Analysis of periodicity in hydrological sequences

Journal of Hydrology 14 (1971) 66-82 © North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission f r o m the publisher

ANALYSIS OF PERIODICITY

IN HYDROLOGICAL

SEQUENCES

J. A N D E L and J. B A L E K * Department of Mathematics and Statistics, Charles University, Academy of Sciences, Prague 6, Czechoslovakia

Abstract: An attempt has been made to provide a series of statistical tests as a basis for the objective analysis of a general hydrometeorological sequence and for the purpose of constructing a "best fit" model. A further aim of the analysis was the identification of the composition of the sequence. The authors anticipated neither the existence or nonexistence of periodicities in the sequence, however, where the existence of one or more periodicities was indicated by tests at various level of significance, these were integrated grated into the sequence simulation.

Periodicity, autoregressive schemes, serial correlation G e n e r a l l y , h y d r o l o g i c a l sequences are c o m p o s e d o f a trend, one or m o r e p e r i o d i c c o m p o n e n t s a n d a r a n d o m variable, t h o u g h one o r m o r e o f these m a y n o t be present. I n m o s t cases, the t r e n d can be described analytically or at least e s t i m a t e d a n d if the p a r a m e t e r s o f the function are n o t k n o w n the m e t h o d o f least squares can be used for their estimation. A n u n d a m p e d p e r i o d i c c o m p o n e n t or a c o m b i n a t i o n o f these m a y be represented b y a summation of superimposed harmonics: Pt = ~

(al cos r.oit -t- b i sin 0)it )

(1)

i=1

where a l , a 2 . . . . a s are real n u m b e r s , s is a n a t u r a l n u m b e r a n d the frequencies col, ~o2,... cos are in the interval 0 to re. The frequency ogk(l~
(2)

T h e r a n d o m c o m p o n e n t is p o s t u l a t e d as having zero mean. It m a y be represented as a n identically d i s t r i b u t e d set o f i n d e p e n d e n t or d e p e n d e n t r a n d o m variables. F o r the f o r m e r case, an estimate o f the variances can be p r o v i d e d , while for the latter a m a t h e m a t i c a l statistical m o d e l can be c o n s t r u c t e d c h a r a c t e r i z i n g the b e h a v i o u r o f the r a n d o m c o m p o n e n t . * Present address: National Council for Scientific Research, P.O.Box RW 166, Ridgeway Lusaka, Zambia. 66

A N A L Y S I S O F P E R I O D I C I T Y IN H Y D R O L O G I C A L SEQUENCES

67

I f the values of the r a n d o m component are Yl... YN, where N is the number of members in the sequence then let Yt = alYt-1 + a2Yt-2 + aryt-r + et

(3)

where a l . . . ar are real numbers ( a t # O) and et are independent r a n d o m variables with zero means and equal variances a z. The representation given by Eq. (3) is called an autoregressive sequence of the r th order. I f all the real and complex roots of the sequence 2' - ~12' - 1 - ~2;t"-z ... ~r = 0

(4)

have absolute value less than unity then the autoregressive sequence is stationary. I f the absolute value of one or more of the roots of Eq. (4) is greater than unity then the sequence is called evolutive or explosive. However, according to Wise 1) the stationarity of the autoregressive scheme can be recognized directly from the coefficients a l . . . ar and it is thus not necessary to solve Eq. (4). I f a model as described by Eq. (3) is to be applied then the order of the autoregressive scheme has to be derived first, then the autoregressive coefficients and finally an estimate of the residual variance provided. For the purpose of extrapolation the trend and the periodic components, if present, may be extrapolated directly while the extrapolation PN+I of the random component may be calculated using the formula Ys+l = a l Y s + a 2 y s - 1 +... a r y s - , + l

(5)

which is valid for one step of extrapolation. For further extrapolation a method as described by Yaglom 2) may be applied. Let the discrete hydrometeorological time series be denoted as X1... Xs where N is an odd number such that N = 2M + 1

(6)

and M here is a natural number. I f this condition is not satisfied, the first member of the sequence may be eliminated. If it appears on a logical analysis that the trend component is a constant value then the members of the sequence fluctuate periodically, randomly or as a combination of both forms about a certain mean value, the estimate of which is the arithmetic mean:

'X N

a° = g

X,.

i=l

By eliminating the constant trend value we obtain a sequence x~=Xt-X

with mean zero.

l <_t<_N

(7)

68

J. A N D E L A N D J. B A L E K

It now remains to determine whether that sequence is composed of independently distributed random variables or whether further analysis of the sequence must be undertaken. F r o m the series of various tests described by Hannan 8) the test based on the circular correlation coefficient N

Z R

=

X t X t + l -- N X 2

t=l

N

(8)

Z )4 - N X 2 t=l

(where xN + 1 ---xl) was chosen. A treatise of the circular correlation coefficient together with a table of critical values was given by R. L. Anderson 4).

