forecast for the solar activity based on the autoregressive desciption of the sunspot number time...
TRANSCRIPT
Forecast for the solar activity based on the autoregressive desciption of the
sunspot number time series
R. WernerSolar Terrestrial Influences
Institute - BAS
In the last year we have learned about some basics of the
time-series analysis by descriptive and inference statistics
Descriptive statistics: We have been acquainted with definitons for the time series, arithmetic mean, variance, correlation and auto-correlation, co-variance and cross-correlation We have decomposed the time series into a trend, a seasonal and a rest component We have examined problems to estimate the trend component and we have learned basic methods such as average moving, linear and polynomial regression, analysis and the harmonic analysis to determine the seasonal component. We have used the phase average method, the periodogram We have learned the Box/Cox transformation as a method to stabilize the variance
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Inference statistics:
We have learned about a very important condition as the stationarity, and the weakly stationarity. We have presented auto-regression models and average moving models and have shown some important characteristics of AR and AM models of first and second order. We have shown properties of the auto-correlation func- tion (ACF) and of the partial auto-correlation function (PACF) We have presented the Yule Walker equation. We have demonstrated how we can determine the auto- regressive model using the ACF and the PACF for the time series of the sunspot number We have learned about the principles of the dynamic regression, of some simple models, the Koyke transformation and the Cochrane-Orcutt method
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Today
We will analyse the sunspot number time series in more detail with the main goal to make forecasts for the next solar cycle activity using the Box/Jenkins methodology for the model identification.
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The forecasts for the solar activity are very important:
for the satellite drag
the telecommunication outages for hazards in connection with the occurrence of
strong solar wind streams producing the
blackout of power plants. for manned space flights, for the prognosis of
the radiation risk High powerful radiation can lead to computer upsets and computer memory failures S
eco
nd
Wo
rksh
op
"S
ola
r in
flu
ence
s o
n t
he
ion
osp
her
e an
d m
agn
eto
sph
ere"
, S
ozo
po
l, B
ulg
aria
, 7-
11 J
un
e, 2
010
Pesnell, Solar Phys. (2008) 252:209-220
Solar activity predictions of the R24
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Regardless of the advance in the application of physical methods for the purpose of forecasting, the results are
very inconsistently spread and substantiate the application of
statistical methods.
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Box/Jenkins method
1. Box/Jenkins model identification
1.1 Stationarity Box/Cox transformation
1.2 Seasonality
1.3 Auto-correlation and partial auto-correlation
plots
1.4 Determination of the type of the process and
its order
2. Estimation of the model parameters
3. Model diagnostics
4. Forecasting
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Yule, 1927
Stationarity?
Sunspot numbers 1749-1924
0
50
100
150
200
1740 1780 1820 1860 1900
Time, years
Su
ns
po
t n
um
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rs Mean=44.8 Std=34.8
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Yule, 1927
The means are not significantly different, however, the standard deviations depend on the means. Therefore, the series is not stationary! Box-Cox
transformation
Sunspot numbers 1749-1924
0
50
100
150
200
1740 1780 1820 1860 1900
Time, years
Su
ns
po
t n
um
be
rs Mean=56.1 Std=39.5
Mean=52.0 Std=37.4
Mean=37.0 Std=27.1
Mean=44.8 Std=34.8
Mean=21.6 Std=16.6
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Square root of the sunspot number data 1749-1924
0
7
14
1740 1780 1820 1860 1900
Time, years
Sq
rt(S
SN
)
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Detrended square root of the sunspot number data 1749-1924
-8
-4
0
4
8
1740 1780 1820 1860 1900
Time, years
De
tre
nd
ed
sq
rt(S
SN
)
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Autocorrelation Function
SSN 1749-1924
Conf. Limit-1.0 -0.5 0.0 0.5 1.00
30 -.298 .2028 29 -.355 .1992 28 -.346 .1958 27 -.279 .1935 26 -.148 .1928 25 +.023 .1928 24 +.179 .1919 23 +.261 .1899 22 +.237 .1882 21 +.122 .1877 20 -.031 .1877 19 -.177 .1867 18 -.282 .1843 17 -.315 .1812 16 -.277 .1788 15 -.155 .1780 14 +.032 .1780 13 +.264 .1758 12 +.467 .1686 11 +.555 .1578 10 +.491 .1489 9 +.292 .1456 8 +.011 .1456 7 -.254 .1431 6 -.421 .1359 5 -.427 .1280 4 -.259 .1250 3 +.051 .1248 2 +.451 .1152 1 +.818 .0754Lag Corr. S.E.
