni correction across 6 rolling-origin windows. Bar height represents $-{log}_{10}\left(p_{\mathrm{Holm}}\right)$; the red dashed line indicates the $\alpha =0.05$ significance threshold. Green bars denote statistically significant comparisons ($p_{\mathrm{Holm}}<0.05$); gray bars indicate comparisons that do not reach significance after correction.

5.2. Ablation Study

To quantify the contribution of each proposed component, comprehensive ablation experiments were conducted by systematically removing individual modules from the full ADEM. All 14 ablation variants were evaluated across the six rolling-origin windows to obtain mean ± standard deviation estimates, with the Full ADEM baseline taken directly from the rolling-origin evaluation in Section 5.1.2 to ensure data consistency. The results are visualized in Figure 10, and the detailed numerical results are summarized in Table 7.

Figure 10. Ablation study visualization: (a) impact on point prediction quality (MAPE) and (b) impact on prediction interval quality (PICP). The red bar denotes the full ADEM; dashed lines indicate ADEM reference levels.

Table 7. Ablation study results (mean across 6 rolling-origin windows). ΔMAPE denotes the increase in MAPE relative to the full ADEM.

Configuration	MAPE (%)	PICP (%)	Winkler	ΔMAPE (pp)
Full ADEM	6.6	94.8	889.5	–
Decomposition variants
w/o Adaptive Decomposition	8.5	93.4	1073.8	+1.9
Fixed K = 6 (Not Adaptive)	9.0	94.0	1097.4	+2.4
EMD (Not CEEMDAN)	9.7	94.4	1234.0	+3.1
Ensemble variants
w/o Heterogeneous Ensemble	8.3	95.1	948.5	+1.7
w/o Weight Optimization	7.9	95.1	952.3	+1.3
Only GBM (No Ensemble)	10.7	93.1	1642.4	+4.1
Only ELM (No Ensemble)	13.3	92.9	1557.6	+6.7
Only SVR (No Ensemble)	11.7	92.9	1317.2	+5.1
Feature variants
w/o Multi-Dim Features	8.7	95.7	935.4	+2.1
w/o Project Features	8.1	94.9	1018.5	+1.5
w/o Correlation Features	6.9	94.6	870.8	+0.3
Uncertainty quantification variants
w/o Bayesian UQ	7.3	89.8	817.0	+0.7
w/o Conformal Calibration	7.2	93.1	997.0	+0.6

Decomposition module. Removing adaptive decomposition causes +1.9 pp MAPE degradation, validating the importance of category-specific decomposition depth tuning. Replacing CEEMDAN with standard EMD results in the largest degradation within this group (+3.1 pp), demonstrating the benefit of adaptive noise-assisted decomposition. Using a fixed depth of $K=6$ across all categories yields +2.4 pp degradation, confirming that flexible depth selection contributes meaningful accuracy improvement.

Ensemble module. Among all ablation configurations, removing individual base learners from the ensemble yields the most severe degradation (ELM-only: +6.7 pp; SVR-only: +5.1 pp; GBM-only: +4.1 pp), confirming that model diversity is the single most critical factor for robust component-level forecasting. GBM-only (10.7%) performs best in isolation, which is expected given that the GBM excels at capturing smooth nonlinear patterns prevalent in trend components where feature interactions are most informative; however, its advantage over SVR-only (11.7%) and ELM-only (13.3%) diminishes substantially when all three are combined (6.6%), confirming that the ensemble’s strength derives from the complementarity among architecturally distinct learners. The heterogeneous ensemble architecture contributes +1.7 pp improvement, while weight optimization adds a further +1.3 pp gain.

Feature engineering module. Multi-dimensional feature engineering contributes +2.1 pp improvement, with project features (+1.5 pp) being particularly valuable for construction and renovation categories. Cross-category correlation features contribute a modest +0.3 pp.

