Wavelet Interpolation Methodによる身長・体重の平均発育速度曲線の検討

(1)

愛知工業大学研究報告

5

第

30

号

A

平成

7

年

T

h

e

I

n

v

e

s

t

i

g

a

t

i

o

n

r

e

g

a

r

d

i

n

g

T

h

e

M

朗

nG

r

o

w

t

h

V

e

l

o

c

i

t

y

C

u

r

v

e

i

n

H

e

i

g

h

t

加

d

W

e

i

g

h

t

b

y

T

h

e

W

a

v

e

l

e

t

I

n

t

e

r

p

o

l

a

t

i

o

n

M

e

t

h

o

d

恥

v

e

l

e

tI

n

t

e

r

p

o

l

a

t

i

o

n

除

t

h

o

d

に

よ

る

身

長

φ

体

重

の

平

均

発

育

J

m

曲

線

の

擬

1

藤井勝紀 Katsunori FUJII

ABSTRACT The purpose of this study is to determine the peak height velocity age( P H V ag巴) of the田ean growth velocity curve approximated by the Wavelet Interpolation Method

which author presented as an efficient procedure of analysis for this purpose. Ninety eight longitudinal data of height and weight in boys (age 6 to 17years old ) were obtained retrospectively from health examination records in1983.When' considering yearly growth distance data y { y = F

(t) }

of height and weight as time series data， the growth distance curve F and gro曹thvelocity curve f (the first derivative of F ) are approximated with Wavelet series by Analysing Wavelet Function of Yves. Meyer. Therefore，

F (t) can be interpolated from longi tudinal time series data of height and weight { (t

i

.

y i) i=l， 2， 3， ".. "，11} by computer simulation ( H -P UNIX Work Station ). As the result， the graph y = f (t) could be descr i bed c 1 ear ly， and the P H V age was identified and the阻ean growth velocity curve of height and weight approximated was investigated. In this paper， the procedure of analysis in the study of the physical growth and development was proposed n巴wlyas the Wavelet Interpolation Method ( W 1 M ).

1. INTRODUCTION

In this study， auther consider the proble問

。

f analysing growth pattern of that as time series data for the longitudinal height and

胃eight datas. there are considerable literatures

on this problem. Tanner1) 2) estimated the peak age(Peak Height Veloci ty age， P H V age)by his graphic method， and he emphasized that the P H Vage could be an index of maturity speed because of the relationship between the P H V age and maturity rate. So regarding the growth pattern of height， author3i4l grouped the P H V age in each that age and has investigated on characteristics of the P H V age groups class-ified.However， because number of times measured on height were once a year， the range for a year has arose in P H V ages， and the analysis accor-ding to the determination of precise P H V age

could not have done. Besides， studies that the growth pattern of height was analysed by fit-ting mathematical function to the growth dis-tance curve have been done by Joossens5)， Preece6l， Largo7l， Berky8) and logistic func-tion of Marubini9l since gomperz function of Deming10l in 1957. In Japan， Matsuura11i in-vestigate with regard to the mid-growth spurt and after-growth spurt by fitting the poly-nomial to the longitudinal growth distance data of individual， and Tahara and Takaishi12i， and Ohno13l analyse on flattening of serial datas by fitting the spline function to the growth veloci ty curve. In ei ther case， however，

it is difficult to identify the peak or the peak age (especially P H V and P H V age ) with accuracy. Therefore， we propose newly the Wavelet Interpolation Method( W 1 M ) by using Analysing Wavelet of Yves. Meyer14i15i (Fig 1)

(2)

5

6

愛知工業大学研究報告，第30号

A

，平成

7

年， Vo

1 .

