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
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
( ) ( ) ( ) ( ) ( ) ( )
,
T k k k k k k
R A A A ξ ϕ = = r r
1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
,
T k k k k k k k k
R P P P ξ ϕ = Λ = Λ r r
( ) ( ) ( )
[1] [ ]
T k k T k
I B J I B B k J I − = − r K K K K r
( ) ( ) ( ) ( ), k k k k
B D ξ ϕ = r r
1 ( ) 1 k k
C C D C − = K K K K
( ) [ ]
,
k k
R B k R = r r
Models of gaussian joint time series
( ) 1 2
( ), ( ), , ( ) ,
T
T T T n n
t t t ξ ξ ξ ξ = r r r r K
1 2
( ) ( , , , ) ,
T
i i i i pi
t ξ ξ ξ ξ ξ = = r r K
( ) 1 2
( ), ( ), , ( ) ,
T
T T T n n
t t t η η η η = r r r r K
1 2
( ) ( , , , ) , 1, ,
T
i i i i pi
t i n η η η η η = = = r r K K
1 1 1 2 ( ) 1 2
... ... , ... ... ... ... ...
n T n n T T n n
R R R R R R R R R R
− − − −
= ,
i i k i i k i i k i i k
k
R R R R R
ξ ξ ξη ηη η ξ
+ + + +
=
r r r r r r r r
where
, 0, , 1
k
R k n = − K 2 2 p p ×
1 1 1 1 1 1 1 1 1 1 2 1 1
( )
... ...
i i i i i i i i i i n i i n i i i i i i n i i i i i i n i i i i i i i i i i n i i i i i i i i i
T n
R R R R R 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 η η η η η η η ξ η ξ η ξ ξ ξ ξ η ξ ξ ξ η ξ ξ ξ η η η η η ξ η ξ
+ + + − + − + + − + + − + + + − + +
=
r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r
2 2 2 1 1 2 2 1 2 1 2
. ... ... ... ... ...
i n i i n i i n i i n i i n i i n i i n i i i i i i n i i n i i i i n i i n i i
T T
R R R R R R R R R R R R R R
η η η η ξ ξ ξ ξ η ξ ξ ξ η ξ ξ ξ η η η η η η η η ξ η ξ η ξ
+ − + − + − + − + − + − + − + − + − + − + −
r r r r r r r r r r r r r r r r r r r r r r r r r r r r
Simulation algorithm of gaussian vector
( ) 1 2
( , , , )
T T T T n n
ξ ξ ξ ξ = r r r r K
1 0 1 2 (1) (1) 1 2 ( 1) ( 1) 1
, [1] , [ 1] ,
T T n n n n n
C B J C B n J C ξ φ ξ ξ φ ξ ξ φ
− − −
= = + = − + r r r r r r K r r r r
1 2 1
, , , : , 0, , [ ] ( [ ],..., [ ]) , 1,..., 1,
T T T T T n k k p k l k
E I E k l B k B k B k k n φ φ φ φ φ φ φ = = ≠ = = − r r r r r r r r K are matrixes, are lower triangular matrixes,
[ ]
i
B k
p p ×
.
T i i i
C C Q =
i
C
Method of conditional distributions functions
( )
, 1,..., 1
p k p
I J k n I = = − K K K K K
p p ×
Calculation of matrix coefficients
( ) [ ]
,
k k
R B k R = r r %
( ) ( ) ( ) ( ), k k k k
R J R J = %
( )
[ ] [ ],
T k k
Q R B k R B k = − r r %
1 1 1 1 1 1 1 1 ( ) 1 1 ( ) 1 1 1 ( ) 1
[1] , [1] , , , ( [ 1], , [ 1]) [ ] [ 1] [ ] , ( [ 1], , [ 1]) [ ] [ 1] [ ] , [ 1] ( [ ]), [ 1]
T T T T T T T T k k k T T T T T k k k T k k k k k k k
B R R B R R Q R Q R B k B k B k B k B k J B k B k B k B k B k J B k Q R R J B k B k Q
− − + + − + + +
= = = = + + = − + + + = − + + = − + = % % % r r % K r r % % % % K r r % % % %
1 1 ( ) 1 1
( [ ]), [ ], [ ], , ( , , ) , ( , , ) , 1,..., 1.
