# A Framework for Market State Prediction with Ontological Asset Selection: A Multimodal Approach
## 3 Theoretical Framework
### Machine Learning Techniques in Financial Time Series Prediction  
Machine learning models have demonstrated exceptional capabilities in capturing temporal dependencies and patterns inherent in financial data. Among these, Long Short-Term Memory (LSTM) networks, an advanced variant of Recurrent Neural Networks (RNNs), are particularly effective for sequential data processing. Given a time series \(\{\mathbf{x}_{t}\}_{t=1}^{T}\), where \(\mathbf{x}_{t}\in\mathbb{R}^{d}\) represents the feature vector at time \(t\), the LSTM updates its hidden state \(\mathbf{h}_{t}\) as follows:  
\[\mathbf{h}_{t}=\sigma(\mathbf{W}_{h}\cdot\mathbf{h}_{t-1}+\mathbf{W}_{x}\cdot \mathbf{x}_{t}+\mathbf{b}), \tag{5}\]  
where \(\mathbf{W}_{h}\) and \(\mathbf{W}_{x}\) are weight matrices, \(\mathbf{b}\) is the bias term, and \(\sigma\) is the activation function. LSTMs are particularly suited for financial time series prediction due to their ability to retain and utilize long-term dependencies, a critical aspect of financial data. Studies, such as [PERSON] and [PERSON] (2019) [22], have validated their effectiveness in this domain.  
In parallel, Bidirectional Encoder Representations from Transformers (BERT) has emerged as a powerful tool for sentiment analysis. In financial applications, BERT processes a corpus \(\mathcal{D}\) of news articles, encoding each document \(d\in\mathcal{D}\) into a semantic embedding \(\mathbf{z}_{d}\):  
\[\mathbf{z}_{d}=\text{BERT}_{\text{encoder}}(d;\Theta), \tag{6}\]  
where \(\Theta\) represents the model parameters. The embedding \(\mathbf{z}_{d}\) encapsulates the contextual semantics of the document, enabling the extraction of valuable insights for predicting market behavior. This capability has been extensively validated in tasks involving stock market prediction, as demonstrated by [PERSON] et al. (2022) [23].  
To achieve a robust prediction framework, the outputs of BERT, Non-Stationary Markov Chain (NMC), and exogenous variables \(\mathbf{Z}(t)\) (e.g., GARCH, VIX) are integrated. These components collectively form the input to the LSTM network, which models temporal dependencies to predict the future market state \(S_{t+1}\). The process is represented as follows:  
\[\hat{S}_{t+1}=f_{\text{LSTM}}(\mathbf{h}_{t}), \tag{7}\]  
where the hidden state \(\mathbf{h}_{t}\) is computed as  
\[\mathbf{h}_{t}=\sigma(\mathbf{W}_{h}\cdot\mathbf{h}_{t-1}+\mathbf{W}_{x}\cdot [\text{BERT}_{\text{output}},\text{NMC}_{\text{output}},\mathbf{Z}(t)]+ \mathbf{b}). \tag{8}\]  
In this configuration,  
* \(\text{BERT}_{\text{output}}\): Encodes sentiment information from financial news articles, capturing the broader market sentiment.
* \(\text{NMC}_{\text{output}}\): Provides probabilistic insights into market state transitions, modeling regime changes over time.
* \(\mathbf{Z}(t)\): Represents exogenous variables such as market volatility (e.g., VIX) and autoregressive conditional heteroskedasticity (e.g., GARCH).  
The weight matrices \(\mathbf{W}_{h}\) and \(\mathbf{W}_{x}\) govern the contributions of the previous hidden state \(\mathbf{h}_{t-1}\) and the current input features, respectively, while \(\mathbf{b}\) introduces a bias term to improve model flexibility. The activation function \(\sigma\), typically a nonlinear function, ensures the model can capture complex patterns in the data.  
This multimodal integration ensures that the contextual insights from BERT, the probabilistic state transitions from NMC, and the exogenous market indicators collectively contribute to the LSTM's temporal modeling. As a result, this approach enhances the accuracy of market state predictions and provides a cohesive methodology for capturing the dynamics of financial markets.