Statistical Methods for River Runoff Prediction

Transcription

1 Water Resources, Vol. 32, No. 2, 25, Translated from Vodnye Resursy, Vol. 32, No. 2, 25, Original Russian Text Coyright 25 by Pisarenko, Lyubushin, Bolgov, Rukavishnikova, Kanyu, Kanevskii, Savel eva, Dem yanov, Zalyain. WATER RESOURCES AND THE REGIME OF WATER BODIES Statistical Methods for River Runoff Prediction V. F. Pisarenko*, A. A. Lyubushin**, M. V. Bolgov***, T. A. Rukavishnikova*, S. Kanyu*, M. F. Kanevskii*, E. A. Savel eva*, V. V. Dem yanov*, and I. V. Zalyain* * International Institute of the Theory of Earthquake Prediction and Mathematical Geohysics, Russian Academy of Sciences, Varshavskoe sh. 79, kor. 2, Moscow, Russia ** Institute of Earth Physics, Russian Academy of Sciences, ul. Bol shaya Gruzinskaya 1, Moscow, Russia *** Water Problems Institute, Russian Academy of Sciences, ul. Gubkina 3, GSP-1, Moscow, Russia Received November 26, 23 Abstract Methods used to analyze one tye of nonstationary stochastic rocesses the eriodically correlated rocess are considered. Two methods of one-ste-forward rediction of eriodically correlated time series are examined. One-ste-forward redictions made in accordance with an autoregression model and a model of an artificial neural network with one latent neuron layer and with an adatation mechanism of network arameters in a moving time window were comared in terms of efficiency. The comarison showed that, in the case of rediction for one time ste for time series of mean monthly water discharge, the simler autoregression model is more efficient. INTRODUCTION River runoff rediction is a difficult hydrological roblem because the hysical rocesses that control it are too comlex to allow adequate descrition by a system of aroriate equations. This is why stochastic methods are widely used for redicting river runoff [1, 8, 14 16]. Several modifications of statistical forecasts roosed in [13] are based on the autoregression analysis of the series in combination with simulation of trends in harmonics. However, all these methods have a significant drawback, since the stochastic comonent is simulated using the stationary autoregression, while it demonstrates ronounced seasonal variations. Reliable runoff forecasts are indisensable in various branches of water use (water reserves, ower, navigation, flood control, etc.. Clearly, the reliability of an efficient runoff forecast deends on the monitoring systems sulying the required data; therefore, such systems need ermanent modernization. The objective of this study is the develoment of forecasting methods that are based only on data on the ast runoff values. The hydrological exerience shows that long-term forecasts are hardly ossible in such formulation. However, forecasts with a lead time of one time ste (one month are ossible. This allows the ower otential of water bodies to be used more efficiently. The theoretical results of this study include conclusions regarding the tye and comlexity of a stochastic runoff model with a monthly samling interval (in other words, a model with a seasonal trend. Such models are of articular interest in the simulation-based studies of comlex water management systems. Examles of seasonal-trend models are known in hydrology ([2] and others; however, their roerties and the quality of forecasts made with their hel have not been adequately studied. A relatively new statistical method the theory of eriodically correlated rocesses (PCP [3] is used in this case. This model is sometimes referred to as a cyclic stationary rocess [9]. The formulas for redicting runoff values with a monthly lead based on the revious monthly runoff values were obtained with the use of the standard least-squares method. The PCP aroach takes into account both deterministic and stochastic comonents of seasonal variations in runoff data and allows the correction of the rognostic coefficients for a secific rediction month. The rediction coefficients vary from month to month. Runoff rediction with the use of an artificial neuron network (ANN [6, 1] was used as a benchmark test of rediction efficiency. This aroach was used for river runoff rediction, for examle, in [5, 7, 11, 12]. Monthly runoff rediction was made using a ercetron neuron network, containing one latent layer, with training on data from the revious eriod. Comarison of the two methods shows that the forecasts made by the PCP method are never worse than those made by the ANN method. The former method is as a rule 1 3% more efficient than the latter; however, for some rivers, the results obtained by using these methods are almost identical. Data on the runoff of nine large rivers in Euroe and Asia (Loire, Elbe, Danube, Glomma, Vistula, Oka, Northern Dvina, Irtysh, and Amur were used in this study. The length of the observational time series for these rivers varied from 84 to 134 years. Monthly runoff values X(t, t = 1,, N, were used to demonstrate the alication of the roosed method. The general form of a runoff time series is shown in /5/ MAIK Nauka /Intereriodica