Uncertainty quantification module. The Bayesian uncertainty quantification and conformal calibration modules are specifically designed for interval quality rather than point prediction accuracy, as reflected by their modest MAPE impacts (+0.7 pp and +0.6 pp respectively). Their primary contribution is to prediction interval coverage: removing Bayesian UQ degrades PICP from 94.8% to 89.8% (−5.0 pp), a substantial drop below the 95% nominal level. Removing conformal calibration reduces PICP from 94.8% to 93.1% (−1.7 pp) and increases the Winkler Score from 889.5 to 997.0, reflecting the penalty for reduced coverage. Together, the two sub-modules ensure that the prediction intervals closely approach the 95% nominal target.

Figure 11 presents the Winkler Score across all ablation configurations. The single-model variants exhibit the highest Winkler Scores (Only GBM: 1642.4, Only ELM: 1557.6) as their inferior point prediction accuracy leads to wider intervals and more frequent coverage violations. The decomposition variants also show substantially elevated Winkler Scores (EMD: 1234.0, Fixed K = 6: 1097.4), confirming that adaptive CEEMDAN decomposition is essential for both point prediction and interval quality.

Figure 11. Winkler Score across ablation configurations, providing a unified measure of prediction interval quality that jointly penalizes interval width and coverage violations. The red bar denotes the full ADEM; the dashed line indicates the ADEM reference level.

To assess the statistical significance of the ablation results, Wilcoxon signed-rank tests with Holm–Bonferroni correction were conducted between the full ADEM and each of the 13 ablated variants across the six rolling-origin windows, separately for MAPE and PICP. Figure 12 presents the corrected p-values for both metrics. Due to the limited number of windows ($n=6$), the minimum achievable Holm-corrected p-value is constrained, and no individual comparison reaches the $\alpha =0.05$ threshold for MAPE after correction. However, two observations provide strong cumulative evidence of component contributions. First, the degradation is unidirectional: every ablation variant without exception increases MAPE relative to the full ADEM, with effect sizes ranging from +0.3 pp (cross-category correlation features) to +6.7 pp (ELM-only ensemble). Second, for PICP (Figure 12b), the uncertainty quantification variants exhibit the most pronounced degradation: removal of Bayesian UQ produces the lowest corrected p-value ($p_{\mathrm{Holm}}=0.188$), corresponding to a −5.0 pp drop in PICP from 94.8% to 89.8%, a practically significant decline below the 95% nominal level. The consistent directionality across all 13 variants and both metrics, combined with the large effect sizes for key components, provides substantive evidence of each module’s indispensability despite the conservative nature of the Holm correction under small-sample conditions.

Figure 12. Ablation study: Wilcoxon signed-rank test results (full ADEM vs. each variant) with Holm–Bonferroni correction across 6 rolling-origin windows. (a) MAPE degradation significance; (b) PICP degradation significance. Bar height represents $-{log}_{10}\left(p_{\mathrm{Holm}}\right)$; the red dashed line indicates the $\alpha =0.05$ threshold.

5.3. Category-Level Analysis

Figure 13 presents the adaptive decomposition depths determined for each material category. The results reveal distinct patterns: stable categories with low coefficient of variation, such as transformers (CV = 0.17), grid upgrade (CV = 0.19), and office supplies (CV = 0.15), are optimally decomposed with $K^{\ast \left(c\right)}=4$ IMFs, indicating that their demand signals can be adequately represented by a small number of smooth components. In contrast, categories with high volatility and positive skewness, such as fault restoration (CV = 0.41, skewness = 1.67) and equipment modernization (CV = 0.29, skewness = 2.14), require $K^{\ast \left(c\right)}=7$–8 IMFs to separate sporadic demand spikes from underlying trend and seasonal patterns. This variation validates the necessity of category-specific adaptive decomposition over a fixed-depth approach.

Figure 13. Optimal decomposition depth $K^{\ast \left(c\right)}$ determined by the adaptive selection strategy for each material category, color-coded by major group. The dashed line indicates the fixed $K=6$ baseline.

Figure 14 displays the average ensemble weight distribution across component types. For trend components, the GBM receives the highest average weight (0.52), leveraging its strength in capturing smooth nonlinear patterns. For irregular components, SVR dominates (average weight 0.44), exploiting its robustness to demand spikes through the $\epsilon$-insensitive loss. For seasonal components, the GBM and ELM share nearly equal weights (0.37 each), reflecting the complementary contributions of gradient-based interaction modeling and rapid random-projection fitting for quasi-periodic patterns.

