Point-Referenced Data Models

Transcription

1 Point-Referenced Data Models Jamie Monogan University of Georgia Spring 2013 Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

2 Objectives By the end of these meetings, participants should be able to: Define stationarity, variogram, and isotropy. Draw image and surface plots in R. Create empirical semivariograms and estimate parametric variograms in R. Forecast the value of a variable given its spatial site. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

3 Reminder: Our Purpose Looking ahead to chapter 5, a basic model of a stationary spatial process is: Y (s) = µ(s) + w(s) + ɛ(s) (1) The elements are: mean structure (µ(s)), spatial residual process (w(s)), and nonspatial residual process (ɛ(s)). Normally we think of mean structure first. But here, we start by modeling w(s). Why do we do this? Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

4 Stationarity Antonym: trending data. (Or unit roots.) Assumptions Spatial process: {Y (s) : s D}, D R r. Mean, µ(s) = E(Y (s)), exists for all s D. Variance exists for all s D. Forms of Stationarity Gaussian: any set of sites is distributed MVN. Strictly stationary: the distribution of sites does not change. For any set of sites {s1,..., s n } and any h R r, the distribution of {Y (s 1 ),..., Y (s n )} is the same as for {Y (s 1 + h),..., Y (s n + h)}. Weakly stationary (second order): the mean does not change and spatial covariance only depends on distance. µ(s) µ and Cov(Y (s), Y (s + h)) = C(h) for all h R r such that s and s + h both lie within D. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

5 Intrinsic Stationarity Assume E[Y (s + h) Y (s)] = 0. Define E[Y (s + h) Y (s)] 2 = Var(Y (s + h) Y (s)) = 2γ(h). If the LHS on the previous line depends only on h and not on s, then the process is intrinsically stationary. Note: Intrinsic stationarity depends only on the difference E[Y (s + h) Y (s)] and says nothing about joint distributions. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

6 Variograms Give us a sense of how the variance between data points changes with distance. (Alternatively, how much do data points covary, contingent on their distance?) Terms 2γ(h) is the variogram. γ(h) is the semivariogram. C(h) is the covariogram. Properties We can determine the variogram if we know the covariogram. We can also do the reverse if the process is ergodic and lim h γ(h) exists. Analogue: The autocorrelation and partial autocorrelation functions from time series. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

7 Isotropy If the semivariogram γ(h) depends on the separation vector only through its length, h, then the process is isotropic. Hence, we can write γ( h ). For simplicity, h = t, which we represent as the only input. Anisotropy There could be directional dependence. Geometric anisotropy. Sill anisotropy, nugget anisotropy, and range anisotropy. Assessment techniques: directional semivariograms and rose diagrams. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

8 Three Prominent Features of Variograms Nugget: Baseline variance in a variable, present even when t 0. Sill: Maximal (asymptotic) variance in a variable, when t is very large. The partial sill is the difference between the sill and nugget. Range: The value R = 1/φ where γ(t) reaches its sill or ultimate level. φ is the decay parameter, but some call it the range parameter as well. The effective range is often used when R is technically infinite. Switching to the covariance function, when the correlation drops below 0.05 the effective range is often declared. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

9 Process for Fitting a Parametric Variogram Similar to fitting an ARIMA model in time series. Plot the empirical semivariogram: ˆγ(t) = 1 2N(t) (s i,s j ) N(t) [Y (s i ) Y (s j )] 2 Grid-up the t-space into discrete intervals. Compare to the empirical footprint/signature we would see from various parametric processes. Check how well the model fits (AIC, etc). Common parametric isotropic models: Linear Rational quadratic Spherical Wave Exponential Power law Powered Exponential Matérn Gaussian Matérn at ν = 3/2 Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

10 Exploratory Approaches for Point-Referenced Data Drop line plots. Image/contour plots. Surface plots. Empirical semivariograms. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

11 For February 13 Read from Banerjee, Carlin, & Gelfand. Answer 1.a from pp. 66. Download the 1974 Nixon thermometer data. Draw: 1 An image plot with contours, 2 a surface plot, and 3 a drop line plot. Which figure most clearly conveys the data? Graphically report an empirical semivariogram for the data. What do you think this semivariogram tells you? Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