2 116 PISARENKO et al. Fig. 1. These time series can areciably differ. A distinct regularity with annual maximums caused by snow melting and sring floods was observed for some rivers, such as the Glomma, Amur, Irtysh, Oka, and Northern Dvina. These time series are tyical examles of PCPs. The Vistula, Elbe, and Loire feature less ronounced regularities. The Danube demonstrates the most chaotic runoff variations. This can be exlained by the small effect of sring flood on the runoff of these rivers. Mean monthly runoff values (annual forms m(β, β = 1,, 12 (hereafter, β will be used only as a number of a month, are given in Fig. 2 along with the resective root-mean-square deviations, which also are different. Some rivers (the Loire, Danube, and Elbe feature regular quasi-harmonic behavior, with can be attributed to a low sring flood. The Glomma, Vistula, Oka, Northern Dvina, and Irtysh each have a single distinct eak corresonding to the sring flood. Two eaks are recorded in the annual dynamics of the Amur; the lower eak corresonds to the sring flood, and the higher eak, to autumn floods caused by cyclonic reciitation. Root-mean-square deviations (Fig. 2 demonstrate a feature tyical of all the rivers: these deviations are roortional to the monthly runoff values. The relative root-mean-square deviations for months with higher runoff values are also relatively large. Power sectra for runoff time series are shown in Fig. 3. A secific feature of all these sectra is the resence of shar eaks searated by equal intervals. These eaks are caused by eriodic deterministic comonents with a eriod of one year and its overtones. The eak magnitudes are closely related to the resective annual curves, since the ower sectra are smoothed by the Fourier coefficients of the annual variation curves. Thus, the annual runoff variation curves for the Oka and Northern Dvina have six sectral eaks with virtually equal magnitudes, whereas the sectra of annual runoff variations of the Loire, Danube, and Elbe each have one distinct eak. PERIODICALLY CORRELATED STOCHASTIC PROCESSES Let us consider the rincial definitions of PCP [3, 9]. A stochastic rocess X(t with discrete time t is called a stochastic PCP with a eriod T if its mean m(t = E{X(t} and covariation function C xx (t, s = E{(X(t m(t(x(s m(s} are eriodic functions with a eriod of T: mt ( + T mt (; C xx ( t + T, s+ T C xx ( t, s. In this case, T = 12, since the runoff time series have a distinct annual seasonal eriodicity, which should be used in forecasting. Thus, the annual trend curves in Fig. 2, which in fact are the mean functions m(t, resent a simle deterministic forecast of monthly runoff. It is more imortant, however, to redict the stochastic comonent of the runoff time series. Thus, let us isolate a centered stochastic comonent Y(t = X(t m(t and solve the forecast roblem for this urely stochastic art of the runoff time series. In order to satisfy the rincile of causal relationshi, we are to construct an estimate m(t, which is used in the forecast made at time moment t and is based only on the revious values of X(s, s < t. Centered time series Y(t are given in Fig. 4. Obviously, the subtraction of samle means Y(t reduces the 2 variance as comared with the samle variances δ X of 2 2 the original time series X(t. The ratio γ m = δ X / δ y can be regarded as an index of deterministic redictability, when the rediction is based only on the cyclic mean m(t. It can be seen that the subtraction of the cyclic mean m(t areciably reduces the samle variance (by a factor of 1.9 to However, the decrease is not the same in different months. The PCP model enables this heterogeneity to be utilized. Thus, the most redictable were found to be the runoff variations in rivers with high seasonal floods (the Irtysh, Amur, Northern Dvina, and Oka. The least redictable are rivers with low sring floods (the Danube, Elbe, Loire, and Vistula. However, this classification will change if we take into account the redictability of the stochastic comonent Y(t. Correlation matrices for Oka and Irtysh runoff values are shown in Table 1. It can be seen that the Oka features relatively low correlation coefficients for consecutive months; whereas, these coefficients for Irtysh are large and can reach.92 (February March. The PCP model takes into account variations in the correlation coefficients from month to month. RUNOFF PREDICTION Suose that the rognostic value X (t of a rocess has the form X (t = m (t + Y (t, where m (t is an estimate of the cyclic mean m(t based on the revious values X(s, 1 s < t. The rediction Y (t for the stochastic comonent Y(t can be found in the form of a linear combination of revious observations: Y( t = a 1 (Y t ( t a (Y t ( t, (1 where [t, t 1] is the time interval used for rediction. It should be mentioned that the rognostic coefficients a j (t really deend on the time moment t of rediction. Two methods for the evaluation of coefficients a j (t in (1 are tested. The first aroach, referred to as the standard autoregression (AR estimate [4, 12] is based on the following rocedure. Let L be the number of values within the time window used for the current time moment t of the forecast Y(t. Now, the standard ARestimate corresonds to the coefficients of linear rediction that minimize the sum of rediction errors squared: WATER RESOURCES Vol. 32 No. 2 25