30吟

A

，Mar. 1995

to analyse this problem. In this method. the growth distance curve F(t) and the growth velocity curve f (t) as th巴 first derivative of F at t were assumed to be continuos and L2₍_R₎_f_u_n_c_t_i_o_n_s_._t_h_e_r_e_f_o_r_e_._{these functions c}_a_n

be approximated by Wavelet series. From time series datas of the mean height and weight growth. and growth veloci ty curves are de decribed. and the peaks and the peak ages are identified by computer simulation. Further. the mean growth velocity curves were examined in height and weight

2. METHOD

SAMPLE. This study is based upon a longitudinal height growth distance data of 250 japanese male subjects aged 6 to 17 years since 1972 un-till 1983. These samples being derived from the Health Examination Table of public high school male junior students in Nagoya city Aichi pre fecture in 1983. The Health Examination Table put down records of height. body weight. chest girth and sitting height from elementary school to high school. These records of physique items have been measured in Apri 1 every year. and obligated by Ministry of Education. Samples used for this time analysis actually were extracted 98 individuals of about average height (164.0cm

~174. 7cm in 17 y巴ars ) which become complete

without missing as the longitudinal data from subjects of 250 male students.

ANALYSIS. There is Fourier analysis in one method of analysing the ti四日 series data.

Espe-cially. “time - frequency analysis = a spectrum analysis“is used in the time series analysis etc. The characteristic of this Fourier analysis is in periodicity and simi-larity. and function of bounded variation F(t) is expanded to the following shape in th巴

Fourier series for the break-uptype

∞

i

n

πt

( 1 - 1)

F

(t)

=

L

:

C

n

e

司

α

コ

The discontinuity of F(t) and the discontinuity of the derivative are reflected in the attenua-tion order of Fourier coefficient a n. b n al-though cannot request precisely the abnormal place t as the pol巴valueof F(t)in this

meth-od. Besides. Yves. Meyer 1心15)is improving this

and showed that arbi trary

L

2 (R) function

F

made possible orthogonal expansion were done like the following shape by using Analysing Wavelet functionゆ(x)

(1-2) F

(t)ニ

L

:

a j・

k

ψ

(2j

t

-

k )

j ，k

where j. k an integar. This is called Wavelet expansion. the

local-ization is made up by using orthogonal corol-lary consisted from Multiresolution Analysis. 16) Accordingly. this Wavelet expansion is becoming extension of Fo臼rier series which had a

charac-teristic decided the peculiar point. Further. when the right of expansion equation ( 1 - 2 ) is possible to differentiate in each term. the first derivative f of F can also show by Wavelet expansion according to differentiating the both. The following equation consist at this

( 1 -3)

f

(

t

)

=

~.

a

j，

.

2

1 ψ

，

(21

t

-

_k)

)，R

Provided that this equation is ゆ

=dO/dt.

o

(t) is consisted to be sufficient smooth function.

In this paper. the analysing method is to interpolated by the Wavelet analysis17)18)with using Analysing Wavelet function ゆ(t) of Yves. Meyer. The method is that when given the ti四eseries data { ( t

i

.

y i) : i =

1 .

2. 3.

n} of n piece. we request the func tion F n (t) • f n (t) which approximate y

=

F (t) and the first derivative f

=

d F (t)/ d t of F (t) from this data and display the graph. so this is called Wavelet interpolation.

(3)

nle I日開sllgatJ叩時制l同日eIlean Growth Vel恥ityCunモl日制ghtand 11eight by刊日恥削etInterpolati叩VRthod ₅₇

(1-5) f.Ct)二(2::' )

a

j， .23

ψ

， (23 _t_-_k) 「一一一一一一-， 1訂 Provid巴d th呂t(L:' ) shows the total ofintegral 1.10

-

1

S仰い

number j k of n piece which satisfy prop巴r .刊目ー

0.110I condition. When the number of the data is a ..70い

O.MIー f巴w，Wavelet coefficient {a j ， k ; j ， k of '.'0ー目

0/10 I→

n piece} is request巴d as a solution of a 0"1一

。

10・ー linear simul taneous equations of n piece 0.10一・

。

伺

い

which correspond to this by substituting( 1-4 ) ~聞い

equation for th日dataof n piece given. ( 1 -6 ) Y i

=

(2::' )

a

;

.ψ

(2j t _{i -} k) i = ，1 2， 0 0 0 ・， n O.lnl-0.301 -OAO 1-0 ..:101 -o.m1 -0.701 -1).1101 -.0開ト y ニ Fn (t) means th呂tthe interpolation becom巴s a curve betwe巴n the street and the data' s point of n piece giv巴n if substitutes the Wavelet coefficient for ( 1-4) equation.