Periodogram analysis The test for the existence of a periodic component was provided by Fisher's test, suitably modified where necessary. Further extensions of the test have been described by Whittle 5) and Hannan 3). The original Fisher's test is applicable when a given sequence is a combination of a periodic component and an independent random variable, the modified test is applied when the random variables are dependent. F r o m the authors' experience the modified test has not proved sensitive enough: only its use in conjunction with an extremely long sequence contribute to increased sensitivity. In accordance with Fisher's test the quantities N-k

IX

Ck = ~

Xt+kXt k -----0, 1 ... N -- 1

(9)

t~l

are calculated. Some authors call the function

B(k)-

C(k)

k=0,1...N-

c (0)

1

the autocorrelation function of the sequence )(1... X N. The periodogram is defined as I ( 2 ) = 2~

Ct cos t2

o+ 2

0 ~< 2 ~< ~.

(10)

t=l

Put 2k =2nk/N; k = 1, 2... M, where M has been defined in Eq. (6). Denote I(2q) = max [I(2l), I(2z), ... l(2ta)] and o -

I

(2,)

ANALYSIS OF PERIODICITY IN HYDROLOGICAL SEQUENCES

69

If there is no periodic component of the type a cos4t+b sin4t ( = a ' cos (4t+fl)) present and the sequence is composed of independent normal variables with equal means and variances, then the distribution of 9 is given by G(x) = P(g

> x) = E (- I) y-I

(13)

(1 - - j x ) M - I

j=l

where [l/x] denotes the integral part of the number 1/x. Therefore, when G(g)<~ the hypothesis on the absence of the periodic component can be rejected at the level ~. In such a case another test may be provided on the presence of a further periodic component. According to Whittle, 4q may be eliminated and new values found: I (4'q) = Max I (41 )... I (4q_l), I (4q +1 )... I (2M)

(14)

and

(4'.) g'--

M

X 1(4,) k=l

The significance of 9' is then tested by Eq. (13) where the value of M has been replaced by ( M - 1 ) . This procedure is then repeated until significant results have been obtained. Suppose that the procedure was applied s times (s ~>1) which means that s values from 41 ... 4M were found to be significant. Then the parameters ai, bi of the periodic component Pi (Eq. 1) are calculated by the method of least squares i.e. the values of (a I coso)it + bi sine)it)

Xt t=l

t=l

should be minimal. If the deviations from the periodic function are dependent the following modification of the Fisher's test is suggested. First an estimate of the spectral density of the sequence X1... XN is provided by Parzen's formula (Granger, Hatanaka6)). n/2

CO f(6i)=2~+Tr

I ~-~ [

( 1--

1--

!)6kZ 1 lrkj nzjCkcOS--n

k=l

2~( !)3 n

+

1 --

k=n/2

where oj-

~j n

j =O, 1.... n,

7~kj

Ck COS- -

(15)

70

J. A N D E L A N D J. B A L E K

and n is an even number chosen between N/6 and N/5. Linear interpolation of the values f (6i) allowsf to be evaluated over the whole interval. To test for the existence of a periodicity further quantities are evaluated I(2j)

I~
kq = max (kl ... kM) and

kq g--M

Z1 kj is calculated. The value of g is then tested for significance by using Eq. (13) in the original test. If the presence of a periodic component has been verified, the random variables Y¢ = X t - Pt are analysed, otherwise Pt = 0 and Yt = xt and the test for independence is provided by Anderson's test based on the circular correlation coefficient calculated from Eq. (8) where xt is replaced by Yr According to Anderson 7) the determination of the significance of the circular correlation coefficient should be slightly modified when the deviations from the periodic component are analysed. For the case where Pt = 0 the circular correlation coefficient of the Yt ( = xt) will already have been calculated. The order of the autoregressive model (3) is determined by Whittle's test 5). For its application denote N-k

(Yt+k -- )7) (y, -- )7),

Dk = N

k = 0, 1.... N - 1

(16)

t=l

1X

where

N

= ~

Yl.

