基于样本的缺失数据极端学习机研究
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本文主要探讨了"基于样本的极端学习机（Sample-Based Extreme Learning Machine, SELM）在处理缺失数据方面的研究"。作者 Hang Gao、Xin-Wang Liu、Yu-Xing Peng 和 Song-Lei Jian 来自国防科技大学的 Science and Technology on Parallel and Distributed Processing Laboratory，他们针对在实际应用中常见的缺失数据问题提出了新的解决方案。极端学习机（ELM）因其高效性和对分类、回归等任务的高度统一性而在机器学习领域受到广泛关注。然而，现有的ELM算法往往无法有效地处理缺失值，这限制了其在处理现实世界数据时的表现。
SELMArticle的主要贡献在于提出了一种新的策略来处理极端学习中的缺失数据。传统方法通常通过插补（imputation），即用某个统计方法或预测模型来填充缺失值，但这可能引入误差并影响模型性能。文章可能介绍了一种创新的策略，例如利用样本聚类或者基于邻域的信息来估计缺失值，或者采用稳健的权重分配方法，使得ELM能够更适应缺失数据的情况。
作者们可能首先回顾了现有文献中关于ELM处理缺失数据的方法，然后详细阐述了他们提出的SELM在处理缺失值时的工作原理。他们可能会讨论如何在训练过程中有效地融合完整样本和含有缺失值的样本，以及如何通过调整算法参数来优化模型对缺失数据的鲁棒性。此外，他们还可能进行了实验验证，展示了他们的方法在各种缺失数据集上的性能优于传统的缺失值处理方法，并分析了影响性能的关键因素。
值得注意的是，该研究发表于2014年，因此可能包含当时的研究前沿和挑战。随着机器学习领域的不断发展，对于处理缺失数据的算法可能会有新的进展，但这篇文章无疑为当时的ELM研究提供了一个重要的补充，尤其是在实际应用中的数据预处理和模型构建方面。
"Sample-Based Extreme Learning Machine with Missing Data"这篇研究论文对极端学习机在处理缺失数据的问题上进行了深入探讨，为提高机器学习算法在实际场景中的适用性和鲁棒性提供了有价值的方法和思路。通过理解并应用这一方法，研究人员和工程师能够在处理实际数据集时更加得心应手，提升预测和决策的准确性。
Mathematical Problems in Engineering
Inequality Optimization Constraints Based ELM. With the
parameter setting of
1
=2,
2
=1, =2, =1,and
inequality constraints, the general optimization formula can
be written as (), which is common in binary classication.
Since this form is applied wildly and has good sparsity, we
use it as the base model of our extension for classication:
min
𝛽
(𝑖)
,𝜉
𝑖
1
2
𝛽
(𝑖)
2
+
𝑛
𝑖=1
𝑖
,
s.t. y
𝑖
∗x
i
⋅𝛽
(𝑖)
𝑇
≥1−
𝑖
,
𝑖
≥0.
()
Applying KKT conditions, () can be transformed into ();
thenitcanbesolvedindualspace:
max
𝛼
𝑛
𝑖=1
𝑖
−
1
2
𝑛
𝑖=1
𝑛
𝑗=1
𝑖
𝑗
𝑖
𝑗
𝑖
,
𝑗
,
s.t.0≤
𝑖
≤, =1,2,...,.
()
Equality Optimization Constraints Based ELM. With the
parameter setting of
1
=2,
2
=1, =2, =1,and
equality constraint, the general ELM optimization formula
is equivalent to () which can be used in regression and
classication:
min
𝛽
(𝑖)
,𝜉
𝑖
1
2
𝛽
(𝑖)
2
+
𝑛
𝑖=1
2
𝑖
,
s.t.x
i
⋅𝛽
(𝑖)
𝑇
=y
𝑖
−
𝑖
, =1,2,...,.
()
e corresponding KKT optimal conditions are shown in
𝛽 =H,
𝑖
=
𝑖
, =1,2,...,,
h x
𝑖
𝛽 −y
𝑇
𝑖
+
𝑇
𝑖
=0, =1,2,...,.
()
Further, the nal output is given in
f
(
x
)
=h
(
x
)
𝛽 =h
(
x
)
H
𝑇
I
+HH
𝑇
−1
T.
()
2.2. ELM for -Insensitive Regression. For regression, ELM
provides general model for standard setting. It achieves
better predictive accuracy than traditional SLFNs []. In
addition, many variants and extensions of ELM regression
algorithms have been proposed. Inspired by Vapnik’s epsilon
insensitive loss function, []proposed-insensitive ELM. Its
optimization formula is as
min
𝛽
1
2
×
𝛽
2
+
2
×
(
x
)
𝛽 −y
2
𝜖
,
()
where is insensitive factor and the error loss function is
calculated by
(
x
)
𝛽 −
2
𝜖
=
𝑛
𝑖=1
𝑖
−
𝑖
2
𝜖
,
with
𝑖
−
𝑖
𝜖
=max 0,
𝑖
−
𝑖
−.
()
Compared with conventional ELM regression, ELM with
-insensitive loss function uses margin to measure the
empirical risk. It controls the sparsity of the solution []and
is less sensitive to dierent levels of noise []. In this paper,
we extend ELM regression algorithm based on this variant.
3. Missing Data Problem in ELM Learning
3.1. Missing Data Problem. Nowadays, with ever-increasing
datavelocityandvolume,missingdatabecomesacommon
phenomenon. Generally, there are two missing patterns, that
is, missing feature and missing label. In this paper, we focus
on the issue of missing feature.
From the causes of missing data, there are two cir-
cumstances. In the rst circumstance, the missing features
exist but their values are unobserved for the reason that
information is lost or some features are too costly to be
acquired []. Examples of such case can be found in many
domains. Sensors in a remote sensor network may be dam-
aged and fail to collect data intermittently. Certain regions
of a gene micro array may fail to yield measurements of the
underlying gene expressions due to scratches, ngerprints, or
dust []. Second is inherently missing. In this circumstance,
dierent samples inherently contain dierent features. For
instance, in packed malware identication, instances contain
some unreasonable values. In the web-page task, one useful
featureofagivenpagemaybethemostcommontopicof
other sites that point to it. If this particular page has no
such parents, however, the feature is null, and should be
considered structurally missing []. Obviously, imputation
for this circumstance is meaningless.
3.2. Traditional Approaches for Missing Data Learning. Gen-
erally, there are three approaches for dealing with missing
features in machine learning. e rst approach is omitting,
which includes sample deletion and feature ltering. Sample
deletion simply omits the samples containing missing fea-
tures and applies standard learning algorithms in the remain-
ing samples. An example is shown in Figure ; 2
with two missing features is deleted. Feature ltering omits
the features that are missing in most samples. Figure
interprets this approach. Obviously, the advantage of omitting
based approaches is simple and computationally inexpensive.
Notably, the key point of omitting is keeping as much as pos-
sible useful information while omitting. But it is dicult to do
that. Both of them inevitably omit some useful information.
When there is massive information retained aer being partly
omitted, this approach can be a better choice. Otherwise,
in the situation of much useful information being omitted
while few being retained, this kind of approaches aects
learning precision seriously. Second approach is imputation.
In data preprocessing phase, missing features are lled with
most possible values []. Simple imputations ll the missing
features with some default value such as zero or average
value of other samples. Complex imputations use some
probabilistic density function or distribution function to
estimate the missing features. e computational complexity
of imputation varies with dierent estimation methods.
Imputation makes sense when the features are known to
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