Multiresolution Models of Time Series

Transcription

1 Multiresolution Models of Time Series Andrea Tamoni (Bocconi University ) 2011 Tamoni Multiresolution Models of Time Series 1/ 16

2 General Framework Time-scale decomposition General Framework Begin with a univariate time series {x t i } i Z For a specified positive integer m, define the process x s (m) by x (m) s = 1 m m 1 i=0 x t+(s 1)m i The x s (m) values are the averages of non-overlapping groups of m consecutive x values Refer to m as the coarsening window, x t as the fine-level process and x s (m) as the coarse-level aggregate process Tamoni Multiresolution Models of Time Series 2/ 16

3 Time-scale Decomposition General Framework Let {x t i } i Z be the observed time series Let us focus on block of length N = 4 (for convenience, we assume N = 2 J for some J) X t = [x t 3, x t 2, x t 1, x t ] Consider the two-level (orthogonal) transform matrix, i.e W 2 = Tamoni Multiresolution Models of Time Series 3/ 16

4 General Framework Time-scale Decomposition - Cont d Two-level decomposition is performed as follows W = W 2 X t where W = (a 2,t, w 2,t, w 1,t 2, w 1,t ) a 2,t = (xt + x t 1 + x t 2 + x t 3 ) 4 w 2,t = (xt + x t 1 x t 2 x t 3 ) 4 w 1,t 2 = (x t 2 x t 3 ) 2 w 1,t = (xt x t 1) 2 It is possible to reconstruct the time series at different levels of aggregation, i.e. x t = a 2,t + w 2,t + w 1,t x (2) t = a 2,t + w 2,t x (4) t = a 2,t Tamoni Multiresolution Models of Time Series 4/ 16

5 General Framework Decomposition in time-scale components - Interpretation Note that the j-th coefficient is in general obtained as follows w j,t = 2 j 1 i=0 x t i }{{ 2 j } [ low pass filter 0, ] 1 2 j 2 j 1 1 i=0 x t 2 j 1 i } 2{{ j 1 } [ low pass filter 0, Overall w j,t is a band-pass filter with [ 1 2 j 1, 1 2 j ] ] 1 2 j 1 Recall that x t = 2 j=1 w j,t + a 2,t and x (2) t = w 2,t + a 2,t and x (4) t = a 2,t Intuitively w 1,t yields changes on scale 1, i.e. [... t, t + 1, t + 2,...] w 2,t yields changes on scale 2, i.e. [... t, t + 2, t + 4,...] a 2,t yields changes on scale 2, i.e. [... t, t + 4, t + 8,...] Tamoni Multiresolution Models of Time Series 5/ 16

6 General Framework Decomposition in time-scale components - Extensions Let X t = [x t N+1,..., x t 2, x t 1, x t ] where N = 2 J Consider the J-level (orthogonal) transform matrix W J and construct W = W J X t We can partition W as follows W = a J W J W J 1. W 1 We obtain J + 1 time series {{W j } j, a J } each containing N 2 j coefficients, i.e. W j = (w j,t 2 J +1,..., w j,t 2 j, w j,t ) The series W j is related to times spaced 2 j units apart Tamoni Multiresolution Models of Time Series 6/ 16

7 Statistical Properties of the coefficients W j Let {x t i } i Z be a wide-sense stationary process Wong (1993) establishes that {w j,k } are wide-sense stationary for a fixed j We can write the following representation (understood to be in mean square sense) where x t = J j=1 k=0 w j,k ɛ j,t k 2 j (1) ɛ j,t = x (2j ) t P x (2j ) Mj,t 2 j t are called Haar innovations and we let M j,t = sp { x (2j ) t k2 j }k N The spectral representation (1) holds for a broader class of processes known as Affine stationary processes (see Yazici, 1996) Tamoni Multiresolution Models of Time Series 7/ 16

8 Statistical Properties of the coefficients W j - Cont d Whenever ɛ j,t }{{} x (2j ) t P Mj,t 2 j x(2j ) t = 2 j 1 i=0 ɛ t i 2 j }{{} 2 j 1 i=0 x t i P Mt i 1 x t i { where M j,t = sp x (2j ) t k2 }k N and M j t = sp {x t k } k N then we obtain the standard wold decomposition { The sp x (2j ) is the set of all linear combinations of t k2 }k N j and therefore is a subset of the sigma-algebra { } generated by x (2j ) t, x (2j ),... t 2 j x (2j ) t k2 j Tamoni Multiresolution Models of Time Series 8/ 16