1

If a periodic component has not been detected then Dk = Ck,

O <~ k <~ N -

I

where Ck are the values calculated from the Eq. (9). Then the determinants CO, C 1 ... Cp

d p = C1, Co

Cp_l

ICp, C~_ 1... Cot are calculated.

p=0,1...30.

(17)

ANALYSIS OF PERIODICITY IN HYDROLOGICAL SEQUENCES

71

Bearing in mind the minimum sequence length available an upper limit of 30 is placed on the order of the autoregressive schemes considered, in order to preserve the statistical validity of the results. Then the values 2p -

Ap+l Ap-1 2 Ap

~2=(N-P-

1 ~< p ~< 29

1) 1 - 2 p ,2p

1~
are calculated. If the tested sequence is autoregressive of order p then if2 is asymptotically distributed as X2 with 1 degree of freedom. Thus, ~k2 can be used to test the hypothesis that the sequence y l . . . yn is autoregressive of order p against the alternative hypothesis that it is of order p + 1. A further test due to Whittle used in the analysis is based on the following principle: when NAk

Uk-k_

1,

k = 1, 2 .... 10,

then #J~ = ( N - p - q) O P - Op+q

Up+q is asymptotically distributed a s ~(2 with q degrees of freedom. The value of #/2 then tests the hypothesis that the autoregression of the order p is significant as opposed to the autoregression of order p +q. The values ~k2 (starting with p = 1) are compared with the corresponding critical values of the Chisquare distribution with q degrees of freedom. The coefficients al... ar in the model are estimated by the method of least squares such that N

(Y -- a l Y - 2 . . . a , Y t - r ) 2

(18)

t=r+l

is minimised. Mann and Wald 15) proved that such estimates are consistent and have asymptotic normal distribution. Even if the value of ~k2 is not significant, an autoregressive model of the first order is evaluated. The application of it depends on a complex analysis of the results and particularly on the significance of the circular correlation R of the sequence Y1... YN. The quantities *-2

O'p

Ap -

-

A p_ 1

1 <~ p <~ 29

are now calculated. If the autoregression of order r has been found significant, then a,2 is an estimate of the variance a 2 of the component et in the

72

J, ANDEL

AND

J. B A L E K

model (3). Finally, the quantity N

s2 = _1

y2

N t=l

is calculated and used as an estimate of the variance of the random variable Yv On completion of the foregoing analysis a further analysis by the socalled method of hidden periodicities can be provided. The theoretical form of the periodogram may be defined as N

1

?

IN(a) = 2nN

x,

e- it). 2

(19)

/..d t=l

If the sequence xl... xu contains a component acos(2o t+fl)+yt,

0<20
a#0,

(20)

such that x, = a cos (2or + fl) + y,

(21)

where Y , . . . Y u are either independent or dependent random variables then the approximation I~ (2~) -

aZN

(22)

8n

is valid. If Yl...YN can be considered as independent random variables with zero means and variances a 2, then Eq. (22) is more accurate for larger values of N, this being equally true for a given N but smaller a z. Equation (22) gives the value of the periodogram at the peak corresponding to the frequency 2~. As already cited the basic criterion for periodogram analysis is Fisher's test based on the value /max g = ~-

(23)

El, l where

l

k

M, M = N - 1 / 2 .

The distribution of g is given by Eq. (13). A more convenient form of calculation suggested by Hannah 3) is given as P ( g > x ) - M(1 -- x) M-1.

(24)

Fisher's test is applicable in the case where only one periodicity is contained in a sequence such as that given by Eq. (21). With an increase in the number

ANALYSIS OF PERIODICITY IN HYDROLOGICAL SEQUENCES

73

of periodicities, the power of the test decreases rapidly. In order to detect several periodic components of the type postulated in Eq. (21) a much longer realization is necessary. Suppose that the sequence contains only one periodic component of the type postulated in Eq. (21) and that the peak of the periodogram denoted by K corresponds with it. If

~Ik-K=G k=l

then K

g=K+G Suppose that for instance this periodicity has been detected by Fisher's test at the 2% level which means that P

g>~

=0.02.

(26)

An example was studied with N = 7 5 and M = 37. According to Eq. (24) it can be shown that Eq. (26) is valid for K

K+G

- 0.189 = b.

