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European Journal of Applied Sciences – Vol. 11, No. 4
Publication Date: August 25, 2023
DOI:10.14738/aivp.114.15349
Tanaka, N., & Fujii, K. (2023). Relational Composition between Periodic Fluctuation and Seasonal Variation in Height Growth.
European Journal of Applied Sciences, Vol - 11(4). 317-324.
Services for Science and Education – United Kingdom
Relational Composition between Periodic Fluctuation and
Seasonal Variation in Height Growth
Nozomi Tanaka
Toukai Gakuen University, Aichi, Japan
Katsunori Fujii
Aichi Institute of Technology, Aichi, Japan
ABSTRACT
The influence of seasonal variations in height growth has been frequently discussed
previously. In recent years, however, Fujii [1] has hypothesized periodic
fluctuations, accordingly the variation is not from the environmental factor of
season; rather, height growth is achieved with periodic variations due to genetic
factors. Based on this logic, wavelet interpolation was applied to observed data
values measured three times a year in this study to investigate the periodic
fluctuations in height and weight growth in elementary school children, and an
attempt was made to verify the periodic fluctuations from the behavior of the
derived velocity curve. The longitudinal time series information on subjects’ height
and weight were measured three times a year from the first to sixth grades. The
data used in the analysis were for 17 boys and 11 girls. The results showed that the
height growth pattern is essentially achieved with periodic fluctuations, and that
there are seasonal stimulating phenomena within that periodicity. If this growth
mechanism also applies to body weight, growth in body weight would occur
together with increases in height, but with additional seasonal stimulating effects
weight would be affected not only by genetic factors as height is, but also by
seasonal environmental factors.
Keywords: Periodic Fluctuation, Seasonal Variation, Dissimilarity, Local peak velocity
(LPV)
INTRODUCTION
In research regarding height growth, seasonal variations in height growth cannot be observed
with once yearly measurements. At minimum, cross-sectional time series information at short
intervals is essential to investigate periodic fluctuations such as seasonal variation in the
process of transitioning from childhood to puberty, which are also affected by individual
differences in physical growth, yet there have been no attempts at such analysis to date.
Togo and Togo [2-3] and Togo [4] conducted monthly measurements of height and weight in
five children over a long period. In one boy monthly measurements were made from age 4 to
29 years, and a time series analysis was conducted. A report by Kobayashi [5], in which that
time series analysis is applied to kindergarten children, presents findings of seasonal variation
during that time. In classic study’s findings on seasonal variations include those of Shima [6],
Ozaki [7], Fukuda and Ishikawa [8], Saito [9], and Nakamura [10]. There are also findings [2-3]
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on the maximum values seen from April to August, when seasonal variations in height growth.
Other studies have found no distinct seasonal variations [8-10]. In recent years, Togo [3],
Kamioka [11], Oyama [12], Yoshioka [13], and Suzuki [14] investigated seasonal variations in
height growth, but no clear conclusions have been reached for seasonal variation in height
growth, since there are problems in the hypothesis of periodic fluctuation with genetically
programmed growth.
In fact, however, the essential meaning of seasonal variations in height growth has not been
clarified. According to Marshall [15], a certain species of animal (weasel) shows mating
behavior based on the time period of light in the eyes, and animals that are completely blind do
not exhibit that behavior. Taking those findings as a hint, a comparison was done of height
growth in blind children, children with low vision, and healthy children by month of age, and a
relationship with season was reportedly seen only in the healthy children. Tanaka et al. [16]
reported affirmatively on the relationship with daylight hours shown by Marshall [15], but
there are no reports that can immediately affirm that seasonal variations are due to hours of
daylight. In any event, even if seasonal variations are considered to be periodic fluctuations,
clarification of the patterns in height growth rhythm are promising for elucidation of the
relationship between those periodic fluctuations and hormones, as well as the genetic
contribution. Fujii et al. [1] applied wavelet interpolation to cross-sectional growth in height
for month of age over one year in older children, and investigated the seasonal variation and
periodic fluctuation. It was found that height growth over one year is not linear; rather there
are months of much growth and months of almost no growth. This phenomenon has
traditionally been understood as seasonal variation, but one wonders if height growth is really
affected by season as an environmental factor. Fujii et al. [1] advocated periodic fluctuation in
that study. In other words, the variation is not from the environmental factor of season; rather,
height growth is achieved with periodic variations due to genetic factors.
