The Optimization of LSTM Model by Wavelet Transform and Simulated Annealing Algorithm

The main objective of this paper is to study the integration of data processing methods and intelligent algorithms to optimize the results of LSTM prediction models. In this paper, continuous wavelet transform is used to clean and preprocess time series data and improve data quality. Wavelet reconstruction is used to restore the results. At the same time, the simulated annealing algorithm is introduced as an intelligent algorithm to search globally for the best solution to achieve the optimal prediction result. The application of this comprehensive approach can also improve the quality and precision of data analysis in various fields, such as the parameter estimation of pulse signals in physics. The core challenge of the research is to optimize the data prediction results, and for this purpose, a multi-level method of continuous wavelet transform, deep learning model (LSTM) and simulated annealing algorithm is adopted.


Introduction
This paper explores how to enhance the performance of Long Short-Term Memory (LSTM) prediction models using data processing techniques and intelligent algorithms.LSTM is a type of deep learning neural network which is designed to effectively handle time-series data [1].However, the performance of LSTM models heavily depends on the quality of the input data.
Time-series data is typically influenced by various factors such as noise, seasonal variations, and outliers, which can lead to instability and inaccuracy in predictive models.To address these issues, this study first utilizes wavelet transform [2][3], a widely used technique in signal processing, for data cleansing and preprocessing.The application of wavelet transform helps reduce noise in the data, improves data quality, and establishes a solid foundation for subsequent LSTM analysis.After data cleansing, wavelet reconstruction techniques are employed to restore the data, ensuring that the processed data can be effectively used in LSTM models.
Furthermore, to further optimize the predictive results of the LSTM model, this study introduces the simulated annealing algorithm [4][5], an intelligent algorithm used for global search of the optimal solution.The simulated annealing algorithm is employed to fine-tune the LSTM model's predictive outcomes to achieve the best possible results.
In conclusion, this research underscores the synergy between data processing techniques and intelligent algorithms in enhancing the performance of LSTM prediction models [6].This approach is not limited to specific application domains such as vegetable sales forecasting but can also improve the quality and accuracy of data analysis in various fields.It achieves the optimization of predictive results through the utilization of the simulated annealing algorithm.

Building the Prediction Model
2.1.Data Preparation 2.1.1.Wavelet Transform.Wavelet transform is a commonly used technique in signal processing and data analysis.Its main objective is to decompose the original data into different frequency components, predict each component separately, and then reconstruct them to obtain the overall prediction result.The advantages of this approach are as follows: (1) Frequency-based prediction allows independent processing of different time scales or frequency components, which is useful for capturing multi-scale features in the data.(2) Frequency decomposition enables the separation of noise from the signal, helping reduce noise interference in predictions and improving the accuracy of the model.These specific advantages will be demonstrated later in the paper.
In order to enhance the robustness of the prediction model, we preprocess the original data by applying continuous wavelet transform.This technique helps to capture periodic patterns in the data.
The formula for (CWT) is as follows [7]: ( Where is the continuous wavelet transform coefficient at scales, is the original signal, is a wavelet function of Daubechies wavelets , e.g.db4 wavelets, and represents scale and location. The formula for Inverse Continuous Wavelet Transform (ICWT) is as follows: ( Where is the original signal, is the CWT coefficient, which is usually obtained by CWT, is the wavelet function of the Daubechies wavelet, and represents scale and location, respectively, and is the normalization constant of the wavelet function.