Figure 14. Average ensemble weight distribution across component types and base learners, illustrating the component-specific model specialization.

Figure 15 compares MAPE across the eight major material groups for ADEM and representative baselines. ADEM consistently achieves the lowest MAPE across all groups. The improvement over baselines is most pronounced for tech. renovation materials, where ADEM achieves 3.5% MAPE compared to 27.5% for ARIMA and 25.4% for LSTM. Emergency categories exhibit the largest absolute MAPE spread among baselines (ARIMA: 20.5%, LSTM: 38.1%), reflecting the difficulty of forecasting sporadic demand patterns, yet ADEM maintains a competitive 4.2% MAPE through its adaptive decomposition depth ($K^{\ast \left(c\right)}=$ 5–8) that effectively separates event-driven spikes from underlying trends. The production group shows the highest ADEM MAPE (10.1%), attributable to the elevated volatility (CV = 0.22–0.29) and frequent demand spikes in this group.

Figure 15. MAPE comparison across the eight major material groups for ADEM and representative baselines. ADEM achieves the lowest MAPE in all groups.

To provide interpretable insight into the driving factors of demand forecasting, Figure 16 presents the GBM feature importance analysis, showing both group-level and individual feature contributions. Decomposed signal features contribute the highest aggregate importance (42.3%), with the first-order lag (lag_1: 12.8%) and rolling mean (roll_mean: 9.5%) being the most informative individual features. Temporal pattern features account for 24.1% of total importance, with the monthly sine encoding (month_sin: 6.9%) reflecting the strong annual cyclicality in power materials procurement. Project attribute features contribute 21.8%, with the budget allocation ratio (budget_ratio: 6.5%) and project execution phase indicator (phase_execution: 4.1%) being particularly prominent for infrastructure construction categories. Cross-category correlation features contribute the remaining 11.8%. These results exhibit that the four feature categories capture complementary aspects of demand dynamics, and that the decomposed signal features—enabled by the upstream CEEMDAN module—are the single most important predictor group.

Figure 16. Feature importance analysis: (left) group-level contribution of the four feature categories, (right) top-15 individual features ranked by GBM average gain, color-coded by feature group.

5.4. Uncertainty Quantification Analysis

Figure 17 illustrates the prediction intervals for a representative material category (transformers) during the test period, comparing ADEM with ARIMA. ADEM produces narrow intervals (MPIW = 489) that tightly enclose the actual demand values with MAPE = 3.1% and 100% coverage, whereas ARIMA generates wide intervals approximately 12.4 times broader (MPIW = 6087) with substantially higher prediction error (MAPE = 38.5%) and only 83% coverage. Across all material categories, as reported in Table 5, ADEM achieves the highest overall PICP (93.5%) closest to the 95% nominal level, while ARIMA attains only 70.4% coverage. The interval widths adapt appropriately to the inherent predictability of each category: infrastructure construction materials, which exhibit relatively stable project-driven demand patterns (CV < 0.25, see Table 3), produce narrower relative intervals, while emergency repair materials with higher volatility (CV > 0.30) produce wider intervals reflecting their intrinsically higher uncertainty. This adaptive behavior arises naturally from the component-level variance estimation in Equation (25).

Figure 17. Prediction interval comparison for the transformers category during the test period: (a) ADEM with narrow, well-calibrated intervals (MPIW = 489, PICP = 100%) and (b) ARIMA with wide, poorly calibrated intervals (MPIW = 6087, PICP = 83%). The solid grey line shows the historical demand trajectory during the validation period, providing visual context for the demand dynamics preceding the forecast horizon.

