Multi-Scale Image Representation-Driven Structural Modeling and Non-Stationary Feature Decomposition for Financial Time Series

2.1 Overall framework design

The financial time series structural modeling and non-stationary feature decomposition framework proposed in this paper is an adaptive end-to-end architecture. Its core logic is to reconstruct one-dimensional financial time series into multi-channel image representations, relying on the technical advantages of the image processing field in feature extraction and global modeling. Through a series of specialized image processing operations, it achieves precise decoupling, effective modeling, and efficient fusion of non-stationary features in financial time series, and finally outputs reliable prediction results. The overall processing flow of the framework follows a progressive logic of raw financial time series input, multi-scale image generation, non-stationary feature decomposition and modeling, dynamic feature fusion and prediction. Each module cooperates and links to form a complete technical chain, ensuring full-process optimization from the original sequence to the prediction results. The core advantage of the framework lies in taking images as the core processing object throughout the entire process, deeply integrating image processing techniques with the modeling requirements of financial time series. It can not only fully exploit the inherent advantages of image processing techniques in complex pattern analysis and multi-scale feature capture to achieve precise characterization and global analysis of non-stationary features in financial time series, but also possesses strong architectural interpretability and adaptability to non-stationary scenarios. It can effectively cope with the inherent dynamic changes and complex structural characteristics of financial time series, providing a unified and efficient technical architecture for image-based modeling of non-stationary signals. The specific architecture is shown in Figure 1.

Figure 1. Framework structure of the proposed method

2.2 Multi-scale image generation module

The core innovation of the multi-scale image generation module lies in breaking through the limitations of existing single image transformation methods. It adopts a design strategy combining dual-path parallelism and multi-scale sampling to achieve dual visual encoding of temporal dependencies and frequency dynamics of financial time series. This design abandons the defect of traditional single mapping methods that cannot fully characterize non-stationary structures. Through two parallel image generation paths, it captures the temporal correlation characteristics and frequency evolution patterns of the sequence respectively. Combined with a multi-scale sampling mechanism that covers intraday, inter-day, and long-term time dimensions of market behavior differences, it finally generates a multi-channel image representation with both richness and specificity, providing solid visual information support for subsequent precise decomposition of non-stationary features in the image domain.

The dual-path image generation strategy is the core technical innovation of the module. It consists of two parallel paths: GAF encoding and CWT time-frequency image generation, which respectively achieve precise visualization of temporal dependencies and frequency dynamics. The GAF encoding path focuses on capturing the temporal dependency relationships of the sequence. First, min-max normalization is adopted to map the original financial time series to the interval [0, π/2], adapting to the polar coordinate angle range. The normalization formula is:

$x_i^{\prime}=\frac{x_i-\min (x)}{\max (x)-\min (x)} \times \frac{\pi}{2}$     (1)

Subsequently, each data point x_i′ in the normalized sequence x′=[x₁′,x₂′,…,x_n′] is mapped to polar coordinates (r_i,θ_i), where r_i is the normalized magnitude reflecting the data value, and θ_i=x_i′ corresponds to the angular feature of the time order. By computing the angle sum of any two time points i and j, θ_i+j=θ_i+θ_j, and using the cosine function to transform the angular correlation into pixel grayscale values, a GAF feature map of dimension R^n×n is generated. The pixel value calculation formula is:

$G(i, j)=\cos \left(\theta_{i+j}\right)=\cos \left(\theta_i+\theta_j\right)$     (2)

A grayscale value closer to 1 indicates a stronger correlation between the two time points. To enhance the recognizability of temporal dependency textures and adapt to subsequent convolutional layer feature extraction, a grayscale enhancement strategy is introduced to optimize the GAF feature map and further strengthen the feature representation capability of the image. The CWT path is used to supplement frequency dynamic information. Considering the transient fluctuation characteristics of financial time series, the complex Morlet wavelet [18] is selected as the basis function, expressed as:

$\psi(t)=\pi^{-1 / 4} e^{j w_0 t} e^{-t^2 / 2}$     (3)

where, ω₀=5 to balance time-frequency resolution. The original sequence x(t) is subjected to CWT to obtain the wavelet coefficients:

$W(a, b)=\int_{-\infty}^{+\infty} x(t) \psi^*\left(\frac{t-b}{a}\right) d t$     (4)

where, a is the scale parameter corresponding to frequency (smaller a indicates higher frequency), b is the translation parameter corresponding to time, and ψ^∗ denotes the complex conjugate of the wavelet basis function. The modulus value |W(a,b)| of the wavelet coefficients is normalized to [0,255] and converted into a CWT time-frequency spectrum image of dimension R^M×n. The pixel grayscale value corresponds to the intensity of the frequency component, achieving intuitive visualization of frequency dynamics. To ensure compatibility between the dual-path images, bilinear interpolation is adopted to resize the GAF feature map and the CWT time-frequency spectrum image to the same size, laying the foundation for subsequent multi-scale fusion and feature decomposition.

The multi-scale sampling mechanism is another important innovation of the module, aiming to accurately capture behavioral differences of financial markets at different time scales. An adaptive time-window sampling strategy is adopted, where the window size is adaptively adjusted by the sequence volatility σ. The window length is calculated as:

$L=k \times \sigma$     (5)

where, k is an empirical coefficient ranging from 3 to 5, ensuring that the window size can dynamically adapt to the fluctuation characteristics of the sequence and avoiding the defect that a fixed window cannot balance different fluctuation intensities. Based on this sampling strategy, the original financial time series is sampled with m different scales of windows. Under each scale, GAF feature maps and CWT time-frequency spectrum images are generated in parallel, finally obtaining m × 2 multi-scale and multi-perspective images. To achieve integrated input of multi-source image information, all multi-scale dual-path images are concatenated along the channel dimension to construct a multi-channel image input tensor:

$X \in \mathrm{R}^{C \times H \times W}$     (6)

where, C=2m is the number of channels corresponding to two types of images under m scales, and H and W denote the height and width of the image, respectively. This tensor integrates temporal dependency and frequency dynamic information under different scales, forming a multi-dimensional and comprehensive image representation, effectively improving the completeness and precision of subsequent image-domain feature decomposition.

Figure 2. Schematic diagram of Gramian Angular Field (GAF) time series-polar coordinate-image pixel mapping principle

Figure 2 shows the schematic diagram of the GAF time series-polar coordinate-image pixel mapping principle. Through the innovative design of dual-path parallelism and multi-scale sampling, the multi-scale image generation module successfully transforms one-dimensional financial time series into multi-channel image tensors rich in temporal and frequency information. It not only breaks through the limitations of traditional single image transformation, but also realizes multi-perspective visual encoding of non-stationary features. The image tensor generated by this module fully preserves the complex non-stationary structure of financial time series, accurately characterizes the temporal dependency relationships and frequency evolution patterns of the sequence, and provides high-quality input for the subsequent image-domain adaptive non-stationary feature decomposition network, ensuring that the decomposition process can accurately capture trend, periodic, and noise components in the sequence. This fully reflects the deep integration of image processing techniques and financial time series modeling, laying a data foundation for the performance improvement of the entire framework.

2.3 Non-stationary feature decomposition and modeling module

The non-stationary feature decomposition and modeling module is the innovative hub of the entire framework. Its core breakthrough lies in abandoning the traditional inherent process of “signal-domain decomposition → imaging” and constructing a new technical path of “direct decomposition in the image domain → feature modeling,” specifically addressing the key problem that traditional signal decomposition methods cannot be directly adapted to image inputs and have difficulty decoupling complex non-stationary components in images. This module realizes precise decoupling of trend, periodic, and high-frequency noise components in multi-scale images through a parameterized, end-to-end adaptive decomposition network. Combined with an improved Vision Transformer [19, 20] to strengthen global correlation modeling of multi-component image features, it can accurately match the non-stationary characteristics of financial time series and provide high-quality structured feature support for subsequent dynamic feature fusion.

