Introduction. Spatial Processes & Spatial Patterns

Transcription

1 Introduction Spatial data: set of geo-referenced attribute measurements: each measurement is associated with a location (point) or an entity (area/region/object) in geographical (or other) space; the domain informed by a measurement is called the sample unit or support Slide 1 Characteristics: spatial arrangement of sample locations can be regular or irregular, i.e., sample locations can be arranged on a raster (regular lattice) or can be scattered in space attribute measurement scale can be continuous or discrete, e.g., chemical concentration, soil types, disease occurrences attribute values measured at nearby supports tend to be more similar than those measured at distant supports; this is the concept of spatia/temporal auto-correlation Objectives of this handout: to provide an overview of spatial process models and how they relate to multivariate statistics and the generation of attribute spatial patterns to illustrate how the case of spatial data without any temporal footprint is particularly difficult to deal with, when trying to infer the parameters of spatial process models from spatial patterns Spatial Processes & Spatial Patterns Spatial process: Set of physical laws, mathematical equations, rules, or statistical models that generate attribute values (hence patterns) in space Deterministic vs stochastic process: deterministic process: mathematical description of spatial patterns; each map location is associated with a single attribute value Slide stochastic process: statistical description of spatial patterns; each map location is associated with multiple attribute values terms stochastic and random do not imply unpredictability; just lack of uniqueness, but in a predictable way often, stochastic process models have a deterministic component, but they are still considered stochastic Analysis objective: from patterns to processes: inverse problem: analyze map patterns to infer the parameters of a stochastic spatial process, which is believed to have generated these patterns; in other words, observed attribute values are viewed as realizations from that spatial process To tackle the inverse problem, however, one must first understand the forward problem, i.e., how can (stochastic) spatial processes generate spatial patterns... Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 3/6

2 Multi-Variate Gaussian Correlated Random Variables Random vector: a collection of correlated random variables (RVs), each associated with a particular support. For N supports {s n,n =1,...,N}, therearen RVs {Y (s n ),n =1,...,N} which can be arranged compactly in a (N 1) vector y =[Y n,n =1,...,N] T with Y n Y (s n ). When the joint distribution of all N RVs in vector y is N-variate Gaussian, y is termed a Gaussian random vector; in this case, each of the N marginal distributions is also Gaussian Slide 3 Parameters of a Gaussian random vector: (N 1) mean vector: μ Y =[μ Y (n),n=1,...,n] T,whereμ Y (n) =E{Y (s n )} (N N) covariancematrixσ Y between all pairs of RVs Σ Y =[σ Y (n, n ),n=1,...,n,n =1,...,N]: σ Y (1, 1) σ Y (1,N) Σ Y = σ Y (n, n ) σ Y (N, 1) σ Y (N, N) σ Y (n, n )=Cov{Y n,y n } = E{[Y (s n ) μ Y (s n )][Y (s n ) μ Y (s n )]}: covariance between two RVs Y n and Y n One cannot populate Σ Y with arbitrary entries; it is always required that Σ Y be positive definite Compact notation: y G(μ Y, Σ Y ), i.e., the random vector y follows a multivariate Gaussian distribution of as many dimensions as the number of its constituent RVs (here N), with parameters μ Y and Σ Y. The particular case of N uncorrelated RVs corresponds to a covariance matrix Σ Y with off-diagonal elements. When, in addition, all diagonal elements of Σ Y are the same and equal to a common variance σ Y,thenΣ Y = σ Y I where I is the (N N) identity matrix Realizations from A Gaussian Random Vector Ensemble of simulated realizations: For N RVs, one can generate L sets of simulated values (realizations) from the N RVs and arrange them in a (L N) matrix Y. The n-th column Y(,n) of matrix Y contains realizations for support s n.thel-th row Y(l, :) of matrix Y contains the l-th simulated N-variate sample, i.e., one snapshot or cross-section of the ensemble of simulated values. Note that in theory L can be very large... 9 Profile #1 9 Profile # 9 Profile #3 4 Pair wise scatter plots Slide Since the N RVs are pair-wise correlated, their realization profiles are also pair-wise correlated; this implies that the N simulatedvaluesinthel-th snapshot, for example, are expected to be similar to each other. Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 5/6

