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Time Series Prediction
Leonid ZotovFulbright Scholar 2008-2009
San Juan, Puerto Rico, 8 May 2009
Sternberg Astronomical InstituteLomonosov Moscow State University
L. Zotov, San Juan, Puerto Rico, 8 May 2009
FROM THE BEGINNING OF THE HISTORY PEOPLE TRIED TO PREDICT FUTURE
“The Past isn't dead. It's not even past.“ W. Faulkner
“Good memory wherewith Nature has endowed us causes everything long past to seem present.” Leonardo da Vinci
“We remember only past events, not future ones.” The Mystery of Time, John Langone
“But I knew, that only that which repeats itself can be grasped by study…The future is immanent in the present.” Citadelle, Antoine de Saint-Exupery
PREDICTIVE ABILITY IS ONE OF THE MOST IMPORTANT CHARACTERISTIC OF THE SCIENTIFIC THEORY
Future of the Universe
L. Zotov, San Juan, Puerto Rico, 8 May 2009
Weather forecast Climate prediction
DETERMINISTIC AND STOCHASTIC COMPONENTS
Functional modeling Probabilistic modeling
0 40 80 120 160 200
-1.5
-1
-0.5
0
0.5
1
1.5
t,c
x,M
Harmonic trends
N.S. Sidorenkov 2009
Polynomial trends
1900 1920 1940 1960 1980 2000year
-0.8
-0.4
0
0.4
0.8
arcs
ec
Y-POLE
L. Zotov, San Juan, Puerto Rico, 8 May 2009
MATHEMATICAL AND PHYSICAL MODELING
Mathematical approximations Dynamic modeling
Auto Regression with Moving Average
Least Squares Methodu yx
“the flow of cause-effect relationshipsfrom the past to the future”Least Squares Collocation
Neural networks Kalman filtering
L. Zotov, San Juan, Puerto Rico, 8 May 2009
Series EOP С04 with 1-day step since 1962 г.
Series EOP С01 with 0.05-year step since 1890 г.
IVS VLBI
Laser ranging of Moon and Satellites
IGS GPS
PREDICTION OF THE ROTATION OF THE EARTHInitial data
DORIS
Observations from
Y, arcsecX, arcsec
Y, arcsec
X-Y POLE
1970 1980 1990 2000 2010year
-0.8
-0.4
0
0.4
0.8
sec
UT1-UTC
L. Zotov, San Juan, Puerto Rico, 8 May 2009
ANALYSIS OF THE EARTH ORIENTATION PARAMETERS (EOP)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2cycles per year
0.001
0.01
0.1
1
10 X-POLE
Fourier-spectrogram
Chandler and annual oscillations
18-year harmonic
Annual and semiannual modes1900 1920 1940 1960 1980 2000
years
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
arc
sec
SSA-decomposition of X-coordinate of the poleChandlerannualtrend
Singular spectral analysis
0 1 2 3 4cycles per year
1E-0061E-0050.0001
0.0010.01
0.11
10100 TAI-UTC
Wavelet-scalogram
frequency
time
0.6 0.8 1 1.2 1.4 1.6 1.8 21900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
L. Zotov, San Juan, Puerto Rico, 8 May 2009
TREND MODELING
Smooth trend – polynomial of the 2-d order
1970 1980 1990 2000 2010year
-0.4
-0.2
0
0.2
0.4
0.6
arcs
ec
Y-POLE
LS-based solution
1970 1980 1990 2000 2010year
-5
0
5
10
15
20
25
sec
UT1-TAI
L. Zotov, San Juan, Puerto Rico, 8 May 2009
51200 51600 52000 52400 52800 53200 53600MJD
-0.04
-0.02
0
0.02
0.04
сек
UT1-UTC
PERIODIC COMPONENTS MODELING AND PREDICTION
Nonlinear LS for phases and frequencies adjustment
∑=
+=n
iiiic tAtf
1)cos()( ϕω
0,0151
0,5118.6
0,149,3
0,0090,5
Amplitude, sec
Period, years0,151,19
