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Minimum Modal Regression
Koichiro Yamauchi
1
and Vanamala Narasimha Bhargav
2
1
Department of Computer Science, Chubu University, Matsumoto-cho 1200 Kasugai, Japan
2
Indian Institute of Technology, Guwahati, Assam, India
Keywords: Modal Regression, Kernel Distribution Estimator, Incremental Learning on a Budget, Kernel Machines,
Projection Method.
Abstract: The recent development of microcomputers enables the execution of complex software in small embedded
systems. Artificial intelligence is one form of software to be embedded into such devices. However, almost
all embedded systems still have restricted storage space. One of the authors has already proposed an
incremental learning method for regression, which works under a fixed storage space; however, this method
cannot support the multivalued functions that usually appear in real-world problems. One way to support the
multivalued function is to use the model regression method with a kernel density estimator. However, this
method assumes that all sample points are recorded as kernel centroids, which is not suitable for small
embedded systems. In this paper, we propose a minimum modal regression method that reduces the number
of kernels using a projection method. The conditions required to maintain accuracy are derived through
theoretical analysis. The experimental results show that our method reduces the number of kernels while
maintaining a specified level of accuracy.
1 INTRODUCTION
The recent development of microcomputers enables
the embedding of complex software into small
devices. Machine learning algorithms are one
example of such software. One of the authors has
previously proposed a learning algorithm for kernel
regression in embedded systems (Yamauchi, 2014),
but this general regression method estimates the
conditional expectation of the dependent variable (Y)
given the independent variables (X=x). In contrast,
modal regression (Einbeck et al, 2006) estimates the
conditional modes of Y given X=x. This strategy
enables the learning machine to predict a portion of
the missing variables from the other known variables
according to the given sample distribution. This
property is quite different from that of other typical
regression methods.
To estimate the conditional modes, partial mean
shift (PMS) is an assured method. At first, the PMS
method attempts to obtain the joint kernel density and
derives it using the gradient ascent. However, if the
number of samples is increasing, minimum modal
regression is proposed, which can estimate the joint
kernel density by projecting the new sample,
replacing the old kernel, or adding the new kernel to
the sample. The equation for PMS is then modified
accordingly.
2 MODAL REGRESSION
Modal regression approximates a multivalued
function to search the local peaks of a given sample
distribution. Modal regression consists of the kernel
density estimator with a PMS method.
2.1 Kernel Density Estimator
The kernel density estimator (KDE) is a variation of
the Parzen window (Parzen, 1962).
Let
be the set of learning samples, and
. The estimator
approximates the probability density function by
using a number of kernels, namely, the support set
.
The kernels used are Gaussian kernels, and
t
Si
x
i
h
Kp
xx
x)(
448
Yamauchi, K. and Bhargav, V.
Minimum Modal Regression.
DOI: 10.5220/0006601304480455
In Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2018), pages 448-455
ISBN: 978-989-758-276-9
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