3 From Gaussian Random Vectors to Gaussian Stochastic Processes Gaussian stochastic process: One of the most important and widely used stochastic process models, linked to a Gaussian random vector in a spatial or temporal context s 1 s s 3 s 4 s 5 cross-sections of process realizations Slide 5 locations In a spatial context, RVs corresponding to neighboring or nearby supports are expected to be more correlated than RVs corresponding to distant supports. The consequence of this expectation is that process realizations at nearby supports are expected to be more similar to each other than realizations at distant supports. Since observed data are assumed to be a particular realization of a stochastic process (one of the many possible snapshots on the right graph), the term spatial auto-correlation is used to convey the notion of similarity in attribute values, often as a function of distance between the corresponding supports. Components of Stochastic Processes Attribute values as outcomes of a stochastic process: A stochastic process for, say, a continuous spatial attribute can generate many attribute values at the same location in space; in other words, there are multiple attribute maps for a given spatial process model Slide 6 First-order effects: influence of external or environmental factors on process outcomes; e.g., abundance of plants within a sub-region could depend on soil type. Note: first-order effects are typically assumed to influence the magnitude of process outcomes at each location, and hence are associated with the mean of all possible process outcomes at each location Second-order effects: influence of process outcomes at one location on possible process outcomes at nearby locations; e.g., non-contagious versus contagious diseases. Note: second-order effects typically express some measure of similarity between possible process outcomes at different locations once the first-order effects have been removed, and are often associated with the covariance or correlation coefficient between different RVs. For a Gaussian stochastic process, there are N N = N(N 1) off-diagonal entries of the covariance matrix Σ Y to be specified under the constraint of positive definiteness. Due to symmetry, this number is reduced to N(N 1)/ entries, but with the diagonal entries goes up to N(N 1)/+N = N(N +1)/. WhenN is large, there are too many entries to be specified, hence the covariance matrix Σ Y needs to be parameterized... Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 7/6

4 Some Characteristics of Stochastic Processes First-order stationarity: particular case, whereby the effect of environmental factors is spatially fixed or independent of position. This characteristic implies that the process is expected to yield the same value on average over many realizations (large L) at any location (constant or flat climatology ). Notation-wise, μ Y (s n )=E{Y (s n )} = μ Y, n Slide 7 Second-order stationarity: particular case, whereby the spatial interaction between process outcomes at two locations does not depend on which two locations are considered but on the distance separating them. This characteristic implies that the association between realizations at a pair of locations is the same as the association between realizations at another pair of locations, as long as the locations of the one pair are the same distance apart as the locations of the other pair. This characteristic also implies that the covariance σ Y (s n,s n ) between any two RVs Y (s n ) and Y (s n ) could be specified as a function of distance d nn between their corresponding supports s n and s n, i.e., σ Y (n, n )=φ(d nn ; θ); that function is typically parametric (e.g., exponential distance decay), and thus replaces the need to specify all N(N 1)/ entries of Σ Y by the need to specify its parameter vector θ Circumventing the above assumptions: develop different process models for different sub-regions of the study area, where such characteristics are more likely to apply include explicitly first-order effects into a model via data on covariates Examples of 1D Stochastic Process Realizations (1) Gaussian random vector realizations: at N = locations arranged on a regular grid with unit spacing: 1 1 Slide left: any RV Y (s n ) has mean and variance 1; RVs are uncorrelated, i.e., Σ Y = I. Thisisa completely random stochastic process model, also termed (in engineering) a white noise model right: RVs in the left half of the transect have mean, whereas RVs in the right half have mean 1; all RVs have variance 1 and are pairwise uncorrelated, i.e., Σ Y = I Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 9/6

5 Examples of D Stochastic Process Realizations (1) Gaussian random vector realizations: at the nodes of a ( ) regular raster, i.e., N =, with unit cell size: Realizations from D stoch. process Realizations from D stoch. process Slide 9 left: any RV Y (s n ) has mean and variance 1; RVs are uncorrelated, i.e., Σ Y = I; thisisa completely random spatial process model, also termed (in engineering) a white noise model right: RVs in the left half of the domain have mean, whereas RVs in the right half of the domain have mean 1; all RVs have variance 5 and are pair-wise uncorrelated, i.e., Σ Y =5I Examples of 1D Stochastic Process Realizations () Gaussian random vector realizations: at N = locations on a regular grid with unit spacing: Slide 1 4 left: all RVs have mean and variance 1; correlation between RVs is specified via an exponential distance decay function whose parameter is a de-correlation length (or range of influence), here set to distance units right: all RVs have mean and variance 1; correlation between RVs is specified via a distance decay function whose rate of decay is different (slower) than the one used on the right (de-correlation length is now 5), leading to more similar values in space ( smoother profiles) Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 11/6

