TitleApplication of Extreme Value Distribution in HydrologicFrequency Analysis Part I. Extreme (Largest) ValueDistribution and Method of Fitting

Author(s) KADOYA, Mutsumi

Citation Bulletins - Disaster Prevention Research Institute, KyotoUniversity (1964), 66: 1-32

Issue Date 1964-03-25

URL http://hdl.handle.net/2433/123739

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

1

DISASTER PREVENTION RESEARCH INSTITUTE

KYOTO UNIVERSITY

BULLETINS

Bulletin No. 66 March, 1964

Application of Extreme Value Distribution

in Hydrologic Frequency Analysis

By

Mutsumi KADOYA

Contents

Synopsis Part I. Extreme (Largest) Value Distribution and Method of Fitting

1. Introduction

2. Extreme (largest) value distributions and their fundamental properties

3. Estimation of population parameters by method of moment

4. Selection of applicable type of asymptotes from view point of hydrologic

frequency analysis

5. Concept of plotting value

6. Plotting values for first and second asymptotes 7. Practical method of estimation of population parameters

8. Expected value for given return period

9. Conclusion

Part. II. Singular Hydrologic Amount and Rejection Test

1. Introduction

2. Definition of singular hydrologic amount

3. Estimation of singular hydrologic amount based on extremes

4. Method of rejection test

5. Conclusion

2

Application of Extreme Value Distribution

in Hydrologic Frequency Analysis

By

Mutsumi KADOYA

Synopsis

It is well known that there are three types of asymptotes for the dis-

tribution of the extremes or largest values, which are expressed as follows :

F(y) = exp (— a-u) ;

y=a(x—u), for the first asymtote or the Gumbel's dis-

tribution,

y=a log (x+b)/(u+b), for the second asymptote or the type A of

log-extreme value distribution,

y=a log (g—u)/(g—x) for the third asymptote or the type B of log-extreme value distribution,

in which a, u, b and g are population parameters.

Several problems have remained unsolved in practical analysis of hy-

drologic frequency by the use of these asymptotes.

(I) In Prat I, first of all, the statistical characters of the three asymptotic

distributions are discussed theoretically and it. is shown that they should be

applicable in limited range of the value of coefficient of skew, Cs, that is

' the second asymptote , Cs = 1.1395•••; for the first asymptote,

\ the third asymptote.

Next, although methods of estimation of the parameters included in the

asymptotic equations have been proposed by Gumbel and others by the help

of method of moment, the results obtained by such methods seem to be not

so good in fitness to hydrologic data. Then , a method of estimation based on the concept of plotting value instead of plotting position , proposed by the author is succesfully developed for the first and the second asymptotes from

a view point of practical application.

(II) Generally, very large or small data are to be contained in a sample ,

3

which is called the singular value. In estimating the population parameters

of asymptotes, the rejection test of such data is essential in the sense of

stochastics. Moreover, evaluation of the singular value is important in the

sense of engineering.

In Part II, first, applying the concept of two-sample theory on normals

the method of evaluation of a singular value is proposed. Next, on the basis

of the binomial distribution, the criterion for rejection of singular data is

defined.

Part I. Extreme (Largest) Value Distribution

and Method of Fitting

I. Introduction

The extreme value distributions are defined generally as asymptotic

forms of the distribution for the largest or smallest value in a sample.

The practical methods of application of these distributions to various engi-

neering problems have been considered by several investigators. But the

introduction of statistics in this field to hydrologic forecasting seems to owe

much to Gumbel, who showed the usefulness of the first asymptote of the

distribution for largest value to the frequency analysis of floods in 1941".

Moreover, he showed that the third asymptote of the distribution for smallest

value is successfully applicable to the frequency analysis of droughts in

19542'.

In estimating the parameters included in these asymptotes, the classical

method of moment and the method based on the concept of plotting position

were adopted by Gumbel"-3). As methods of estimation of the parameters in

such asymptotes, besides the above methods, there are useful ones proposed

by Thom" (1954), Lieblein5) (1954) and Jenkinsona (1955).

Since 1952, the statistics for the largest values of the hydrologic data

have been studied by the author, who investigated (1) the statistical proper-

ties of three asymptotes for the largest value (1955)7', (2) the method of

estimation of their parameters by the classical method of moment (1955)7',

and (3) the one based on the concept of plotting position (1953, 1954)8'9'.

As a result, it was clear that the first and the second asymptotes were

available to the frequency analysis of the hydrologic amount. But the

4

results of estimation of their theoretical distribution by the classical method

of moment did not prove very good in fitness to hydrologic data.

Afterwards, a reasonable method",ss' of fitting based on the concept of

plotting value, which was proposed by the authors' instead of plotting

position, was developed for these asymptotes. In this part, the statistical properties of the extreme (largest) value

distributions and the method of estimation of their parameters and so on,

obtained by these studies are summarily presented.

2. Extreme (largest) value distributions and

their fundamental properties

It is well known that there are three types of the asymptote for the

distributions of the extreme (largest) value in a sample, which are practically

expressed as follows :

F(y) = exp (—e-u) ; (1.1)

1 st ; y=a(x—u), —00<x< 00 (1.2)

2 nd ; y=a log (x+b)/(u+b)=k lg (x+b)/(u+b), —b<x< 00 (1.3)

3 rd ; y=a log (g—u)/(g— x)= k 1g (g—u)/(g—x), —00<x<g (1.4)

In the above equations, k=a log e=0.4343a, u, b and g are population

parameters, y is the reduced variate of actual extreme x and is called the reduced extreme, F(y) is the asymptotic probability in which the extreme

variate will not exceed a certain fixed variate, and log and lg stand for the

common and natural logarithms, respectively.