Based on this logic, wavelet interpolation was applied to observed data values measured three
times a year in this study to investigate the periodic fluctuations in height and weight growth
in elementary school children, and an attempt was made to verify the periodic fluctuations from
the behavior of the derived velocity curve.
METHODS
Subjects
The subjects were students at an elementary school in Shizuoka City, born between April 2002
and March 2003. Longitudinal growth data on height was obtained from grades 1 to 6. Height
was measured three times a year. This study used the longitudinal time series information on
height and weight measured three times a year from the first to sixth grades. The data used in
the analysis were for 17 boys and 11 girls.
Analytical Techniques
To approximate the true growth curve from the supplied growth data with the wavelet
interpolation method (WIM), the data are interpolated from the wavelet function of the data
and a growth distance curve is drawn. This distance curve is differentiated to arrive at the
growth velocity curve, and the growth distance value of the age at menarche and the puberty
peak is examined. The WIM can sensitively read local phenomena, and has an extremely high
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Tanaka, N., & Fujii, K. (2023). Relational Composition between Periodic Fluctuation and Seasonal Variation in Height Growth. European Journal of
Applied Sciences, Vol - 11(4). 317-324.
URL: http://dx.doi.org/10.14738/aivp.114.15349
level of approximation accuracy. The theoretical background and grounds for its efficacy have
been described in previous reports ((Fujii and Yamamoto [17]; Fujii and Matsuura [18-19]). In
the present paper we shall describe the method of analyzing data using the WIM.
In this study was applied to height values from 6 to 11 years of age in the Japanese elementary
school boys and girls for WIM. First, WIM was applied to the age distance value of height for
each individual. Then, the age distance curves for height in both groups were differentiated,
and the age at LPV of height (age at local peak velocity in school age) from the velocity curve
was specified through this process. Please note that the derived velocity value from the
differentiation of the current function here is expressed as the annual growth rate in
centimeters per year.
Analytical Procedures
1) Wavelet interpolation was applied to the growth distance values in longitudinal time
series data for height from April of first grade to April of sixth grade in 17 elementary
school boys and 11 elementary school girls. In this case it is necessary to construct an
age axis with one year divided into three equal parts, taking April of first grade as 6.0
years old and then August as 6.33 years old and December as 6.66 years old, April of
second grade as 7.0 years old, and April of sixth grade as 11.0 years old. In other words,
to obtain correspondence month of age, age was set to be approximately 6.0 years old
around April, approximately 6.33 years old around August, and approximately 6.66
years old around December.
2) Growth distance values and velocity curves were drawn with computer simulation. It
should be noted that in this case velocity is cm/year and kg/year (growth in height and
weight in one year).
3) The peak states in the drawn velocity curves were investigated.
4) The correspondence between the velocity peak and the age axis (month of age) was
determined and the relationship between the periodic fluctuations and seasonal
variation was investigated.
Local Peak Velocity (LPV)
LPV is the name given to local velocity peaks with respect to MPV. Fujii[20] detected the
appearance of 2–3 peaks during growth spurts in a process in which mid-growth spurts were
verified with application of wavelet interpolation to height growth. However, the essential
meaning of this method is that it reveals sudden increases and static phenomena in growth and
development by showing locally detected maximum velocity values. Based on the
characteristics and theoretical grounds of the wavelet interpolation method, the size of the LPV
is a projection of sudden increases and slowdowns in growth. It is very useful in elucidating
growth phenomena.
