1
Deparatment of Statistics~ University of North CaroUna~ Chapel Hill~ Nonh
Caro Una. The research of this autho'P and the pzrepaMtion of the manusc'Pipt tDel'6
suppo'Pted by National Science Foundation Gnmt GP-23420.
2
Department of Biomathematics and Med:loal Engineering~ University of Nonh
Chapel [f{,ll~ NoI'th Ca:l'o'Lina~ and Division of Ca:l'dio"Logy~ Veten:ms Administmtion Hoapi tal~ Du'Pham~ No'Pth Caro Una. The Nseareh of this autho'P lJQS
suppo'Pted by NationaZ Institute of Health TMining Grant # 6T-OI-GM-Ol-S04-03.
Ca'POUna~
Estimation of the mode with an application
to cardiovascular physiology
by
Edward J. wegman1
D. Woodrow Benson, Jr. 2
Depazotment of Statistics
Unive'Psity of No'Pth Ca:l'olina at Chapel Rill
Institute of StatiStics Mimeo Series No. 702
August. 1970
Estimation of the mode with an application to cardiovascular physiology.
by
1
Edward J. Wegman
and
Department of Statistios
University of North Caro"tina
Chapel, Hill" North Carolina
D. Woodrow Benson, Jr.
2
Department of Biomathematios
and Medioal. EngineePing
University of Nozrth Caro lina
Chapel, Hill" Nozrth Carolina
and
Division of Cardio Zogy
Veterans Administra:tion Hospital.
Durham" North Caro"tina
Abstract. This paper reviews several proposed methods for estimating the
mode of a probability density function. The method of Chemoff [2] is examined
in greater detail and is shown to be a strongly consistent estimate of a somewhat generalized mode under relaxed conditions on the distribution. Finally, an
application to determining blood flow characteristics is given.
KeywoTds:
Mode
Es timation of Mode
Consistency
Strong Consistency
Aortic Flow
Cardiac Cycle
Cardiovascular
Electromagnetic Flowmeter
Subject Classification Numbers:
60.30
62.10
62.15
62.70
62.97
1
The research of this author and the preparation of the manusoPipt lJere supported by National, Science Foundation Grant GP-23520.
2
The researoh of this authop 1.11GB suppozrted by National Institute of Health
Training Grant # ST-Ol-GM-Ol-504-03.
Estimation of the mode with an applicati.on to cardiovascular physiology.
by
Edward J. Wegman
1
1. Introduction and Review.
D. Woodrow Benson. Jr.
2
The estimation of the mode of a probability dis-
tribution has received the attention of several authors, [2], [5]. [ 6] , [ 8] ,
[10], [ll}, [13] and [14] recently.
Without exception, to demonstrate consis-
teney, these authors at least assume continuity of the density function,
f,
and, of course, define the mode,
That
is,
f(M) > f(x)
for every
M,
x rJ M.
as that number which maximizes
f.
We shall call this type of distribution a
unimodal distribution of Type I.
One could define the mode in several other cases.
infinite
discontin~ty 1n
continuity is the mode.
e
UmX+M f(x) • co,
then
If there is exactly one
the density function, then the location of this dis-
That 1s, 1f there is exactly one M such that
M is the mode of
modal distribution function.
f.
We shall call this a Type II un1-
Even more generally, if the distribution function
may be written as the sum of an absolutely continuous distribution and a discrete
distribution with isolated mass points, then the mass point with largest probability is the mode.
This distribution we shall call a Type III unimodal dis-
tribution.
In the remainder of this section, we shall discuss various proposed estimates of the mode.
1
In Section 2, we shall provide some strong consistency
Department of StatistiC8~ University of North Caro'tina~ Chapel BiZl~ North
Carolina. The I'esea.rch of this author and the preparation of the manusoript ""ere
supported by National, Science Foundation Grant GP-28~20.