6 Examples of D Stochastic Process Realizations () Gaussian random vector realizations: at the nodes of a ( ) regular raster, i.e., N =, with unit cell size: Realizations from D stoch. process Realizations from D stoch. process Slide 11 left: all RVs have mean and variance 1; correlation between RVs is specified via a D exponential distance decay function with range along any direction (isotropic variation) right: all RVs have mean and variance 1; correlation between RVs is specified via distance decay function whose rate of decay is different (slower) than the one used on the right (de-correlation length is now 5), leading to more similar values in space ( smoother images) Examples of 1D Stochastic Process Realizations (3) Gaussian random vector realizations: at N = locations on a regular grid with unit spacing: Slide left: all RVs have variance 5; RVs in the left half of the transect have mean, whereas RVs in the right half have mean 1; correlation between RVs is specified via an exponential distance decay function with range right: all RVs have variance ; the mean of any RV is a linear function of its position (linear trend); correlation between RVs is specified is specified via an exponential distance decay function with range Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 13/6

7 Examples of D Stochastic Process Realizations (3) Gaussian random vector realizations: at the nodes of a ( ) regular raster, i.e., N =, with unit cell size: Realizations from D stoch. process Slide 13 left: all RVs have variance 5; RVs in the left half of the domain have mean, whereas RVs in the right half have mean 1; correlation between RVs is specified via a D exponential distance decay function with range along any direction right: all RVs have variance ; each RV has a mean specified as a function of its position (linear D trend from West to East); correlation between RVs is specified via a D exponential distance decay function with range in any direction Examples of 1D Stochastic Process Realizations (4) Gaussian random vector realizations: at N = locations on a regular grid with unit spacing: 1 Slide left: RVs in the left half of the transect have mean and variance 1, whereas RVs in the right half have mean 1 and variance 1; RVs are uncorrelated, i.e., R Y = I, wherer is the correlation matrix right: all RVs have mean and variance 1; correlation between RVs is specified via a periodic function of distance (cosine model) with a given amplitude and wavelength (plus a small component of white noise) Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 15/6

8 Examples of D Stochastic Process Realizations (4) Gaussian random vector realizations: at the nodes of a ( ) regular raster, i.e., N =, with unit cell size: Realizations from D stoch. process Realizations from D stoch. process Slide 15 left: all RVs have mean and variance 1; correlation between RVs is specified via a D sine wave function with range 5 along any direction right: all RVs have mean and variance 1; correlation between RVs is specified via a D sine wave function with range 5 along EW and 5 along NS; this is the case of anisotropic spatial variability, whereby the correlation between any two RVs is both a function of the magnitude and orientation of the vector separating their respective supports What we have done so far: Back to the Real World we studied the forward problem; that is, given a particular process model (here assumed multi-gaussian) with a known mean vector μ Y and covariance matrix Σ Y,wesawsome examples of how such a model can be used to generate attribute spatial patterns Slide 16 this is fine; just keep in mind that not all real-world patterns can be modeled by such a process, particularly those involving patterns with clear geometrical characteristics Remember our objective, however: we want to solve the inverse problem; that is, from the observed attribute patterns, we would like to select a spatial process model and then estimate its parameters. In all generality, the Gaussian model has N + N(N +1)/ parameters to be estimated: the N entries of vector μ Y (one mean value per RV at each location) and the N(N +1)/ entries of, say, the upper diagonal of the the covariance matrix Σ Y this would not have been a daunting task had we seen L realizations of each RV at the N different locations, but very unfortunately this is not the case in a purely spatial data analysis context... Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 17/6

