Significant advances have been witnessed in the financial time series forecasting domain, which are driven by deep learning architectures (Ge et al., 2022; Wu et al., 2024; Torres et al., 2021). Despite the notable advancements achieved by deep learning models such as Transformers and Temporal Convolutional Networks (TCNs) in financial time series forecasting, significant challenges remain in fully capturing the inherent complexity of financial data (Khan and Khan, 2024; Smith and Doe, 2024). For instance, Transformer models suffer from quadratic computational complexity, rendering them inefficient for processing complex financial data. Although TCNs are effective in modeling long-range dependencies, they struggle with capturing nonlinear dynamics and adapting to exogenous shocks. Recently, State Space Models (SSMs) have attracted increasing attention for their strengths in long-sequence modeling (Nguyen and Chen, 2025). However, existing SSMs–such as Mamba–still fall short in capturing the hierarchical structure and cross-variable interactions that are critical in financial time series (Kumar and Lee, 2023; Zhang and Wang, 2024). These limitations form the theoretical foundation for the development of CMDMamba. However, three fundamental challenges still remain: nonstationary data distributions, ultrahigh-frequency noise contamination, and nonlinear cross-asset dependencies. Traditional linear statistical models and their regularized variants, although maintaining computational efficiency and interpretability through shrinkage techniques, are essentially restricted by their dependence on linear additive assumptions Johnson and Martinez (2023). This limitation seriously limits their ability to capture the intricate non-linear dynamics that is inherent in financial markets (Luo et al., 2021). Empirical studies have shown that these methods have crucial deficiencies in modeling sudden regime shifts in complex financial situations.

Innovations in deep learning architectures (Zhang and Hua, 2025; Yu et al., 2024) have brought about a profound transformation in the methodological framework for financial prediction. Architecturally, multilayer perceptron (MLP)—based systems have improved gradient propagation. They achieve this through stacked fully - connected layers with residual connections, as demonstrated by DeepAR's excellent performance in low—frequency return prediction benchmarks (Salinas et al., 2020). Temporal convolutional networks (TCNs) utilize causal and dilated convolution operations to build exponentially expanding temporal receptive fields (Yao et al., 2023). Researchers have achieved significant improvements in high—frequency trade signal detection by stacking multi—layer convolutional modules, especially in market microstructure analysis (Hao and Gao, 2020). To overcome the quadratic complexity bottleneck in Transformer—based sequence processing, optimized solutions incorporating sparse attention mechanisms have been developed. The Informer (Zhou et al., 2021) architecture uses probabilistic sampling strategies. Although this approach greatly reduces computational resource demands, the remaining complexity is still not practical for real—world financial systems. Despite the use of enhanced techniques leveraging time—frequency domain transformations (such as fast Fourier transforms) (Qin et al., 2025), spectral analysis capabilities are still insufficient when dealing with ultra—low—frequency signals contaminated by strong noise (Zhou et al., 2022).

Current methodologies continue to exhibit critical deficiencies in modeling dynamic coupling relationships among multivariate series. The dual constraints of computational complexity and prediction accuracy present urgent challenges for real-time deployment, particularly pronounced in high-frequency trading environments, where the synergistic mechanisms between market microstructure noise and macroeconomic policy shocks remain insufficiently resolved (Song et al., 2024; Chomicz-Grabowska and Orlowski, 2020). CMDMamba adopts the Mamba architecture, which ingeniously reduces computational complexity to a linear level. The model also introduces a dual-layer Mamba design, dynamically adjusting the state decay rate according to different sensitivities to microstructure and macroeconomics, thereby effectively weakening noise interference and significantly enhancing overall model performance.

State Space Models (SSMs) (Gu et al., 2021a,d) have experienced a renaissance in sequential modeling, primarily attributed to the groundbreaking advancements in Structured State Space Sequence Models (S4) (Gu et al., 2021e,b). The innovative High-Order Polynomial Projection Operators (HiPPO) Gu et al. (2020) initialization framework established by S4 has created a new paradigm, demonstrating unprecedented capabilities in long-range dependency modeling. Subsequent studies have continuously enhanced computational efficiency while preserving the theoretical foundation of continuous system modeling. Notably, the diagonal parameterized SSMs (S4D) Gu et al. (2022) and data-dependent SSMs (DSS) have laid crucial groundwork for modern SSM applications across diverse domains (Gu et al., 2021c).

