京 都 大 学 防 災 研 究 所 年 報 第 50 号 B 平 成 19 年 4 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 50 B, 2007

Suitability of Sample Size for Identifying Distribution Function
in Regional Frequency Analysis

Binaya Kumar MISHRA*, Yasuto TACHIKAWA* and Kaoru TAKARA*Graduate school of Urban and Environment Engineering, Kyoto University
Synopsis
Estimation of environmental extremes using regional frequency analysis
needs fitting of an appropriate frequency distribution function representing ahomogeneous region. Best fit frequency distribution depends on the computed
moment coefficients using either conventional method of moment or recently
developed method of L-moment. These moment coefficients depend on thelength of observed data record. This paper examines the deviation of moment
coefficients with the length of record in terms of root mean square error (RMSE).
It shows that percentage deviation is about 50% for sample size of 20 anddecreases to near or below 20% for sample size of 50.
Keywords: Distribution function, L-moments, Sample size, Moment coefficients

1.

Introduction

Reliable estimation of the hydro-metrologicalextremes for the given return period is of particularimportance in planning and design of hydraulicstructures. Frequency analysis relates the magnitude
of extreme events to their frequency of occurrence
through the use of probability distributions. It needs alarge number of historical observed data at the placeof interest. In practice, either there is no data or is ofvery short length. In such cases, regional frequency
analysis can be an effective tool.
Estimation of frequency distribution function
representing a homogeneous region is one of theimportant major steps in regional frequency analysis.
Best fitting frequency distribution function
representing the regional data can be determinedfrom the plot of skewness versus corresponding
kurtosis using either conventional method ofmoments or recently developed L-moment method
against various frequency distribution function
representing lines.

A diagram based on Cs and Ck, such as that inFigure 1, can be used to identify appropriate
distributions in conventional methods. A diagram
based on LCs and LCk, such as that in Figure 2, can beused similar to conventional moment ratio diagrams
to identify appropriate distributions in the method ofL-moment. The location of sample estimate withrespect to the distribution gives an indication of thesuitability of a distribution of the data. A suitableparent distribution is that which averages thescattered and around which the points spreadconsistently.
However, if the sample size is small, the bias in
the values of higher moments may be large enough togive misleading results (Rao & Hamed, 2000).
Misleading of best fitting distribution function takes
place since moment ratios (coefficient of variance,skewness and kurtosis) varies largely with the lengthof data or sample size. Therefore, it is very much
important to have the suitable length of data (samplesize) for identifying best-fit distribution function inregional frequency analysis.

－ 69 －Fig. 1 Cs-Ck moment ratio diagram (Rao & Hamed,2000)

2.1 Data screening
Annual maximum daily rainfall values ofHirakata, Gojyou, Kameoka and Katada rain-gauge
stations lying in the Yodo River basin region, Japan
have been used to carry out the job. All these datawere made available from the data bank of Innovative
Disaster Prevention Technology and Policy ResearchLab, Disaster Prevention Research Institute, Kyoto
University, Japan. At every considered rain-gauge
station, sub-samples were formed from the wholesample considering continuous dataset of different
sample size varying from 20 to 80 at an increment of
10. Test for independence and homogeneity(assumption that the whole set data come from thesame distribution) of the data was performed as
discussed by Mann-Whitney (1947) and checked at
5% significance level. Test for outlier (an observationthat deviates largely from the bulk of the data) was
performed as discussed by Grubbs (1969) andchecked at 5% significance level.
2.2 Moment coefficients
The computation relationships for mostly used
moment ratios namely coefficient of variance (Cv orLCv), coefficient of skewness (Cs or LCs) andcoefficient of kurtosis (Ck or LCk) have been
discussed in following subsections.

Fig. 2 LCs-LCk moment ratio diagram (Rao & Hamed,1994)
In the present paper, it is intended to show thefluctuation pattern of moment coefficients (skewnessand kurtosis) in terms of root mean square error(RMSE) with the change in sample size using
conventional method of moments and recently
developed method of L-moment. This fluctuation
pattern will provide guidance for determining sample
size in regional frequency analysis.
2.

Methodology

2.2.1 Conventional moments
Moment coefficients, particularly coefficientof skewness and coefficient of kurtosis, are used tocharacterize regional probability distributions. If xi isobserved rainfall data at station i and n is the no. of
rainfall records, these moment coefficients arecalculated as:
Sample mean, m = x− = 1 n x∑1n i =1in−Variance, m2 = 1 ∑ ( xi − x ) 2n − 1 i =1

Standard deviation, σ = (variance)1/2
Data screening of the collected data to check forindependence, homogeneity, outliers etc., calculationof moment coefficient for data set of different sample
size and finally calculation of root mean square error(RMSE) are the major steps involved to carry out thepresent study. These steps have been discussed in thefollowing sections.