RESULTS
Pattern of Height LPV Appearance and the Pattern Appearance Rate
With regard to the appearance of LPV in height, it was shown that there are cases in which LPV
appears in 3 places, 4 places, and 5 places in the limited span of elementary school. As shown
in Figures 1 and 2, in boys LPV appeared in 3 places in 1 boy (5.9%), 4 places in 7 boys (41.2%),
and 5 places in 9 boys (52.9%). In girls, it appeared in 3 places in 1 girl (9.1%), 4 places in 4
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girls (36.4%), and 5 places in 6 girls (54.6%). Judging from these appearance rates, it seems
that LPV appears in 4 or 5 places in nearly all boys and girls both. In elementary school this kind
of periodic pattern is formed in which this LPV appears in 4 to 5 places.
Figure 1: Height frequency distribution of % number in appearance patterns of LPV in boys
Figure 2: Height frequency distribution of % number in appearance patterns of LPV in girls
Relationship between Appearance Pattern of LPV in Height and Seasonal Month
To investigate the month in which LPV of height appears, the statistics for LPV age in the cases
of LPV in 4 or 5 places was calculated. It was found that in boys with LPV in 4 places, the first
LPV was around mid-May in 7-year-old children, the second around mid-June in 8-year-olds,
the third around late June in 9-year-olds, and the fourth around late June in 10-year-olds. With
LPV in 5 places, the first was around late November in 6-year-olds, the second around late
August in 7-year-olds, the third around early July in 8-year-olds, the fourth around early July in
9-year-olds, and the fifth about mid-June in 10-year-olds. In girls with LPV in 4 places, the first
was around early July in 7-year-olds, the second around mid-June in 8-year-olds, the third
around mid-June in 9-year-olds, and the fourth around early August in 10-year-olds. With 5
places, the first appearance was detected around late November in 8-year-olds, the second was
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Tanaka, N., & Fujii, K. (2023). Relational Composition between Periodic Fluctuation and Seasonal Variation in Height Growth. European Journal of
Applied Sciences, Vol - 11(4). 317-324.
URL: http://dx.doi.org/10.14738/aivp.114.15349
around early August in 7-year-olds, the third around mid-June in 8-year-olds, the fourth around
mid-July in 9-year-olds, and the fifth around early December in 10-year-olds. From this it is
seen that LPV appears periodically from early summer through summer in both boys and girls
when it appears in 4 places. When it appears in 5 places, it appears once in the winter in boys
and twice in the winter in girls, and then appears 4 times periodically from early summer
through summer in boys, and 3 times periodically from early summer through summer in girls.
In patterns with the appearance of LPV in 4 or 5 places in boys and girls, nearly all LPV
appearance points agree and no sex difference is seen when looking at the sex differences in
the same appearance patterns. In both boys and girls, periodicity was seen from early summer
through summer (Figures 3 and 4).
Figure 3: A sample pattern of growth of the case which the LPV appear in four places during
elementary school
Figure 4: A sample pattern of growth of the case which the LPV appear in five places during
elementary school
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Relationship Between LPV Appearance Pattern Periodic Variations and Seasonal
Variations
To In both boys and girls, the LPV appearance pattern in elementary school shows LPV in 4 or
5 places in nearly all cases. The peak age for LPV closely agrees with those respective patterns.
As shown in Table 1-1, 1-2 the standard deviation against the mean is very small, ensuring the
degree of coincidence. These peak time points often appear from early summer through
summer, and may seem to be seasonal variations but are in fact showing periodic fluctuations.
Thus, it may be that periodically fluctuating energy is displayed from early summer through
summer, a season when there are many hours of daylight. Such phenomena have traditionally
been viewed as seasonal variations, but the results of analysis with the application of wavelet
interpolation showed periodic fluctuations. If one considers the hereditary strength of height
growth in particular, hereditary periodic variation would seem to be more valid than seasonal
variation as an environmental factor.
