Probabilistic Parametric Curves for Sequence Modeling: Karlsruher Schriften zur Anthropomatik

This work proposes a probabilistic extension to Bézier curves as a basis for effectively modeling stochastic processes with a bounded index set. The proposed stochastic process model is based on Mixture Density Networks and Bézier curves with Gaussian random variables as control points.

The key advantage of this model is the ability to generate multi-mode predictions in a single inference step, thus avoiding the need for Monte Carlo simulation. This is particularly useful in scenarios where the underlying distribution of the stochastic process exhibits multiple modes, as the model can capture this complexity and provide accurate predictions without the computational overhead of sampling-based approaches.

The use of Bézier curves as the foundation of the model provides a flexible and intuitive way to represent the stochastic process. Bézier curves are defined by a set of control points, and in this case, those control points are Gaussian random variables. This allows the model to capture the uncertainty and variability inherent in the stochastic process, while still maintaining the smooth and continuous characteristics of Bézier curves.

The Mixture Density Network component of the model is responsible for learning the parameters of the Gaussian random variables that define the control points of the Bézier curves. This allows the model to adapt to the specific characteristics of the stochastic process being modeled, ensuring that the generated predictions accurately reflect the underlying distribution.

One of the key advantages of this approach is the ability to generate multi-mode predictions in a single inference step. Traditional Monte Carlo simulation techniques often require a large number of samples to capture the full complexity of the underlying distribution, which can be computationally expensive and time-consuming. In contrast, the proposed model can directly output a multi-modal prediction, providing a more efficient and accurate representation of the stochastic process.

This work has a wide range of potential applications, including predictive modeling in fields such as finance, engineering, and climatology, where stochastic processes with bounded index sets are commonly encountered. The ability to capture the complexity of these processes while avoiding the computational burden of sampling-based approaches makes this model a valuable tool for researchers and practitioners working in these domains.

In conclusion, this work presents a novel probabilistic extension to Bézier curves that provides an effective and efficient way to model stochastic processes with bounded index sets. The combination of Bézier curves, Gaussian random variables, and Mixture Density Networks enables the generation of multi-mode predictions in a single inference step, offering a significant advantage over traditional Monte Carlo simulation techniques. The versatility and effectiveness of this approach make it a promising tool for a wide range of applications in fields where accurate and efficient modeling of stochastic processes is of critical importance.

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