APPLICATION OF RUN-LENGTHS

TO HYDROLOGIC SERIES

by

J . Saldarriaga and V. Yevjevich

April 1970

40

April 1970

APPLICATION OF RUN-LENGTHS

TO HYDROLOGIC SERIES

by

J . Saldarriaga and V. Yevjevich

HYDROLOGY PAPERS COLORADO STATE UNVIERSITY

FORT COLLINS, COLORADO 80521

No . 40

ACKNOWLEDGEMENTS

The financial support of the U.S. National Science Foundation

under grant GK-11444 (Hydrologic Stochas tic Processes) and grant

GK- 11564 (Large Continental Droughts) for the research leading

to this Hydrology Paper is gratefully acknowledged.

Acknowledgement is also made to the Nati onal Center for Atmospheric

Research, sponsored by the National Science Foundation, for some use

of its CDC 6600 computer in the investigations leading to this paper .

Acknowledgement goes to Dr. M. M. Siddiqui, Professor , and Dr.

P. Todorovic, Associate Professor, at Colorado State Univer s ity for

their advice in connection with some mathematical problems of run

theory.

The cooperation of two M. S. graduate students in the computing

phase of the study, Mr . V. K. Gupta and Mr. P. C. Tao, is appreciated.

The bulk of computations was done on the CDC 6400 computer at the

Colorado State University Computing Center.

iii

Abstract

1

II

TABLE OF CONTENTS

Definition of Problems Investigated 1.1 Stationary hydrologic series 1.2 Practical significance ..... 1.3 Two problems related to the application

to hydrologic processes . . . . . . Methods for investigation of hydrologic Autocorrelation analysis . Variance spectrum analysis Ranges . . Runs .......... . Comparison of four techniques. Runs as the technique

of the theory of run-lengths

sequences 1.4 1.0 2.0 3.0 4.0 5.0 6.0 1.5 1.6 1.7

Two approaches to investigations of stochastic sequences . Objectives for determining properties of run-length Uefinition of runs ......... .

Summary and Status of Knowl edge on Discrete Runs 2.1 Introductory stat ement 2.2 . Distribution theory of the number of various runs of independent

random variab les. 2.3 Distribution theory of run-lengths of indepent random variables . 2.4 Distribution theory of run-lengths of dependent random variables 2. 5 The 11ulti variate normal integral

III Probabilities of Run-Length of the First-Order linear Autoregressive

IV

v

Model 3.1 3 . 2 3.3 3.4

3.5

Runs of 4.1 4.2 4.3 4.4 4.5 4. 6 4.7

of Normal Variables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General notations and expressions for probabilities of run- l ength ...... . Stationary and erogodic multidimensional gaussian processes .......... . ~fultivariate nomal probability density function ........ . ... . General expression for joint probability of at least k subsequent values

below truncation level, followed by at least j subsequent values above truncation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Probabilities of runs for truncation level. . . . . . . . . . . . . . . . . . Probabilities P (2+) andP (1-, 1+) .... ... . . .... . ... . . . . Probability P (3+) ................. . . .... ..... . Probabilities of the type P (j +) . . . . . . . . . . . . .• . . . . . . . . . . . Probabilities of the type P ( 1- , j+) .... . .. .. . . ...... . . . . . Probabilities of the type P (k- , j+) Distribution of Nr D1stribut1on of Ni . Joint distribution of Ni, Ni, N2, N2 . Distributions of NK and Nk ..

Stationary Dependent Gaussian Processes . . . . . . . . . . . . . . . . . . . . Firs t-order linear autoregrcss i ve process . . . . . . . . . . . . . . . . . . . Probability mass function , and moments of run-lengths N+ and W ....... . Properties of total run-length, N = N+ + N- . . . . . . . . . . . . . . . . . . . General procedure for evaluating properties of runs . . . . . . . . . . . . . . Probabilities of the non-normal case . . . . . . . . . . . . . . .. Properties of runs of the fir~t-order, autoregressive linear process Properties of runs of the first-order , autoregressive linear process

obtained by the data generation method. · · . . . . . . . . . . .

Application of Run-Lengths to Investigation of Series 5 .1 Introduction . . . . . . . . . . . . . 5.2 Using runs for investigation of series ... 5.3 Properties of run-length for sequences of independent

distributed random variables . . . . . . . . . . . 5.4 Run-lengtt test for stationary independent variables. 5.5 Two-levels run-l ength test for stationary independent

iv

identically

variables

1 1 1

l 1 1 2 2 2 2 3 3 4 4

6 6

6 7 8 8

10 10 10 11

11 11 11 12 12 13 13 15

15

15

15

16 16 16 16 16 17 17

20

23 23 23

24 24 26

VI

VII

VIII

Examples 6.1 6.2 6.3

6.4 6.5

6 .6

6 . 7

6.8

Examples 7.1 7. 2

7 . 3

TABLE OF CONTENTS (Continued)

of Investigation of Stationary Hydrologic Series by Run-Lengths Int roduction. . . . . . . . . . . . . . . . . . . . . . . . Application to investigation of annual precipitation series Examples of invest i gation of annual precipitation series by the

mean run- length of the medi an. . . . . . . . . . . . . Application to investigation of annual river flows series examples of investigation of annual river flows series by the

mean run-length of the median . . ...... . Examples of investigation of annual precipitation series by the

relation of mean run-length to values of q Examples of investigation of annual runoff series by the relation

of mean run-length to values of q . . . . . . Examples of investigation of annual precipitation and r unoff series

by N* (p) for values of p . . . . . .

of Computation of Probabilit ies of Run-Lengths .. Introduction. . . . . . . . . .... Determination of run-length probabilities of stationary

and i ndependent series . . . . . . . . . . . . . . . Determination of run- l ength probabilities of stationary

dependent series

Conclusions .

References

Appendix .

v

29 29 29

29 31

31

31

31

37

40 40

40

45

49

50

51

1. 1

4 .1

4.2

4 . 3

4.4

5 .1

5.2

5. 3

5.4

5. 5

5 . 6

5 .7

6 .1

6 . 2

6 . 3

6 .4

6.5

6. 6

6.7

6 . 8

6 . 9

6.10

LIST OF FIGURES AND TABLES

Definition of positive and negative runs ..... .

Probability distribution of positive run-lengths of the first-order autoregressive process for q • 0 . 3 .. .

Probability distribution of positive run-lengths of the first­order autoregressi ve process for q = 0 .4, 0 . 5 , 0 . 6 , 0 .7

Mean run-lengths of the first-order autoregressive process .

Differences between theoretical probabilities of run-lengths and those obtained by t he data generation method . . .. ... .

Expected correlogram and 95% tolerance limits of an independent series . . . . . . .

Expected variance spectrum and 95% tolerance limits of an independent series . . . .

Mean run-lengths of independent series for various values of q

Two-sided run- length test with q = 0 . 50 for independent variables.

Tolerance region for Nk' of an observed independent time series.

Graph paper for the analysis of time series by using N(q) .

Graph paper for the analysis of time series by using N*(q)

Investigation of t ime series independence by using N(q) , for four ~omogeneous annual precipitation series . . . . . . . . . . ...

Investigation of time series independence by using N(q) , for the non-homogeneous annual precipitation series at Natural Bridge, N.~ .• Arizona . . . . . . . . . . . . . . . . . . . . . . . . .

Investigation of time series independence by usin.g N(q), for the annual r unoff series of the Rhine River . . . . . . . . ...

Investigation of time series independence by using N(q) , for four annual runoff series . . . . . . . .

Investigation of independence of whitened series, under the assumption of the first-order autoregressive process as a time dependence model , by using N(q), f or f our annual runoff series .......... .

Investigation of time series independence by using N*(q) for four non-homogeneous annual precipitati on ser ies . . . . . . . . . . .

Investigation of time series independence by using N*(q) for the homogeneous annual precipitation series at Natural Br idge, ~ . M .,

Arizona . . . . .. . ...... . .......... .

Investigation of time ~eries i ndependence by using ~*(q) for the annual runoff series of the Rhine River . . . . . . . . . . . . .

Investigation of time series independence by using K(q) for four annual runoff series . . . . . .. ..

Investigation of independence of whitened series , under the assumption of the first-order autoregressive process as a time dependence model, by using N*(q) for four annual runoff series ...... ... ... .

vi

5

18

19

20

21

23

24

25

25

25

27

28

32

33

33

34

35

36

37

37

38

38

7.1

7 . 2

7 . 3

7 . 4

7.5

7. 6

7 . 7

4 . 1

4 . 2

4.3

5. 1

6 . 1

6 . 2

6 . 3

6 . 4

6 . 5

6 . 6

LIST OF FIGURES AND TABLES (Cont inued)

Estimat ed and expected probabilities of run-lengths of t he annual precipi tation series at Ord, Nebraska ..

Estimat ed and expected probabi l ities of run-lengths of the annual precipitation series at Ravenna, Nebraska ..

Estimated and expected probabilities of run-lengths of the annual precipitation series at Antioch F. Mills, California

Estimated and expected probabi l ities of run-lengths of the annual runof f ser i es of the Rhi ne River ....

Estimated and expected probabi li ties of run- l engths of the annual runoff series of the Gate River .. ...

Estimated and expected probabilities of run-lengths of the annual runoff series of Ashley Creek ..... .

Estimated and expected probabilities of run-l engths of the annual runoff series of Trinity River .....

Equati ons for the evaluati on of properties of runs . . . . . .

Expected values of ~ of the first-order linear autoregressive process ..

Variance of N+ of the first-order linear autoregressive process

Properties of run-l engths for independent identically distri buted variables . .

Runs of five annual precipitation seri es .

Properties of run- lengths of the five annual precipitat ion series

Properties of run-lengths of the five annual rive r flow series .

Values of 1/pq and (p3 + q3)1/2 . ..

Tolerance limits of N at the 95% level .

Mean and 95% confidence limits of N* . Appendix Tables . . . .

vii

41

42

43

44

46

47

48

17

18

18

25

30

30

33

33

33

37

52- 56

N

a

R n

K

X 0

q

p

N+ j

Nj Nj

r: J

r; r ( r)

LIST OF SYMBOLS

k ·lag serial correlation coefficient

Sample size

k-lag autocorrelation coefficient (population value)

Population mean

Population standard deviation

Of the order of p 3

Surplus

Deficit

Range

Sequence of independent variables with a common distribution

Number of runs of kind i of length j

Number of r uns of kind i

Number of total runs

Number of elements of kinds 0 and 1 respectively in a binomial population

NO + Nl

N0 /N1 . Also level of significance

Truncation level

Probability of drawing an element of kind 1 in a binomial population . Also F(x

0)

Probability of drawing an element of kind 0 in a binomal population. Also l-F(x

0) • 1-q

j -th positive run in discrete time

j-th negative run in discrete time

j-t h total run in discr ete time

j-th positive run in continuous time

j-th negative r un in conti nuous time

Gamma function of r

Correlation coefficient between Xi and Xj

Probability that x1

, x2

, ... , Xj are s imultaneously positive for a

probability truncation level q• .S

Same as previous one for any q

Probabil ity that x1

, x2

, ... , Xk are simultaneously negative and

~+l , Xk+2 ; · ·• ~j are simultaneously positive for a probability

truncation level qeO .S

Same as previous one for any q .

viii

E

var

y(K)

f(X)

F(X)

R

k

k

- *

LIST OF SYMBOLS (continued)

Meaning

Expectation

Variance

Autocorr e l ation function

Probability density function of X

Probability distribution function of X

Matrix of correlation coefficients

Hermite polynomial of order i .

Number of positive runs

Number of negative runs

Number of total runs

(k+ + k-1/2.

Mean positive run-length

Mean negative run-length

Mean to~al run-length

N Mean run- length when two levels q and p • 1-q are considered.

ix

ABSTRACT

A method is developed for investigating time series structure

by using the mean run-length parameter . This method is distribution­

free. Applications to selected annual precipitation series and annual

runoff series demonstrate the feasibility of this method.

Analytical expressions are developed by which the probabilities

of sequences of wet and dry years of specified lengths can be calculated

when the basic hydrologic time series is either an independent or a

dependent stationary series of a var iable which follows the first-

order l inear autoregressive model.

Numerical values of probabilities of run-lengths are obtained by

the digital computer integration of expansion equations for run-length

probabilities of the first-order l inear autoregressive model. A set

of tables and a set of graphs are presented to make the numerical values

readily useable. Probabilities of run-lengths of dependent variables

with a common distribution are also distribution free.

The significance of this investigation, and several applications

in the text, are based on the premise that run-lengths, as statistical

properties of time series, represent attractive parameters in studying

droughts and surpluses .

X

APPLICATION OF RUN-LENGTHS TO HYDROLOGIC SERIES

by

Jaime Sal darriaga* and Vujica Yevjevich**

Chapter I

DEFINITION OF PROBLE~!S INVESTIGATED

1.1 St ationary Hydrol.ogic Ser ies . Annual precipitation, annual effective precipitation (pre­cipitation minus evaporation) , and annual r iver flow vary from year t o year. This variation is generally r eferred to as the sequence of wet and dry years . These sequences are hydrologic time processes . For all practical purposes in water resources development, they can be assumed to be approximately stationary time series [1,2] . The hydrologic stationary pro­cesses of annual river flow and annual effective pre­cipitation arc dependent time series . This means that successive values are linked in some persistent manner, or sequences of annual river flow and annual effective precipitation are stationary dependent processes [2]. Sequences of annual precipitation are very near t o being stat ionary and independent st o­chastic processes [2].

Hydrologic continuous time processes , such as ri v·er flow discharge, intensity of precipitation and similar variables, and hydrologic discrete t ime series of time i ntervals, which are fractions of the day or year, or multiples of the day or the month, usually are non-stationary. They are periodic­stochastic processes with various weights of periodic and stochastic components [3,4, 5). Therefore , they are non-stationary processes .

The theory and properties of run-lengths , either already known or developed in this paper, are appli­cable only to stat ionar y processes of annual time series of various hydrologic variables . The applica­tion of the theory of run-lengths of periodic­stochastic processes to hydrologic complex periodic­stochastic time series is not feasible at the present time f or the simple reason that this theory has not yet been developed in the form to be applicabl e to discrete hydrologic time series composed of periodic and stochastic components .

1.2 Practical Significance. Sequences of annual values of many hydrologic variables have several practical connotations . The behavior of sev·ere and prolonged droughts, with thei r properties , may not be known with sufficient accuracy to allow the probabilistic prediction of their occurrence , duration and areal coverage with a suffi cient degree of reliability. Statistical properties of runs of time series may represent one of the best ways for an objective definition of drought [6) . This inves­tigation of run-lengths and their application to series of wet and dry years is related to some sig­nificant problems of hydrology and water resource development.

Apart from determining the probabilities of droughts of various durations and severity at one

poi nt ?r o~er a_region, the probability of droughts occurr1ng 1n adJacent regions have si2nificant economic impl ications . If two or more regions produce an important crop, or are supplying water to the producers of the same industrial product, then the conditional probabilities of droughts covering simultaneously these regions may be of importance to various plans.

The probability of an extended period of wet years is similar to the problem of the probability of droughts. It may be important for restoration of biological cover in semi-arid or arid regions , or for the fight of prolonged pollution produced during dry years in soils and various water environments .

1. 3 Two Problems Related t o the Application of the Theor of Run- Len ths to H drolo ic Processes. A run is de ined, in proba i l i t y theory, as a succes­sion of similar events preceded and succeeded by different events. The number of elements in a run is usually referred to as its length. Therefore, the successions are called run-lengths . Two ~ypes of events must be appropriately defined , either as greater , or smal ler values than a given value.

The application of the theory of run-lengths to hydrologic stationary processes may be viewed from t wo basic standpoints :

(1) Some parameters of the run- lengths, as functions of another parameter, may be used for the investigation of stochastic hydrologic processes , particularly whether the series are stationary or not, and if so, whether they are serially independent or dependent . If found to be dependent , the interest is, what are the best mathematical model s to descr ibe this dependence .

(2) To det ermine, in the most r eliable way, the properties of run- lengths of a hydrologic series whenever it is found to be stationary, independent, or dependent, and the mathematical model of depen­dence is found to describe we ll the empirical dependence , if the series is dependent .

Before these two standpoints are discussed in detai l, the two c lassical methods and the two new potential methods , including runs, are briefly reviewed in order t o better define the problems in­vestigated in this paper .

1. 4 ~let hods f or Invest i gati on of Hydrolog ic Sequences. Four methods based on specific statisti ­cal parameters , as they change wi th other parameters,

1. Autocorrelation analysis . Parameters involved are the a~tocorrelation coefficients , Pk , as a function of the lag k between the correlated

*Former Ph.D. Graduate of Colorado State Universi t y , Civil Engineering Department, Fort Collins , Colorado, now temporary Research Associate, Civil Engineeri ng Department, Col orado Stat e University, Fort Collins, Colorado .

**Professor of Civil Engineer i ng and Professor-in-Charge of Hydrology and Water Resources Program , Civil Engineering Department , Colorado State University,

1

are or may be effectively used for the investigation of hydrologic processes:

values, or Pk = f(k) , with pk defined by

f(k)

for a discrete time series . The values estimated by the sample values rk .

(1. 1)

are

The use of autocorrelation analysis as an inves­tigative technique of hydrologic time series is based on the concept of analogy . One should know the cor­relograms of particular processes , and then by sta­tistical inference determine whether a computed correlogram of a hydrologic process is well approxi­mated by the correlogram of a kno~<~n process . To read the type of process that results from a correlogram, the alphabet of correlograms must be known.

2. Variance spectrum analysis . Basically this is the Fourier series analysis where an i nfinite num­ber of elementary periodic components, with a con­tinuous distribution of frequencies, is fitted to an observed series. The parameters involved are the variance densities , Vf , of various harmonics fitted to this series, represented against the fre­quencies f as the parameter. The variance of a harmonic is equal to the half of its squared amplitude . This type of analysis is a representation of the pro­cess in frequency domain,

( l. 2)

while the autocorrelation is a representation in time domain, or any other dimension on which the process occurs (say, the length). It might be noted that the variance density spectral function is the Fourier transform of the correlogram. The variance densities vf arc estimated by the sample variance densities, vf.

The usc of variance spectrum analysis as an i nvestigative technique of hydrologic processes is also based on the concept of analogy. Statistical inference should be performed to find whether a com­puted variance spectrum of a hydrologic process is well approximated by the variance spectrum of a known process. A reading knowledge of the alphabet of variance spectra should be known to advance hy­potheses on the kind of mathematical model for the process investigated.

3. Rafges. The ranges, Rn , are defined in terms of di ferences between maximum and minimum on the cumulative sums of departures of values from the average, or from any other value, for given subseries sizes , n . The expected ranges, E(Rn) , or similar parameters, as random variables, are related to the subsample size, or

(1. 3)

Let {xi; i•l, ... , N} be the observed sequence, and let x0 be a specified truncation level which in general represents the reference level. Then the sum is

i s. . r (xi - xo) (1.4)

l i=l

for i•l, 2, ... , n The surplus is defined by

2

sn +

{O ,Si} for i: l J 2' • • • 1 • max n ( 1. 5)

and the deficit by

s n " min {O,Si} i=l,2, ... , n ( 1. 6)

where from

n represents the size of a subsample taken {xi} .

The range is defined by

R n s + n sn = max{O,Si} - min{O,Si} (1. 7)

for i•l ,2, ... , n

As in the case of the autocorre l ation and vari­ance spectrum analyses, the use of the expected range (or of a similar parameter), as a function of n , may be conceived as an investigative technique of hydrologic series. It should be based also on the concept of analogy . The parameters E(Rn) are estimated by the sample mean ranges, ~n The com-parison of the function Rn = f(n) with the function of the same parameter of a known process allows the advancement of hypotheses about mathematical models describing dependence of a stochastic process. The statistical inference of the goodness of fit of these theoretical and hypothetical models decides whether they should be accepted or rejected. The alphabet of these range functions for various types of processes should be known before hypotheses are advanced.