Table 1-1: Appearance pattern of LPV and appearance rate of the pattern (male)
Number of
LPV
The first LPV
age
The second
LPV age
The third LPV
age
The fourth LPV
age
The fifth LPV age
3 (1) 6.66 8.13 9.26
4 (7) 7.12(SD=0.29) 8.23(SD=0.10) 9.25(SD=0.05) 10.22(SD=0.10)
5 (9) 6.69(SD=0.14) 7.41(SD=0.13) 8.26(SD=0.07) 9.27(SD=0.05) 10.21(SD=0.14)
Table 1-2: Appearance pattern of LPV and appearance rate of the pattern (female)
Number of
LPV
The first LPV
age
The second LPV
age
The third LPV
age
The fourth LPV
age
The fifth LPV age
3 (1) 8.13 9.40 10.23
4 (4) 7.25(SD=0.15) 8.21(SD=0.07) 9.29(SD=0.08) 10.35(SD=0.23)
5 (6) 6.66(SD=0.12) 7.36(SD=0.10) 8.21(SD=0.07) 9.30(SD=0.10) 10.68(SD=1.25)
DISCUSSION
Application of wavelet interpolation to height growth at measurement intervals of three times
a year in elementary school is an approach used in this study for the first time. However, the
time axis of age has a slightly extended projection and an attempt was made to investigate it on
the same dimension as an age axis by smoothing of the one-third year time axes. Therefore, the
phenomenon of sudden increases in velocity in this age axis differs from the pubertal peak
(maximum peak velocity; MPV), and the implication is thought to be that, as with the mid- growth spurt local peak velocity (LPV) proposed by Fujii (2002), hormone released prior to
puberty contribute to the continuous changes. That is, the velocity peak shows increased
energy to increase height, for which a hormonal contribution is naturally suggested. It is
conjectured that this may express a fluctuating periodicity in the growth coded in genes. This
is the growth fractal phenomenon that Fujii and Yamamoto [17] and Fujii and Matsuura [19]
explained in the proposed theory for the wavelet interpolation method, and is significant in
leading to chaos phenomena.
In this study the patterns of LPV appearance in height in elementary school were investigated,
and patterns were seen with local peaks in 3, 4, and 5 places. Of them, the patterns with 4 or 5
local peaks accounted for more than 90%. This shows that a periodic LPV pattern is shown in
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Tanaka, N., & Fujii, K. (2023). Relational Composition between Periodic Fluctuation and Seasonal Variation in Height Growth. European Journal of
Applied Sciences, Vol - 11(4). 317-324.
URL: http://dx.doi.org/10.14738/aivp.114.15349
nearly all boys and girls, and is thought to be evidence suggesting periodic fluctuations in
growth. Looking only at height growth, when a pattern with 4 or 5 local peaks in LPV age is
derived, a local peak appears from about June to August in most boys and girls. This finding is
thought to suggest that much of the growth in height during a year occurs from early summer
through summer. Marshall [15] hypothesized a relationship with daylight hours and Tanaka et
al. [16] presented findings that supported such a relationship. Park and Kobayashi (1997) also
showed a relationship with daytime hours. These findings are understood as seasonal
fluctuations, and if the environmental factor of daylight hours acts on growth in height, the
variation may well be understood as a seasonal fluctuation. However, Ooki[21], Ooki and
Asaka[22] pointed out the strong hereditary nature of height growth in monozygotic twins
from a study of twins, and if height growth is controlled by genetic factors the influence of
environment should be very small. Therefore, from the fact that the appearance of LPV is seen
in similar time periods it may be more reasonable to consider the growth to be due to a periodic
fluctuation as a growth mechanism, rather than from a relationship with the season.
Judging from the LPV appearance pattern, it is conjectured that seasonal elements have a
facilitatory action in the growth pattern due to periodic fluctuations. This kind of seasonal
facilitatory action may have previously been seen as seasonal variations. It may be that the
height growth pattern is essentially achieved by periodic fluctuations, with a seasonal
acceleration phenomenon acting in this periodicity.
CONCLUSION
In this study, based on the hypothesis that height growth basically does not fluctuate from the
environmental factor of season but is achieved by periodic fluctuations due to genetic factors
to investigate the periodic fluctuations in height growth in elementary school. For that purpose,
we applied wavelet interpolation to height measurements taken three times a year to verify the
periodic fluctuations from movements in the derived velocity curves. The results showed that
the height growth pattern is essentially achieved with periodic fluctuations, and that there are
seasonal stimulating phenomena within that periodicity.
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