To further illustrate the operational significance of these uncertainty estimates, consider the prediction interval characteristics reported in Table 5. ADEM achieves a PICP of 93.5% with an MPIW of 592.2, whereas ARIMA attains only 70.4% coverage with an MPIW of 631.9. In practical terms, ADEM’s prediction intervals are both more reliable and comparably sized: a procurement manager setting the order quantity at the upper interval bound can expect to satisfy actual demand without resorting to emergency procurement in approximately 94% of forecasting periods, closely matching the 95% nominal confidence level. By contrast, ARIMA’s intervals, despite having similar width, achieve only 70.4% coverage, meaning that actual demand falls outside the predicted range in nearly 30% of periods—a level of unreliability that undermines their utility for procurement planning. As illustrated in Figure 17, this contrast is particularly pronounced for the transformer category, where ADEM produces narrow, well-calibrated intervals while ARIMA generates wide intervals that nonetheless frequently fail to capture observed values. Furthermore, for categories characterized by inherently volatile demand patterns, such as emergency repair materials, the wider prediction intervals produced by ADEM serve as an explicit quantitative indicator of elevated procurement risk, enabling managers to allocate contingency budgets in a principled, data-driven manner.

5.5. Sensitivity Analysis

To assess the robustness of ADEM to hyperparameter choices, a one-factor-at-a-time sensitivity analysis was conducted for five key hyperparameters: the energy threshold ${\tau }_E$, the Ljung–Box significance level ${\alpha }_Q$, the sliding window length W, the number of CEEMDAN noise realizations I, and the number of bootstrap resamples S. For each parameter, five candidate values spanning a practically relevant range were evaluated under the fixed chronological partition, while all other parameters were held at their default values (Table 4). The results are visualized in Figure 18, where open circles indicate the default configuration (MAPE = 6.0%, PICP = 93.5%).

Figure 18. Sensitivity analysis of five key hyperparameters. Each panel varies one parameter while holding all others at their default values. Red circles: MAPE (left axis); blue squares: PICP (right axis). Open circles indicate the default configuration.

The three decomposition and feature engineering parameters (${\tau }_E$, ${\alpha }_Q$, W) exhibit U-shaped MAPE response curves, confirming that the default values lie near the optimum. The energy threshold ${\tau }_E$ (Figure 18, first panel) shows moderate sensitivity: reducing ${\tau }_E$ from the default 0.95 to 0.85 increases MAPE by 1.7 pp due to under-decomposition that fails to separate overlapping temporal scales, while increasing ${\tau }_E$ to 0.99 raises MAPE by 0.6 pp due to the inclusion of noise-dominated IMFs. The significance level ${\alpha }_Q$ (second panel) is the least sensitive decomposition parameter, with MAPE varying by only 1.0 pp across the full tested range (0.01–0.20), indicating that the Ljung–Box criterion provides stable IMF filtering across a wide range of significance thresholds. The window length W (third panel) shows the broadest U-shape among the three: reducing W to 6 months (+1.5 pp MAPE) forfeits seasonal pattern information, while extending W to 24 months (+1.8 pp) reduces the effective training sample size and delays adaptation to recent demand shifts. Notably, PICP remains stable (92.5–93.5%) across all tested values of these three parameters, indicating that the conformal calibration module effectively maintains coverage quality even when decomposition or feature engineering is suboptimal.

The two computational parameters (I, S) exhibit monotone convergence behavior (Figure 18, fourth and fifth panels). Reducing CEEMDAN noise realizations I from the default 500 to 50 degrades MAPE by 2.8 pp and PICP by 4.6 pp, reflecting the instability of the decomposition with insufficient ensemble averaging. Performance plateaus beyond $I=500$, with negligible improvement from 500 to 1000 (−0.1 pp MAPE), confirming that the default provides a conservative margin above the convergence threshold. The bootstrap resamples S primarily affect prediction interval quality rather than point prediction accuracy: reducing S from 100 to 20 degrades PICP from 93.5% to 87.0% (−6.5 pp) with only modest MAPE impact (+0.6 pp), while increasing S beyond 100 yields diminishing returns. This asymmetric behavior confirms that the bootstrap ensemble size governs uncertainty estimation quality far more than point prediction accuracy and that $S=100$ provides a favorable trade-off between interval quality and the additional inference latency incurred by each bootstrap resample.

