BASEMENT
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.M414
OeNNeV
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ASYMPTOTIC PROPERTIES OF UNIVARIATE
SAMPLE K-MEANS CLUSTERS
Anthony Wong
Massachusetts Institute of Technology
M.
Working Paper #1341-82
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ASYMPTOTIC PROPERTIES OF UNIVARIATE
SAMPLE K-MEANS CLUSTERS
M. Anthony Wong
Massachusetts Institute of Technology
Working Paper #1341-82
Key words and phrases:
k-means clusters, wi thin-clusters sum of squares,
cluster lenghths, non-standard asymptotics.
AMS 1980 subject classification:
Primary 62H30.
ABSTRACT
A random sample of size
N
is divided into
k
clusters that mini-
mize the within clusters sum of squares locally.
Some large sample
properties of this k-means clustering method (as
k
N)
are obtained.
In one dimension,
approaches
°°
with
it is established that the sample
k-means clusters are such that the wi thin-cluster sums of squares are
asymptotically equal, and that the sizes of the cluster intervals are
inversely proportional to the one-third power of the underlying density at
the midpoints of the intervals.
The difficulty involved in generalizing
the results to the multivariate case is mentioned.
0745^7^
INTRODUCTION
1.
Let the univariate observations
a
distribution
x-,,
with density function
F
tions are partitioned into
X2,
f.
.
.
,
x^,
be sampled from
Suppose that these observa-
groups with means
k
.
z,,
...,
z.
no movement of an observation from one group to another will
such that
reduce the
within groups sum of squares
WSS., =
^
min
Z
x.
i=l l<j<k
"
This method for division of
in groups sum of squares
k-means.
as
-
z
.
•
I
J
^
a
'
sample into
k
groups to minimize the with-
locally is known in the clustering literature
In one dimension,
the partition will be specified by
k-1
cutpoints; the observations lying between common outpoints are in the
same group.
See Hartigan (1975) for a detailed description of the k-means
method, and see Hartigan and Wong (1979) for an efficient computational
algorithm.
This method has been widely used in various clustering
applications (see Blashfield and Aldenderfer, 1978).
properties (when
N
^
for fixed
<=°)
Its asymptotic
have been studied by MacQueen (1967),
k
Hartigan (1978), and Pollard (1979).
Here, the sampling properties of
k-means clusters when
°°
k
approaches
with
N
are presented.
The properties of the univariate population k-means clusters when
k
->
CO
are given in Wong (1982a).
It
is
shown that for large
k,
the
optimal population partition is such that the wi thin-cluster sums of
squares are equal, and that the sizes of the cluster intervals are
inversely
proportional
to the one-third power of the underlying density
at the midpoints of the intervals.
In
-1-
this paper, non-standard asymptotics
k
^
°°
k-means clusters for samples from
a
general popula-
are used to obtain the asymptotic properties (when
the locally optimal
tion
on
F
[0,1];
in particular,
with
N)
of
it is shown that the locally optimal
partition approaches the population optimal partition under certain re-
gularity conditions.
are:
(1)
The special difficulties in showing this result
the number of clusters
approaches
k
the length of each cluster interval
°°
with
(such that
H
approaches zero while the number of
observations in each cluster approaches infinity), and (2) the locations
and sizes of the cluster intervals are determined by an optimization pro-
cedure (so all
for all
results concerning the clusters need to hold uniformly
clusters)
Theorem
1
.
and Theorem 2, respectively, gives the asymptotic expres-
sion for the lengths and the within-cluster sums of squares of the
locally optimal k-means clusters.
in Section 2, and Theorem 2
is
The result of Theorem
proved in Section
3.
1
is
obtained
Some concluding
remarks are given in Section 4, in which the difficulties involved in
generalizing these univariate results to many dimensions are also mentioned.
-2-
<
ASYMPTOTIC LENGTHS OF THE LOCALLY OPTIMAL
SAMPLE K-MEANS CLUSTERS
2.
Let
Xp,
X-,,
.
.
.
be
X|M
,
random sample from
a
density
a
positive and has four bounded derivatives in [0,1].
derivative of
=
b
f(x).
