Research Interests
Peihua Qiu's Research on Jump Regression Analysis
Nonparametric regression analysis provides a statistical tool for
estimating regression curves or surfaces from noisy data. Conventional
nonparametric regression methods (e.g., kernel, splines), however, are
appropriate for estimating continuous regression functions only. When
an underlying regression function has jumps, its estimators by the
conventional methods would not be statistically consistent at the jump
positions. More intuitively, the jumps would be smoothed out by the
conventional smoothing methods. In practice, regression functions are
often discontinuous. For instance, the equi-temperature surfaces in high
sky or deep ocean usually have jumps and discontinuities, caused by
turbulence or other climate phenomena. The image intensity surface of
a typical image has jumps at the outlines of image objects. Jump regression
analysis (JRA) is specifically for estimating jump regression curves/surfaces
from noisy data.
I started my research on nonparametric regression when I was a graduate
student in China. At that time, most existing nonparametric regression
methods assumed that the curves or surfaces to estimate were continuous.
In my opinion, this assumption was faulty as a general rule because curves
or surfaces could be discontinuous in some applications, and thus I decided
to investigate this problem in my MS thesis at Fudan University in China.
As I began my graduate studies in the United States during the early 1990s,
several methods were proposed in the statistical literature for
estimating jump curves or surfaces. However, these methods often imposed
restrictive assumptions on the model, making them unavailable for many
applications. Another limitation of the methods was that extensive
computation was required. In addition, the existing methods in the image
processing literature did not have much theory to support them. A direct
consequence was that, for a specific application problem, it was often
difficult to choose one from dozens of existing methods to handle the
problem properly. Therefore, it was imperative to suggest some methods that
could work well in applications and have some necessary theory to support
them; this became the goal of my Ph.D. thesis research at Wisconsin, and I
have been working in the area since then.
For one dimensional (1-D) JRA models, my coauthors and I proposed the
so-called difference kernel estimators (DKEs) of the jump and continuity
parts of the regression function when the number of jumps is known (Qiu
1990, Qiu 1991, Qiu, Asano and Li 1991, Qiu 1993). When the number of
jumps is unknown, my coauthors and I proposed the difference apart kernel
estimator (DAKE) of the jump regression curve and two modified versions
(Qiu 1994, Qiu and Yandell 1998, Joo and Qiu 2009). We also proposed some
methods for estimating regression curves without detecting jump points
explicitly (Qiu 2003, Gijbels, Lambert and Qiu 2007). Recently, we proposed
a new model selection criterion, called Jump Information Criterion
(JIC) for estimating the number of jumps (Xia and Qiu 2015).
In two-dimensional (2-D) cases, jump locations become curves, which I
termed jump location curves (JLCs). When the number of JLCs is known, I
proposed the rotational difference kernel estimators (RDKEs) of JLCs
(Qiu 1997). However, the assumption that the number of JLCs is known is
too strong for many applications. To avoid such restrictive assumptions,
I proposed the idea to approximate JLCs by a pointset, instead of by
curves. By that idea, my coauthors and I proposed a jump detection
procedure for detecting arbitrary JLCs (Qiu and Yandell 1997). In that
paper, we also proposed two jump modification procedures, pointed out
certain limitations of the well-known Hausdorff distance for measuring
the goodness of a set of detected jump points, and proposed a new metric
for measuring the performance of the detected jumps. In Qiu (2002), I
generalized the well known Sobel edge detector in image processing,
and provided both theoretical and numerical arguments concerning
circumstances when such procedures would perform well. In Sun and Qiu
(2007), a jump detection procedure based on both first- and second-order
derivatives of the regression surface was proposed, which performed
better than the popular Canny's edge detector in certain cases. Recently,
Hall, Qiu and Rau (2008) proposed a flexible tracking procedure for
estimating JLCs using local partial likelihood estimation. For
jump-preserving surface estimation, I proposed an indirect method after
jumps were first detected and the JLC in a neighborhood of a given point
was approximated by a principal component line (Qiu 1998). A number of
direct estimation methods without explicitly detecting jumps were
proposed rather recently (Qiu 2004, Gijbels, Lambert and Qiu 2006, Qiu
2009, Mukherjee and Qiu 2011).
In summary, my coauthors and I have proposed a sequence of concepts,
terminologies, and procedures for jump detection and jump regression
curve/surface estimation. Most of them were summarized in the first
five chapters of my book Qiu (2005, Wiley).
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