Akım serilerinin modellenmesi

Abstract

Bu çalışmada, Kolmogorov ve Kolmogorov-Gabor modeli aylık akım serilerinin türetilmesinde kullanılmış olup sonuçlar, yaygın olarak kullanılan Thomas-Fiering modeli sonuçları ile karşılaştırılmıstır. Uygulamada, 30 yıllık bir akım serisi kullanılarak bil gisayar ortamında, Thomas-Fiering, Kolmogorov ve Kolmogorov- Gabor modelleriyle 30 vıllık sentetik seriler türetilmiştir. Thomas-Fiering modelinden beklenen sonuçlar elde edilmiş ancak, Kolmogorov ve Kolmogorov-Gabor modellerinin akım serilerinin türetilmesinde kullanılabilmesi için bir rastgele değişkenin modele ilave edilmesi gerektiği ortaya çıkmıştır. Bu rastgele değişkenin analitik yolla belirlenmesinin güçlüğü nedeniyle, ampirik yaklaşım uygulanmış ve iyi sonuç veren bir esas geliştirilmeye çalışılmıştır. Sonuçlar, yaygın olarak kabul ve uygulama Fiering modeli sonuçlarına oldukça yakındır. Bazı istatistik parametreler açısından da veni model avantajlı görünmektedir.