In the field of hydrologic statistics in Japan, the first asymptotic dis-

tribution is usually called as the Gumbel's distribution in honour of his

pioneering and fruitful work, and the second and third asymptotic distribu-tions are called the type A and B of logarithmic extreme value distributions,

respectively, since the author's proposal in 19557', these three asymptotes

are generally called the extreme (largest) value distribution.

Since the mathematical or statistical properties of these asymptotic

distributions have been studied by a number of investigators and are dis-

cussed in detail in the masterpiece "Statistics of Extremes" by Gumbel"),

it is not necessary to discuss them again here. But the following proper-

ties, among which several unpublished ones are included, should be noticed.

(1) The first asymptote : There are simple relations between the

5

population moments vi(x—u) and vi(y) about origin of order i, and between the population central moments gi(x) and pi(y) of order i, that is

Vi(x —U) =14(y) / al 1 (1.5) Pi (X) = Pi(Y)/ai

The population moments vi(y) and pi(y) are easily calculated by using the

moment generating function Q(t) and the semi-Invariant as follows' :

1g Q(t)-=1g F(1—t), 111<1

=rt+ES(r)tr/r r=2 where

Y = 0.5772.- is the Euler's constant, S(r)= lim (1+2-T-F3-r±•••+n-'),

n—>co

Therefore, the mean my and mz, the variance 6,2 and o-.2, and the other

moment Pet(y) and Pi(x) are expressed, respectively, as follows :

MY = V1(Y) mz=u+r/a

0-52=P2(1) =S(2) =n2/6, o-x2=S(2)/a2= 7r2/6a2 (1.6)

Ili(y) =2S(3), P8(x)=2S(3)/a3

It should be noticed that the coefficient of skew of the extremes

themselves, equal to that of the reduced extremes, becomes

Cs = PO-123/2 = 2S(3)/S(2) 3/2 = 1.1395... (1.7)

(2) The second asymptote : The population moment vi(x+b) about

origin of order i is expressed by

vi(x+b)=S:(x+b)idF(x) After several calculations,

vi(x+b)=(u+b)ir(1—i/k) (1.8)

The mean and the variance and the other moments are easily obtained

by putting i=1, 2, 3,-- in above equation. And the coefficient of skew Cs

becomes

/23 7'(1 —3/k) — 3T'(1-2/k)r(1 —1/k) +27-'3(1 —1/k) Cs=(1.9) P23/2 Cr (1 — 2/k) — [2 (1 — 1/k) 3/5

It becomes clear after some examinations that

dCs d(l/k) >0

and

6

lira G=2S(3)/S(2)8/2=1.1395••• 1/k—A

That is, the second asymptote is the extremely skewed distribution as its

coefficient of skew Cs is greater than 1.1395•••.

(3) The third asymptote : The population characters of the third

asymptote can also be easily examined as well as is done for the second

asymptote, and the following relations are obtained,

vi(g — x) = (g t)''1r (i±i / k) (1.10)

r (1 ± 3/k) — 3P (1 + 2/k)/' (1 +1/k) +2r3(1 + 1/k) (1.11) — —

G

cr (1+2/ k) —T2(1+1/k)D3ia f oo Moreover, it can be

o.8 let Asymptote, de1.1395 made clear that the

I\0.6 2nd Asymptote, 1/k=0.2, Os.3.535 coefficient of skew Cs `\ of this asymptotic

..4 3rd Asymptote, 1/k.0.5, distribution is less

than 1.1395•••.

• , From the facts

-3 -2 -,0 I2 3 4 X mentioned above, the

Fig. 1.1 Shapes of asymptotis distribution, provided conclusion is obtained

m,=0 and 6,=1. that the three as-

ymptotic distributions of the largest value

„ are characterized by the coefficient of a

skew, or that they should be applicable

6

in limited range of the value of coefficient

of skew Cs, that is 4 f

or the second asyptote,

2 1st Asymptote Cs =1.1395 b Cs = 1.1395•••; for the first one,

g—co 0 o.5 1.0for the third one.

3rds'Lk— -2j'inPtoteFigures 1.1 and 1.2 show this relation.

-4 3. Estimation of population

parameters by method Fig. 1.2 Relation between

Cs and 1/k. of moment

A method of estimation of the population parameters included in these

asymptotes by a method of moment is easily derived from the results in the

7

preceding section.

(1) The first asymptote : Using Eq. (1.6), two parameters a and u, which are scale and location parameters of the first asymptotic distribution,

respectively, can be obtained by

cry/ax

= 7/ ^ 6 ex -t- 1/0.7797a. (1.12) u=mx— mu/a

=mx—r/atmx— 0.45000.

A so-called classical method of moment has been adopted by Gumbel in

his earliest work'', in which the population values crx and mx in Eq. (1.12)

are directly replaced by the sample values S. and _I", respectively.

(2) The second asymptote : Three parameters 1/k, b and u, which

are skewness, location and scale parameters of the second asymptotic dis-

tribution, respectively, can lead to the following expressions after several

calculations based on Eq. (1.8),

Cs= F(1 — 3/k) — 3P (1 — 2/k)P (1 — 1/k) + 2P3 (1 — 1/k) Cr (1 — 2/k) — F2(1 — l/k) D372 (1.9)'

1/a= 0.4343/k

(1.13) u=mx—Biax

u+b—Ciax

where

.417.---1"' (1 — 1/ / (1 — 2/k) — P2(1 — 1/ k)1/2

B1------CT (1 —1/k) — 1)/CP(1— 2/k) — r2(1 —1/k))112 (1.14) — B1—=1/CP (1— 2/k) — r2 (1 - 1/k))1/2

In the above equations, it will be noticed that the values of Cs, Al, B1

and C1 depend only upon the value of parameter 1/k. In Table 1.1, those

values as a function of 1/k are shown for the practical facilities, which are

originally prepared with six decimal places'. If an adequate method of

estimation of the population values Cs, ax and mx from the sample values is

found out, the parameters may by easily estimated from Eq. (1.13) by using

Table 1.1.

