COGITO Smart Journal Ae Vol.
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December 2025.
P-ISSN: 2541-2221.
E-ISSN: 2477-8079
ISSN: 8-1520
Forecasting the Highest and Lowest Prices in Financial Markets Using a VMD-LSTM Hybrid Model I Made Adi Purwantara*1.
Kusrini2.
Arief Setyanto3.
Ema Utami4 Department of Informatics Doctorate.
Universitas Amikom Yogyakarta.
Indonesia e-mail: *1adipurwa@students.
id, 2kusrini@amikom.
id, 3arief@amikom.
u@amikom.
1,2,3,4
Abstract Accurate forecasting of the lowest and highest prices in financial markets poses a considerable challenge due to the inherent nonlinear behaviour, non-stationarity, and high noise levels of financial time series data.
Most prior studies focus only on closing prices, with limited attention to the simultaneous prediction of high and low prices.
Yet, predicting the lowest and highest prices is essential for investors to make informed trading decisions.
To address this gap, this study proposes a hybrid DL framework that integrates VMD and LSTM networks for predicting daily high and low prices simultaneously.
This study used 12 years of daily data from three diverse assets: AUD/USD.
TLKM, and XAU/USD.
The data underwent preprocessing.
VMD-based decomposition, and were input into the LSTM.
The dataset was split 80% for training and 20% for testing.
Experiments varied the number of decomposition modes (K = 7, 10, .
and sliding window sizes .
, 15, 30, 45, 60, .
Results show that the VMD-LSTM model exhibits improved performance in most of the tested scenarios compared to traditional LSTM.
These findings underscore that the use of VMD decomposition can help enhance the accuracy of forecasting the highest and lowest prices in the financial market.
KeywordsAi VMD.
LSTM.
Decomposition.
Deep Learning.
Financial Market
INTRODUCTION
Recent advancements in machine learning (ML) and deep learning (DL) have significantly improved the forecasting accuracy of financial time series.
Studies consistently show that ML and DL approaches outperform traditional statistical modelsAisuch as ARMA.
ARIMA.
VAR, and GARCHAiin predicting market behavior .
, .
, .
These traditional models are often constrained by rigid statistical assumptions, making them less effective in capturing the nonlinear and multiscale patterns inherent in real-world financial data .
contrast, artificial intelligence-based models, particularly DL methods, overcome these limitations by adapting to nonlinearity and non-stationarity in financial signals .
The superior performance of ML over classical models like the Random Walk and ARIMA has established it as a powerful tool for financial forecasting .
Among DL architectures.
Long Short-Term Memory (LSTM) networks are widely recognized for their ability to model complex dynamics and long-range dependencies in sequential .
, and have become increasingly popular in stock price prediction tasks .
While many statistical and intelligent models have been developed, relying solely on a single model often proves insufficient in effectively denoising financial data .
The introduction of neural network-based architectures such as LSTM has greatly enhanced model accuracy .
, leveraging past market activity to make informed predictions .
The LSTM model demonstrated superior performance compared to the SVR model in prediction accuracy .
] and remains one of the most effective DL methods for learning from sequential time series data .
, particularly in the financial domain .
, .
, .
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Nevertheless, predicting financial market prices remains a major challenge due to the strong presence of nonlinearity, non-stationarity, and noise .
, .
, .
, .
, .
This highlights the need for a robust denoising technique during preprocessing before applying DL models .
Several signal decomposition methods have been used, including Empirical Mode Decomposition (EMD), wavelet transform, and VMD .
While wavelet transform requires prior identification of signal componentsAioften resulting in inaccurate frequency analysisAi adaptive methods like EMD and VMD allow data-driven decomposition based on local signal characteristics .
However.
EMD suffers from issues such as mode mixing and lacks solid theoretical foundations .
To address these limitations.
Dragomiretskiy and Zosso .
introduced VMD as a theoretically grounded and non-recursive decomposition technique with strong robustness to noise and sampling .
VMD has demonstrated strong performance in enhancing feature extraction and reducing noise in non-stationary time series, including financial market data.
It decomposes complex sequences into simpler, more stationary Intrinsic Mode Functions (IMF.
, making meaningful patterns more accessible to predictive models .
, .
, .
Recent studies have begun to explore hybrid forecasting frameworks that combine signal decomposition methods like VMD with DL models to enhance prediction accuracy .
The integration of signal decomposition and AI models significantly impacts forecasting performance .
, .
, .
For example, the CNNTLSTM hybrid model successfully predicted the next-day closing price of the USD/CNY exchange rate .
Likewise, a hybrid LSTM-DNN model produced consistent and accurate stock price predictions across multiple datasets .
