Numerical stochastic models of joint non-stationary and - PowerPoint PPT Presentation

Numerical stochastic models of joint non-stationary and non-gaussian time series of weather elements for solving the statistical meteorology problems Ogorodnikov V.A., Khlebnikova E.I., Derenok K.V Institute of Computational Mathematics and Mathematical Geophysics SB RAS A.I. Voeikov Main Geophysical Observatory

General simulation algorithms of gaussian sequences with a given correlation matrix r r T R A A , A = ξ = ϕ ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k r 1 r T R P P , P 2 = Λ ξ = Λ ϕ ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k r r B D ( ) , ξ = ϕ ( ) k ( ) k ( ) k k C 0 0 0 I 0 0 0 0 r 0 C 0 0 T B [1] J I 0 0 − 1 = K D ( ) k B = ( ) k K K K ( ) k K K K K r 0 0 0 C − T B [ ] k J I − k 1 ( ) k r r R B k ( ) [ ] R , = k k

Models of gaussian joint time series r r r r T ( ) T T T K ( ), t ( ), t , ( ) t , ξ = ξ ξ ξ ( ) n 1 2 n r r r r T ( T T T ) K ( ), t ( ), t , ( ) t , η = η η η ( ) n 1 2 n r r K ( ) t ( , , , ) , T ξ = ξ = ξ ξ ξ i i 1 i 2 i pi r r K ( ) t ( , , , ) , T η = η = η η η i i 1 i 2 i pi K i 1, , n = R R ... R 0 1 n 1 − R R r r r r T R R ... R ξ ξ ξη R i i k i i k , + + 1 0 n 2 = R − , = k R R ( ) n r ... ... ... ... r r r ηη η ξ i i k i i k + + T T R R ... R n 1 n 2 0 − − 2 p 2 p K R , k 0, , n 1 × = − where - are matrixes k

R R R R R R r r r r r r r r r r r r ξ ξ ξ η ξ ξ ξ η ξ ξ ξ η i i i i i i 1 i i 1 ... i i n 1 i i n 1 + + + − + − R R R R R R r r r r r r r r r r r r η η η η η η η ξ η ξ η ξ i i i i 1 i i n 1 i i i i 1 + i i n 1 + − + + − T R R R R R R r r r r r r r r r r r r ξ ξ ξ η ξ ξ ξ η ξ ξ ξ η i i 1 i i 1 i i i i ... i i n 2 i i n 2 + + + − + − R R R R R R R . = r r r r r r r r r r r r ( ) n η η η η η η η ξ η ξ η ξ i i 1 i i i i n 2 i i 1 + i i i i n 2 + − + + − ... ... ... ... T T R R R R R R r r r r r r r r r r r r ξ ξ ξ η ξ ξ ξ η ξ ξ ξ η i i n 1 i i n 1 i i n 2 i i n 2 ... i i i i + − + − + − + − R R R R R R r r r r r r r r r r r r η η η η η η η ξ η ξ η ξ i i n 1 i i n 2 i i + − + − i i n 1 i i n 2 i i + − + −

Simulation algorithm of gaussian vector r r r r T T T T K ( , , , ) ξ = ξ ξ ξ ( ) n 1 2 n Method of conditional distributions functions r r C , ξ = φ 1 0 1 r r r r T B [1] J C , ξ = ξ + φ 2 (1) (1) 1 2 K r r r r T B [ n 1] J C , ξ = − ξ + φ n ( n 1) ( n 1) n 1 n − − − r r r r r r r r K T T T T T , , , : E I , E 0, k l , B k [ ] ( B k [ ],..., B k [ ]) , k 1,..., n 1, φ φ φ φ φ = φ φ = ≠ = = − 1 2 n k k p k l 1 k p p p p C B k T [ ] are matrixes, are lower triangular matrixes, × C C Q . × = i i i i i K 0 I p K K K J , k 1,..., n 1 = = − ( ) k K I 0 p

Calculation of matrix coefficients r r r r % % T R B k ( ) [ ] R , Q R B k R B k [ ] [ ], = = − k 0 ( ) k k k % R J R J ( ) , = ( ) k ( ) k ( ) k k Recurrent algorithm % T T 1 T 1 B [1] R R , B [1] R R , − − = = 1 1 0 1 1 0 % % Q R , Q R , = = 0 0 0 0 r r % T T T T T K ( B [ k 1], , B [ k 1]) B [ ] k B [ k 1] B [ ] k J , + + = − + 1 k k 1 ( ) k + r r % % % % T K T T T T ( B [ k 1], , B [ k 1]) B [ ] k B [ k 1] B [ ] k J , + + = − + 1 k k 1 ( ) k + r r % % 1 T B [ k 1] Q − ( R R J B k [ ]), + = − k 1 k k 1 k ( ) k + + r r % % % 1 T T B [ k 1] Q ( R R J B k [ ]), − + = + − k 1 k k 1 k ( ) k + r r r r % % % T T Q R R B k [ ], Q R R B k [ ], = − = − k 0 k k 0 k r r % T T K T T K T C C Q , R ( R , , R ) , R ( R , , R ) , k 1,..., n 1. = = = = − k k k k 1 k k 1 k