2
Depanment of Biomathematics and MedicaZ EngineeI'ing~ University of North
Caro'tina~ Chapel BiZZ, No1't:h Caro'tina.. and Division of CardioZogy, Veterans Administration B08pitaZ~ Duman~ Nonh Carolina. The research of this author ""as
support:ed by National. Institute of Health Training Grant # ST-OI-GM-Ol-504-08.
2
results when the underlying distributions are of Type II and Type III.
~e
In
Section 3, we give an interesting physiological application.
Estimates of the mode are obtained directly or indirectly.
The indirect
estimates are found as by products of density estimation procedures and include
estimates found in Parzen [10], Nadaraya [8], and Wegman [14].
Parzen and Nadaraya use the kemal-type density estimation and Wegman uses
a maximum likelihood technique.
The indirect estimate of the mode is chosen as
the mode of the density estimate.
There are several direct estimates of the mode.
sequence
an
converging to
0
Chemoff [2] chooses a
sufficiently slowly and picks as his estimate
the center of the interval of length
2a
which contains the most observations.
n
This estimate is identical to the estimates of Parzen and Nadaraya when the
kernal is chosen to be the uniform kema!.
e
available for this estimate.
sistency and
the
~Tadaraya
Several consistency results are
Chemoff shows consistency.
Parzen obtains con-
strong consistency as by products of similar results for
densityestimates~
Venter [13] and Dalenius [51 apparently independently propose the same a1temative estimate.
Whereas Chemoff fixes the length of the interval and
chooses the interval containing the most observations, Venter and Dalenius fix
the number of observations and choose the shortest interval containing this
fixed number of observations.
Venter also provides a proof of strong consistency.
This estimate is related to the "nearest neighbor" density estimate of
Loftsgaarden and Quensenberry [71.
A refinement of the technique proposed by Venter and Dalenius is found in
Robertson, Cryer and Hogg [11].
They define a function
shortest interval containing ken)
observations.
choose the smallest interval containing
k[k(n)]
ken)
and choose the
Within this interval they
observations, repeating until
3
a smallest interval contains something close to a fixed number of observations.
These authors also provide strong consistency results.
Finally, Grenander [6] provides several estimates. one of which
is consistent.
he shows
Grenander's estimate is based on the fact that raising a density
to a power makes the mode more and more pronounced.
Grenander's estimate is ap-
pealing because it uses all of the data, whereas the other estimates throw more
or less of the data away.
Dalenius [5]
concludes from a Monte Carlo study that
the Chemoff-type and Venter-type estimates are on the average closer to the
mode than the Grenander-type but the fonner have larger variances than the latter.
Finally. we remark that Cherooff. Venter and Grenander each develop the asymptotic distribution theory for their respective estimates.
4
>e
2.
Strong Consistency.
We shall lim\t our attention to strong consistency
results for the Chernoff-type estimate.
any type unimodal distribution.
scribed later and let
(in,r )
n
Let
Xl' ~, ••• , Xn
Let
an
be a sample from
be a sequence of numbers to be de-
be the interval of length
2a
n
containing the
most observations.
We shall prove two theorems: one applying to Type I and Type II distributions and the second to Type III distributions.
The first theorem shall re-
quire a somewhat unusual condition on the dis tribution which we shall call
is a sequence of intervals with
converging to
or to
+~
·ao
and with
that the probability measure of
c
- b
n
(b, c)
n n
converging to
n
(a, a+c -b)
n
or of the form
n
I
f
(b, c)
n n
0,
c
n
both
it is clear
In addition to this
eventually be less
is either of the form
n where I n
(a-c +b , a).
n
n
f (x), eventually decreases monotonically to
shall assume that
o.
converges to
we shall require that the probability measure of
than the probability measure of
band
n
This condition is met if the density,
o
as
x ....
+~
or
-ao.
Also we
is either left- or right-continuous at each of its finite
discontinuities (if any).
