A new covariance function for spatio-temporal data analysis with application to atmospheric pollution and sensor networking

Transcription

1 A new covariance function for spatio-temporal data analysis with application to atmospheric pollution and sensor networking György Terdik and Subba Rao Tata UofD, HU & UofM, UK January 30, 2015 Laboratoire des Signaux et Systemes, Supelec, Paris, Gif-sur-Yvette

2 Spatio-temporal observations New York City air pollution data, PM2.5 measure is one of six primary air pollutants and is a mixture of fine particles and gaseous compounds such as sulphur dioxide (SO2) and nitrogen oxides (NOx)

3 PM2.5 In 2002, every 3 days and during the first 9 months, 91 equally spaced days, observed at 15 monitoring stations, X (s, t), see [SM05], {Z (s, t) = t X (s, t) : (s, t) : s R 2, t Z} Days

4 Longitudes Missing data: mean by locations and fixed in time, but 3/4 of data at Location #11 is missing, namely from days 24th to 91st, location s 0, {Z(s 0, t); t = 1, 2, 3...n}. Sample {Z (s i, t) ; s i = 1, 2,...m; t = 1,..., n}. Bronx, Brooklyn, Manhattan, Queens, Staten Island X = Y = Latitudes

5 X = Y =

6 We assume that the random process is spatially and temporally second order stationary, homogeneous and isotropic, i.e. E [Z (s, t)] = µ, Var [Z (s, t)] = σ 2 Z < Cov [Z (s, t), Z (s + h, t + u)] = c (h, u) = c ( h, u), We note that c (h, 0) and c (0, u) correspond to the purely spatial and purely temporal covariances. Spatio-temporal variogram for {Z (h, t)} 2γ (h, u) = Var [Z (s + h, t + u) Z (s, t)]. 2γ (h, u) = 2 [c (0, 0) c (h, u)], for an isotropic process, γ (h, u) = γ ( h, u).

7 DFT Frequency domain in time and Spatial domain in space, {Z (s i, t) ; t = 1,..., n} DFT at ω k = 2πk n, k = 0, ±1,..., ± [ n 2]. J si (ω k ) = 1 2πn n t=1 Z (s i, t) e itω k, periodogram I si (ω k ) = J si (ω k ) 2.

8 J si (ω k ) asymptotically independent and Gaussian Isotropy E (J si (ω k )) = 0 Var (J si (ω k )) = E (I si (ω k )) g si (ω k ). Cov ( J si (ω k ), J si ( ωk )) 0, k = k. g si (ω k ) = g (ω k ), for all i

9 Fourier T and Spectral Repr Introduce white noise Laplacian [ 2 J s,e (ω) = s s 2 2 [ ] n e is λ dz e (λ, ω). 2π c (ω) 2 ] ν J s (ω) = J s,e (ω). ( λ 2 1 λ 2 2 c (ω) 2) ν dzz (λ, ω) = dz e (λ, ω),

10 Spectral density Covariance f z (λ, ω) = σ 2 e (2π) 2 ( λ λ c (ω) 2) 2ν Cov (J s (ω), J s+h (ω)) ( ) = σ2 e h 2ν 1 K 2ν 1 ( c (ω) h ) 2π 2 c (ω) Γ(2ν) K 2ν 1 : modified Bessel function of the second kind of order 2ν 1.

11 General, R d, ν 2 integer, correlation function is given by ρ ( h, ω) = ( h c d (ω) )2ν 2 2 2ν d 2 1 Γ ( )K 2ν d 2ν d ( h c (ω) ), 2 2 and C (0, ω) = (2π) d 2 2 d 2 σ 2 e ( c (ω) 2) 2ν d 2 Γ ( 2ν d 2 Γ (2ν) ) = g (ω).

12 Parameters C (0, ω) = σ 2 e 2 (2ν 1) ( c (ω) 2) 2ν 1 = g (ω). c (ω) 2 and common spectral density g (ω), ARMA, FARMA etc.. Estimation of parameters

13 Correlation ρ ( h, ω) = C ( h, ω) C (0, ω) = 1 2 2ν 2 Γ (2ν 1) ( h c (ω) ) 2ν 1 K 2ν 1 ( c (ω) h ). Special case ν = 1, ρ ( h, ω) = h c (ω) K 1 ( h c (ω) ).

14 Spatio-temporal prediction J s0 (ω) = and by inversion, we have 1 (2πn) n t=1 Z (s 0, t) e itω, Z (s 0, t) = n 2π π π J s0 (ω) e itω dω. In other words given {J s0 (ω), for all π ω π}, we can uniquely recover the sequence {Z (s 0, t) ; t = 1,..., n}.

15 Given We note J m (ω) = [J s1 (ω), J s2 (ω),..., J sm (ω)]. E [J m (ω)] = 0 E [J m (ω) J m (ω)] = F m (ω), F m (ω) = (C ( s i s j, ω) ; i, j = 1, 2,..., m), and each element C ( s i s j, ω) is given above.

16 J m+1 (ω) = [J 0 (ω), J m (ω)], which has zero mean, and variance covariance matrix = = E [J m+1 (ω) J m+1 (ω)] [ C 0 (0, ω) E (J 0 (ω) J m (ω)) E (J m (ω) J0 (ω)) E (J m (ω) J m (ω)) [ C0 (0, ω) G ] 0 (ω), G 0 (ω) F m (ω) ]

17 C 0 (0, ω) = E (J 0 (ω) J 0 (ω)) = C (0, ω), G 0 (ω) = E [J 0 (ω) J m (ω)] = [C ( s 0 s 1, ω),..., C ( s 0 s m, ω)] E [J 0 (ω) J m (ω)] = G 0 (ω) F 1 m the minimum mean square error (ω) J m (ω) σ 2 m (ω) = C (0, ω) G 0 (ω) F 1 m (ω) G 0 (ω). Ĵ 0 (ω) = Ĝ 0 (ω) ˆF 1 m (ω) J m (ω).

18 Air pollution data, common spectral density g (ω), ARMA(1,1), parameters estimation by Whittle method, see [ST13] for details Predicted Measured & means Gy. Terdik & S. Rao T. (UofD, 0 HU & UofM, 20UK ) Spatio-Temporal 40 Fields

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20 S. K. S K. V. M. A bayesian kriged kalman model for short-term forecasting of air pollution levels. Journal of the Royal Statistical Society: Series C (Applied Statistics) 54(1), (2005). T. S R G. T. A space-time covariance function for spatio-temporal random processes and spatio-temporal prediction (kriging). ArXiv e-prints (November 2013).