,
p.
at
f
Suppose that the
and let
Denote the
=
B
q^J^P^
i
is
th
and
f(x)
observations are grouped into
N
z,<Z2 <...<
clusters with means
k^i
f^^^x),
by
x
which
f
so that the within clusters
z^
um of squares of this locally optimal
k^,-parti tion cannot be decreased
by moving any single observation from its present cluster to any other
cluster.
Denote the
pth
order statistic by
X/
and let
>
be the
n-
i
number of observations in the
Then
cluster.
jth
x
t=0
x
are the observations in the jth cluster, where
.
n
2
(
-^
,
(Jp
.
.
.'
n,+l)
^
And the
n|-,=0.
)
t=0 ^
length
and the midpoint
e.
[x
J
X/
.
(
n^n)
^z
of the jth cluster are defined to be
m.
J
.
,
^J-^
-
]
n
and
1/2
[x,
ly, where
x^^^
=
t=0
and
x^j^^^^
=
.
,
^
1.
Theorem
There may be many locally optimal k-maans partitions.
states that for any such partition, if
max
Ik^ e. f(m.)^/^
N
J
J
V-J^k^
To show the result of Theorem
consequence of
a
-
1,
AU
respective-
]
(^l^ n^)
t-0
n +1)
i
)
t=0
t=o
+ x
.•
^
k^,
=
f{x)^^^ dx
we need
o([N/logN]
|
=
1/3
),
1
then
(1).
o
p
a
few lemmas.
Lemma
1
is
a
theorem of large deviations given in Feller (1971).
direct
It
in proving Lemma 2 which states
useful
is
that if the
are large
n.'s
enough, the cluster means are suitably close to the cluster midpoints.
This result is then used in the proof of Theorem
1
to determine the
relationship between the lengths of neighboring clusters.
Lemma
1
:
Let
density
Put
...,
be a random sample from
X|u
J^^
fix)
,
and
b
=
^''^
,
contained in an open subinterval
E[z.-Uj]
let
and
=
E[(z.-Uj)^]
oj
Proof
(or
Zp
of [0,1] by
C,
observations
n
z^j
•
• •
and
D,
and
not depending on
N
-1
as
n
1/2
^
^0
n~
^1 +
.
1
n^N(log N/N)^/^b/16,
..+^n
-
as
uJ
n
>
->
we have
°°,
(41ogN)^/2}/(2ll)"^/^(41ogN)"^/2N"2-l
f
has bounded derivatives and
Ot
n
1
a
distribution not depending on
follows.
contains at least
k^,
=
o
N(logN/N)
and
Uj
(In the proof of Theorem 1,
large enough and if
of this
549),
p.
°=.
Now since
has
Uj
:
since (41ogN)
;
and
z„>
,
From the theorem of large deviations given in Feller (1971,
D
with
F
a^.
=
N^N^
such that if
I)
Denote the
f(x).
I
Then there exist constants
and
distribution
a
which is positive and has four bounded derivatives in [0,1].
f
=
B
Xp,
X-,,
a.
it will
(or I),
'-^
"
I
^^t
the lemma
be shown that when
N
is
([N/logN]
1/3
'
1/3
'
observations, and hence the result
lemma can be applied.)
b/16
),
I
'
n
'
then each of the
k^,
clusters
the application of Lemma 1,
In
tions in an open subinterval
mately
NF(I),
where
=
F(I)
will be the number of observa-
n
I
of [0,1].
/r
dF.
Therefore,
More precisely, using Donsker's
theorem for empirical processes (see Billingsley 1968,
^^P] "j
where
ny
is
-
p.
141), we have
(N^/2)
=
NF(I)
is approxi-
n
(2.1)
I
the number of observations in
over all open subintevals of
used in the proof of Lemma
I.
and the sup is taken
I,
Both Lemma
and Equation (2.1) will be
1
to give a uniform estimate of the deviations
2
between cluster means and midpoints for those clusters that are large
enough.
Lemma
2
:
Let
Xo,
.
.
.
,
be
x^,
random sample from
a
whose density
F
=
^"P.
0<Xil
f(x), b = n"""^, f(x), and
Oi:Xi:l
=
F(I)
Let
dF.