(3) The third asymptote : Three parameters 1/k, g and u, which are

skewness, location and scale parameters of the third asymptotic distribution,

respectively, are also obtained from Eq. (1.10), as follows :

8

Table 1.1 Population values of Cs, A1, B1 and Ci for 1/k in second asymptote.

b=Aia-nvx, xo=mx-Biax, xo+b=Cui.,

1/k Cs Al B1 Ci

0.001 1.1455 779.13 0.450 2 778.68

2 1515 389.28 4 388.83 3 1576 259.33 6 258.88

4 1636 194.35 8 193.90

5 1697 155.37 9 154.92

0.006 1.1758 129.38 0.451 1 128.92

7 1819 110.81 3 110.36 8 1881 96.89 5 96.44

9 1943 86.06 6 85.61

10 j 2005 77.39 8 76.94

0.011 F 1.2067 70.30 0.452 0 69.85 2 2130 64.40 1 63.94

3 2193 59.40 3 58.95

4 2256 55.11 5 54.66

5 2319 51.40 6 50.95

0.016 1.2383 48.15 0.452 8 47.70

7 2447 45.28 , 9 44.83 8 2512 42.74 0 .453 1 42.28 9 2576 40.46 2 40 .00

20 2641 38.40 4 37.95

0.021 1.2706 36.54 0.453 5 36.09

2 2772 34.86 7 34 .40 3 2837 33.32 8 32 .86 4 2904 31.90 0 .454 0 31.45

5 2970 30.60 1 30.15

0.026 1.3037 29.40 0 .454 3 28.95 7 3604 28.29 4 27.84

8 3171 27.26 6 26.80 9 3239 26.30 7 25.84 30 3307 25.40 8 24.95

9

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18

r (1+ 3 / k) — 3ra +2/ k)r (1+1/k) + 2r3a+ 1/ k) Cs= (1.11)' Cr (1+2/ k) -rza+1/k) .Dzn

1/a--=0.4343/k

g=mzd-A2az (1.15) u= mx —

g—u—C2az where

(1 +1/k)/Cr (1+2/k) — r2 + k))112

B2= (1 (1+ 1/ k) )/cr(1 +2/k) — P2 (1 +I/k)]1/2 (1.16)

C2-=-A2 -PB2 =1/Cr(1 +2/ k) — r2(1+1/k)Din

The values of G, A2, B2 and C2 can be tabulated as a function of 1/k

as well as that for the second asymptote. These values, however, correspond

to the ones for the third asymptote of the smallest value prepared by Gumbel

in his book, as follows :

For the third asymptote of For the third asymptote of largest value (the author)" smallest value (Gumbel)1c'

— pi (k)

A2 B(k) — A (k)

B2 A (k)

C2 Em7 B(k)

Since there is no difference between the two values in essence, it will be

unnecessary to show such a table in this paper.

4. Selection of applicable type of asymptote from

view point of hydrologic frequency analysis

It has been shown above that the three types of asymptote for the

largest value should be applicable in limited range of the value of coefficient

of skew G. But such a discussion for the population value is not always

realistic for the sample value. The reason is that, for example, the value

of the coefficient of skew C's of the first asymptote sample, which means a

sample taken from the population of the first asymptote, is not always equal

to Cs=1.1395-• for the population, but there are various cases where it will

be larger or smaller than Cs=1.1395-• from a view point of sample theory.

Otherwise, one of the most important problems in relation to the

hydrologic largest variates is how to estimate a bigger future value. In

estimation of such a value by basing on a sample of small size obtained by

hydrologic observation, it may be often happen that the third asymptote is

19

wrongly applied to the first or the second asymptote sample , or that either

the first or the second asymptote is wrongly applied to the third asymptote

sample. However, from the view point of the prevention of disasters , the

former mistake seems to be more serious than the latter.

Under the above considerations, y-Rsia the following course of treatment in--9r

.9 the hydrologic frequency analysiss9.I

. should be adopted.... -99.5 5 i) If the series of plotted points _99.

of hydrologic data on the extremal 4- guier12aignium - 1.probability paper is scatterd about a3- 95."3"WAB straight line or a curve, denoted as 2 -111W11111111 amL

IFAUMMEM G or B in Fig.1.3, the first asymptoteEL9111111111 ' miroz

ellmm -=MAO.=MENEM must be applied. o -5rah=2 - A _ iz•,•••m• • B ii) If the series of points is 10 otan-E,

scattered about a curve, denoted as - _2_ 0.1 Ain Fig. 1.3, the second asymptote Hydrologic Amount

is usefully applicable. Fig. 1.3 Condition of application for three asymptotes.

iii) The third asymptote must

not be applied, except for a family of data of which the plotted points are

arranged about the curve B with extremely large curvature.

Therefore, a discussion of the third asymptote will be omitted in the

following sections.

5. Concept of plotting value

The simplest method of estimation of the parameters of asymptotes for

the distribution of the largest value is the so-called classical mothod of

moment in which the population moments described in section 3 are directly

replaced by the sample moments. The population moments are obtained by

integration over the whole domain of variation, while the sample moments

are based on the sample of limited and small size N, generally. The results

obtained by the classical method of moment seem to be not so good in

fitness to hydrologic data, because of the bias between two moments,

although it may not be a sufficient reason. In order to eliminate this bias,

an approximate method is required, by which the population moments may

20

be evaluated as a function of sample size N.