The emergence of the Mamba architecture has catalyzed a paradigm shift in this field, representing a quantum leap in SSM research. This seminal work innovatively incorporates time-varying parameterized state matrices combined with hardware-aware parallel scanning algorithms. This dual innovation achieves linear computational complexity while maintaining global dependency capture capabilities, attaining positional awareness levels comparable to Transformer architectures. Such breakthroughs have precipitated the proliferation of derivative models, finding extensive applications in language modeling, audio processing, computer vision, and temporal sequence analysis. Notwithstanding these advancements, current SSM architectures confront substantial optimization challenges. The S-Mamba (Wang et al., 2025) model, while implementing decoupled modeling through channel-temporal state space tensor decomposition (Feng et al., 2022), theoretically exhibits deficiencies in dynamically representing high-order cross-feature interactions within temporal data. The bidirectional gating mechanism in MambaMixer (Behrouz et al., 2024b), though capable of synchronously capturing inter-sequence and intra-sequence dependencies, suffers from significant information loss during forward-backward feature selection processes. TimeMachine's (Ahamed and Cheng, 2024) quad-scale architecture integrates channel mixing mechanisms, yet its static statistical feature-based policy selector proves inadequate in adapting to distributional drift phenomena inherent in non-stationary time series. The parameter-sharing scheme in Bi-Mamba (Liang et al., 2024) enhances temporal modeling efficiency but introduces bidirectional state coupling interference. Moreover, its reverse scanning mechanism requiring full-sequence input conflicts with the low-latency demands of streaming inference, creating critical deployment barriers that highlight systemic architectural contradictions in balancing temporal correlation and real-time processing.

Despite notable progress, critical scientific challenges remain unresolved: how to develop novel architectures capable of capturing cross-variable correlations, modeling complex nonlinear interactions, and maintaining linear computational complexity. This bottleneck constrains the applicability of the Mamba-series models in complex scenarios such as financial forecasting and industrial IoT. Our proposed DconvFFN module effectively captures these correlations, thereby enhancing the predictive performance of the model.

Financial time series forecasting is formally defined as follows. Let X*={x1,x2,…,xN} denote a multivariate financial time series, where each observation xt∈ℝd represents a d-dimensional vector of financial variables at time step t. Given a fixed-length lookback window T, the forecasting task at any time point t involves predicting the future τ-step sequence Xt+1:t+τ={xt+1,xt+2,…,xt+τ} based on the historical observations within the window Xt-T+1:t={xt-T+1,xt-T+2,…,xt}. This formulation captures the temporal dependencies and multivariate interactions inherent in financial data, enabling the prediction of future trends and patterns.

State Space Models (SSMs) describe recurrent processes through latent states h(t) ∈ ℝ^N governed by a first-order differential equation. Inputs x(t) ∈ ℝ^D drive state evolution, while outputs y(t) ∈ ℝ^M are generated via a mapping from h(t). The dynamic process is illustrated in Equation 1:

where A ∈ ℝ^N×N denotes the state transition matrix governing temporal dynamics, B ∈ ℝ^N×D the input projection matrix, and C ∈ ℝ^M×N the output projection matrix. All three matrices are learnable parameters optimized during training. By employing zero-order hold (ZOH) techniques, the SSM can be discretized, as shown in Equations 2:

where the state matrix A¯ and input matrix B¯ are derived from continuous parameters (A, B, Δt) through a discretization method. The specific transformation formulas (Equation 3) are given as follows:

Where the invertibility of ΔtA enables efficient discretization through linear recurrence operations, the Structured State Space model (S4) introduces HiPPO-initialized constraints on matrix A. This integration establishes S4 as a computationally efficient hybrid architecture that combines classical state-space formulations with deep learning principles, achieving superior long-range sequence modeling capabilities compared to conventional recurrent networks.

Mamba innovatively employs a data-driven approach to parameterize critical matrices while integrating a dynamic selection mechanism into the S4 architecture. As illustrated in Figure 1, the proposed hardware-aware parallel algorithm significantly enhances training efficiency. In contrast to self-attention-based Transformer models, this architecture achieves linear computational complexity while preserving global receptive fields. This breakthrough enables the model to efficiently capture long-range dependencies in time series forecasting without incurring the high computational costs associated with conventional methods, thus achieving an optimal balance between computational efficiency and temporal modeling capability.