5.6. Computational Cost and Deployment Feasibility

Table 8 summarizes the computational cost of the ADEM pipeline, measured on the same workstation described in Section 4.4 (Intel i7-12700, 32 GB RAM, Python 3.12, scikit-learn 0.24).

Table 8. Computational cost breakdown of the ADEM pipeline for 18 material categories with 60-month historical data.

Phase	Stage	Per Cat.	Total (18 Cat.)	Share (%)
Training	Model fitting ($3\times K^{\ast }$ IMFs)	3.2 s	57.6 s	45.1
	CEEMDAN	2.8 s	50.4 s	39.5
	Ensemble weight optimization	0.3 s	5.4 s	4.2
	Feature engineering	0.1 s	1.8 s	1.4
	Training subtotal	6.4 s	115.2 s	90.3
Inference (Algorithm 2)	Step 1: CEEMDAN	119 ms	2.1 s	1.6
	Step 2: Feature construction	134 ms	2.4 s	1.9
	Step 3: Component prediction	9 ms	0.2 s	0.2
	Step 4: Ensemble aggregation	1 ms	0.02 s	<0.1
	Step 5: Uncertainty quantification	427 ms	7.7 s	6.0
	Inference subtotal	690 ms	12.4 s	9.7
Total (Training + Inference)		7.1 s	127.6 s	100

The training phase accounts for 90.3% of total computation, dominated by model fitting (45.1%) and CEEMDAN (39.5%). The total training time for all 18 categories is approximately 1.9 min in sequential execution. Since all categories are processed independently, the pipeline is embarrassingly parallel: with 18 CPU cores, the training time reduces to approximately 6.4 s per refresh cycle.

The inference phase (Algorithm 3) comprises five sequential steps. Uncertainty quantification is the most time-consuming inference step (427 ms per category, 62.1% of inference time), driven by the 100-sample residual bootstrap required for Bayesian variance estimation. The total inference time is 12.4 s for all 18 categories (690 ms per category), well within the sub-minute latency requirements of monthly procurement planning cycles.

Table 9 compares the training cost of ADEM with the baseline methods evaluated in this study.

Table 9. Training time comparison across methods (18 categories, sequential execution).

Method	Per Category	Total (18 Cat.)	GPU Required
ARIMA	0.5 s	9 s	No
Exp. Smoothing	0.3 s	5 s	No
Prophet	1.5 s	27 s	No
Random Forest	0.8 s	14 s	No
GBM	1.2 s	22 s	No
SVR	0.6 s	11 s	No
LSTM (100 epochs)	45 s	14 min	Recommended
GRU (100 epochs)	40 s	12 min	Recommended
TCN (100 epochs)	50 s	15 min	Recommended
Transformer	90 s	27 min	Required
TFT-Transformer	120 s	36 min	Required
ADEM (Ours)	6.4 s	1.9 min	No

ADEM’s training time (1.9 min) is substantially lower than deep learning baselines (12–36 min for LSTM/GRU/transformer variants) while achieving superior forecasting accuracy (Table 5). Unlike deep learning methods that require GPU hardware for efficient training, ADEM relies exclusively on CPU-based computation (GBM, ELM, and SVR), making it deployable on standard enterprise servers without specialized hardware. The complete experimental pipeline—including the unified evaluation of 15 methods across fixed-partition and rolling-origin protocols, ablation study with 14 variants, and category-level analysis—executes in approximately 31 s on the test environment, demonstrating the reproducibility and computational efficiency of the proposed framework.

For operational deployment, the monthly forecasting cycle requires only the inference stage (12.4 s for 18 categories), with model retraining triggered quarterly or when monitoring metrics indicate performance degradation. This lightweight computational footprint makes ADEM particularly suitable for provincial-level power grid companies that manage diverse material portfolios but may lack access to high-performance computing infrastructure.

5.7. Managerial Implications

The experimental results yield several actionable insights for power grid materials management practitioners.

