Solar Extreme Events 2005 prediction by Singular spectrum analysis and neurofuzzy models
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Solar Extreme Events 2005 prediction by Singular spectrum analysis and neurofuzzy models 1- locally linear neurofuzzy 1-1 utputs of locally linear models is as follows p i i i i i u u u y p 2 1 2 1 0 ˆ M i i i u y y 1 ˆ ˆ M j j i i u u u 1 2 2 2 1 2 1 1 2 2 2 1 2 1 1 2 1 exp 2 1 exp 2 1 exp ip ip p i i ip ip p i i i c u c u c u c u u where 1-2 least square optimization Mp M p ... ... ... 0 21 20 1 11 10 1 2 ; ... M X X X X N u N u N u N u N u u u u u u u u u u u X i p i i i p i i i p i i i 1 1 1 2 2 2 2 2 1 1 1 1 1 1 ˆ ˆ ˆ ; ; T T y X X X X y
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Solar Extreme Events 2005 prediction by Singular spectrum analysis and neurofuzzy models. 1- locally linear neurofuzzy. 1-1 outputs of locally linear models is as follows. where. 1-2 least square optimization. Figure1- Illustration of LOLIMOT algorithm for two dimensional input space. - PowerPoint PPT Presentation
Transcript of Solar Extreme Events 2005 prediction by Singular spectrum analysis and neurofuzzy models
Solar Extreme Events 2005 prediction by
Singular spectrum analysis and neurofuzzy models
1- locally linear neurofuzzy
1-1 outputs of locally linear models is as follows
piiiii uuuyp
21 210ˆ
M
iii uyy
1
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M
jj
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uu
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2
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2
1exp
2
1exp
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1exp
ip
ipp
i
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ip
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where
1-2 least square optimization
MpMp ......... 0212011110 1 2; ... MX X X X
NuNuNuNuNu
uuuuu
uuuuu
X
ipii
ipii
ipii
i
1
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22222
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1ˆ ˆˆ; ; T Ty X X X X y
Figure1- Illustration of LOLIMOT algorithm for two dimensional input space
2- Learning Algorithm: Locally Linear Model Tree (LOLIMOT)
1- Start with an initial model: start with a single LLM, which is a global linear 2- Find the worst LLM3- Check all divisions: The worst LLM is considered for further refinement. Divisions in all
dimensions are tried, and for each of the divisions the following steps are carried out:3-1- Construction of the multi-dimensional membership functions for both generated
hyper rectangles;Construction of all validity functions.3-2- Estimation of the rule consequent parameters for newly generated LLMs.3-3- Calculations of the loss function for the current overall model.
4- Find the best division: The best of the alternatives checked in step 3 is selected, and the related validity functions and LLMs are constructed. The number of LLM neurons is incremented.
5- Test the termination condition: If the termination condition is met, then stop, else go to step 2.
3- Singular Spectrum Analysis
NttX ,,1: 1,,,1,1,,1,~ MNNNtMtXtXtXtX
1- producing M-dimensional vectors from time series
DDN
C TX
1
2- covariance matrix is calculated as
3- corresponding principal component (PC) are:
M
jkk jjtXtPC
1
1
4- time series is reconstructed by combining the associated principal components:
Kk
U
Ljkk
tK
t
t
jjtPCM
tR 11
,1, , 1 1
; , , ,1, ,
min( , 1), , , 1
t t t
t t t M
M L U M M M t N
t N t t N M M N t N
Figure2-Block Diagram of SSA+LOLIMOT method for time series prediction
4- SSA+LOLIMOT method
M principle component extracted and then for each PC a LLNF model should train; then next value prediction of each PC obtained; finally predicted PCs combined for achievement to prediction of main series.
5- Proton Events 2005 prediction
Figure3- one-step prediction of proton density with LOLIMOT+SSA method: (a)-16 July; (b)-7 May 2005
Thank you for your attention
