FEATURE WEIGHTING FOR NEAREST NEIGHBOR ALGORITHM BY
BAYESIAN NETWORKS BASED COMBINATORIAL OPTIMIZATION
Iñaki Inza (
ccbincai@si.ehu.es)
, post

graduate student.
Dpt. of Computer Sciences and Artificial Intelligenc
e. Univ. of the Basque Country. Spain.
Abstract

A new approach to determining feature weights for nearest neighbor classification is explored.
A new method, called kNN

FW

EBNA (Nearest Neighbor Feature Weighting by Estimating Bayesian
Network Algorit
hm), based on the evolution of a population of a discrete set of weights ( similar to
Genetic Algorithms [2] ) by EDA approach ( Estimation of Distribution Algorithms ) is presented.
Feature Weighting and EDA concepts are briefly presented. Then, kNN

FW

EB
NA is described and
tested on Wave
form

21 task, comparing it with
the classic K

NN unweighted version.
Nearest neighbor basic approach involves the storing of training cases and then, when given a
test instance, finding the training cases nearest to th
at test instance and using them to predict the class.
Dissimilarities among values of the same feature are computed and added, obtaining a representative
value of the dissimilarity between compared pair of instances. In basic nearest neighbor approach,
dis
similarities in each feature are added in a ‘naive’ manner, weighing dissimilarities equally in each
dimension. This approach is handicapped, allowing redundant, irrelevant, and other imperfect features to
influence distance computation. However, when feat
ures with different degrees of relevance are present,
the approach is far from the bias of the task. See Wettschereck et al. [13] for a review of feature
weighting methods for nearest neighbor algorithms.
A search engine, EBNA [5], based on Bayesian net
works and EDA [11] paradigm, is the basis to
state the algorithm within search parameters: searching for a set of discrete weights for the nearest
neighbor algorithm.
Evolutionary computation groups a set of techniques
–
genetic algorithms, evolutionary
programming, genetic programming and evolutionary strategies

inspired on the model of organic
evolution, which constitute probabilistic algorithms for optimization and search. In these algorithms, the
search in the space of solutions is carried out by me
ans of a population of individuals
–
possible solutions
to the problem

which, as the algorithm is developed, evolves through more promising zones of the
search space. Each of the previous techniques requires the design of crossover and mutation operators
in
order to generate the individuals of the next population. The manner in which individuals are represented
is also important, because depending on this and on the former operators, the algorithm will take into
account, in one implicit way, the interrela
tions between the different pieces of information used to
represent the individuals.
An attempt to design evolutionary algorithms for optimization based on populations that avoid
the necessity to define crossover and mutation operators specific to the p
roblem, and which are also able
to take into account the interrelations between the variables needed to represent the individuals, is the so
called Estimation of Distribution Algorithms (EDA). In EDA there are no crossover nor mutation
operators, the new p
opulation is sampled from a probability distribution which is estimated from the
selected individuals.
This is the basic scheme of EDA approach:
D
o
Generate
N
individuals ( the initial population ) randomly.
Repeat for
l = 1,2,...
until a stop cr
iterion is met.
DS
l

1
Select
S <= N
individuals from
D
l

1
according to a selection method.
p
l
(x) = p(x  DS
l

1
)
Estimate the probability distribution of an individual being
among the selected individuals.
D
l
Sample
N
individuals ( the new population ) from
p
l
(x).
The fundamental problem with EDA is the estimation of
p
l
(x)
distribution. One attractive
possibility is to represent the n

dimensional
p
l
(x)
distribution by means of the factorization provided by a
Bayesi
an network model learned from the dataset constituted by the selected individuals. Etxeberria and
Larrañaga [5] have developed an approach that uses the BIC metric ( Schwarz [12] ) to evaluate the
fitness of each Bayesian network structure in conjunction w
ith a greedy search ( Buntine [4] ) to obtain
the first model. The following structures are obtained by means of a local search that starts in the model
found in the previous generation: this approach was called EBNA ( Estimating Bayesian Network
Algorithm
). We will use this EBNA approach as the motor of the search engine that seeks for an
appropriate set of discrete feature weights for the nearest neighbor algorithm.
After Feature Weighting ( FW ) problem and EBNA algorithm are presented, the
kNN