T T k k k T T k k k k T T T T T k k k k k k k
R R J B k Q R R B k Q R R B k C C Q R R R R R R k n
− + −
= − = − = = = = − r r % r r r r % % % r r % K K
Recurrent algorithm
Models of periodically collerated random processes
( ) ( ),
i i
E t T E t ξ ξ + = ( ) ( ),
i i
D t T D t ξ ξ + =
( , ) ( , ).
i j i j
R t T t T R t t
ξ ξ
+ + =
( )
( ) 1 2
( ), ( ), , ( ) ,
T
n N
t t t ξ ξ ξ ξ = r K
Conditions for periodically colleration of random process :
ξ r
1 1
, , ( 0, 1),
N i i
T p t T t t t t t t
+
= Δ ≤ Δ = − = Δ =
Numerical simulation of gaussian periodically collerated sequences
1 2
( , , , ) , 1, ,
T
i i i pi
i n ξ ξ ξ ξ = = r K K
For infinite periodically collerated sequences
1 1 1 ( 1) 1
[ 1] [ 1] .
T T t t n t n n t
B n B n C ξ ξ ξ ϕ
− − − − −
= − + + − + r r r r K
we can use a many-dimensional autoregression model of the order n-1:
1 2
, , , ,
n
ξ ξ ξ r r r K K
1 2 1 2 2 ( 1) 1 ( 1) 2
, , , , , , , , , , , ,
p p p p n p n p np
ξ ξ ξ ξ ξ ξ ξ ξ ξ
+ + − + − +
K K K K
Scalar periodically collerated sequence
Method of inverse distribution function
1
,
t t t
F ξ η
−
= Φ
– standard normal distribution function ,
t
η
,
i j ij
M g ηη =
. 1 =
t
Dη
1 1
( ), ( ) ( ( )) ( ( )) ( , , ) ,
i j i j
i j i j ij F F ij i j F F ij i j ij
M M M r R g D D R g F x F y x y g dxdy ξ ξ ξ ξ ξ ξ ϕ
∞ ∞ − − −∞ −∞
− = = = Φ Φ
2 2 2 2
2 ( , , ) 2 1 exp( ) 2(1 )
ij ij ij ij
g xy x y x y g g g ϕ π ⎡ ⎤ − − = − ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦
.
1 ( )
i j
ij F F ij
g R r
−
=
2
1 ( ) 2
x u
x e du π
− −∞
Φ =
∫
– gaussian sequence,
0,
t
Mη =
distribution is estimated
( ) F x
ξ
* * * 1 2
, , ,
n
η η η K
2
* 1 *
1 ( ( )), ( ) 2
x u i i
F x e du
ξ
η ξ π
− − −∞
= Φ Φ =
is estimated.
* * * 1
, , ,
N N N n
R R R K
function is constracted.
1 2
, , ,
n
η η η K
* * * 1
, , ,
N N N n
R R R K
constracted.
i
η
1( (
))
i i
F
ξ
ξ η
−
= Φ
Simulation method based on the normalization of a real time series
1 2
, , ,
n
ξ ξ ξ K
* * * 1 2
, , ,
n
ξ ξ ξ K
The correlation function is determined by used transformations
Aproximation of distributions
Mixture of two gaussian distributions:
Temperature , 1 ≤ ≤
t
θ
( ) ( ) ,
, , , ,
1 , 2 , * , 2 , 2 2 , 1 , 2 , 1
=
− =
k i t i t i t i t t t t t
p p p F σ σ µ µ θ
–––––––––––––––––––––––– А. С. Марченко, Л.А. Минакова. Вероятностная модель временных рядов температуры воздуха. // Метеорология и Гидрология, 1980, № 9, с. 39-47. ( )
( )
( )
2 2 1, 2, 2 2 1, 2,
2 2 1, 2,
1 ( ) , 2 2
t t t t
x x t t t t t
p x e e
µ µ σ σ
θ θ πσ πσ
− − − −
− = +
( )
* 1 , t t
m µ =
( )
( ) .