In the field of hydrologic statistics, the population moments are often

calculated by using the plotting position. However, the concept of the

plotting position must be more fully considered in applying to calculation of the population moments, because it may be originally used for construction

of the empirical distribution function.

Suppose that x1, x2,• • • x.67 are a set of observations of size N, and

xi<x2� • < xN. And let them be a sample of size N from a population,

having the continuous cdf F(x) of which only the type and, therefore, the

value of coefficient of skew Cs, is known. Then, as is well known, the

probability element dp(xt)dxt for the i-th order statistics xi in such a sample is given by

dP(xt)dxi=r (N+ 1)P(i)r (N F(xi)N-Y.(x i)d x (1.17)

If the parameters included in F(x)=z f(x)dx are not larger than

three in number, and if the distribution functions of such asymmetrical

types as are applicable to the hydrologic frequency analysis are supposed,

the unknown parameters included in Eq. (1.17) must be two in number,

since the value of coefficient of skew is known. Now, let the linear reduced

variate z be defined as

z= a(x —b) (1.18)

where, a and b are numerical constants.

Then, the probability element dP(zi)dzi for the i-th order statistics zi is

given by

dp(zi)dzi=(N +1) (F(zi)Di-1(1—F(zi)PN-tf(ZOdzi (1.19) P (i)P (N —i+1)

and the unknown parameter must be not included in this equation.

Since the unknown parameters a and b should be estimated as follows,

E Czi — a(xi— b)D = m i nim um (1.20)

under the condition

E(zi)= aCE(x I) —b), (1.21)

the problem of estimation of the parameters is reduced to estimating the

value zi corresponding to xt.

A schematic figure of the distribution of zi corresponding to xi is shown

in Fig. 1.4, where xi are regarded as the fixed variate. Since the probability

21

element of zt is given. by Eq.

(1.19) as the functions of order

i and sample size N, the de- : • / : termination of zt is almost

equivalent to estimating the •. .

value of Zt satisfying the follow- • • : : •

ing condition ..• • : E (Zij— 202= minimum. (1.22) •.•0; Expectation of

The solution of Eq. (1.22) is,•P.• Plotting Value

•

clearly, • • :Mode of Plotting

•

zt—E(zt) (1.23)Value ( max. spot )

The line connecting eachSample Value X( attend to order ) Fig. 1.4 Schematic figure for distribution

value of E(z1) is not always of plotting value. equivalent to the one which

satisfies the condition of Eq. (1.20) in a theoretical sense, but the difference

between the two lines seems to be so small that it may be ignored in a

practical sense. In this paper, the value of E(zt) obtained from this idea is named

the (expected) plotting value, to distinguish it from the plotting position.

In addition, if the discussion of the plotting position is made, the dis-

tribution of the value of Ft instead of the value of xt should be considered

in Eq. (1.17). P(N+1) dp(F

i)dFi=(1.24) r (or - +1)

This equation is originally parameter-free, differing from Eq. (1.17),

and the function of i and N. The polotting position fri must satisfy the

following condition,

E (F1j—ft)=minimum. (1.25)

And its solution is, evidently, -P

,=E(F1)=i/(N+1) (1.26)

This result differs in no way from the plotting position adopted by

Thomasi8), Gumbel2'3) and the others.

6. Plotting values for first and second asymptotes

Although the plotting position is distribution-free, the plotting value is

22

neither distribution-free nor parameter-free, except the special type of dis-tribution with two parameters. In this section, the plotting values for the first and the second asymptote samples will be discussed.

(1) The first asymptote :

F(y) = exp(-e--20

f(y)= exp(- y - e-v) y=-= z = a(x-u)

Since the reduced extreme y itself is linear to the actual variate x, the value of E(yt) has only to be estimated, that is, the plotting value for the first asymptote is parameter-free. In this case, Eq. (1.19) becomes

rov+1) dp(YOdyt=(i)T (N - + 1) Cexp{-(i-1)e-VIDC1 - exp( -YODN-1 x

Cexp ( - Therefore, E(yt) is

E(Yi) = (N+1) - r(i)r(N -i+1)3 _y exp(-y-ie-v)C1 -exp(-e-Y)Div-idy

71) 1;-,-'( =riorcAr-i+1)to•1)riv-zC, y exp{ - y - (i+ r)e-Y}dy

After several calculations, it becomes

(N+1)1 E( .30=E( 1)riv-tC,{r+1g(i+ r)).(1.27) r(orcAr -i+1)r =oi+r

where, r is a so-called Euler's constant and C means the symbol of combi-nation.

(2) The second asymptote :

F(y) = exp(-e-20,--exp(-z-9

y=klgz z= (x+b)/(u+b)

The plotting value E(zt) in the case of the second asymptote is

( - E(zt) = N +1) .1z T(i)r(N -i+1)oexp(- iz-19Cl-exp(-z-9j1v-'dz

r(v+1) -i exp{ - (i+r)z-}clz (i)r(AT-i+.1) kE_0(-1)''N-2Cro

Then, it is expressed as follows :

(N+ 1)N E(zi)= r(or,(N _i+1)r(1-k)E(--N-LC,(i+r)--1+1/1G (1.28)

r =0

23

(3) A practical method of computation of the plotting values : The

plotting value E(.311) or E(zi) of the i-th order statistics in the first or the second asymptote sample must be able to be calculated strictly by Eq.

(1.27) or Eq. (1.28) as a functions of the order i and the sample size N.