L一一一一ーし一一一一一J一一一一一一一ーし一一一一一一一L一一一一ー一一

ln addi tion， f n ( t ) which approximate the

growth velocity curve as well as Fn (t)is d巴ー termined by substituting( 1-5)巴quation. This curve can be displayed as the graph by computer ( Fig2 ).So the peak and the P H V age c昌nbe identifi巴dby this graph of f n (t)

3. RESULTS

The interpolation problem ofy F (t) in relationship betw巴巴n the growth di stance (y)

and the ag巴(t) was sol ved by using Analysing Wavelet of Yves. Meyer in each on日 of98 indi -viduals， and two exampl巴s graph which simu-lated it by computer is shown in Fig 2. ln this figur巴， the graph drawn with solid lin巴sho百S

growth distanc巴 curve， and which drawn wi th dotted line shows growth velocity curve which diff日T巴ntiatedgrowth distance curve by time. As shown in Fig 2， a maximum peak( P H V ) was displayed very clearly， P H V in oth日r96i n dividuals was also very cl巴丘r. Wha t the P H

V

旦ge was ascertain巴d by such as the procedure，

means that it can apply the procedure to th巴間ean growth distancies of height and weight Then growth distancies and growth amount during a year of height and weight were calculated in

.~，OO

，

.

。凹 2回 4叩

Pig 1

Y

v

eSoMey哩r's Analrsing Isvelet Ileight:cm 2

冊

。

18臥00 160.00

_ν

-

-い/

140.00

レ/

司‘ 、 120.00 l

冊

。

BO，OO

曲

。

...司'.・， ‘ ‘ ... 40.00 20.00 t守・・，."-、 0.00 守20，00 6.00 B.OO 10.00 12.00 14.00 16.0

。

H町ght:cm 2

開

。

180.00 l回開

/

〆

-

-1"""" 140.叩 120.00 l凹凹 "‘一 80.00

"

，.ー‘. ・、、‘..1" 60.00 40凹

.

20.00 、、、、 0.00 、' -20.00 6明白田 10.00 12凹 14凹 16凹 3色 det96.!I. dat96.b 泊百'6.e- -A" d1l163.1!. liiiiii3]; dei6i，e-ー A" 98 individuals， and mean growth dist丘nCl巴sand

veloci ties were done. Therefore， in this paper _Fig_{2 Height growth distance curv}_日_{s an d} velocity curves simulated by

W

1 M

(4)

58 愛知工業大学研究報告，第30号

A

，平成

7

年，

Vo

.130-

A

，

Ma

r.1995

the relationship betw巴巴n the mean growth

dis-tanc巴 (y) and th巴age (t) was sol ved by the W 1 M in height and weight， and graphs of height and weight which simulat巴d by computer

is shown in Fig 3， 4 . The P H V age was 12.21 years and the velocity at that time was 8.08 cm/yr in height， and in weight was 12.60years and was 7.20kg/yr. The peak age of mid-growth spurt was 6.68years and the velocity at that time was 6.20cm/yr in height， and in weight was 7.86years and was 3.40kg/yr.