3 STATISTICAL METHODS FOR RIVER RUNOFF PREDICTION 117 Q, m 3 /s (a 2 8 (b 4 2 (c 1 4 (d 2 16 (e 8 3 (f 1 16 (g 8 2 (h (i Year Fig. 1. Mean monthly discharges for nine rivers in Russia and Euroe (here and in Figs. 2 4: a i are Loire, Elbe, Danube, Glomma, Vistula, Oka, Northern Dvina, Irtysh, and Amur, resectively. WATER RESOURCES Vol. 32 No. 2 25

4 118 PISARENKO et al. (e (a (f (b (g (c (h (d (i Jan. Mar. May July Se. Nov. Jan. Mar. May July Se. Nov. Month Fig. 2. Mean annual runoff variations m(β, β = 1,, 12 in relative units with standard deviations. L Y( t n a k Y( t n k 2 min (2. a k n = 1 k = 1 Unlike the PCP aroach, the rognostic coefficients a k in (2 are not suosed to deend on the time t. The solution to the roblem (2 must satisfy the system of linear equations ( C kj k = 1 (a t k ( C j = (, t j = 1,,, (3 where ( C kj L ( t = Y( t n ky( t n j. n = 1 ( a k (4 Suose that (t is the solution to linear system (3 (in this form, a deendence of a k on t still ersists. It should be mentioned that in the case when an infinite set is averaged (L, this deendence for a stationary time series Y(t disaears. However, some deendences of a k on t can still exist for finite L values. WATER RESOURCES Vol. 32 No. 2 25

5 STATISTICAL METHODS FOR RIVER RUNOFF PREDICTION 119 (e (a (f (b (g (c (h (d (i Frequency, year 1 Fig. 3. Estimated ower sectra of seasonal runoff variations, relative units. The second aroach referred to as AR-PCP-model reresents a modification of the least-squares roblem (2 that takes into account the eriodic correlation roerty. Suose that q is the integer art of the relationshi (L /T: q = [(L /T]. The integer q = q (t, generally seaking, deends on t. Now we have to solve the following roblem: qt ( n = 1 Y( t nt a k Y( t nt k 2 min. a k k = 1 (5 WATER RESOURCES Vol. 32 No. 2 25

6 12 PISARENKO et al. Q, m 3 /s (a (b (c 1 2 (d 2 8 (e 8 2 (f 2 8 (g (h (i Year Fig. 4. Deviations form current estimates of cyclic means (signals Y(t for mean monthly water discharges. From (5, we obtain the following system k = 1 ( 1 C kj (a t k ( 1 C j = (, t j = 1,,, (6 ( 1 C kj qt ( ( t = Y( t nt ky( t nt j. n = 1 ( 1 a k WATER RESOURCES Vol. 32 No (7 Let us denote the solution to system (6 by (t. Thus, we have two different redictors of the th order:

7 STATISTICAL METHODS FOR RIVER RUNOFF PREDICTION 121 Table 1. Matrices of correlation coefficients between different months for deviations from the cyclic mean (signal Y(t for the Oka (to number and Irtysh (bottom number β ( α ( = a k ( ty ( t k, α = 1., Y ( α t k = 1 (8 If we comare (2 with (5, it can be easily seen that the number of terms in the sum (2 is greater than that in the sum (5 by a factor of T = 12, and the sum (2 ensures more reliable averaging. However, if the time series examined is eriodically stationary rather than stationary, such averaging can worsen the forecast. On the other hand, averaging (7, which takes into account the rincial roerty of eriodicity of PCP, can enable a more efficient forecast. Thus, the advantage of the PCP forecast deends on the extent to which the time series in question is eriodically correlated, that is, its correlation varies eriodically. Let us consider both these methods and their ossible combinations used to reduce the rediction error. A time series of rediction errors can be constructed for both methods e ( α ( t = Y( t Y ( α ( t, α = 1,, t = L + 1,, N. (9 Here, L + 1 is the first time moment for rediction, N is the number of values in the time series considered. The first L values are used to initialize the algorithm. In the case of AR-PCP-estimate, the calculation of all time series is more convenient to start from January. Since some time series do not meet this requirement (Fig. 1, the first values ( 11 were not used in order to make all the series to start from January. Again, the length of initialization L or the length of the moving time window L was taken equal to an integer number of years: L or L = Tn = 12n, where n is an integer. In all cases, n = 3 years, L = L = 36 values. Thus, the time series of rediction errors (9 also started from January. Let N Y be the integral art of N/12, that is, the number of comlete years of observations. Note that the time series Y(t can be determined only starting from the second year of observations. WATER RESOURCES Vol. 32 No. 2 25

8 122 PISARENKO et al. The time series Y(t can be divided into 12 artial series corresonding to months: Y( τβ = Y( 12( τ 1 + β, (1 β = 1,, 12; τ = 2,, N Y, where β denotes months, and τ denotes successive observational years. Thus, Y(τ β, τ = 2,, N Y, is the time series of deviations from the cyclic mean value that corresonds to the month with the number β. Similarly to formula (9, we can divide the time series of rediction errors (8: ( α e β ( τ = e ( α { 12[ τ ( n + 1 ] + β }, (11 τ = n + 1,, N Y. We can introduce an index to characterize the efficiency of rediction for month β: N Y µ ( α ( β Y 2 ( α = ( τβ/ ( e β ( τ 2. (12 τ = n + 1 τ = n + 1 In addition to that, we can introduce two indices to characterize the overall efficiency of the forecast averaged over all months: γ γ 1 = = N t = L + 1 N t = L + 1 (13 Both moving and increasing time windows were used for linear AR-redictors. The increasing time series always yield the best result. This is why the results resented below were obtained only with the use of increasing time windows. Table 2 gives the results of exeriments with ARrediction for all nine time series. The rediction algorithm based on AR-model with = 1 12 was tested for each series. An increasing time window beginning from the 361st value (January that follows the first 3 years used for the initialization was used. The results of rediction with a standard AR-estimate and with an estimate using AR-PCP (α = were comared. For each month β = 1,, 12 and for each method α =, 1, the efficiency of the monthly rediction µ (α (β was evaluated. Suose that α*(β and *(β is a solution of the simle maximization roblem: (14 We will refer to the series of airs (α*(β, *(β as a rediction scenario, because it determines the choice law of the tye α of AR-estimate for any AR-order and month β. Prediction scenarios for each river are given in Table 2. The lines marked by a symbol µ contain the rediction efficiency for each month and the N N Y X 2 (/ t e 2 (, t t = L + 1 N Y 2 (/ t e 2 (. t t = L + 1 α* ( β, * ( β : µ ( α ( β max. α, overall efficiency γ and γ 1 obtained as a result of the use of the given rediction scenario. The last column of Table 2 contains the values of γ m for comarison with γ /γ 1. In this case, γ ~ γ m γ 1, since the ratio γ m was calculated for the time interval [τ + 1, N], while equation (13 was written for time interval [L + 1, N]. The value γ 1 shows the rediction efficiency for the stochastic comonent. It is worth mentioning that scenarios (14 in Table 2 were obtained by rocessing the entire observational intervals for each river. Therefore, such scenarios can be of use for rediction in the future (rediction efficiency estimates are available for them. At the same time, it would be of interest to test the forecast algorithm using the standard method: dividing the observational interval into two equal arts and using one of them to find a scenario and the other for its examination. The rincial obstacle for such exeriments is the short length of the observational intervals. The duration (the number of observations used for the AR-PCP-estimate is the number of years N Y. To initialize AR-redictor (i.e., to evaluate AR-coefficients, we will use the initial time interval with a length of n = 3 years, which seems not sufficient for a reliable AR-PCP-estimate, esecially, in the case of large values. This is why n cannot be reduced. However, if we divide the interval in half, we will have only (N Y /2 n observations for the search for a scenario in the first art and for studying its quality in the second art. Taking into account that N Y = years, we obtain observations for the latter number rather than (N Y n for the results given in Table 2. Nevertheless, a validation exeriment was made for five rivers (the Irtysh, Northern Dvina, Oka, Elbe, and Loire with the longest observational intervals. The observational intervals were divided into two aroximately equal arts (not strictly equal, because the first and second arts have to begin from January. Table 3 gives the results of validation. The alication of rediction scenarios derived from the first half of the observational interval to the second half yielded analogous results for all months and for all full efficiencies. Comarison of Tables 2 and 3 shows that the scenarios and efficiencies sometimes can radically differ (because of the small number of samles, though the efficiencies, esecially full ones, are almost the same. This fact demonstrates certain stability of the rediction results, given in Table 3. ARTIFICIAL NEURON NETWORK MODEL Let us aly the method of artificial neuron network to the roblem of rediction of time series Y(t a stochastic comonent of river runoff. We will make an ANN-rediction in a moving time window with a length of L and comare it with an ordinary AR-rediction in the same time window (288 observations or 24 years. The need to use moving time window is WATER RESOURCES Vol. 32 No. 2 25