4. ~· Various properties of runs, clearly defined, have parameters o , which may be used as function of another parameter B , so that o = f(B) is a characteristic of a process of independent or dependent sequences . for the purposes of this paper, the run is identical to the concept of run-l ength . Basical ly, both are the number of consecutive posi­tive or negative departures from a specified constant value called here the truncation level . In this narrower definition of runs, positive runs are asso­ciated with positive departures and negative runs with negative departures . The structure of a series may be analyzed by studying the properties of runs at different truncation l evels . Parameters of runs have practical meanings in hydrology , because a positive run can be associated with the duration of a wet period or with a water surplus inteTval, while a negative run can be associated with the duration of a drought, or with a water deficit interval .

S. Comparison of four techniques: The two classical techniques for the investigation of time series are autocorrelation analysis and variance spectrum analysis . The way they are used in explor­ing the internal structure of a process depends to some extent on the purpose of inquiry and prior knowledge of the generating system of the process. The correl ograrn tells something about the linear relation between the consecuti ve values of a series . The spectrum exhibits the extent the series is in step with certain fundamental rhythms, measured at various frequencies [7] . These two techniques offer no particular advantage over other parameters for the task of investigating the properties of various sequences . One fact seems clear, name ly that it is difficult to use the t1~0 functions pk " f(k) or

vf ~ ~(f) , respect i vely for t hese two t echniques, directly in the solution of various water resources problems .

Ranges and runs are tt.•o t echniques that can be used advantageously in water r esources problems and at the same time, t hey may be used to investigate hydrologic processes. They can be readily associated with concepts of storage and drought, or with concepts of surplus and deficit, which are of int erest to the solution of various water t"esources problems. This is one of the main r easons for investigating proper­ties of run-length f or both objectives: the inves­tigation of hydrologic stochastic processes, and the direct computation of properties of runs , from the information in samples of these processes .

6 . Runs as the technique. If a truncation l evel is specified, the run-length associated with a nega­tive run represents the duration of a deficit rela­tive to this level . The probability of length of the deficit periods is r el evant f or t he planning, design, and operation of water resources systems.

The structure of a stochastic process is reflected in the properties of runs that it generates at speci­f ied truncation levels . For example, i ndependent variables wi'th a common distribution arc characterized by a mean run-length equal to two for a truncation level equal to the median of the distribution of variables. Identically distributed variables with a highly positive first serial correlation coefficient are char act erized by a mean run- length greater t han two at t he same level . On the other hand, identi ­cally distributed var iables, with a highly negative first serial correlation coeffi cient, are character­ized by a mean run-length smaller than two at the same l evel. These properties, which are investigated in detail i n t he fo llowing chapters, shoul d just ify t he use of runs not only in the making of water resources decisions, but also as a technique for the investiga­tion of s eries , and more specifically for the testing of stationarity and of mathematical dependence model s of hydr ol ogic processes.

1.5 Two Approaches to Investigations of Sto­chastic Sequences. Regardless of which of the four methods of investigation of hydrologic sequences is used, a sequence of a parameter as a f unction of anot her parameter characterizes a stochast ic process, like the functions Pk = f(k) , vf = $(f ) , E~ = f(n) , or ENq c f(q) . This last case is an example

of r uns , where ENq is the expected va lue of run­

l ength, estimated by the sample mean run- l ength Nq , as it changes with the probability q of all values of a variable not greater than the truncation level. These four functions, related to autocorrelation co­effic i ents , spectral densit ies , expected r anges , and expected run-lengths, should have we ll -defined math­ematical expressions for various stochastic dependence models, or for processes composed of the periodic and stochastic components. Particularly, these four f unct ions for t he population of a stochastic station­ary and independent process ar e well defined.

Two approaches for investigating time series may be used. The first approach consist s of the analysis of or i ginal data. It is here referred to as the use of the origi nal sample series . In t his case , anyone of the four above functions is computed from the

3

sampl e series, and compared with the f amily of corresponding population functions for various mathe­matical dependence models. Then a model is selected, its parameters estimated, and the population function compared with the sample function in such a way that their differences are or are not statisticall y sig­nificant. If t hey are significant, new models are selected as hypotheses, their parameters estimated, and the comparison repeated. The knowledge of shapes of above functions, pk = f(k) , vf = ~(f) , ERn

f(n) , or EN = f(q) , for various hydrologic q

mathematical dependence models i s a prerequisite, so that sight comparison with the sample function of any of the above four functions may l ead to the most l ikely hypotheses for the population models.

The second approach assumes a mat hematical model for the dependence of a process that is composed of a systematic dependence component(s), and an indepen­dent stochastic component. A residual series is obtained by separating the systematic dependence com­ponent(s) from the original series. Under the hy­pothesis that the assumed model is an adequate repre­sentation of the process, the residual series after this separation should be a sequence of independent stochasti~ variables. The independence of t he resi ­dual series is t hen tested. The assumed dependence model is accepted or rejected, depending on whether the independence of the residual series was accepted or rejected. This procedure is here referred to as "whitening," meaning that the residual series is expect ed to be a "white noise," or independent series .

It is perhaps interesting to emphasize a basic difference between these two approaches. The first approach does not assume a model a priori for the process, but rather the curve of t he sampl e funct ion leads to the hypothesis about the structure of the process, so that eventually a mathematical dependence model can be fitted to it. The second approach may start a priori by assuming a dependence model for the process, without computing the sampl e function, and aft er t he model parameters are estimated, the supposed independent stochastic component (white noise) is computed and tested. Logically, the sample function in any of the four above methods helps advance a more realistic hypothesis about t he model . However, if previous knowledge about these models is already available for the similar processes in a region , t he hypothesis can be advanced a priori, and the whiten­ing and testing performed in an appropriate way .

In order t o use the methods of run- length for investigating hydrologic sequences, run properti es should be known for various mathematical dependence models of hydrol ogic sequences, regardless of the two approaches used. Therefore, the objective of investigation in this paper is to add knowl edge about the properties of run-lengths for some mathematical dependence models of stationary hydrologic processes.

As an example, let the hypothesis be that {X.) is a first-order linear autoregressive process in 1

the f or m

with X ,

(1.8)

~ t he expect ed value and o2 t he variance of ci is a s t andardized st ationary independen t

variable (0,1), while p1

is the first autocorrela-

tion coefficient. The parameters,

are estimated by sample parameters X , $2

The "whitened" series is

£. ].

and

(1. 9)

Under the given hypothesis, {£i} is a sequence

of standardized, independent random vari ables. Then the whitened series is tested for independence.

1.6 Objectives for Determinating Properties of Run-Length . The first objective of this study is t o develop a method for investigating stationary inde­pendent and dependent hydrologic time series by using statistical parameters of runs. Four phases must be involved in this investigation:

(a) ~1athematical formulation of the problem;

(b) Selection of suitable parameters for testing hypotheses of stationarity and time dependence;

(c) Statistical inference for stationarity and time dependence models, and

(d) Tests of application of the method to some selected time series .

The second objective of this study is to develop, in an approximate analytical procedure, the proper­ties of run- lengths of the stationary, first -order, and linear autoregressive mathematical model of time dependence, as defined by Equation (1.8). This objective has a significant, practical aspect, as shown by the follow ing example .

For a river with large storage capacJ.tJ.es, what is the probability of a drought to occur with a dura­tion of n or more years, if the drought is defined as a run of all annual inflows into reservoir of above capacity, which are not greater than a given annual runoff. In this case, it is possible to deter­mine the truncation level of the series of annual runoff and from it the probability q . If the dependence in the series of runoff can be well approxi­mated by the model of Equation (1.8), and ~ , o , and pl are esti mated from it, then the results of

investigations in this study should answer readily and accurately the above classical problem. The available runoff series may not include even a drought of the duration of n/2 or of a shorter duration, so that the current empirical methods cannot give an answer to this problem. There are two reasons for concentration on the model of Equation (1 . 8): (1) It is often the most appropriate model for dependence of series of annual river flows, and (2) It is simple for an analytical treatment.

4

1 . 7 Definition of Runs . A series of the variabl e x is cut at many places by an arbitrary horizontal truncation level x , and the relation of this constant x to'all0 other values of x of the pro­cess serv-es0 as a basis for the definition of runs in this study. Basically, there must be t wo processes intersecting each other in order to define runs. Because these two processes cross each other, the theory of runs is often called the crossing theory. The term "theory of runs" is used in the case of discrete series [7], and the term "crossing theory" in the case of conti nuous series [8] .

One of the t wo processes must be the original process. The second process may be a constant x

0 ,

the process of a random variable y , or any other type of deterministic, combined deterministic -stochastic, or pure stochastic process. When this second process is not a constant, the development of properties of runs becomes compl ex . In the case of runs to be used in this study, the main assump­tions are:

1. Only discrete series are invest igated, so that the expression "runs" is used;

2. The variable x may have discrete, contin­uous, or fixed probability distribution;

3. The second process is a constant x0

, or any constant value in the range of fluctuation of the variable x ;

4. The probability P(x ~ x0 ) = q may replace the constant x , in order to make some properties of runs indepen~ent of the type of distribution of x.

The number of values of a discrete sequence bet ween an upcrossing of the truncation level and the follo·wing downcrossing is defined as a positive run-length, or briefl y, for this study, the positive run. Similarly, a negative run-length, or the nega­tive run, is defined as the number of values of a discrete series between a downcrossing and the next upcrossing. They are shown in the upper graph of Figure l. 1, and are designated by

+ N. J

for the length of the j-th positive run, and

by N~ for the J

l ength of the j-th negative run.

The j -th total run is defined as

Nj = N; + Nj , with j=l,2, ... , where is

counted from the origin of a time series .

These may be extended to definitions T; , Tj ,

and Tj , as the positive, t he negative and the total run of a continuous process, respectively. This is analogous to the definitions of runs of dis­crete time series, as shown in the lower graph of Figure 1.1.

n u

Fig. 1.1 Definition of positive and negative runs for a given truncation level. Upper graph refers to a discrete series and lower graph to a continuous series.

Other parameters used in literature as defini­tions of various runs of discrete time series,

besides N: , N~ , and N. , are: J J J

1. Sum of deviations associated with positive runs, as the positive run-sum, or the run-surplus,

5

2. Sum of deviations associated with negative runs, as the negative run-sum, or the run· ciefici t,

3. Number of positive runs for a given series of size N .

4. Number of negative runs for a given series of size N ,

5. Number of total runs for a given series of size N

For the continuous ~ime ~eries, the following parameters other than T. , T. , and T. are used:

J J J +

1. Area above truncation level for T j , as the positive run-sum, or the run-surplus;

2. Area below truncation level for T~ , as the negative run-sum, or the run-deficit; J

3. Number of positive runs for a given series length, T ;

4. Number of negative runs for a given series length, T ;

s. Number of total runs for a given series length, T ;

6. Time interval between successive peaks;

7. Time interval between successive troughs.

Al l of these runs are random variables, and are functions of the process (xi} and the truncation level x

0 ,

Properties of runs relating to these functions can be directly used in many water resources problems. If x0 determines the level of demand, and if this level is not reached, a drought occurs . If a flooded area begins for x > x0 , and the flood damage is a function of the time during which x > x0 , then the distribution of positive run-length and/or run-sum determines t he character of flooding. If a given type of run is regionalized, or shown over an area with its isolines, the regional phenomena of drought, flood , and similar phenomena may be studied for their probabilities of recurrence [6).

Chapter II

SUMMARY AND STATUS OF KNOWLEDGE ON DISCRETE RUNS

2.1 Introductory Statement. Two main aspects are reviewed, the distribution t heory of runs for both independent and dependent random variables, and the multivariate normal integral which serves as a base for the mathematical developments in Chapter I II. The summary is related only t o those properties of runs , which are relevant to investigations in this paper.

2.2 Distr ibution Theory of the Number of Various Runs of Independent Random Variables. The classical distribution theory of runs has been mainly concerned wit h independent arrangements of a fixed or a random number of several kinds of elements . This is not particularly relevant for this study , but is summa­rized for t he sake of completeness. In the case of two different kinds of elements, it is assumed that the number of e l ements of each kind are N0 and N1 , and that they are al l randomly dr awn without replace­ment. This is equivalent to sampling a binomial popul ation, with probabilit ies of· elements, p and q = 1 - p , respectively . Let K~ denote the number

of r uns of kind (o) of the l ength i , and l et K~ 1

denote the number of runs of kind (1) of t~e length

i . Finally, let K0 = r K~ des ignate 'the number i 1

of all runs of elements N K1 = r K~ the number 0 i 1

of all runs of elements Nl and K = Ko + Kl the

total number of runs , and N No + Nl the tot al

number of e l ements, or t he sampl e size, with i = 1,2 ' 0 ••

Wishart and Hirshfeld (9) obtained and t abulat ed the joint probabilities of the number of r uns

and

N 0

(2 .1)

(2 .4)

6

In Equations (2.1) t hrough (2.4) , the capi tal letters designate t he random variables, and the small letters t he values those variab l es can take.

As t he sample size N increases to infinity, K is asymptotically normally distributed, with

EK = 2npq + p2 + q2 = 2(n- l )pq + 1 (2 . 5)

and

var K = 4npq(l-3pq) - 2pq(3-10pq) (2 . 6)

However , Cochran (10) gives expressions for the expected values of t he number of positive and nega­tive runs as

EK0 = p + (n-l)pq (2 0 7)

EK1 q + (n-l)pq (2 0 8)

and

EN 0

" np , and EN1 = nq (2 0 9)

with . N0

and N1 being also the random variables in t h1s case .

Stevens (11) gives the distribution of the total number of runs, without a regard to their l ength, from the arrangements of two kinds of ele­ments . He develops a x2- criterion for the test of significance. Wald and Wolfowitz [12) study the same distribution as Stevens [11], and show t hat it is asymptotical ly normal. The conditional distribu­tions of K are

(2 0 10)

and

(2 . 11)

where n0

and n1 are values that N0

and N1 can

take . These probabilit ies are i ndependent of t he parameter p . For n

0 = an1 , wi th a > 0 , and

n0

-+ oo , Wal d and \~olfowitz [12] give t he above

distributions of Equati on (2 . 10) as a normal asymp­tot ic distribution wi th

EK

For o = 1

z

2n 0

l+o var K

the statistic

K-n 0

,r;

4on 0

is a standard normal variable.

(2. 12)

(2. 13)

Mood [13) derives distributions of the number of runs of a gi ven length for t he independent arrange­ments of the fixed number of elements of two or more ki nds of the binomial and multinomial populations. He shows these distributions as asymptotically normal with an increase in the sample size . Their expected values arc :

0 i EKi = p q [(n-i-l)q + 2) (2. 14)

and

1 i EK. q p [(n-i - l)p + 2) 1

(2 .15)

The statistic

K- EK X:--

/varK

K-2npq (2. 16) 21npq( l -3pq)

is asymptotically normal with the mean of zero and the variance of unity. Comparing Equations (2 .5) and (2.6) with the mean 2npq , and the variance 4npq(l-3pq) of Equation (2.16) , the mean and variance given by Mood [13), and the mean and variance given by Wishard and Hirshfeld [9] , are different. Parameters in Equat ion (2.16) are approximations to those of Equations (2.5) and (2.6). Bendat and Piersol [14) give tables for the conditional dis t ribution of K when N

0 ., N1 ., N/2 .

2.3 Distribution Theory of Run- Lengths of

Independent Random Var iables . Let N: and N: J J

denote the positive and negative j - th run- length for the given truncation l evel, x0 . Also let {X} be the sequence of independent r andom variables of the common distributi on, F(x), with F(x

0) = q, and

l-F(x0

) ., p , and let

{N_} : {N: + N: } J J J

be the random sequence of the total j-th run-length.

The probability mass function of N1 by feller [15) as

k k pq - gp P(N1 = k) = q-p

for k=2,3, . .. , with

is given

(2. 17)

7

1 pq

(2 . 18)

The distribut ion of the number of total runs, k(N) , in a discrete time series of length N has the follo,•ing parameters

Ek (N) = (N- l)pq for N > 1 (2 .19)

and

1 5 vark(N) = Npq (l-3pq - N + N pq) , for N ~ 4; (2.20)

this distribution is asymptoticall y normal. Downer, Siddi qui and Yevjevich [16] studied the distrihution of positive and negative run- lengths for a sequence of independent identically distributed random vari­ables , and applied it to the normal variabl e . They

have shown that {N: } is also a sequence of i ndepen-J

dent identically distributed r andom variables with the probability mass f unctions

P(N: = k) J

k - 1 = qp ' and P(Nj k) k-1 pq (2.21)

and their moments are

+ : l EN: 1 EN. J q J p (2 . 22)

+ ..E... N: =_g_ var N. var J 2 J 2

q p ( 2.23)

For t he case p = q = 1/2

+ PCNj

l P(Nj = k) = = k) k

2 (2.24)

EN: J

EN: J

2 (2.25)

and

+ N: var N. ; var ; 2 J J

(2 . 26)

Llamas ( 17) studied t he case of standard, one­parameter Gamma random variables, with the probabil­ity distribution function

F(x) X

f -10

r et-1 Ct(et+t~Ct)

f(et) -a-t /a

e dt (2 . 27)

For x0

= 0 , he obtained p = F(O) = P(a,a) , and

q = 1- P(et,et), where P(et,et) is the i ncomplete Gamma function, or

l 0. 1 P(a,a) = r (a) f e-t t a- dt

0

(2. 28)

Llamas and Siddiqui [18] studied the case of a sequence of a two-di mensional random process {x,y} , where the two variables are independent and have a common distribution funct ion , F(x,y) . Given the t wo levels, x

0 and y

0 , such that o < F(x

0,y

0) < 1 ,

the four possible events are defined as

Both sequences are associated with the sign mi nus if A occurs, and wi th the sign plus if D occurs. The sequence of k consecutive A events followed and preceded by any other event is a negative r un of the length k . The sequence of k consecutive D events fol lowed and preceded by any other event, is a positive run of the length k , and for the initial run the requirement of ·~receded b~' is dropped . If

Ac is the complement set of A , then

Llamas and Siddiqui have shown [18) that

with

- k-1 P (N. = k) = p q

J

1 p

, and

(2.29)

( 2. 30)

the analogous relati ons hold f or sponding values of p and q .

+ N. f or i ts corre-

2 . 4 Dependent two states transition

J

Distribution Theory of Run-Lengths of Random Variables. For a Markov chain with

(0) and (1) , Cox and Miller [19], give the probabi lity matrix of this chain, which is

p 0 ~1-a al ~ 1-~

(2. 31)

They give t he distribution of t he r ecurr ence time of state (0), designated by N° , which is equal to t he run-length of state ( 1) plus unity, as

0 k-2 P(N =k) = aS (l-6) for k=2 , 3, ... , (2 . 32)

and

1 - a , for k=l (2 .33)

The mean recurrence time of the state 0 is then

EN° = a+e (2 . 34) s

Simi lar relati ons hold for the recurrence time of the state (1), which is equal to the run-length of state (0)

8

plus unity, designated by N~ , by interchanging a and 13

Heiny [20) defines the two states wi th their transition probabilit ies of the Markov chain as

P (xJ. > x I x. 1 > x0

) = r , and 0 J-

P(xJ. < A lx. 1

> x0

) = s - 0 J-

with r + s = 1 The following relations are valid for this Markov Gaussian process {x}:

.. I k-1 2 P(N = k x1 > 0) = sr [1 + O(p )], k=1 ,2,3, ... ,

(2.35)

wi th

(2. 36)

and

(2.37)

where O(p2) indicates an expression that becomes negligible for small values of p . He also found an approximation for the conditional joint proba­bility mass f unction of the first j positive and t he f i rst j negative runs, given x1 > 0 , as follows:

.. N~=m.,

.. .. N~=m1 1 x1>0) P(N .=n . , N. 1 =n. 1' ... ' Nl =nl,

J J J J J- J-

m - 1

=

n1-l

sr tv 1 srn2-l m2-1 nj-l tv ... sr

m. 1 tv J - [l+O(p2)] ,

where

t = P(xJ. > X lx . l < X ) 0 J - - 0

and

t + v = 1

v

(2. 38)

P(xJ. < X jx. l< X ) - 0 J-- 0

This t reat ment, however, has t wo disadvantages : (a) it is based on a conditional probability that x1 > 0, and (b) it is appl icable only to very small

values of p , since the errors O(p2) may be sig­nif icant for larger val ues of p

2 . 5 The Multivariate Normal Integral. Gupta [21] presents an exhaustive bibliography on the multi normal integr al and related t opics, and gives a review (22] of these works. Onl y works that do not overl ap with references in [21] and [22], but are related to mathemat ical developments in the following chap·cers are reviewed here.