The architecture of the CMDMamba model, as illustrated in Figure 2, comprises four critical processing stages: Initially, the embedding layer divides the input time-series data into feature blocks with high information density. Subsequently, the DconvFFN module models local correlations among features to achieve dynamic fusion of multi-feature representations. Then, a dual cascaded Mamba processing module is designed, where the high-sensitivity Mamba block captures microscopic fluctuation characteristics within sequences, while the low-sensitivity Mamba block focuses on modeling long-term dependencies across time steps. Finally, the TNF Encoding module blends information from the Mamba-processed temporal features, generating a final feature vector with robust representational capabilities.

In this experiment, to evaluate the model performance, we selected four publicly available financial time series datasets. These datasets encompass historical trading data from diverse financial markets and were chosen based on their broad applicability and the significant challenges they present in the financial domain. Specifically, these data sets include the German DAX index, the Dow Jones Industrial Average (DJI), the Hang Seng Index (HSI), and Standard & Poor's 500 Index (S&P500). All data sets cover the period from January 2, 2014, to December 12, 2022, with daily observations collected throughout this timeframe. Each dataset was partitioned into training, validation, and test sets using a 7:1:2 ratio. Detailed characteristics of the data sets are provided in Table 1, In addition, Figure 3 illustrates the price trends of various variables in each financial dataset over the past two years.

All experiments were conducted using the following configuration: an L2 loss function with Adam optimizer (Kingma and Ba, 2014), initial learning rate of 5 × 10⁻⁵, and batch size of 8. The Transformer-based architecture employed an attention factor of 3, while the Mamba-based model utilized a state expansion factor of 2. Weight decay was consistently set to 0.1 throughout all trials. Training procedures implemented early stopping with a patience period of 10 epochs. To ensure statistical reliability, each experimental condition was independently repeated three times with random initialization. The complete implementation, developed using PyTorch (Paszke et al., 2019), was executed on a dedicated NVIDIA Tesla V100 GPU (32GB memory) (Markidis et al., 2018) to maintain computational consistency across trials.

In time series forecasting, the accuracy of the model is crucial for the reliability of decision-making. Mean squared error (MSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) are commonly used evaluation metrics. They can quantify the deviation between the predicted values of the CMDMamba model and the actual observed values from different perspectives, providing intuitive data for evaluating the model's performance. As shown in Equation 9, the calculation is performed as follows:

In these formulas, y_i represents the actual observed value, ŷ_i is the predicted value of the model, and n denotes the total number of data points. To accurately compare model performance, we normalize the values. With metrics such as MSE and MAE, we can intuitively assess the prediction accuracy of the CMDMamba model. The lower the values of these two metrics, the closer the predicted values are to the actual values, indicating better prediction performance of the model.

Since models applicable to general time series forecasting are equally suitable for financial time series prediction, we comprehensively adopt state-of-the-art models from the time series community as baseline references. These include transformer-based models: Autoformer (Wu et al., 2021), Informer Zhou et al. (2021), Reformer (Kitaev et al., 2020), Pyraformer (Liu et al., 2022b), Crossformer (Liu et al., 2023b), and iTransformer (Li et al., 2019); MLP-based models: TiDE (Das et al., 2023); linear-based models: DLinear (Zeng et al., 2023); convolution-based models: SCINet (Liu et al., 2022a); recurrent-based models: LSTM; Statistical modeling-based methods:GARCH; and Mamba-based models: S_Mamba (Shehzad et al., 2020).

