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Sternberg Astronomical Institute Lomonosov Moscow State University

Time Series Prediction
Leonid Zotov
Fulbright Scholar 2008-2009

San Juan, Puerto Rico, 8 May 2009


FROM THE BEGINNING OF THE HISTORY PEOPLE TRIED TO PREDICT FUTURE

"We remember only past events, not future ones." The Mystery of Time, John Langone "Good memory wherewith Nature has endowed us causes everything long past to seem present." Leonardo da Vinci "The Past isn't dead. It's not even past." W. Faulkner "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
L. Zotov, San Juan, Puerto Rico, 8 May 2009


PREDICTIVE ABILITY IS ONE OF THE MOST IMPORTANT CHARACTERISTIC OF THE SCIENTIFIC THEORY
Weather forecast Climate prediction

Future of the Universe

L. Zotov, San Juan, Puerto Rico, 8 May 2009


DETERMINISTIC AND STOCHASTIC COMPONENTS
Functional modeling
Polynomial trends
0.8 0.4
0.5

Probabilistic modeling
x,M
1.5 1

Y-POLE

arcsec

0 -0.4

0

-0.5

-1

-0.8 1900 1920 1940 1960 year 1980 2000

-1.5 0 40 80 120 160 200

t,c

Harmonic trends

N.S. Sidorenkov

2009

L. Zotov, San Juan, Puerto Rico, 8 May 2009


MATHEMATICAL AND PHYSICAL MODELING
Mathematical approximations
Least Squares Method

Dynamic modeling

u

x

y

Auto Regression with Moving Average

Least Squares Collocation Neural networks

"the flow of cause-effect relationships from the past to the future" Kalman filtering

L. Zotov, San Juan, Puerto Rico, 8 May 2009


PREDICTION OF THE ROTATION OF THE EARTH
Initial data Series EOP 04 with 1-day step since 1962 . Series EOP 01 with 0.05-year step since 1890 .
X-Y POLE

Observations from IVS VLBI Laser ranging of Moon and Satellites IGS GPS DORIS
0.8
UT1-UTC

0.4

sec
X, arc sec

0

-0.4

c rcse Y, a

-0.8 1970 1 98 0 1990 year 2000 2010

L. Zotov, San Juan, Puerto Rico, 8 May 2009


ANALYSIS OF THE EARTH ORIENTATION PARAMETERS (EOP)
Fourier-spectrogram
10 1 0.1 0.01 0.001 0 0.2 0.4 0.6
X-POLE

2000 1990 1980 1970

Wavelet-scalogram

time
cycles per year
0.8 1 1.2 1.4 1.6 1.8 2

1960 1950 1940 1930 1920 1910 1900 0.6 0 .8 1 1.2 1. 4 1.6 1. 8 2

Chandler and annual oscillations
100 10 1 0.1 0.01 0.001 0.0001 1E-005 1E-006 0 1 2
TA I-U T C

frequency

Singular spectral analysis
0.3 0.2 0.1 arc sec 0 -0.1

SSA-decomposition of X-coordinate of the pole Chandler annual trend

cycles per year

3

4
-0.2 -0.3 1900 1920 1940 1960 years 1980 2000

Annual and semiannual modes 18-year harmonic

L. Zotov, San Juan, Puerto Rico, 8 May 2009


TREND MODELING
25 20 15 10 5 0 -5 19 70 1980

UT 1-T AI

0.6

Y-POLE

0.4

arcsec
1990 2000 20 10

0.2

sec

0

-0.2

-0.4

year

1970

1980

year

1990

2000

2010

Smooth trend ­ polynomial of the 2-d order LS-based solution
L. Zotov, San Juan, Puerto Rico, 8 May 2009


PERIODIC COMPONENTS MODELING AND PREDICTION
0.04 0. 4

UT1-UTC
0.02 0. 2

X-p o l e

0

arcsec 51200 51600 52000 52400 MJD 52800 53200 53600

sec

0

-0.02

-0. 2

-0.04

-0. 4 51200 51600 52000 52400 MJD 52800 53200 5360

Period, years 18.6 9,3 1 0,5

Amplitude, sec 0,51 0,14 0,015 0,009

fc (t ) =


i =1

n

Period, years

Amplitude, arcsec 0,15 0,09

Ai cos(it + i )

