# A Framework for Market State Prediction with Ontological Asset Selection: A Multimodal Approach
## 3 Theoretical Framework
### Stochastic Models and Non-Stationary Markov Chains
The dependency structure among states and inputs is modeled using \(t\)-copulas \(C_{t}(u_{1},\ldots,u_{k};\
u)\), where \(u_{i}=F_{i}(S_{t})\) is the marginal distribution of state \(i\) and \(\
u\) represents the degrees of freedom:  
\[C_{t}(u_{1},\ldots,u_{k};\
u)=\int_{0}^{u_{1}}\cdots\int_{0}^{u_{k}}f_{t}(x_{1 },\ldots,x_{k};\
u)dx_{1}\ldots dx_{k}. \tag{4}\]  
The \(t\)-copulas choice is appropriate to the nature of the financial data, which are often nonlinear, dependent, and heavy-tailed, and the co-occurrence of extreme events represents the same salient feature of a stressed market. In this respect, while the Gaussian copulas are unable to capture them, \(t\)-copulas allow for modeling the tail dependency and the asymmetric relationships. By doing so, the model improves its robustness to stress the use of the model in extreme cases. This is particularly valuable in the face of market turbulence, where it would be very difficult for a model to describe the occurrence of simultaneous extreme events in the variables. Nevertheless, this approach has its drawbacks. First, the computational complexity of the model increases with the use of \(t\)-copulas, especially if the dimension of the data is high. The second challenge is the reliance of the model on the exogenous input data. For instance, missing or noisy data in the SI or ILLIQ may lead to less reliable predictions. The third challenge is related to the assumption that the dependence structure between exogenous variables and transition probabilities is constant over time, which can be restrictive in highly volatile markets or during structural breaks. Future work could address these challenges incorporating Bayesian inference or machine learning techniques to enhance robustness and scalability. By integrating \(t\)-copulas into a Non-Stationary Markov Chain (NMC) framework, this approach combines the temporal modeling strengths of Markov Chains with the ability of copulas to capture complex dependency structures. As highlighted by [PERSON] et al. (2003) [20] and [PERSON] et al. (2009) [21], this hybrid approach offers a powerful tool for forecasting market behaviors under varying conditions. The dynamic adjustment of transition probabilities based on exogenous variables such as the Sentiment Index (SI) and Illiquidity Index (ILLIQ) ensures the model remains responsive to evolving market environments, making it highly relevant for real-world applications. Despite its strengths, this approach is not without limitations. First, the use of \(t\)-copulas increases computational complexity, particularly when dealing with high-dimensional datasets. The need for extensive processing power may pose challenges in real-time applications or when scaling the model. Second, the accuracy of predictions is closely tied to the quality and availability of exogenous inputs. Missing or noisy data in SI or ILLIQ indices can significantly undermine the reliability of the model's outputs. Finally, the framework assumes a stable relationship between exogenous variables and transition probabilities over time--a condition that may not hold during periods of extreme market volatility or structural changes. To overcome these challenges, future research could explore the incorporation of techniques such as Bayesian inference or machine learning to improve both robustness and scalability. Bayesian methods could enhance the model's ability to handle uncertainty in parameter estimation, while machine learning algorithms might enable the detection of evolving relationships between variables and transition probabilities. These enhancementscould pave the way for a more adaptable and computationally efficient framework, capable of managing the complexities of modern financial markets more effectively.