The image-domain adaptive non-stationary feature decomposition network is the core innovation of this module. It adopts an encoder-decoder architecture design, specifically adapted to multi-channel image tensor input, to achieve end-to-end precise decoupling of three types of non-stationary components. The network input is the multi-channel image tensor X output from the multi-scale image generation module, and the outputs are the trend component map T, periodic component map P, and high-frequency noise component map N. The three strictly satisfy the component orthogonality constraint and reconstruction constraint, namely X=T+P+N. The encoder part adopts a 3-layer proprietary grouped convolution design. The number of groups is consistent with the input image channel number C. The convolution kernel size is set to 3 × 3, step is 1, and padding is 1. While retaining the image spatial structure to the greatest extent, it precisely extracts dedicated features of different channels. After each grouped convolution layer, a batch normalization layer and a LeakyReLU activation function are connected. The slope of LeakyReLU is set to 0.2, effectively alleviating the gradient vanishing problem and enhancing the nonlinear fitting capability of the network. The decoder part corresponds to the encoder output dimensions and adopts 3 layers of transposed convolution to gradually restore the image size. The last layer connects a sigmoid activation function to normalize the three component feature maps to the interval [0,1], ensuring that each component has clear physical meaning and interpretability. To retain multi-scale image detail information and improve decomposition accuracy, U-Net-style skip connections are introduced, concatenating the outputs of each encoder layer with the corresponding decoder layer inputs to achieve complementary fusion of detail features and global features. To ensure decomposition effectiveness, a composite decomposition constraint loss function is constructed to guide network training. The total decomposition loss is expressed as:

$L_{\text {decomp}}=\alpha \cdot L_{\text {rec}}+\beta \cdot L_{\text {orth}}+\gamma \cdot L_{\text {period}}$     (7)

where, $\alpha=1.0$, $\beta=0.5$, and $\gamma=0.3$ are weight coefficients determined by cross-validation to balance each constraint objective. The component orthogonality constraint loss L_orth ensures that the three component feature maps are independent and non-overlapping, expressed as: L_orth=||T·P^⊤||₂+||T·N^⊤||₂+||P·N^⊤||₂, where (⋅) denotes matrix dot product and || ||₂ denotes the L2 norm. The decomposition reconstruction loss L_rec adopts the L₁ norm design, expressed as: L_rec=||X−(T+P+N)||₁, which effectively reduces the risk of gradient explosion and improves image reconstruction accuracy. The periodic consistency loss L_period ensures that the periodic component is consistent with the periodic characteristics of the original sequence, expressed as: L_period=|f_P−f_x|, where f_P is the dominant period of the periodic component map and f_x is the dominant period of the original financial time series. The multi-channel image tensor X is input into the decomposition network, and network parameters are trained end-to-end by minimizing L_decomp through backpropagation. Finally, three types of multi-channel component feature maps T∈R^C×H×W, P∈R^C×H×W, and N∈R^C×H×W are output, completing precise decoupling of non-stationary features in the image domain. Figure 3 intuitively explains the mathematical principle of component reconstruction and orthogonality constraints in the image-domain decomposition network.

Figure 3. Mathematical principle of component reconstruction and orthogonality constraints in the image-domain decomposition network

The core innovation of multi-component feature modeling based on the improved Vision Transformer lies in optimizing the image patch partition strategy and self-attention mechanism, strengthening the global correlation modeling capability of multi-component image features while reducing computational complexity to adapt to the modeling requirements of multi-scale component features. Considering the global correlation characteristics and “time-frequency” semantic characteristics of component feature maps, the image patch partition strategy of Vision Transformer is improved. The three types of component feature maps T, P, and N are partitioned into image patches according to “time-frequency” semantics. The patch size is set to 4 × 4. Each image patch corresponds to a local time window and frequency interval of the original financial time series, ensuring that each image patch contains clear physical semantics and overcoming the limitation that traditional image patch partition lacks semantic specificity. In terms of the self-attention mechanism, a time-frequency attention weight factor is introduced to perform weighted adjustment of the original self-attention matrix of Vision Transformer, enhancing the feature correlation in the time and frequency dimensions. The adjustment formula is:

$A^{\prime}=A \cdot W_{t f}$    (8)

where, A is the original self-attention matrix and W_tf is the time-frequency weight matrix, generated by a fully connected layer, used to quantify the feature importance of different time and frequency dimensions. During the modeling process, the three types of component feature maps are input into the improved Vision Transformer separately, each outputting corresponding global feature vectors t∈R^D, p∈R^D, and n∈R^D, where D is the feature dimension. To preliminarily integrate multi-component features, component attention weights are introduced. The weight value is determined by the variance of each component feature vector. A larger variance indicates stronger effectiveness of the component feature and thus a higher weight. Through weighted aggregation, a preliminarily fused component feature vector F_decomp∈R^D is obtained, providing structured global component features for subsequent dynamic feature fusion.

Through the collaborative design of the image-domain adaptive decomposition network and the improved Vision Transformer, the non-stationary feature decomposition and modeling module realizes integrated processing of “precise decoupling-efficient modeling” of non-stationary features, fully reflecting the deep integration of image processing techniques and financial time series modeling. This module not only breaks through the technical bottleneck that traditional signal decomposition methods cannot be directly adapted to image inputs and achieves precise decoupling of three types of non-stationary components in multi-scale images, but also strengthens global correlation modeling of multi-component features through semantic image patch partition and weighted self-attention mechanism, effectively capturing feature interaction relationships in different time and frequency dimensions. The structured global feature vector it outputs not only retains the independent characteristics of each component, but also realizes preliminary integration of multi-component features, providing high-quality feature input for the subsequent dynamic feature fusion module and further improving the adaptability and structural analysis capability of the entire framework for the non-stationary characteristics of financial time series.

2.4 Dynamic feature fusion and prediction module

The core innovation of the dynamic feature fusion and prediction module lies in breaking through the limitations of existing static feature fusion mechanisms. In view of the heterogeneity of multi-source image features and multi-component features, as well as the fluctuation of feature importance caused by dynamic changes in financial market states, an attention-based gated fusion network is constructed to realize adaptive dynamic aggregation of multi-source heterogeneous features. At the same time, a lightweight prediction head and a global composite loss function are designed to complete end-to-end efficient prediction and collaborative optimization of the entire framework, taking into account model accuracy, efficiency, and robustness, and further strengthening the deep integration of image processing techniques and financial time series forecasting. Figure 4 shows the gradient backpropagation flow of the global composite loss function and the principle of end-to-end collaborative optimization.

Figure 4. Gradient backpropagation flow of the global composite loss function and principle of end-to-end collaborative optimization

The dynamic attention gated fusion network is the core technical innovation of this module. Its design core is to realize dynamic weight allocation of multi-source and multi-component features, adapting to the dynamic non-stationary characteristics of financial markets. The input of this network contains two types of heterogeneous features: one is the original features extracted by convolutional layers from multi-scale dual-path images, denoted as F_img∈R^C×D, where C is the number of channels and D is the feature dimension; the other is the multi-component global feature output by the improved Vision Transformer, denoted as F_decomp∈R^D. To realize effective fusion of the two types of features, a fully connected layer is first used to align their dimensions, uniformly mapping F_img and F_decomp to dimension D, obtaining aligned features F_img′∈R^C×D and F_decomp′∈R^D. Subsequently, a gating unit is designed to compute a dynamic weight matrix to achieve adaptive adjustment of the importance of each channel image feature. The gating unit is expressed as:

$G=\sigma\left(W_g \cdot\left[F_{i m g} ; F_{d e c o m p}\right]+b_g\right)$     (9)

where, σ is the sigmoid activation function, W_g and b_g are learnable parameters of the gating unit, and [;] denotes the feature concatenation operation. The output weight matrix is G∈R^C×1. The aligned image features F_img′ are multiplied element-wise with the dynamic weight matrix G to obtain weighted image features F_img,weighted∈R^C×D, realizing dynamic selection of different channel features. The weighted image features are further concatenated with the aligned multi-component global features, and global fusion weights are calculated through an attention mechanism to complete deep aggregation of multi-source features, obtaining the final joint feature F_joint∈R^D. To avoid model overfitting and improve the robustness of fusion features, a Dropout layer and a batch normalization layer are introduced during the fusion process. The Dropout rate is set to 0.3, effectively suppressing redundant feature interference and strengthening the generalization capability of fusion features.

The design of the lightweight prediction head and the global composite loss function achieves dual improvement in prediction efficiency and collaborative optimization of the entire framework, which is another important innovation of this module. The prediction head is constructed using a two-layer multilayer perceptron, taking into account prediction accuracy and computational efficiency. The first layer takes the joint feature F_joint as input, with output dimension set to D/2, and connects a ReLU activation function to enhance nonlinear fitting capability. The output dimension of the second layer is set to 1 or K according to prediction requirements, corresponding to future single time-step prediction or multi-step prediction of K time steps, respectively, and finally outputs the predicted value $\hat{y}$. To realize end-to-end collaborative optimization of the entire framework, a composite loss function is constructed that takes into account prediction accuracy, decomposition effectiveness, and fusion rationality. The total loss is expressed as:

$L_{\text {total}}=L_{\text {pred}}+\lambda \cdot L_{\text {decomp}}+\mu \cdot L_{\text {fusion}}$     (10)

where, $\lambda=0.4$ and $\mu=0.1$ are weight coefficients determined by cross-validation to balance different optimization objectives. The prediction error loss L_pred adopts root mean square error to measure, accurately quantifying the deviation between predicted values and true values. It is expressed as:

$L_{\text {pred}}=\sqrt{\frac{1}{N} \sum_{i=1}^N\left(\hat{y}_i-y_i\right)^2}$      (11)

where, y_i is the true value of the original financial time series, $\hat{y}_i$ is the corresponding predicted value, and N is the number of samples. L_decomp follows the composite decomposition constraint loss of the image-domain decomposition network described above, ensuring the effectiveness of feature decomposition. The fusion regularization loss L_fusion is used to constrain the sparsity and correlation of fusion features, avoiding feature redundancy and fusion bias. It is expressed as:

$L_{\text {fusion }}=\left\|W_{\text {fusion }}\right\|_1+\left\|F_{\text {joint }} \cdot F_{\text {joint }}^{\top}-I\right\|_2$     (12)

where, W_fusion is the global fusion weight matrix, I is the identity matrix, || ||₁ is the L1 norm, and || ||₂ is the L2 norm.

Through collaborative innovation of dynamic attention gated fusion, lightweight prediction head, and global composite loss function, the dynamic feature fusion and prediction module effectively solves key problems such as insufficient fusion of multi-source heterogeneous features, inability of static fusion to adapt to dynamic market changes, and difficulty in balancing prediction efficiency and accuracy. This module not only realizes adaptive dynamic aggregation of multi-scale image features and multi-component decomposition features, improving the robustness and specificity of feature utilization in non-stationary scenarios, but also ensures synchronous improvement of prediction accuracy, decomposition effectiveness, and fusion rationality through end-to-end collaborative optimization of the entire framework. Its lightweight design takes into account model efficiency and practicality, and the dynamic fusion mechanism fully fits the non-stationary characteristics of financial time series, further improving the integrated technical chain of “image generation-feature decomposition-dynamic fusion-prediction,” providing core support for the high-performance performance of the entire framework, and enriching the application ideas of image processing techniques in feature fusion and cross-domain prediction.