/.
z.
be the mean
1
I
=
xdF/F(I)
be the
of the observations in an open interval
I,
conditional mean of
be the size of the interval
Then there exists
P^{ ^'f
^i~'^^^\~^i
taken over all
a
-
F
on
constant
^i\
^
and
I,
^0
Sj
lit
(logN/N)^''^} =
statistics) containing at least
/t
I
1
-
o(l),
where the sup
(whose boundary ooints are order
N(logN/N)
1/3
'
b/16
observations.
Proof:
For any
J
N^N
,
I.
such that
C
open intervals
f
Put
positive and has four bounded derivatives in [0,1].
is
B
x-|,
consider an interval
-5-
I
of the form
is
.
(X(p), x^p^^
where
^^p,
Using Lemma
n^
2
(log N/N)^/\/16.
N
(first conditioning on the two order statistics and
1
then integrating out), we obtain
p/aj^
Zj
rij^/^l
-
Uj
C(log N)^/^}
>
DN"^ (log N)"^/^
<
I
where
2
a. = fr
(x -
Ur)
2
dF/F(I)
conditional variance of
=
Now, by the Taylor series expansion of f,
2(2)
1
+ T"(x-mj) f
(fir)
of
I
and
is between
q^^
=
F(I)
^^^
^^'^
(q^^
x
f(mj) Sj
and
^
^I
^
^T2
aj
=
y^
S^
[1
P^
^
{
the midpoint
is
2
•
'l
4
* 0(^1^'
^"^
term depends on the bound of the second
f)
It follows
that (2.2) can be written as:
vl2
Sj^
[1
'
+ 0(s^)]
(Note that the constant in the
deri vati ve of
f^
Therefore,
m..
r
f(mj
•
m.
I.
[l+0(s^)]
1
=
on
+ (x - mr)
fim-r)
where
^'
^'^
^
=
f(x)
F
(2.2)
+
0(s^)]
nj^/2
I
5j
-
ujl
:.
CdogN)^/^}
DN"^ (logN)"^/^
Since the number of possible intervals
I
is
bounded by
N
2
,
we have
2
P,
^JPsjl
(
[1.0
n^l/2
(s2)]
-uj
l-zj
<
JLc(logN)l/2j
I
/1
^
D(1ogN)'^/^
-
1
=
1
-
o(l).
Now from (2.1), we have uniformly in I,
Ht
>-2NF(I)
>
j
with
Nb Sr
probability tending to one.
Therefore,
{Sup 3jl/2
where
=
C„
form
—
2
b
-1/2
^^1
^
^^ (logN/N)^/^}
^
1
as N
and the sup is taken over all intervals of the
C,
X(p^^
(x^p),
_
|-^
p^
with
^^j)
The result of Lemma
n^
^
N
(
logN/N)
^^^
b/16.
shows that if the k-means clusters are large
2
enough, the cluster means are suitably close to the cluster midpoints.
When combined with some well-known properties of locally optimal k-means
partitions, this result is useful in establishing the relationship between
The main difficulty to be overcome
the lengths of neighboring clusters.
in the proof of Theorem
1
showing that all the clusters are large
is
enough.
Theorem
1
Let
:
X-,
Xp,
,
.
whose density
f
Put
f(x)
B
-
Q^JJP^
.
is
be the length of the
the sample.
.
,
be
X[u
a
random sample from
distribution
a
F
positive and has four bounded derivatives in [0,1].
and
jth
b
=
Q^^^^f(x),
cluster of
Then, provided that
k^,
-7-
e^
(
j
locally optimal
a
=
let
o
([N/logN]
1/3
),
=
1,
2,
.
.
k^^^-parti
we have
.
,
k^)
tion of
max
where
m.
I
is
n1/3
r,
1
.1
the midpoint of the
n1/3
r,
,1
,
,s
cluster.
jth
J
Proof
:
Consider
a
locally optimal
(whose boundary points
Denote the open interval
containing the
kj^-parti tion with
cluster by
jth
k^,
([N/logN]
=
1/3
).
order statistics)
?i.rQ.
Then, as before, we have
I..
^^^^(^•)
=
u.
/
.
X
dF
/
F(I
.)
