Multivariate spatial modeling
|
|
- Norma O’Connor’
- 6 years ago
- Views:
Transcription
1 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Chapter 7: Multivariate Spatial Modeling p. 1/21
2 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Chapter 7: Multivariate Spatial Modeling p. 1/21
3 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. Chapter 7: Multivariate Spatial Modeling p. 1/21
4 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed Chapter 7: Multivariate Spatial Modeling p. 1/21
5 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income Chapter 7: Multivariate Spatial Modeling p. 1/21
6 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements Chapter 7: Multivariate Spatial Modeling p. 1/21
7 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements at a particular location Chapter 7: Multivariate Spatial Modeling p. 1/21
8 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements at a particular location across locations Chapter 7: Multivariate Spatial Modeling p. 1/21
9 Basic issues Y(s) denotes a p 1 vector of random variables at s Chapter 7: Multivariate Spatial Modeling p. 2/21
10 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Chapter 7: Multivariate Spatial Modeling p. 2/21
11 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Crucial object: the cross-covariance, C(s, s ) = Cov(Y(s), Y(s )) a p p matrix that need not be symmetric, i.e., cov(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) Chapter 7: Multivariate Spatial Modeling p. 2/21
12 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Crucial object: the cross-covariance, C(s, s ) = Cov(Y(s), Y(s )) a p p matrix that need not be symmetric, i.e., cov(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) C(s, s ) is not positive definite except in a limiting sense: C(s, s) is the covariance matrix associated with Y(s). Chapter 7: Multivariate Spatial Modeling p. 2/21
13 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Crucial object: the cross-covariance, C(s, s ) = Cov(Y(s), Y(s )) a p p matrix that need not be symmetric, i.e., cov(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) C(s, s ) is not positive definite except in a limiting sense: C(s, s) is the covariance matrix associated with Y(s). our primary focus: Gaussian processes and valid specification for C(s, s ) Chapter 7: Multivariate Spatial Modeling p. 2/21
14 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix Chapter 7: Multivariate Spatial Modeling p. 3/21
15 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix Chapter 7: Multivariate Spatial Modeling p. 3/21
16 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Chapter 7: Multivariate Spatial Modeling p. 3/21
17 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Chapter 7: Multivariate Spatial Modeling p. 3/21
18 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Σ Y is positive definite since H and T are Chapter 7: Multivariate Spatial Modeling p. 3/21
19 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Σ Y is positive definite since H and T are Σ Y is convenient since Σ Y = H p T n and Σ 1 Y = H 1 T 1. Chapter 7: Multivariate Spatial Modeling p. 3/21
20 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) Chapter 7: Multivariate Spatial Modeling p. 4/21
21 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Chapter 7: Multivariate Spatial Modeling p. 4/21
22 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: Chapter 7: Multivariate Spatial Modeling p. 4/21
23 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) Chapter 7: Multivariate Spatial Modeling p. 4/21
24 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) µ(s) = (µ 1,µ 2 ) is coordinate-free. Chapter 7: Multivariate Spatial Modeling p. 4/21
25 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) µ(s) = (µ 1,µ 2 ) is coordinate-free. Then p(y (s) X(s)) N(β 0 + β 1 X(s),σ 2 ), where β 0 = µ 2 T 12 T 11 µ 1, β 1 = T 12 T 11, and σ 2 = T 22 T 2 12 T 11 Chapter 7: Multivariate Spatial Modeling p. 4/21
26 Bivariate spatial regression (cont d) Rearrangement of the components of Z to Z = (X(s 1 ),X(s 2 ),...,X(s n ),Y (s 1 ),Y (s 2 ),...,Y (s n )) yields ( X Y ) N (( µ 1 1 µ 2 1 ), T H (φ) ), Chapter 7: Multivariate Spatial Modeling p. 5/21