4.DISCUSSION

Ma them呂tical functions in the past CLogistic function， Gomperz function， Spline function and polynomial) have b巴巴n investigated the precise

degree and fitted the function assu四ing that

the growth distance cueve was imagined. In thes巴methods， how巴ver，it is difficult to explain the fitting to thos巴curves

theoreti-cally. However，by using Meyer Wav巴l巴twith the

similarity， localization and smoothness， author

the growth distancies and slightly smoothed th日 drawn curve. By this graphic m巴thod，he investigated th巴m日an growth velocity curve of

height and weight based upon cross-sectional datas of10，000children in the 1966-67London County Council Surv巴y. In his paper， boys had a diminution of deceleration or relative spurt，

lIcighl /¥ VE. Ilcighl:cm r一一一←ー下一一一一一「一一一一一I一一一一一一「一一一一一一 1---'~~-lJ而 ij~ (j~iïüfii 2凹uul--1一一一一ー1-一一一一1---1---1一一一一一l一一一←!dltlioI.c

j

[

二

日

と

に

100.00 !←一一一一l一一一一一一→一一一一一一一l一一一一一一→一一一一一一!一一ー一一一l

j

十

日

FEU7

0.00 _{一一一一一}l一一一←ト←一一一￨一加山ト一一一￨一一一一l一一←一一l一一一一l一←一一一一一一一一一l ι

。

B.OO IU.UO 12.00 14.0一一ーし一一一一」0 16.00 ^，巴

approach can be inv巴stigated the growth model Fig 3 Mean growth distance curve and velocity

by the simpl日and unified m巴thod. curve in height approximated by W 1 M

By the Wavelet Interpolation Method， author

巴xamined to approximate th日皿巴an growth and Wdghtkg

Wcighl /¥ VE，

veloci ty curve in height and weight ( Fig 3，4 ) The att日mpting to approximate the mean growth

curve has been done by Count1計 _i_n₁₉₄₃_. _{Fig 5} is the graph 胃hich Count fitted three succes-slve curv日s to the means for height in a cross-sectional study of Chinese childr巴n from birth to age 21. He basically fitted differential equation to th巴mean growth curve， and calcuー lated as the velocity curve for which differ-entiat巴d the mean growth curve approximated by i t.However， h巴wrot巴 that he wasted a great

deal of ti冊巴 to calculate thes巴procedure， and they were very complicated. Then Tanner20) attempted to examine the growth velocity curve

100.uul-ート一一」ー 1--1--1

三工仁コ罰

Z

(rniioi.c-一 90 凹，--，---，一一一一一-，一一一一←，~~-，一一一一一一 l 一一一ー一一別UUI--I-←ー→一一一l一一一一一一一叶ー←一一一一-，---，-一一一一一l 70.00 I ト一一一一~，一一一一一一一，---，一一一一一ー，---，一一一一一一 10.00 6肌} 8.00 10.00 12肌 14.0U JG肌j

^

"

with simple method. Th巴simplemethod is that Fig

4

Mean growth distance curve and veloci ty

it drew dots plotted at successive half-year curve in weight昌pproximated by

W

1 M age centers in accordance with raw datas of

(5)

TheInvestigatiOllr可制l略恥蜘nGrowth VeJ舵ityCI山田m脳出t制Weightby恥恥削etJnterpoJation IIethod

₅

₉

from about 6 to about 7 in height， in weight there was a clear increase of velocity occur -ing from 7 to 8. So he labeled Mid-Growth Spurt for these phenomena. The mid-growth spurt was ascertained in also the mean growth velocity curve of height and weight approxi -mated by the W 1 M in this paper. Though Tanner and Count did not mention regarding the pre -cise peak age of the mid-growth spurt. Author showed that the precise peak age of that was ag巴6.68 in height and was age 7.86 in weight， furthermore the peak velocity at the peak point of that was 6.2 cm/yr in height and was 3.4 kg/yr in weight. Of course， the precise P H V age and the velocity at that point in height and weight could be requested.