LSTM Model LSTM (Long Short-Term Memory) is a variant of recurrent neural networks (RNN) proposed by Sepp
Hochreiter and Jürgen Schmid huber in 1997.The main advantage of LSTM over traditional time series analysis methods like ARIMA and some shallow machine learning algorithms like linear regression or decision trees is its ability to effectively capture and utilize long-term dependencies in time series data.This allows LSTM to perform well in complex, nonlinear, and irregular time series data.
Step  Based on the above predicted data and Figure 1, we can observe that the predicted data exhibit similar periodic variations and fluctuation curvature as the original data, which shows a high level of consistency in overall trends and volatility.This indicates that the LSTM model used in predicting the wavelet coefficients for the upcoming week demonstrates excellent effectiveness.
Step 2: Data Reconstruction Inverse Continuous Wavelet Transform (ICWT) [7] is used to reconstruct the data predicted by the LSTM model.The predicted results generated by LSTM are a series of wavelet coefficients.By applying ICWT, we can transform these wavelet coefficients back into the original time series data.This process allows us to restore the original form of the time series, enabling further analysis, visualization and applications.The following graph shows the reconstructed predicted data for the flower leaf category (similar results are obtained for other five categories): According to the Figure 2, it can be concluded that the restored sales volume for the next week shows a close match with the cyclical variation of the sales volume data in the past two months, leading to a close fit with the actual sales situation.
Step 3: Sensitivity Analysis  According to the Figure 4, the R-square of the continuous wavelet transform is 0.85.This is because wavelet transforms generally have denoising effects.If there are outliers or noise in the original data, the wavelet transform may weaken or remove them.We can also observe from the graph that the error points are some data values that are too large.Therefore, considering the reduction of outlier effects, our R-square may be higher in practical applications.

Function Construction
The following is for the category of flower leaves, and the results for the other five categories are similar.

Simulated Annealing Model
Step 1: Building a Planning Model Planning Model: The above planning model takes sale (initial sales volume) and pricing (initial pricing) as inputs."sturnover" is the objective function, "Ensemble Prediction" maximizes the objective function, 1 n is the exponent of the loss function for replenishment quantity, 2 n is the exponent of the loss function for purchase price, rv is the initial replenishment quantity, op is the initial purchase price, rvs is the generated new replenishment quantity, pricings is the generated new pricing, ops is the generated new purchase price, and sales is the generated new sales volume.
Step 2: Solving with Simulated Annealing Simulated Annealing: Simulated annealing has unique advantages in solving planning models.Firstly, it can globally search the problem space, helping to avoid getting trapped in local optima, especially in complex multimodal problems.Secondly, simulated annealing is an adaptive algorithm that can dynamically adjust its search strategy, balancing global exploration and local optimization.It is also suitable for continuous and discrete problems.Its simple implementation and relatively few parameters make it a powerful tool for solving planning problems.
The results are shown in Table 1

Conclusion
The core challenge of this study lies in optimizing the data prediction results, which we address through a multi-level approach.Firstly, we employ continuous wavelet transform to enhance data quality and stability, which can enable us to accurately capture data features and provide a solid foundation for the prediction model.Secondly, we incorporate a deep learning model (LSTM) to effectively model time series data, achieving accurate future predictions.Subsequently, we perform model validation.The LSTM model exhibits outstanding fitting and generalization performance with R-square values of approximately 0.92 and 0.93 on the training and testing sets, respectively.Additionally, the R-square value of the continuous wavelet inverse transform is approximately 0.85, indicating the denoising effect of the wavelet transform on the data.These experimental results emphasize the reliability of the research and the effectiveness of the model.Finally, we apply the simulated annealing algorithm to address the optimization challenges of data prediction results through global search.This process ensures the identification of the optimal strategy to optimize the prediction results to the maximum extent.In summary, our method tackles the optimization challenges of data prediction results through a multilevel approach, and its effectiveness is validated through model validation, providing a comprehensive solution to address this problem.This study highlights our profound considerations and logical completeness in data processing and analysis.
1: Data Prediction The wavelet coefficients obtained from the Continuous Wavelet Transform (CWT) are used as inputs to the LSTM model.(Other detail coefficients: similar processing) This combination take advantage of the feature extraction capabilities of wavelet transform and the sequential modeling abilities of LSTM, thus improving the overall robustness of the model.The predicted data for the flower leaf category is shown in Fig 1. (similar results are observed for other five categories):