More refined techniques, such as GA-VMD, further optimize the number of decomposed components using genetic algorithms, offering a deeper understanding of financial signals .
, .
Most of the existing studies have focused on predicting opening and closing prices .
, .
, .
, .
, .
, .
, .
, .
, .
However, .
, .
argue that the highest and lowest prices provide a more comprehensive representation of market characteristics and risk levels compared to closing prices.
Forecasting these price extremes is crucial for effective risk management and offers critical insights to both market participants and regulators .
Despite their importance, few studies address the simultaneous prediction of both highest and lowest prices in financial time series.
This paper fills that gap by proposing a hybrid deep learning modelAiVMD-LSTMAidesigned to forecast both the daily high and low prices VMD is used to decompose the raw price series into a set of IMFs, reducing noise and isolating multiscale patterns.
These refined components are then used as inputs for the LSTM network, which captures long-term dependencies in the data.
This dual-stage design leverages the strengths of both VMD and LSTM to provide more accurate, reliable forecasts.
This study makes four key contributions: .
it presents a unified VMD-LSTM model for simultaneous prediction of high and low prices, offering richer insights than single-target models.
it leverages VMDAos denoising and multiscale decomposition capabilities to improve input .
it applies LSTMAos sequential learning capabilities to capture temporal dynamics.
it validates the model using three diverse datasetsAiAUD/USD .
TLKM .
, and XAU/USD .
Aidemonstrating the modelAos adaptability and robustness across Practically, this approach offers substantial value to stakeholders.
For traders and investors, accurate high and low price forecasts improve entry and exit strategies.
For regulators and policymakers, they offer better visibility into market volatility and systemic risk .
, .
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E-ISSN: 2477-8079
RESEARCH METHODS
Data Collection To ensure the robustness and real-world relevance of the developed forecasting model, this study uses historical time series data from three different financial asset classes, each representing unique market characteristics.
The assets were carefully selected to encompass
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variations in volatility levels, market participant types, and trading volume.
This strategy aims to test the model's generalizability across a range of complex and dynamic market conditions.
The three assets used in this study are:
AUD/USD, a currency pair from the foreign exchange .
TLKM, a telecommunications company stock from the Indonesian domestic stock XAU/USD, a gold commodity traded in US dollars on the global market.
These three assets were selected based on three main criteria:
High volatility, to evaluate the model's resilience to extreme price fluctuations.
Availability of complete and continuous daily historical data.
Representation of different types of financial markets .
orex, stock, and commodity market.
to allow for testing the model's performance in different economic contexts.
All datasets were retrieved from Investing.
com, a widely used and reliable financial data The data span a comprehensive 12 years, from early 2013 to the end of 2024.
leveraging real and high-resolution daily data over a long horizon, this study ensures both temporal depth and diversity in the input sequences used for model development and testing.
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Data Pre-Processing To prepare the datasets for effective modeling, a structured data pre-processing workflow was implemented.
Missing Value Handling Each dataset was first cleaned to handle missing values and ensure time alignment across the Outlier Detection and Adjustment Outliers were identified and smoothed where necessary to avoid distortion in model Data Normalization All data were scaled using Min-Max normalization to transform the price values into a standardized range between 0 and 1.
Data Splitting An 80:20 ratio was employed to divide the normalized dataset into training and testing The initial 80% was utilised for training purposes, while the remaining 20% was set aside to assess the model's performance.
This temporal split guarantees that the model is evaluated on unseen data, reflecting real-world forecasting scenarios where future prices must be predicted based solely on past observations.
Sliding Window Formation To generate time series inputs that can be utilized by the model, a sliding window approach is used.
Several window lengths are used, including 5, 15, 30, 45, 60, and 90 days, to capture both short-term and long-term price patterns.
This technique also enriches the feature representation provided to the model, thereby improving accuracy and generalization Varational Mode Decomposition (VMD) VMD algorithm proposed by .
VMD was developed as a non-recursive and theoretically grounded approach that decomposes signals into adaptive modes simultaneously, offering greater robustness to noise and sampling compared to previous methods .
Let the input signal be f t , and assume we want to decompose it into K modes ya .
}ya yco=1 with corresponding center frequencies{Oyco }yco=1 .
Each mode is expected to be bandlimited after demodulation to baseband.
COGITO Smart Journal Ae Vol.
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E-ISSN: 2477-8079
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ya OeycO min {Ocyco=.
OCyc .
UI yce cyco },{Oyco } Subject to Ocyayco=1yc yco yc = yce yc This formulation minimizes the total bandwidth of all modes by penalizing the gradient of each demodulated mode in time .
hich corresponds to concentration in frequenc.