The mul tinormal integral is involved in the t heory of runs of dependent normal variables because it is directly related to the prob lem of h auto­correlated random variab l es, z1, z2, ... ,Zh . If

these variables have a standar d multivariat e normal distribution, the problem to solve i s the probability

that all h variables are simult aneously positive . A new sequence of random variables {X) is defined as follows:

1 for z > 0

X

- 1 f or Z < 0

The probability that all h variabl es are simulta­

neous ly positive is P (h+) , where the index m indicates that the truNcation level x

0 of the ran-

dam process {X) is the median of t he distribution of {Z} For r.. EX.X . , Mcfadden [23] gi ves,

lJ l J f or any h > 4

For

(2 . 40)

9

If Equation (2 .40) is substituted into Equat ion (2.39),

2-h[ l + ~ L arc sinpij j >i ?_l

Obvi ously, and for the univariate case , Equation (2.41) becomes

2 (2 . 42)

For the bivariate case, the resu l t is known as Sheppard ' s theorem [24) of the median dichotomy; i t i s

+ 1 1 Pm(2 ) = 4 + z; arc sinp (2 . 43)

This equation is tabulated in t he Tables of Ma the­matical Functions of the National Bureau of Standards [28] for P , which varies from 0 t o 1, with incr e ­ments of 0 .01. For the trivariate case, the follow­ing result is given by David [26)

+ 1 1 . Pm(3 ) = 8 + 4" (arcs1np12 + arcsinp13 + arcsinp23) .

(2.44)

Chapter III

PROBABILITIES OF RUN- LENGTH OF THE FIRST-ORDER LINEAR AUTOREGRESSIVE ~10DEL OF NORMAL VARIABLES

3 . 1 General Notations and Expressions for Pro­babilities of Run-Length. For purposes of simplici ty, the following notation is adopted:

P(Xl~xo,X2~xo•· .. ,Xk~xo,Xk+l>xo,Xk+2>xo•···• Xk+j >xo}

-= P(k ,j+) ,

and

with k=l,2, . .. and j =l ,2, .. . The probability of the firs t positive run-length

from the beginning of a series being equal to or greater than j , is

00

P (N~ ~ j) = p (j +) + ~ P(k ,() (3 .1) k=l

The probabil ity mass function of + Nl is

P(N~ = j) = P(N~ ~ j) - P(N+ 1 ~ j

+ 1) (3.2)

The computation of joint probabilities . P(k-,j+) requires the joint probability distribution of the variables x1 ,x2,... This j oint distribution for

the purposes of this study is assumed multivariate normal.

3 . 2 Stationary and Ergodic ~1ul tidimensional Gaussian Processes. An arbitrary Gaussian random process {xi} , or x1 ,x2, ... ,xn' where i =l,2, ... ,n

at arbitrary or equally spaced positions in time , has the multivariate normal distribution in n dimensions. This process is completely described by the param­eters of this distribution: the expected values E(xi) , i=l,2, ... ,n, and the covariance matrix,

cov(x . , x .) as a function of i and 1 J

if, the are Ex.

1

A multivariate Gaussian process is stationary and only if, the expected value is constant and covariances depend only on the lag Jj -iJ , and independent of i For any stationary process is equal to ~ and cov(xi,xi+k) is equal to

C(k). In particular, C(o) is equal to var x and C(k) is a constant independent of i The function

is the autocovariance function, while

-~ p(k) - C(o) (3 . 3)

is the autocorrelation function . It specifies the GO~relation coefficient between values of the process , which are k intervals apart, and it is the k-th autocorrelation coefficient.

Let {x} be a stat ionary Gaussian process with zero expected value and variance unity. Its probabil ­ity density function is

10

f(x) (3 .4)

The bivariate probability density function of xi

and xJ. , with Ex. Ex. = 0 , and var x. l J ~

var x. = 1 , is J

f(x . ,x.)= 1

exp[- -21 (x~-2p .. x.x.+x~)J , (3.5)

l J 2r.~ 1 l.J 1 J J l.J

where pij is the correlat ion coefficient between

si and xj The multivariate normal probability

density f unction of x1,x2 , .. . ,xn takes a more com­

p l ex form, but is analogous to Equation (3.5) and given by Equat ion (3 . 9). In this case , the correla­tion matrix of random vairables x1,x2, ... ,xn is the

n by n matrix with the elements pij representing

the correlation coefficients between any two variables x. and x., i=l ,2, . . . ,n and j=l,2, . .. ,n. It is a

l. J symmetrical matrix since pji = pij , and all e l ements

of the main diagonal are one. For a stationary pro-cess

Pij = Pjj-iJ = pk (3.6)

with k = Jj-i l ; therefore, all elements of any diagonal are i dentical. The correlation matrix of a stationary process is

~-2

7r=

~-2

f.:_ , 1.:-2

If the random process {x} is second-order stationary, as described above , and if the expected values and crossproduct f uncti ons def ined by averages of indi­vidual realizations (sampl e functions) as

and

1 N lim - ~ x. N..,... N i= l 1

(3 .7)

(3.8)

then the process is ergodic. A second- order station­ary and ergodic Gaussian process is also strictly stationary and ergodic, or higher-order stationary and ergodic. This means that all ensemble averaged statistical properties are equal to the corresponding time averages. Hence, the verification of self­stationarity for a single time series justifies the assumption of stationarity and ergodicity.

3.3 Function. is

Multivariate Normal Probability Density The normal distribution of n variables

1 tln n Jn dF= n/ 2 exp - 2 L L a.kx.xk n dx. (211) IJRT j=l k=l J J j =l J

where t he variables x1, x2, . .. ,xn ' have expect ed

valu·es of zero and variances of unity . Also, IRI is the determinant of the corre l at ion matrix of these variables, while ajk arc the el ements of the in-

verse of the correlation matrix. The characteristic function of this distribution is not expressed in terms of the inverse of the correlation matrix, but in terms of the elements of the correlation matrix itself. This property helps in computing pr obabili­ties of run-lengths. The characteristic function is

~(t) .. exp (- t r r p .. t.t. l l i .. lj=llJlJj (3.10)

3. 4 General Expression for Joint Probability of at Least k Subsequent Values Below Truncation Level, Followed by at Least j Subsequent Values Above Tr uncation Level . In order to find an expr ession for t he joint probabilities, P(k- ,j•), involved in Equation (3.1), the following assumptions are made:

1. The hydrologic time series of annual preci­pitation and annual runoff are second-order stationary. Some of these series may have , however , a smal l degr ee of non-stationarity, which comes from either man-made changes in river basins and around the pr ecipitation gauging stations , or f r om the i nconsistency in data [27] . These series should be made stationary by cor­rections before t he theory of runs , as discussed here, is app lied.

2. The process of annual values is a Gaussian process or approximately so. This assumpti on is justified from the point of view that some runs are distribution free, or independent of the underlying distributions of {Xi) . It is also just i f i ed from

t he point of view that many non-Gaussian hydrologic processes can be reduced to Gaussian processes through appropriate transformations. This point will be treated in detail in Chapter IV.

3. The stationary Gaussian processes are stan­dardized for a simpler treatment of various problems.

With the above three assumptions, the joint probabilities P(k-,j+) can be expressed as

11

X 0

f X

0 00

J f f X~

j

dF (3 . 11)

Substituting dF by its equivalent into Equation (3.9) gives

X X 0 0 00

J .. . f f .. . f X X ~0

j

{ 1 n n J n

• exp - 2 L L ajkx.xk n dxj j•l k• l J j=l

(3.12)

where n • j + k . Equation (3.11) is the multi­normal integral. No explicit expression exists for the general solution of the multinormal i ntegral. Efforts are devoted to finding expressions for seve­ral cases of this mul t inormal i ntegral i n this s tudy , so that specific numbers can be as signed to probabi l ­ities in Equation (3 . 12). These probabilities wil l be called jo~nt probabilities t o distinguish t hem from the probabilities of runs.

3. 5 Probabilities of Runs for Any Truncation Level. Throughout this subchapter concern is with the evaluation of probabilities of the type

where q • F(x0

) • To simplify notation, the sub­

index q is dropped, and it will be used only when it is necessary to refer to it .

Probabilities P(2+) and P( l- , 1+) . In the univariate case, the following expression obviously holds ..

(3. 13)

where r(x0

) is t he standard normal distribution

function . In t he bivariate case (xi,xi +l) ,

and

f 21T/J-p2 X

0

{3 . 14)

(3.15)

These two probabil ities are related as X X

0 0 - + : J J f J P(l , 1 ) dF dF -oo X

0

_ .. xo

-J J dF=l-F(x0

) - P(2+) X X

0 0

(3.16)

Bivariate tables are given by the National Bureau of Standards [28) for +p = from 0 to .95, wit h i nter­vals 0.05; and from 0.95 to 1, with intervals, 0.01; and variates in the range from 0 to 4, with intervals 0.1, to 6 or 7 decimal places . Zelen and Severo [29] give charts for the bivariate norma! i ntegral with an error of 1 percent or less.

Probability P(3+). For three variables,

"' "' .. P(3+) = f f f dF

X X X 0 0 0

The integral of this equation has been evaluated in terms of the tetrachoric series expansion by Kendall [20). It is

I j,k,2

where f(x0

) is the standard no·rma1 probability

density function, Hr(x) is the rth Hermite poly­

nomial defined by

(- ddx)r f(x) = (-D)r f (x)

and j, k , 2 can take t he values 0, 1, 2, .. . . The first three Hermite polynomials are H

0(x) =l,

H1(x)=x and H2(x)=x2-l .

Probabilities of the type P(( ) . The tetrachoric series expansion for the trivariate case [30] can be generalized to t he multivariate case by the fol lowing procedure. As discussed previously, the multinormal probability density function can be expressed in terms of elements of the inverse of the correlation matrix. A direct i ntegration of the mul­tinormal p .d.f. would impl y an inversion of this correlation matrix, if the integral is evaluated in t erms of the correlation coefficients. This can be avoided, if the Fourier transform of the multinormal characteristic function is expressed in terms of the correlation coefficients themselves, and this expres­sion integrated. This is a parallel procedure to the one followed by Kendall (30) for the trivariate case . By definition

f dF X X ~

j

12

.. J <l>(t) _..,

(3.20)

where

. (3.21)

Also, ~(t) can be rewritten as

<l>(t) = exp[- .!. ( i t~ + 2 i=l l

(3.22)

In using the exponential series expansion

L -,- I p.kt.tk "' ( -l) r ( )r r=O r · k> i ~.1 1 1

(3 . 23)

Substituting Equation (3 . 23) i nto Equation (3.22),

<l>(t) = exp [-} f t~] I (-l( [ i p.kt.tk]\3 . 24) i=l 1 r =O r. k>i>l 1 1

where

[ Ji>/ik\ tkr = [cpl2tl t2+. · .+plntl tn)+(p23t2t3+

. . . +p2 t2t + ... +p 1 nt lt )1 r n n n- , n- nj

Substituting Equation (3 . 21) and Equation (3 . 24) i nto Equation (3 .20) gives

.. .. .. .. [ ~ ] P(()=(2!). f dx1 .. . f dx. f ... f exp - } ) t~

J X X J -ao •"" 1 = 1 0 0

(3.26)

By adopting the notation

" A(p,i) ,

(3. 27)

Equation (3.26) becomes

.. .. f dx1 ... j dxj J ..• f X X -CD -00

o_____s ~ J J

(3 . 28)

This is the product of j integrals, the first of which is

i~ ~"" dx1_£exp[-} t~] exp(-it1x1)dt1 0

' (3.29)

and the remaining j-1 integrals are similar expres ­sions in Xi and ti . Since

exp(- } t 2) exp(-itx) dt

13

Equation (3.29) is

.. ( i)r f dxH (x)f(x) (-i/H

1Cx ) f(x ) ,

- xo r r- o o (3.31)

and Equation (3 . 28) becomes

P(() fj(x0

) ! A(o,i)H51

_1(x0

) .. . H53

__ 1

(x0

)

r=O { i }

i i 12 n-l,n

fj(x) y ~12 1 ' ' '~n-1 n o 0112" .. 1 1 I r= n- ,n

{i } (3. 32)

It is important to notice at this point that the definition of the Hermite polynomials applies only to r"'0,1,2,... . For rs-1 , H_

1 (x) is defined by

means of Equation (3.31) as

.. H_ 1(x0 )f(x

0) • I H (x)f(x)dx c 1- F(x

0)

xo 0

For I l:i , and

Equation (3 .32) becomes

.. P(j+) = fj(x) r A(p,i)a(H)

0 I=O

Probabilities of the type definition,

X 0 ..

P(l-,j+) = f I I dF X X L.._-P

j

IDCIO 00 00 GO

- + P(l,j).

= J I .. . J dF- f ... I dF = P(j.)-P((j+l)+] X X

0 0 ..........._., j

(3 . 33)

By

(3.34)

The probabilit ies P(j+) and P((j+l)+) can be evaluated by Equation (3.33).

Probabil ities of the t ype similar procedure ,

X 0

X 0 ""

f f · · · f dF - 00 X X

,______,____. ~0

k j

X X 0 0 "'

- + p (k 1 j ) •

(21!) k+j f f f · · · J $(t)exp(- i t'X)

- 00 X X '--.....---' ~0

k j

By a

Using the expansion of the mul tinormal characteristic function given by Equation (3 . 24) ,

X 00 0

1 L (-l)r LA( P, i) f dx1 (2n)k+j r =O -"'

"' ( 1 j +k ) f dxk+l' . . f dxk+J' exp - - I t~ exp(it' X) X X

2 i =l 0 0

(3. 36)

This is the product of k integrals of the type

and integrals of the type

Taking into account Equation (3.30), the product of k int egr a l s i s

X 0

(-i)r f dxHr(x)f(x)

and t he product of integr als is

(-i) r J dxHr(x)f(x) - ar(x0

)

xo

14

1. ~s Equation (3.36) becomes

P(k- , ( ) = I A(p ,i)a~ (x ) .. . a~ (x )as (x ) I=O 1 ° k

0 k+ 1 °

(3 . 37)

The sequences ( a: (x0

)} and {ac(x0

) } can be

expressed in f unctions of Hermite.polynomials as a0 (x0 ) = l - F(x

0) and ar(x

0) =Hr - l (x

0) f (x

0) , f or

c c r gl ,2, .. . ; and a0

(x0) =F(x

0) and ar (x

0) =-Hr_

1Cx

0)

f(x0

) , for r =l,2, ... . In Equation ( 3 . 37),

I=l:i

then

For

Let us define

c c c CXS (X ) .. . aS (X) TI (u)

l 0 k 0

as (x ) .. . as (x ) 1T (a) k+l 0 k+j 0

I = 0 I

L A(p,i)1Tc(a)rr(a) 1=0

A(p ,i) = 1

(a (x ))j 0 0

(3. 38)

k . F (X ) [ 1-F(x ))1 + L A(p,i)nc(a) n(a)

0 0 1=1,2, .. .

(3 . 39)

Equation (3 . 40) is an i nfini t e seri es . However , i n practice it is only necessary to include a finite number of terms of this series t o compute numeri cal

values of P(k- ,j +) . A truncation of this seri es

after I=2 implies that terms containing pi or

higher powers of p 1 are neglect ed . For values of

p 1 less than 0 . 30 , the error int roduced by this

t r uncat ion is negligible . However for values of p 1

great er t han or equal t o 0 . 40 , t his t runcati on may i ntroduce a significant error . In this case i t is necessary to include more terms in Equation (3.40), and truncate the series at a higher value of I .

For values of p equal to or greater than 0.50, the truncation of this series after 1=3 implies that

terms containing p~ or higher powers of pl are

neglected. The error introduced in this case may be negligible . In other words, the higher p 1 the

larger power m of p1 should be included.

+ Distribution of N1 . This distribution

can be obtained for any truncation level, provided the expressions are avai l able for the joint proba-

bilities P{k- ,j+), so that

P{N~ ~ j) = P(() + L P(k- ,/) k=l

(3 .40)

Probabilities P{j+), P(l-,j+), and P(k- ,j+) for k=2,3, .. . may be evaluated by Equations (3. 32), (3.34), and (3.39), respectively. Equation (3.40) combined with Equation (3 . 2) gives the probability mass f unc-

t ion of

Distribution of Nl. By definition

P(N~ ~ j ) (3 .41)

where

P(j (3.42)

and

+ -P(k , j ) =P{x1>x , ... , xk>x ,xk 1<x , ... , xk .<x} 0 0 + 0 ' + J 0

(3 . 43)

Consider

where c x0

and p are defined by P = F(x~) = l -F(x0

)

"' 1- q Because of t he symmetry of the normal distri-bution,

+ -p (k • j )

q

The distribution of N~ is

P(N~ ~ j)

(3.44)

{3 . 45)

15

+ - + - .f. and probabilities Pp(j ) , Pp( l ,j ) , and Pp(k , j)

for k=2,3, ... may be also evaluated by Equations (3 . 32) , (3.34) , and {3 . 39), respectively .

The pr obability mass function of N~ is then

P(Ni = j) = P(N~ ~ j) - P(N~ ~ j+l) (3 . 46)

J oint dist ribution of

By definition

(3. 4 7)

The result expressed in Equations (3.37), (3 .38), and (3 . 39) can be extended to joint probabilities involved in Equation (3.47) . In t his case

c

' llsj l +i l+l .. ,(lsj l +il+j2 asjl+i l+ j2+i2

· a sj l•il+j2•i2+1

( 3 . 48)

A completely anal ogous result is obtained for

P(ii, j ;, - • +

1 -) of .+ and i2' J 2 ' If the places Jl .+ are interchanged in Equation (3. 48)' the resul t J2 does not vary because al l possible permut ations of {i} arc considered . This implies that the marginal

distribution of N~ also holds for Ni

+ and N2 are identical. This

and N; + Distributions of Nk and N~. The approach

in the previous subsection also applies to the se­

quence {N~ ; i=l, ... ,kl; the result is a sequence l

of identical ly distributed random variab l es , with distribution function of Equation (3 . 40). The same

is true for the sequence {N~ ; i= l , ... ,k}. It is

a l so a sequence of i dentically distributed r andom variables with distribution function of Equation (3. 45). For a more detai l ed analysis of this chapter and particularly for a special treat ment of the case of a truncation level of the median, the readeT is referred to Saldarriaga [6] .

Chapter IV

RUNS OF STATIONARY DEPENDENT GAUSSIAN PROCESSES

4.1 First-Order Linear Autoregressive Process . The usual linear regression prediction models, namely the moving averages and the autoregressive models, can be shown to be Gaussian processes when the inde­pendent component is normally distributed. Among these models, the first-order linear autoregressive processes of normal variables are considered only, because of its broad application in hydrology, and its simplicity .

Suppose that the process {~i} is defined by the recurrence relation

( 4. 1)

for Eti• O and Dti• var t 1=1 . It can be solved

formally by successive substitutions , and be rewritten as ..

pj x . . L £ . l. j=O l. -j

(4 . 2)

Then for Ex. l.

: 0

Ox. 1 = var X.

