Time series estimation in a spiked signal regime

Singular Spectrum Analysis, Time Series, Spiked Covariance Model, Missing Data

Eng, Benjamin C. ; Waldrop ; Lara S.

关键词: Singular Spectrum Analysis, Time Series, Spiked Covariance Model, Missing Data;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/100962/ENG-THESIS-2018.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Working with missing or incomplete data is a universal problem in all sciences. In meteorology, temperature data streams can contain missing values due to sensor malfunctions. In geophysical remote sensing, missing data can may be attributed to irregular global sampling by an orbiting spacecraft. In a collaborative filtering application, like the Netflix Challenge, data is incomplete since it is not possible for all users to provide a recommendation on all items. Though we do not have access to complete data, it is still quite possible to forecast weather, and to recommend good movies on Netflix. The development of estimation algorithms that properly handle missing data make data imputation and forecasting possible.The design of any estimation algorithm depends on the assumptions one can make on a given set of data. This thesis addresses the problem of estimating a noisy, incomplete time series of a dynamical system with unknown state evolution. The technique presented is TSCC (Transformed Spiked Co- variance Completion), a matrix completion algorithm for signal estimation that leverages the spiked signal model, an assumption that holds true for many high-dimensional datasets. The TSCC technique exploits this assumption to develop an estimator that is resilient to noise and accurately fills in missing data.This thesis first addresses the specific estimation problem and the signal model that it follows. It then presents a survey of both standard and the state-of-the-art techniques in addition to an analysis of TSCC. These methods are used to solve the problem of estimating the state of dynamical system, with partial, noisy observations. Standard textbook techniques are not reliable in state estimation due to their inability to handle missing data and to generalize dynamical models. TSCC is an algorithm which addresses this estimation problem and accounts for the deficiencies. Concluding this thesis, several numerical experiments on both synthetic and real data demonstrate that TSCC outperforms these other techniques by forming a time-lagged embedding and estimating the dynamical modes of the system.TSCC has an advantage over other techniques as it does not require knowledge of the state dynamics and that it leverages the asymptotic behavior of noisy, low-rank matrices to perform imputation and denoising. The TSCC technique assumes that a system can be represented by several dynamical modes which is analgous to a matrix having a low rank. Overall, TSCC is a state estimation algorithm that performs estimation on noisy and incomplete data without prior model assumptions. Numerical experiments show that TSCC is an enhancement of the current, accepted techniques which address the same estimation problem.