Partial Autocorrelation Function
SSN 1749-1924
(Standard errors assume AR order of k-1)
Conf. Limit-1.0 -0.5 0.0 0.5 1.00
30 -.008 .0754 29 -.160 .0754 28 -.028 .0754 27 +.043 .0754 26 -.049 .0754 25 -.050 .0754 24 -.117 .0754 23 -.129 .0754 22 +.023 .0754 21 +.041 .0754 20 -.028 .0754 19 -.006 .0754 18 -.175 .0754 17 -.046 .0754 16 -.154 .0754 15 +.011 .0754 14 +.109 .0754 13 -.023 .0754 12 +.002 .0754 11 +.063 .0754 10 +.042 .0754 9 +.171 .0754 8 +.125 .0754 7 +.163 .0754 6 +.116 .0754 5 -.064 .0754 4 -.022 .0754 3 -.120 .0754 2 -.656 .0754 1 +.818 .0754Lag Corr. S.E.
AR(2)-model
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t2211 a ttt zzz
AR(2)-model
at error term: white noise
φi have to be determined by the Yule-Walker equations
Yule In this work
φ1 1.3425 1.3571
φ2 -0.6550 -0.6601
Plot of variable: VAR5
ARIMA (2,0,0) residuals;
-20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320
Case Numbers
-4
-3
-2
-1
0
1
2
3
4
5
VA
R5
-4
-3
-2
-1
0
1
2
3
4
5
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Sunspot numbers 1749-2008
0
50
100
150
200
1740 1780 1820 1860 1900 1940 1980
Time, years
Su
ns
po
t n
um
be
rs Mean=67.3 Std=49.7
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Partial Autocorrelation Function
SSN 1749-1924
(Standard errors assume AR order of k-1)
Conf. Limit-1.0 -0.5 0.0 0.5 1.00
15 +.011 .0754 14 +.109 .0754 13 -.023 .0754 12 +.002 .0754 11 +.063 .0754 10 +.043 .0754 9 +.171 .0754 8 +.125 .0754 7 +.163 .0754 6 +.116 .0754 5 -.064 .0754 4 -.022 .0754 3 -.120 .0754 2 -.656 .0754 1 +.818 .0754Lag Corr. S.E.
Partial Autocorrelation Function
SNN 1749-2008
(Standard errors assume AR order of k-1)
Conf. Limit-1.0 -0.5 0.0 0.5 1.00
15 -.032 .0620 14 +.125 .0620 13 -.006 .0620 12 -.012 .0620 11 -.016 .0620 10 +.022 .0620 9 +.281 .0620 8 +.165 .0620 7 +.192 .0620 6 +.172 .0620 5 -.071 .0620 4 -.030 .0620 3 -.147 .0620 2 -.678 .0620 1 +.812 .0620Lag Corr. S.E.