The constraint ensures that the sum of the modes reconstructs the original signal.
To solve this constrained optimization, an augmented Lagrangian approach is applied:
ya ya ya Ee(.
cyco }, {Oyco }, ) = O.
OCyc \ycaycnyci.
UI yce OeycOycoyc\ ycaycnyc.
Oe Oc ycyco .
, yce.
Oe Oc ycyco .
yco=1 yco=1 yco=1 .
Where:
A yu is a penalty parameter controlling the bandwidth constraint.
A yc is the Lagrange multiplier In this study.
VMD was used to decompose the daily high and low price time series into IMFs.
These IMFs were then used as multivariate inputs for an LSTM model.
The VMD parameters used in this experiment were as follows:
Number of modes (K): 7, 10, and 12 Convergence tolerance (A): 1e-7 This value is used as the stopping criterion in the VMD iterative process.
Alpha penalty (): 2000 This is the default value used to maintain consistency between modes.
Initialized center frequencies (Ini.
: 1 Represent the starting frequency estimates around which each mode is formed.
The choice of VMD over other decomposition methods such as Wavelet and EMD is based on the following advantages:
Robustness to noise VMD exhibits stable denoising performance even on highly volatile financial signals.
Avoids mode mixing Unlike EMD, which often produces mixed modes.
VMD explicitly limits the bandwidth of each mode .
Robust mathematical formulation VMD is built on variational theory, making it more controllable and analytically tractable .
Compatibility with Deep Learning VMD decomposition results are well-suited for use as input features in DL architectures such as LSTMs, as they simplify temporal complexity and improve prediction accuracy .
LSTM (Long Short-Term Memor.
LSTM is a specialized type of RNN intended to learn long-term dependencies and address the limitations of traditional RNNs, especially issues associated with vanishing and exploding gradients .
LSTM is a widely used and highly effective DL approach within RNNs for time series and sequence prediction tasks .
, .
, .
LSTM introduces a memory cell and a series of gates that control the flow of information, enabling it to retain relevant information over extended time steps.
The architecture of LSTM is shown in Figure 1.
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Figure 1.
LSTM Architecture Each LSTM cell contains three core gates: input, forget, and output gates .
, which manage the retention, update, and transmission of information across time steps.
Let ycuyc be the input vector at time step yc, yaycOe1 the previous cell state, and EaycOe1 the hidden state from the previous time step.
The computations of an LSTM cell are defined as follows:
Forget Gate Determines which information to remove from cell state:
yceyc = yua ycOyce UI Ea ycOe1 , ycuyc ycayce .
Input Gate Updates the cell state with new candidate values:
ycnyc = yua ycOycn UI Ea ycOe1 , ycuyc ycaycn yayc = tanh ycOya UI EaycOe1,ycuyc ycaya Cell State Update Combines the previous cell state and new information:
yayc = yceyc O yaycOe1 ycnyc O yayc Output Gate Determines the next hidden state:
ycuyc = E ycOycu UI EaycOe1,ycuyc ycaycu Eayc = ycuyc O tanh yayc In this study, the LSTM model is applied to learn temporal patterns from historical financial time series data to predict the next-day lowest and highest prices of a financial market.
The sequential nature of LSTM enables the model to capture both short-term and long-term dependencies in the price fluctuations, providing a robust framework for accurate forecasting in volatile financial environments.
The Proposed Model (VMD-LSTM) This study proposes a hybrid DL framework combining VMD and LSTM networksAi referred to as VMD-LSTMAito simultaneously predict daily lowest and highest prices in financial This approach is motivated by the need to enhance the model's ability to capture both the nonlinear and non-stationary characteristics of financial time series, which are often difficult to model using conventional methods.
VMD functions as an effective signal decomposition technique that breaks down a time series into a collection of IMFs, each capturing oscillatory behavior within distinct frequency ranges.
These IMFs reflect different underlying components in the original data, such as trend, noise, or short-term fluctuations.
By decomposing the input data before training the model.
VMD helps isolate meaningful features and reduce noise, allowing the LSTM network to focus on relevant temporal patterns.
LSTM networks are then employed to learn the sequential dependencies within the decomposed signals.
Their internal memory mechanism allows them to retain valuable historical information, making them particularly effective for time series prediction.
This hybrid framework leverages the strengths of both VMD and LSTM: VMD enhances signal clarity, while LSTM captures long-term temporal COGITO Smart Journal Ae Vol.
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The proposed model architecture, as shown in Figure 2, follows these sequential steps:
Input Data The process begins with financial time series data of daily highest and lowest prices from the AUD/USD.
TLKM, and XAU/USD datasets.