2 1 * 2 t t
s = σ
T
0.00 0.10 0.20 0.30P
1 2T
0.00 0.10 0.20 0.30P
1 2Fig 1. Empirical -1 и model -2 density.
( ) ( ) ( ) ( )
( )
15055 , , , , , 00035 , , 64687 , , 64652 ,
2 , 2 2 , 1 , 2 , 1 1 1 * 1 1 *
= = − = =
t t t t t t t t t
F А А А А σ σ µ µ θ
.
( ) ( ) ( ) ( )
( )
09424 , , , , , 29382 , , 00000 , , 29382 ,
2 , 2 2 , 1 , 2 , 1 1 1 * 1 1 *
= = − = =
t t t t t t t t t
F А А А А σ σ µ µ θ
Fig 2. Empirical -1 и model -2 density.
Mixture of two gamma distributions:
Module of wind speed
( )
( ) ,
, , , ,
1 , 2 , * , , 2 , 1 , 2 , 1
=
− =
k i t i t i t i t t t t t
p p p F λ λ ν ν θ
–––––––––––––––––––––––– А. С. Марченко, А. Г. Семочкин. Модели одномерных и совместных распределений неотрицательных случайных величин. // Метеорология и Гидрология, 1982, № 3, с. 50-56.
( ) (
)
( ),
1 ) (
, 2 , 2 1 , 1 , 1 1
, 2 , 2 , 2 , 1 , 1 , 1
t t x t t t x t t
Г e x Г e x x p
t t t t t t
ν λ θ ν λ θ
ν λ ν ν λ ν − − − −
− + =
, 1 ≤ ≤
t
θ
( ),
2 * t t
m = µ
( )
( ) .
2 2 * 2 t t
s = σ
0.00 4.00 8.00 12.00 16.00
V, м/с
0.00 0.05 0.10 0.15 0.20 0.25
Р
1 2( ) ( ) ( ) ( )
( )
15055 , , , , , 00035 , , 64687 , , 64652 ,
2 , 2 2 , 1 , 2 , 1 1 1 * 1 1 *
= = − = =
t t t t t t t t t
F А А А А σ σ µ µ θ
Aproximation of distributions
Fig 3. Empirical -1 и model -2 density.
1 1 2 2
1 1 1 1 1 1 2 2
1 ( ) (1 ) (1 ) ( , ) ( , )
H
f x x x x x B B
ν µ ν µ
θ θ ν µ ν µ
− − − −
− = ⋅ ⋅ − + ⋅ ⋅ −
1, θ ≤ ≤
1 θ θʹ″ + =
:
Approximation of one-dimensional distribution of relative humidity by mixture of two beta distributions
µ µ µ µ
, , 1 1
1 1 ( )( ) ,
i i k
m n k j j T i n m i k n m j i
R E E m n
ζ ζ
ζ ζ ζ ζ
+
− + = =
⎛ ⎞ = − − ⎜ ⎟ ⎝ ⎠
u r u r u r u r
µ µ
, 1 1
1 1 .
m n j n m i j i
E m n ζ ζ
= =
⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
u r u r
Estimation of the matrix covariation function
Real data: 8 times per day measurements of air temperature, Astrakhan, 34 years.
Air temperature, January 1967 (red line) и 1987 (blue line) years.
50 100 150 200 250
5 Номер измерения
Температура
Correlation function of daily average temperature (- - -) and the correlation function of the temperature with a step equal 3 hour (----), calculated in the assumption of stationarity of the process, January (Astrakhan)
Correlation function of time series of air temperature (the stationary approximation)
Let is periodically collerated sequence with zero mean and block Toeplitz covariance matrix
is value of linear function at the point passing through the points
and weighted average of the elements
Statement
If for all then
Consequent
Let At for all the inequality takes place.
