Multiresolution Models of Time Series Andrea Tamoni (Bocconi University ) 2011 Tamoni Multiresolution Models of Time Series 1/ 16
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
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. 1 4 1 4 1 4 1 4 W 2 = 1 4 1 1 1 4 4 4 1 1 2 2 0 0 0 0 1 1 2 2 Tamoni Multiresolution Models of Time Series 3/ 16
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
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
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
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
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
Time series and shocks at different time scales Tamoni Multiresolution Models of Time Series 9/ 16
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 + 1 + (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
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
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
Classical vs. Multiscale modeling Illustrative Example Correlation functions on different time scales Tamoni Multiresolution Models of Time Series 13/ 16
Classical vs. Multiscale modeling Illustrative Example Correlation functions on different time scales - Cont d Tamoni Multiresolution Models of Time Series 14/ 16
Literature Time-scale decomposition Classical vs. Multiscale modeling Illustrative Example Tiao, G.C., 1972, Asymptotic behavior of temporal aggregates of time series, Biometrika 59, 525-531. Brewer, K.R.W, 1973, Some Consequences of Temporal Aggregation and Systematic Sampling for ARMA and ARMAX Models, Journal of Econometrics, 1, 133-154 Wong, P.W., 1993, Multiresolution Decomposition of Harmonizable Random Processes, IEEE Transactions on Information Theory, pp.7-18. 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. 259-263, Princeton NJ. March Tamoni Multiresolution Models of Time Series 15/ 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