9 Time series and shocks at different time scales Tamoni Multiresolution Models of Time Series 9/ 16

10 Classical vs. Multiscale modeling Illustrative Example ARMA time series model for fine-level process In this section we not only aggregate data but we also aggregate the model! Let x t be an ARMA(p,q) process Brewer (1973) shows that the time series model governing x s (m) is an ARMA(p,q ) where p = p and q = p (q p 1) m for any finite m Assume x t follows an AR(1) process with autoregressive coefficient equal to ρ The pattern of autocorrelations of the temporally aggregated variable x s (m) is determined by the autoregressive coefficient ρ m Tamoni Multiresolution Models of Time Series 10/ 16

11 Multi-scale time-series modeling Classical vs. Multiscale modeling Illustrative Example What if instead the autocorrelation function at horizon h has a decay governed by ρ h > ρ? The question is how can we make the autocorrelation functions at the fine- and coarse-level compatible We use the approach suggested by Dijkerman (1994) and define a multiresolution model on the {W j } series. The multi-scale autoregressive process is defined by an AR(1) process at each level j, i.e. w j,t+2 j = ρ j w j,t + ɛ j,t+2 j for all j, where ɛ j,t+2 j is Gaussian noise with variance equal to σ j independent across scales Tamoni Multiresolution Models of Time Series 11/ 16

12 An example Time-scale decomposition Classical vs. Multiscale modeling Illustrative Example The fine scale interval is 1 year. Suppose J = 3. a 3,t = ρ 4 a 3,t 8 + ɛ 3,t+4 w 3,t = ɛ 3,t w 2,t = ɛ 2,t w 1,t = ρ 1 w 1,t 2 + ɛ 1,t Recall that x (8) t = a 3,t thus a 3,t pins down the autocorrelation of the process at coarse level. Moreover x t = w 1,t + w 2,t + w 3,t + a 3,t. The decay of the autocorrelation function at the fine-level is controlled by the parameters σ j j = 1,..., J (only through the ratio) Tamoni Multiresolution Models of Time Series 12/ 16

13 Classical vs. Multiscale modeling Illustrative Example Correlation functions on different time scales Tamoni Multiresolution Models of Time Series 13/ 16

14 Classical vs. Multiscale modeling Illustrative Example Correlation functions on different time scales - Cont d Tamoni Multiresolution Models of Time Series 14/ 16

15 Literature Time-scale decomposition Classical vs. Multiscale modeling Illustrative Example Tiao, G.C., 1972, Asymptotic behavior of temporal aggregates of time series, Biometrika 59, Brewer, K.R.W, 1973, Some Consequences of Temporal Aggregation and Systematic Sampling for ARMA and ARMAX Models, Journal of Econometrics, 1, Wong, P.W., 1993, Multiresolution Decomposition of Harmonizable Random Processes, IEEE Transactions on Information Theory, pp R.W. Dijkerman and R.R. Mazumdar, 1994, Wavelet Representations of Stochastic Processes and Multiresolution Stochastic Models. IEEE Transactions on Signal Processing, Vol. 42. No. 7 B. Yazici and R. L. Kashyap, 1996, Affine stationary processes, Proceedings of Conference on Information Sciences and Systems, pp , Princeton NJ. March Tamoni Multiresolution Models of Time Series 15/ 16

16 Affine Stationary Processes Classical vs. Multiscale modeling Illustrative Example A stochastic process x j,t is called affine stationary if it satisfies the following conditions: ( j E[x j,t x j,t ] = R j, 1 ) j (t t ) Within the same scale, i.e. j = j an affine stationary process reduces to an ordinary wide sense stationary process Across the scales for a fixed shift index, i.e. t = t the process reduces to a class of self-similar processes known as scale stationary process. Example: the increments of the fractional Brownian motion Wold decomposition Tamoni Multiresolution Models of Time Series 16/ 16