In multivariate predictive analysis, multiple time series are considered simultaneously to evaluate the model's ability to capture interdependencies and mutual influences among various features. The experimental results in four data sets demonstrate that CMDMamba consistently outperforms most baselines and prediction horizon configurations, as shown in Table 2. CMDMamba significantly reduced the mean squared error (MSE) by 11. 6% in DAX (0.570 → 0.504), 1. 5% in DJI (0.200 → 0.197), 0.4% in HSI ( 0.563 → 0.561), and 10.2% in S&P500 (0.498 → 0.447) compared to the previous best results. Compared to the previous best models, the performance of CMDMamba in this configuration improved by 10.4%. As forecast horizons expand, modeling complex long-range dependencies becomes increasingly demanding. In this context, the CMDMamba model distinguishes itself through its exceptional capability to efficiently integrate temporal information and intricate cross-variable dependencies. Its robust performance has been thoroughly validated across multiple real-world datasets, showcasing both outstanding prediction accuracy and broad applicability. However, the performance improvement of CMDMamba on the DJI and HSI datasets is relatively less pronounced compared to other datasets. This may be attributed to the fact that both datasets represent comprehensive stock indices characterized by relatively stable macro trends, rendering their temporal patterns less responsive to the dynamic state transition mechanisms that CMDMamba relies on.

Univariate analysis is based solely on the historical data of a single time series to predict future values. In this experiment, we present the univariate results for four datasets (as shown in Table 3). Compared to other models, our model (CMDMamba) achieved superior performance in the prediction task. The model uses 96 historical data points to predict 58 future data points), our model reduced the Mean Absolute Error (MAE) on the DAX dataset by 4. 2% (0.522 0.500), on the DJI dataset by 8. 7% (0.344 0.314), on the HSI dataset by 1. 5% ( 0.394 0.388), and on the S& P500 dataset by 5. 4% ( 0.185 0.175). Moreover, as the forecasting horizon extends, most baseline models exhibit a noticeable increase in prediction errors–specifically in terms of MSE and MAE–across various datasets. This trend is particularly evident for models such as Autoformer and Informer, which experience significant error escalation at longer time steps. In contrast, our proposed model, CMDMamba, demonstrates comparatively smaller error growth in most scenarios, highlighting its superior stability in long-term forecasting. However, in terms of MAPE, CMDMamba does not exhibit the same level of dominance as observed in multivariate forecasting tasks. This suggests a potential limitation in univariate settings, where limited input information may constrain the model's ability to effectively capture temporal dependencies.

To evaluate the impact of the dual-layer Mamba and DconvFFN on CMDMamba's performance, we conduct an ablation study examining three model variants: tFFN/oMamba, which integrates the DconvFFN module alongside a single Mamba layer; oFFN/tMamba, which excludes the DconvFFN module but incorporates two Mamba layers; and oFFN/oMamba, which omits both the DconvFFN module and Mamba layers. We compare these variants against Reformer and Transformer models across four datasets, utilizing MSE and MAE as evaluation metrics. As presented in Table 4, the results indicate a progressive enhancement in model performance with the inclusion of additional components. This confirms the substantial contribution of both the dual-layer Mamba and DconvFFN module to the overall effectiveness of CMDMamba.

To rigorously assess the parameter sensitivity of CMDMamba, we systematically examined two pivotal hyperparameters through cross-market evaluations using datasets representing diverse economic environments: (a) dropout rate and (b) SSM state expansion factor. Our experimental protocol maintained fixed non-target parameters while executing five independent training runs per configuration, each spanning 30 epochs with early stopping monitoring validation loss (patience threshold = 5 epochs). Model performance was rigorously evaluated using MSE metrics across four distinct forecast horizons O∈{12,36,58,96}.. This approach aligns with the rigorous evaluation methodologies used in recent studies on hyperparameter tuning for deep learning models in time series forecasting, ensuring that our findings are comparable and robust.

Based on the high accuracy demonstrated by the CMDMamba model in predicting closing prices (Close), we further evaluated its generalization capability through experiments on two distinct market economy datasets: the regional HSI and the globally representative S&P500 Index. Our study extended the model's application to four additional crucial price variables—opening price (Open), highest price (High), lowest price (Low), and adjusted closing price (Adj Close)—which represent various aspects of financial market behavior. As illustrated in Figure 6, the CMDMamba model exhibits superior performance in these diverse stock price prediction tasks.