VMD Decomposition Using VMD, the input sequence is separated into multiple IMFs, each corresponding to a unique frequency element of the initial time series.
Concatenation of IMFs The resulting IMFs are concatenated in a predefined order .
, from highest to lowest frequenc.
, forming a new input sequence enriched with decomposed features.
LSTM Layer The concatenated sequence is fed into the LSTM layer, which learns the temporal patterns and dependencies within the multi-resolution data.
Dense Layer The LSTM layer's output is subsequently fed into one or more fully connected .
layers, which transform the learned features into the target prediction domain.
Output Layer The final prediction consists of the next-day highest and lowest prices, estimated from the processed temporal signals.
Figure 2.
VMD-LSTM Model Architecture Figure 3.
VMD-LSTM Model Technical Configuration To provide a comprehensive overview of the proposed prediction system.
Figure 3 presents the block diagram of the technical configuration of the VMD-LSTM model.
This diagram outlines the end-to-end process, including preprocessing, signal decomposition via VMD, sequence construction using sliding windows, model architecture, and training parameters.
This hybrid VMD-LSTM model is designed to improve forecasting accuracy by integrating signal decomposition and deep sequential learning in a unified framework.
The layered architecture ensures that both low-level fluctuations and high-level trends are captured effectively, enabling the model to adapt to the complex nature of financial markets.
The LSTM model was constructed using a sequential architecture, starting with an input layer derived from the VMD decomposition signal.
All IMFs of the highest and lowest prices are arranged and consolidated into a multivariate input.
The model architecture consists of one LSTM layer with 50 hidden units, which is responsible for capturing temporal dependencies in financial time series data.
After the LSTM, a Dense output layer with two units is used to simultaneously predict the highest and lowest prices.
This model was compiled using the Adam optimizer, a widely used adaptive gradient-based optimization algorithm known for its efficiency in training deep learning models.
The training process was conducted over 100 epochs with a batch size of COGITO Smart Journal Ae Vol.
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Evaluation Metrics To evaluate the predictive performance of the proposed model in forecasting the highest and lowest prices in financial markets, several widely accepted statistical metrics are employed.
In this study, the performance evaluation is conducted using MSE.
RMSE.
MAE.
MAPE, and RA.
Each metric captures different aspects of prediction quality, and using a combination of them ensures a comprehensive assessment of the modelAos forecasting capability.
MSE (Mean Squared Erro.
By averaging the squared differences between predictions and actual outcomes .
MSE places greater emphasis on larger errors, thereby increasing its sensitivity to outliers.
MSE = ycu Ocycuycn=1.
cycn Oe ycCycn )2 .
RMSE (Root Mean Squared Erro.
As the square root of MSE.
RMSE expresses error in the predicted variableAos unit, allowing for clearer interpretation .
RMSE = Oo Ocycuycn=1.
cycn Oe ycC) ycn MAE (Mean Absolute Erro.
MAE measures the mean of the absolute deviations between predicted and actual values .
In contrast to MSE, it assigns equal weight to all errors and demonstrates greater robustness to outliers.
MAE = ycu Ocycuycn=.
ycycn Oe ycC| ycn MAPE (Mean Absolute Percentage Erro.
As a percentage-based metric.
MAPE allows consistent evaluation of prediction accuracy across datasets with different value ranges .
MAPE = 100\% ycu yc OeycC Ocycn=1 | ycn yc.
RA (Coefficient of Determinatio.
The RA value spans from 0 to 1, where 0 indicates that the model fails to explain any variance in the target variable and 1 indicates perfect prediction .
Ocycu .
c OeycC )2 ycn ycI 2 = 1 Oe Ocycn=1 ycu .
c OeycI)2 ycn=1 ycn These metrics collectively offer insights into the model's effectiveness regarding accuracy, error magnitude, relative error, and explanatory strength.
By analyzing these indicators, the reliability and generalization ability of the proposed VMD-LSTM model for forecasting financial market prices can be effectively validated.
To determine the effect of applying the VMD decomposition method, the performance of the proposed model was compared with that of a baseline model .
single LSTM) across various window size parameters and K values.
To determine whether the effect was significant, a statistical test was performed using a paired t-test between the models.
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RESULT AND DISCUSSION
Decomposition Results The selection of the number of modes (K) in VMD is a critical factor that directly influences the quality of the decomposition process and the subsequent predictive performance of the model.
Since VMD requires the value of K to be predefined .
, an inappropriate choice can degrade model effectiveness.
A small K may lead to under-decomposition, conversely, a value of K that is too large can cause over-decomposition .
To address this.