As Tanner21lhas been indicated since 1947， the existence of mid-growth spurt was ascer -tained in this paper. The P H V age and the peak age of mid-growth spurt， and the velocity at those point could be computed precisely by the W 1 M， however， the difference between the peak velocity at the P H V age in the mean growth velocity curve of height and weight approximated by the W 1 M， and those in the mean age of the P H V age calculated individ -ually in each 98 occurred 2.26 cm/yr in height and 1.62 kg/yr in weight. This phenomenon is defined Phase Difference Effect， and shows the distinction of statistical procedure bet官een cross-sectional and longitudinal datas. The statistical significance of the phenomenon is still not made clear. It should be discussed with regard to the phase difference effect in the future. In height velocity curve approximated by the

W

1 M ， a slight onset of spurt was shown after the P H V age， but was not shown in weight velocity curve. This phenomenon of slight spurt in height velocity curve is labeled After-Growth Spurt by Matsuura22l23). Though this phenomenon like final spurt of last parts in height growth was shown as the result which Matsuura fitted polynomial to the height growth， was not almost shown in other literatures except Gasser' s study24l25l 26) In this paper， the after-growth spurt was displayed by the

W

1 M also， however， the significance of the existence is not made clear and it is considered that it should be more investigated with regard to the aft巴r -growth spurt in the future. 同一

.

，e ，~

メきと二一一

・:正広三

ι

;

.

-

_

-J

L

:

山

合一ー「←一吋γ一寸白

主

3区【・.t・t A~curve: y=α+bx+clogx， i .e. )'=492・71+2'26x+682'4Iogx B.curve: )'=k(A.curve)+q， wherek= 1'5，q= -547-25 AH-curve:_¥'=v= 10"8， -24'5 erev _ +33・5 1+ 10・4365antilog( -6'7529fJ) fB-curve -1330¥ f J=logJ一一一一一一一一l ¥ 10 J IJ= -0，003981

B

=

(

巴干旦

.

r

Fig 5 Count' s representation of the growth distance and velocity curve fitted by mathematical function in cross-sectional data on height in Chinese children. Above， distance; below， velocity

(6)

6

0

愛知工業大学研究報告，第30号A，平成7年， Vo

.

1

30-A， Mar. 1995

5. CONCLUTI0NS

When considering the gro官th proc巴ss of height as the concept of time series， the relationship between growth distances (y) and time (ages， t)， y = F (t) and the f (t)

{the first d日rivativeof theF

(t) }

，官hat is called ， the problem of interpolation 官as solved by using Analysing Wav巴let of Yv巴S M巴yer，and the graph was displayed by the simu -lation. The r日sult1はsconcluded as th巴fol lowing. P H V ag日 was identifi巴d clearly in height and weight， the mid-growth spurt was also ascertained in both. The after-growth spurt was ascertain巴d in height， but was not in W巴ight. This phenomenon should be discussed in the future. Auther propos巴d n巴wly as the Wavelet lnterpolation Method for the procedure of analysis in this paper.

6. LITERATURE CITED

1) Tanner， J. M.，官hit日house， R. H. and Takaishi， M. :St旦ndard from birth to maturity for height， weight，height v巴locity and weight velocity of British children in 1965. Arch Dis. in Child. 41-219 :454-471， 41-220 613

635. 1966

2) Tanner， J. M. :Growth at Adolesc巴nt. Blackw巴II Sci巴ntific Publication. 1962.

3) Fuji1. K. :Gro胃th patterns of height in boys -analysis based on velocity curv巴 Bull of Aichi lnstitute of Technology. No 20 :39 44. 1985

4) Fuj i 1. K. and Matsuur昌， Y. : Gro百th patterns classified by difference of height in boys -analysis bas巴don growth distance curve呂nd velocity curveー Japan J. Phys. Educ. 39 213-224， 1994 5) Jooss日ns，J. V. and Brerns， H巴yns. E. High pow巴rpolynomial regression for the study of distanc巴， velocity and accel巴ration of growth. Growth. 39・535-551. 1975 6) Preece， M. A. and Bain巴s，M. J.: A new family of mathematical rnodels describing the hurnan growth curv巴. Annals of Human Biology. 5: 1 24. 1978.

7) Largo， R. H. et， al. Analysis of the adol巴scent growth spurt using smoothing spline function. Annals of Human Biology. 5: 421-434， 1978

日)Berk巴y，C. S.， Reed， R. B. and Valadian， 1. Midgrowth spurt in height of Boston child-r巴n. Ann. Human Biology. 10 25-30， 1983. 9) Marubin

i

.