T~
2.1.
dition I and if
If the distribution
a
n
converges to
0
F
is of Type I or II and satisfies Con-
more slowly than
l£og (£og(n)} In,
~~d L:: ;:er;::;:nE:;iD:~ =w~ P:::i::Y.~~
F
n
is the empirical distribution function.
probability 1.
then
IFn(xl-F(xlI and
Smirnov [12] shows that
0'
has
A proof in English may be found in Chung [3] or in Csaki [4].
W
It is evident that
0' c [ n....
Pim
..;-..ao ....!!.
a
= 0].
We shall restrict our attention to
n
points in
e
If
Q'.
in
fails to converge to
M with probability one, by the extended
Bolzano-Weierstrauss Theorem, there are points in
0'
for which one of three
5
things may happen:
.e
1.
.tim
.6Ltp
l n • +ao
2.
tun
.in6
r
3.
There is a subsequence
n
._CD
lnj
inj
such that
~ M.
l
"
The cases 1 and 2 are similar so we shall investigate Um..in6 r
is a subsequence
f(M)
r nj
~ f(x».
(or
F
n
j
IF
n
Dividing by
n
j
(r
n
j
2au
j
j
(tn ) .
Choose
a < M so that
There
0 < f(a-) <
~
+
IF
j
-)-F(r
n
j
)I
j
and letting
F
-.
-CD.
Now
-) - F
(r
n
(2.1)
diverging to
•
n
(r
n
n
j"
n
j
)
I+
F (l )
-) -
IF(r
n
j
)-F(£.
n
j
)
I.
j
CD,
F(r
.
nj Ilj
n j nj
Um .6up --~..oe-2':""a---n
)-F(t
(t
j
s Urn
) - F(£'
n
.6Up
j
n
j
2an
)
j.
j
By Condition I,
F(a) - F(a-2an )
eventually •
2a
n
Thus
Thus we may obtain a subsequence of
n
j
(for convenience let us relabel it
such that
tun
j ..
CD
~
f(a-).
n )
j
6
On the other hand, let
.e
center at the mode,
M.
(in'll, r *)
n
2a
with
n
If the Type I distribution has a j\IDP at the mode, a
slight chaug~ in the choice of
F(r
D
j
IF(r
~j
*)-F(t 'II)
u
be the interval of length
(.t:,r *)
n
is uecesoary.
Consider
~
j
+
*)-F (r"*->I
Dj u j
IF(t *)-F (t
nj
n j nj
*>1
As before
6
D:_ •
(2.2)
F(r
nj 'II)
"""'" ..(.n
nj 'II>
- F(t
2a
n
Urn ln6
(r
n
l
j
*-) - Fn (tn
2a
j
f(M)
(t *,r if)
infinitely often.
n
(t t r )
n n
But
i. o.
is less than the number of
was chosen to be the
(t,r)
n n
interval with the most observations so that we have a contradiction.
Um in6 r
n
.,. -co.
in t:h~
In either case we have
That is to say, the number of observations in
n
j
in the Type I distribution or -
Fn(rn -) - Fn(tn > < Fn(rn *-) - Fn(tn'II>
observations
j
'II)
nj
But the left-hand-side is either
Type II distribution.
~
Fn
Similarly
.tim
~up
I
Thus
.,. +me
n
tn j
Suppose then there is a subsequence
.lnj
such that
-+
t .,.
M.
By anal-
or
f(i+),
ysis similar to that we used to obtain (2.1), we obtain
F(r
~ .tUn
(2.3)
) - F(t
11
.6Up
)
11
j
2a
j
D
j
The right-hand side is less than or equal to the maximum of
both of which are less than
f(M) •
(which as before we relabel
11
Fn
tim
j -+
00
j
Thus we may obtain a subsequence of
) such that
(r
j
11
f(l-)
Fn (t11 )
-) -
j
j
2a
11
j
j
<
f{M).
n
j
7
By arguments similar to those leading to (2.2). we obtain
.e
(2.4)
As before. this leads us to a contradiction. so that we must conclude that
t n+2a.
n
tn
can converge only to M.