But in solving these equations for the various values of i and /V, it may be

seen that the smaller i is, and y F9) 7 -99.9 the larger N, the more difficultIIM.ra^IN

EMOM:=21ENNME...„.1,e--NS the calculation becomes. There- 99.8 ^updimmil.opor....g 6-.ipprop .FmEssad fore, the following method of gimm ,a17,miiii,li,o,.a.-E 99.5 nimmez,,, ....1reha \i*mem computation may be used in a

5-MS121.;2111110--mi'l,. practical sense. 99.mm .r........„..^MallINCAMM egw^Ems.n...^.:ASMEN NEASM•Ilamil INIMMINWA'AMMENNIGMMENS.

0i= N : Thelottin wolurdomumvxmom•,wirm^^Iim pgvaluesInma74^00.11. ,"=^•••ni NCIW,"Affiri• 3"4-te.:.= for the two asymptotes are ob-NwT-41.linc.?jiAmiI wilm.trmori

tained by puttingi= N inEqs.^3 _ 95friAmt,sm.....g.... ^stramaasTioll (1.27) and (1.28), respectively.maimmoomini =mramponhuiIII 90a^ipimmeitMailME For the first asymptote ; 2 - 6-E'smilleII ik" E ( yIv) =r+lgN (1.29)90 pwasilma141i,r,',0,.3H i) I- For the second asymptote ;

E (z.67) = N1m l' (1 — 1/k) (1.30) so o o- i=i

ii) i= 1 ; The plotting values jiq 0 1 20 MIEIIIIIIII - 9-2- for the two asymptotes are cal-

I--I °----==.7.---g-S.REInI. PI/.k - 0 culated by a method of numerical 5Wa",AM ..5ti°M-7Edel,,,,AZk0 3 #111MIEM111111....gra • 0.2

integration. Several results oh- I IIIIMMIIIMMELLMIlliii ','o.1111=1111111111MMEN*..: -2 - 10 20 30 50 70 100 200 300 500 tained by such calculation are Sample Size N

shown in Fig. 1.5, where the Fig. 1.5 Plotting values for i=N and 1. Where T.P. =Thomas Plot =i/(N+1),

confidence limit121 defined by Eq. H.P. =Hazen Plot= (2i-1)/2N and G.P. means Gumbel Plot proposed in

(1.31) is also shown. Ref. 19).

P(yi�vor zi�z).�131=r(N+1) F131P-1(1— F)N-IdFi 0, Pi T(i)r(N—i+1)50 (1.31) P(N+1) 1

P(Yt>31or zi>z )<132=5Fi-l(1—F)iv-ic/Fi fiz ——132 r(i)P(N—i-F1)Fg2

iii) 1<i<N: The plotting values for the two asymptotes are not easily

obtained as long as the troublesome calculation of numerical integration is

not performed. So, the values which satisfy the following equation will be

24

adopted as the approximations of plotting value, after a model of the method

proposed by Gumbe119) in his paper "Simplified Plotting of Statistical Observations".

i-1 F)(1 .32) Pt=F(— 1iv _F.1171

where F1=-=-F{E(y1)1 or F{E(zi)}

Fiv—=-F{E(y1,7)} or F{.E(ziv)}

Then, the estimates of plotting value are obtained by

for the first asymptote ; yi= — Ig ( — ig (1.33) for the second asymptote ; zi=(-1g FL) -1/k

Therefore the plotting value yi is expressed as a function of the sample

size N and the order i, and z as a function of the skewness parameter 1/k

and N and i.

7. Practical method of estimation of population parameters

(1) The first asymptote : It has already been explained that the fundamental equations for estimation of the parameters in the first asymptote

are expressed as follows : a = ay/ax

u=mz—my/a

If the sample size N is very large, the classical method of moment

must be valid, in which the following values are adopted as already shown

by Eq. (1.12). ay=7r/ ,/ 6

But, since the number of hydrologic observations is usually 101 in order, it

cannot but be considered that m„�r, a5_-.2r/ -V 6 . Therefore, the population

values y and Sy as a function of sample size N should be adopted instead

of the values of which m5=1- and au= ir/ 's/ 6 , although some difficult points

remain to be discussed. In this case, the parameters a and u are rewritten

as follows : 1/a = sx/s,

u=36-- (1/a)3)(1.34)

where

25

Tablel.2PopulationvaluesofラandSyfor/Vi豆firstasymptote.

1/a=s出/Sy,%=-x一 夕/α

2VラSy

__..____一 一__「_______「_1

200.56921,1825

296895

499956

60.57021.2009

805055

1

1300・57071・2095209130

411162[1 ・i6i13'192

815219

400.57171,2244

218266

420287　

621306L

823323

500.57241.2338

225352

4263661 」627379

828391

600.57291.24021 1

230413!ミ

4314231

632433

833442

700,57341、2451

7536472

8037490i

8539506

9040520

954153210042542

26

1—1 7x.=Nx2=NEx2

12(1.35) Sex=NE (x —79)2 =

The population values ij and sy calculated by basing on the plotting value

as a function of sample size N are presented in Table 1.2.

(2) The second asymptote : In estimation of the parameters of the

second asymptote, the problem of bias between the population values arises

as well as in the first one. Speaking generally, this bias seems to become

large with the increase of the order of moment, because it must be caused

by the difference between the domains of variate under consideration.

Therefore, if the bias of moment of the highest order which is needed at

least in the calculation is satisfactorily settled, the ones of the other moment

of lower order may be so small that they can be ignored in a practical

sense. That is, if the skewness parameter 1/k has only to be evaluated

successfully, Eq. (1.13) must be usefully available. The theory of plotting

value described in section 6 will be utilized for such a purpose.