1,19 1

Nonlinear LS for phases and frequencies adjustment
L. Zotov, San Juan, Puerto Rico, 8 May 2009


AUTOREGRESSION MODELLING
0.8

xi = -


j =1

N

a j xi

-j

+ ni

UT 1- T AI

Parameters are estimated from the minimization of the sum of squares of the discrepancies between the forward and 0 backward predictions for the time series counts
-0.4
1. 5 1 0. 5 0 -0. 5 -1

0.4

Burg method

N=50

Yule-Walker equations solution
Parameters are estimated from the first counts of the autocovariance function (ACF)

se c

1 Rxx (m) = N

N - m -1


n =0

* xn + m xn

3 2 1

1 Rxx (m) = N -m

N - m -1


n =0

x

* n+ m n

x

ACF estimation
-0.8

AR model of the 50 order solution with Burg algorithm
-1 -2 0 1000

0

1970

biased

1980 2000

3000

year

1990

0

20002000 1000

2010 3000

unbiased

L. Zotov, San Juan, Puerto Rico, 8 May 2009


LEAST SQUARES COLLOCATION Signal ACF and its forecast
0.03

N points
0.02 0.01

l = x+n x = Hl

Interpolation Filtering Prediction

0

-0.01

f = Q fl Q l
0 1 2

-1 ll

N-N1 points

-0.02

delay

3

4

5

N

1

N-N

1

Q
ext xx

Q

xx

Q

ext xx

fx
2 n

Qxx = Q fx
N
1

N

1

N-N

1

White noise assumption

Qnn = I Q

=Q

ext ll

- Qnn

- f = Q fx Qll 1l

L. Zotov, San Juan, Puerto Rico, 8 May 2009


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-Marquardt algorithm is used for weights W and bias b tuning
L. Zotov, San Juan, Puerto Rico, 8 May 2009

"Amoeba"


RESULTS OF EOP PREDICTION CAMPAIGN

L. Zotov, San Juan, Puerto Rico, 8 May 2009


PHYSICAL MODELLING - INPUT EXCITATION RECONSTRUCTION Corrective smoothing, regularization

=F
Jefferson-Wilson filter

-1

(F

(Wreg ) F (m)

)
c

m (t ) = c t ie
if c t
2

t t+ 2

-e

i c t

m

t t- 2



=h

reg

u = hreg ( - t )u (t ) dt
it

1 hreg (t , ) = e t
0.15 0.1



c

X-POLE

0.04

Excitation reconstructed for the components of SSA-decomposition of X-coordinate of the pole Chandler component annual component

- i cos



X-POL E

1

0.02
0.05



0

arc sec

0


0 -0.05 -0.1

-1

-0.02

-2 1900 1920 1940

-0.04

1960 1980 2 00 0

1970

1980

1900

1920

1940

1960 years

1980

2000



1990

2000

L. Zotov, San Juan, Puerto Rico, 8 May 2009


EXCITATION AND POLE TRAJECTORY FORECAST

Excitation forecast
-0.2

Trajectory forecast by Kalman filter
0.6

Y-POLE

Y-POLE
-0.25

0.4
-0.3



-0.35


0.2 0
1980 1 99 0

-0.4

-0.45



2000

2010

1980

1990



2000

2010

L. Zotov, San Juan, Puerto Rico, 8 May 2009


15-NEURON BRAIN FORECASTS
North Atlantic foraminiferal oxygen isotope data
-2 -2.5 -3 -3.5 -4 -4.5 -5
Site 980

-1000

-800

-600 -400 thousand years

-200

0

Dow Jones Industrial Average
16000 14000

12000

10000

8000

6000

4000

Jul-98

Apr-01

Jan-04 time

Oct-06

Jul-09

Jul -11

L. Zotov, San Juan, Puerto Rico, 8 May 2009


Muchas gracias para atencion

San Juan, Puerto Rico, 8 May 2009