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 Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Spatio-temporal observations New York City air pollution data, PM2.5 measure is one of six primary air pollutants and is a mixture of …ne particles and gaseous compounds such as sulphur dioxide (SO2) and nitrogen oxides (NOx) Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

In 2002, every 3 days and during the …rst 9 months, 91 equally spaced days, observed at 15 monitoring stations, X (s, t), see [SM05], fZ (s, t) = ∆t X (s, t) : (s, t) : s 2 R2, t 2 Zg 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 Days PM2.5 Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Missing data: mean by locations and …xed in time, but 3/4 of data at Location #11 is missing, namely from days 24th to 91st, location s0 , fZ(s0 , t); t = 1, 2, 3...ng. Sample fZ (si , t) ; si = 1, 2, ...m; t = 1, . . . , ng. Bronx, Brooklyn, Manhattan, Queens, Staten Island -74.15 -74.1 -74.05 -74 -73.95 -73.9 -73.85 -73.8 40.55 40.6 40.65 40.7 40.75 40.8 40.85 40.9 40.95 X = -73.96 Y = 40.75 Latitudes Longitudes Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

-74.2 -74.15 -74.1 -74.05 -74 -73.95 -73.9 -73.85 -73.8 -73.75 -73.7 40.5 40.55 40.6 40.65 40.7 40.75 40.8 40.85 40.9 40.95 41 X = -73.84 Y = 40.77 Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

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)] = σ2 Z < ∞ Cov [Z (s, t) , Z (s + h, t + u)] = c (h, u) = c (khk , u) , We note that c (h, 0) and c (0, u) correspond to the purely spatial and purely temporal covariances. Spatio-temporal variogram for fZ (h, t)g 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 (khk , u). Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

DFT Frequency domain in time and Spatial domain in space, fZ (si , t) ; t = 1, . . . , ng DFT at ωk = 2πk n , k = 0, 1, . . . , n 2 . Jsi (ωk ) = 1 p 2πn n ∑ t=1 Z (si , t) e itωk , periodogram Isi (ωk ) = jJsi (ωk )j2 . Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Jsi (ωk ) asymptotically independent and Gaussian E (Jsi (ωk )) = 0 Var (Jsi (ωk )) = E (Isi (ωk )) ' gsi (ωk ) . Cov Jsi (ωk ) , Jsi ωk0 ' 0, k 6= k0. Isotropy gsi (ωk ) = g (ωk ) , for all i Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Fourier T and Spectral Repr Introduce white noise Js,e (ω) = Z eis λ r n 2π dZe (λ, ω) . Laplacian ∂2 ∂s2 1 + ∂2 ∂s2 2 jc (ω)j2 ν Js (ω) = Js,e (ω) . λ2 1 λ2 2 jc (ω)j2 ν dZz (λ, ω) = dZe (λ, ω) , Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Spectral density fz (λ, ω) = σ2 e (2π)2 λ2 1 + λ2 2 + jc (ω)j2 2ν Covariance Cov (Js (ω) , Js+h (ω)) = σ2 e 2π khk 2 jc (ω)j 2ν 1 K2ν 1 (jc (ω)j khk) Γ(2ν) K2ν 1 : modi…ed Bessel function of the second kind of order 2ν 1. Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

General, Rd, ν 2 integer, correlation function is given by ρ (khk , ω) = (khk jc (ω)j)2ν d 2 22ν d 2 1Γ 2ν d 2 K 2ν d 2 (khk jc (ω)j) , and C (0, ω) = σ2 e (2π)d 2 2d 2 jc (ω)j2 2ν d 2 Γ 2ν d 2 Γ (2ν) = g (ω) . Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Parameters C (0, ω) = σ2 e 2 (2ν 1) jc (ω)j2 2ν 1 = g (ω) . jc (ω)j2 and common spectral density g (ω), ARMA, FARMA etc.. Estimation of parameters Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Correlation ρ (khk , ω) = C (khk , ω) C (0, ω) = 1 22ν 2Γ (2ν 1) (khk jc (ω)j)2ν 1 K2ν 1 (jc (ω)j khk) . Special case ν = 1, ρ (khk , ω) = khk jc (ω)j K1 (khk jc (ω)j) . Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Spatio-temporal prediction Js0 (ω) = 1 p (2πn) n ∑ t=1 Z (s0 , t) e itω, and by inversion, we have Z (s0 , t) = r n 2π π Z π Js0 (ω) eitωdω. In other words given fJs0 (ω) , for all π ω πg, we can uniquely recover the sequence fZ (s0 , t) ; t = 1, . . . , ng. Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Given J0 m (ω) = [Js1 (ω) , Js2 (ω) , . . . , Jsm (ω)] . We note E [Jm (ω)] = 0 E [Jm (ω) Jm (ω)] = Fm (ω) , Fm (ω) = (C (ksi sj k , ω) ; i, j = 1, 2, . . . , m), and each element C (ksi sj k , ω) is given above. Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

J0 m+1 (ω) = [J0 (ω) , J0 m (ω)] , which has zero mean, and variance covariance matrix E [Jm+1 (ω) Jm+1 (ω)] = C0 (0, ω) E (J0 (ω) J 0 m (ω)) E (Jm (ω) J0 (ω)) E (Jm (ω) Jm (ω)) = C0 (0, ω) G0 0 (ω) G0 (ω) Fm (ω) , Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

C0 (0, ω) = E (J0 (ω) J0 (ω)) = C (0, ω), G0 0 (ω) = E [J0 (ω) J 0 m (ω)] = [C (ks0 s1 k , ω) , . . . , C (ks0 sm k , ω)] E [J0 (ω) jJm (ω)] = G0 0 (ω) F 1 m (ω) Jm (ω) the minimum mean square error σ2 m (ω) = C (0, ω) G0 0 (ω) F 1 m (ω) G0 (ω) . ˆ J0 (ω) = ˆ G0 0 (ω) ˆ F 1 m (ω) Jm (ω) . Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

Air pollution data, common spectral density g (ω), ARMA(1,1), parameters estimation by Whittle method, see [ST13] for details 0 20 40 60 80 100 -20 0 20 40 60 80 100 120 Predicted Measured & means Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

-73.9 -73.8 -73.7 -73.6 -73.5 -73.4 40.5 40.6 40.7 40.8 40.9 41 2 4 6 8 10 12 Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19

S. K. Sahu and K. V. Mardia. 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), 223–244 (2005). T. Subba Rao and Gy. Terdik. A space-time covariance function for spatio-temporal random processes and spatio-temporal prediction (kriging). ArXiv e-prints (November 2013). Gy. Terdik & S. Rao T. (UofD, HU & UofM, UK ) Spatio-Temporal Fields January 30, 2015 Laboratoire des Signaux et S / 19