INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.

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1 INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.

2 Summary of the previous lecture Regression on Principal components x ij ( X ) ij xj s j y = Y y = i i Z = X A Y = ZΒ 2

3 MULTIVARIATE STOCHASTIC MODELS 3

4 Stochastic models discussed for single site in relation with the auto correlations and auto covariance. Thomas Fiering models Stationary and non-stationary models ARMA models Box Jenkins models 4

5 First order Markov process: Random component X t+ = µ x +ρ (X t µ x ) + ε t+ Deterministic component ε Mean 0 and variance σ ε 2 2 X = µ + ρ X µ + u σ ρ t+ x t x t+ x 5

6 First order Markov model with non-stationarity, for stream flow generation: σ X X t ρ j is serial correlation between flows of j th month and j+ th month. t i, j+ N(0, ) j+ 2 i, j+ = µ j+ + ρ j ij µ j + i, j+ σ j+ ρj σ j 6

7 ARIMA Models ARMA (p, q) Residuals of order q X t = φ X t- + φ 2 X t φ p X t-p + θ e t- + θ 2 e t θ q e t-q AR of order p + e t {e t } is the residual series Assumptions : {e t } has zero mean with uncorrelated terms p Xˆ = φ X + θ e t+ j t j j t j j= j= q 7

8 Data generation (or forecasting) on a random variate depending on two or more sites is usually required. For example, in the design of a reservoir, the flow from all the streams fed to the reservoir must be considered. If the time series for the random variables are independent, then the generation techniques for single site can be used. When they are not independent, it is important to consider the simultaneous behavior of the random variables. 8

9 Correlation of a random variable between two sites is cross-correlation. Lag zero cross-correlation is the correlation of a random variable at two points in the same time period. Lag k cross-correlation, r j,h (k) is the correlation between random variable at site j with the random variable at site h, with lag time k. 9

10 r jh, ( k) = n ( x ) ji, xj xhi, + k xh i= ( n ) k s s j h where n is the total number of pairs of observations on X j and X k, x j,i is the i th observation on X j xj, sj are the mean and standard deviation of the observations on X j 0

11 Example Obtain the lag one cross correlation of annual rainfall data in mm at two sites A and B. Year Annual rainfall at site A (mm) Annual rainfall at site B (mm) Year Annual rainfall at site A (mm) Annual rainfall at site B (mm)

12 Example (Contd.) Site A B Mean Std.dev lag one cross correlation of sites A and B is given by r AB, = n ( x ) Ai, xa xbi, + xb i= ( n ) s s A B 2

13 S.No. Annual Annual xai, xa (i) rainfall at A rainfall at B x x Bi, x Ai, xa + B xbi, + xb Σ

14 Example (Contd.) n ( x ) Ai, xa xbi, + xb i= = r AB, = = = n ( x ) Ai, xa xbi, + xb i= ( n ) s s A B 4

15 Multisite Markov model (Two sites): Model preserves mean, variance, skewness, lag one serial correlation and lag zero cross-correlation (Haan 977). One site is to be selected as key site. Selection may be based on the length of the data and the quality of the record. Consider j as the key site and h as the subordinate site to key site j. A sequence of observations is generated for site j using single site generation technique. Ref.: Haan, C.T. (977) Statistical methods in Hydrology, Iowa State University Press 5

16 A cross-correlation model is used to generate values of site h based on generated values at site j. s X = x + r 0 X x + us r 0 h 2 ht, h jh, jt, j t h jh, s j j and h refer to two sites, in this model First order Markov model with non-stationarity (single site) σ X X t j+ 2 i, j+ = µ j+ + ρ j ij µ j + i, j+ σ j+ ρj σ j i is year j is month in this model 6

17 where u t is a standardized random variate adjusted to incorporate the serial correlation at site h. u t ( X x ) ht, h 2 tt ζ sh = ζ + t t is a standardized random variate ζ = r r r 0 2 h j j, h r 0 2 jh, 7

18 Multisite Markov model: Multisite generation requires simultaneous generation of data at several sites while preserving the correlation between the data at various sites. Consider x j,t x jt, = ( x ) jt, xj s j 8

19 The first order Markov model for site h is x = ρ x + ε ρ 2 ht, + h ht, ht, + h The first order Markov model for site j is x = ρ x + ε ρ 2 jt, + j jt, jt, + j 2 X = µ + ρ X µ + u σ ρ t+ x t x t+ x µ= 0 and σ = because it is standardized data The equations are written in matrix form 9

20 X EX Gε = + t+ t where X t is a p x vector of standardized values of the variable generated at time t, E is a p x p diagonal matrix whose j th diagonal element is ρ j (), G is a p x p diagonal matrix whose j th diagonal element 2 is ρ j ε is a p x vector of random variates 20

21 ε is defined to preserve the first order serial correlation (auto correlation) of the x j s and the lag zero cross-correlation between x j and x h. ε is made of elements that are ε j,t+ ; each ε j,t+ is independent of x j,t ; ε j is N(0,) The cross correlation between ε j and ε h is ρ* j,h (0), { ρ } * j ρh ρj, h 0 ρ jh, ( 0) = 2 2 ρ ρ { }{ } j h ρ* j,h (0) reproduces the desired ρ j,h (0), which is the lag zero cross correlation between x j and x h. 2

