Time Series Prediction - msu.rulnfm1.sai.msu.ru/~tempus/english/science/fulbright/Time Series... ·...

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


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


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


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


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


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


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


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


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

QQ

N1

N-N1

N1

xxQ

N1

N points

fxQN-N1

lQQf llfl1−= N-N1

points


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”


RESULTS OF EOP PREDICTION CAMPAIGN


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


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


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


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

Time Series Prediction - msu.rulnfm1.sai.msu.ru/~tempus/english/science/fulbright/Time Series... ·...

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