M.
This completes the proof of Theorem 2.1.
But
rn •
so that
r
n
also converg,es to
We remark here that the proof just given does not require continuity of the
density.
TtEOREM 2.2.
converging to
PROOF:
e
Let
If
0,
F
then
is a Type III distribution and
tn
n' = [~
4Up
n co x
and
r
n
an
is any sequence
-+- M with probability one.
IF (x)-F(x)I = 0].
n
fl'
has probability one by the
well-known Glivenko-Cante1U Theorem, we restrict our attention to
we:n'.
As in
the proof of Theorem 2.1. one of three things may happen:
1.
tim 4up l n •
2.
.urn .in6
3.
there is a subsequence
r
n
+00,
._,
Again cases 1 and 2 are similar.
There is a subsequence
F'(a-) < P(X=M).
r nj
lnj
such that
diverging to
(l *, r Ie)
n
n
triM.
-.
Pick
(!nj,rn ) c (-co.a),
j
(2.5)
Let
-+-
We will investigate case 2.
But eventually since
Also eventually by the
lnj
G1iv~-Cantel1i Theorem,
be as in Theorem 2.1 so that
a
so small that
8
(2.6)
.e
Combining (2.S) and (2.6) we have the contradiction that
more observations than
Urn
.i.n6
r
n
rI
-OD.
(in' r )
n
infinitely often.
(i *.r *)
n
n
contains
Thus we have
Urn 4up r n wi +aD.
converges to i rI M. If l is a mass point of
Similarly, we may conclude
Let us suppose that
lnj
the discrete part of the distribution, then
othexwise
On the other hand
Since
P(X-M) > P(X-l)
servations in
(l,
r)
n n
we again have the contradiction that the number of ob-
(ln *, r n *)
is less than the nUDDer of observations in
infinitely often.
Thus we can only conclude that
for all points in
n'.
in
and
r
n
converge to
M
In Section 1, we pointed out that this estimate of the mode suggested by
Chernoff is related to the kernal estimators of the density, where the kemal is
The type of analysis found in Theorem 2.1 can
chosen to be the uniform k.ernal.
be extended to show a simple proof of strong consistency for the kernal estimate
with uniform kernal.
THEOREM 2.3.
more slowly than
F (x+h )-F (x-h )
If
f (x) =
n
n
Ilog (lOg (n» In,
n
2h
n
n
n
'
where
hn
converges to
then at every continuity point
x
of
f,
0
9
f (x) + f(x)
with probability
n
-e
PRooF:
Let
0'
1.
be defined as in Theorem 2.1.
Fix (1)£0'.
By noting that
F (x+b ) - F (x-h )
n
n
n
n
2hn
IF (x-h) - F(x-h)1
n
n
n
+
2hn
and letting n
F(x+b) - F(x-h )
n
n
2hn
diverge to infinity, we have
F (x+b ) - F (x-h )
tim 4up n
n
n
n
2h
~
f(x) •
n
Similarly
F(x+b ) - F(x-h )
n
n
2hn
IF(x-h ) - F (x+h)/
n
2h
n
Letting
n
diverge to infinity, we have
which is sufficient to complete the proof.
n
n
F (x+h ) - F (x-h)
+n
n
n
n
2h·
n
10
3. An Application. An interesting application of estimates of the mode occurs
.e
in determining the baseline calibration for ascending aortic flow as obtained
with an electromagnetic flowmeter.
The physiological details may be found in
[1].
Records of ascending aortic flow are illustrated in Figure 1.
For purposes
of this discussion, each cycle of the flow record can be discussed in terms of
two intervals:
systole and diastole.
The prominent, positive deflection in the
flow reeord corresponds to the ejection of blood by the heart into the ascending
aorta during systole.