(27)

Now another situation can be tested where the sequence contains two periodicities of the type a cos (2~t +flo) and a cos (2~ t +ill) where 2~ ~ 2~. Then the periodogram produces two peaks each of them approximately equal to K. The existence of a periodicity may be tested by g : K

g-

2K+ G because of the value of G has been changed only slightly by omitting one of the values Ik. From Eq. (27)

bG K -

l-b

is calculated and thus

K K+G

b • 0.15896. 1 +b

According to Eq. (24) K

P{g > - -

2K +

G~J = P{g > 0.15896} - 0.0739.

(28)

74

J. A N D E L A N D J. B A L E K

Similarly, if these periodicities exist in the sequence of the form

acos(2'it+fll)

1~
0~<).'~#2~-#2~
then, similarly, g -

K

3K + G

-

b l+2b

- 0.13716

and P{# > 0.13716} - 0.1825. Thus in the case where two or three periodicities are contained in the sequence, Fisher's test does not verify the existence of even one of them at the 5% level of significance. For the case where a single periodicity exists, significance at the 5% level would only certainly be obtained. Obviously the power of the test depends on the ratio of the amplitudes of the periodic components. Numerical tests were carried out on the model 3

Xt = Z a, cos (21t + [31) + 10 + Yt, f=l N

x,=X,-~,

1
x~ 1

where Yr... Y75 are independent normally distributed random variables with mean zero and variances a 2 =25, 21 =0.3, 22 = 1.0, 23 =2.2, fit = 1.0, f12 =2.0, fl3=4.0. The values Y1 ... YTs were constructed using tables of normal ACTUAL SEOUENCE MODELLED SEOUENCE

? Fig. l a. Artifical sequences actual and modelled. Amplitudes 3:3:3.

ANALYSIS OF PERIODICITY IN HYDROLOGICAL SEQUENCES

75

random numbers (BuslenkoS)). Two variants have been considered (Fig. la, lb): A. a 1 = a 2 =a3 =3. In this case the maximum of the periodogram was found at the point 2=1.0053, with a corresponding value of g=0.1189, P (g > 0.1189) = 0.351. Thus a periodicity was not detected by Fisher's test even at the 10~ level of significance. Further the value of the circular correlation coefficient, equal to 0.0108, was also found to be not significant. According to Whittle's test X t = a x t _ 3 + b x t _ 4 + c x t _ l o + et

would be the best autoregressive representation. ACTUAL SEQUENCE MODELLED SEOUENCE

Fig. lb. Artifical sequences actual and modelled. Amplitudes 1:3:4.

B. a I 1, a 2 3, a3 = 4. The maximum of the periodogram occurred at the point 2=2.178, whence g =0.1548, P(g >0.1548)=0.086. Here the dominant period was detected by Fisher's test at the 10~ level. A value of - 0 . 1 2 7 resulted for the circular correlation coefficient. According to Whittle's test the models =

=

Xt ~ a x t - 4 "t- ~t or xt = a x t - 4 'I- b X l - l O x

+ ~,t

can be used as autoregressive representation. Regardless of the absence of a statistical test capable of detecting several periodicities a harmonic analysis was applied even where Fisher's test did not yield a significant result.

76

J. ANDEL AND J. BALEK

Application of the method toward a set of various rivers of four continents

For a set of rivers from four continents the frequencies corresponding to the major peaks in the periodogram were noted, while the behaviour of the periodogram in the neighbourhood of peaks was also analysed. The subsequent approach was to maintain statistical stability through using a representation involving a minimum number of frequencies while simultaneously seeking to minimise the value of the residual variation. The construction of a model was undertaken bearing in mind that for the set of rivers described in Table 1 the existence of a periodic component has been proved for the rivers Nile, Niger and Vah. Further, similar influences producing these periodicities though not quite so dominant, might be expected for the other rivers; perhaps in combination with local effects other types of periods would be produced. The method was applied to the sequences of mean annual discharge of 15 rivers from various climatic regions (Table 1). The criterion for chosing some of the most representative rivers was limited by the availability of data. The information in Table 2 was obtained by application of the method. Rows 1-4 summarise the data giving the name of the river, units of measurement, length and period of record respectively. These are followed by the means, variances and circular correlation coefficients (rows 5, 6, 7). In the 8th row the probabilities of the absence of a single periodic component are presented. The frequencies corresponding to peaks in the periodogram are given in row no. 9. These are followed by the residual variances obtained after application of the "hidden periodicities" model (row no. 10) and the autoregressive model of the first and twenty-eight order (rows 11 and 12). Finally, the lags of the optimal autoregressive models which have been found as significant are given in row 13. For the seven rivers representing the climatic region of mid-Europe, Vah, Main, Morava, Danube, Labe, Rhine, Vltava, a significant period of 5.15 years was found. This supports evidence of a period of 5.6 years found by Bratranek °,1°) for Czech rivers. In the sequences for the rivers Labe, Danube and Main a period of 15.95 years was revealed. Periods of 2-2.5 years were found for rivers having upper basins in rather high mountains. This poses the question as to whether this period could be related to similar ones found by Clough 11) for the oscillation of the atmosphere. From Table 2 it can be seen that the "hidden periodicities" models generally yield better results than the autoregressive schemes for a given number of parameters. Only the river Gota among the European rivers yielded better results for the latter representation.