Piecewise-linear envelope correlation function of periodically correlated process
( ) n
R
. n → ∞
1 2 1 2 2 ( 1) 1 ( 1) 2
, , , , , , , , , , , ,
p p p p n p n p np
ξ ξ ξ ξ ξ ξ ξ ξ ξ
+ + − + − +
K K K K
1 1 1 2 ( ) 1 2
... ... , ... ... ... ... ...
n T n n T T n n
R R R R R R R R R R
− − − −
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
1 1 1 2 ( 1) ( 2)
,
p k k k p k k k k p p k k k
r r r r r r R r r r
− − − − − − −
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ L L L L L L K
h
ρ
* lp t
ρ
+
( ) lp t +
1
( ) ( 1) , , ( 1) ,
l l
n l pr n l pr lp l p np np
+
⎛ ⎞ ⎛ ⎞ − − − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠
0, 2, 0, 1, l n t p = − = −
, i k
,
i k k
r r ≥
* , lp t lp t
ρ ρ
+ +
≤
. n h np −
,
T k k
R R ≠
n → ∞
1
1 .
np h h t t h t
np ρ ξ ξ
− + =
=
∑
* lp t lp t
ρ ρ
+ +
≤ , i k
Example
1 2 1 2 2 ( 1) 1 ( 1) 2
, , , , , , , , , , , ,
p p p p n p n p np
ξ ξ ξ ξ ξ ξ ξ ξ ξ
+ + − + − +
K K K K
20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0
( )
,
n p n
R G R = ⊗
( ), exp( ), , 0, , 1, ( ), exp( ), , 0, , 1, =0.9, =0.4, =8.
ij ij p p p kl kl n n n
G g g i j i j p R r r k l k l n p α β α β = = − − = − = = − − = − K K
Let be periodically correlated sequence with block -Toeplitz covariation matrix where Correlation function of sequence calculated with the help of the model samples in the stationary approximation (red curve) and piecewise-linear envelope curve (blue curve).
Numerical simulation of air temperature time series
Correlation functions of real (left part) and model (right part) time series.(Periodically
correlated, january, Astrakhan) Correlation functions of actual (red line) and simulated in stationary approximation (blue line) time series (january, Astrakhan)
50 100 150 200 250
0.0 0.4 0.8 1.2
Номер измерения Корреляция
20 40 60 0.7 0.8 0.9 1.0 r (i, i+1) r (i, i+2) r (i, i+4)20 40 60 0.7 0.8 0.9 1.0
r (i, i+2) r (i, i+1) r (i, i+4) Statistical characteristics of real and simulated time series
LºС 3 hours 6 hours РД СМ СМКЛ ПКМ РД СМ СМКЛ ПКМ
0.6509 0.6512 0.6516 0.6508 0.5412 0.5415 0.5419 0.5410
0.0364 0.0362 0.0371 0.0365 0.0347 0.0342 0.0350 0.0348
0.0312 0.0315 0.0319 0.0311 0.0301 0.0307 0.0311 0.0302
0.0282 0.0283 0.0284 0.0280 0.0225 0.0229 0.0232 0.0224
0.0000 0.0001 0.0001 0.0001 0.0000 0.0001 0.0001 0.0003 t ºС 6 hours 9 hours РД СМ СМКЛ ПКМ РД СМ СМКЛ ПКМ 5 0.1310 0.1520 0.1582 0.1339 0.2325 0.2397 0.0241 0.2334 10 0.0112 0.0201 0.0012 0.0109 0.0330 0.0313 0.0311 0.0330 15 0.0020 0.0009 0.0009 0.0018 0.0040 0.0037 0.0036 0.0041 20 0.0004 0.0002 0.0002 0.0003 0.0007 0.0011 0.0012 0.0007 25 0.0002 0.0001 0.0001 0.0003 0.0002 0.0002 0.0001 0.0002
Table 1. Probabilities of the events: the air temperature decreases below of the certain level during the given period Table 2. Probabilities of the events: temperature changes on the certain value during the given period
Probability that the temperature changes by С = 1,2,3,4,5,10,12 (°С) during 6,12,18,24 hours. Model data - (-----) and real data - (- - -) ( May-June, Sverdlovsk)
( )
ip l jp m
P C ξ ξ
+ +
− ≥
Statistical characteristics of real and model time series
(periodically correlated model)
Average values of air temterature and levels С = 0, 6, 10 (°С) by day (Sverdlovsk).
