1. Introduction

2. Related Works

3. Methodologies

3.1. Autoencoder (AE)

An autoencoder is a neural network architecture designed for unsupervised learning, consisting of an encoder and decoder. It aims to learn a compact representation of input data by reducing the reconstruction error between the original and reformed data. Figure 3 is an illustration of the autoencoder architecture.

Figure 1. Autoencoder architecture.

Figure 1. Autoencoder architecture.

$$x\stackrel{\mathrm{Encoder}}{⟶}h=e\left(x\right)\stackrel{\mathrm{Decoder}}{⟶}\widehat{x}=d\left(h\right)$$

The architecture of an autoencoder is designed to extract the hidden representation of input data through its encoder component. This representation, which exists in a reduced dimensional space, allows the model to discern non-linear characteristics of the input feature vector. In forecasting time series, these non-linear characteristics can offer valuable insights into the non-linear relationships among data points, which are crucial for forecasting future values. In the given equation, h signifies the hidden representation of the input data as obtained by the encoder component. The decoder component then reconstructs the data, denoted by $\widehat{x}$, by minimizing the reconstruction error. This process of reconstruction eliminates noise or outliers that are not correlated with the dataset, leaving behind only the significant features that are relevant to the context of the dataset. Autoencoders have proven to be effective tools for denoising across various fields [40,41].

3.2. Stacked AE-LSTM and AE-GRU Architecture

In the realm of financial time series forecasting, the process of extracting non-linear features from a dataset is of paramount importance. Financial time series data is unique in that it often exhibits non-linear characteristics. This means that the relationships between variables in the data are not simply proportional, but can change in complex ways. These non-linear relationships can be influenced by a multitude of factors, making them challenging to model accurately.

Algorithms designed for prediction tasks, such as those used in machine learning, are capable of learning from these inherent patterns within the data. They do this by adjusting their parameters to minimize the difference between their predictions and the actual data. Over time, this learning process allows the algorithm to improve its predictive accuracy. However, one of the challenges these algorithms face is the presence of outliers in the dataset. Outliers are data points that deviate significantly from the other observations. They could be the result of noise, errors, or genuinely exceptional observations. Regardless of their source, outliers can distort the learning process of the algorithm, leading it to make inaccurate predictions.

LSTM and GRU are commonly employed in the prediction of stocks and cryptocurrencies. However, these basic LSTM and GRU structures do not take into account the significance of non-linear features. In Natural Language Processing (NLP), both LSTM and GRU encounter difficulties in predicting words when the sentence length increases significantly. Consequently, these algorithms struggle to remember the context over a very long term. To address this issue, the Transformer Network (TN) was introduced, which uses an attention mechanism in an encoder-decoder architecture to remember the long-term context [42]. However, the encoder-decoder architecture of TN doesn’t compress the input vector for learning intricate features from the latent representation. The encoder component of the Transformer handles the processing of the input sequence, while the output sequence is produced by the decoder. However, the implementation of such a complex model is computationally demanding. In the prediction of financial time series, it is crucial to remember the long-term context of past significant market events. This aids in identifying similar trends and predicting future values based on these trends. Therefore, the basic LSTM and GRU architecture is not the optimal solution for this task. Typically, for a one-day-ahead forecast for stocks and cryptocurrencies, the training dataset is usually very large. Consequently, running such a large dataset with a TN architecture becomes computationally prohibitive. Encoder-decoder-based Recurrent Neural Network (RNN) architectures demonstrated effective performance when an Autoencoder-based LSTM was proposed [18]. To a certain degree, the encoded latent representation of the input data preserved the long-term contextual information about past observations, which helped to enhance the predictive accuracy of the basic LSTM architecture. This research found that significant improvements in predictive accuracy could be achieved through extensive experimentation with the model’s hyperparameters. However, in some studies, the GRU outperformed the LSTM in terms of achieving higher predictive accuracy. Therefore, this research also examines the accuracy of such an encoder-decoder architecture with GRU and proposes a new model with improvised performance.