9 Table 2. Prediction scenarios and its efficiency STATISTICAL METHODS FOR RIVER RUNOFF PREDICTION 123 River β γ m γ 1 γ Amur α* * µ Irtysh α* * µ North. Dvina α* * µ Oka α* * µ Vistula α* * µ Elbe α* 1 1 * µ Danube α* * µ Loire α* * µ Glomma α* * µ caused by the comutational comlexity of the assessment of ANN arameters for an increasing time window (the moving window will require only a slight modernization of the current arameters of ANN. We use for comarison only the standard AR-estimate with a constant AR-order = 12. AR-PCP-estimate is not used for this case because the number of observations (288/12 = 24 is too small to allow a reliable statistical estimate of AR-coefficients. However, 288 observations enable us to make a standard AR-estimate. Moreover, the maximum AR-order is used ( = 12. Such choice does not make AR-rediction worse. We will try to use a simle ANN architecture with one latent level comrising neurons to make a forecast of the rocess for time moment t. For this urose, we consider q revious terms of the time series, calculate their weighted sum, and use this scalar value as an inut for every neutron. Thus, this rocedure is based on the formation of scalar signals: q z j ( t = w ji Y ( t i + c, j = 1,,. j i = 1 (15 Each signal (15 is assed through the nonlinear activation function f(s. The same function ξ j (t = f(z j (t, f(s = s/( s + 1/2, (16 where ξ j (t are neuron outut signals, is used for any neutron. We define the forecast value Y(t as the weighted sum of neuron signals lus the shift arameter: Y ( n t ( = α j ξ j ( t + β, j = 1 (17 where (n is ANN redictor. Thus, the comlete vector of ANN arameters has the form θ = ( βα, j, c j, w ij, j = 1,, ; i = 1,, q. (18 WATER RESOURCES Vol. 32 No. 2 25