2 l. l. 1-p (4. 3)

where I o I < 1 is required for the process to be stationary. It is a well known result that

k ( 4.4) pk = p

It is apparent from Equation (4.2) that the first-

EN+ = L j s l

p (N+ ~ j) (4. 7)

By definition, the second moment of N+ is

E(N+) 2 L j2P(N+ j) (4.8) j=l

Substituting Equation (4 .5) into Equation (4.8) gives

E(N+) 2 = L (2j-l) P(N+ ~ j) j=l

In general, the r-th moment of X is

..

(4 .9)

L jrP(N+=j) • j=l

L [/-(j-1/]P(N+~ j).(4.10) j • l

Equations (4 .5) to (4.10) analogously apply to N-.

+ -4.3 Properties of Total Run-Length, N=N +N . Statistical properties of this parameter are rather

complex because N+ and N are not independent in autoregressive models, and their bivariate distribu­tion is unknown. The only property of N that can be calculated on the basis of a univariate distribu-

tion of N+ and N- is the mean,

EN = EN+ + EN ( 4. 11)

order linear autoregressive model is a moving average This equation can be written also in the form scheme of an infinite extent, with monot onically de-

creasing weights, p , 0 2 , 0 3 , ... {ci ) are normal variables, {x1 l

Therefore, if as a linear com-

bination of {E. . } is also a sequence of normal l. -J

variables and is a Gaussian process.

4.2 Probability Mass Function, and Moments of

Run-Lengths N+ and N- . The following relation

holds between the probability mass P(N+aj) of a

given run-length N+=j , and the probability distri­

bution functions P(N+~) of run-lengths N+

(4 . 5)

By definition , the first moment of N+ is .. EN+ L jP(N.=j) (4 .6)

j=l

Substituting Equation (4.5) into Equation (4.6) gives

16

(4 .12)

where q F(x0

) • Because

( 4 . 13)

due to t he symmetry of normal distributions,

4.4 General Procedure for Evaluating Properties of Runs. Statistical properties of run-lengths of stationary Gaussian processes can be evaluated for any truncation level, x

0 , 'by using the relations

obtained in this chapter and in Chapter 11!. The general procedure of this evaluation can be made in four steps. Equation numbers to be used for various expressions in these four steps are given in Table 4.1. These steps are:

1. St arting at an arbitrary time, probabi lities

P(j+) for at least the first values of X being above the t runcation level specified by q , are first computed. For these probabilities, nothing is specified about the values of X preceding or following the occurrence of these j values . They may be either above or below the truncation level;

P(j+) are not probabilities of runs but are needed for their computation .

2. Starting at an arbitrary time, probabilities

P(k- , j+) for the first k values of X , being belo.,.., and the j subsequent values of X being above the truncation level specified by q , are next com­puted. For these probabilities, nothing is specified about the values of X preceding or following the occurrence of these k+ j values. They are not prob­abilities of runs but they are necessary for the com­putation of these probabilities or runs.

3. The probability distribution and the proba­bility mass function of the run-length are calculated by using probabilities obtained in the two previous steps.

4 . Moments of run- lengths may then be calculated, when needed, by using the computed probability dis ­tribution . It might be noted that the probabilities

P(() (l.nd P(k- ,/) also have practical meaning by themselves, besides being used for the computations

of probabilities of runs. In fact, P(j+) is asso­ciated with the probability that starting at an .arbi­trary year at l east the f irst j years are wet .

Similarly, P(k-,j+) is associated with the proba­bi l ity that starting at an arbitrary year t he first k years are dry and are followed by at l east j wet years. This analogous ly applies to P(j-) and P(k+,j -).

TABLE 4 .1

EQUATIONS fOR THE EVALUATION OF PROPERTIES OF RUNS

Step Expression Equation

P(l +) (3. 13)

P(2+) (3.14)

P(()' > 3 (3.33)

2 - + P(l ,j ) (3. 34)

P(k-,() (3.39)

3 P(N+ ?_ j) (3.40) P(N+ = j) (3 . 2)

4 EN +

(4 . 7) E(N+) 2 (4 .9) E(N+) r ( 4 . 10)

17

4.5 Probabilities of the Non-Normal Case. Let F1(y) be the distribution function of a non-normal

variable Y , while F2

(x) for the variable X is

normal. In this case, probabilities of multivariate events are

- + P (k ,j ;y ) =P (Y1~ , ... ,Yk~ ,Yk 1>y , . .. ,Yk .>y) r o o o + o +J o

( 4.15)

and

- + PX(k ,j ;x )=P(X1<x , ... ,Xk<x ,xk 1>x , ... ,Xk .>x) , 0 - 0 - 0 + 0 +J 0

( 4 .16)

respectively, for F1(y) and F2(x) . For strictly

increasing distribution functions, F1(y) and

F2(x) I it is always possible to find such unique

values Yo and X 0

that satisfy

F 1 (yo) = F2(xo) = q ( 4 .17)

If the probabilities of multivariate events are equal, with Equation (4 . 17) satisfied, then

( 4 .18)

Equation (4.18) implies that the joint probabili­ties are dependent only on the probability level q for given x

0 and y

0 and not on the underlying

distribution .

As an example, consider the case of {y} log­normally distributed with

( 4. 19)

By definit ion of the log-normal distribution the following relation holds,

( 4. 20)

The probability P (k-,j+) can be expressed in terms as Y

=P(~nY1<~nx , . . . • ~nYk<~nx .~nYk 1<~nx , .. . ,tnYk .<~nx \ - 0 - 0 + - 0 +]- o'

=P(X1<tnx , . . . ,Xk<inx ,Xk 1<£nx , . . . ,Xk .<inx) - 0 - 0 + - 0 +J- 0

This last expression is P (k- ,() , so that q

( 4. 21)

It follows that all properties of run-lengths depend only on q

4.6 Properties of Runs of the First-Order , Autoregressive Linear Process. Positive Runs . Pro­perties of run- l engths of the f irst-order linear autoregr essive model are computed on a digital com,­puter by using the appropriate equations of Table 4.1. Five values of the probability truncation l evel q = F(x

0) , and five values of p wer e used, as

shown by Tables 4.2 and 4.3. For each possibl e com­bination of q and p , the probabilities P(j +)

and P(k-,j+) were first calculat ed for j=l,2, ... , 10 and k=l,2, .. . ,10 . Theoret ical ly, probabili ties

- + of runs are given by an infinite series of P(k ,j ) , as shown by Equation (3 . 40). Actually these terms become very smal l for suffi ciently large values of k and only a finite number of terms need t o be con­sidered. Tables of computed probabilities and parameters of runs of the first-order linear auto­regressive process are given in Appendix A; they are :

P(k- ,j+) , P(N+ > j), and P(N+ = j), with k=l,2, ... , 10 and j:ol, 2,-:-.. ,10. The parameters of distribu-

tions are EN+ , var N+ . Figures 4 . 1 and 4. 2 give these probability distributions of positive run­lengths of the f i rst -order linear autoregressive pro­cess for values of p , varying from 0 t o 0.50, with increment s of 0 .10, and for values of q , from 0 . 30 to 0 . 70, with increments of 0.10 . In t hese figures, points are used for computed probabilities of dis­tributions of discrete (integer) random events, while curves are used for t he visualization of these dis ­tributions. Tab l es 4.2 and 4.3 summari~e the results for the first two moments of N+ •

TABLE 4 . 2

EXPECTED VALUF.S OF N+ OF THE FIRST-ORDER LINEA1 AUTOREGRESSIVE PROCESS

~~ .3 .4 .5 .6 . 7

~~ .3 .4 .5 .6 . 7

0 .1 . 2 .3 .4

3.33 3. 45 3.65 3.87 4. 29 2.50 2.66 2.84 3 .05 3.32 2 .00 2 .14 2. 29 2 . 47 2.66 1.67 1. 78 1.90 2. 05 2. 20 1. 43 1. 51 1.61 1.72 1. 86

Ti\Bl,E 4. 3

VARIANCE OF N+ OF THE FIRST-ORDER LINEAR AUTOREGRESSIVE PROCESS

0 .1 • 2 .3 . 4

7 . 77 7. 77 8 .74 9 .83 12.78 3 .75 4 . 32 5.10 5 .98 7 . 34 2.00 2.42 2 .94 3 .53 4 . 24 1.11 1.38 1. 71 2.08 2.50 0 . 61 0.79 0 . 99 1. 22 1.47

.5

4.69 3.60 2.88 2.33 2.00

.5

15.46 8.66 4.98 2. 94 1.72

18

Negative Runs. Properties of negati ve runs are

found by means of the equation P (k+ ,j -) =P (k- ,/). - + q - p +

It then follows EN (q) =EN (p), var N (q)=var N (p),

and , i n general , E(N- (q)]r = E[N+(p) ] r .

has

EN~

Total Runs . The process of the total r un-length

been defined as {N.} ={N: + N:} . Then EN. = J J J J

+ EN: J

three J

is its expected value . Figure 4.3 shows

dif f erent sets of curves :

EN. plotted against q for p J

0 . 5 with increments of 0 .1.

.80

.70

.60

-::. Ill . .50 z i'

.40

,30

.20

.10

0 ~ I

+ -EN . , EN . , and J J

val ues from 0 to

q = 03

10

Fig . 4.1 Pr obability distributions of posit ive run­l engths of the f i rst-order linear auto­regressive process for q = 0.3

4. 7 Propert ies of runs of the f irst- order l inear autoregressive Gaussian process obtained by the data generation method . Probabilities of run- lengths are given by an infini t e series as shown by Equation (3.40). At t he same time, each term contained in this series is give·n by an expansion of t he tetrachoric series

.80

.7

.60

-=. /\I

z .50 cr

.4-0

.:3.0

.z

.10

0 I

1.00

.70

.60

-::. /\I

~ 50 Q.

.40

. .30

.20

10

0 I

I.

q :0.4 q : 0.5

.80

.70

.60

-=. /\I . z Q.

.40

. .30

.20

.I

1.00

q =0 6 .90 q =0 7

.eo

,7

~ .ErO

-=. I /\I

"z so cr \

.40

~0

.20

10

0 I

Fig . 4. 2 Probability distributions of positive run-lengths of the first-order linear autoregressive process for q • 0.4, 0.5, 0 .6, 0 . 7

19

10

10

7.0 .------,r---------,-------r-------,.-------.-------, EN

6.0 Total Runs 0 .4

0 .3

0 .2

5.0 0 .1 0.0

0.5

4.0 1-----,P~c-~~---f----====---.... --=-----t---~=----~ 0.4

0 .3

0.2 0 . 1 0 .0

1.0 L_ _ ___l _____ _..L _____ ...l_ _____ .L.-_ ___ ____,~, ___ ____,~

0.3 0.4 0.5 0.6 0.7 q = F( u )

Fig. 4.3 Mean positive, negative, and t otal run-lengths of the first-order x

0 linear autoregressive process

20

0.03 8

0.02

0.01

0 .01

0.01

0

-0.01

0.01

-0.01

0.01

-0.01

- 0.02

2 3

q = 0.3

q = 0.5

p =0.5 -------p =0.4

p =0.3 - --- ---p =0 .2 - · - · - · ­

p =0. 1 -------------·

p = 0---- ---

, --------~--­---------

---------/___ ----

Fig. 4.4 . Differ ences between the probability mass of run-lengths obtained by the data generation method and that obtained by the analytical approximation from the truncated series .

4 5 6 7 8 9 10

21

which is also an infinite series . Both of these ser ies are convergent. However, the rate of conver­gence of the series used for calculating P(j +) and

P(k-,j+) , given by Equation (3 .33) and Equation (3 . 39) respectively, varies with q , p , k , and j The rate of convergent~ of the series given by Equa­t ion (3.40) varies also with q , p , and j . In other words, the problem of the rate of convergence of each of these two series is a complex mathematical problem in itself, and to the writer's knowledge, to dat e, it has never been solved.

In order to assess the accuracy of the computed probabilities of run- l engths, obtained in this study by using the truncated series, the data generation method (Monte Carlo method) i s used to experimentally compute t he properties of r un-lengths, and to compare them wi th the probabilities obtained from the trun­cated series . The procedure is the fol l owing .

(a) Normal random number s were generated following the fi rst-order autoregressive model: xi = pxi-l + ti , where ti are standard normal

random numbers and p is given the values 0, 0 . 1, 0.2, 0 . 3, 0. 4, and 0.5.

(b) The probability truncation level , q , was given the values 0.3, 0 .4 , 0 . 5 , 0.6 , and 0 . 7, with corresponding x

0 values equal to -0.544002,

-0.252933, 0 , 0.252933, and 0 . 524002, respectively.

(c) First a value of p was selected , then for each value of q , xi ' s are generated until

30,000 positive run-lengths were obtained. Absolute frequencies are calculated for the runs having run­lengths 1, 2, 3, .. . , and the probability mass of each run-length is estimated by computing the rela­tive frequency as the absolute frequency divided by the sample size of 30 ,000 .

(d) Then another value of p is selected, and step (c) is repeated until all values of p are used .

A comparison of the probabilit y mass of run­length, obtained from the truncated series, as an ana lytical approximation of the data generat ion method and probabilit ies obtained by the analyt ical approximation from t he truncated series are shown in Figure 4.3.

22

Chapter V

APPLICATION OF RUN-LENGTHS TO INVESTIGATION OF SERIES

5.1 Introduction. The hydrologist is concerned with two basic types of variables, namely serially inde­pendent and serially dependent variables. He is inter­ested in first testing whether they are stationary or not. If they are stationary, further tests arc used to determine the goodness of fit of various mathematical models of serial dependence. The mathematical model is a representation of the process. It reflects sta­tistical characteristics of the sequence in terms of parameters related to the physical properties of the system.

Before discussing the application of run-lengths to the investigation of series, the application of autocorrelation coefficients and the variance densities to the investigation of series is briefly reviewed. This shows an analogy, and points to the necessity of carrying out various tests.

In the case of investigation of series by auto­correlation, the parameters involved are the serial correlation coefficients, rk, as estimates of popula~

coefficients, pk . A comparison of the computed sample

correlogram rk a f(k) is made with correlograms of

theoretical models, or with the expected correlograms, provided these models are good after the model parame­ters are estimated. This comparison allows making in­ference about the mathematical structure of this dependence. For an independent series

2of size N,

Erk = -1/(N-k), var rk = (N-k-1)/(N-k) , and the con-

fidence limits at 95\ probability level for the distri­bution of r 1 are given by (r1) 1 2 • -1/N ~ 1.96

[(N 3- SN2+6)/N2 (N2-l)]~. For r~, N is replaced by

N-k+l, or by similar expressions [2). Figure 5.1 shows the expected correlogram and the 95% tol erance limits of an independent series with N • 30.

In order to test the structure of an observed series, the null hypothesis, that the observed time series is an independent sequence, is used. If the value of rk fall within the 95% tolerance interval,

this hypothesis is accepted. Or if 5% of k values of rk

are outside the limits, but 95% are inside, the hypothe­sis is accepted; otherwise it is rejected.

In the case of investigation of series by spectral analysis, the parameters involved are the spectral densities vf = ~(f), with the frequencies f. These vf

values are estimates of population spectral densities, vf. Again a comparison of the observed variance den-

sity spectrum with spectra of theoretical models permits an inference about the matheamtical structure of de­pendence in a series. For a discrete series with equal intervals 6t, in the case of time series, the maximum frequency is fmax = l/26t, and for finite series of

size N the minimum frequency is fmin = l /N6t, which is

usually extended to f = o. The spectrum has the prop­erty that the area under the variance density graph

23

represents the total variance of the variable. The expected spectrum of an independent stochastic series is a straight horizontal line between fmin and fmax'

or between o and l/26t. The distribution of tho variance density for independent variables is approximated by a x2-distribution with the number of degrees of freedom given by v a 2N/m - ~ , where m is the number of serial correlation coefficients used in computing the variance densities. For N = 30, m = 4, and 6t • 1, the number of degrees of freedom is v = 7. The maximum frequency is 0.50, assuming fmin = 0. The mean variance

density is vf • 2a2 Then vvf f ja2 is a x2 random max v variable. The 95% tolerance limits are

2 x0.025 <

vvf fmax 2

1.0

0.5

a

ll---·-·-·-· I

' ' ' 5 10

~~!===~o~'=~\~~~======~---k I

I I E(~)=-1/(N-k)

l ( ___ _ ~ --·-·-

0339

-{).5

-1.0

Fig. 5.1 Expected correlogram of an independent series, of the sample size N = 30, with tolerance limits at the 95\ level.

For v - 7 x2 1.69, and x02 _

975 = 16.0 , the 95% - ' 0.0.25

confidence limits are 0.483o2 < vf < 4.57la2. Figure

5. 2 shows the expected values and the 95% tolerance limits for vf' for independent variables.

5.2 Using Runs for Investigation of Series . In the use of investigation of series by using runs. the basic parameter selected here is the run-length, selected for the following reasons:

4.571r:r2 -------1 I I I I I I I

2.0C)Or:r2t-----+-~r-~

0.483r:r

0 .0 0.5

Fig. 5. 2 The expected values and the 95\ tolerance limits, of variance densities in. a spectrum of an independent series with the sampl e size N = 30.

(a) If a given series is cut at many truncation levels, and for each level the sequences of positive and negative run-length are obtained, theoretically it is possible to reproduce the original time series, at least at a finite number of points, by using these sequences. The larger the number of levels selected, the larger the number of points of the original time series that can be reproduced. If all sequences of positive and negative runs at all possible truncation levels are known, the whole original time series can be reproduced.

(b) Properties of run-lengths based on a proba­bility truncation level are distribution-free, both for independent variables and linearly dependent processes. This is an important property, because the results ob­tained for Gaussian processes can also be applied to other types of stationary processes.

(c) The physical significance of positive and negative run-lengths is obvious in hydrology. They can imrnedia~ely be associated with periods or durations of deficit and surplus, or with duration of droughts and floods.

(d) A parallel technique to autocorrelation analysis and analysis by the variance density spectrum can be developed for run-lengths to investigate hydro­logic series .

(e) A comparison of properties of observed run­lengths with the corresponding properties of run-lengths of population theoretical models is similar to other techniques of investigation. The mean positive run­length~+, the mean negative run-length~-. and the mean total run-length N, are attractive parameters for this comparison. Because ~ contains information of both N-and N+, this parameter~ is used i n this study as a basic parameter for investigating hydrologic time series by runs. The parameter N+ is also used as the alternative .

The technique by runs as developed in this study compares the statistic N as a function of q of ob­

q served time series with the expected value E(N ) of the

q total run-length of theoretical models, or the statistic N~ is compared with its expected value E(N~).

24

5.3 Proper ties of Run-Length for Sequence of Independent Identically Distributed Random Variables. Let Xn be a sequence of independent random variables

with a common distribution, and N. be the associated J

process of the total run, then N. is a renewal process J , ~d a: such, it is also a sequence of independent 1dent1cally distributed, random variables.

For a given q,

(5 .1)

then by the central limit theorem for large k, Nk is

asymptotically normally distributed with

EN (5. 2)

(5. 3)

Substituting Equations (2.22) and (2 . 23) into Equations (5 . 2) and (5.3) gives , for a given q and p = 1-q,

1 pq

~ ... 2 2 .. p q

(5 . 4)

(5 .5)

This result holds when k is a fixed number. In the case the series length n a being a fixed value, the number k of total runs in the interval (o,n) is a random variable, k(n), and the mean Nk(n) is

Nl + ••• + Nk(n) Frk (n) • k (n)

Feller (15) shows that the ratio k(n)/n is asymptoti­cal ly normal with the mean equal to the mean recur­rence time of the complet ion of a total run. It con­verges in probability to a positive-valued random variable. In vir tue of the central limit theorem for the sum of a random number of independent random variables (31], the result obtained for N also holds

- k for Nk(n).

Table 5.1 gives values of the mean and variance of N+, N-, and N, respectively, for a range of values of q between 0.10 and 0.90. Figure 5.3 shows a graph of EN+, EN-, and EN versus q for the independent case. It is apparent from this graph that t hese functions are symmetrical about the line q • 0 .5.