Third moment,m3 =n−n( xi − x )3∑(n − 1)( n − 2) i =1

Fourth moment, m =4n−n2( xi − x ) 4∑(n − 1)( n − 2)( n − 3) i =1

Moment coefficients i.e. moment ratios are calculated
as given below:−Coefficient of variation, Cv = σ/ mean = m12 / 2 / x－ 70 －

Coefficient of skewness, Cs =m3/ σ3
Coefficient of Kurtosis, Ck = m4/ σ42.2.2 L-moments
L-moments (Hosking and Wallis, 1997) based
method is a simpler and effective alternative ascompared to conventional method to select thebest-fit distribution function for a region. For theordered sample xj=1≤xj=2 ≤ ---- ≤ xj=n at any station,
the first four L-moments are calculated as:
l1 = b0, l2 = 2b1 − b0l3 = 6b2 − 6b1 + b0
l4 = 20b3 − 30b2 + 12b1 − b0
1 n
Where, bo = ∑ x jn j =1n1
( j − 1)x j∑n ( n − 1) j =2n1b2 =( j − 1)( j − 2) x j∑n( n − 1)( n − 2) j =3

b1 =b3 =n1( j − 1)( j − 2)( j − 3) x j∑n(n − 1)( n − 2)( n − 3) j =4

L-moment ratios are expressed as follows:
L-coefficient of variation, LCv=l2/l1
L-coefficient of skewness, LCs=l3/l2
L-coefficient of kurtosis, LCk=l4/l2
2.3 Error calculation
Bias and root mean square error (RMSE) arecommon measures of the performance of an estimator(Hosking & Wallis, 1997). Bias is expressed as meanof the difference between parameter obtained fromavailable sample and the parameter representing thepopulation data set. But RMSE is expressed as meanof ((θs-θp)2)1/2 where θs and θp represents sample
parameter and population parameter respectively.
Both the bias and RMSE of parameter have the sameunits of measurement as the parameter. It isconvenient to express bias and RMSE as ratios withrespect to the parameter itself and hence, we obtain
dimensionless measures, the relative bias and relative
RMSE. Contributions of negative and positive biases
may cancel out to give a misleadingly small value ofthe bias. Hence, the present study has been dealt interms of RMSE only as the average relative RMSE
measures the overall deviation of sample quantiles
from population quantiles.
3.

Results and discussion

As discussed in previous section, data werefirst analyzed to check for independence,homogeneity and outliers. Data sets of all theconsidered stations were found independent andhomogeneous at 5% significance level. For example,
value of z, an indicator for independence andhomogeneity test by Mann-Whitney (1947) method,
was found as 1.94. This value is lesser than 1.96
which is the critical value for sample size greater than
20 at 5% significance level. Hence, the population
data set was assumed as independent andhomogeneous. The presence of outliers in the datacauses difficulties when fitting a distribution to thedata. Presence of even a single outlier, may be on
upper or lower side, affects the moment coefficientmuch. For example, deviation in moment coefficientvalues in presence and absence of outlier at Hirakata
station has been compared and shown in table 1. The
fall in LCs and LCk was found as 23% and 30%;
whereas in Cs and Ck was found as 64% and 69%
after deleting the single outlier in upper side. The
table 1 shows that conventional moment coefficientsare much deviating with the presence of outlier ascompared to L-moment coefficients. In other words,
conventional moment coefficients are more sensitive
to the presence of outliers as compared to method ofL-moments.
Figures 3-8 shows variation pattern of moment
coefficients (skewness and kurtosis) at Hirakata as anexample. In general, similar variation pattern wereobtained at all considered rain-gauge stations.
Considering all the results, error in both conventionaland L-moment coefficient of skewness and kurtosis isvery large (points are deviated largely from one
another) for sample size of 20. Error in moment
coefficients is reduced largely to a sample size of 50
or more and hence, deviations in between points are
smaller as shown in Figures 3, 4, 5 and 6.
As shown in Figures 7 & 8, points plotted forskewness versus kurtosis for dataset of smaller
sample size deviated very much from each other aswell as from the point of whole sample size and got
concentrated as the sample size increased to 50.
Deviation in moments coefficients for different
sub-samples size from that of the whole sample (94years) has been shown numerically in table 2 as an
example at Hirakata in terms of RMSE.