27 Bivariate spatial regression (cont d) Rearrangement of the components of Z to Z = (X(s 1 ),X(s 2 ),...,X(s n ),Y (s 1 ),Y (s 2 ),...,Y (s n )) yields ( X Y ) N (( µ 1 1 µ 2 1 ), T H (φ) Priors: Wishart for T 1, vague but proper normal for (µ 1,µ 2 ), uniform or other suitable choice for φ ), Chapter 7: Multivariate Spatial Modeling p. 5/21
28 Bivariate spatial regression (cont d) Rearrangement of the components of Z to Z = (X(s 1 ),X(s 2 ),...,X(s n ),Y (s 1 ),Y (s 2 ),...,Y (s n )) yields ( X Y ) N (( µ 1 1 µ 2 1 ), T H (φ) Priors: Wishart for T 1, vague but proper normal for (µ 1,µ 2 ), uniform or other suitable choice for φ Full conditionals for Gibbs sampler: again Wishart for T 1, bivariate normal for (µ 1,µ 2 ); nonconjugate for φ (so need Metropolis sampling here) ), Chapter 7: Multivariate Spatial Modeling p. 5/21
29 Example with dew-shrub data 1129 locations with UTM coordinates Chapter 7: Multivariate Spatial Modeling p. 6/21
30 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s Chapter 7: Multivariate Spatial Modeling p. 6/21
31 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Chapter 7: Multivariate Spatial Modeling p. 6/21
32 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh Chapter 7: Multivariate Spatial Modeling p. 6/21
33 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region Chapter 7: Multivariate Spatial Modeling p. 6/21
34 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region (µ 1,µ 2,T 11,T 12,T 22 ) updated directly; φ updated via Metropolis Chapter 7: Multivariate Spatial Modeling p. 6/21
35 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region (µ 1,µ 2,T 11,T 12,T 22 ) updated directly; φ updated via Metropolis (β 0,β 1,σ 2 ) samples automatically determined as functions of the others Chapter 7: Multivariate Spatial Modeling p. 6/21
36 Dew and shrub data sites Eastings Northings Chapter 7: Multivariate Spatial Modeling p. 7/21
37 Parameter estimation, dew-shrub data Parameter 2.5% 50% 97.5% µ µ T T T φ β β σ T 12 / T 11 T surprising significant negative association between dew duration and shrub density! (also emerges with a nonspatial model) Chapter 7: Multivariate Spatial Modeling p. 8/21
38 Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Chapter 7: Multivariate Spatial Modeling p. 9/21
39 Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Limitations: Symmetry in cross-covariance matrix Imposes same spatial range for every measurement (only one correlation function) Chapter 7: Multivariate Spatial Modeling p. 9/21
40 Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Limitations: Symmetry in cross-covariance matrix Imposes same spatial range for every measurement (only one correlation function) One possible solution: Coregionalization models!... Chapter 7: Multivariate Spatial Modeling p. 9/21
41 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). Chapter 7: Multivariate Spatial Modeling p. 10/21
42 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p Chapter 7: Multivariate Spatial Modeling p. 10/21
43 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T Chapter 7: Multivariate Spatial Modeling p. 10/21
44 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T In general, the cross covariance matrix is C(s s ) = p j=1 ρ j (s s )a j a j where a j is the jth column of A. Chapter 7: Multivariate Spatial Modeling p. 10/21
45 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T In general, the cross covariance matrix is C(s s ) = p j=1 ρ j (s s )a j a j where a j is the jth column of A. Approach is constructive" and hence immediately valid, still stationary, and provides a distinct covariance function for each component Chapter 7: Multivariate Spatial Modeling p. 10/21
46 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! Chapter 7: Multivariate Spatial Modeling p. 11/21
47 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Chapter 7: Multivariate Spatial Modeling p. 11/21
48 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): Chapter 7: Multivariate Spatial Modeling p. 11/21
49 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): T(s) = g(x(s)) T Chapter 7: Multivariate Spatial Modeling p. 11/21
50 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): T(s) = g(x(s)) T T(s) is a spatial process (e.g., T 1 (s) is a spatial Wishart process) Chapter 7: Multivariate Spatial Modeling p. 11/21
51 Other Approaches Moving average or kernel convolution of a process: Y j (s) = k j (u)z(s + u)du = k j (s s )Z(s )ds where Z(s) is a univariate spatial process and k j are kernel functions, j = 1, 2,...,p. Yields the cross covariance C ij (s s ) = k i (s s + u)k j (u )ρ(u u )dudu Chapter 7: Multivariate Spatial Modeling p. 12/21