9 The spatial data situation: The Single Cross-Section Predicament in a purely spatial analysis context, one has access to N measured attribute values at the N supports where data have been collected. These N data are viewed as one process realization out of L possible ones, i.e., one row of the matrix Y of realizations of the random vector y from this single process realization we would like to: (i) select a spatial process model, and (ii) estimate its parameters; good luck... s 1 s s 3 s 4 s 5 Slide 17 observed data = a single process realization From the single cross-section with N measured data, we need to somehow dream of the N profiles of the corresponding N RVs to find their means and variances. Even worse, we need to dream of N(N 1)/ scatter plots of realizations between RVs so we can find their pair-wise correlations Random Fields: An Even Tougher Situation Definition: an infinite collection of spatially correlated RVs {Y (s), s A}, oneperlocations, is called Random Field (RF); here s is the coordinate vector of an arbitrary location, and Y (s) is the RV defined at s. This conceptual framework corresponds to spatial attributes that vary continuously in space (e.g., elevation). Spatial law: M-dimensional multivariate (multi-point) distribution of constituent RVs: Slide 18 F Y1,...,Y M (y 1,...,y M )=Prob{Y (s 1 ) y 1,...,Y(s M ) y M } M = infinite number of locations within study area A y 1,...,y M = arbitrary thresholds (y-attribute values) Data: Asinglejoint realization of N RVs defined at N locations where attributes have been measured {y(s n ),n=1,...,n}; when sample data are sparse, N<<M Big problem: how to model the multi-variate distribution of all M constituent RVs {Y (s m ),m=1,...,m}, when at best only one realization {y(s n ),n =1,...,N} of the N RVs {Y (s n ),n=1,...,n} is available at the data locations? Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 19/6

10 Stationarity / Homogeneity Objective: model distribution of any RV Y (s) without repetitive realizations (samples) {y (l) (s),l =1,...,L} at location s One solution: substitute unavailable realizations {y (l) (s),l =1,...,L} at location s by N data realizations at sample locations {y(s n ),n =1,...,N} y(s ) = 1. y(s ) = 5. 5 Frequency Stationary density function Slide 19 y(s ) = 7. 3 y( s ) =? Attribute Value y(s ) = y(s ) = 8. 4 A In other words, copy sample distribution (or only its mean μ Y and variance σ Y ) to all locations in study region now μ Y = μ Y 1,where1 is a vector with M 1s, and the diagonal entries of Σ Y are all equal to σ Y Requisite: prior definition of zone/area (possibly a subset of the entire study region A), within which data are pooled together for inference stationarity is a model (not data) characteristic; cannot formally test decision regarding appropriate area!!! now all statistics pertain to the particular area/zone used as replication domain for pooling sample data together for inference; what about extrapolation? Bivariate Stationarity / Homogeneity (1) Objective: evaluate joint distribution of any two RVs Y (s) and Y (s ), without repetitive realizations {[y (l) (s),y (l) (s )],l =1,...,L} at locations s and s Problem: at best, if locations s and s coincide with two sample locations s n and s n,onlya single joint realization {y(s n ),y(s n )} of these two RVs Y (s n ) and Y (s n ) is available Slide A Solution: 1. compute separation vector h = s s between two locations s and s ; Note: a vector h has both magnitude and orientation; Note: locations s and s = s + h are called, respectively, the tail and head of vector h. substitute unavailable joint realizations {[y (l) (s ),y (l) (s)],l =1,...,L} with realizations (data pairs) {[y(s n + h),y(s n )],n =1,...,N(h)} separated (approximately) by the same vector h; N(h) = number of sample location pairs separated by h Note: There are other possible ways to group data together; grouping based on Euclidean distances, however, is by far the most widely used For areal data, unless one is prepared to work with centroids (not recommended), grouping is typically achieved by specifying which units are neighbors of any areal unit (units that share common boundaries are typically called neighbors) Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 1/6