^
The proof is in three parts.
of length bounded by
^
.
part
In
-^^
1
.
nij
neighboring clusters contain at least
In part
II,
l/(2k^),
N(logN/N)
•
(
e.
I
and
is given by
-rr-^)
f(m.)
^
k^,
1
+ k./
(1).
p'
N
-^
II
clusters satisfies
1/3
observations.
b/16
a
bound of the ratio
Since at least one of
qualification:
1/3
/k|^
^
e- ^
applying parts
l/(2k|^),
By (2.1), the
statements are to be read as if they included the
"with probability tending to one as
Suppose that
vations.
2(B/b)
repeatedly gives the result of this theorem.
To avoid wordiness,
[I]
then both it and its
J
vj
the
(2.3)
using the result of Lemma 2, the relationship between the
lengths of neighboring clusters is established;
—^
.] . 0(e])
it is shown that if a cluster is
I,
and
2{B/b)'^^^/k^
4
2
.
2(B/b)
jth
^'''^/k^
^
e^ ^
l/(2k^).
cluster contains at least
N
Then
Nb/(2k|^)
Therefore, the number of observations in the
exceeds
Nb/4k^,
exceeds
N(logN/N)^/^ b/4.
eventually.
Since
-8-
k^,
=
([N/logN]
approaches infinity'
F(Ij)
-
jth
1/3
),
^
b/(2k^).
(N^^^)
obser-
cluster
this number
Applying Lemma
-
\~z.
where
the
Ujl
<
Cq
(logN/N)^/2 e}^^,
(2.4)
the mean of the observations in the jth cluster.
is
z-
to the jth cluster, we have
2
cluster,
(j-l)st
a
cluster adjacent to the jth cluster.
(j-l)st cluster is
largest observation in the
x
._
smallest observation in the jth cluster is
x
.
optimality, the midpoint
local
.
and
.
X
(^^
"t^
^
t=0
.
.
n. +
4=0
i^_^
=
z.
,
and
z-
5.
n
+
1)
''
.M
=
i. - X
"t
must
lii
t=0
^
.Op ([logN/ND,
^_^
^
since the largest gap between successive order
statistics is
It
Then by
1)
,
(^'^
ej.i^M-
between
M
\^'
,
{^z
t=0
The
and the
,
(t=0
Consider
([logN/N]
follows from (2.3) that
^J-1
^'j " "j^
'
And from (2.4)
,
^
hj
^ Op([logN/N])
we obtain
-j-1 '
i^j
^
|ej
-
Co [logN/N]^/2
^
l/(8k^), since
^_l/2
e^.
.>
Therefore, it follows from (2.1) that the
at least
[II]
Nb/(8k^)
0p(Nl/2^
Now, applying Lemma
^J-1
Since
-
M
-
I._^
-
"j-ll
=
i.
-
'
2
,
N
to the
^0
^
g^ ([logN/Ni:
l/(2kj^).
(j-l)st
[logN/N]^/^ b/16
(j-l)st
contains
observations eventually.
cluster, we
(1ogN/N)l/2^._^l/2.
m, we have
interval
h
ave
(2.5]
+
[nij_i
-^j.i
+ Op([logN/N])]
z._^
-
=
~z.
-
[m.
-
^^
+ Op( [logN/N]
) ]
(2.6)
From (2.3), (2.6) can be written as
f^^^("ii
1
t^j-1 - 12
f^j -
Let
T2
•
-fT^
0(ej^)]
=
ej
•
^j
[fVf(mj)]"^/^
f* = f(mj)
sion of
about
f
-
\ f^^^m^)
-
-
0(^?
-
i^j^
-
\
2
•
-
-
1
^^J-l^ ' 2 ^J-1^
°
-
^J-1
^J
-
20p([logN/N])
^
m.
and
+ Op
(logN/N)],
e^ + 0(ej)
+ Op
(
[logN/N]
(2.7)
m,-_-i-
+ O(ej^)
[1
=
Then
by the expan-
)
m..
Similarly, by the expansion of
e,-itl
4
'U
•
be the density at the midpoint between
f*
since
?
i)
f(ni.;j
•
^^(^YT^
f
0(ej.,)l
about
=
ni-j_i'
we have
e..,[fVf(.,..,)]-^/^
^ 0(e^.,)
1
.