During diastole, there is a small retrograde (negative)
flow associated with closure of the aortic valves.
During the remtrlnder of dias-
tole, flow is virtually zero.
The electromagnetic flowmeters are stable for short periods of time, say
1 minute, but over longer periods, say 10 - 20 minutes, the baseline corre-
e
sponding to zero flow may be expected to drift.
In order to process the flow
data with a digital computer, it is essential to identify zero flow.
It is
known physiologically that the interval of virtually zero flow, diastole, is
longer than systole.
Therefore, when discrete equally spaced samples of the
flow record are available, the most frequently occurring value should correspond
to zero flow. This suggests that the problem of determining the zero flow baseline somehow corresponds to estimating a mode.
Since a parametric form of the
true flow curve is not known, exact arguments cannot be given.
We present, how-
ever, a heuristic discussion.
Denote the flow curve without noise by
a derivative, so that it is measurable.
one cycle will be denoted by
in this interval.
[to,to+r].
fl(T).
Thus,
T
[to,tO+r];
fl(t);
we will assume
fl(t) "has
The time interval required to complete
we will choose a point,
T,
at random
is a random variable distributed uniformly on
We wish to determine the density function of the random variable
The exact form of the density depends on the nature of
fl(t),
but we
11
observe that
.e
ft(t)
has zero derivative at the peak positive flow, at the peak
negative flow, and throughout the zero flow portion of the cycle.
Jacobian of the transformation.
ft,
Thus. the
will be infinite in these places, and the
density will have a shape as in Figure 2.
In addition. the point corresponding
to zero flow will have probability mass proportional to the length of time there
is zero flow.
Thus,
ft(T)
has a distribution of Type III.
According to
Theorem 2.2. we could consistently estimate the zero flow baseline by choosing
points,
T1' '1'2' ••• , Tn'
estimate of the mode to
In practice, samples
to digital converter and
at ran~ from
[to,to+r]
and applying the Chernoff
ft(T ). ft(T ) ••••• ft(T ).
l
2
n
ft(T ),ft(T ), ... ,ft(T )
I
2
n
ft(T)
are obtained with an analog
is contaminated by electrical noise, errors in
conversion. electrical artifacts arising from artifical or spontaneous activation of the heart, and other unidentified noise.
None the less, this technique
has proved successful in accurately locating the zero flow baseline.
A frequently used alternative is the average diastolic flow; O'Rourke and
Taylor [9] have used this approach to obtain a baseline calibration.
A disad-
vantage with this method is that the beginning and end of diastole must be accurately known.
Furthermore, the presence of large artifacts during diastole
(see Figure I-B) would reduce the effectiveness of a calibration based on an
average diastolic value.
the mode is used.
These problems are not eneo\Jlltered when an estimate of
AORTIC FLOW
(em 3/sec.)
N
o
o
o
o
.
.• ·..···1•
1
·
.11 ... :
.
I
.. ,.":
.......
.
~
..
.'~ ~
:
"
.
t
·
~ ....
.
~
,.
.
,i
j
t
.
_ _~---...-.:.::,._••i....:i
· i,
....
--
"
......:.. 'm
.
rn
.....
n
o
fI
'1
t
•
.~.":.;
_...
.
"
i·
.:.
.
..
.
. '. .......
,
.... ~
._·,t····_·.
t··__·-.
I
,t,.- .. -
.
,j
U)
o
z
.\.;
·
t· ._-
. .' ~- , i
'.
; :, i
·
I
: . .i
-~;-':'-+-""""""-L:i'
..
,
..
i
I
"
I
!
r
1.
,
I
"
I
I
,t.
_ •.
_U··~"
'<-
'.•, ' ...- , ..I
_.,""
-.~"-
,
..... ~
.
i
_........
. j
!I
_._.-4.
!
"
i
1"-
. ! •
:. t
l···
-~
... -.