ANALYSIS O F P E R I O D I C I T Y I N H Y D R O L O G I C A L SEQUENCES

zzz~

0

¢11 m

[--

~

~

77

78

J. ANDEL AND J. BALEK TABLE2 Labe ma/s 107 1851-1957 299.6 7 813 0.277

Morava m3/s 66 1901-1966 113.7 1 969 0.292

Vah ma/s 46 1921-1966 151.8 1 166 0.106

Danube m31s 132 1829-1960 1 728.7 59 825 0.149

0.25 0.294 0.411 1.233

0.14 0.286 1.238 1.809

0.04 0.956 1.503 1.776 2.595

1599

5 955

1 259

508

0.18 0.286 0.381 0.666 1.476 1.809 41 509

0.41 0.204 0.490 0.857 1.469 3.101 20 655

1960

7 221

1 794*)

853*)

58 495

25 731

770*)

1636

6 266

1 197")

546*)

48 692

22 665

652*)

1 24

1 15 19

1 8 19

6*)

1. 2. 3. 4. 5. 6.

River Vltava Unit ma/s No. of members 142 Period 1825-1966 Mean Q. 136.5 Variance 2 173 7. Cyc. cor. coeff. 0.305 8. Probability of the absence of period. comp. 0.45 9. Frequency as indi0.044 cated by periodogram 0.266 0.443 1.239 10. Residual variance (hidden per.) 11. Residual variance (autoreg. 1st. or.) 12. Residual variance (autoreg. 28th or.) 13. Lags significant at 10 % level

1

Rhine Main m~ts mS/s 154 111 1808-1961 1845-1955 1 026.4 100.0 25 949 825 0.093 0.263

1t

5 6 11

*) related to the random variable component.

The results for the river D n e p r which can be considered as a representative river for the c o n t i n e n t a l climatic region also indicate a significant p e r i o d o f 5.15 years; m o r e i m p o r t a n t still a p e r i o d o f 11.95 years was found. Analysis o f the Mississippi r e c o r d yielded a significant p e r i o d o f 22.6 years; similar p e r i o d s were identified for the rivers D a n u b e , Nile, a n d Niger. It is also r e m a r k a b l e t h a t p e r i o d s o f 11.4 years a n d 34.3 years were discovered; these m a y well be related to the latter m e n t i o n e d period. F o r the river O t t a w a the m o s t n o t i c e a b l e p e r i o d was t h a t o f 5.15 years; even if the c o r r e s p o n d i n g p r o b a b i l i t y o f its absence was higher t h a n in previous cases. The rivers Nile a n d N i g e r represent the A f r i c a n continent. H u r s t 12) f o u n d a r e l a t i o n s h i p between the occurrence o f sunspots a n d the fluctuations o f the level o f l a k e Victoria. In the Nile sequence a p e r i o d o f 84 years was revealed, similar to the p e r i o d f o u n d for sunspots by WilletlZ). Here also further significant p e r i o d s o f 22.6 years a n d 7.3 years were traced. F o r the river Niger a highly significant p e r i o d o f 25.5 years was f o u n d ; however, this could

0.13 0.396 1.245 1.472

625

1

19

ANALYSIS OF PERIODICITY IN HYDROLOGICAL SEQUENCES

G6ta cfs

Dnepr mS/s

Mississippi cfs

9.45 x 106

0.25 0.228 0.501 0.683 1.093 2.777 1.58 x 105

0.08 0.183 0.305 0.549 0.732 1.647 1.38 x 109

1.05 × 108

9.26 x 106

2.08 x 105

1.48 x 109

2.1 x 10s

7.63 × 106

1.81 x 105

1.04 x 109

1.84 x 108

150

1807-1957 18 921 11.7 x 106 0.469

0.22 0.545 0.880 1.215

79

Ottawa Nile Niger Murray Goulburn cfs mld.m3 cfs cfs cfs 138 103 89 84 54 74 73 1818-1955 1861-1967 1871-1959 1871-1954 1906-1957 1877-1950 1881-1954 1741 174308 69167 92.7 54381 5003 3004 2.2 × 105 2.36 x 109 1.48 x 108 391 1.7 × 106 6.26 x 106 1.95 x 106 0.236 0.338 0.069 0.362 0.506 0.203 0.249