, 1,
i
E i p ξ = K
Periodically correlated model with linear trend
Probability of temperature decreasing below the level С during the given period
, ( , ) i l
P c L
3 , 10 . L hour C C = = 3 , 6 . L hour C C = =
3 , . L hour C C = = 6 , . L hour C C = =
Numerical models of non-stationary time series of air temperature and wind (Astrakhan)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 1 1 2 2 1 2 2
,
T
R R R M M M R R
ξ ξ ξ ξ ξξ ξ ξ ξ ξ
ξ ξ ξ ξ ⎛ ⎞ ⎜ ⎟ = − − = ⎜ ⎟ ⎝ ⎠
r r r r rr r r r r
r r r r
( )
( )
( ) ( )
( )
( ) ( )
( )
* * * * * 1
1 [ ] [ ] . 2 1 1
L N T ij i i j j ks k l k s l s l L n
R n m n m L N ξ ξ
+ + =− =
= − − + − ∑∑ r r r r
, 1, , , k s K = K
N - years, L – smoothing step,
, 1,2 i j =
0.00 100.00 200.00 300.00
Сроки
0.00 5.00
Т
100 200 300Сроки
0.00 4.00 8.00 12.00V, м/с
Empirical characteristics of air temperature and wind speed
Mean ¡ Аsymmetry ¡ Variance ¡
2 4 6 8
t
2.50 3.00 3.50 4.00 4.50 5.00
V, м/с
1 2 3 4Means of module of wind speed for 2 days: 1 – mean for 23 of january; 2 – mean for 24 of february; 3, 4 – limits of statistical estimation error 1 и 2.
V T
Structure of correlation matrix
0.00 100.00 200.00 300.00 0.80 0.84 0.88 0.92 0.96 1.00 0.00 100.00 200.00 300.00 0.50 0.60 0.70 0.80 0.90 1.00
The first collateral diagonal of the correlation matrix of air temperature The first collateral diagonal of the correlation matrix of module of wind speed
Precision of reproduction of the entrance parameters of models
Mean Standart deviation
1 – real data, 2 – model data
0.00 100.00 200.00 300.00 4.00 5.00 6.00 7.00 8.00 1 20.00 100.00 200.00 300.00
0.00 0.40 0.80 1.20
1 2Line of correlation matrix
0.00 100.00 200.00 300.00 Results of joint modeling of air temperature and module of wind speed
Wind speed 2,5 m/s - 4,5 m/s
1 – model data, 2 – real data.