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1749-1924 1749-2008
φ1 1.211202 1.203376
φ2 -0.458256 -0.493566
φ3 -0.124498 -0.108303
φ4 0.149170 0.206869
φ5 -0.131423 -0.223224
φ6 -0.020802 0.057670
φ7 0.098886 0.098426
φ8 -0.103147 -0.176114
φ9 0.189920 0.286768
1848-2008
1.043363
-0.336989
-0.176310
0.198759
-0.215978
0.060787
0.061170
-0.245499
0.401278
1909-2008
1.095090
-0.416573
-0.159289
0.219609
-0.243758
0.088986
0.105020
-0.298126
0.435216
Yule In this work
φ1 1.3425 1.3571
φ2 -0.6550 -0.6601
AR(9) modelS
eco
nd
Wo
rksh
op
"S
ola
r in
flu
ence
s o
n t
he
ion
osp
her
e an
d m
agn
eto
sph
ere"
, S
ozo
po
l, B
ulg
aria
, 7-
11 J
un
e, 2
010
Forecast (ex-post-prognosis, prognosis of known values of the past)
2211 ttt zzz
12213ˆ zzz One-step prognosis
22314ˆ zzz
For an AR(2)
Two-step prognosis 12213ˆ zzz
22314 ˆˆ zzz
mean squared forecast error:
22 )ˆ(1
1ii zz
pns
p: model order
Which is the optimal model? For example, minimization of the
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Forecast Sunspot numbers 1749-2008
0
50
100
150
200
250
1740 1780 1820 1860 1900 1940 1980Time, years
Su
nsp
ot
nu
mb
ers
forecast SSN - one step prognosis
original SSN
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Sunspot number forecast 1749-2008
SSNforecast = 0.9891SSN + 0.8854
R2 = 0.8749, s=15.83
0
50
100
150
200
250
0 50 100 150 200
Original sunspot numbers
Su
nsp
ot
nu
mb
er f
ore
cast
, o
ne
-ste
p p
rog
no
sis
- - - prognosis interval (α/2=0.05)
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Forecast sunspot numbers 1749-2008
0
50
100
150
200
250
1740 1790 1840 1890 1940 1990Time, years
Su
nsp
ot
nu
mb
ers
forecast SSN, two-step prognosis
original sunspot numbers
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Sunspot number forecast 1749-2008
SSSforecast = 0.8445SSN + 8.3413
R2 = 0.6575, s=26.67
0
50
100
150
200
250
0 50 100 150 200Original sunspot numbers
Su
nsp
ot
nu
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er
fore
cast
, t
wo
-ste
p p
rog
no
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- - - prognosis interval (α/2=0.05)
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Prognosis
horizon
1749-2008
1848-2008
1908-2008
1 15.8 15.4 17.4
2 26.7 23.8 27.7
3 33.3 29.4 33.2
4 36.4 31.0 34.2
The standard deviations for the 1848-2008 series are the smallest ones, unfortunatelly the deviations in the solar activity maxima during this period are greater than the ones for the 1749-2008 series.
Standard deviationsS
eco
nd
Wo
rksh
op
"S
ola
r in
flu
ence
s o
n t
he
ion
osp
her
e an
d m
agn
eto
sph
ere"
, S
ozo
po
l, B
ulg
aria
, 7-
11 J
un
e, 2
010
2211 ttt zzz For an AR(2)t=1,…,n
1211 nnn zzz
nnn zzz 2112 ˆ
12213 ˆˆ nnn zzz
2211 ˆˆ hnhnhn zzz h: horizon prognosis
Forecast (ex-ante-prognosis)
Prognosis of the future value, based on the last
and the next to last series value, and so on
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Sunspot number forecast 1749-2008
0
50
100
150
200
1940 1950 1960 1970 1980 1990 2000 2010 2020Time, years
Su
ns
po
t n
um
be
rs
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Sunspot number forecast 1749-2008
0
40
80
120
2000 2005 2010 2015 2020
Time, years
Su
nsp
ot
nu
mb
ers
Forecast from 2008
Forecast from 2007
Forecast from 2006
Forecast from 2005
Forecast from 2009
I would like to acknowledge the support of this work bythe Ministry of Education, Science and Youth under the DVU01/0120 Contract
Acknowledgement
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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10
2010