Chapter 5: Spectral Domain From: The Handbook of Spatial Statistics. Dr. Montserrat Fuentes and Dr. Brian Reich Prepared by: Amanda Bell

Transcription

1 Chapter 5: Spectral Domain From: The Handbook of Spatial Statistics Dr. Montserrat Fuentes and Dr. Brian Reich Prepared by: Amanda Bell

2 Background Benefits of Spectral Analysis Type of data Basic Idea Representation of Transformation Mathematical Considerations Spectral Representation Theorem Brochner s Theorem Definition of Spectral Density Aliasing Examples of Spectral Densities Estimation Periodogram and properties Whittle Approximation to likelihood Data Taper Correction for Aliasing Lattice Missing Values Data Application in Text Outline

3 Background Information

4 Benefits of Spectral Analysis Computationally efficient for large datasets using FFT (OO(nnllllll 2 nn)) Modeling is intuitive in spectral domain Guarantees positive definite covariance function Some operations become easier once they are transformed

5 Type of Data Equally-Spaced Lattice Little missing data Stationary and Isotrophic

6 Basic idea in Time Series Setting Series generated with Matern Covariance with range = 1, Smoothness = 1.5, scale =1.

7 Continuous Fourier Transform Suppose g is a real or complex-values function that is integrable over RR dd. f is the Fourier transform of g when for ωω εε RR dd : f(ω) = RR dd gg ss exp iiωω tt ss dddd If f is integrable over RR dd, g has representation: gg ss = 1 (2ππ) dd RR dd ff ωω exp iiωω tt ss ddωω

8 Mathematical Considerations

9 Spectral Representation Theorem ZZ ss = ee iissttωω dddd(ωω) RR 2 The Y process is called the spectral process associated with a stationary process Z. The random spectral process Y has the following properties: EE YY ωω = 0 EE YY ωω 3 YY ωω 2 YY ωω 1 YY ωω 0 = 0 ωω 3 < ωω 2 < ωω 1 < ωω 0 EE{ dddd ωω 2 } = FF(dddd) wwwwwwwww FF ddωω < aaaaaa FF iiii aa pppppppppppppppp ffffffffffff mmmmmmmmmmmmmm

10 Brochner s Theorem CC ss = RR dd exp iiss tt ωω FF(dddd) A continuous function C is nonnegative definite if and only if it can be represented in the form above where F is a positive finite measure

11 Spectral Density ff ωω = 1 (2ππ) 2 RR 2 exp iiωω tt xx CC xx dddd Defined as the Fourier transform of the autocovariance function

12 Aliasing exp iiωω tt zz 1 = exp ii ωω + zz 22ππ tt zz 1 = exp iiωω tt zz 1 exp(iiiiizz 2 tt zz 1 )

13 Examples of Spectral Densities

14 Triangular Model CC h = σσ(aa h) + α = 1 α =.9 ff ωω = σσππ 1 1 cos αααα ωω 2

15 Squared Exponential (Gaussian) Model CC h = σσee ααh2 α =.5 α = 1 ff ωω = 1 2 σσ(ππππ) 1 2 ee ωω 2 (4αα)

16 Matern Class CC h = ππ dd 2ϕ 2 ν 1 Γ ν + dd 2 αα2ν ααα ν KK ν (ααα) KK ν is a modified Bessel function of the third kind αα = ν = ϕ = 1 ff ωω = ϕ(αα 2 + ωω 2 ) ( ν dd 2 ) Φ, ν, α > 0 Φ is scale parameter α is the inverse of the autocorrelation range ν is the smoothness parameter

17 Scale Parameter ff ωω = ϕ(αα 2 + ωω 2 ) ( ν dd 2 ) φ = 1 φ =.75 φ =.5

18 Range Parameter ff ωω = ϕ(αα 2 + ωω 2 ) ( ν dd 2 ) α = 1 α =.75 α =.5

19 Smoothness Parameter ff ωω = ϕ(αα 2 + ωω 2 ) ( ν dd 2 ) ν = 1 ν =.75 ν =.5

20 Estimation

21 Periodogram nn 1 nn 2 2 II NN ωω 0 = δδ 1 δδ 2 (2ππ) 2 (nn 1 nn 2 ) 1 ss 1 =1 ss 2 =1 ZZ ss exp( ii ss tt ωω) Is the Fourier transform of the sample covariance The expected value of the periodogram, II NN ωω, is asymptotically ff (ωω) The asymptotic variance of II NN ωω is ff 2 (ωω) The periodogram values II NN ωω and II NN ωω for ωω ωω, are asymptotically independent

22 Periodogram Example

23 Whittle Approximation to the Gaussian Negative Likelihood Representation: NN (2ππ) 2 RR 2 {log ff ωω + II NN ωω ff(ωω) 1 } dddd Estimated by: Asymptotic Covariance of MLE Estimates:

24 Tapering

25 Tapering

26 Correction for Aliasing ff ωω = QQ ZZ 2 ff(ωω + 2ππππ ), ωω εε ππ 2 = [ ππ, ππ ] 2 nn = qq 1 = nn nn qq 2 = nn ff( ωω 1 + 2ππqq 1, ωω 2 + 2ππqq 2 )

27 Lattice Data with Missing Values

28 Summary of Analysis Take out any obvious mean trends Taper the data and re-adjust variance Take the FFT of the data Estimate periodogram Choose a spectral covariance model (Matern, Gaussian etc.) Write a function to estimate the density corrected for aliasing (slow) Minimize the Whittle Likelihood for the estimates of the parameters (leave out 0 frequency)

29 Data Application

30 Goal of Analysis Wish to estimate the spatial structure of sea surface temperature fields in the northeast Pacific Ocean using Tropical Rainforest Measuring Mission (TRMM) microwave imager (TMI) satellite data

31 Motivation Sea surface temperature fields are the main factor to identify phenomena such as El Nino and La Nino One of the main climate factors to identify tropical cyclones (hurricanes) Used as an oceanic boundary condition for numerical atmospheric models Used as a diagnostic tool for comparison with SSTs produced by oceanic numerical models

32 Trend Removal

33 Exploration of Isotropy

34 Parameter Estimation