40 80 120
t
0.00 0.05 0.10 0.15 0.20 0.25
P
1 2
40 80 120 160
Сроки
0.00 0.10 0.20 0.30
Р
1 2
Wind speed 4,5 m/s - 6,5 m/s
Probabilities of joint realization of events:
1 2
15 , T C V V V < − ≤ <
Results of joint modeling of air temperature and module of wind speed
8 16 24 32 40
Сроки
0.00 0.04 0.08 0.12
Р
1 2
0.00 10.00 20.00 30.00 40.00 0.00 0.10 0.20 0.30 0.40
1 2
1 – model data, 2 – real data. Probabilities of joint realization of events:
2.5 / , ,
i
V m s T T − < 7, 4, 1, 0, 2, 8, 11, 14, 17, 20 , T C = + + + − − − − − −
0C
(Astrakhan)
8 16 24 32
Сроки
0.00 0.10 0.20 0.30 0.40 0.50
Р
1 2 3
8 16 24 32
Сроки
0.00 0.10 0.20 0.30
Р
1 2 3
Results of joint modeling of air temperature and module of wind speed
1 – model data, 2 – real data. Probabilities of joint realization of events:
2.5 / , ,
i
V m s C T T − < 7, 4, 1, 0, 2, 8, 11, 14, 17, 20 , T C = + + + − − − − − −
(Astrakhan)
Conclusion ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡The numerical stochastic parametrical models of joint time series of
various weather elements (air temperature , speed of a wind, relative humidity etc.), taking into account one-dimensional distributions and matrix correlation functions of real processes are constructed. The approximation of periodically correlated process is used. According to this approximation the daily periodic character of parameters of one-dimensional distributions and correlation functions is taken into account. On the basis of these models the statistical properties of the adverse meteorological phenomena (for example, long adverse temperature phenomena, adverse combinations of meteorological elements etc.) are investigated.
References
mics of Oceanology Processes. Gidrometeoizdat, Leningrad, 1987 [in Russian].
Inter-year variability of sea heaving. In: Problems of Study and Mathematical Modelling of Wind Agitation. Gidrometeoizdat, St. Petersburg, 1995, pp. 446-454 [in Russian].
Algorithms and Applications, Utrecht: VSP. The Netherlands, 1996.
Russian].
1987 [in Russian].
Meteorology and Hydrology (1980), No. 9, 39-47 [in Russian].
A.S. Marchenko and V.A. Ogorodnikov, The modelling of very long stationary gaussian sequences
with an arbitrary correlation function," USSR Comput. Math. Math. Phys., 1984, Vol. 24, № 10, pp. 1514-1519 [in Russian] S.M.Prigarin, Numerical modeling of random processes and fields, ICM&MG Publisher, Novosibirsk, 2005, 259p. [in Russian] Semochkin AG An effective way of estimating the parameters of the mixture of two biparametric distributions / / Meteorology and Hydrology, 1984, № 3, pp. 108-110 [in Russian].
Derenok K.V., Ogorodnikov V.A. Numerical simulation of significant long-term decreases in air temperature //Russ. J. Numer. Math. Modelling. (2008) 23, № 3, pp. 223-237.
Climate Change 2007: Impacts, Adaptation, and Vulnerability, edited by M.L. Parry, O.F. Canziani, J.P. Palutikof, P.J. van der Linden, and C.E. Hanson (Cambridge U.P., Cambridge, 2007). M. A.I. Evstafieva, E.I. Khlebnikova, V.A. Ogorodnikov (2005) Numerical stochastic models for complexes of time series of weather elements. Russ. Journal of Numerical Analysis and Mathematical Modelling, V.20, No. 6, pp. 535-548. Yu.A.Izraehl, N.D.Sirotenko (2003) Modeling of the influence of changes climate changes on the productivity of agriculture in Russia. Meteorology and Hydrology, 6, pp. 5-17 (in Russian). E.I. Khlebnikova (2008) Numerical stochastic models of meteorological time series and their applied possibilities. Trudy of Main Geophysical Observatory, v. 557, pp 34-52 (in Russian).
weather generator: effects of interdiurnal and interannual variability reproduction. Int. J. Climatol., 2005, pp.251-269. Marchenko A.S., Minakova L.A.(1982) Probabilistic model of air temperature time series}. Meteorology and Hydrology, 3, pp. 51-56.. I.M. Shkolnik, E.K. Molkentin, E.D. Nadezhdina, E.I. Khlebnikova , I.A. Sall (2008) Extreme features of thermal regime in Siberia and dynamics of fire danger conditions in in 21 century: estimations using the MGO regional climatic model. Meteorology and Hydrology, 3, pp. 5-15 (in Russian). Stardex, 2005: Statistical and regional dynamical downscaling of extremes for European regions. Final Report to the European Commision. http://www.cru.uea.uk/cru/projects/reports/stardex).