Models of periodically collerated random processes Scalar periodically collerated sequence K K K K , , , , , , , , , , , , ξ ξ ξ ξ ξ ξ ξ ξ ξ 1 2 p p 1 p 2 2 p ( n 1) p 1 ( n 1) p 2 np + + − + − + r Conditions for periodically colleration of random process : ξ E ( t T ) E ( ), t D ( t T ) D ( ), t R t ( T t , T ) R t t ( , ). ξ + = ξ ξ + = ξ + + = i i i i i j i j ξ ξ Numerical simulation of gaussian periodically collerated sequences r K K ( , , , ) , i 1, , n T ξ = ξ ξ ξ = i 1 i 2 i pi r T K ( ), ( ), t t , ( t ) , ( ) ξ = ξ ξ ξ ( ) n 1 2 N T p t , T t , t t t ( t 0, t 1), = Δ ≤ Δ = − = Δ = N i 1 i 1 + r r r K K , , , , For infinite periodically collerated sequences ξ ξ ξ 1 2 n we can use a many-dimensional autoregression model of the order n-1: r r r r T T K B n [ 1] B [ n 1] C . ξ = − ξ + + − ξ + ϕ t 1 t 1 n 1 t ( n 1) n 1 t − − − − −

Method of inverse distribution function 1 ( ) F , − ( ) ξ = Φ η t t t x 1 2 u ( ) x e du – standard normal distribution function , − Φ = ∫ 2 π −∞ D η 1 . M g , M η = 0, η – gaussian sequence, = ηη = t t i j ij t M M M ξ ξ − ξ ξ i j i j r R ( g ), = = ij F F ij D D i j ξ ξ i j ∞ ∞ 1 1 R ( g ) F ( ( )) x F ( ( )) ( , , y x y g ) dxdy , − − = ∫ ∫ Φ Φ ϕ F F ij i j ij i j . −∞ −∞ 2 2 2 g xy x y ⎡ ⎤ − − ij 2 ( , , x y g ) 2 1 g exp( ) ϕ = π − ⎢ ⎥ ij ij 2 2(1 g ) − ⎢ ⎥ ⎣ ⎦ ij 1 ( ) g R r − = ij F F ij i j

Simulation method based on the normalization of a real time series * * * K , , , 1. On the basis of the real data the one-dimensional ξ ξ ξ 1 2 n distribution is estimated F ( ) x ξ * * K * , , , η η η 2. The normalized sequence is constructed : 1 2 n x 1 2 * 1 * u ( F ( )), ( ) x e du − − η = Φ ξ Φ = ∫ i i ξ 2 π −∞ 3. On the basis of the normalized sequence the covariance function N * N * K N * R , R , , R is estimated. 0 1 n K , , , η η η 4. Gaussian sequence with the covariance 1 2 n N * N * N * K R , R , , R function is constracted. 0 1 n K , , , 5. The sequence with the help of transformation ξ ξ ξ 1 2 n 1 ( ( F )) − of elements of Gaussian sequence is ξ = Φ η η i i i ξ constracted. The correlation function is determined by used transformations

Aproximation of distributions Temperature Mixture of two gaussian distributions: 2 2 ( x ) ( x ) − µ − µ 1, t 2, t 1 ( ) − − θ − θ 2 2 2 2 σ t σ 0 1 , p x ( ) t e e , 1, t 2, t ≤ θ ≤ = + t t 2 2 πσ πσ 1, t 2, t 2 ( ) , * 1 , * p p k − ( ) ( ) . 2 m ( ) 2 * ( ) 1 2 2 i , t i , t µ = s F , , , , ∑ σ = θ µ µ σ σ = t t t t t 1 , t 2 , t 1 , t 2 , t p 0.30 P i 1 = i , t 0.30 1 P 2 1 2 0.20 0.20 0.10 0.10 T T 0.00 0.00 -20 -10 0 10 -10 0 10 20 Fig 1. Empirical -1 и model -2 density . Fig 2. Empirical -1 и model -2 density . . * ( ) 1 ( ) 1 * ( ) 1 ( ) 1 А 0 , 64652 , А 0 , 64687 , А А 0 , 00035 = = − = * ( ) 1 ( ) 1 * ( ) 1 ( ) 1 А 0 , 29382 , А 0 , 00000 , А А 0 , 29382 t t t t = = − = t t t t ( ) 2 2 F , , , , 0 , 15055 ( ) θ µ µ σ σ = 2 2 F , , , , 0 , 09424 t 1 , t 2 , t 1 , t 2 , t θ µ µ σ σ = t 1 , t 2 , t 1 , t 2 , t –––––––––––––––––––––––– А . С . Марченко , Л . А . Минакова . Вероятностная модель временных рядов температуры воздуха . // Метеорология и Гидрология , 1980, № 9, с . 39-47.