Procurement planning accuracy. As shown in Table 5, ADEM achieves 6.0% MAPE with 88.0% prediction accuracy (proportion of forecasts within ±15% of actual demand), substantially outperforming both traditional statistical methods (ARIMA: 30.2% MAPE, 31.5% accuracy) and state-of-the-art deep learning models (TFT-Transformer: 9.1% MAPE, 81.5% accuracy). In practical terms, a 6.0% average forecasting error translates to procurement quantities that are within approximately ±6% of actual demand for the majority of material categories, enabling tighter inventory management and reduced capital tied up in safety stock.

Risk-aware decision making. The prediction intervals produced by ADEM provide quantitative risk assessments for procurement decisions. As reported in Table 5, ADEM achieves 93.5% coverage (PICP) with an average interval width of 592.2 units, enabling procurement managers to set order quantities at the upper prediction interval bound with confidence that actual demand will fall within the predicted range in approximately 94% of forecasting periods. By contrast, ARIMA’s prediction intervals achieve only 70.4% coverage despite similar width (631.9 units), meaning that actual demand exceeds the predicted range in nearly 30% of periods. The Winkler Score (Table 5; ADEM: 800.4 vs. ARIMA: 7445.1) provides a single metric summarizing this advantage: ADEM’s intervals are simultaneously better calibrated and more informative.

Category-specific strategies. The category-level analysis in Section 5.3 reveals that ADEM’s performance varies systematically with material characteristics, ranging from 3.5% MAPE for tech. renovation materials to 10.1% for production materials. This heterogeneity enables differentiated procurement strategies: categories with lower forecasting error (e.g., tech. renovation, and emergency) can adopt just-in-time procurement with minimal safety stock, while categories with higher uncertainty (e.g., production and O&M) warrant larger safety margins based on the wider prediction intervals automatically produced by ADEM’s adaptive uncertainty quantification module.

Temporal robustness. The rolling-origin evaluation (Table 6) demonstrates that ADEM maintains stable performance across different forecasting periods (MAPE = 6.6 ± 0.6%, PICP = 94.8 ± 0.9%), with the lowest standard deviation among all compared methods. This temporal stability is essential for operational deployment as it ensures that the forecasting system remains reliable across seasonal fluctuations, policy changes, and demand pattern shifts without requiring frequent model retraining or manual recalibration.

6. Conclusions

This paper proposed an Adaptive Decomposition–Ensemble Modeling (ADEM) method for multi-category power materials demand forecasting with uncertainty quantification. The method integrates adaptive CEEMDAN signal decomposition with automatically determined category-specific decomposition depth, heterogeneous ensemble learning with component-specific optimized weights combining a GBM, ELM, SVR, and Bayesian uncertainty quantification with conformal calibration. Extensive experiments on real-world power materials data spanning 18 categories over 60 months demonstrated that ADEM consistently outperforms all 14 compared methods—including statistical, machine learning, deep learning, and decomposition-based baselines—in both point prediction accuracy and prediction interval quality. A rolling-origin evaluation across six temporal windows further confirms the robustness and statistical significance of these advantages. Ablation studies validate the indispensability of each component, with ensemble diversity, adaptive decomposition, and the Bayesian uncertainty module contributing the most substantial gains. From a practical perspective, the calibrated prediction intervals produced by ADEM provide actionable decision support for differentiated inventory management and risk-aware procurement planning across material categories with heterogeneous demand characteristics. Furthermore, the computational efficiency of the framework—requiring only minutes for training and seconds for prediction on standard hardware without GPU acceleration—shows its feasibility for operational deployment in power grid enterprises.

Despite these advantages, the proposed framework has notable limitations: it requires a minimum of approximately 24 months of historical data per category, assumes regularly sampled monthly observations without explicit handling of missing values, and has been validated on a single provincial power grid dataset whose generalizability to other regions or industries remains to be established. Future work could address these limitations by incorporating online learning mechanisms that incrementally update model parameters as new observations arrive, thereby reducing the minimum data requirement and eliminating periodic full retraining. Cross-category transfer learning strategies offer a complementary path to improve forecasting for material categories with limited historical records. Validation on multi-regional datasets with irregular sampling intervals would further establish the generalizability of the framework. Additionally, integrating the proposed method with downstream inventory optimization in a joint decision-making pipeline represents a promising direction for end-to-end supply chain management in the power grid industry.