In this paper, we introduce an innovative framework comprising stacked Autoencoder-based Long Short-Term Memory (AE-LSTM) and Gated Recurrent Unit (AE-GRU). The foundational structure of the proposed algorithm is elucidated in Figure 4. While both AE-LSTM and AE-GRU share a common architecture, distinct hyperparameters and activation functions tailored to each architecture were employed for optimal performance after strenuous investigation.

3.2.1. AE-LSTM

AE-LSTM, or Autoencoder-based LSTM, is a hybrid neural architecture that combines the capabilities of an autoencoder with LSTM units. Autoencoders are employed to learn a compressed representation of input data, and this encoded information is then fed into LSTM layers, enabling the model to capture and leverage long-term dependencies in sequential data. AE-LSTM is particularly effective in tasks involving time-series data, such as stock price prediction, where both feature learning and sequential context preservation are crucial for accurate forecasting. Figure 2 represents the model structure of the proposed improvised AE-LSTM architecture.

$$\mathrm{Encoder}:x\stackrel{\mathrm{LSTM}200}{⟶}{h}_1\stackrel{\mathrm{LSTM}100}{⟶}{h}_2\stackrel{\mathrm{LSTM}80}{⟶}z$$

$$\mathrm{Decoder}:z\stackrel{\mathrm{LSTM}80}{⟶}{h}_3\stackrel{\mathrm{LSTM}100}{⟶}{h}_4\stackrel{\mathrm{LSTM}200}{⟶}\widehat{x}$$

In the given equation, it’s evident that the encoder structure employs several stacked LSTM layers to identify latent features at various compression levels. With each level of compression, the architecture learns more prominent features. However, additional compression leads to information loss. Such a deep architecture results in a higher reconstruction loss and is susceptible to the vanishing gradient problem. The initial compression occurs when the number of units in the second LSTM layer is reduced from 200 to 100. Over time, each layer experiences a reduction in units until the latent representation z is finally extracted from the LSTM layer with 80 units.

Similarly, the decoder layer incrementally increases the units in the subsequent LSTM layers. The entire structure is trained by minimizing the reconstruction loss. The output of the decoder is the reconstructed data that encapsulates the internal non-linear complexities of the financial time series. Ultimately, a Dense layer is employed to obtain the predicted output.

Figure 2. AE-based LSTM and GRU architecture.

Figure 2. AE-based LSTM and GRU architecture.

Algorithm 1 shows an algorithmic view of the proposed novel improvised encoder-decoder architecture of LSTM which outperforms the existing models.

Algorithm 1 AE-LSTM

1: $data\leftarrow parse("stock\_name")$  2: $normalized\_data\leftarrow normalize\left(data\right)$  3: $train\_data\leftarrow normalized\_data[start:end]$  4: $test\_data\leftarrow normalized\_data[start:end]$  5: $preprocessed\_train\leftarrow preprocess\left(train\right)$  6: $encoder\leftarrow LSTM(units=200,activation{=}^{\prime }activatio{n}^{\prime })(preprocessed\_train)$  7: $encoder\leftarrow LSTM(units=100,activation{=}^{\prime }activatio{n}^{\prime })\left(encoder\right)$  8: $encoder\leftarrow LSTM(units=80,activation{=}^{\prime }activatio{n}^{\prime })\left(encoder\right)$  9: $decoder\leftarrow LSTM(units=80,activation{=}^{\prime }activatio{n}^{\prime })\left(encoder\right)$  10: $decoder\leftarrow LSTM(units=100,activation{=}^{\prime }activatio{n}^{\prime })\left(decoder\right)$  11: $decoder\leftarrow LSTM(units=200,activation{=}^{\prime }activatio{n}^{\prime })\left(decoder\right)$  12: $model\leftarrow Model(encoder,decoder)$  13: $model.compile\left(\right)$  14: $predict\leftarrow model.predict(test\_data)$

3.2.2. AE-GRU

AE-GRU, or Autoencoder-Gated Recurrent Unit, is a hybrid neural network architecture that combines the capabilities of an autoencoder with a Gated Recurrent Unit (GRU). This innovative model integrates the feature learning abilities of autoencoders with the sequential modeling strengths of GRU, aiming to enhance the representation and prediction capabilities in various applications such as time series analysis and sequential data processing. The AE-GRU architecture involves the use of a stacked autoencoder for feature extraction, feeding the encoded features into GRU cells for capturing temporal dependencies and patterns. This combination leverages both unsupervised learning and recurrent neural network architectures to achieve improved performance in tasks requiring sequential data understanding and prediction. To our knowledge, this hybridization is a novel approach to financial asset price prediction. Figure 2 represents the model structure of the proposed novel AE-GRU algorithm.

