Download Nearest Neighbors by Neighborhood Counting

Nearest Neighbors by
Neighborhood Counting
Author : Hui Wang
Reporter : Tze Ho-Lin
2006/11/13
PAMI, 2006
1
Outline

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

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Motivation
Objectives
Method (NCM)
Evaluation
Conclusion
Personal Comments
2
Motivation

Finding nearest neighbors is a general idea that
underlies many artificial intelligence tasks.

Complex real-world applications may generate
new types of data or new combinations of existing
types of data.

This may call for new distance/similarity
functions.
3
Objectives

We propose using the number of
neighborhoods as a generic measure of
similarity which can then serve as a
methodology.
4
Method: NCM
Assumption:
the domains of
attributes a1 and a2
are both {1,2,3,4,5}
the domain of c is {+,-}
Hypertuples of numerical case
t(ai)
Count: 8
5
4
3
2
1
ai
(max(ai)-t(ai)+1)*
(t(ai)-min(ai)+1)
Hypertuples of categorical case
 Domain
:{+,-}
 All hypertuple

(5-2+1)*(2-1+1)
=4*2=8
{Ø} {+} {-} {+,-}
 All

Hypertuples of 1
mi  2
hypertuple of {+}
{+} {+,-}
2
2
mi
mi 1
5
Method: NCM
Ex.
t1
x1
5
4
1
(5-5+1)*(4-1+1)
=4
5
t2
x2
x3=t3
4
1
2
21
2
cov(t3 , t4 )  4 * 8 * 2  64
(5-4+1)*(4-1+1)
=8
6
Experimental result
Fig. 4.
With weighting
•NCM generally performed
well under relatively large k.
•NCM consistently
outperformed HEOM when
k> 1 without weighting and
when k> 11 with weighting.
Table 3
Runtime, in Seconds, where k = 11 and There Is No Weighting
PS.
HEOM: Heterogeneous
Euclidean-Overlap
Metric
7
Conclusion
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Uniform approach for both numerical and
categorical attributes
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It can be used for classification and clustering.
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Efficient–computational complexity is O(n), in
the same order as Euclidean distance, where n is
the number of attributes.
8
Personal Comments
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Application

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Advantage



A lot of nearest-neighbors-required algorithms.
It can be used for classification and clustering.
This function has a simple, easy-to-implement formula.
Disadvantage

This function can’t really measure different categorical data.
9
Hypertuples of numerical case
t(ai)
Count: 8
5
4
3
2
1
ai
(max(ai)-t(ai)+1)*
(t(ai)-min(ai)+1)
(5-2+1)*(2-1+1)
=4*2=8
Hypertuples of 1
10
Hypertuples of categorical case
 Domain
:{+,-}
 All hypertuple

{Ø} {+} {-} {+,-}
 All

mi  2
hypertuple of {+}
{+} {+,-}
2
2
mi
mi 1
11