11 Bivariate Stationarity / Homogeneity () Dependence between any two RVs Y (s ) and Y (s) (two-point or bi-variate dependence): fully characterized by h-specific bivariate PDF: f h (y,y)=prob{y (s + h) =y ± dy,y(s) =y ± dy} 1. Scatterplot for lag h y(s +h) n Slide 1 8. ρ(h) =.85 y y y(s n ) bivariate PDF is not a function of the individual pair of locations s and s bivariate PDF is now a function of (norm and possibly orientation) of vector h partially characterized by h-specific moments of PDF f h (y,y): e.g., γ(h) = 1 E{[Y (s + h) Y (s)] }, σ(h) =E{[Y (s + h) μ(s + h)][y (s) μ(s)]}, σ(h) ρ(h) = ; σ(s+h) σ(s) μ(s + h) =E{Y (s + h)} = head data mean; σ(s) =Var{Y (s)} = tail data variance Distance-Specific Spatial Correlation 1. construct scatter plots of attribute data pairs separated by various lag vectors h. compute and plot summaries of bivariate distributions, e.g., correlation coefficient ρ(h), versus h possibly along different directions: Scatterplot for lag h=1 m Scatterplot for lag h=3 m Head ρ(h=1).85 Head ρ(h=3).56 Slide Tail 1. Sample Correlogram Tail.8 Scatterplot for lag h>1 m ρ(h).6.4 Head ρ(h>1) = Tail h. Scatterplot for lag h=5 m Scatterplot for lag h=7 m Head ρ(h=5).31 Head ρ(h=7) Tail Tail Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 3/6

12 Variants of Stationarity Strict stationarity: spatial law invariant under translation: F Y (s1 +h),...,y (s M +h)(y 1,...,y M )=F Y (s1 ),...,Y (s M )(y 1,...,y M ) same multivariate distribution when M locations are shifted by vector h Second-order stationarity: only first two moments are translation invariant: Slide 3 mean = constant: E{Y (s + h)} = E{Y (s)} = μ, s, h covariance = function only of separation vector h: Cov{Y (s ),Y(s)} = Cov{Y (s + h),y(s)} = σ(s s) =σ(h) σ(s, s) Intrinsic stationarity: second-order stationary applies to increments or spatial differences: mean of differences: E{[Y (s + h) Y (s)]} =, s, h variance of differences = function of separation vector h: Var{[Y (s ) Y (s)]} = E{[Y (s + h) Y (s)] } =γ(h) Isotropy: γ(h) = γ(h), σ(h) = σ(h), andρ(h) = ρ(h), h = h =normofh Non-Stationary RF Models Accounting for Covariates Random field: infinite collection of spatially correlated, geo-referenced RVs {Y (s), s A} RV decomposition: into trend and stochastic residual component: Y (s) =μ(s)+r(s), s A y Elevation values along a transect Slide 4 μ ( s) y ( s) original elevation profile trend s μ(s): deterministic function modeling large-scale average spatial variation first-order or environmental effects, trend or drift (typically function of covariates) R(s): stochastic RF modeling small-scale spatial variation not explained by μ(s) second-order effects, spatial interaction Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 5/6

13 More on Non-Stationary Models General RF decomposition: Y (s) =μ(s)+r(s), s A mean of RV Y (s) at any location s is the trend component at that location: E{Y (s)} = μ(s), s A residual RF {R(s), s D} with constant zero mean: E{R(s)} =, s, and covariance being a function of separation vector: σ R (h) =Cov{R(s ),R(s)} = E{R(s + h)r(s)} Slide 5 Key concepts: subjective decomposition into trend and residual components; no data exist on either μ(s) or R(s) proportion of spatial variability not modeled by trend component μ(s) is absorbed into (modeled by) residual component R(s) under the scale-dependent decision of stationarity: Y (s) =μ + R(s) σ Y (h) =σ R (h) Note: Most of the concepts presented in this handout apply to area (lattice) data, too; the important difference is the way in which the covariance of the residuals is specified for such data: σ R (s, s) σ R (h), i.e., covariance is not a direct function of distance and is not stationary From spatial processes to spatial patterns: Recap forward problem: given a stochastic process model (here assumed multivariate Gaussian) generate realizations from such models these process realizations exhibit spatial patterns that bear the imprint of the model form (here multi-gaussian), the first-order effects, the specification of the covariance matrix (say via a distance decay function) encapsulating second-order effects, and the magnitude of the parameters (e.g., range of influence) parameterizing the covariance function Slide 6 From spatial patterns to spatial processes: inverse problem: given a set of measurements analyze its spatial patterns to: (i) select an appropriate process model, and (ii) estimate its parameters the data are assumed to be a single realization from a spatial process; this renders the inverse problem under-determined or ill-posed. To resolve the under-determinacy one needs to first select a process model and then estimate its parameters... Shortcuts for inference: stationarity / homogeneity and sometimes isotropy can incorporate data on covariates in the model by relating them typically to the mean component (first-order effects) Phaedon C. Kyriakidis Elements of Spatial Stochastic Processes page 6/6