12 Kriging Goals of Data Analysis Inference Prediction Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

13 Prediction in OLS E(Y X) = Ŷ = Xˆβ Given a certain exogenous situation, what will we expect our endogenous variable to be? Spatial Prediction Given observations Y = (Y (s 1 ),..., Y (s n )), how do we predict the variable s value at an unobserved site s 0? Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

14 A Linear Spatial Predictor li Y (s i ) + δ 0 Our best predictions will minimize the following equation over δ 0 and the l i : E[Y (s 0 ) ( l i Y (s i ) + δ 0 )] 2 We assume intrinsic stationarity, which causes δ 0 to drop out and allows us to instead minimize: E[Y (s 0 ) l i Y (s i )] 2 where l i = 1. A constrained minimization for the l i will yield λ(s 0 ), which allows us to make our predictions. Ultimately, the following equation provides our projections: Ŷ (s 0 ) = λ(s 0 ) y. But what is λ? Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

15 Ordinary Kriging No covariates Y = µ1 + ɛ, where ɛ N (0, Σ). Σ = σ 2 H(φ) + τ 2 I, where: (H(φ))ij = ρ(φ; d ij ), dij = s i s j, and ρ is a correlation function, perhaps defined by a model in BCG Table 2.1. The only feature of this model that can change our projection is the semivariogram; hence, that is all we need to determine λ. Intuition: more correlated values carry greater weight. Note: if we have no nugget effect (i.e., all unexplained variance is spatially-dependent), then τ 2 = 0 Note: R refers to µ as β Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

16 Ordinary Kriging Equation λ 1. λ n η = 2γ(Y 1, Y 1 )... 2γ(Y 1, Y n ) 1 2γ(Y 1, Y (s 0 )) γ(Y n, Y 1 )... 2γ(Y n, Y n ) 1 2γ(Y n, Y (s 0 )) In this estimator: λ i is the weight that the observation at location i is given. η is a Lagrange multiplier used in the computation of the forecast error. γ is the semivariogram function (and hence 2γ the variogram makes-up the component cells). Y i is shorthand for Y (s i ). The value of λ(s 0 ) is contingent on the location of s 0. Once a unique λ(s 0 ) is computed, λ(s 0 ) y will return your forecast of Y at location s 0. Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

17 Universal Kriging Covariates are present. Y = Xβ + ɛ, where ɛ N (0, Σ) and Σ is the same as before. A model of the mean surface influences predictions, as well as the semivariogram. Rethink the prediction problem E[(Y (s 0 ) f (y)) 2 y]. The predictor that minimizes this is f (y) = E[Y (s 0 ) y]. MVN theory leads us to: E[Y (s0 ) y] = x 0 β + γ Σ 1 (y Xβ), and Var[Y (s0 ) y] = σ 2 + τ 2 γ Σ 1 γ. Provided we know population parameters and x 0 (large caveats). Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

18 Using Estimates f (y) = x 0 ˆβ + ˆγ ˆΣ 1 (y Xˆβ), where: where ˆγ = (ˆσ 2 ρ( ˆφ; d 01 ),..., ˆσ 2 ρ( ˆφ; d 0n )), and ˆβ = (X ˆΣ 1 X) 1 X ˆΣ 1 y. In our original terms of Ŷ(s 0 ) = λ(s 0 ) y, we can now define: λ(s 0 ) = ˆΣ 1ˆγ + ˆΣ 1 X(X ˆΣ 1 X) 1 (x 0 X ˆΣ 1ˆγ). We recognize all of the elements individually, but simple subsitution would not account for uncertainty. Do we know x 0? If not, try an EM algorithm. Expectation step: ˆx 0 = X λ(s 0 ). Maximization step: Calculate λ(s 0 ). Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19

19 For February 20 Read from Banerjee, Carlin, & Gelfand. Download the 1974 pilot data. Study the variable V Recode values of 0, 8, & 9 as missing. Graphically report an empirical semivariogram for the variable. Estimate, a parametric variogram for the variable. Krige the predicted values of V for two locations. Meades Ranch in Kansas (0,0). Oregon countryside ( , ). Jamie Monogan (UGA) Point-Referenced Data Models Spring / 19