Transcript
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
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1-2 least square optimization
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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 LLM 3- 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.
Figure1- Illustration of LOLIMOT algorithm for two dimensional input space
3- Singular Spectrum Analysis 1- producing M-dimensional vectors from time series
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2- covariance matrix is calculated as
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3- corresponding principal component (PC) are:
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4- time series is reconstructed by combining the associated principal components:
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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.
Figure2-Block Diagram of SSA+LOLIMOT method for time series prediction
5- Proton Events 2005 prediction
Figure3- one-step prediction of proton density with LOLIMOT+SSA method: (a)-16 July; (b)-7 May 2005