FW

EBNA approach can be explained:
kNN

FW

EBNA is a feature weight search engine based on the
‘wrapper idea’
[7]: the search is
guided through the space of discrete weights using 5 fold
–
cross validation error of the nearest neighbor
algorit
hm in training set as evaluation function. In order to learn the n

dimensional distribution of
selected individuals by means of Bayesian networks, populations of 1,000 individuals are used. Half of
the best individuals, based on the value of the evaluation
function, are used to induce the Bayesian
network that estimates
p
l
(x)
. The best solution found through the overall search is presented as the final
solution when the next stopping criterion is reached: the search stops when in a sampled new generation
of
1,000 individuals no individual is found with an evaluation function value improving the best
individual found in the previous generation.
D
o
Generate 1,000 individuals randomly and compute their 5 fold
–
cross validation error.
Repeat for l = 1,2,
... until the best individual of the previous generation is not improved.
DS
l

1
Select the best 500 individuals from
D
l

1
according to the evaluation function.
BN
l
(x)
induce a Bayesian network from the selected individuals.
D
l
Sample by PLS [6]
1,000 individuals ( the new population ) from
BN
l
(x)
and
compute their 5 fold
–
cross validation error.
Experiments with 3 (0,0.5,1),5 (0,0.25,0.5,0.75,1) and 11 (0,0.1,0.2,...,0.9,1) possible weights
were run on the Waveform

21 task ( see Breiman et a
l. [3] for more details ): a 3 class and 21 feature task
with features with different degrees of relevance to define the target concept. In the learned Bayesian
network each feature of the task was represented by a node, its possible values being the set o
f discrete
weights used by the nearest neighbor algorithm. Except for the weight concept, dissimilarity function
presented in Aha et al. [1] was utilized, using only the nearest neighbor to classify a test instance (1

NN).
10 trials with different trainin
g and test sets of 1,000 instances were created to soften the random starting
nature of kNN

FW

EBNA and reduce the statistical variance. This sample size was vital to work with a
standard deviation of the evaluation function lower than 1.0 %: higher levels
of uncertainty in the
estimation of the error can overfit the search and evaluation process, obtaining high differences between
the estimation of the error in the training set and the generalization error on unseen test instances.
We hypothesized that t
he unweighted nearest neighbor algorithm would be outperformed by
kNN

FW

EBNA in each set of possible weights. Average results on test instances are reported in Table 1,
instances which do not participate in the search process.
Accuracy level
CPU time
Stopped generations
Size of search space
Unweighted approach
76.70 %
0.30
%
30.5
6


3 poss楢汥lw敩ghts
77.72
┠
0.37
%
㠵
,
5
68.0
3
,3,3
ⰳ
ⰴ
ⰲ
ⰲ
ⰲ
ⰳ
ⰳ
3
21
5 possible weights
77
.
85
%
0.50
%
103,904
⸰
4,4,3,3
ⰳ
,4,3,3
ⰴ
ⰳ
5
21
11 possible weights
77
.
76
%
0.5
4 %
ㄱ1
,
128
⸰
4,4,4,4,4,4,3,3
,4,4
ㄱ
21
Table 1: average accuracy levels, standard deviation
s and average running times in seconds for a SUN
SPARC machine from 10 trials for Waveform

21 task are presented. Generation where each trial stopped
( initial population was generation ‘0’ ) and search space cardinalities are also presented.
A t

test wa
s run to see the degree of significance of accuracy differences between the found
discrete set of weights and the unweighted approach: thus, differences were always significant in a
= 0.01
significance
level.
On the other hand, differences between feature weighting approaches were
never statist
ically significant:
a bias

variance trade

off analysis of the error [
9] respect the number of
possible discrete weights (3,5 or 11) which will give us a better understanding of the problem. High cost
of kNN

FW

EBNA must be marked, increased with the size of the search space.
The work done by Kelly and Davis [8] with gene
tic algorithms and that done by Kohavi et al.
[10] with a best

first search engine to find a set of weights for the nearest neighbor algorithm can be
placed near kNN

FW

EBNA. This work can be seen as an attempt to join two different worlds: on the
one hand
, Machine Learning and its nearest neighbor paradigm and on the other, the world of the
Uncertainty and its probability distribution concept: a meeting point where these two worlds collaborate
with each other to solve a problem.
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