$$\mathrm{Encoder}:x\stackrel{\mathrm{GRU}200}{⟶}{h}_1\stackrel{\mathrm{GRU}100}{⟶}{h}_2\stackrel{\mathrm{GRU}80}{⟶}z$$

$$\mathrm{Decoder}:z\stackrel{\mathrm{GRU}80}{⟶}{h}_3\stackrel{\mathrm{GRU}100}{⟶}{h}_4\stackrel{\mathrm{GRU}200}{⟶}\widehat{x}$$

In a manner akin to AE-LSTM, AE-GRU follows a similar architecture. Multiple encoder layers are layered to condense the feature vector into a representation in the latent space. After compressing to a feature vector, z, with an GRU layer consisting of 80 units, it is observed that further compression leads to a loss of information.

Several GRU layers, each with varying units, are stacked in a descending order to construct the encoder component of the AE-GRU architecture. This compression into the latent space z preserves valuable non-linear features of the financial time series. This latent feature vector encapsulates the long-term context derived from past observations.

Similarly, the decoder component incrementally increases the units in the GRU layers, resulting in an accurate reconstruction of the input data. This process eliminates outliers and extracts the non-linear contextual features from the financial data. Subsequently, a dense layer comprising a single unit is employed to predict future values.

Algorithm 2 shows an algorithmic view of the proposed novel encoder-decoder architecture of GRU, which outperforms the existing models and even the improvised AE-LSTM designed in this study. This encoder-decoder-based GRU architecture is novel and tested rigorously under varying market conditions. This proposed novel model overperforms existing architectures by retaining long-term non-linear contextual features.

Algorithm 2 AE-GRU

1: $data\leftarrow parse("stock\_name")$  2: $normalized\_data\leftarrow normalize\left(data\right)$  3: $train\_data\leftarrow normalized\_data[start:end]$  4: $test\_data\leftarrow normalized\_data[start:end]$  5: $preprocessed\_train\leftarrow preprocess\left(train\right)$  6: $encoder\leftarrow GRU(units=200,activation{=}^{\prime }activatio{n}^{\prime })(preprocessed\_train)$  7: $encoder\leftarrow GRU(units=100,activation{=}^{\prime }activatio{n}^{\prime })\left(encoder\right)$  8: $encoder\leftarrow GRU(units=80,activation{=}^{\prime }activatio{n}^{\prime })\left(encoder\right)$  9: $decoder\leftarrow GRU(units=80,activation{=}^{\prime }activatio{n}^{\prime })\left(encoder\right)$  10: $decoder\leftarrow GRU(units=100,activation{=}^{\prime }activatio{n}^{\prime })\left(decoder\right)$  11: $decoder\leftarrow GRU(units=200,activation{=}^{\prime }activatio{n}^{\prime })\left(decoder\right)$  12: $model\leftarrow Model(encoder,decoder)$  13: $model.compile\left(\right)$  14: $predict\leftarrow model.predict(test\_data)$

3.2.3. Hyperparameter Consideration

The process of tuning hyperparameters is a critical step in the development of machine learning or deep learning algorithms. Hyperparameters are distinct from other parameters in that their values are determined before the learning process begins. These are not derived from the data during training but are manually set to guide the learning process. The selection of these hyperparameters significantly influences the functioning of the algorithm. Incorrect choices could lead to sub-optimal performance, even for a model with high potential.

In this study, the Adam optimizer was chosen due to its widespread use in adaptively adjusting the learning rate while taking momentum into account. The first and second-moment exponential decay rates, represented by ${\beta }_1$ and ${\beta }_2$, were identified as the primary hyperparameters in this research. Additionally, this study explored three major activation functions to select the most suitable ones to improve the performance of the models. Other than that, rigorous investigation has been conducted to select the number of layers for designing the novel architecture.