52 Other Approaches Moving average or kernel convolution of a process: Y j (s) = k j (u)z(s + u)du = k j (s s )Z(s )ds where Z(s) is a univariate spatial process and k j are kernel functions, j = 1, 2,...,p. Yields the cross covariance C ij (s s ) = k i (s s + u)k j (u )ρ(u u )dudu Convolution of Covariance Functions: Suppose C 1,C 2,...C p are valid covariance functions. Define C ij (s) = C i (s t)c j (t)dt. Then the p p matrix C(s) = {C ij (s)} is a valid cross covariance function. Chapter 7: Multivariate Spatial Modeling p. 12/21
53 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Chapter 7: Multivariate Spatial Modeling p. 13/21
54 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units Chapter 7: Multivariate Spatial Modeling p. 13/21
55 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units As in univariate case, often handled using spatial random effects φ ij, where again i = 1,...,n indexes region but now j = 1,...,p indexes variable (e.g., cancer type) within region Chapter 7: Multivariate Spatial Modeling p. 13/21
56 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units As in univariate case, often handled using spatial random effects φ ij, where again i = 1,...,n indexes region but now j = 1,...,p indexes variable (e.g., cancer type) within region Suppose we observe Y i = (Y i1,y i2,...y ip ), Then g(e(y ij )) = x T ij β j + φ ij, where φ i = (φ i1,...,φ ip ) and φ = (φ 1,...,φ n ). Link function g useful for modeling rate (e.g., Poisson) or survival (e.g., Weibull or Cox model) data. Chapter 7: Multivariate Spatial Modeling p. 13/21
57 Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Chapter 7: Multivariate Spatial Modeling p. 14/21
58 Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Example: For breast cancer control model in county i, let Y 1i = observed age-adjusted mortality rate / 100,000 Y 2i = observed age-adjusted incidence rate / 100,000 Y 3i = observed percent of late diagnoses Y 4i = % of surveyed women over the age of 40 who have not had a mammogram in two years Chapter 7: Multivariate Spatial Modeling p. 14/21
59 Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Example: For breast cancer control model in county i, let Y 1i = observed age-adjusted mortality rate / 100,000 Y 2i = observed age-adjusted incidence rate / 100,000 Y 3i = observed percent of late diagnoses Y 4i = % of surveyed women over the age of 40 who have not had a mammogram in two years ind Assume Y ki N(θ ki,σki 2 ), k = 1,...,4, i = 1,...,I = 87, and set σki 2 = σ2 k /n ki, where n ki is the number of persons at risk for event k in county i. Chapter 7: Multivariate Spatial Modeling p. 14/21
60 Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). Chapter 7: Multivariate Spatial Modeling p. 15/21
61 Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). We then borrow strength across cancer indicators: λ k iid G(a,b) and σ 2 k iid IG(a,b), where G and IG denote the gamma and inverse gamma distributions, respectively. Chapter 7: Multivariate Spatial Modeling p. 15/21
62 Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). We then borrow strength across cancer indicators: λ k iid G(a,b) and σ 2 k iid IG(a,b), where G and IG denote the gamma and inverse gamma distributions, respectively. Parameter of interest: For a set of weights α k such that 4 k=1 α k = 1, form the breast cancer control variable, η i = 4 k=1 α k θ ki. Chapter 7: Multivariate Spatial Modeling p. 15/21
63 Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Chapter 7: Multivariate Spatial Modeling p. 16/21
64 Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Using α k = 1/4 for all k, compare the posterior medians of the η i with naive averages: Chapter 7: Multivariate Spatial Modeling p. 16/21
65 Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Using α k = 1/4 for all k, compare the posterior medians of the η i with naive averages: Note smoothed control variates cover a narrower range, and clarify the overall spatial pattern in the state! Chapter 7: Multivariate Spatial Modeling p. 16/21
66 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: Chapter 7: Multivariate Spatial Modeling p. 17/21
67 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) Chapter 7: Multivariate Spatial Modeling p. 17/21