0.53 0.706 1.721 1.906 2.753

0.001 0.075 0.299 0.823 1.496 2.244 202

0.001 0.246 0.862 2.094

0.39 0.170 0.425 0.849 1.698

0.89 0.516 1.033 1.205 1.721

0.58 × 108

4.17 x 106

1.31 x 106

278*)

0.81 × I06.) 6.0 x 106

1.83 x 106

225*)

0.59 x 108*) 4.8 x 106

1.37 x 106

1

1

1

2

8*)

2 14 21

13

13

9 15

4*)

4*)

1

1

11

8

assume a n y value in the interval 20.4-34.2 years, taking into a c c o u n t the relatively short record a n d the shape of the periodogram. Thus, the period of 22.6 years could be considered significant for b o t h African rivers. The A u s t r a l i a n c o n t i n e n t is represented by the rivers M u r r a y a n d G o u l b u r n ; a period of 3.7 years has been d e t e r m i n e d for b o t h sequences. This being more significant in the case of M u r r a y . I n this sequence a p p r o x i m a t e multiples of the period, n a m e l y 7.3, 15.9, a n d 30 years were identified. O n a lower level of significance further periods of 5.2, 6.1, a n d 12.2 years were f o u n d for the Goulburn. I n Fig. 2 the intervals belonging to the calculated periods a n d determined according to the points in the n e i g h b o u r h o o d of the peaks of the periodograms, have been plotted for all the rivers a n d from the sums of intervals identified the most frequently occurring periods, these being 22.6 a n d 5.15 years. The former can perhaps be related to the periodicity for sunspots categorized by some a u t h o r s within the interval 22___2 years. Willet 13) a n d

80

J. ANDEL AND J. BALEK

S ©

O

o

d

(

O

O

_o

I '

I

~

~

..

~4

I

o

~ ~ z ~

,.Q

co

SIN3A3

~

~1- cxl ~ O !AINS

ANALYSIS OF PERIODICITY IN HYDROLOGICAL SEQUENCES

81

Kopecky 14) proved this period to be more significant in the sunspot analysis than the eleven year cycle.

Conclusion Acknowledging the fact that the hydrological cycle is highly influenced by solar energy a similar influence of the sun can be expected on the time formation of hydrological phenomena. However, various combinations of this effect with other local either random or periodic effect produces time series varying from watershed to watershed. On the smaller watersheds the local influence may play a more important role, while the influence of the basic factors might be expected to prevail on the larger basins. Similarly the lag between the sunspot periodicity and the complex hydrological periodicity cannot be expected to be equal for all sequences. A mathematical statistical analysis for the detection of hidden periodicities and the provision of optimal autoregressive schemes has been provided. This provides the information which aids in the construction of a "best fit" model of a hydrological sequence. Tests on 15 different rivers suggested the most frequently occurring periods to be 3.4, 4.3, 5.15, 7.3, 9.35, 11.95, 15.9, 22.6, 30.5 years. The existence of longer periods was indicated among some of the longer sequences. The existence of a periodicity in the sequence for the rivers Vah, Niger, and Nile was proven at the 5 ~ level of significance while this level was almost attained for the Mississippi. Only in the case of the Gota did the autoregressive scheme prove more suitable than the periodical or combined model. With regard to the autoregressive models, only for two rivers was the first order scheme found to be optimal at the 5 ~ level of significance. The optimal autoregressive models for the other rivers were formed by two or more components or by one component only, this being different from the first order scheme.

Acknowledgement The authors are indebted to Mr. E. O'Connell of the Imperial College, London, for his critical comments and for reviewing of the manuscript of this paper.

References 1) J. Wise, Stationary conditions for stochastic processes. Biometrica 43 (1965) 2) A. M. Yaglom, General theory of stationary time series (Czech Edition), Sov~tsk~i v~da No. 5 (Prague, 1955) 3) E.J. Hannan, Time series analysis, Chap. 4 (Russian edition, 1964) 4) R. L. Anderson, Distribution of the serial correlation coefficient. Annals Math. Star. XIII (1942)

82

J. ANDEL AND J. BALEK

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