Also upon considering multiple time windows, this research has decided to use 120 days past observation for predicting the one-day ahead forecast for stocks, cryptocurrencies, and stock index.

3.3. Activation Functions

This study examined three primary activation functions suitable for integration into the proposed hybridized models, namely: (1) Rectified Linear Unit (ReLU), (2) Exponential Linear Unit (ELU), and (3) Hyperbolic Tangent Function (tanh). These three activation functions are mostly used with DL architectures. The choice of activation function depends on the dataset and model architectures. Improper choice might result in sub-optimal performance.

Figure 3. Exponential Linear Unit.

Figure 3. Exponential Linear Unit.

3.3.1. ReLU

The Rectified Linear Unit (ReLU) activation function is extensively utilized in neural networks, providing non-linearity by outputting the input directly for positive values and zero for negative ones. Its popularity stems from its simplicity, computational efficiency, and effectiveness in dealing with the vanishing gradient problem during deep neural network training. ReLU addresses the issue of vanishing gradients that arise due to the saturation of neurons when sigmoid functions are used. Improper weight initialization can lead to the "dying ReLU" problem, especially due to the negative saturation.

The formula for ReLU is:

$$f_{Relu}\left(h_{i,k}\right)=max(0,h_{i,k}).$$

Figure 5(a) presents a graphical representation of ReLU1.

3.3.2. ELU

The Exponential Linear Unit (ELU) is an activation function commonly used in artificial neural networks. It brings non-linearity to the network by permitting the inclusion of negative values, unlike traditional activation functions such as ReLU. ELU has a smooth curve for negative inputs, avoiding the issue of dead neurons and enabling better convergence during training. It exhibits the ability to capture information from both positive and negative input ranges, contributing to improved learning in deep neural networks. ELU addresses the issue of the "dying ReLU" problem that occurs due to negative saturation but bad initialization still can cause underperformance. The ELU activation function is defined as: f(k)$=\left\{\begin{array}{cc}k\hfill & \mathrm{if}k>0\hfill \\ \alpha ·(e^k-1)\hfill & \mathrm{if}k\le 0\hfill \end{array}\right.$, where $\alpha$ is a positive hyperparameter controlling the slope of the function for negative inputs. Figure 5(b) presents a graphical representation of ELU 2.

3.3.3. Tanh

The hyperbolic tangent function, often denoted as $tanh\left(x\right)$, is a common activation function in neural networks. It squashes input values to the range of $(-1,1)$, making it zero-centered and aiding in mitigating vanishing gradient problems during training. The tanh activation is particularly useful in scenarios where zero-centered outputs are desired and has applications in various layers of neural network architectures. The tanh activation function delivers output centered around zero, but it can lead to the vanishing gradient problem due to neuron saturation caused by improper weight initialization. The formula for the tanh activation function:

$$tanh\left(k\right)=\frac{e^k-e^{-k}}{e^k+e^{-k}}$$

Figure 5(c) presents a graphical representation of tanh 3.

4. Experiments and Results

4.2. Exploratory Data Analysis

Figure 4 depicts the Bollinger bands for the financial instruments analyzed in this study. Both Figure 4 and Table 1 collectively indicate that AAPL and JPM display elevated volatility, while JNJ demonstrates comparatively lower volatility, and CVX exhibits a moderate level of volatility. S&P registers notably low average volatility, whereas the cryptocurrency bitcoin experiences the highest volatility between 2022 and 2023 among all the assets considered.

Figure 4. Bollinger plots for financial assets from 2022 to 2023.

Figure 4. Bollinger plots for financial assets from 2022 to 2023.

Figure 5 illustrates the change in the volatility trend in the considered timeframe. Notably, Apple stock exhibits a significant rise in volatility. Conversely, Bitcoin’s volatility displays abrupt fluctuations. The S&P, reflecting a blend of diverse financial assets, exhibits a high magnitude of volatility but lower average volatility. On the other hand, JPM, JNJ, and CVX consistently experience growth in volatility throughout the years.

Figure 5. Change in volatility.

Figure 5. Change in volatility.

5. Conclusions

6. Future Works

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