A Time-Varying Model for Disturbance Storm- Time (Dst) Index Analysis Presentation: Yang Li (Phd student) Supervisor: Prof. Billings S.A. and Dr. Hua-Liang Wei Department of Automatic Control and Systems Engineering, the University of Sheffield

1. Background 1) Empirical modelling approach ----some physical insights or assumptions 2) Time-invariant system identification approach ----family of NARMAX ----advantage and disadvantage 3) Time-varying system identification approach ---- wavelet basis function approximation (a) model coefficients approximated by wavelet basis function (b) multi-resolution wavelet decompositions

2. Time-Varying ARX Model 1) TVARX process: (1) where time-varying parameters :, model orders :, model residual :

Expanding and by wavelet basis function (2) where expansion parameters :, set of basis functions :.

Substituting (2) into (1), yields, (3) new variables can be defined: (4) Substituting (4) into (3), yields, (5)

model (5) in the form of matrix: (6) where regression vector: coefficient vector: The definition of time-dependent spectrum: (7)

3. The Multi-Wavelet Basis Functions B-spline function of m-th order: (8) with (Haar funciton) (9) fourth-order B-spline: (10) where for and for

coefficients expression and by B-spline function: (11) where recursive coefficient: and.

4. Model Identification and Parameter Estimation 1) Identification and parameter estimation algorithm normalized least mean square (NLMS), 2) model order determination: Bayesian information criterion (BIC), (12) where mean-squared-error (MSE): (13)

5. Dst index Modelling and Analysis Fig,1 The input (the solar wind parameter VBs) and the output (the Dst index) measured, with the sampling interval of 1-h. A total of 1176 observation, with a time resolution of -1-h, were involved. 1) Modelling data

2). Result analysis Fig. 2. The estimates of the six time-varying coefficients a i (t) (i=1,2,3,4) output estimated parameters and b k (t) (k=0,1,2) input estimated parameters for the magnetosphere data.

Fig. 3, on the left hand side: A comparison of the recovered signal (one-step-ahead (1-h-ahead) predictions) from the identified TVARX (4, 2) model and the original observations for the magnetosphere data. Solid (blue) line indicates the observations and the dashed (red) line indicates the signal recovered from the TVARX (4, 2) model. On the right hand side: the 3-D topographical map of the time-dependent spectrum estimated from the TVARX (4, 2) model for the magnetosphere system data.

Figure 4 The overlap of the transient power spectra calculated at different time instants from the limited 1176 observation data for the problem described by (6).

6. Conclusions 1) modelling approach are locally defined more flexible and adaptable 2) time-dependent spectrum of transient properties the global dynamical behaviour

Thank you!