10 124 PISARENKO et al. Table 3. Verification of rediction scenarios for rivers with longest observational series River β γ 1 γ Irtysh α* * µ North. Dvina α* * µ Oka α* * µ Elbe α* * µ Loire α* * µ Vector θ has the dimension q Thus, (17 can be written as Y ( n t ( = Y ( n ( t θ. Vector θ is determined by minimizing the quadratic criterion of the adjustment quality: τ J( θ = ( Y( t Y ( n ( t θ 2 min, t = τ L + 1+ q (19 where τ takes the value L for the first time window of initialization. Hereafter, τ is the time moment corresonding to the right-hand end of the current moving time window, τ = L,, N. Let us suose that θ (τ is the vector of arameter estimates obtained by solving roblem (19. It should be mentioned that function J(θ has, as a rule, a number of local minimums because of the high dimension of θ and a nonlinear character of this function. The roblem of searching for the global minimum is difficult to solve. To find the resective minimum (19, let us consider 1 3 random initial values θ from a aralleleied, containing the zero oint, with a side of For every initial value θ, we imlement the gradient method to search for a local minimum with a maximum ste of gradient of 1 4, so that when the cost function increases, the ste of gradient is reduced by 1/2. The total number of stes of the gradient method is limited either by the number of 1 4 stes or by the moment when the ste becomes less than 1 8. The global minimum is chosen from all the local minimums obtained. Once the estimate θ (τ is obtained for the first time window, this window starts moving from the left to the right with a ste of 1. In this case, the revious vector θ θ is used as the initial oint. Thus, we can determine a one-ste-forward rediction for each time window: ( n Y F (τ + 1 = Y ( n (n(τ + 1 θ (τ for τ = L,, N 1. In this case, the error of the ANN-redictor δ ( n ( n ( τ = Y( τ Y F ( τ, τ = ( L + 1,, N, (2 and the samle estimate of its variance in the moving time window of the same length L deends on the righthand end of this window: σ F 2 τ ( τ = ( δ ( n 2 (/ t ( L q, t = τ L + 1+ q τ = 2L,, N. (21 For the ANN-redictor, we have L = 288, = 2, q = 3; the total number of ANN arameters is 11, that is, aroximately the same as that for the AR-redictor. Now we can comare the four estimates of variance in the moving time window: for the initial time series X(t, for the stochastic comonent Y(t, for forecast errors of standard AR-estimate with an order of = 12; and for ANN rediction errors (21. Four lots of rediction variance for the Irtysh, Amur, Danube, and Elbe are shown in Fig. 5. Note that both AR- and ANN-redictors exhibit a decrease in variance, but AR-redictors are the best in all cases. This conclusion holds for other rivers as well. Thus, the statements, regarding the advantage of the ANNaroach to the formation of time series, which can be found in some ublications, are not confirmed. Simler and faster classical methods roved to be more efficient. It aears reasonable to continue discussion of this roblem. WATER RESOURCES Vol. 32 No. 2 25

11 STATISTICAL METHODS FOR RIVER RUNOFF PREDICTION 125 Variance Q 1 4, (m 3 /s 2 1 (a 1 1 (b 1 (c (d Months Fig. 5. Estimated rediction variance in moving time window for (a the Irtysh, (b Amur, (c Elbe, and (d Danube. Full thick lines are for the variance of the initial time series X (t, dashed lines are for variance estimates for the stochastic comonent Y(t, dashand-dot lines are for the error of ANN rediction (21; full thin lines are for the errors of standard AR-rediction in a moving time window with an order of = 12. All the curves are given as functions of the right-hand end of the moving time window. Time is given in months from the beginning of observations. CONCLUSIONS Statistical analysis allows the assessment of some arameters characterizing the features of the concrete hydrological regime. First, we should mention that the cyclic mean can be used for rediction. The nine rivers can be divided into two grous in accordance with the value of index γ m. The first grou has high seasonal (sring floods. Such rivers can be called seasonally controlled. Contrary to that, the second grou consists of the rivers the regime of which is only slightly deendent on snowmelt floods. The next characteristic of river regime is determined by the shae of its sectra (Fig. 2. The number of harmonics in the ower sectra is different for different rivers. In accordance with this characteristic, the rivers can be divided into three grous. The first grou (Loire and Danube has a single basic harmonic. Maybe, the Elbe can also be included in this grou, although, a weak second overtone can be seen in the ower sec- WATER RESOURCES Vol. 32 No. 2 25