. 5.4 Run-Length Test for Stationary Independent Var~ables. Properties of "k derived in the previous

subchapter allows the construction of a test. The null hypothesis is that {Xn} is a sequence of independent,

identically distributed, variables, either for the original or "whitened!' series . Then Nk is approximately

TABLE 5 .1

PROPERTIES OF RUN-LENGTHS FOR INDEPENDENT IDENTICALLY DISTRIBUTED VARIABLES

N+ N- N q Mean Variance Mean Variance ~lean Variance

0.1 10.00 90.00 1.11 .12 ll.ll 90 .12

0.2 5 . 00 20.00 l. 25 . 31 6 . 25 20.31

0.3 3.33 7.78 1.43 .61 4. 77 8.39

0. 4 2. 50 3. 75 1.67 1.11 4.17 4.86

0.5 2. 00 2.00 2.00 2.00 4.00 4 .00

15.0'~----------------------------------~

£ 10.0 01 c:: Q)

-l I c:: :l cr

5 5.0 Q)

~

+ EN = I /p EN = I /q

0 0.2 0.4 0.6 1.0 Probability Level q

Fig. 5.3 Mean run-length for independent variables, with a common distribution) versus q.

normally distributed for large k with the mean and the variance given by Equations (5.4) and (5.5) . At the 1-a t olerance l evel, the region of acceptance of the hypothesis is, f or a two-sided test,

or

1 pq

t a/2 3 3 ' 1 t r:./2 (p3+kq3)~ pq <9-) 'S !. R'k pq + pq

Now, for a median as the truncation l evel, or for p = q = 0. 50,

EN 4

var N 4, and var Nk = I The 95% t olerance limits , with a =.OS and t a/2 1. 96 , are

(S. 6)

(5. 7)

(S .8)

(5 .9)

25

4 _ 3.92 < Nk ~ 4 + 3.92 lk - lk

(5 .10)

If Nk falls outsi de the limits of Equation (5. 10) the

hypothesis is rejected. The test is illustrated by Figure 5.4 for the case of the tolerance l evel 1-a ~ 1.95 and the truncation level being the median , q = 0.50.

For a truncation l evel q ~ 1/2, the tolerance limi ts are different than those given in Figure 5 .4, as shown in Figure 5 .5. For any value of q , the r ight-sided run-length test, for the 95\ tolerance limit, is

(5 .11)

Region of Acceptance of Null Hypothesis

1· I

3.92/IK 3.92/IK

Fig. 5.4 Two-sided run-length t est for the truncation level of the median, q = 0.50, for i ndependent variables.

I I I I I I \ \

' \ \ ' ' ',

', ....... _________ _ ,

_ ____ , ,'"

I I

I I

I I

I I

I I

'

0 0.1 0 .2 0 .3 OA 0 .5 0.6 0.7 0.8 0.9 1.0 q

Fig. 5.5 Tol erance region for Nk' of an observed

independent time series.

The left-sided run-length test, for the 95% tolerance limit, is

1 -+ pq

(5.12)

Figure 5.6 shows a graph with EN and the 95% tolerance limits of N(q) for 0.2 ~ q ~0 . 8 and k = 10, 15, 20 , 25 , 30, 35. The q values are given as abscissas, and the N(q) are given a.s ordinates in the upper graph. The k values are given by the ordinates downwards versus q in the lower graph. The upper graph shows a family of curves. The central curve is EN(q) = 1/pq . The upper and lower sets of curves are the 95% tolerance limits of N(q). This is a convenient plotting graph that can be used readi l y for the analysis of t ime series by the mean total run­length N(q) = f(q) . The sample function N(q) is calculated from the observed time series and plotted i n the upper graph. The sample function of k is plotted i n the l ower graph. Finally, by using the upper and lower families of curves of the upper graph and the k values of the lower graph, the upper and lower 95% tolerance limits of N(q) are drawn i n the upper graph . If the whole sample function is confined inside the tolerance region, the analyzed series is considered as not being significantly different from an i ndependent series at the 95% l evel .

5.5 Two-Levels Run-Length Test for Stationary Independent Variables . An alternative s tatist ic to

N(q) is also considered in this study, and is defined as

(5 . 13)

For k+ + k

k * = ;.;_....,2~- (5 .14)

rewriting Equation (5.13) in the form

- l[k+ _+ k--- ] N*(q) = 2 F N (q) + k* N (p) (5 . 15)

and taking into account Equation (2.22) and Equation (4.13), the expected value of Equation (5 .13) becomes

l [k+_+ k-- ] 1 EN*(q) = 2 FEN (q) + k* EW(p) = q (5.16)

Figure 5.3 shows EN*(q) as a function of q for independent variables. The random variables Ni(q)

and N~(p) are independent f or q ~ 0 . 5 . Further-J

more, it is assumed here that they are a l so indepen-dent for 0.2 ~ q ~ 0.5 . Equation (5.17) gives

var N* (q)

However,

+ var N'+ (q) var N ~g) ___E_

k+ k+q2 (5 . 18)

and

var N- (p) var N- 1E~ ___E_

k k-q2 (5 .19)

since {N+} and {N-} are sequences of independent variables with variances given by Equation (2.23) . Substituting Equation (5 . 20) and Equation (5.21) into Equation (5.19) gives

26

var N* (q)

Final ly, taki ng into account Equation (5 . 16) ,

var N*(q) = ___..E_ 2k*q2

(5.20)

(5 .21)

Since the central l imit theorem applies under rather broad conditions, it is evident that it al so applies in the case of Equation (5.17), so that N*(q) is approximately normally distributed for large values of k* For a two- sided test, the 1-~ tolerance region for independent variables is

~ (1- t 012 ~)~N* ~~ (1 + t 012 ~) ·

(S . 22)

Figure 5. 7 shows a practical graph that can be used readily for the analysis of time ser ies by using N*(p) , with p = 1 - q . This graph is constructed and used in a manner similar t o the gr~h of Fig. 5.6 . The di fference is that the statistic N*(q) is used instead of N(q) .

The parameter N*(p) is used inst ead of N~(q) only with the purpose of having the curve N(p) =f(p) increasing from the left t o the right, instead of from the right t o the l eft which would be the case for N* (q) = f(q) .

9.0 1\

1\

8.0 1\ 1\

L\\1\

~\ i\ ~\ , \ 1\

7.0

1\\

~

6.0

1\

5.0

4.0

3.0

2.0 0.2

Fig . 5 .6.

k

10 lJ

15

20 25 30

lll llr.JJ. 35 lL (J_

IL 1/

' v w

1\ 1\ 1\ v N

r,r,

1/,

"" I'/ ~t\:

1'\: ~t'. 1'- ~ 1-- I':

r--. ~

" ,.... v

35 LL 30

25 k 20

r-- v 15 "' 11":

v r'v 10 ..... ,....

...... 1--

1- ......

q :: F ( X0

)

0 .3 0.4 0 .5 0 .6 0 .7 0.8

Gr aph paper for ~lotting the mean total run-length, N(q) . The upper graph serves for plotting N(q) versus q, the truncation level, as well as its tolerance limits at the 957. level versus q. The l ower graph serves for plotting the number of total run-lengths k versus q, needed for computing the tolerance limits of the upper graph.

27

0

10

20

30

k

- · N 7.0

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5 ...... -1.0

0.2

Fig . s. 7 .

I

'

r7

[7 w 1/ [fi

I/ I/ ri.

f? 1/

7-: I/ /~

1/ ~ ~~ 17

~~ ~ ~ ~ ~~

v v ~~ ..<:~ / ~ ~

...... ~ ~~ 1.--

1-- _....

""" 1-'

p = I - F ( x0 )

0.3 0.4 0.5 0.6 0.7

Y!. rli

r li r!.

II

17

w ~

[6

7

1/1/ 'h TlJ Ill

' I 'II

rt1 r11

1/

1/

1/

1/. r7J '}

~rJ 7

k'

10

15 20 25 30 35

35 30 25 20 15

[71 0

/ I/

0

10

20

30

Graph paper for plotting t he mean run-length N*, as defined in the text. The upper graph serves for plotting N*(p) ve~sus p • 1 - q , as well as its tolerance limits at the 95% level versus p. The lower graph serves for plotting the number k*, as defined in the t ext , ver sus p, needed for computing the tolerance limits of N*(p) of the upper graph .

28

Chapter VI

E~~PLES OF INVESTIGATION OF STATIONARY HYDROLOGI C SERIES BY RUN-LENGTHS

6 . 1 Introduction. There is a two-fold application of runs to stationary hydrolog~c series. First, runs are used to investigate the structure of a series by testing whether or not a particular series is a sequence of independent random variables with a common distribution. If this is not the case, then the linear autoregressive models are assumed, the in­dependent component in the original series is computed by using these models, and tests are performed to determine whether this component is a sequence of independent variables with a common distribution .

Three approaches are used for the investigation in this test:

1. Only the median truncation with q = 0.50 and the corresponding value N(0.50) are used. This case is analogous to using only the first serial correlation coefficient, r 1 , in the autocorrelatiqn

investigation of whether a series is independent or dependent .

2. The total run parameter, N(q) for various values of q , and a curve is obtained.

.Lis used, N (q) =f(q)

3. of p . to have

The parameter N* (p) is used as a funct ion N*(p) is used instead of N*(q) in order

an increasing function as p increases.

Examples are given for each of these three approaches for investigating hydrologic series by run-lengths, using the supposed stationary time series of annual precipitation and annual river flow .

The second application of runs to stati onary hydrologic series is for the prediction of durations of periods of surpluses and deficits for a variable . Once the structure of the series is known, i. e. , whether it is stochastically independent or dependent of the first-order, or the second-order linear auto­regressive models , or similar models, the derived properties of runs are used to make probability statements about durations of periods of surpluses and deficits. The truncation level in each case is specified by the probability q .

Chapter VI is concerned with f irst application, the investigation of series . Chapter VII treats the second application, the prediction of duration of surpluses and deficits (run-lengths) of stationary hydrologic series.

6.2 Application to Investigation of Annual Precipitation Seri es . Computed values of annual precipitation have random errors, systematic errors (inconsistency), and nonhomogeneity . Inconsistency is caused primarily by changes in instruments, methods of measurements, and so forth. Inconsistency comes mainly from two sources : moving a precipitation station a substantial distance, and changes in the environment around a station, such as growth of trees, building of houses, or any other major environmental change which affects the flow pattern of air around the station . Nonhomogeneity comes mainly from cloud

29

seeding operations. Precipitation data must be considered, therefore, as often having relatively large random errors and inconsistency in their annual series [2, 27] .

The presence of inconsistency and/or non­homogeneity in an observed series i mplies that it does not come from a sequence of variables with a common distribution. When applying the run-length t est to an observed precipitation series wi th incon­sistency and/or nonhomogeneity in data, this null hypothesis may be rejected, though the series without these factors present may show the acceptance of a null hypothesis.

The presence of inconsistency and/or non­homogeneity in data is reflected in the run-length test when N is greater than the expected value, EN = 4, for q = 0.5, and may be outside the 95% tolerance limits of the distribution of N . It is therefore necessary to first remove inconsistency and/or nonhomogeneity in data .

On the other hand, if no significant inconsis­tency and/or nonhornogeneity are present in the data, and the observed time series is stochastically independent, the null hypothesis is accepted in applying the run-length test. In this case N is inside the region of acceptance of the null hypothe­sis at a given level.

6.3 Examples of Investigation of Annual Precipitation Series by the Mean Run-Length of the Median. To sho"'' the method of run-lengths for the investigation of whether annual precipitation series are independent, identical ly distributed variabl es (null hypothesis) or not, five series are selected from around the United States. The appli cation of run-length is developed on the assumption that the number of total runs is large . As a consequence, the observed series should be long enough to satisfy this assumption. For this reason, long precipitation series were selected .

The five annual precipitation series are:

1. Chico Experimental Station, California, for the period 1871-1965, N = 95;

2. Ord, Nebraska, 1896-1965, N = 70;

3. Natural Bridge N.M., Arizona, 1890-1960, N 71;

4. Antioch F. Mills, California, 1879-1965, N 87 years, and

5. Ravenna, Nebraska, 1878-1965, N = 88 years.

Table 6 . ] gives the sequence j = 1, 2, . . . of runs, with run-lengths of Nj and Nk for these

five series and for q 0.50. The incompl ete first and last runs are included. They introduce a small bias, but their effects may be neglected for series with N > 70 .

TABLE 6.1 RUNS OF FIVE ANNUAL PRECIPITATION SERIES

Chico Ord Natural Bridge Antioch F. Mills Ravenna Station Station Station Station Station

+ + j N~

J N.

J N. N~

+ N. J J J

1 1 1 2 2 1 2 1 1 4 1 5 3 3 2 2 1 1 4 9 1 1 4 8 5 2 5 1 1 2 6 2 1 1 4 2 7 2 1 5 1 2 8 1 4 1 2 2 9 1 1 2 4 2

10 3 4 1 6 3 11 3 1 1 1 1 12 1 1 1 1 2 13 3 2 6 1 2 14 7 4 1 3 1 15 l 6 1 2 16 2 1 1 1 17 1 3 1 2 18 1 2 19 1 2 20 1 1 21 1 2 22 23 24

r 47 46 32 37 34

Table 6.2 gives the fol lowing parameters of these

N'+, N-, N = N'+ + N- , and the two tolerance limi ts on t he 95\ level about N These limits are computed by

2t N 2 a EN + ~

1, - II<

where EN = 4 f or q = 0.50 cal ly distributed variables , confidence probabi lity level,

(6 . 1)

of independent, identi­~/2 is the one-tail t~12 is the normal

standard variate for the given ~/2 , and k is the largest number j of run-lengths in Table 6.1, or k = jmax . Using the 95% tolerance level and the

two-sided test, o/2 • 0.025 , then t~12 = 1.96.

N~ J

N: J

N~ J

N: J

+ N. J

14 1 1 1 1 1 1 3 1 3 1 6 4 1 5 1 1 1 1 2 2 1 2 3 2 2 1 1 1 1 3 1 2 1 6 3 2 1 1 1 2 1 1 4 1 2 1 1 1 1 1 3 2 2 3 1 1 1 2 1 1 2 2 2 1 1 2 3 1 1

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

35 44 41 42 43

In the above case, the tolerance limits are

Nl 2 • 4 .00 + 3.92 ' - II<

(6. 2)

Table 6.2 shows that all five computed N-values of the precipitation series are confined within the tolerance limits N1 and N2 .

For a stronger test, with a = 80% level , and t a/2 = 1.28, all computed N-values of five stations

except Natural Bridge, which has a nonhomogeneous series, are still within these new limits, as shown in the last two columns of Table 6.2 .

TABLE 6 .2 PROPERTIES OF RUN-LENGTHS OF THE FIVE ANNUAL PRECIPITATION SERIES

a = 95\ a = 80%

Station if> N'- N k N1 (95%) N'2 (95\) N'1 (80%) N' (80~•) 2

Chico 2.190 2. 238 4 . 428 21 4.858 3 .142 4.560 3 . 44'0

Ord 1.882 2 . 176 4 . 058 16 4.980 3.020 4.640 3 .360 Natural

Bridge 2.429 2 . 500 4 .929 14 5.050 2.950 4.685 3. 315 Antioch

F. Mills 1. 833 1. 708 3. 541 24 4.800 3.200 <\. 522 3. 478

Ravenna 1 . 826 1. 870 3.696 23 4 . 820 3. 180 4.535 3. 465

30

It can be concluded that four out of the five annual precipitation series are independent variables, as it relates to the use of total run-length param­eter, N , for the median truncation level with q = 0 . 50, as the invest igation parameter.

6.4 Application to Investigation of Annual River Flows Series . River flow essentially integrates the precipitation received over large areas, but it also includes the effects of evaporation and storage as other important physical factors. The water carry­over from year to year, and especially the change in this carryover from one year to another, usually introduces time dependence into the sequence of annual flows .

To investigate the annual flow series either the original or the whitened series is used. By applying the same procedure to the original annual river flow series, as for the annual precipitation series, the hypothesis that the series is independent is accepted or rejected . If the hypothesis is rejected, the dependence models are assumed and the series is whitened. Then the same procedure used for the annual precipitation series is applied t o the whitened series of annual river flow . If the hypothesis of the whitened series being independent is accepted, the postulated dependence model is al so accepted. Equa­tions (6.1) and (6.2) are used for these investiga­tions and tests just as they are used for the annual precipitation series.

6.5 Examples of Investigation of Annual River Flows Series by the Mean Run-Length of t he ~1edian . Investigations and tests are limited to long series for the same reason they are limited in the case of annual precipitation series of annual river f l ows with long records [1} were selected as examples. They are:

1. The Mississippi River at Saint Louis, Missouri 1862-1957, N = 96;

2. The St. Lawrence River at Ogdensburg, New York, 1861-1957, N = 97;

3 . 1957,

The Mississippi River at Keokuk, Io"•a, 1879-N = 79;

4 . The Gota River at Sjotorp, Vanersburg, Sweden, 1808-1957, N = 150, and

5 . 1957, N

The Rhine River at Basle, Switzerland, 1808-150 .

Table 6.3 shows the results of this investiga­tion. The analysis of five original series of annual river flows shows that only the Rhine River at Basle, Switzerland is an independent time series, or rather the null hypothesis is accepted for the 95% tolerance l evel. The whitened series are obtained by the hypoth­esis of t he first-order, linear, autoregressive model of dependence, or

Ei = xi - r l xi-1 (6 . 3) where xi and xi- l are elements of a standardized

series of annual river flows and r 1 is the first

serial correlation coefficient, also given in Table 6.3 .

Table 6.3 shows the results of the analysis of whitened series of the first four rjvers by using the mean run-length of the median, or Equations (6.1) and (6.2). Al l four whitened series are shown to be independent series, or the null hypothesis for the whitened series is accepted for the 95% tolerance lev.el. This finding means also that the hypothesis

31

of the first-order linear autoregressive model des­cribing well the series dependence is accepted for all four rivers .

6.6 Examples of Investigation of Annual Precipitation Series by the Relation of Mean Run­Length to Values of q . In the case of using any value of q as the truncation leve l, the 1-o tolerance limits for a two-sided test are given by

(6.4)

By using the 95% tolerance level, 0 . 05 , t 012 = t 0 .025 = 1.96 , the limits are

N' = _!_ [1 + ~ Cp3+q3)~] 1,2 pq - lk (6. 5)

Table 6.4 gives values of 1/pq and (p3+q 3)~ for values of q ranging between 0.20 and 0 . 80 with the increment of 0.05. Table 6.5 gives t he tolerance limits of N at the 95% level.

The mean and the 95% tolerance limits of N are shown in Fig. 6.1 and 6 . 2 for values of k s 10 , 15, 20, 25, 30, and 35, and q - 0.20 to q • 0 .80 wi th in­crement of 0.05. Th~ examples of annual precipitation series are analyzed by computing N(q) for_values of q = 0.3, 0.4, 0.5,10.6, 0.7, and plotting N(q) against q, as shown in Figure 6.1. For e!ch q and the corres­ponding k, the~olerance limits , N1 2 are computed and plotted on the same graph .. ' '

If the analyzed series is a sequence of independent, identically distributed variables, the sample func­tion N(q) should be inside the tolerance region. As can be seen from Figure 6.1 , four out of the five series of the above examp!'es have N(q) inside the tolerance region, but one series has only a point of N(q) outside the tolerance region. This particu­lar series is a nonhomogeneous precipitation series .

6.7 Examples of Investigation of Annual Runoff Series by the Relation of Mean Run-Length to Values ~· The same procedure used for the precipitation ser1es is appl ied ·to investigating the examples of the original series of annual runoff. The results are given in Figure 6.3 and 6.4 . . ~tis apparent from this figur e that .at least three out of the five series are not independent, i dentically distributed random variables, since their N(q) are not completely inside the tolerance region. The Mississippi River at St. Louis seems to be a case of weak serial depen­dence and/or of some nonstationarity, since its N(q) is so close to the upper tolerance limit. Finally, the annual runoff of the Rhine River is accepted as a sequence of independent, identically distributed variables. This result agrees with the autocorrelation analysis made on this series in a previous study [2] .