Guo et al.
conducted empirical evaluations using K values ranging from 7 to 13 and found that models with K = 9, 10, and 12 yielded superior forecasting accuracy compared to other configurations .
Informed by these findings, this study explores K values of 7, 10, and 12 to identify effective decomposition settings across multiple financial time series.
Figure 4 presents a comprehensive visualization of the VMD decomposition results for three distinct financial datasets: AUD/USD.
TLKM, and XAU/USD, with varying values of the mode parameter K set to 7, 10, and 12.
Each image illustrates the resulting modes for both the highest and lowest price series, offering a detailed view of how different values of K influence the granularity and frequency separation of the extracted components.
This visual representation serves as a foundational step for understanding the intrinsic structures within each dataset, enabling a deeper exploration of price dynamics through signal decomposition.
Each subplot illustrates one of the IMFs.
This decomposition aims to separate the original signals into distinct frequency bands, enabling a clearer representation of hidden patterns and reducing noise prior to modeling.
By isolating these components, the model gains access to more structured and meaningful features that may enhance forecasting performance.
By examining these multi-resolution modes, researchers and practitioners are better equipped to design more adaptive and insightful forecasting models that can accommodate the unique characteristics of diverse financial instruments.
This structured breakdown helps in uncovering latent temporal dynamics that may be otherwise obscured in the raw data, facilitating a more nuanced analysis of market behavior.
Figure 4.
VMD Results on Dataset Upon closer inspection, increasing the number of modes leads to a finer separation of signal components.
For instance, when K=7, most of the informative structure appears concentrated in the first few modes, while the remaining modes carry residual noise.
As K COGITO Smart Journal Ae Vol.
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increases to 10 and 12, the later modes become increasingly granular and attenuated, suggesting that the additional components capture less significant, high-frequency fluctuations or noise.
This observation underlines the importance of selecting an optimal K, where too few modes risk underrepresentation of complex features, while too many introduce redundancy and potential Training Performance Evaluation Table 1 presents an evaluation metric of model performance during the training phase across three financial datasets: AUD/USD.
TLKM, and XAU/USD.
These evaluations compare the conventional LSTM model with the VMD-LSTM hybrid model across different configurations of mode decomposition (K) and window sliding parameters.
Each configuration is assessed using several performance metrics: RMSE.
MSE.
MAPE.
MAE, and RA, offering a multidimensional perspective on prediction accuracy and consistency.
By providing a systematic breakdown of these results, the tables serve as a foundational reference for understanding the impact of VMD pre-processing and parameter tuning on model learning behavior across diverse market characteristics.
This comparison allows for a deeper understanding of how decomposition and temporal windowing impact model performance.
Table 1.
Training Performance Evaluation AUD/USD Dataset
Model
LSTM
LSTM
LSTM
LSTM
LSTM
LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
Dataset
Sliding Window
AUD/USD
RMSE
0,004442515
0,004031258
0,004208575
0,004177324
0,00398611
0,004096135
0,002409807
0,002327581
0,002360489
0,002125096
0,002261227
0,002297384
0,001965094
0,002372911
0,001861668
0,002362396
0,001904868
0,001825123
0,002322122
0,00173972
0,001815474
0,001745456
0,001775374
0,001785629
TLKM
RMSE
53,71704294
54,15265478
54,60715476
55,41750242
53,99657907
57,45842819
25,22548282
28,41748175
26,43825719
28,35296239
27,9316685
28,77239484
21,78152356
21,67915819
19,73209835
20,46459545
21,70196694
23,24276674
19,09675989
17,04845191
17,99433365
17,29201142
14,76778659
15,76554868
XAU/USD
RMSE
12,19181067
11,97839877
11,69768836
12,13575095
12,70419147
11,28761051
6,378307003
6,306351768
6,948995959
6,633423255
6,193917283
6,154127654
5,697663474
6,399940615
5,568940101
4,99930748
5,085203781
6,579942954
5,94250028
5,008176365
4,823725969
5,198055305
5,03297987
4,722103917
The comparative evaluation of the LSTM and VMD-LSTM models across the AUD/USD.
TLKM, and XAU/USD datasets suggests a general trend in which the integration of VMD tends to improve predictive accuracy.
In the AUD/USD dataset, the VMD-LSTM model with K=12 and a sliding window of 15 consistently achieved the lowest error rates.
MSE .
,02662E-.
RMSE .
,00173.
MAE .
,001341.
, and MAPE .
,172350.
, along with the highest R-squared .
,99962.
Similarly, for the TLKM dataset, the model with K=12 and window size 60 produces the lowest MSE .
,0875.
RMSE .
,76778.
MAE .
,34078.
, and MAPE .
,344460.