12 126 PISARENKO et al. trum. A single convex sectral eak means that the trend resembles the harmonic, which confirms the analysis of Fig. 3. In other cases, the ower sectrum can contain the maximum ossible number of eaks (in this case, when the cyclic mean for 12 months is exanded in the Fourier series, there are six eaks. Again, all these sectral eaks have almost equal magnitudes. This grou of rivers includes the Oka, Northern Dvina, and maybe the Irtysh and Glomma. One eak means that the cyclic mean is somewhat similar to a delta function, that is, one eak out of the 12 monthly runoff values dominates over all others. Finally, there are an intermediate number of eaks in the Fourier transformation of the cyclic mean. This is characteristic of the Vistula and Amur. This characteristic of river regime can be called the sectral comlexity of the cyclic mean. The Monthly cyclic mean was simulated in [13] with the hel of trigonometric olynomials of the 2 4 orders. Thus, these rivers have an intermediate comlexity. As can be seen from the examles considered above, cases with different degree of comlexity are also ossible. In general, the redictability of monthly runoffs, estimated with the use of the PCP model and other statistical characteristics gives useful data for both theoretical studies of the hydrological regime of rivers and for ractical use. ACKNOWLEDGMENTS This study was financially suorted by INTAS, roject no , National Scientific Foundation EAR, roject no , and ISTC, roject no REFERENCES 1. Privalsky, V.E., Panchenko, V.A., and Asarina, E.Yu., Modeli vremennogo ryada s rilozheniyami k geofizicheskim issledovaniyam (Time Series Models with Alications ot Geohysical Studies, Leningrad: Gidrometeoizdat, Ratkovich, I.A. and Bolgov, M.V., Stokhasticheskoe modelirovanie kolebanii komonentov vodnogo balansa rechnogo stoka (Stochastic Modeling of Variations in Water Balance Comonents of River Runoff, Moscow: IVP RAN, Rytov, S.M., Vvedenie v statisticheskuyu radiofiziku (Introduction to Statistical Radiohysics, Moscow: Nauka, Box, G. and Jenkins, G., Time Series Analysis, Forecasting and Control, San Francisco: Holden-Day, Camolo, M., Soldati, A., and Andreussi, P., Forecasting River Flow Rate During Low-Flow Period Using Neural Networks, Water Resour. Res., 1999, vol. 35, no. 11, Cheng, B. and Titterington, D.M., Neural Networks: A Review from a Statistical Persective, Statistical Sci., 1994, vol. 9, no. 1, Dawson, C.W. and Wilby, R., Artificial Neural Network Aroach to Rainfall-Runoff Modelling, Hydrol. Sci. J., 1998, vol. 43, no. 1, Fernandez, B. and Salas, J.D., Periodic Gamma Autoregressive Processes for Oerational Hydrology, Water Resour. Res., 1974, vol. 22, Gardner, W.A. and Franks, L.E., Characterization of Cyclostationary Random Signal Processing, IEEE Trans. Inf. Theory, 1975, vol. 21, no. 1, Haykin, S., Neural Networks, a Comrehensive Foundation, New Jersey: Prentice Hall, Jayawardena, A.W. and Fernando, A., Use of Artificial Neural Networks in Estimation of Evaoration, Proc. Int. Conf. on Water Resour. Environ. Res., Kyoto, 1996, vol. 1, Jayawardena, A.W., Fernando, T.M.K.G., Chan, C.W., and Chan, W.C., Comarison of ANN, Dynamical Systems and Suort Vector Aroaches for River Discharge Prediction, 19th Chinese Control Conf., Hong Kong, vol. 2, Kashya, R.L. and Rao, A.R., Dynamic Stochastic Models from Emirical Data, New York: Academic, Loucks, D.P., Stedinger, J.R., and Haith, D.A., Water Resource System Planning and Analysis, Englewood Cliffs: Prentice Hall, Moss, M.E., Bryson, M.E., Autocorrelation Structure of Monthly Stream Flows, Water Resour. Res., 1974, vol. 1, Vecchia, A.V., Periodic ARMA (PARMA Model with Alication to Water Resources, Water Resour. Bull., 1985, vol. 21, WATER RESOURCES Vol. 32 No. 2 25