Dissertation
New functional depths and applications
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2020
DOI: 10.17077/etd.005379
Abstract
The statistical depth has become increasingly popular as a powerful tool to analyze complex data over the past decades. It ranks the data from center-outward based on the data “centrality” or “outlyingness”. The statistical depth has been applied in a lot of tasks, such as data visualization, outlier detection, robust estimation, etc. The functional data are data of curves or trajectories and widely used in a wealth of applications. In this thesis, we study the statistical depths for functional data and explore their applications.
In Chapter 2, we review the state-of-the-art statistical depths for functional data, i.e., the functional depths, and summarize them into three types based on their characteristics. We conclude that most existing functional depths in the literature are not sensitive to the important curve shape feature, which motivates our work.
In Chapter 3, we propose new depths for univariate functional data where each data observation corresponds to one functional curve. We first propose a novel concept named shape distance to measure the curve shape differences. With the incorporation of the curve shape feature via the shape distance, the proposed univariate functional depths are strongly sensitive to a wide range of curve shape outlyingness while maintaining the ability to detect location outlyingness effectively. We establish the theoretical properties for the proposed depths, which include invariance, monotonicity, and vanishing at infinity property.
Functional data analysis can be seriously impaired by abnormal, outlying observations, and detecting those outliers is an important topic that receives intensive interests. In Chapter 4, we propose a depth-based outlier detection procedure and implement it with our proposed functional depths. Given a clean or slightly contaminated functional dataset, the proposed procedure uses the smoothed depth-based weighting/ trimming bootstrapping to obtain an outlier detection threshold. A new functional curve will be classified as an outlier if it has a depth value smaller than the threshold. This procedure is robust to the potential outliers in the initial dataset and has a low computation requirement.
In Chapter 5, we extend the study of the functional depth to multivariate functional data where each data observation corresponds to multiple related functional curves. We propose the general notion of multivariate functional shape distance to measure the shape difference between two multivariate functional curves. Then we propose two new multivariate functional depths that are sensitive to both the shape outliers and the location outlier. The proposed shape distance and functional depths in Chapter 3 can be viewed as special cases here when the functional data dimension is one. We study the theoretical properties of the proposed depths and demonstrate the performance of the proposed depth using both the simulated datasets and a real-world dataset.
Details
- Title: Subtitle
- New functional depths and applications
- Creators
- Xudong Zhang
- Contributors
- Yong Chen (Advisor)Stephen Baek (Committee Member)Matthew Bognar (Committee Member)Andrew Kusiak (Committee Member)Xuan Song (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Industrial Engineering
- Date degree season
- Spring 2020
- Publisher
- University of Iowa
- DOI
- 10.17077/etd.005379
- Number of pages
- xiii, 126 pages
- Copyright
- Copyright 2020 Xudong Zhang
- Language
- English
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (pages 118-124).
- Public Abstract (ETD)
The statistical depth has become increasingly popular as a powerful tool to analyze complex data over the past decades. It ranks the data from center-outward based on the data “centrality” or “outlyingness”. The statistical depth has been applied in a lot of tasks, such as data visualization, outlier detection, robust estimation, etc. The functional data are data of curves or trajectories that are widely used in a wealth of applications, such as environmentology, healthcare, economics, and biology. In this thesis, we study new functional data depths and explore their applications. Comparing to existing functional data depth, the proposed functional depths are strongly sensitive to a wide range of curve shape outlyingness while maintaining the capability of detecting functional location outliers. We study both the theoretical properties of the proposed functional data depths and their performances in applications such as outlier detection and robust estimation using simulated and real-world data sets.
- Academic Unit
- Industrial and Systems Engineering
- Record Identifier
- 9983949492002771
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