68 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) Chapter 7: Multivariate Spatial Modeling p. 17/21
69 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n Chapter 7: Multivariate Spatial Modeling p. 17/21
70 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Chapter 7: Multivariate Spatial Modeling p. 17/21
71 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Simplification: B ij = b ij I,b ij = w ij /w i+, Σ i = ( 1 w i+ )Σ Chapter 7: Multivariate Spatial Modeling p. 17/21
72 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Simplification: B ij = b ij I,b ij = w ij /w i+, Σ i = ( 1 w i+ )Σ To make proper, add ρ or perhaps ρ j, j = 1,...,p Chapter 7: Multivariate Spatial Modeling p. 17/21
73 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. Chapter 7: Multivariate Spatial Modeling p. 18/21
74 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. Chapter 7: Multivariate Spatial Modeling p. 18/21
75 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years Chapter 7: Multivariate Spatial Modeling p. 18/21
76 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years δ ijk, the corresponding death indicator Chapter 7: Multivariate Spatial Modeling p. 18/21
77 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years δ ijk, the corresponding death indicator x ijk, a vector of patient-specific covariates Chapter 7: Multivariate Spatial Modeling p. 18/21
78 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years δ ijk, the corresponding death indicator x ijk, a vector of patient-specific covariates z ijk = ( z ijk1,z ijk2,...,z ijkn ) T, where zijkl = 1 if patient ijk suffers from cancer type l, and 0 if not. Chapter 7: Multivariate Spatial Modeling p. 18/21
79 Application to spatial frailty modeling Hierarchical model: The likelihood is p i=1 n sij { ( )} δijk j=1 k=1 h tijk ; x ijk ( ) ( )} exp { H 0i tijk exp x T ijk β + zt ijk θ + φ ij, where h ( t ijk ; x ijk ) = h0i ( tijk ) exp (x T ijk β + zt ijk θ + φ ij H 0i ( tijk ) = tijk 0 h 0i (u) du. φ i = (φ i1,φ i2,...,φ in ) T ( ) T and Φ = φ T 1,...,φ T p MCAR (α, Λ). ) Chapter 7: Multivariate Spatial Modeling p. 19/21
80 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Chapter 7: Multivariate Spatial Modeling p. 20/21
81 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Chapter 7: Multivariate Spatial Modeling p. 20/21
82 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Chapter 7: Multivariate Spatial Modeling p. 20/21
83 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Not residuals, but the effects of the presence of the primary cancer on the death rate Chapter 7: Multivariate Spatial Modeling p. 20/21
84 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Not residuals, but the effects of the presence of the primary cancer on the death rate Strong spatial pattern: SW Iowa counties have high fitted values for pancreatic and stomach cancer, while SE counties have high rates of colorectal and small intestinal cancer. Chapter 7: Multivariate Spatial Modeling p. 20/21
85 Fitted spatial effects, Iowa SEER data Colorectal Cancer Stomach Cancer Pancreas Cancer Small Intestine Cancer Chapter 7: Multivariate Spatial Modeling p. 21/21
Hierarchical Modelling for Univariate and Multivariate Spatial Data
Hierarchical Modelling for Univariate and Multivariate Spatial Data p. 1/4 Hierarchical Modelling for Univariate and Multivariate Spatial Data Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota
More informationFrailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Mela. P.
Frailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Melanie M. Wall, Bradley P. Carlin November 24, 2014 Outlines of the talk
More informationHierarchical Modelling for Multivariate Spatial Data
Hierarchical Modelling for Multivariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Point-referenced spatial data often come as
More informationHierarchical Modeling for Multivariate Spatial Data
Hierarchical Modeling for Multivariate Spatial Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department
More informationModels for spatial data (cont d) Types of spatial data. Types of spatial data (cont d) Hierarchical models for spatial data
Hierarchical models for spatial data Based on the book by Banerjee, Carlin and Gelfand Hierarchical Modeling and Analysis for Spatial Data, 2004. We focus on Chapters 1, 2 and 5. Geo-referenced data arise