, with a strong RA value of 0,999568616.
Meanwhile, for the XAU/USD dataset, the model with K=12 and window size 90 produces the lowest MSE .
,29826.
RMSE .
,722103.
MAE .
,55485.
, and MAPE .
,247810.
, with a 304 COGITO Smart Journal Ae Vol.
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strong RA value of 0,999659819.
These findings illustrate the effectiveness of VMD in decomposing complex and erratic patterns into more interpretable subcomponents, which enhances the LSTMAos ability to generalize across highly volatile time series.
Notably, the results also suggest that during model training, the VMD-LSTM configuration with K=12 consistently yields the best predictive performance among the tested settings.
Testing Performance Evaluation Table 2 presents an evaluation metric of model performance during the testing phase for three different financial datasets: AUD/USD.
TLKM, and XAU/USD.
This table systematically summarizes the performance metrics of two predictive modelsAistandard LSTM and the hybrid VMD-LSTMAiacross various values of K .
umber of decomposed mode.
and window sizes.
For each configuration, performance is evaluated using five key indicators: RMSE.
MSE.
MAPE.
MAE, and RA.
This structured presentation allows for a clear and consistent comparison of how well each model captures the patterns within different market instruments under varying experimental setups, setting the stage for a deeper analysis of predictive accuracy and generalization capability.
The evaluation results across the three datasetsAiAUD/USD.
TLKM, and XAU/USDAi indicate that the VMD-LSTM hybrid model generally performs better than the conventional LSTM model under the tested conditions.
For the AUD/USD and TLKM datasets, the best performing configurations are VMD-LSTM with K=12 and a sliding window of 60.
For the AUD/USD dataset, the VMD-LSTM with K=12 and a sliding window of 60 achieves the lowest MSE .
,07056E-.
RMSE .
,001438.
MAE .
,001097.
, and MAPE .
,165546.
, along with the highest R-squared .
,99313.
Similarly, for the TLKM dataset, the model with K=12 and window size 60 produces the lowest MSE .
,3456.
RMSE .
,54051.
MAE .
,64432.
, and MAPE .
,29567.
, with RA value of 0,999360302.
Meanwhile, in the XAU/USD dataset, the configuration with K=12 and window size 45 results in the lowest MSE .
,4699.
RMSE .
,0195.
MAE .
,326852.
, and MAPE .
,42259.
, with the highest R-squared of 0,998409082.
Table 2.
Testing Performance Evaluation AUD/USD Dataset
Model
LSTM
LSTM
LSTM
LSTM
LSTM
LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
VMD-LSTM
Dataset
Sliding Window
AUD/USD
0,939114227
0,956511775
0,948308884
0,95039475
0,954434364
0,953293655
0,981232805
0,986102654
0,985834085
0,98628833
0,985942706
0,985152042
0,98548931
0,988471665
0,989985191
0,989317073
0,989078113
0,990840136
0,99274339
0,991651702
0,985636443
0,989352367
0,99313337
0,989701724
TLKM
0,991540903
0,991475363
0,990575183
0,990113165
0,991498244
0,990628635
0,997820724
0,996644742
0,997940737
0,997064178
0,99739871
0,997715056
0,997630342
0,99813777
0,998825598
0,998728667
0,998272042
0,998454507
0,997574532
0,998714009
0,998718297
0,99922435
0,999360302
0,998766653
XAU/USD
0,996798535
0,996829899
0,989006788
0,996481738
0,993272673
0,993566515
0,990755804
0,993941185
0,99817653
0,997109787
0,989493295
0,992981576
0,994046246
0,996070642
0,997370753
0,997693018
0,996357696
0,997159577
0,994603621
0,995828544
0,993816056
0,998409082
0,988004503
0,994646076
COGITO Smart Journal Ae Vol.
No.
December 2025.
P-ISSN: 2541-2221.
E-ISSN: 2477-8079
ISSN: 8-1520
This suggests not only a potential improvement in predictive precision but also a more stable model fit, highlighting the advantages of decomposing complex price signals into IMFs prior to time-series modeling.
In comparison, the standard LSTM models tend to produce lower RA scores and higher error values, which may reflect the challenges of modeling raw, unprocessed sequences when capturing detailed price dynamics.
These findings are further illustrated in Figure 5.
Figure 6, and Figure 7, which clearly demonstrates the superior and more consistent R-squared performance of the VMD-LSTM models across different sliding window sizes.
Figure 5.
RA Performance on the AUD/USD Dataset Figure 6.
RA Performance on the TLKM Dataset Figure 7.
RA Performance on the XAU/USD Dataset These findings provide supportive evidence for the role of VMD in improving LSTMAos ability to learn from multi-scale features in financial data.