For the four other series where the hypothesis that the series are independent, identically dis­tributed variables is rejected, the first-order autoregressive model is assumed, and consequently the series are whitened. Then, the same procedure is used with both the whitened and the original series. The results are shown in Figure 6.5 . The first-order autoregressive model for annual series is accepted for these four series .

;;

9.0

ii

9.0

a.o

7.0

6.0

5.0

4.0

3.0

Fig. 6.1.

10 90 10

15

20 25 30 ~

6.0

5.0 3~ 30 25 20

15 15

10

qaFt ~.)

0 2!l 0 03 0.4 05 0.6 0.7 08 02 03 0.4 0 .5 0.6 07 0.8

4 10 A 10

zo 20

30 3()

ii

10 10

15

zo 20 25 25 30 35

30 ~

~ ~ 30

25 25 20 20

15 15

10

0 0

10 10

20 20

30 30

Investigation of time series independence by the mean total run-length, N(q), as a function of the probability, q, of the truncation level, for four annual precipitation series: (I) Chico Experimental Station, California; (II) Ord, Nebraska; (III) Antioch F. Mills, California, and (IV) Ravenna, Nebraska; and for each st~tion : (1) the expected mean total run-length of inde­pendent series EN(q) = 1/q(l-q); (2) the observed N(q) ~ f(q); (3) the 957. tolerance limits for given k as function of q, and (4) the number of total run-lengths, k .

32

TABLE 6.3 PROPERTIES OF RUN-LENGTHS OF THE FIVE ANNUAL RIVER FLOW SERIES

Original Series, x. l.

Whitened Series, xi-rlxi-1

Station N+ N- N k N1 (95\) N2(95\) r1 N+ w N k N"1 (95%) N"2 (95\ )

l. Mississippi River 2 .389 2.500 4.889 18 4. 775

2 . St . Lawrence 3. 583 3.917 7 . 500 12 4.950 River

~- Mississippi River 2 . 500 2 . 857 5 . 357 14 4.878

~- Gota River 2.704 2. 778 5. 482 27 4.634

~- Rhine River 2.027 2 .027 4 .054 37 4.540

TABLE 6 . 4 . Values of 1/pq and (p3 + q3) 1/2

p 1 (p3+q3)~ q pq

.20 .80 6.25 . 7211

.25 .75 5 . 33 .6614

.30 .70 4.76 .6083

.35 .65 4 .40 . 5635

.40 .60 4.17 .5292

.45 . 55 4.04 . 5074

.so .so 4.00 . 5000

.55 .45 4.04 . 5074

.60 . 40 4.17 .5292

.65 . 35 4.40 . 5635

.70 . 30 4.76 .6083

.75 . 25 5.33 .6614

.80 . 20 6 . 25 . 7211

Fig. 6. 2. Invest igation of time independence by the mean total run-length , N(q), as a function of q, for the non- homogeneous annual precipitation series at Natural Bridge , N.M., Arizona ; (1) the expected mean Eotal run-length EN(q) • 1/q(l- q); (2) the observed N(q) s f{q) ; (3) the 95% tolerance limits for given k and (4) the number of total run-lengths , k.

33

3.225

3. 050

3 . 122

3.365

3.450

0 . 301 l. 783 2.130 3.913 23 4.816

. 707 2 .045 2. 136 4.181 22 4.835

0 . 407 1. 714 l. 619 3.333 21 4 . 854

0.445 2.202 2.212 4.054 33 4 .682

--- --- --- --- -- ---

TABLE 6 .5 TOLERANCE LIMITS OF N AT THE 95% LEVEL

~ 0.20 0.25 0.30 0 . 35 0.40 0.45

10 9.04 7.51 6.55 5.91 5.54 5 . 32 3.46 3.15 2.97 2.89 2.80 2. 76

15 8.53 7.11 6.21 5.66 5.29 5.08 3.97 3.55 3.31 3.14 3.05 3.00

20 8.22 {i.88 6.02 5.48 5.14 4.95 4.28 3.78 3.50 3.32 3 . 20 3.13

25 8 . 05 6.71 5.88 5.38 5 . 04 4.85 4.45 3 . 95 3. 64 3.42 3.30 3. 23

30 7.85 6.58 5.78 5.29 4. 96 4. 78 4 . 65 4.08 3.74 3. 51 3.38 3. 30

35 7.75 6 . 50 5. 72 5.22 4.91 4. 72 4.75 4.16 3.80 3.58 3. 43 3. 36

3 .184

3 . 165

3.146 . I

3.317 I ---

0.50

5 . 24 2.76

5. (}2 2.98

4 . 87 3.13

4.78 3 . 22

4. 71 3.29

4 .66 3.34

10

Fig. 6. 3. Investigation of time independence by the mean total run-length , N(q), as a f unction of q , for the annual r unoff series of the Rhine River at Sasle, Switzerland: (1) the expected mean total run-length, EN(q) • 1/q(l-q); (2) the observed N(q) • f(q); (3) the 95% tolerance limits for given k, and (4) t he num­ber of total run-lengths, k .

N

Fig. ' 6. 5.

0 03 04 ~ 06 07 0

~ ~

4 w w

~ ~

N

~ 90 ~

~ ~

w w u ~ ~

~ ~

ro

60

~ ~ w ~

~ ~

Investigation of independence of whitened series, under the assumption of the first-order autoregressive process as a t ime dependence model, by using the mean total run-length , N(q), as a function of the probability, q, of the truneation level, for four annual runoff series: (XI) Mississippi River at St. Louis, Missouri; (Xli) St. Lawrence River at Ogdensburg, New York; (XIII) Mississippi River at Keokuk, Iowa, and (XIV) Gota River at Sjotorp, Vauersburg; and for each station: (1) the expected mean total run-length of independent series, EN(q) • 1/q(l-q); (2) the observed N(q) • f(q) of whitened series; {3) the 95% tolerance limits for given k as function q , and (4) the number of total run-lengths, k of whitened series .

35

7.0 'N '

6.0

10

,. 7.0 iii '

,.

l5

"' 20 20 I!

10

Fig. 6. 6. Invest igation of time series independence by the mean run-l ength , N(q), as a function of the prob­ability , q, of the truncation level, for four annual precipitation series : (XV) Chico Experimental Station, Californi a; (XVI) Ord , Nebraska; (XVII) Antioch F. Mills, California and (XVIII) Ravenna, Nebraska; and for each station : (1) the expect ed mean run-l en%th, EN*(q) - 1/q; (2) the observed N*(q) • f(q ); (3) the 95% tolerance limits for given k* = (k+ + K- )/2.

36

6.8 Examples of Investigation of Annual Precipitation and Runoff Series by N*(p) for Values ~ . The 1-a tolerance region for independent identically distributed variables is given by Equa­t ion (5.22). Using the 95% tolerance level, a= 0.05, t a/2 = t 0 _025 1.96 , the tolerance limits are

(6.6)

Tab l e 6.6 gives the mean and the tolerance l imits of N* at the 95% level.

The sampl~ of precipitation series was analyzed by computing N*(p) for values of q = 0.3, 0.4, 0.5, 0.6, 0 . 7 and plotting N*(p) against p as shown in Fi gures 6. 6 and 6. 7. The resul~s ·are shown in these figures.

In a similar manner the sample of · the original runoff series was analyzed. The results are shown in Figures 6.8 and 6.9. r1na11y, the whitened runoffseries using a first-order autoregressive model were also analyzed in a simi l ar manner. The resu1 t .s are shown in Figure 6.6. From these figures 1t is apparent that all precipit ation ser1es are accepted as independent time ~~ries,whereas all runoff series except Rhine Kiver are accepted as first-order autoreRressive pro cesses .

- · 7.0 N

6.0

5.0

4.0

3.0

0.3

XIX

0.4 05

k'

10

15 20 25 30 35

35 30 25 20 15

Fig. 6. 7 •. Invest!gation of time independence by the mean run-length, N*(q), as a function of q, for the non-homogeneous annual precipitation 3eries at (XIX) Natural Bridse, N. M. , Arizona: (l) the expected mean run-length EN*(q) • 1/q; (2) the observed N*(q) ~ f(q); (3) the 95% tolerance limits for given k* as function of q, and (4) k* • (k+ + k-)/2

37

TABLE 6.6 MEfu~ AND 95% TOLERANCE LIMITS OF N*

k* 10 15

p EN* --- ---

0.20 2.500 2.696 2 .660 --- 2.304 2 . 340

0.25 2 .666 2.885 2 .845 --- 2.447 2 . 487

0.30 2.858 3.098 3.054 --- 2.618 2.662

0.35 3 . 076 3.335 3.288 --- 2.817 2 . 864

0 . 40 3.334 3.611 3.560 --- 3.057 3.108

0.45 3.636 " 3 .930 3 . 876 --- 3.342 3.396

0.50 4.000 4.310 4 . 253 --- 3 .690 3.747

0.55 4.444 4.769 4.614 --- 4 . 119 4.274

0.60 5.000 5.340 5.277 --- 4.660 4. 723

0.65 5.714 6.067 6.003 --- 5.361 5.425

0. 70 6.667 7.034 6.966 --- 6 . 300 6 . 368

0.75 8.000 8 . 380 8 . 310 --- 7.620 7.690

0.80 10.000 10 . 392 10.320 --- 9.608 9.680

1.o N.

20 25

--- ---

2.639 2. 624 2.361 2 .376

2.821 2 . 805 2.511 2 . 527

3.028 3.010 2.688 2 . 706

3.259 3.240 2.893 2.912

3.513 3.509 3.155 3.159

3. 844 3.822 3.428 3.450

4 . 219 4 . 196 3.781 3.804

4. 674 4.650 4.214 4 . 238

5.240 5 .215 4 . 760 4.785

5 .964 5.938 5.464 5.490

6.926 6.899 6.408 6.435

8.268 8.240 7. 722 7.760

10.277 10 . 248 9. 723 9 . 752

4

30

---

2.613 2.387

2.793 2.539

2.997 2. 719

3.226 2.926

3.494 3.174

3. 806 3.466

4.179 3 .821

4. 632 4 . 256

5.196 4 . 804

5.918 5.510

6.879 6. 455

8.219 7.781

10 . 226 9. 774

.. 10

10

:>5

- - -

2.605 2.395

2 . 783 2.549

2.986 2.730

3.215 2 .937

3.482 3.186

3 . 793 3.479

4 . 166 3.834

4.618 4.270

5.182 4. 818

5.903 5.525

6.863 6.471

8.203 7.797

10. 210 9.790

Fig. 6.8. Investigation of time independence by the mean run-length. N*(q), as a function of q, for the annual run-off series of (XX) Rhine River at Basle, Switzerland: (1) the expected mean run-length, EN*(q) • 1/q; (2) the observed N*(q) • f (q); (3) the 95% tolerance limits for given k* as funct ion of q, and (4) k* • (k+ + k-)/2

7.0 N" ' ,.

1.0 N" '

5. ~-

35 30 25 20

0.3 04 05 0.6 07 0

4 10

@)

N" ' ,. N" ' 7.0

10 7.0

l 15 I ao

25 30 35

Fig . 6. 9 . Investigation of time series independence by the mean run-length , N*(q), as a function of the probability, q , of the truncation leve l, for four annual runoff series : (XXI) Mississippi River at St. Louis, Missouri ; (XXII) St . Lawrence River at Ogdensburg , New York; (XXIII) Mississippi River at Keskuk, Iowa, and

I I I

,. 10

15 20 as so )5

,. 30 25 2.0 I~

10

0

10

20

30

40 k'

,. 10

15 ao as 30 35

10

•o k'

(XXIV) Gota River at Sjotorp, Vanersburg ; and for each station : (1) the expected mean run-length of independent series EN*(a) = 1/q; (2) the observed N*(q) = f (q): (3) the 95% tolerance limits for given k* as a function of q , and(4) k* • (k+ + k-)/2

38

7.0

6.0

~.o

4.0

7.0

6.0

5.0

4 .0

3 .0

z.o

'N ,. 7.0 N

10

30

m "

·~ Yl 25 2C IS

z.

0.3 0.4 0~ 06 07 0 o.• 05 06 0 7

4 10

20

'30

•o ..

'N " ,. 7.0 iii "

10

/ IS l 20

2~ 30 6.0

" s.

4.0

3.0

z.o

03 Q,4 0~ 06 07 0 06 07

Fig. 6.10. Investigation of independence of whitened series, under the assumption of t he firs t-order auto:*~ressive process as a time dependence model, by using the mean run-length N (q), as a function of the probability, q, of the truncation level, for four annual runoff series: (XXV) Mississippi River at St . . Louis, Missouri; (XXVI) St . Lawrence River at Ogdensburg , New York; (XxVII) . Mississippi River at Keokuk, Iowa, and (XXVIII) Gota River at Sjotorp, Vanersburg; and for each s tation: (1) the expected mean run-length of independent series , EN*(q) = 1/q; (2) t he observerl N*(q); (3) the 95% tolerance limits for given k as function of q, and (4) k* ~ (k+ + k-)/2

39

' 10

,.

M >0 2S 20 I~

10

0

30

40 ..

,. 10

I ~

20 2S 30 ,

,. "" 2S 20

/I" 10

0

10

40

k"

Chapter VII

EXANPLES OF COMPUTATION OF PROBABILITIES OF RUN- LENGTHS

7.1 Introduction . Present empirical techniques for determining probabilities of drought or low-flow durati ons , or probabilities of water surplus dura­tions, for a well defined drought or water surplus level use sample data to derive the necessary i nfor­mation. The empirical procedure is as follows . Drought or surplus level is first defined; then a series of t he hydrol ogic variable is plotted, with this l evel as t he truncation level. Next, all dura­tions as run- lengths equal to or greater t han the truncation level are counted, and the relative fre­quency of run-lengths that are greater than the critical duration are computed. These frequencies are estimates of probabilities. Alternatively, the longest drought or water surplus duration is selected as the design drought or the design surplus. It is often difficult in practice to assign meaningful probabilities to a drought or a surplus because of large sampling fluctuations of these rel ative fre­quencies. Much confusion is often unavoidable in assigning probabilities to results empirically obtained.

It is not surprising then that the est imat es of probabilities of historical droughts in some river basins or regions sometimes vary from 1:100 (one in a hundred years) to 1:3,000 (one i n t hree thousand years) by di fferent empirical approaches .

The method of using the run- length properties of the independent or of the f irst-order linear autoregressive series helps solve these important practical problems and also helps to avoid some con­fusion . Two aspects of these probabilities current ly are of interest, probabilities of a given duration (say, probability of a 5-year drought), and proba­bi l i ties of al l events equal to or greater than a given duration (say, probabi l ity of al l droughts of 3-years or more). The purpose of t he t echniques studied and developed in t his paper are designed to solve these types of problems. In this study, how­ever, only two cases are analytical ly approached: independent s t ationary time series , and dependent stationary time series of the first-order l inear autoregressive model. For more complex models, such as the second-order , third-order, or higher-order linear autoregressive mode l s , or for models with periodicity present in some paramet ers of a t ime series , t he Monte Carl o simulat ion technique seems best suited at present as the alternative to t he analytical met hod.

7 . 2 Det erminat i on of Run- Length Probabilities of St at i onary and Independent Series . The annual seri es of precipi tation and runoff of the following four gauging stations , found to be i ndependent time series in Chapter VI , are used as examples for det er­mining probabilities of run- l engths equal to or greater than a given length, for a given truncation level and its probabil ity , q :

40

A. Precipitation Stations

1. Ord, Nebraska

2. Ravenna, Nebraska

3. Antioch F. ~li lls , California

B. Runoff Station

1. The Rhine River at Basle, Switzerland .

The probabi l ities of run- lengths being not less than a given value j are obtained as follows from Equa­tion (2 . 21),

P(N+ = k)

Hence,

k-1 qp

j-1 P(N+=k) 1 - I

k=l

1 - (1 -pj - 1 ~ q 1-p

j -1 k- 1 1 - I qp k=l

j-1 p (7 . 1)

By using Equation (7 .1) for probabilities of run- lengths of independent series , which are functions of q , or p = 1 - q , comparison is made between the relative frequencies of r un- l engths, empirically determined, and t he probabilities of the same r un­lengths, analytically determined.

Figures 7. 1 t hrough 7.4 gi ve these comparisons for the three annual precipitation series and th·e annual runoff series of t he Rhine River, and for the run- lengths Nj , with the following j values :

2, 3, 4, 5 , 6 , 7, 8, 9 , and 10, and in each case for f ive values of q , 0 . 3, 0 . 4, 0.5, 0.6, and 0. 7.

As expected, probabilities of r un-lengt hs P(N ~ j) determined analytically by Equat i on (7.1) depart from the relative frequencies empirical ly determined from samp l e data . These deviations i n­crease both with an increase of j and an increase of absolute departure of q values from q = 0.50. The unreliability of t hese relative frequencies of run- lengths N ~ j empiri cally determined for extremes of q and for l arge values of j is the primary reason for the cont roversy between t he vari ous empirical methods of estimat ing probabilities of droughts or water surp luses of given durations for t he prescribed levels of droughts or surpluses . These high sampling errors, associ ated with empirical methods current ly used in practice, justify using the analytical method for computing probabilities P(N ~ j) ins t ead of using various empirical methods .

1.0

.80

\

\ \ \ \ \ \

.70 \

.60

.50 p

.40

.30

.20

. 10

0 1.0

1.00

\ \ \ \ \ \ I \ \ \ \ \ \

\ I \ \ \ I \ \ I '-----7--~ ------------, -\ ' ... ,

P(N.:!:j ; q•03 '- ''-....

2 .0 3.0 4 .0 5.0

. ' ' ' ., '• ·- ·- ·- ·- ·-·-·...... '

6.0 7.0 8 .0

'· '· 9-0 10.0

3.0 4.0 5.0 6,0 7.0 8-0 9.0 10.0

1.00

.40

. 30

.20

l

"\,, P(N"~j;q=0.3l

;\:'·---,., _______ " P(N"~j·,q•O. -n-: --,, ' ·,,

' '-,

.10

0 7.0 8.0 9 .0 1.0 2 .0 3.0 4 .0 5.0 60 10.0

41

1.00

10.0

1.00

8.0 9.0 10.0

Fig. 7.1 Estimated probabilities by sampl e relative frequencies (dashed l ines) of positive run-lengths, P(N+ ~ j ; q) , and negative run-lengths, P(N- ~ j ; q) , for the annual precipitation series at Ord, Nebraska (1896-1965) as compared with expected probabilities (solid lines) of positive and negative run-lengths of independent series for five truncation values: q = 0.3, 0.4, 0 .5, 0.6, and 0 . 7 .

1.00

.80

.70

.60

.50 p

.40

.30

.20

.10

I

' ' \ •

\~. \~ ' \ ' ' ' ' ' ' ' I

'""' \ '-\. "'~~~~7·<:j, q=0.31 ' ~. ''---~ P(N-<:1; q =0 .7) \ ,,

\__ ', ................. .....

'-::--+::---f-=--~-~,.--'-_.;.:""·-·--'-·--'--.::..0 2.0 3.0 4.0 5.0 6.0 7.0 8 .0 9.0 I 00

o. 1.0

1.00

.20

.10

o. 1.0

.60

.50

.40

.30

.20

.1 0

o. 1.0

p

20 3.0

--------.... ',

' '------- -~,,

4.0 7.0 8 .0 90 10:0

2.0 3.0 4 0 5.0 6. 0 7. 0 8 .0 9.0 10.0

42

1.00

-----·-100

1.00

.so

.70

.60

.50 p

. 40

.30

.20

.10

0 . 1.0 2 .0 3.0 9.0 10 .0

Fig. 7.2 Estimated probabilities by sample relative frequencies (dashed lines) of positive run-lengths, P(N+ ~ j ; q) , and negative run- lengths , P(N- > j ; q) , f or the annual precipitation series at Ravenna, Nebraska (1878-1965) as compared with expected probabilities (solid lines) of positive and negat ive run-lengths of i ndependent series for five truncation values: q = 0.3, 0. 4, 0 .5, 0.6, and 0.7.