More informationNearest Neighbor Gaussian Processes for Large Spatial Data
Nearest Neighbor Gaussian Processes for Large Spatial Data Abhi Datta 1, Sudipto Banerjee 2 and Andrew O. Finley 3 July 31, 2017 1 Department of Biostatistics, Bloomberg School of Public Health, Johns
More informationApproaches for Multiple Disease Mapping: MCAR and SANOVA
Approaches for Multiple Disease Mapping: MCAR and SANOVA Dipankar Bandyopadhyay Division of Biostatistics, University of Minnesota SPH April 22, 2015 1 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR
More informationModelling Multivariate Spatial Data
Modelling Multivariate Spatial Data Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. June 20th, 2014 1 Point-referenced spatial data often
More informationBayesian Areal Wombling for Geographic Boundary Analysis
Bayesian Areal Wombling for Geographic Boundary Analysis Haolan Lu, Haijun Ma, and Bradley P. Carlin haolanl@biostat.umn.edu, haijunma@biostat.umn.edu, and brad@biostat.umn.edu Division of Biostatistics
More informationHierarchical Modeling for Univariate Spatial Data
Hierarchical Modeling for Univariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Spatial Domain 2 Geography 890 Spatial Domain This
More informationAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information
Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information p. 1/27 Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information Shengde Liang, Bradley
More informationHierarchical Modelling for Univariate Spatial Data
Hierarchical Modelling for Univariate Spatial Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department
More informationHierarchical Modeling for Spatio-temporal Data
Hierarchical Modeling for Spatio-temporal Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of
More informationMultivariate Spatial Process Models. Alan E. Gelfand and Sudipto Banerjee
Multivariate Spatial Process Models Alan E. Gelfand and Sudipto Banerjee April 29, 2009 ii Contents 28 Multivariate Spatial Process Models 1 28.1 Introduction.................................... 1 28.2
More informationBayesian Linear Regression
Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective
More informationSpatio-Temporal Threshold Models for Relating UV Exposures and Skin Cancer in the Central United States
Spatio-Temporal Threshold Models for Relating UV Exposures and Skin Cancer in the Central United States Laura A. Hatfield and Bradley P. Carlin Division of Biostatistics School of Public Health University
More informationGaussian predictive process models for large spatial data sets.
Gaussian predictive process models for large spatial data sets. Sudipto Banerjee, Alan E. Gelfand, Andrew O. Finley, and Huiyan Sang Presenters: Halley Brantley and Chris Krut September 28, 2015 Overview
More informationMultivariate Survival Analysis
Multivariate Survival Analysis Previously we have assumed that either (X i, δ i ) or (X i, δ i, Z i ), i = 1,..., n, are i.i.d.. This may not always be the case. Multivariate survival data can arise in
More informationOn Gaussian Process Models for High-Dimensional Geostatistical Datasets
On Gaussian Process Models for High-Dimensional Geostatistical Datasets Sudipto Banerjee Joint work with Abhirup Datta, Andrew O. Finley and Alan E. Gelfand University of California, Los Angeles, USA May
More informationGibbs Sampling in Linear Models #2
Gibbs Sampling in Linear Models #2 Econ 690 Purdue University Outline 1 Linear Regression Model with a Changepoint Example with Temperature Data 2 The Seemingly Unrelated Regressions Model 3 Gibbs sampling
More informationGeneralized logit models for nominal multinomial responses. Local odds ratios
Generalized logit models for nominal multinomial responses Categorical Data Analysis, Summer 2015 1/17 Local odds ratios Y 1 2 3 4 1 π 11 π 12 π 13 π 14 π 1+ X 2 π 21 π 22 π 23 π 24 π 2+ 3 π 31 π 32 π
More informationAreal data models. Spatial smoothers. Brook s Lemma and Gibbs distribution. CAR models Gaussian case Non-Gaussian case
Areal data models Spatial smoothers Brook s Lemma and Gibbs distribution CAR models Gaussian case Non-Gaussian case SAR models Gaussian case Non-Gaussian case CAR vs. SAR STAR models Inference for areal
More informationHierarchical Modeling for Spatial Data
Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.