•~ .. : .j
·1
·
'f
... '.1
I
I
FIGURE 1. This figure illustrates portions of two records of ascending aortic
blood flow obtained from dogs with chronically implanted electromagnetic flowmeters. The record in Panel B demonstrates a large artifact during diastole;
this corresponds to application of a pacing voltage to the heart.
e
e·
. -
,
I
(I
~:T':GATIVE
PEAK OF
AORTIC PRESSURE
CYCLE
I
I
I
,
t
I
<
:1
I
I
I
ZERO FLOW BASELINE
,J
I
I
,
POSITIVE P&\K OF
AORTIC PRESSURE
CYCLE
t
t
I
1
I
,
,
I
•I
I
,
t
I
I
t
t
FIGURE 2. This figure illustrates the approximate shape of the density of
is uniformly distributed.
fl(T),
where T
14
4. Conclusions. The results of Section 2 are really not surprising. All of the
e.
estimates of the mode rely on the fact that probability is most concentrated
around the mode.
This is even more true in distributions of Types II and III
than in distributions of Type I.
One ean speculates therefore s that results
analogous to Theorems 2.1 and 2.2 should hold for the other estimates of the
mode.
We remark also on the choice of
an.
It is desirable to choose
an
as
small as possible to have the interval estimate of the mode as short as possible.
Choice of too short an interval s however. may lead to spurious results.
2.1 and the result of Nadaraya [8] both indicate
o(n(l/f(n»
where
f(n)
f(n)
n
should be chosen to be
is some slowly increasing function of n
The consistency result of Chernoff suggests that
where
l/a
is as above and Q < 1.
lla
n
This allows
a
n
can be
on
the density •
and
(I
< ~.
o(nQ/f(n»
to be somewhat shorter
but the consistency result is correspondingly somewhat weaker.
Theorem 2.1 is less res trictive
Theorem
Of course,
f.
Finally, we remark that the proof of Theorem 2.2 holds when the distribution is a discrete distribution with isolated mass points.
15
References
e
.
..
[1]
Benson, D.W., Jr. (1970), A Relationship Betrveen Aome Pressure (JJld
F7,OliJ~
a Ph.D. dissertation submitted to the faculty of Biomathematics and Medical Engineering, University of North Carolina at
Chapel Hill.
[2]
Chemoff, H. (1964), "Estimation of the mode,"
Stat. Ma:th.~ 16, pp. 31...41.
[3]
Chung, K.L. (1949), "An estimate concerning the Kolmogorov limit distribution," Tr-ans. Amer. Math. Soc.~ 67, pp. 36-50.
[4]
CsW, E. (1968), "An iterated logarithm law for semimartingales and
its application to empirical distribution ftmction, It Studia Sm,ent.
Math. BUng.~ 3, pp. 287-292.
[5)
Da1enius, Tore (1965), ''The mode-a neglected statistical parameter,"
JRSS(A)" 128, pp. 110-117.
[6]
Grenander, Ulf (1965), "Some direct estimates of the mode,"
Statist.~ 36, pp. 131-138.
[7]
Loftsgaarden, D.O. and Quensenberry, C.P. (1965), "A non-parametric estimate of a multivariate density function," Ann. Math. Statist."
38, pp. 1261-1265.
[8]
Nadaraya, :f.A. (1965), "On nonparametric estimates of density functions
and regression curves," Theorry Probe App7,.~ 10, pp. 186-190.
[9)
O'Rourke, H.F. and Taylor, M.G. (1967), "Input impedance
circulation," CircuZation Res.~ 20. pp. 365-380.
[10]
Parzen , E. (1962), "On estimation of a probability density function and
its mode," Ann Math. Stati8t.~ 33, pp. 1065-1076.
[11]
Robertson, T1m; Cryer, J.D; and Hogg, R.V. (1968),
estimation of distributions and their modes, fI
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