The multi-scale image-driven financial time series structural modeling and non-stationary feature decomposition framework proposed in this paper achieves multiple technological breakthroughs in the interdisciplinary field of image processing and time series analysis, and has important academic value and application potential. The multi-scale dual-path image generation strategy breaks through the limitation of traditional single image transformation and provides a new idea for the image-based representation of non-stationary signals. Through dual visual encoding of temporal dependency and frequency dynamic, it solves the key problem that a single image cannot fully characterize the multi-dimensional characteristics of non-stationary signals. This strategy is not only applicable to financial time series, but can also be extended to other non-stationary signals such as electrocardiogram and meteorology, enriching the technical system of image-based representation of non-stationary signals. The image-domain adaptive non-stationary feature decomposition network breaks the separation between traditional signal decomposition and image processing, and innovatively proposes a new paradigm of direct decomposition in the image domain. Through customized constraint losses and encoder-decoder architecture design, it achieves precise decoupling of complex components in multi-scale images, further enriching the research results of image processing technology in the field of feature decoupling. At the same time, the framework deeply integrates advanced image processing technologies such as Vision Transformer and dynamic attention fusion with financial time series modeling, constructing an end-to-end collaborative optimization architecture of “image generation-feature decomposition-dynamic fusion”. It provides a reusable new idea and new solution for the application of image processing technology in cross-domain non-stationary signal analysis, effectively expanding the application boundary of image processing technology and promoting technological integration and innovation in interdisciplinary fields.

Compared with existing image processing related research, the proposed method shows significant advantages in multiple dimensions, further highlighting the innovation and pertinence of this research. Compared with existing time series image-based methods, this paper does not simply perform time series to image transformation and feature extraction, but realizes deep integration of multi-view image information through multi-scale sampling and dual-path fusion strategy, and combines image-domain feature decomposition technology to fully utilize the advantages of image processing technology in complex pattern analysis, improving the utilization efficiency of non-stationary features. Compared with existing image-domain feature decomposition methods, the decomposition network designed in this paper is specifically optimized for the dynamic characteristics of non-stationary signals. By introducing customized constraint losses such as component orthogonality and periodic consistency, it not only improves decomposition accuracy, but also ensures clear physical meaning of each component. More importantly, it can achieve end-to-end collaborative optimization with subsequent feature modeling and dynamic fusion modules, forming an integrated technical chain, and solving the problem of insufficient adaptability between existing decomposition methods and modeling processes. Compared with the application of general Vision Transformer in image processing, this paper strengthens the pertinence of global feature modeling by improving image patch partition and self-attention mechanism and combining the “time-frequency” semantic characteristics of financial time series. It effectively avoids the overfitting problem that general Vision Transformer is prone to in small-sample and non-stationary scenarios, and improves the robustness of the model in complex scenarios.

Although the proposed framework shows excellent performance in experiments, there are still certain limitations, which point out clear directions for subsequent research. The window size of multi-scale image generation currently relies on empirical coefficient adjustment, and its adaptive adjustment ability needs to be further improved. It is difficult to fully adapt to financial time series with different fluctuation characteristics. In the future, a multi-scale window optimization method based on adaptive evolutionary algorithm will be studied to realize dynamic adaptive adjustment of window size and improve the autonomy and adaptability of image generation. The image-domain decomposition network adopts multi-layer grouped convolution and transposed convolution structure, and its computational complexity is relatively high, which makes it difficult to directly adapt to real-time processing requirements of ultra-large-scale financial time series. In the future, lightweight technologies such as distillation learning will be combined to design an efficient and simplified image-domain decomposition network, reducing computational cost while ensuring decomposition accuracy. In addition, the current framework is only designed for univariate financial time series and does not consider the correlation characteristics among variables in multivariate series. Future work will focus on exploring multi-channel image generation and fusion methods for multivariate financial time series to achieve accurate analysis of multivariate non-stationary features. At the same time, the application scenarios of the framework will be further expanded to other non-stationary signal processing fields, promoting the in-depth application of image processing technology in cross-domain non-stationary signal analysis and providing stronger technical support for interdisciplinary research and development.