1.00

.70

.60

.50 p

.40

.30

.zo

. 10

o. 1.0 2 .0 3.0 4.0 5.0 6.0 7.0 8.0 90 10.(

LOO

.60

.50

.40 p

.30

.zo

.10

0 . 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 .0

1.00

0. 5.0 6 .0 7.0 8.0 9 0 10 .0

43

1.00

10

j ·::-:·-·-·-· .......... 0.

1.0 2 .0 30 4.0 5.0 6.0 70 8.0 9 .0 10.0

1.00

6 .0 7.0 8 .0 9.0 10.0

Fig. 7.3 Estimat ed probabilities by sample relative frequencies (dashed lines) of positive run- lengths, P(N+ > j ; q) , and negative run-lengths, P(N- ~ j ; q) , for the annual precipitation series at Antioch F. Mills, California (1879-1965) as compared with expected probabilities (solid lines) of positive and negative run-lengths of independent series for five truncation values: q • 0.3, 0.4, 0.5, 0 .6, and 0.7.

.40

.30

. 20

.10

o. 1.0 2.0 3.0 4.0 50 6.0 7.0 8.0 9.0 10.0

1.00

60

.50 p

.40

.30

.20

.10

0. 1:0 2 .0 3.0 4.0 5.0 6 .0 7.0 8 .0 9.0 10.0

1.00

60

.50

.40

. 30

.20

. 10

0. 1.0 20 5 .0 6.0 7.0 8.0 9.0 10.0

44

1.0

.20

.10

o. 1.0 20 3.o 4.0 s.o e..o 7.0 ao 9.0 10.0

1.00

1.0 2.0 3.0 4 .0 5.0 6.0 7.0 8.0 9.0 10.0

Fig. 7. 4 Estimated probabilities by sample rel ative frequencies (dashed lines) of positive run-lengths, P(N+ ~ j ; q) , and negative run-lengths, P(N- > j ; q) , for the annual runoff series of the Rhine River (1808-1957) as compared with expected probabilities (solid lines) of positive and negative run-lengths of independent series for five truncation values: q = 0.3, 0.4, 0.5, 0.6, and 0.7 .

Because of large sampling fluctuati ons in series of limited size, either a drought of more than 3-4 years may not have been experienced in a period of 50-60 years, or a drought of nine years may have occurred, though no drought of 4-8 year duration was recorded. Many similar sampling biases are unavoidable in using current empirical methods in estimating probabilities of run-lengths.

7.3 Determination of Run-Length Probabilities of Stationary Dependent Series. The annual runoff series of the following three rivers, which are known to be dependent time series, are used as examples for determining probabilities of run-lengths equal to or greater than a given length, for a given truncation level and its probability, q :

5. The Gota River at Sjotorp, Vanersburg, Sweden , with r 1 = 0.463 , where r 1 is

the est imate of the first autocorrelation coefficient pl .

6. The Ashley Creek near Vernal, Utah, with r 1 s 0.274 , and

7. The Trinity River at Lewiston, California, with r 1 ~ 0 . 180.

Probabili ties of run-lengths being equal to or greater than a given value j are obtained by using

45

the values P(N+ ~ j) of the Appendix, for values of p1 = 0.1, 0.2, 0.3, 0.4, and 0.5. A l inear

interpolation is used for the r1

values, which

are between the P1 values of the Appendix. These

probabilities and the relative frequenci es of run­lengths empirical l y determined are compared. Similarly, as in the case of independent series, probabilities of run-lengths P(N ~ j) of dependent series depart from the r elative frequencies empiri­cal l y determined from the sample data. The Trinity River is analyzed i n two ways: ( 1) by using the first serial sample correlation coefficient r

1 and

the first-order autoregressive model for annual flows, and (2) by using ri = r 1 - r 1(P) , where r 1(P) is

the first basin. This is done to remove the s ampl ing fluctuation of r 1 (P) i ncluded in r

1 of runoff,

because E(r1) for precipitation series should be

close to zero. For the area of the Trinity River Basin, r 1 (P) has been calculated in a previous

study [2) as r 1 (P) = 0.10 , so that the corrected

value ri = 0.08. Figures 7.5 through 7 .8 show the

results and comparisons of relative f requencies and expected probabilities for the corresponding cases.

0 .80

0.70

0.60

0.50

0.40

0.30

0 .20

\ \

\ \

\ \

\ ' ' ' ' ' ..........

P(W 0? j ; q=0 .7)>---......2'· , 0.10 ---- ' ... ~ ....... -,, 0~~~~--~--~~~~--~--~--J

1.0 2 .0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

1.00 p

0.6

0.5

P(N- ?; j;q =0.5)

0 20

0.10

1.00 p

0 .

\ ~ P( N- ?; j ; q ~ 0 . 3 ) 0 .60 v . 0.50 \ . 0.40

0.30

020 q =Q7)

0. 10

46

1.00

0.70

0.60

0.50

0.40

0 .3 0

0.20

0.10 P(W O?j ; q·=0.6)

1.00 p

0.6

0.50

0 .40

0.30

0.

0 . 10

o L-~--L-_L __ L_~~~~=-._~ 1.0 20 3.0 40 5.0 6.0 70 8.0 9.0 10.0

Fig. 7 . 5 Estimated probabi l ities by sample re lative frequencies (dashed lines) of positive run-lengths , P(N+ ~ j ; q) , and negative run-lengths , P(W ~ j ; q) , f or t he annual runoff series of the Gota River {1808-1957) as compared with expected probabilities (solid lines) of positive and negative run-lengths of a first -order autoregressive process with p 1 = r 1 • 0 . 450 .

. '----, /'·

P( N-> j; q=0.7) \ - \

\ \,,

o~~--~--~--~~---L--~--~~ 1.0 2.0 3.0 4.0 5.0 6 0 7.0 8.0 9.0 10.0

1.00

0 .20

0. 10

oL--L_J __ ~-L~L-~~~~~ LO 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

1.00 p

0.60

0.50 \ \ ........ -·P(N+ ~j; q=0.7)

0.40 \ \

0.30 P(W>j; q=0.3 )

0.20

0.10

0~~~~~~~~~~~~~ 1.0 6.0 7.0 8.0 9.0 10.0

47

I.

0.70

0.60

0.50 p

0.40

0.30

020

0. 10

--- - ·-•• ~ p ( w ? j ; q = 0.4)

\ • . . . .

• . \

o~~--~--~--~--~~L---~~--~

1.0 2.0 3.0 4.0 5.0 6.0 7.0 ao 9.0 10.0

1.00

0.60

0.50

0.10

' \ \ \ .----P( N+ ~ J; q=0.6)

Fig. 7.6 Estimated probabilities by sample r el ative frequencies (dashed lines) of positive run- l engths, P(N+ ~ j ; q) , and negat ive run- l engths, P(W ~ j ; q) , for the annual runoff series of Ashley Creek (1914-1957) as compared with expected probabili ­ties (solid lines) of positive and negative run-lengths of a first-order autoregressive process with pl = r 1 = 0 . 274.

1.00

0.10

1.00 p

0.7

1.00 p

0 .90

0 .80

0.70

0.60

/P( N+2: j; q =0.3)

- '· I I .

I . J

P(N-?j; q=0.7)

----" -, ''-·-

P( W2:j i q =0.5)

2 0 30 4.0 5.0 6.0 10 8 .0 9.0 100

48

1.00

1.00 p

0 .90

Fig . 7.7 Estimated probabilities by sample relative frequencies (dashed lines) of positive run-lengths, P(N+ ~ j ; q) , and negative run-lengths, P (W ~ j ; q) , for the annual runoff series of Trinity River (1911-1956) as compared with expected probabilities (solid lines) of positive and negative of a first-order auto­regressive process with (l) P1 = r 1 = 0.180 , and (2) pl = ri = 0.080.

Chapter VIIl

CONCLUSIONS

A methodology is developed in t his study for using the mean run-length as the parameter

for investigating hydrologic series . The basic statistical parameters used are the mean total

run-length and the mean positive or negative run-length, as they change with the probability

of the truncation level of a series. This study leads to the following conclusions:

(1) The method is effectively used to investigate whether or not a time series of

annual precipitation a.t a point is a sequence of stochastical ly independent variables with

a common distribution .

(2) The method is al so effectively used to investigate whether or not a time series of

annual river flows is independent or first-order linear autoregressive dependence model.

(3) The method does not depend on the underlying distribution of the variables that are

being investigated.

(4) Autoregressive linear models, widely used in hydrology, usually are referred to as

stationary Gaussian processes, if their independent stochastic component is norma.lly distributed.

Properties of runs of these models are rel evant for t he invest igations of multiannual periods

of surplus and deficit, and for the study of hydrologic droughts.

(5) An analytical approximation is developed for probabilities of a sequence of a given

length of wet and dry years, when hydrologic time series are stationary, either independent,

or first-order linear autoregressive process, and the truncation level is specified. Numerical

values of these probabilities are obtained on a digital computer for the range of p1 , the

first autocorrelation coefficient between 0 and 0.50 , with i ncrements of 0.10, and the r ange of

the probabi l ity of truncation level, q , between 0.30 and 0.70 with increments of 0. 10, all

for the first - order l inear autoregressive model. These probabilities can be readily used for

probability statements about the multiannual periods of water surplus or deficit, with respect

to a specified truncation level that defines the surplus or the deficit.

(6) Probabilities of run-lengths of linearly dependent variables, with a common distri­

bution, do not depend on the underlying, univariate distribution of the variable. They depend

only on t he probabili ty q of the truncation level and the series dependence model .

(7) Examples of the investigation of stationary ser ies by the run-length technique and

examples of the computation of probabilities of lengths of surplus or deficit periods show the

relevance of run theory for various applications in hydrologic and water resources investigations.

49

REFERENCES

1. Yevj evich, V., 1963, Fluctuat ions of wet and dry years, Part I. Research data assembly and mathe­matical models . Hydrology Paper No. 1, Colorado State University.

2. Yevjevich, V., 1964 , Fluctuations of wet and dry years , Part II , Analysi s by serial correlation . Hydrology Paper No. 4, Colorado State University.

3. Pinkayan, Subin, 1966, Conditional Probabilities of occurrence of wet and dry years over a large continental area. Hydrology Paper No. 12, Colo­rado State Universi ty .

4. Quimpo, Rafael G. , 1967, Stochastic model of daily river flow sequences , Hydrology Pap er No . 18, Colorado State University .

5 . Quimpo, Rafael G. , and Yevjevich, V., 1967, Sto­chastic description of daily river flows. Pro­ceedings of the International Hydrology Symposium 1967, Fort Collins , Colorado

6. Yevjevich, V., 1967 , An objective approach to definitions and i nvestigations of continent a l hydrologic droughts, Hydrology Paper No . 23, Colorado State University .

7. Saldarriaga, J ., 1969, Investigation of wet and dry years by runs , unpubl ished Ph .D. dissertat ion , Colorado State University .

8 . Cramer , H. and Leadbetter , M. R. , 1967, Stationary and related stochastic processes, Wiley and Sons, New York.

9 . Wishart, J. and Hirshfel d, H.O., 1936 , A theory concerning the distribution of joins between line segment s , London Mathematical Soci ety Journal, Vol . 11, p. 227 .

10 . Cochran, W. G., 1936 , The statistical analysis of field counts of diseased plants, J . R. Statistical Society Suppl . , 3: 49-67 .

11. Stevens, W. L., 1939, Distribution of groups in a sequence of alternatives, Annals of Eugenics, Vol . 9, p. 10 .

12 . Wald , A. and Wolfowi tz, J ., 1940 , On a test whether two sampl es are from t he same population, Annals of Mathematical Statistics, Vol. I!.

13. Mood , A.M., 1940 , The distribution theory of runs , Annals of Mathematical Statistics , p. 367 .

14. Bendat, J . S. and Pier sol, A. G., 1966, Measurement and analysis of random data , Wiley and Sons, New York .

15. Feller , W., 1957, An introduction to probability theor y and its applications, Wiley and Sons, New York.

16 . Downer, R.N., Siddiqui , M.M., and Yevj evich, V. , 1967 , Application of runs to hydrologic droughts, Proceedings of the Int ernat ional Hydrology Sym­posium, Fort Collins , Colorado.

so

17. Ll amas, J., 1968 , Defi cit and surplus i n precipi­tation series , Unpublished Ph . D. Dissert ation , Colorado State University .

18 . Llamas, J. and Siddiqui, M.M., 1969, Runs of pre­cipitation series, Hydrology Paper No . 33 , Colo­rado State University .

19. Cox , D. R., Miller, H. D., 1965, The theory of sto­chastic processes, Methuen and Co . Ltd., London.

20 . Heiny, R., 1968, Stochastic variables of surplus and deficit, Unpublished Ph. D. Dissertation, Colo­rado State University.

21. Gupta , S.S. , 1963, Bibl i ography on t he multi­variate normal integrals and related topics, Annals of Mathematical Statistics, Vol. 34, pp . 829-838 .

22 . Gupta, S. S., 1963 , Probability integrals of multi­variate normal and multivariate t ., Annals of Mathematical Statistics, Vol. 34, pp . 792-828 .

23 . McFadden, J.A., 1955 , Urn models of correlation and a comparison with the multivariate norrnal integral. Annals of Mathematical Stat ist ics Vol . 26, pp. 478-489 .

24. Sheppard, W.F., 1900 , On the calculation of the double integral expressing normal correlation, Transactions of the Cambri dge Philosophical Soci ety . Vol. 19, pp . 23-66 .

25 . National Bureau of Standards, 1959 , Tables of the bivariate normal distribut i on and related func­tions, Applied Mathematics Series 50, U. S. Gov­ernment Printing Office, Washington.

26 . David, F.N., 1953 , A note on the evaluation of the multivariate normal integral, Biometrika 40, pp. 458-459 .

27. Yevjevich , V. , and Jeng, E. I ., 1969, Properties of non-homogeneous hydrologi c series, Hydrology Paper No . 32, Colorado State University .

28 National Bureau of Standar ds, 1959, Tables of the bivariate normal distribution and related func ­t ions , Applied Mathematics Series SO , U. S. Gov­ernment Pr inting Office , Washington.

29 . Zalen , M. and Severo, N. C., 1960 , Graphs for bi­vari ate normal probabilities, Annals of Mathemat­ical Statistics , Vol. 31, pp . 619-624 .

30 . Kendall, M. G., 1941, Proof of relations connected with the tetrachoric series and its generaliza tion, Biometrika, Vol. 32 , pp . 196-198.

.31 . Blum, J.R. Hanson , D.L. and Rosenblatt, J.I. , l963 On t he central limit theorem for the sum of a random number of independent random variables , z. Wahrscheinlichkeitstheoric, 1, pp. 389-393 .

APPENDIX

This appendix gives the properties of run-lengths of the first-

order linear autoregressive model for five values of the first auto-

correlation coefficient, p1, 0 .1, 0.2, 0 . 3, 0.4 , 0.5, and for five

values of the probability, q = F(x0), of the truncation level, x0

,

or q equal 0.3, 0.4, 0.5, 0.6, and 0.7 . The first column gives

these parameters: + + q = F(x0), EN (q), and var N (q); the second

column gives the discrete value j of the run-length; the third column

are probabilities, P(j+), of at l east j consecutive values being

above the truncation level, and subsequent ten columns give probab-

ilities of at least k consecutive values being below the truncation

level, x0 , followed by at least j consecutive values being above the

truncation level, fork= 1,2, ... , 10, and j = 1,2, .. . , 10. Finally,

the last two columns give probabilities of run-lengths being greater

than or equal to j, or being exactly equal to j, respectively . By

using the values given in the subsequent five tab l es, it is feasible

to make probability statements about durations of droughts of a river

basin, with annual runoff series following the first-order linear auto-

regressive process, of p1 estimated by the sample r1

, and by finding

P(N+ ~j) values for a selected q in this appendix.

51

"' ...,

\ P, = 0. 1 EN .

var N+ j P(j+) I P(l~ /> I P(2~ /> I P(3~ t> _l P(4~ t> J P(S~ t> j P(6~ j+) I P(7~ t> I P(8~ j+) I P(9~ j+) ~(10~ j+) P(N+?! j+) P (N+ = j)

1 .7oouoo .197744 .o67781 . o22~47 . o o770S , 0 0 2601 . oo osr7 . ooo295 . oouo99 . ooooJJ .oooo11 1 . oooooo . 283162 •j!_ ..5022.5.!1 . . ..14.14.93 ....• 048273 . 016'+03 . • 0 05569 . 001886 . o00637 .000214 .00007~ o00il024 .ll00008 .7lo838 . 201899

o . l ·J . 360763 .101655 . o34698 .oll78S . oo3998 . ool353 .ooo4s 7 . ooolS4 . oooosi .ooov11 .oooooo . 514939 . 145oao 4 . 2S'H07_. __ .0730.18 __ ..,02493.6 . -.008465 .002870 . 0 0 0971 . 000327 . 000110 .oooo37 . 000012 .OU0004 o3b9860 .104217 5 . 186089 .052448 . 0 17918 . 006079 .002060 . 0 00696 oOU023~ . 000079 . OOOOZb . 000009 . 000003 o 265b43 .074863

3.45 ".6. . L336't.l __ . QJ7~7.2 ..• 012873 . 004365 . 001 4 78 . ooo499 . o00168 . 000056 .000019 . 000006 . oooooc! o190780 .oS377S ·7 . o95969 ,027058 .009247 . 003134 . 001060 .0 0 0358 o000120 .000040 .00001~ .000004 . 000001 . 137006 .0~8625 .tt ...Ob.8.9l.l_.....D..l.2~-.....0.06b.H. ___ . 002249 . 000760 . 000256 o000086 . 000029 oOOUOlO oOOUOOJ . 00000 1 o 098380 , 027742

7.77 9 · , 0 49477 . 013956 . oo-'+769 , ool.6i'+ .oooS45 . oco 184 ·000062 . 000021 . ooooot .0000 02 . 00000 1 o0706 38 .oi 9924 110 . 035521 .910022 . 003424 . • ool1!:i8 . UQ 03 91 .o oo 132 o000044 o000Q15 . OOOUO!:I oOOOU02 .OUOOO! . o50713 .01 4308

l . • 600000 .... .. 225.D.2a_._ • .D9B5S'i . o.043056 , Ol8f:Ol .008211 o 0035S4 . 001563 .000681 oOOOc:!96 . 00012e loOOOOOO , 375750 ~ . 374978 .14026 9 . 061300 . 026 847 . 0 11750 . o o5137 o00224 2 . 000977 . 000~2~ . OOU!84 . OUOOKO .624250 .23345~

0.4 .3 . .. 2.347.1.0 .. _. _OB_78.28 . __ _.Q.3B't06 . • 01oBi2 .007353 . oo32i2 o001401 . 000610 o CJ0026~ . 00011 5 .. • 000050 .39.0198 . 1~6214 4 .146881 . 05 4~72 .oz4oS5 . o 1os24 . ~4599 .oozoo1 . ooo874 . ooo38o . ooo i 65 . oooon . ooooJ1 .244~84 .o91S29

• .::i . • 09190.9..._ • .l13!+_40L_ . ,015.0.6l . • coc 5S5 • 0 ~875 ,oo1Z54 • 0005't6 ,0002H ,I,IOQlO~ o00004't ,U QOOl'i! o lS30S5 ,Q:)729'+ 2.66 6 .057502 .021534 o009426 o004118 o 01797 o0 0 0783 o000340 o000l48 o000064 .000028 0 000012 o 095761 0 035861