Now, since the plotting value z is linear reduced variate of actual

variate x, the sample value of the coefficient of skew C's(z) about z must

be equal to C's(x) about x,

C's (z) = C' s(x) (1.35)

The calculated values of C's are shown in Fig. 1.6 as a function of the

skewness parameter 1/k and the sample size N. Moreover, if the relation

between the population value G given by Eq. (1.9) and the sample value

C's is expressed by Cs= C's(1-Fgs) (1.37)

where g, ; the additional coefficient of skewness

the values of ,Qs are shown in Fig. 1.7.

Therefore, if the sample value of the coefficient of skew C's is calculated

from a sample of size N by

3

C's= 1 E (x —3-+ N sx3 Sz

2 (1.38)

=

.32x="i2

the value of skewness parameter 1/k must be able to be estimated from

Fig. 1.6, or from Table 1.1 by using the value of G presumed by Eq . (1.37)

27

C's

3 . 0

IIIII

0 ,

_ -C4;50

1

prime;40 APAMIN __ApagpmE30

2,0 AMMEMMAM

1111°warieripvq, merPgpomdp4i 0dp!a 144014. ,A6111111

ERMONMEMEN MEEPWROMMIEM

Lo_EMMESOMMEmmm 11111111 gE100:01E211 openg

•

0 0.1 0.2 0.3 I/ k

Fig. 1.6 Relation between C's and 1/k for given N.

13s 1..71.8 ." 2 • 2.0 Cd=1..

I i IIIIMMIWII

111=MILIIIMI 1111=1111111E= 1.5 • 5^SILMEEMBM 11111111E=LIMill=

11111111E=WAILME AlLIMMMIll^en IM

IMILIIIM=INIM -3=MMIIIMMOIAM . 1 . 0IWIMIIMIMMAIL .

. 41^.=11^11.^^^1=1111. =MAIIIMINEMMOSII.\. . °W ,MI1NIMliniMIMI^ 0 .9 ffIRMMM\MM=b11. — 0.5,wro jmwim,--_m___w..^

E...mingsramm..= immEMEINNE

o ME= 20 30 40 50 60 N

Fig. 1.7 Values of Qs provided for Cs=Cs'(1+0s).

28

and Fig. 1.7. And the other parameters tt and b can be easily estimated

from Eq. (1.13) and Table 1.1, using the estimates

rx= ^NAN-1) Sx }(1.39) nzz=i-

Besides, there is a case where the location parameter b may be assumed

to be zero, which happens when the series of points of log x on the extremal

probability paper is scattered about a straight line. In this case, it is needless to say that the other parameters may be estimated by

1/a = S zog x/S, }(1.40) logii= log x— (1 /a):y-

in which S„ and .3 ; see Table 1.2

s2 ioa z = (log x)2 — log x (1.41) 1 (log x) /=—NE (log x)i

8. Expected value for given return period

If all parameters are reasonably estimated, the expected value of hy-

drologic amount for the desired return period T can be easily calculated by

y - Fm) 1000 2 000 300010000 T _99.95

MIPIIIIMPIEF4111111_,__, 7 -99 .91000

6-99.8111111111111111.11111111500 PIO•MOVA11111111113 =MMII/=1=M1,1=M.

INIIONWARM=1.1.1=M 99.5 5-_WILEIREESE200 _wagenra=--- 99 .0:EMI...FM:GM100

4-111.WORIIIIP.1511' 4,50 .1.921^111°SU'zi 3 - 95orrifir „..enn-a-::20

2

41-4411.. -90 ,-,17011-11to ,t,911 ,,,'4MEMh•MIIII '''P. 42111 f MIME 1° ,.'

1- 4.1:11:11 ...-ITAM11111 .. 2 2 - 50a P,,e.-..-m. %Mail INFM..=.=2=== !, 0- trzeizamar;= .7. g ,I ..m.-Em mum =I.= 11%•••=11 -1-1° orison..I1

_ 1MINIM I 0.1 111111I

0 500 1000 0 5000 10000 m3/s

Fig. 1.8-(1) Frequency curves for annual maximum flow, Yahagi River and Miya River.

29

1蕪 轟 熱 譲ll毒 裁 暴 賢憂 驚無 厳 憂薦墨茎吾 置 難 轟 ミ蕪1ミ 罷罷

ぎ 漏onbDC£00022292-ooO

一 円,

1鴇

歪o≧ 。,9m暮 黛 器 遷 鋸 等 δ1に 臼 誌N禦 露 留 認 雷$饅

b遍 毘 も 門 『 『 帆 笹qα 「 の σ1ρ 州 卜 笹 内 叉『 尉 卜 『 尉 『Ooり 一 目o「 司Hoo「 →-〇 一 「→-Oo一 ひa一 目 σミ

・U　.+　Mも8i

妾 婁 能 郭}"』 皿 『 … 魯頃 頃 ひ_。 寸 ξ.。1喜。,。 。 一一

罷 肇懸 羅 艶 轡3調 鱒 聲醜 翻 黙 欝

海 轟 黙 轟 単 §隙1・1き§§爵沼1-一 一㎝… 一 一.一 一一一一…一一一 一一一… 一 一㎝ 一_口!Φ

零 讐,曳 潟m自om尊m$等 需m露 誘m專 專C　mO　mO　m

,V(nの軒の ロ 　 　　　　 　　 　 　　 　 　 　'=1

①w口oo肖 頃 の 頃 ㎝ 」 頃 頃m頃oo恥 頃 頃 の の

llll龍1餐1鎖ll輩llll籠りoω.cう ひ司 一〇-HFく σ㌔aF→ooo(バ ーr→Hσ 潮 ひaひv山 ■⊇oひ ひ ◎ ひ ひ ◎つhひ ひ ひo◎ つ ひ ひ ひoひ ひ