E.， Resele， L F.， Tann巴r，J M. and Whitehouse， RH. The fit of Gompertz and Logistic curves to longitudinal data during adolesc巴nceon height， si tting height and biacrominal dia田eter in boys and girls of the Harpend巴nGrowth Study. Human Biology. 44

511-524， 1972

10)Deming， J. :Application of th巴Gomperze curv日 to the obs巴rved pattern of growth in length of 48 individual boys and girls during the adolescent cycle of growth. Human Biology 29 83-122， 1957

l1)Matsuura， Y. A study on physical growth and developm巴nt through investigating the polynomial fitted their distance curves - in the term over 6 and 18 years old. Bul1 of Health and Sport Scienc日s， UniV of Tsukuba 14 : 211-222， 1991

12)T旦hara，K.， Tatara， H.， Murata， M.， Takaish

i

.

M. and Funakawa， H. :Mathematical Analysis of Adolescent Growth Accel巴rationPh巴nomenonー part 1 -Adoleseentology. Vol.4:51-58， 1986. 13)Ohno， Y.， Ishijima， Y. and Murata， M. :Devisal of standard physical growth curve by using spline function. Math and Scienc巴s of statistics. 34 222-231. 1986. 14)Meyer， Y. :Ond巴l日tte et Operateur 1. Hermann， 1990. 15)Meyer， Y. Ondelelettes. Hermann， 1993 16)Mallat， S. Multiresolution Approximations and Wavelet Ortonomal Bas巴s ofL (R ) . Trans of Am巴rMath Soc. 315 69-87， 1989 17)Frage， M.， Hunt， J.C. and Vassilicos， J. C. Wavelets， Fractals and Fourier Transforms

(7)

恥Inv邸tigationI'egal~ing 恥 M釧 Growth 恥locityCIII開inHe，岨htand Weigllt hyThe恥削etIntel'r~lalion 怯thud ₆₁

Clarendon Press， 1993.

18)Strang， G. :Wavelet Transforms versus Fourier Transforms. Bull. A. M. S. Vol. 28: 288-305， 1993.

19)Count， E.

W

.

， Growth patterns of the human physique an approach to kinetic anthro-pometry. Human Biology， 15 1-32， 1943. 20)Tanner， 1.M. and Cameron， N.， Investigation

of the冊id-growth spurt in height， weight and limb circumferences in single-year velo-city data from the London 1966-67 growth survey. Annals of Human Biology， 7 565-577. 1980.

21)Tanner，

J

.

M.， The morphological level of personality.Proceedings of the Royal Society of Medicine， 40 301-303.

22)Matsuura，

Y

.

and Kim， M. :Analysis of physical growth by fitting the polynomial to its growth distance data - Girl' s stature and body weight - . J of Korean Public Health Association. 17 130-148， 1991.

23)Matsuura， Y. and Kim， M. 1991.Analysis of

growth by fitting the polynomial to the longitudinal growth distance data of indi-vidual - age 6 to 17 - . Research Monograph， Growth and Developmental Research， Institute of Health and Sport Sciences， Univ of Tsukuba pp 1-153， 1991.

24)Gasser， T.， Kohler， W.， Muller， HG.， Kneip， A.， Largo， ~， Molinari ，L. and Prader， A Velocity and acceleration of height growth using kernel estimation. Annals of Human Biology. 11 397-411. 1984.

25)Gasser， T.， Muller， HG.， Kohler，

W

.

， Prader， A.， Largo， R. and Molinar

i

.

L. An analysis of the mid-growth and adolescent spurts of height on acceleration. Annals of Human Biology. 12 129-148. 1985.

26)Gasser， T.， Kohler， W.， Muller， HG.， Largo， R.， Molinari， L. and Prader， A. Human height growth correlational and multi-variate structure of velocity and accelaration Annals of Human Biology. 12 501-515. 1985.