Unfolding Latent Tree Structures using 4th Order Tensors

Discovering the latent structure from many observed variables is an important yet challenging learning task. Existing approaches for discovering latent structures often require the unknown number of hidden states as an input. In this paper, we propose a quartet based approach which is \emph{agnostic} to this number. The key contribution is a novel rank characterization of the tensor associated with the marginal distribution of a quartet. This characterization allows us to design a \emph{nuclear norm} based test for resolving quartet relations. We then use the quartet test as a subroutine in a divide-and-conquer algorithm for recovering the latent tree structure. Under mild conditions, the algorithm is consistent and its error probability decays exponentially with increasing sample size. We demonstrate that the proposed approach compares favorably to alternatives. In a real world stock dataset, it also discovers meaningful groupings of variables, and produces a model that fits the data better

Essays on price dynamics, discovery, and dynamic threshold effects among energy spot markets in North America

Given the role electricity and natural gas sectors play in the North American economy, an understanding of how markets for these commodities interact is important. This dissertation independently characterizes the price dynamics of major electricity and natural gas spot markets in North America by combining directed acyclic graphs with time series analyses. Furthermore, the dissertation explores a generalization of price difference bands associated with the law of one price. Interdependencies among 11 major electricity spot markets are examined in Chapter II using a vector autoregression model. Results suggest that the relationships between the markets vary by time. Western markets are separated from the eastern markets and the Electricity Reliability Council of Texas. At longer time horizons these separations disappear. Palo Verde is the important spot market in the west for price discovery. Southwest Power Pool is the dominant market in Eastern Interconnected System for price discovery. Interdependencies among eight major natural gas spot markets are investigated using a vector error correction model and the Greedy Equivalence Search Algorithm in Chapter III. Findings suggest that the eight price series are tied together through sixlong-run cointegration relationships, supporting the argument that the natural gas market has developed into a single integrated market in North America since deregulation. Results indicate that price discovery tends to occur in the excess consuming regions and move to the excess producing regions. Across North America, the U.S. Midwest region, represented by the Chicago spot market, is the most important for price discovery. The Ellisburg-Leidy Hub in Pennsylvania and Malin Hub in Oregon are important for eastern and western markets. In Chapter IV, a threshold vector error correction model is applied to the natural gas markets to examine nonlinearities in adjustments to the law of one price. Results show that there are nonlinear adjustments to the law of one price in seven pair-wise markets. Four alternative cases for the law of one price are presented as a theoretical background. A methodology is developed for finding a threshold cointegration model that accounts for seasonality in the threshold levels. Results indicate that dynamic threshold effects vary depending on geographical location and whether the markets are excess producing or excess consuming markets

Computational methods for nonlinear dimension reduction

Issued as final reportNational Science Foundation (U.S.

Two-way bidiagonalization scheme for downdating the singular-value decomposition

AbstractWe present a method that transforms the problem of downdating the singular-value decomposition into a problem of diagonalizing a diagonal matrix bordered by one column. The first step in this diagonalization involves bidiagonalization of a diagonal matrix bordered by one column. For updating the singular-value decomposition, a two-way chasing scheme has been recently introduced, which reduces the total number of rotations by 50% compared to previously developed one-way chasing schemes. Here, a two-way chasing scheme is introduced for the bidiagonalization step in downdating the singular-value decomposition. We show how the matrix elements can be rearranged and how the nonzero elements can be chased away towards two corners of the matrix. The newly proposed scheme saves nearly 50% of the number of plane rotations required by one-way chasing schemes