In this experiment, we comprehensively evaluated model efficiency from three dimensions: (a) predictive accuracy, (b) memory usage, and (c) training speed. For the prediction task, we set L = 96 (input length), O = 12 (forecast horizon), and B = 8 (batch size), with results presented in Figure 7. All other parameters were fixed to ensure modeling consistency. A direct comparison was made between the self-attention mechanism and CMDMamba. In HSI data tests focusing on regional markets, iTransformer demonstrated marginally superior computational efficiency compared to CMDMamba. However, CMDMamba achieved better prediction performance and memory optimization, showing particular effectiveness in capturing temporal patterns in relatively stable sequences. For the S&P500 dataset representing global multi-market dynamics, both CMDMamba and self-attention-based models exhibited strong predictive capabilities in volatile market conditions. Notably, CMDMamba outperformed all counterparts across both datasets. This advantage stems from the linear complexity characteristics of SSM, enabling CMDMamba to achieve an optimal balance between prediction accuracy, training speed, and memory consumption. This finding is consistent with recent advancements in efficient sequence modeling, where linear complexity models are increasingly preferred over quadratic complexity models for large-scale applications. Overall, compared to self-attention mechanisms in Transformers with equivalent embedding dimension D, CMDMamba demonstrates significantly lower computational costs, particularly evident in training speed and GPU memory utilization. With superior predictive performance, accelerated training efficiency, and reduced memory requirements, Mamba-like models exhibit great potential for financial time series forecasting applications.

The selection of an appropriate review window size is critical for time series forecasting models, as it determines the extent to which historical dependencies can be captured. A proficient forecasting model should be capable of learning long-term dependencies by expanding the review window, thereby enhancing predictive accuracy. In our experimental design, we maintained a fixed forecast horizon (= 12) across two heterogeneous datasets while systematically varying the review window sizes ( ∈ {12, 36, 58, 96}), ensuring that all other model parameters remained constant to facilitate a reliable comparative analysis. As illustrated in Figure 8, the CMDMamba model consistently outperforms across all review window configurations, demonstrating its robust forecasting capability. This empirical advantage can be attributed to two key architectural innovations. First, the DconvFFN module efficiently captures multivariate feature dependencies by leveraging dilated convolutional operations. Second, the dual-layer Mamba architecture integrates differentially sensitized modules: the shallower layer specializes in detecting local fluctuations within shorter review windows through high-frequency filtering, while the deeper layer focuses on extracting global dependencies in extended sequences via low-frequency pattern analysis. These design principles collectively enable CMDMamba to achieve superior forecasting performance across diverse time series scenarios.

In this study, we systematically evaluated the performance of four mainstream architectures–state-space models, Transformer, temporal convolution, and MLP–in financial time series forecasting using datasets such as HSI and S&P500. As shown in Figure 9, compared with the iTransformer, which also has the ability to model global dependencies, the CMDMamba based on Mamba not only demonstrates superior predictive performance but also optimizes the computational complexity to linear. When compared with SCINet, which has a convolutional module, CMDMamba can better capture global dependencies through its selective attention mechanism, showing significant performance advantages over temporal convolutional models. Compared with the MLP-based TiDE, CMDMamba employs a dual-layer Mamba framework that can adaptively enhance attention weights for recent abnormal fluctuations, while the static processing mode of the MLP lacks flexibility in responding to market dynamics. These findings indicate that Mamba, as an emerging deep learning framework, shows great potential and broad application prospects in financial sequence prediction by balancing computational efficiency and the ability to model global dependencies.

This study systematically evaluates the robustness of the CMDMamba model against noise interference in financial time series data, addressing the pervasive challenges of multidimensional game noise and market information friction. Experiments applied 50% and 100% Gaussian white noise to the HSI and S&P500 datasets, simulating variations in the signal-to-noise ratio (SNR) in real-world markets. The results (see Figure 10) show that CMDMamba outperforms Transformer-based architectures in both dynamic noise testing and cross-market validation. The model's dual-layer Mamba mechanism can effectively suppress noise while maintaining linear computational complexity. Compared to self-attention mechanisms, this approach reduces redundancy and achieves precise noise-signal separation in complex financial environments.

This study employs a univariate forecasting approach to systematically assess the impact of six key financial indicators on model performance across varying input sequence lengths L ∈ {12, 36, 58, 96}, as shown in Figure 11. The results indicate that the five price-related indicators demonstrate relatively consistent prediction accuracy and exhibit limited sensitivity to changes in input length. In contrast, trading volume presents significant variability. Owing to its high trading activity, pronounced volatility, and the absence of clear trends or structural stability, its prediction errors are notably sensitive to input sequence variations. Furthermore, to address the high degree of correlation among price-based features, each variable was modeled independently to evaluate changes in model performance when using only a single price indicator.