22 ε = 2 AD e λ where D 2 is a p x p diagonal matrix whose j th λ diagonal element is the square root of the j th largest eigenvalue of the p x p correlation matrix whose elements are ρ* j,h (0) A is a p x p matrix consisting of eigenvectors of correlation matrix, e is p x vector of independent observations from N (0,) 22

23 Matalas (967) has given a multisite normal generation model that preserves the means, variances, lag one serial correlation, lag one crosscorrelations and lag zero cross-correlations. X = AX + Bε t+ t t+ where X t and X t+ are p x vectors representing standardized data corresponding to p sites at time steps t and t+ resp. Ref.: Matalas, N.C. (967) Mathematical assessment of synthetic hydrology, Water Resources Research 3(4):

24 ε t+ is a form of N(0,) with ε t+ independent of X t. A and B are coefficient matrices of size p x p. X t+ (, t+ ) ( 2, t+ ) x x.. = x( i, t+ ).. x( p, t+ ) X t (, t) ( 2, t) x x.. = xit (, ).. x( p, t) ε t + (, t + ) ( 2, t + ) ε ε.. = ε ( it, + ).. ε ( pt, + ) 24

25 The equation form is where p = + + x a x i, t b ε i, t it, + i, j i, j j= j= a i,j and b i,j denote the (i, j)th elements of the matrices A and B. B is assumed to be lower triangular matrix. i 25

26 Coefficient matrices A and B: The expectation of X t X t is denoted by M 0 M = E X X ' 0 t t If m 0 (i, j) is a element of M 0 matrix in the i th row and j th column, m0 ( i, j) = x( i, ) x( j, ) x( i,2 ) x( j,2 )... x( i, n) x( j, n) n + + n m0 i j = x i t x j t (, ) (, ) (, ) n t = 26

27 m 0 ( i, j) X x X x = n s s n it, i jt, j t= i j i.e., m 0 (i, j) is correlation coefficient between the data at sites i and j at time t. Therefore M 0 is the cross-covariance matrix of lag zero 27

28 The expectation of X t X t- is denoted by M M = E X X ' t t If m (i, j) is a element of M matrix in the i th row and j th column, m i j x i x j x i x j x i n x j n n (, ) = (,) (,0) + (,2) (,) +... (, ) (, ) n (, ) = x( i, t) x( j, t ) m i j n t= 2 28

29 (, ) m i j X x X x = n s s n it, i jt, j t= 2 i j i.e., m (i, j) represents lag one cross correlation between the data at sites i and j. Therefore M is the cross-covariance matrix of lag one 29

30 Considering the model, X = AX + Bε t+ t t+ Post multiplying with X t on both sides and taking the expectation,. ' ' ' E Xt X t AE XtXt BE ε + = + t+ X t M A 0 = = AM + M M

31 Post multiplying with X t+ on both sides and taking the expectation,. X = AX + Bε t+ t t+ ' ' ' E Xt X t AE XtXt BE ε + + = + + t+ X t+ M 0 3

32 M M = E X X ' t t { } ' ' E XtX t = ' { } ' ' = E XtX t ' = E Xt Xt or ' ' M = E XtX t + 32

33 ε ' X = ε { AX + Bε } ' t+ t+ t+ t t+ ε = XA+ ε ε ' ' ' ' t+ t t+ t+ Taking expectation on both sides, ' ' ' ' ' E εt X t E εt XtA ε ε + + = + + t t B + + ' ' ' ' = E εt XtA E ε ε + + t+ t+ B = 0 + IB = B ' ' B 33

34 Substituting in the equation, ' ' ' E Xt X t AE XtXt BE ε + + = + + t+ X t+ M = AM + BB ' 0 M = M M M + BB ' ' 0 0 ' A = M M 0 BB = M M M M ' ' 0 0 If C = BB C = M M M M '

35 B does not have a unique solution. One method is to assume B is a lower triangular matrix. BB ' b, b, b 2,... b p, b 2, b 2, b 2, 2... b p, 2 = b( p, ) b( p,2 )... b( p, p) b( p, p) (,) (, 2) (,3 ).. (, ) ( 2,) ( 2, 2) ( 2,3 ).. ( 2, ) c c c c p b b b b p C = b( p, ) b( p,2 )... b( p, p) 35

36 The diagonal elements of the B matrix are obtained as, b, = c, { } b 2, 2 = c 2, 2 b 2,... (, ) = (, ) (, ) (, 2 )... (,) { } bkk ckk b kk b kk b k

37 The elements in the k th row are obtained as, (, ) b k j b k, b k,2... = = c k, b, c k,2 b 2, b k, b 2, 2 (, ) (,) (,) (,2) (,2)... (, ) (, ) b( j, j) c k j b j b k b j b k b j j b k j = 37

38 If the model is to fit the data, the matrices M 0 and BB should be positive definite. This condition is used to check the inconsistency in the data. Assumption is that the model is multivariate normal. 38