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Publications

Understanding the Complexities of the Fine Structure of Interest Rates: A Wasserstein Barycenter Learning Approach

• Authors: Carlo Mari & Cristiano Baldassari 

• Publication date: 03 August 2024

• Abstract: A novel methodology to investigate the fine structure of interest rates based on Machine Learning techniques is discussed. The aim is to capture in an unsupervised way the common stochastic structure that drives the dynamics of interest rates of different maturities. The proposed approach is based on the Wasserstein barycenter, a powerful tool of analysis that allows us to construct, from a set of assigned probability distributions, a single probability distribution that captures the essential features of the whole set. To identify common stochastic factors, a Gaussian Mixture Model is fitted to the Wasserstein barycenter by maximum likelihood using the Expectation-Maximization algorithm with an initialization strategy based on Graph Machine Learning techniques. A fine-tuning of single-maturity interest rates is discussed in an attempt to capture maturity-specific stochastic factors. The proposed analysis also gives us the opportunity to test the hypothesis of a market segmentation into a short-term segment, the money market, and a long-term segment, the capital market, each with its own segment-specific stochastic factors. The methodology is tested on the US zero-coupon Treasury yield curve. The results obtained seem to show that most of the stochastic nature of the dynamics of the US zero-coupon yield curve can be captured by a three-component Gaussian Mixture Model describing the Wasserstein barycenter of the short-term segment of the yield curve.

• Authors: Carlo Mari & Cristiano Baldassari 

• Publication date: 30 March 2023

• Abstract: A fully unsupervised graph-based superframework is proposed to handle the EM initialization problem for estimating mixture models on financial time series. Using a complex network approach that links time series and graphs, the graph-structured information derived from the observed data is exploited to produce a meaningful starting point for the EM algorithm. It is shown that structural information derived by complex graphs can definitely capture time series behavior and nonlinear relationships between different observations. The proposed methodology is employed to estimate Gaussian mixture models on US wholesale electricity market prices using two different configurations of the superframework. The obtained results show that the proposed methodology performs better than conventional initialization methods, such as K-means based techniques. The improvements are significant on the overall representation of the empirical distribution of log-returns and, in particular, on the first four moments. Moreover, this approach has a high degree of generalization and flexibility, exploiting graph manipulation and employing functional operating blocks, which can be adapted to very different empirical situations.

• Authors: Carlo Mari & Cristiano Baldassari 

• Publication date: 30 May 2023

• Abstract: We propose a fully unsupervised network-based methodology for estimating Gaussian Mixture Models on financial time series by maximum likelihood using the Expectation-Maximization algorithm. Visibility graph-structured information of observed data is used to initialize the algorithm. The proposed methodology is applied to the US wholesale electricity market. We will demonstrate that encoding time series through Visibility Graphs allows us to capture the behavior of the time series and the nonlinear interactions between observations well. The results reveal that the proposed methodology outperforms more established approaches.

• Authors: Carlo Mari & Cristiano Baldassari 

• Publication date: December 2022

• Abstract: An efficient initialization of the Expectation-Maximization algorithm to estimate mixture models via maximum likelihood is proposed. A fully unsupervised network-based initialization technique is provided by mapping time series to complex networks using as adjacency matrix the Markov Transition Field associated to the time series. In this way, the optimal number of mixture model components and the vector of initial parameters can be directly obtained. An experiment conducted on financial times series with very different characteristics shows that our approach produces significantly better results if compared to conventional methods of initialization, such as K-means and Random, thus demonstrating the effectiveness of the proposed method.