－ 71 －Table 1 Comparison of moment coefficient due to presence of outliers at Hirakata, Japan
Moment Coeff.
Without outlier
With outlierError% error
LCs
LCk
Cs
Ck

0.158
0.109
0.709
3.004

0.194
0.141
1.158
5.071

0.370
0.032
0.450
2.067

Conventional skewness vs Sample size

0.35

1.8

0.3

1.6
1.4

0.25

Skewness (Cs)

L-skewness (LCs)

L-skewness vs Sample size

23.33229.59563.52668.791

0.2
0.15
0.1
0.05

1.210.8
0.6
0.4
0.2
00-0.2

-0.05
020

4060

80

0100

2040

60

80

100

Sample size (N)Sample size (N)Fig. 4 Conventional moment skewness versus sample
size at Hirakata, Japan

L-kurtosis vs Sample size

Conventional kurtosis vs Sample size

0.3

70.25

60.2

Kurtosis (Ck)

L-kurtosis (LCk)

Fig. 3 L-moment skewness versus sample size atHirakata, Japan

0.15
0.1
0.05

54
32
1

00

-0.05
020

40

60

80

0100

20406080

100

Sample size (N)Sample size (N)Fig. 5 L-moment kurtosis versus sample size atHirakata, Japan

Fig. 6 Conventional moment kurtosis
versus sample size at Hirakata, Japan

Table 2 Moment coefficient for different sample size dataset at Hirakata, Japan
Sample
size
2030
40
50
60
70
80

L-moment
RMSE in RMSE in
LCs
LCk
0.115
0.087
0.066
0.048
0.033
0.023
0.009

0.091
0.051
0.037
0.030
0.027
0.019
0.008

Conventional moment
RMSE in RMSE in
Cs
Ck
0.574
0.444
0.343
0.253
0.160
0.143
0.039

1.747
1.178
0.783
0.522
0.312
0.257
0.062

－ 72 －L-moment% error in %ge error
LCsin LCk
48.58
32.54
25.89
18.62
11.82
6.77
4.79

52.10
36.77
27.37
19.93
11.78
9.78
6.79

Conventional moment% error in % error inCsCk
59.93
48.33
38.32
22.28
17.89
16.43
9.064

44.12
25.99
18.05
12.07
8.17
7.36
5.067

Skewness vs Kurtosis

L-skewness vs L-kurtosis
0.30

n=20

0.25
0.20

n=40

0.15

n=50

0.10

n=70

0.05

n=80

n=60

n=94

5

n=30

n=20
n=40

4

n=50
n=60

3

n=70
n=80

2

0.00
-0.05-0.10

6n=30

Kurtosis (Ck)

L-kurtosis (LCk)

7n=94

1

0.00

0.10

0.20

0.30

0-0.5

0.40

L-skewness (LCs)

00.5

1

1.5

2Skewness (Cs)

Fig. 7 LCs-LCk for different sample size at Hirakata,Japan
While calculating the value of root mean squareerror (RMSE) for each data set, population
coefficients were assumed as that resulted from thewhole data set which has length of more than 90

Fig. 8 Cs-Ck for different sample size at Hirakata,
Japanyears. It shows that average % deviation is about 50%
for sample size 20 and decreases near or below 20%
as the sample increases to 50.
References

4.

Conclusions

Findings of the paper are as follows:
1. Conventional moment coefficients are moresensitive to outlier observation as compared to
L-moment coefficient.
2. Deviation pattern of moment coefficients(skewness and kurtosis) with respect sample size issimilar for both methods showing there is not muchcomparative advantages of L-moments method overconventional in terms of sample size.
3. Identified frequency distribution function with thehelp of data set of sample size 40 or less may not bebest-fit frequency distribution representing a region.

Grubbs, F.E. (1969): Procedures for detecting
outlying observations in samples, Technometrics,
Vol. 11 (1), pp. 1-21.
Hosking, J.R.M. and Wallis, J.R. (1997): Regional
Frequency Analysis, Cambridge University
Press, United Kingdom.
Mann, H.B. and Whitney, D.R. (1947): On a test ofwhether one of two random variables isstochastically larger than the other, The annualsof mathematical statistics, Vol. 18, pp. 50-60.
Rao, A. R. and Hamed, K.H. (1994): Frequency
analysis of upper Cauvery Flood data byL-moments, Water resources management
8:183-201.
Rao, A. R. and Hamed, K.H. (2000): Flood
Frequency Analysis, CRC Press LLC, Florida.

－ 73 －地域頻度解析における分布関数同定のためのサンプルサイズの適切性

Binaya Kumar MISHRA*・立川康人*・宝 馨* 京都大学大学院工学研究科都市環境工学専攻

要旨

地域頻度解析において極値水文量を推定するためには，同質と考えられる地域を代表する適切な頻度分布関数を決定する必要がある。頻度分布の母数推定には，積率法やL積率法が用いられ，推定された母数の値は同定に用いる時系列データの期間に依存する。本研究では，観測時系列データの長さによって現れる母数推定値の違いをRMSEを用いて分析する。その結果，20サンプルを用いた場合には50％近い母数推定値の違いが現れ，50サンプ
ルとするとそれが約20％に減少することがわかった。

キーワード: 分布関数，L 積率法，サンプルサイズ，モーメント係数－ 74 －