_]_ .. 0..3.5.9..6B. ___ .JH3!t7.~---•-D-05ll'77. .. _. Q02575 . 001122 .0004S8 o 000212 . 00009;:! .000040 o00111Jl 7 .QOOOQ1 . •. 059900 . OZ244~ ~ .022493 . ooa431 .oo3b&a . ool609 . o oo7o1 . ooo3o5 . ooo132 .oooos7 .oooo2~ .oooo11 . ooooo~ . oJ74S9 . oi 4 o41

4. 32 .. !.t . ...Oli.Mi-2-~0.052.11t __ _.o.oz.Jos. ...... o.o1oos . ooo'+37 . • ooo19u . oooos2 . ooo036 . oooo1~ . ooooo7 . • il.oooo;L . o ~3itl7 . o0 8 78J 10 .008788 .003?9.8 . ppl440 . O.UQt>27 . o oo273 .00 01 1 8 oOOOOsl . 000022 . 000010 oOOOOO't .000002 . o1 4 b35 . 005492

1 .soooo~ ~2340B~ . 1242~4 . o 6 6 22& . o35288 .o1eaoz .o1o014 . oo~331 . oo2~3 7 . oi1~os .oooao t .999990 , 4687<+2 o . 5 -'- .2b5.9.l...S__.J.2..~t,lo.f+ __ _..oli.S.2.3.6 ____ . _o 3.5157 .o1873 o .oo 9 9 7a .oos310 . 002824 . oo1soo . ooo79o . ooo'+22 .531247 .2~8117

3 .141711 .o66226 . oJS1S7 . n lB736 . oo9978 . oo 5 3 1 0 .oo282'+ . 0015oo ~ooo796 . oo0422 . ooo224 . 283131 . 132J l8 4 . 07548& . 035288 . 0 18736 ,009978 . 005310 . 0 0 28Z4 .001500 . 000796 . 000422 . 000224 . 00011~ o15 0813 . 07051 0

2.14 5 . 0 4 0197 .018802 . 009978 , 005310 . 002824 .00 1500 . 000796 .000422 . 00022 4 o000l 1 8 , OOOOo3 o0 8 0302 . oj7S67 ~- ...Jl2.1.3'I5 .010014 . oo.S_-ll.o_ ___ ._ooZ!}~4 . 001500 .000796 o0004c2 . 000224- oOOOllB o000063 . OOOOJ3 o042735 . ozoooa 7 . 011381 .005331 . 002824 , 0 0 1500 . 000796 .ooo422 o 00022~ . 000118 . oooooJ . ooooJJ . ooool7 . 022 728 . 010650

2 . 42 .6 .Jl0.6.0SQ __ _._O.Q21i37 __ .. 0_0150.0 .• 0 00796 . • C00422 o 00022~ o 000118 o000063 o OOOO~~ o000017 . 000009 . o 12078 . 005 665 9 .003213 .001508 . ooo796 . 0 00422 . 00022! . ooo l l8 o000063 . 000033 .00001? . oouoo~ .ooooo~ . 006413 . 003011

iO . 00170 5 o00080l • . OOO"-Z2 . 0 00224 . 000118 .00 0063 o000033 .000011 .000009 . 000005 .OOOOOJ .003 402 . 001599

l . 't_\I.OJl.OO. ___ . _ii:,~JI.~--•.lJ~~.'t_6 , __ _. QIH7tiZ _ ._0550 10 o03't477 o0Zl604 oU1353 4 o00847o oOO!:)JQ7 , 00 3JZl , 999796 , 562698 2 .174978 .098238 . 061300 . n38406 . 024055 .0150o 1 o009426 . 005897 .003688 .002~05 . 00 1440 o'+37098 . 245534

0 . 6 ;3_1 .. o.J6.H.l ___ , O.'t:U2l ___ ,_Q2.ftB4_7 . • 01 6812 oOlOS24 . 006585 o0041lll .002575 o00160'! . 00 1005 .OOOo27 • 191564 .107 699 4 .033619 . 018904 . 011750 . 0 07353 . 004599 oOC2875 e001797 . 001122' .000701 .000!37 . 000273 o0838o5 . 047 186 Ji o.!ll.lt715 .008283 , oo51J.l __._Q__fi_J2l2 o002007 . oo1254 o000783 . 000488· .OOOJO!:I oOOCJi'iU .llOOU.tL ._oJ.6~79 .0206 59

1.78 b . OOM32 .003625 .002242 . ttOi40l -- .0008i4 .00054b o000340 . 000212 oOOOiJc o000082 . 000051 .016020 , 009 035 i

.1 ._o.l.J.2.80L _ _.aJU..a.6.~ -__._0Q .0.97L. .• !! Q0.6l0 oOU0380 oOOOZ37 oOOOl46 . 000092 . 000 0 5 ·7 oOOOUJ6 , OOOOZ2 . 006985 . 00394 5 8 .oo}223 . 000691 . 000425 .000265 .000165 . 000103 o000064 . 000040 . 000025 .000015 . 000010 . 003040 , 001720

1. 38 !i_ ._QJI.O.i.~Z.. __ ._i)_Q.O.J.O.l. ___ .!l.OC.Ht'± ....... QO Q U ~ , o 0 o o 71 • oo o 044 o o 0 0 0 28 • 0 00 017 • 0 0 0 011 • 0 00 CJ 0 7 • 00 0004 , oo 1321 • 000 748 1 0 . • ooo 2Jl .ooo131 . ooo oso . ll oouso . oooo31 . ooooi9 . oooo12 .ooooo1 , ooooo!:i . • oouoo3 . uooooc:! . ooo572 . ooo3 2!l

1 . Jooooo .19714<+ . 141008 . lo1459 . o72983 . o52495 .oJ77so . 0271 5 2 .01952 4 . 014037 . ol o09l . 997467 . 657 289 .Z... .. 11l2.2..5.6. ........... il.6lZ9_1._ ....Jl...l.tR273. ___ • .D3~o98 .. o02493o o 0179 i8 o012873 . 009247 .006o4i . 004769 . 00.3 424 . 34017 8 . 224278

0 . 7 3 , 034959 . 023040 .016403 . 0 11785 . 0 08465 . 006079 o0043.t,5 , 003134 . 002249 oOO l ol4 . OOll5tl .115900 . 0764 93 -~- .o.Ol..l9_l_9 __ ._.Q..Q.7.86fL • ...o.Dll5.569 ___ , Jl 03998 _ o.002870 o0020 6 0 o0014 78 o00 1 060 . 0007o0 . 000545 , 000391 o039407 . 026 0 it 0 5 . • 004053 . 002680 .0018 8 6 . 0 01353 . 000971 . 000&9~ o000499 . 000358 . 00025~ .000 184 .oool3c . o l 3367 . 008847

1. 51 .6. ..JlJl.UH . 000..2.10 . ooo637 . 000 4.5.1_ . 00.D32.J . • ooOZ3'+ .o00168 . 000120 . 00008o oOOO U6~ ..• OO.OQ't!t . . ..o.004 S 19 . 002997 7 . 000464 .000306 . 000214 . 00015'+ . C00110 .ooooy9 o000056 . 000040 .00002~ o000021 , 000015 e001522 . 001012 e ....OD.U l .56 ... ..o..OO.Ol04 .. ~o..oou7.z. __ .• JlO..Il.051 . 000037 .oooo26 . 00001 9 .000013 . 000010 .000007 , UUOOOS oOOOS ll . 000 340

o. 79 9 . ooo o5z .oooo 3 s . ooo o 24 . oooo11 . oooo12 . o o ooo9 . ooooo6 .ooooo 4 .ooooo3 . ooooo2 . oooooc . ooo111 .000 1 1~ 10 .oOo017 . 00001 2 o0000 08 . 000006 o OO .OOO~ o 000003 _ . 000,(10_2_ . 00000!_ oOOOOO l o OOOOOl .00000_1 ~ •_000057 oOQOo38

"' (,>

\ P,=0.2 EN

+ var N j

l .7Q0000 .185157 . o72536 oQ26685 . ~09846 .003653 .001350 . oboi9~ . 000 179 2 .514843 . 134716 .051845 . o19683 .oo7430 . oo277?. .001021 .oo0371 . oo0133

0. 3 I 3 .380127. .o99662 . a038395 .. • Q.L45~2 ... 00.5~64 _ .• Ooao.3.i __ . O.O.IlHS. ._ . 0.00269. . ooO.I,)96 4 . 28o464 . 073585 . 028402 .o1o7t6 .oo4014 . oo1486 .ooo54~ . o0019~ .oooo7o 5 . 2o6879 . 054332 ·020986 oQ07892 .0029~5 . 001086 . 000395_ .000142 . oooo5t

3.65 1 6 .152547 . 040111 o0l~489 o005806 . 002159 o0 00793 o000288 .00010~ oOOD037 7 o1 l243~ . 0 2~605 e 0ll4~0 _ o Q04~61 . 001581 o000579 o000209 .00001~ o000026 8 o08?.830 0 021843 o008410 o003133 o001156 o000422 o000152 0 000054 o000019

8 . 741 9 . 060987 . o1 6i10 o006l88 o.002297 . 000845 o000307 oOOOllL __.0000 39 oOJl.OOH 11f .044877 . o11a7s .oo4548 . oo1683 .ooo617 . ooo224 .oooo8o .oooo2.- . oooo1o

1 . 6ooooo .209948 . 1oo3I9 . o4747:f .o22381 . olo566 .oo4975 - .oo233o . oo l o85

1 2 . 390052 . l3Sio2 . • 064040 . o3o6.25 . 014575 .oo6888 .. ..003230 --o-Do.l502 . oo0693

0. 4 3 . 2!i495tf . 088551 o042 01.0 • 020Q44 . oo9498 o004468 o00208t. .00096#. . 1100444 4 .1~6399 .os7860 ---Q~7~~~ .. 0LJoa.4 _ _ .oo617J __ • .ao2B91 _ . o.o1JH _ _..ooo62o . • ooo?84

I 5 . 1o8539 .o37811 . o180t7 . oo8520 .004003 . 001867 .000864 .000397 .000181 2. 84 6 .07o728 . o24697 • Ol175o .oo5534 .oo2589 .001202 .ooo554 . o00254 .oooil5

1 . 046031 . o l~ i17 . gq76'4 • OQ3S~6 . oo1671 .ooo773 . ooo355 .ooo16?. . oooo 73

1 8 . 029914 .o1o5os . oo4962 . oo2318 ..• oo1o76 .ooo496 . oDo227 .oool oJ . oooo47

5. 0 9 .9 o0t9408 . 006837 oOQ3?,l3 • 00l495 . 000691 oDOO:H7 . 000145 . 00006~ . Q000 30 10 .012571 .004443 . oo2n76 . noo963 .000443 o0002DJ . 000092 _.OD0042 . 000019

0 . 5

1 2 3 4

2 . 291 5 6

2. 94

0.6

1.90

1.11

7 8 9

10 1 2 3 4 5 6 7 8 9

10 1 -

0 . ; I !I l.~l l ; o. 9QI :

u •

. sooooo . 21816~ .121817 .o69445 . ~039.396 . 022:138 ~•01~64.2 . oo7133 .oo4oll o28i83 1 .12l817 o06A288 oQ38919 o022100 o012497 o0 07035 . 00 394~ o002200 o16Q01~ _ -.069445 o0389}9 oQ22.foo .012n1 .007035 .0039H __ . 002200 of!O l222 . 09o57o . o39396 . o221oo . o12497 . oo7o35 . oo3943 .oo22oo . oo1222 . ooo676 . ostl74 . o2?.338 • Oi?~97 .oo7o35 . oo394J .oo2~oo .oo1222 . ooo67~ . ry ooJ72 .0?.8836 . 0 12642 o 0070~5 .ooJ943 . 002200 .001222 .000676 . 00037~ o00020 4 . Oi6194 .oo7i33 ·003943 . 002200 o001222 o000676 .000372 .000204 . 00011 2 o009061 . 0040 11 . 002200 . ooi2~2 . 000676 o 000~72 . 000204 .000112 o000061 .oo5o~i . ooi246 • 001222 .~oo676 . ooo372 . ooo2o4 . ooo112 .oooo6) . oooo33 o002B04 .001253 o000676 o000372 .• 000204 o000112 o00006} . 0001133 . 000 018

o400000 .209948 ol337Q1 .g88243 . 058023 .038117 .025006 .016377 oOl070 6 .190052 0 098918 o064040 o042060 . 027561 o018017 o01175o .007644 1 004962 o0911~; .047734 •Q~06~5 •020044 o013084 o 008~20 o00 5534 0 00358~ oQ02318 .043400 .022833 o014575 o00 9498 . 006173 . 004003 .002589 . 00 1671 .oolo76 . ozo566 .010~86 · 006888 · 0 04~68 . 002891 o001867 . 001202 .000773 .o00496 . 009680 . 0051 58 o003230 · 002086 .oo1344 . 000864 .ooo554 .000355 o000 227 o004522 . 00?426 . 001502 •Q00966 . 000620 . ooo~9~ . 000254 .000162 . ooOiOJ o0 Q2096 . 001132 oOQ0623 • 000444 . 000284 o000181 . 000115 .00001~ o000047 o000964 . OOQ524 • 000~18 •QQQ203 . 000129 o000082 . 000052 .000033 oQ00021 .ooo441 .000241 o000144 o000092 . 000058 . oooo 37 .000023 . 000015 . 00 00 09 .3qoooo . • 1BS1S7 ___ ....1J2692- - • 998.767 __ .013.U.8_ __ • .054536 o04.0it79 _ . • 030 02Q . 0222U .1}4843 .o7o421 · 9~1845 oQ38395 . 028402 .0~098~ .ol5489 .o1142Q . 008410 .044422 . o275o9 . o19683 . g14532 . o1o716 . • oo7892 . oos8o6 . oo4267 .oo31 33 .ot&9I2 . o 10552 . pQ7430 . gos~64 .oo4014 .o o2~4S .oo2159 .oo158i . oo11S6 oD06l60 .00~003 • QQ277?. • QQ?031 . 001486 oO Ol086 .ooo793 .000579 . 000~22 o0ftZ357 . 001496 • 001021 • 0011745 . 000543 o000395 .000288 . 000209 . 0001 52 .~ul&l .ooo551 .ooo37l . ooo269 .oo.o196 . oool42 .ooo103 .oooo7s_ .oooo54 . oooll~ . ooo200 . oon1~3 . oooo96 . oooo7o .oooo51 .000037 . 000 026 .oooo19 .ooollo .oooo71 •QO~Q47 . oooo~4 . oooo2s . oooo1s .oooo13 .oooooq . qoooo7 -~~~~39 .oooo25 . oooo16 . nooo12 .ooooo9 . ono11o~ .ooooo4 . ooo oo' . noooo2

. 000064

. 000047 • 000.034 . 0000'.5 . oooo i e . oooop . ooooo9 .ooooo7 .ooooos . ooooo3 . ooo5ot o0003i8 . l'I 00?.03 . 0.00129 . 000082 .oooo52 . 000033 .oooo2i .oooo\3 •. 000008 . 00 2246 . 001 222 . 000676 .000372 . ooo224 .o00112 .00006l . oooo~3 .OOOO} B .000010

~ 006985 . 0032t3 .00149~ .000691 . ooo~i7 . 000145 . 000066 . 000030 .ooooiJ . 000006 . 016467 . 006188 .002297 o000845 . 00030! .000111 o.OOOo39 . oooni4 . ooooi)s . ooooo 2

. 000 023

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. ooooo9 o000 006 . 000004 . ooooo3 . ooooo~ . 000002 . oooooi .ooo?3n .000144 . oooo9~ . oo oo5e . oooo37 oOOOQ2~ . Q00015 oQ0000 9 . Q OOOO~ .000004 . on l253 . 000676 .00037~ . 000204 . oooi12 , oooo6i . oooo3J o00001A . ooool o . o.oooo c;

. on454R

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.000443 o000203 . oooo92 . oooo42 .Q00019 o000008 . 000004 • 01217.9 .o ~454A ,0 0 168~

. ono6t7

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. 000028

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Key Words: Hydrology , Time Series , Theory of Runs, Run-Lengths, Stochastic Processes in Hydrology, Droughts, Precipitation, Run-Off

A method is developed for investigating time series structure by us ing the mean run-length parameter. This method is distribution­free . Applications to selected annual precipitation series and annual runoff series demonstrate the feasibility of this method. Analytical expressions are developed by which the probabilities of sequences of wet and dry years of specified lengths can be calculated when the basic hydrologic time series is either an independent or a dependent stationary series of a var iable which follows the first ­order linear autoregressive model. Numerical values of probabilities of run-lengths are obtained by the digital computer integration of expansion equations for run-length probabilities of the first-order linear autoregressive model. A set of tables and a set of graphs are presented to make the numerical values readily useabl e. Probabil­ities of run- l engt hs of dependent variables with a common distribution are also distr ibution free . The significance of thi s i nvestigation , and several applications i n the text, are based on the premise that run-lengths, as statis tical properties of time series, represent attractive parameters in studying droughts and surpluses .

Key Words : Hydr ology, Ti me Series, Theory of Runs, Run-Lengths, Stochastic Processes in Hydrology, Droughts , Precipitation, Run-Off

A method is developed for investiga·ting t ime series structure by using the mean run-length parameter. TI1is method is distribution­free. Applications to selected annual precipitation series a~d annual runoff seri es demonstrate the feasibility of this method. Analytical expressions are developed by which the probabilities of sequences of wet and dry years of specified lengths can be calculated when the basic hydrologic time series is either an independent or a dependent stationary series of a variable which f ollows the first­order linear autoregressive model. Numerical values of probabilities of run-lengths are obtained by the digital computer integration of expansion equations fpr run- l ength probabilities of the first-order linear autoregressive model. A set of tables and a set of graphs are presented to make the numerical values readily useable. Probabi l­ities of run-lengths of dependent variables 1d th a common distribution are also distribut ion free. TI1e significance of this investigation , and several applications in the t ext, are based on t he premise that run-lengths, as statistical proper ties of time series, r epresent attractive parameters in studying droughts and surpluses.

Key Words: Hydrology , Time Series, Theory of Runs , Run-Lengths, Stochastic Processes in Hydr ology , Droughts, Precipitation, Run-Off

A method i s developed for i nvestigating time series structure by using the mean run-length parameter . This method is dist ribution­free. Applications to selected annual precipitation series and annual runoff series demonstrate t he feasibility of this method. Analytical expressions are developed by which the probabilities of sequences of wet and dry years of specified lengths can be calculated when the basic hydrologic time series is either an independent or a dependent stationary series of a variable which follows the first­order linear autoregressive model . Numerical values of probabilities of run-lengths ar e obtained by the digit al computer integration of expansion equations for run- l ength probabilities of the first-order linear autoregressive model . A set of tables and a set of graphs are presented to make the numerical values readily useabl e. Probabil ­ities of run-lengths of dependent variables with a common distribution are also distribution free . The significance of this investi gation , and several applications in the text, are based on the premise that r un- lengths, as statistical properties of time series, represent attractive parameters in studying droughts and surpluses.

Key Words: Hydrology, Time Series, Theory of Runs, Run- Lengths, Stochastic Processes in Hydrology, Droughts , Precipitation, Run-Off

A method is developed for investigating time series structure by using the mean run-length parameter. This method is distribution­free . Appl ications to selected annual precipitation series and annual runoff series demonstrate the feasibility of this method. Analytical expressions are developed by which the probabilities of sequences of wet and dry years of specified lengths can be calculated when the basic hydrologic time series is either an i ndependent or a dependent stationary series of a variable which follows the first­order linear autoregressive model . Numerical values of probabilities of run-lengths are obtained by the digital computer integration of expansion equations f or run-length probabilities of the first-order linear autoregressive model . A set of tables and a set of graphs are presented to make the numerical values readily useable. Probabil­ities of run-lengths of dependent variables with a common distribution are also distribution free . · The significance of this investigation , and several applications in t he text , are based on the premise that run-lengths, as statistical properties of time series , represent attractive parameters in studying droughts and surpluses.