Introduction It is well appreciated that planning and design of the most efficent reservoirs for hydropower, water supply, irrigation and other water use systems necessitate the use of runoff data which covers a long span of time in which occurs the most unfavourable combinations of extreme drought and flood periods or at least combinations of droughts and floods in a given return period. Neverthless, available records do not almost generally fulfill this requirement and artificial data is generated by means of statistical models. First and second order Markov Models and The Thomas and Fiering Model which is a special form of the first order Markov model are widely used in hydrology. A nonlinear auto-regressive model which is an extended form of the general Markov model is investigated in this study. Linear Auto-regressive Model Markov discovered that a stationary time series can be simulated by the linear autoregressive models. M x = I a x + 6 (1) i j=l j i-j i where xi are the terms of the time series, aj are the autoregressive coefficents and ei are the members of an N(0,1) independent stationary and ergodic stochastic process, and M denotes the order of the model. The first and second order linear Markov model, i.e. the models having M=l and M=2, are of particular interest in hydrology. Sometimes the third order model is used, but higher order linear Markov models are rarely used, because they require the estimation of a large number of coefficients to In the first order linear Markov model a is equal, the first autocorrelation coefficient, estimated from the sample as the first serial correlation coefficent, r In the second order linear Markov model a and a are usually computed from the equuations that relate them to the first two autocorrelation coefficients, and, both estimated in turn by the sample serial coefficients, r and r The first and second order linear Markov models are not capable of simulating the monthly runoff series and hence Thomas and Fiering developed a periodic form of the first order linear Markov Model. The model referred to as Thomas and Fiering Model, is given below. x, J x + b (x -x )+ s ( 1-r2 ) ? j j i - 1, j - 1 j - 1 j j i (2) This equation represents a set of 12 linear regression equations each of which gives the generated runoff for the month in turn. In this equation x and s are the mean and standard deviation of the j'th month, b and r are the regression and correlation coefficients between the runoff values of the (j-1) and j'th months, e are normally distributed random numbers whose mean is zero and standard deviation is equal to one. Thomas and Fiering model preserves the periodicity and statistical properties of the time series and widely used for simulating the monthly runoff data. Kolmogorov, that the equation the great probability theorist, proposed x = i M - y İ = l a x j i-j (3) could be used for the interpolation and extrapolation of stationary time series and gave the least square derivation of coefficients a. As will readily be recognized, this is an M'th order linear Markov model having no random compo nent. Nonlinear Models It was concieved.in 1954, by D.Gabor.The Nobel Prize winner that the generalized form of the Kolmogorov' s equation namely, VI M x = 2 M+l i=l h x i i M M + 2 2 h xx i=l,j = i i,,i i j M M M + 2 2 2 h xxx i=l j=i k=j i,j,k i j k (4) the discrete form of the functional series or Volterra Integral, could be used as a predicting model. Where x is the predicted value, x are the present and (M-l) past values of the time series, h,h,h.... are the linear, quadratic, cubic and so on, autoregessive coefficients. These coefficients reflect the first and higher degree auto- correlaton properties of the time series of interest and hence have a great capacity to store information about it. Müftüoğlu carried out the application of the quad ratic Kolmogorov-Gabor model for monthly runoff prediction. In this study, the model was investigated as a real-time predictor. Therefore, each prediction was made from 1:he preceeding recorded values included in the active past which we shall call here as "memory", i. e. the predictions were made on the basis of the M previously observed elements of the random process of interest. As the degree and the memory length of the Kolmogo rov-Gabor polynomial increases the information storage capacity of the model about the past of the process also increases. This in turn increases the amount of computations neccessary for model calibration. Present Model It was thought that the capacity of the Kolmogo runoff generator, i.e. as an previously generated values t theless, in order to reduce requirements, the model is term and applied a small modi linear memories do not have c model has the form: it would be interesting to know rov-Gabor model as a monthly operatör which operates on the o produce further values. Never- the computation time and space truncated from the second order fication that the linear and non ommon elements. Thus the present L x =2 M+C+l i=l h x + i i+C N N S 2 i=l j=i l.J X X i+L+C j+L+C VII where L, N and M are the linear, nonlinear and total memory lengths, respectively, C is the shift counter to facilitate the representation of the successive generations. It is easily seen that M=L+N. As stated above the longer the memory and the higher the degree of the model, the greater its capacity to store information about the occurences of past observations. Having a long total memory and involving quadratic operations, though on one part of the total memory, the present model can be considered as a good compromise between the Kolmogorov and the Kolmogorov-Gabor models. Preliminary investigations indicated that the Kol mogorov as well as the modified Kolmogorov-Gabor Models depicted reasonable adaptation to the calibration data but produced a monotonically decreasing series when used as synthetic data generators. In order to overcome this short- come a random component (e) is added to the model as follows L N N x = 2hx+2 2h x x +e (6) M+C+l i=l i i+C i=l j=i i,j i+L+C j+L+C The difficulty of the analytical determination of the random component, e, leads to the application of an emprical approach. The approach is based on the determination of the random component from the data used for calibration. For the purpose, the model (Eq.5) is first calibrated on the available data, then the data is regenerated by means of the calibrated model and then the differences between the synthetic and real data are taken. These differences, in fact form a set of random numbers which in turn is used in synthetic data generation by Eq.6. Model Calibration Calibration of the model involves the determination of the memory parameters and autoregressive coefficients. The objective is to have a model which is capable of generating data which posesses the statistical properties of the given data. This end can be achieved by regenerating a given data with minimum deviations. The coefficients which fulfil this requirement can be determined by minimizing the sum of squared deviations: VIII K L N N 2 E = S [ x - ( 2 h x + 2 2 h x x )] (7 C=M M+C+l i=l i i+C i=l j=i i,j i+L+C j+L+C where K is the number of observations for calibration. The values of h and h which minimize the objective function, E,can be obtained by equating to zero the partial derivatives of E with respect to these coefficients. The following are the resulting normal equations: K-M-l L N N 2 (2hxx+2 2hx x x ) C=0 i=l i i+C a+C i=l j=i i,j i+L+C j+L+C a+C K-M-l = 2 x x,a=l,2.. L (8 ) C=0 M+C+l a+C K-M-l L V (Shx x x C=0 i = l i i+C J3+L+C e + L+C N + S h x x x x ) j=i ' i,j i+L+C j+L+C 6+L+C a+L+C K-M-l = S x x x, (3 = 1,2,..N C=0 M+C+l 0+L+C e+L+C ?=fi,fi+l,..N The normal equations represent a set of linear equations, called 'regression equations', in the L+N(N+l)/2 unknown coefficients. The longer the span data of used for calibration the more representive the model will be. The minimum number of observations required to obtain a determine set of coefficients is 2M+1. A representative data must, howewer, have 6M-10M observations. IX Aplications The applications were carried out on the monthly data of River Tigris of various L and N values was produced by the The results produced by Model and by the Thomas the years 1946-1975. The model with have been applied. The best result model with LpIO N=2 and L=ll N=l. the Mth order Markov, by the Present and Fiering Model are given together in TABLE-1 for comparison. TABLE-1 Conclusions The comparison of the present model with the well known and widely used Thomas and Fiering model revealed that the present model is promising.