OH目HHHF司PIHHド → 円Hp峠 目H剛 一{一η

図 一.由一一一一 一d

3「.-Gd郎 日.一

憩 壱..§.自 誤5壱臼1塞1量 鐸 嚢1ξ 言肇 藤1繧

擁 舞 薯 一 ∴)∵ 一「oo一 い→一 いのe

沼.ヒ 面 お 邸 お騨 ε

墨 曽z話 壽ゴ ・§ 屠り 署.磐 §1盤 量 盤

.轡 薯 旨§ox寸ox

lE.一......一 一 一 一一一一..一一...一一..一一....一.一一..一一.一...........一 一...-L-..一..一..一.一..…...…一 一一一..一一I

Ispool3噛xBWIBnuuVilBJuIBHJoyanouzy"xBKIBnuuV

l

30

the following relations, as is well known,

for the first asymptote ; x=u+ (1/a)y,

}(1.42) for the second asymptote ; log (x+b) = log (u+b) + (1/a)y,

where usually

y= -1gClg T/(T -1)) (1.43)

Figure 1.8 shows several examples of application of the proposed method

y - FM) 500 1000 1500 -99.95 .__.__.._1 _._____4__..._1_,_ ..EEE

...ME2^=-.7.==.,.......-7-7==..= 7--99.9I=,_FNIIIIMEItri, L,linams,100 vBIMINI ^BM,AMExfr„tillialif•.5 000 6-99.8 i Ziommeno^., 4-.0-,,..amonE-- -1..---.., -4=,,,gswas.w_er.- 99.5 _Fur__o..gt."'iictmana'11100

°200 5 - -:-DOM3E7"ru."--.z:.0...4Efam-m-- _ 99.0 :MIMIIII,ME-w."0121111101M

0.74:521000

395. 4 - 4'MEM..^'AMMWAVAMEMM.MME

:

__-_,JAMMISOMPABEIMIgniffil IIIMPINIMINNIKAKMEM .NE.----..znprapmzos...Tege

.: 4'rh,......alEzumi.. - 90---savirialfillifli,4 2 - origraiiso

11172P2IFM11111 g, P .t• I - owleumq.. 2 ..im -IIRAMM.Tarall& II

50 :=M:',...'""•''''""'i 0 - MUNIP511111••••^ - .`3gWIIICIIMEMEN

I I - 1 0 grilingiENEM . I 1 -I - 1 NET -E • 1 i

-2- 0.1 III 0 500 1000 1500 m3/s

Fig. 1.8-(2) Frequency curves for annual maximum flow, Yodo River.

y - F66 100 200 300 200 300 400 300 400 500600 800T _99.95 IEnE: ini ^riummirdl mum: - in r.

7 -99 .9

11 -E1011111111^111P""""milmIllri,""' m"""'"'"""" 100° 99.8

111111E110111 1111111.11111711 ..111111!11111M111111500 6-rillMIIMP prOTPAPOMPim r!11211111,7 MMMMa........III ..., ,.....,. %

11.11 '4

ld 99.5'' &man'am .01:1- - 200 5-

^gjunriallingAlIAM ":...e ''lir 99'°11,111111!li111110,111111111HOMMIIII4/proiliuriiii_. -loo. <4'MP11111111111101111M1111111 ..,1^iIII1Rid011111 .4.-1- 50

AUCIL."1"rp:11111"Pmml.•?'pilogipiMum 95:1 •1...1...,' ......... -""iiirjes4..:15:.nr,1 5.-1:,.1351 e/,,7ir,lit- 20

,......... -,. dus1., ,<,;c'- mmlykir......J. -IIP.. .....6,.... ,..-.........`'• 90,,•"711h1 18.rii-10 2 -'unflmorii.•irMr-r.ffra. .

,''-.i.l....51I—hapg'.2;".r's - mnmrmnni e nmiriumnuiriummienurnimin I-mum .,, •^ nm/1on,inutim1 3 1-ICIIIIINIrmu01111111{ .airirmi : umMr'

1 -50milamil.urnhium.norminum"cra:2; ,., ranweingu: ..ISZVOr•"•••.nir gliIIIIMMIII ..' I 0 - ruessraneneeroar—.ellI1111MHIIIIIIIIIM !I 8 4 4 nespeurni-imamMIIIIIMIIIII1111F11:1 I -r=liallEr."1 1-11rilimummLurfi / .....

01111mn iliummtAillmu"mumPrimo-1

1

- 2 -aIIMIM11111MIIMIlifilivIIIIMIIII 1 i , , 0 100 0 100 0 100 200 0 200 400 600 800 Min /day

Fig. 1.8-(3) Frequency curves for annual maximum amount of rainfall, (1).

31

to hydrologic data in Table 1.3, where the observed data are plotted as

F=i/(N+l).

From a stochastic viewpoint, however, various points remain to be

discussed concerning the expected value of hydrologic amont for the desired

return period as expressed by Eq. (1.43). These problems will be discussed

Y - Ft%) 100 200 300 400 400 500400600 1000 T -99.95

7 - IlliiiiiMPiiiiiiiirM7AMIE'idnahlrfil50011-'-'1111111 1;rill;ILI' 111111151MEI ','.'ir1000 pri 99.9999119'

riff'1111T/11muniimiann1 1,1iiNEI 99.8m500 IIIIIIIMJriLFABE14.al 6III6'ahnill e.r__,Iiiiali1.../upir 1,.11+1111191Lard

.-1-1P.,ii1"4-Ern._••••,:UMWII,),4.-1 1eFL-200 99.5MU.7MOM NNNNNN4,1MrApr MI 5-ME47orig .i9""r '''''i-FIr}'.1---''urigikir.11titA,--'u100 99.0.1!URI-?I-I--,---1..Y1eAl.ne.4-,-4rnp-• 4I I,*won_;II0111'-'-!warm44pm/IILL- -,A<,.?-11,,,,,[UMWII-;^,11,•-II:2 50• '''Ira11',Iw

- oo,

prgiIRj,,,r-.IF,rr_iii-,,.'.liillk. 0-'.,-,1,41'0---_ .t 3 - 95 M o 4-m'IR:+1=:AITT',,1115.4.11:1, .-.-20

.t;Holirldilit''•,e-r1:'rlior (09. „ -

'

goILI.1,..1:IMES....----01 2 -EllumnomIli,!!wl;_.'Imomoupliv,,‘,.-, ,i, numitumIII'milNIKEInfilllundrmie'+I>., -opigrAgig

-, i111F,i-',IIIIINDMIIu1111111r1IM" 4P g

P

I' I -WAN= -'r-I ON.j,-'-''1MUM .11101111M9."'.4..