Op(logN/N)].
Therefore, (2.7) becomes
-
("j.l
\
Zj-i) +
^j
\
ej.i
[f*/f(mj_i)]"^^^[l
[^'Vf(m.)]"^''^[l + 0(ej)] +
10-
2
+
0(ej_^)l
=
(i,
'J
Op (logN/N).
-
uj
+
J
(2.8)
'
Combining (2.4), (2.5), and (2.8), we can first show the ratio
is
e.
,/e.
bounded, and then
-
,'j-
[f(mj_^)/f(mj)]^/^
•
-
^
li
[fVf(nij)]^/^(logN/N)^/2ej-^/2
4 Cq
^J
+ 2ej"^ Op(logN/N) +
<
k~/
[4
/2
(e^)
(2.9)
CQ(B/b)^/\^2/2(logN/N)^/2
+ 4k^ Op(logN/N) + 0(kj;;^)]
^n' °p(^)
and this bound does not depend on the intervals involved.
From the first inequality in (2.9), we can show by contradiction
[Ill]
that at least one of the
:j
e.
i
k|^
cluster intervals satisfies 2(B/b)
1/3
'
comparisons of adjacent clusters, we obtain
(e^./e.)
•
uniformly in
[f(m.)/f(mj)]^/^
<
1
=
+
[1
i, j
<
f (m
^^^ ^
/I f(x)
^
k'^Opd)]
-
1
+Op(l)
k|^.
k^
Since
z
I
/k^
Then using the bound in (2.9) and carrying on at most
l/(2k^,).
•J
k^
e
.
J
.)
J
^^"^
^
fol lows.
•
11-
dx
as
N
^ -,
the theorem
ASYMPTOTIC WITHIN-CLUSTER SUMS OF SQUARES OF THE
LOCALLY OPTIMAL SAMPLE K-MEANS CLUSTERS
3.
As in Section 2,
let
x
•_
<...< x
-,
(t=0 "t '
observations in the jth cluster of
of
a
sample from
and let
f,
'^
locally optimal k-means partition
a
be the jth cluster mean.
z-
-
(x
Y,
-
,
.
Z-)
2
In this
•
WSS.,
First,
defined to be
is
k^,
clusters are asymptotically
direct consequence of Feller's theorem on large devia-
a
tions (Lemma 3)
The within-
section, we will show that the
wi thin-cluster sums of squares of the
equal.
n-
(t=o "t)
cluster sum of squares of the jth cluster,
"i
denote the
j
used in the proof of Lemma 4 to obtain
is
estimate of the within-cluster sum of squares, which is
a
uniform
a
function of the
length of the cluster interval and the density at the midpoint of the
interval.
The result of Theorem 2 then follows from Theorem
1
and Lemma
4.
Lemma 3:
Xp
Let
density
Put
B
...,
be
Xju
Q^,JJP^
and
f(x)
b
=
contained in an open interval
E[z.
-
Uj]
=
0,
E[(z.
there exist constants
if
N^ N*
random sample from
a
a
distribution
F
with
which is positive and has four bounded derivatives in [0,1].
f
=
Xp,
and
P^ (Yr^^^n"^/^
-
q^"^^ f(x).
by
I
Uj)^]= Qj,
C*, D*,
z,
,
and
Denote the
z^,
.,.,
E[(z.
and N*,
-
z
observations
n
and let
,
Uj)^] =
not depending on
-,
^
I
<
-.
Then
such that
n^N(logN/N) ^^^b/16,
1
y
(z.
-
~z)^ -
na?
i
>
C*(logN)^/^}
(Since the proof is similar to that of Lemma
12-
1,
it will
^
D* N"^(logN)"^/^
not be given here,
Lemma 4
:
X,, x„,
Let
Xj^
be a random sample from
mean of the observations in an open interval
and
be the size of the interval
Sr
constant
P^
whose density
C*
(SUP
WSSr
I,
cluster sum of squares of the observations in
I,
F
positive and has four bounded derivatives in [0,1],
is
f
...,
Then
I.
be the wi thinbe the midpoint of
m^
I,
Put
there exists a
such that
Sj-5/>SSj
-
^N
f(mj)sj[l + 0(s^)]l
^ C*
N(logN/N)^/^}
1-0(1),
=
where the sup
taken over all open intervals
is
are order statistics)
(whose boundary points
I
N(logN/N)
containing at least
1/3
b/16
observations.