By decomposing the original series into orthogonal sub-signals.
VMD may help the LSTM process cleaner and more informative inputs, potentially contributing to reduced prediction errors.
Nonetheless, the results also show that model performance is sensitive to the choice of decomposition parameters such as the number of modes (K) and window size.
Therefore, while the hybrid approach appears promising, careful tuning of hyperparameters remains essential for achieving optimal performance.
However, the testing results on the XAU/USD dataset indicate that the VMD-LSTM model does not consistently outperform the baseline LSTM, particularly under configurations with shorter sliding windows .
, where the conventional LSTM exhibits better predictive This discrepancy may be due to the non-stationarity of the XAU/USD series, which poses greater challenges for modeling.
Nonetheless, the VMD-LSTM model achieves better performance across most other sliding window configurations.
The experimental results indicate that the sliding window size is vital in determining the predictive performance of both LSTM and VMD-LSTM models across different financial time On the AUD/USD and TLKM datasets, which exhibit relatively stable and periodic behavior, window sizes of 60 yielded the best performance, particularly for VMD-LSTM with K = 12.
Conversely, on the XAU/USD dataset, a shorter window .
, window = .
performed better for VMD-LSTM with K = 12.
This study reaffirms that window size must be aligned with the temporal granularity of the signal, where medium windows are optimal for detecting seasonal patterns, and shorter windows offer better responsiveness in environments with rapid shifts.
Thus, sliding window tuning should be considered a critical hyperparameter in the design of hybrid DL forecasting frameworks for financial applications.
COGITO Smart Journal Ae Vol.
No.
December 2025.
P-ISSN: 2541-2221.
E-ISSN: 2477-8079
ISSN: 1978-1520
To assess whether the integration of VMD into the LSTM model significantly enhances predictive performance, we conducted a paired T-test.
This statistical method compares the prediction accuracy of the baseline LSTM model to the VMD-LSTM model across multiple commonly used performance metrics.
The comparison was carried out on three different financial time series datasets, namely AUD/USD.
TLKM, and XAU/USD.
The results, including p-values and interpretations, are presented as Table 3 below.
Table 3.
Paired T-Test Results AUD/USD TLKM
XAU/USD
Metric
p-value Interpretation Metric
p-value Interpretation Metric
p-value Interpretation MSE
2,24E-01
Significant MSE
1,06E-05
Significant MSE
Not Significant RMSE
2,02E-03
Significant RMSE
1,70E-01
Significant RMSE
Not Significant MAE
2,76E-03
Significant MAE
1,95E-01
Significant MAE
Not Significant MAPE
2,80E-03
Significant MAPE
2,38E-01
Significant MAPE
Not Significant RA
2,06E-01
Significant RA
9,96E-05
Significant RA Not Significant For the AUD/USD and TLKM datasets, across all performance metrics, the p-values are significantly below the 0.
05 threshold.
This indicates that the VMD-LSTM model for the AUD/USD and TLKM datasets outperforms the standard LSTM model with statistically significant improvements.
In contrast, the results from the XAU/USD dataset show a different All p-values are greater than 0.
05, indicating that the VMD-LSTM model for the XAU/USD dataset outperforms the standard LSTM model with statistically not significant A potential reason for this lack of statistical significance in the XAU/USD dataset lies in the stationarity characteristics of the data.
According to our Augmented Dickey-Fuller (ADF) test, the XAU/USD highest price series has an ADF statistic of 1.
21 with a p-value of 0.
indicating strong non-stationarity.
Similarly, the lowest price series shows an ADF statistic of 61 with a p-value of 0.
In contrast, the AUD/USD dataset shows borderline stationarity in the highest price series (ADF statistic = -2.
81, p-value = 0.
and weak non-stationarity in the lowest price series (ADF statistic = -2.
72, p-value = 0.
The TLKM dataset also exhibits nonstationarity, with ADF statistics of -2.
= 0.
for the highest prices and -2.
= 0.
for the lowest prices.
Discussion The experimental results suggest that the VMD-LSTM hybrid model tends to provide improved predictive performance compared to the traditional LSTM in most tested scenarios.
The integration of VMD effectively decomposes complex financial time series into smoother IMFs, which help the LSTM model focus on more informative temporal features.
This finding supports the hypothesis that preprocessing with VMD reduces noise and highlights essential patterns, ultimately resulting in more accurate and reliable predictions .
, .
, .
, .
, .
Support for the reliability of the proposed model is indicated by its relatively consistent performance across three distinct asset classes: currency (AUD/USD), domestic stocks (TLKM), and commodities (XAU/USD).