- =Mitliln ,-- WA-dint1111V 1111111111=-H IA,A 50 Mensuratunei H i 1 Irisrmilranur gememmu-

0 - geninium -mural: 1U1111'4111119 1111111Vd1 ; .i.4' ri IiiiintiiMi , -mum moron% 11111111MINIMI

- ,0 EliP1111111 ri-IF.111=119:96991991!1 IIMIIININIMII1 -' -tl-'

,Iiilti trII-VAlii11111111111 IIIIHNIIIII111' . _ 2 1al 11 1110IM IIIIIIMMII 1111111M11111 11111101111111 III

0 100 0 100 0 100 0 200 400 600 800 1000 mm/day Fig. 1.8-(4) Frequency curves for annual maximum amount of rainfall, (2).

Y - F(%) 100 200 300 200 300 300 400 400 500 _ 99 95T

7-99.9I--r''I i:: i . ;7,,'LoompArSHts„7grunqumor lidliell,1-' d1•IMME1g..No_INEE ..-rn11,---,An-,'„,"i000 99 111111111011NI min'IMMIIIIIMIIMENTRMiiiiiiiiiiiiiiiiiiii5,,„ 6 -.8 EV11111111/1 11llllram"lllllll VAPHI111 INIVPIIIIIPPAFIMII i161Jing01111111.,..nasiPIL-arifflo -99.5Ln:1E01HW•IIIMMIa ..^••• ...,..........1...1:ll211Erwk11:911:::• 200 5- Illr ,1i..insto erivlludt,I." _99.i/Ell MEINwilIniinumumr"Mil 00 0.11111111111111111 111111111111111410111111 IIIIIII1111111111 -

4-NiN1111111141111111MUNI=v11111111 1111.11111111111111111- wil ,y1111111111IK1111111II,11 • /112111 1111111515filMr111 I••50 -lin .:.'..lllll rii12 Seri-'..1111511MH''! ... .,,...........1..,pp/ 1•••••^^^••••••••••^ z o 3 - 95."-..,F.'"."".".•'•'.%rOgilI ii .1 6 mum .am79,"'S _pg11?:..-...n,i4N,Ilff .:A.::...I1„4iliiiiiiii. 90•`•.c,••.'"""'"•,".,.,-.‘.=OP 4:-,!.MI- 2 -II 1•Lum,,.Amisr_c-7luifilll,,,,4. 10 .ia iiio'sHUM1',Gf')111111 I • L."-- ,i,A - I oAmu^.1&_11,11111HI !MIL17IllIIMg ... P

.,,,mum , _ ,i",...191111i.,P.!IIIIrIIp, .-.1m-r-_#.6'-mow,-,il- --11111=1111.. g .2111'I4.",MR,.-n1W.1111 A i -- ....-!sumEIETT-44:',•••••III,.%11111- §num1 0 -"....11111'n:1 --',-. 'Iril.::• :11 MUM o, E1111111 ' ;:i.1

a wan,rlrdisisimmun I -.A111111 -/ 0 IVIIITAII demi; mural= Ifin1'I _I -orrmuumEmpEmpulgm

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0 100 0 100 0 100 0 100 200 300 400 500 mm

Fig. 1.8-(5) Frequency curves for annual maximum amount of rainfall, (3).

32

Y - FO'd I 00 200 300 400 300 400 400 T .......

71999511111111111FAMMIPduIllidPII6Pnm 99.9 -IIMEM11111111111111111011101111111INFIRM5°,000 5.,....rem.C) 6 :99"8^miorgrunirjunmparm

99.5MWIIIIPPIT.. ::::::;;;;;;;; .r'.......—:m200 mai 5 -111111111111111111r: _1111•••.4—MBIE--100_99,0 4 -1111111111111H1 IMIIHM"-firiilliii•'-,-,._ 50 1f4'!ilmIllii_ liIlillimili7rwor..f: 3 - 5-NAT:eldygi....2.0.iiEnni-.3--':, ,, _2 0 90 ,'H.IL IIf4-.Rje....6T,Iiiir

,

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- 0Lllnalu111ipIipl.pillI lllllll111lllllllllllllll 1

.I W11111111111 111115/11111111111111111111 .I

_2_ 0.11111111111M M11111111111 MUM 11 0 100 0 100 0 I 00 200 300 400 mm/day

Fig. 1.846) Frequency curves for annual maximum amount of rainfall, (4).

in Part II.

9. Conclusion

In this part, first, the statistical properties of three types of extreme

(largest) value distribution were examined. As the result, it was disclosed that the applicable range for them must be discriminated by the population

value of the coefficient of skew. Next, a practical method of estimation of

the parameters included in them was successfully developed by using the

concept of the plotting value.

Since these studies were already made in 1954-4956 and 1959, this

publication may seem to be too late. But the author believes that this

paper is still useful in the field of hydrologic frequency analysis.