KoulMde - Koul's Minimum Distance Estimation in Regression and Image
Segmentation Problems
Many methods are developed to deal with two major
statistical problems: image segmentation and nonparametric
estimation in various regression models. Image segmentation is
nowadays gaining a lot of attention from various scientific
subfields. Especially, image segmentation has been popular in
medical research such as magnetic resonance imaging (MRI)
analysis. When a patient suffers from some brain diseases such
as dementia and Parkinson's disease, those diseases can be
easily diagnosed in brain MRI: the area affected by those
diseases is brightly expressed in MRI, which is called a white
lesion. For the purpose of medical research, locating and
segment those white lesions in MRI is a critical issue; it can
be done manually. However, manual segmentation is very
expensive in that it is error-prone and demands a huge amount
of time. Therefore, supervised machine learning has emerged as
an alternative solution. Despite its powerful performance in a
classification problem such as hand-written digits, supervised
machine learning has not shown the same satisfactory result in
MRI analysis. Setting aside all issues of the supervised
machine learning, it exposed a critical problem when employed
for MRI analysis: it requires time-consuming data labeling.
Thus, there is a strong demand for an unsupervised approach,
and this package - based on Hira L. Koul (1986)
<DOI:10.1214/aos/1176350059> - proposes an efficient method for
simple image segmentation - here, "simple" means that an image
is black-and-white - which can easily be applied to MRI
analysis. This package includes a function GetSegImage(): when
a black-and-white image is given as an input, GetSegImage()
separates an area of white pixels - which corresponds to a
white lesion in MRI - from the given image. For the second
problem, consider linear regression model and autoregressive
model of order q where errors in the linear regression model
and innovations in the autoregression model are independent and
symmetrically distributed. Hira L. Koul (1986)
<DOI:10.1214/aos/1176350059> proposed a nonparametric minimum
distance estimation method by minimizing L2-type distance
between certain weighted residual empirical processes. He also
proposed a simpler version of the loss function by using
symmetry of the integrating measure in the distance. Kim (2018)
<DOI:10.1080/00949655.2017.1392527> proposed a fast
computational method which enables practitioners to compute the
minimum distance estimator of the vector of general multiple
regression parameters for several integrating measures. This
package contains three functions: KoulLrMde(), KoulArMde(), and
Koul2StageMde(). The former two provide minimum distance
estimators for linear regression model and autoregression
model, respectively, where both are based on Koul's method.
These two functions take much less time for the computation
than those based on parametric minimum distance estimation
methods. Koul2StageMde() provides estimators for regression and
autoregressive coefficients of linear regression model with
autoregressive errors through minimum distant method of two
stages. The new version is written in Rcpp and dramatically
reduces computational time.