The proof of this lemma is similar to that of Lemma 2; therefore,
only the differences between the two proofs are outlined here.
In this
proof, the Taylor series expansion is carried out to the fourth order
And after some series manipulations, we obtain
terms.
and
al
=
/j(x
-
Uj)^dF/F(I)
=
^ s^
Yj
=
/i(x
-
Uj)^dF/F(I)
=
y^
sj
[1
[1 +
13-
I
(3.2)
0(s^)].
Applying (3.1) and (3.2) to the result of Lemma
because the number of possible intervals
(3.1)
+ 0(Sj)],
is
3
will
give this lemma
again bounded by
2
N
.
Theorem
2
:
Let
X2J
X-,,
whose density
Put
B
.
f
.
is
be a random sample from
x^i
,
b=
f(x).
J^J^^
Let
be the wi thin-cluster sum of squares of the
optimal
a
distribution
F
positive and has four bounded derivatives in [0,1].
and
Q^JJP^ f(x)
=
.
VISS.
=
(j
cluster of
jth
The provide that
k|^-partition of the sample.
1,
k,^
=
2,
..., k^)
locally
a
o([N/logN]
1/3
we have
max
12N"^k„^ WSS,
-
[A
f(x)^/2dx)2
I
oJl)
Proof:
Consider
a
locally optimal
in Theorem
It is shown
tending to one,
G =
1
/„
f(x)
e."^/^
1/3
WSS.
and
(ii)
From
dx.
-
that for all
1
tion with
k^ =
o([N/logN]
1/3
large enough, with probability
N
(i),
^Nf(m.)
where
k"^ f(m.)"^^^ G[l+o (1)],
=
e-
we can apply Lemma 4 to obtain
e^ [l+0(e^)]
|
^
C* N(logN/N) ^/^,
uniformly
I
in
IWSSj
-
from
It follows
l^j^k^,.
^ Nk-2
G^
that, we have uniformly in
(ii)
[l+0p(l)]| ^
2
where
C**
=
-
2C* b
G^
<
I
'° G
Op(l) + C** k^^''^ (logN/N)^/2
'
.
l<j<k^,.
C* N(logN/N)^/2^j^-5/2^-5/6g5/2^
Therefore,
12N"^k^ WSSj
).
the number of observations in each cluster exceeds
(i)
N(logN/N)^'^\/16,
kj^,-parti
=
And the theorem is proved.
•14-
0p(l),
),
Since the globally optimal k-means partition of the sample is
necessarily locally optimal, the results of Theorem
apply to the globally optimal
k^j-partition.
1
and Theorem
2
also
Moreover, the generaliza-
tion of these results to densities with finite support [a,b] is immediate.
-15-
4.
the properties of univariate sample k-means clusters
In this paper,
when
k
DISCUSSION
approaches infinity with the sample size
are presented.
N
This study is of interest in its own right because we want to examine
the sampling properties of this widely-used clustering method.
result in Section
tion
a
also indicates that the k-means method would parti-
2
sample from
The
a
distribution with density
f
on
[a,b]
way that the sizes of the cluster intervals are adaptive to
in such a
the
f;
intervals are large when the density is low while the intervals are small
where the density
is
high.
Therefore, k-means can potentially be used
as a method for constructing variable-cell
gram estimate of
f
histograms.
Indeed,
a
histo-
based on the k-means method is proposed in Wong
(1982b); and it is shown to be uniformly consistent in probability.
The multivariate case requires further investigation as the generalization of univariate results to many dimensions is not straightforward.
An important first step is to determine the configuration of the optimal
population k-means partition in several dimensions.
indication that the best partition in
R
is
There is some
into regular hexagons (Wong
1982c); but it is not clear that the best partition is into regular polytopes for higher dimensions.
16-
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BKs^-^srl
Date
Due
Lib-26-67
HD28IV1414 no 1341- 82
Wonq, M. Antho/Asymptotic properties
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