These results suggest that the VMD-LSTM model shows potential to perform well across different types of financial data, indicating a degree of adaptability to varying volatility patterns and structural characteristics in financial markets.
These findings are in agreement with earlier studies by .
Unlike most prior studies that concentrate solely on predicting closing prices, this study focuses exclusively on the prediction of highest and lowest prices, which represent the daily price range and are crucial for risk management, volatility analysis, and intraday trading strategies.
Previous works such as .
demonstrated improvements in closing price prediction using hybrid VMD-ICSS-BiGRU and 1D-CapsNet-LSTM models, yet none of them explicitly addressed or evaluated the predictive modeling of high and low price levels.
This research fills that gap by presenting a DL framework that is specifically optimized for predicting price extremes COGITO Smart Journal Ae Vol.
No.
December 2025.
P-ISSN: 2541-2221.
E-ISSN: 2477-8079
ISSN: 8-1520
rather than closing trends.
By focusing on the lowest and highest prices, the proposed model contributes new insights into market behavior that are frequently neglected in conventional forecasting approaches.
Research conducted by .
proposed a combined DL model to predict the highest and lowest prices of the S&P 500 Index and the SSEC Index, using a dataset spanning approximately 12 years.
The performance of the proposed model in that study yielded the lowest MAPE of 3105, while in this study, the lowest MAPE for a representative stock market (TLKM) was Nevertheless, there are several limitations worth noting.
First, the determination of the optimal number of VMD modes (K) in this study was based on experiments to get the best value.
Developing a more systematic and adaptive method for selecting K remains an important direction for future research.
Second, while LSTM networks are powerful in modeling sequential dependencies, they often struggle to capture fine-grained local patterns and are susceptible to noiseAiespecially when dealing with non-stationary and highly volatile financial signals.
address this limitation, integrating CNNs as a complementary component presents a promising avenue for enhancing the model's ability to extract localized temporal features more effectively.
Future Research Recommendation Building on the promising results of the VMD-LSTM hybrid approach, future research is encouraged to extend this architecture by integrating additional DL components to further enhance predictive performance and model robustness.
Although VMD-LSTM effectively captures temporal dependencies in decomposed financial time-series data, it presents several LSTM models, while powerful for sequential learning, often struggle to extract finegrained local patterns and are susceptible to noise, especially when handling non-stationary and volatile financial signals.
Furthermore.
LSTMs may face computational challenges and gradient degradation over long sequences.
To address these limitations, incorporating CNNs as a complementary component is a promising direction.
CNNs are well-suited for capturing local temporal-spatial features and filtering noise from VMD-derived sub-signals through their hierarchical feature extraction capability.
Therefore, adding a CNN model to the proposed model can provide a more comprehensive framework, where VMD handles signal decomposition.
CNN captures local patterns, and LSTM models sequential dependencies across time.
Additionally, future studies may also explore the integration of Transformer-based attention mechanisms to capture long-range dependencies and dynamic temporal interactions beyond what LSTM can achieve.
Researchers are also encouraged to investigate adaptive decomposition strategies, in which VMD parameters such as the number of modes (K) are automatically optimized based on the data characteristics, potentially via metaheuristic or learning-based optimization.
Finally, it is essential to validate the generalizability of the proposed model across a wider range of financial instruments, including cryptocurrencies, commodities, and sector-specific indices.
CONCLUSION
This study has developed a hybrid model combining VMD and LSTM to forecast the lowest and highest prices in the financial markets.
VMD is used to decompose complex price signals into IMFs before feeding them into the LSTM, while LSTM learns patterns from these Across three distinct datasetsAiAUD/USD.
TLKM, and XAU/USDAithe VMD-LSTM model exhibits improved performance in most of the tested scenarios compared to traditional LSTM, although its effectiveness may vary depending on the dataset's characteristics.
These findings underscore that the use of VMD decomposition can help enhance the accuracy of forecasting the highest and lowest prices in the financial market.
However, there are several limitations worth noting.
The number of modes (K) in VMD was selected empirically, rather than using automated methods.
Furthermore, the model is not always consistent, especially when applied to highly non-stationary data, such as XAU/USD.
COGITO Smart Journal Ae Vol.
No.
December 2025.
P-ISSN: 2541-2221.
E-ISSN: 2477-8079
ISSN: 1978-1520
Because this study only covered three asset classes, the results cannot be generalized to all types of financial instruments, such as cryptocurrencies or other assets.
In summary, the VMD-LSTM model has potential, but requires more extensive testing.
Future research could attempt to optimize parameters automatically, or add other components, such as CNNs or attention mechanisms, to improve performance in more dynamic market
REFERENCES