latent structure in non-stationary time series - Department of ...

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S26.Low (1) (2) (3) (4) alpha waves fast waves 0 1000 2000 3000 4000 time Figure 7.2: Some of the estimated latent components in the decomposition of EEG series S26.Low. 110
components in Ictal-19 seizures. The main dierence is due to the high levels of high frequency activity that have a considerable contribution in terms of the amplitude to the observed 2-channel series for both treatments. This may be due in part to individual dierences between the subjects studied, dierences between types of treatments administered and possibly to the recording techniques. In terms of dierences between the two treatments we can see that the post-ictal suppression in the low frequency components is greater in the moderate treatment than in the low treatment. Figure 7.3 and 7.4 display estimated frequency trajectories of the latent components for both series. The dotted horizontal lines roughly demark the delta, theta, alpha and beta frequency ranges of neural rhythms according to Dyro (1989) (see also Chapter 2). The \switching" behavior apparent at high frequencies is due in part to the lack of identication of components with very high frequencies, relatively low moduli and similar amplitudes and also to the fact that at some time intervals a pair of complex roots is \exchanged" by two distinct real characteristic roots or vice versa; this is seen for instance in Figure 7.3. Just before t = 500 one of the components whose frequency lies in the beta range vanishes, being replaced bytwo real roots for a very short time and then, these two real roots are substituted by a pair of complex roots whose frequency lies in the alpha range. The frequency components in the alpha and beta bands in S26.low also experience a lot of \switching" activity, particularly after t = 2000. Figures 7.5(a) and 7.5(b) show the trajectories of the estimated amplitudes for dominant frequency components in the decompositions of S26.Mod and S26.Low respectively. The lowest frequency component dominates in amplitude the S26.Low EEG series during the whole time period, but the same is not true for the moderate treatment; component (2) of the decomposition (i.e. the second lowest frequency component) dominates most of the seizure period while component (1) starts rising 111
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LATENT STRUCTURE IN NON-STATIONARY
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Abstract The class of time-varying
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Acknowledgements I wish to thank Pr
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5.2 Factor models and autoregressio
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List of Tables 5.1 Approximate 95%
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5.1 (a) Estimated frequency traject
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Chapter 1 Introduction Time series
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-300 -200 -100 0 100 200 0 500 1000
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analyses are summarized in Chapter
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al. (1992) propose a dynamic factor
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EEG signals are characterized by pe
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across patients and are also inuenc
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Fp1 Fp2 F7 F3 Fz F4 F8 T3 C3 Cz C4
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200 0 voltage -200 -400 -600 0 1000
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2.3 The 2-channel data A second app
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the seizure ends and the post-ictal
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Chapter 3 Time series decomposition
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Consider again model (3.1) and supp
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equivalently, if the characteristic
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where t;j is a real, zero-mean AR(
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a TVARMA(p 2 ,(p , 1)p) whose coeci
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their ability to predict the future
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the same results in terms of predic
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Changes in these parameters over ti
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Finally, model completion requires
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choices for the initial quantities
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AR models are good approximations f
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It is worth commenting at this poin
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channel Cz (1) (2) (3) (4) 0 1000 2
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! t;i and modulus r t;i , the argum
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frequency (cycles per second) 15 10
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t=100 t=1000 t=1500 20 40 60 80 20
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time is obtained, namely x t = rX j
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t=400 t=600 t=800 t=1000 t=1200 -
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Across the 19 channels the time-var
Page 71 and 72: amplitudes 0 1 2 3 4 0 1000 2000 30
Page 73 and 74: have been applied in the past to th
Page 75 and 76: of this factor, x 2;t = x t,1 ;:::
Page 77 and 78: for 0 . A single system discount f
Page 79 and 80: •••• ••• ••••
Page 81 and 82: are not able to capture. Estimates
Page 83 and 84: O 1 in the portion that corresponds
Page 85 and 86: i;t are independent zero mean gauss
Page 87 and 88: t k = ,2 k = ,1 k =0 k =1 k =2 m =
Page 89 and 90: ecients across channels. These grap
Page 91 and 92: 79 Figure 5.8: Estimated posterior
Page 93 and 94: t=400 •• • •• • ••
Page 95 and 96: prob.lags.12.99999[1, p(k=0|data) ]
Page 97 and 98: Chapter 6 Time series decomposition
Page 99 and 100: ewriting (6:2) in terms of the new
Page 101 and 102: sucient stationarity condition for
Page 103 and 104: y the same frequencies and moduli a
Page 105 and 106: x(1,t) x(2,t) (1) (1) (2) (2) (3) (
Page 107 and 108: x(1,t) x(2,t) (1) (1) (2) (2) (3) (
Page 109 and 110: to the data. In the multivariate fr
Page 111 and 112: C z O 1 i (r 1;i ;! 1;i ) (r 2;i ;!
Page 113 and 114: are generally overparametrized and
Page 115 and 116: C z O 1 i (r 1;i ;! 1;i ) (r 2;i ;!
Page 117 and 118: with characteristic modulus and wav
Page 119 and 120: Chapter 7 Further data analyses: la
Page 121: S26.Mod (1) (2) (3) (4) alpha waves
Page 125 and 126: frequency (cycles per second) 20 15
Page 127 and 128: 30 25 S26.Mod amplitudes 20 15 10 5
Page 129 and 130: Chapter 8 Closing remarks and futur
Page 131 and 132: presented in Chapter 3 assumed that
Page 133 and 134: Appendix A Bounding forecast dieren
Page 135 and 136: Appendix B Posterior Sampling Algor
Page 137 and 138: (a) sampling X from p(X j Y; B; ;v;
Page 139 and 140: B.1.2 Sampling from p(jX; s 1 ;:::
Page 141 and 142: (b) sampling l i from p(l i j Y i ;
Page 143 and 144: Bibliography Aguilar, O. (1996) Wav
Page 145 and 146: Molenaar, P.C.M, De Gooijer, J.G an
Page 147: Biography Raquel Prado was born in
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