0,091
Amplitude, arcsecPeriod, years
51200 51600 52000 52400 52800 53200 5360MJD
-0.4
-0.2
0
0.2
0.4
сек дуги
X-pole
sec
L. Zotov, San Juan, Puerto Rico, 8 May 2009
arcs
ec
AUTOREGRESSION MODELLING
N=50Burg method
Parameters are estimated from the minimization of the sum of squares of the discrepancies between the forward and backward predictions for the time series counts
Yule-Walker equations solution
Parameters are estimated from the first counts of the autocovariance function (ACF)
ACFestimation
i
N
jjiji nxax +−= ∑
=−
1
unbiased
∑−−
=+−
=1
0
*1)(mN
nnmnxx xx
mNmR
0 1000 2000 3000
-2
-1
0
1
2
3
biased
∑−−
=+=
1
0
*1)(mN
nnmnxx xx
NmR
0 1000 2000 3000
-1
-0.5
0
0.5
1
1.5
L. Zotov, San Juan, Puerto Rico, 8 May 2009
1970 1980 1990 2000 2010year
-0.8
-0.4
0
0.4
0.8se
cUT1-TAI
AR model of the 50 order solution with Burg algorithm
0 1 2 3 4 5delay
-0.02
-0.01
0
0.01
0.02
0.03
LEAST SQUARES COLLOCATION
Signal ACF and its forecast Interpolation
Filtering
Prediction
nnextll
extxx QQQ −=White noise
assumptionIQ nnn
2σ= lQQf llfx1−=
⎥⎦
⎤⎢⎣
⎡=
fx
xxextxx Q
N1
N-N1
N1
xxQ
N1
N points
fxQN-N1
lQQf llfl1−= N-N1
points
L. Zotov, San Juan, Puerto Rico, 8 May 2009
nxl +=
Hlx =
NEURAL NETWORK
Signal p(k) sequentially comes to the Time Delay Line
At every iteration vectorial signal pd(k)comes to the neurons of the input layer
Neuron with the linear activation function predicts the next value of the signal
Neural network is trained with use of the signal of comparisons (taken from the past interval)
Levenberg-Marquardtalgorithm is used for weights Wand bias b tuning
“Amoeba”
L. Zotov, San Juan, Puerto Rico, 8 May 2009
RESULTS OF EOP PREDICTION CAMPAIGN
L. Zotov, San Juan, Puerto Rico, 8 May 2009
PHYSICAL MODELLING - INPUT EXCITATION RECONSTRUCTION
1970 1980 1990 2000годы
-0.1
-0.05
0
0.05
0.1
0.15
сек дуги
X-POLEФильтр ВилсонаРегуляризация
Corrective smoothing, regularization
( ))()(1 mFWFF reg ⋅= −χ
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
ασ
αα σ c
it
regi
et
th c cos1),(
∫ −=∗= dttuthuh regreg )()(ξχ
1900 1920 1940 1960 1980 2000годы
-2
-1
0
1
2
сек дуги
X-POLE
Jefferson-Wilson filter
⎟⎟⎠
⎞⎜⎜⎝
⎛−
Δ= Δ
−
ΔΔ
+
Δ
22
)( tt
titt
c
tfi
memt
iet cc
σπ
σχ
L. Zotov, San Juan, Puerto Rico, 8 May 2009
1900 1920 1940 1960 1980 2000years
-0.04
-0.02
0
0.02
0.04
arc
sec
Excitation reconstructed for the componentsof SSA-decomposition of X-coordinate of the pole
Chandler componentannual component
EXCITATION AND POLE TRAJECTORY FORECAST
1980 1990 2000 2010годы
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
сек дуги
Y-POLE
Excitation forecast
L. Zotov, San Juan, Puerto Rico, 8 May 2009
Trajectory forecast by Kalman filter
1980 1990 2000 2010годы
0
0.2
0.4
0.6
сек дуги
Y-POLE
-1000 -800 -600 -400 -200 0thousand years
-5
-4.5
-4
-3.5
-3
-2.5
-2 Site 980
15-NEURON BRAIN FORECASTS
L. Zotov, San Juan, Puerto Rico, 8 May 2009
North Atlantic foraminiferal oxygen isotope data
Dow Jones Industrial Average
Jul-98 Apr-01 Jan-04 Oct-06 Jul-09time
4000
6000
8000
10000
12000
14000
16000
Jul -11
Muchas gracias para atencion
San Juan, Puerto Rico, 8 May 2009