More informationModelling geoadditive survival data
Modelling geoadditive survival data Thomas Kneib & Ludwig Fahrmeir Department of Statistics, Ludwig-Maximilians-University Munich 1. Leukemia survival data 2. Structured hazard regression 3. Mixed model
More informationAreal Unit Data Regular or Irregular Grids or Lattices Large Point-referenced Datasets
Areal Unit Data Regular or Irregular Grids or Lattices Large Point-referenced Datasets Is there spatial pattern? Chapter 3: Basics of Areal Data Models p. 1/18 Areal Unit Data Regular or Irregular Grids
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee September 03 05, 2017 Department of Biostatistics, Fielding School of Public Health, University of California, Los Angeles Linear Regression Linear regression is,
More informationMarginal Survival Modeling through Spatial Copulas
1 / 53 Marginal Survival Modeling through Spatial Copulas Tim Hanson Department of Statistics University of South Carolina, U.S.A. University of Michigan Department of Biostatistics March 31, 2016 2 /
More informationBayesian data analysis in practice: Three simple examples
Bayesian data analysis in practice: Three simple examples Martin P. Tingley Introduction These notes cover three examples I presented at Climatea on 5 October 0. Matlab code is available by request to
More informationCTDL-Positive Stable Frailty Model
CTDL-Positive Stable Frailty Model M. Blagojevic 1, G. MacKenzie 2 1 Department of Mathematics, Keele University, Staffordshire ST5 5BG,UK and 2 Centre of Biostatistics, University of Limerick, Ireland
More informationExample using R: Heart Valves Study
Example using R: Heart Valves Study Goal: Show that the thrombogenicity rate (TR) is less than two times the objective performance criterion R and WinBUGS Examples p. 1/27 Example using R: Heart Valves
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference Sudipto Banerjee University of Minnesota July 20th, 2008 1 Bayesian Principles Classical statistics: model parameters are fixed and unknown. A Bayesian thinks of parameters
More informationStat260: Bayesian Modeling and Inference Lecture Date: February 10th, Jeffreys priors. exp 1 ) p 2
Stat260: Bayesian Modeling and Inference Lecture Date: February 10th, 2010 Jeffreys priors Lecturer: Michael I. Jordan Scribe: Timothy Hunter 1 Priors for the multivariate Gaussian Consider a multivariate
More informationChapter 4 - Fundamentals of spatial processes Lecture notes
TK4150 - Intro 1 Chapter 4 - Fundamentals of spatial processes Lecture notes Odd Kolbjørnsen and Geir Storvik January 30, 2017 STK4150 - Intro 2 Spatial processes Typically correlation between nearby sites
More informationPackage SimSCRPiecewise
Package SimSCRPiecewise July 27, 2016 Type Package Title 'Simulates Univariate and Semi-Competing Risks Data Given Covariates and Piecewise Exponential Baseline Hazards' Version 0.1.1 Author Andrew G Chapple
More informationPart 6: Multivariate Normal and Linear Models
Part 6: Multivariate Normal and Linear Models 1 Multiple measurements Up until now all of our statistical models have been univariate models models for a single measurement on each member of a sample of
More informationHierarchical Modelling for Univariate Spatial Data
Spatial omain Hierarchical Modelling for Univariate Spatial ata Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A.
More informationSpatial Survival Analysis via Copulas
1 / 38 Spatial Survival Analysis via Copulas Tim Hanson Department of Statistics University of South Carolina, U.S.A. International Conference on Survival Analysis in Memory of John P. Klein Medical College
More informationMultivariate Normal & Wishart
Multivariate Normal & Wishart Hoff Chapter 7 October 21, 2010 Reading Comprehesion Example Twenty-two children are given a reading comprehsion test before and after receiving a particular instruction method.
More informationThe linear model is the most fundamental of all serious statistical models encompassing:
Linear Regression Models: A Bayesian perspective Ingredients of a linear model include an n 1 response vector y = (y 1,..., y n ) T and an n p design matrix (e.g. including regressors) X = [x 1,..., x
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics, School of Public
More informationBayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Andrew O. Finley Department of Forestry & Department of Geography, Michigan State University, Lansing
More informationLecture 3. Truncation, length-bias and prevalence sampling
Lecture 3. Truncation, length-bias and prevalence sampling 3.1 Prevalent sampling Statistical techniques for truncated data have been integrated into survival analysis in last two decades. Truncation in
More informationIntroduction to Geostatistics
Introduction to Geostatistics Abhi Datta 1, Sudipto Banerjee 2 and Andrew O. Finley 3 July 31, 2017 1 Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, Baltimore,
More informationBayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Alan Gelfand 1 and Andrew O. Finley 2 1 Department of Statistical Science, Duke University, Durham, North
More informationHypothesis Testing. Econ 690. Purdue University. Justin L. Tobias (Purdue) Testing 1 / 33
Hypothesis Testing Econ 690 Purdue University Justin L. Tobias (Purdue) Testing 1 / 33 Outline 1 Basic Testing Framework 2 Testing with HPD intervals 3 Example 4 Savage Dickey Density Ratio 5 Bartlett
More informationMCMC algorithms for fitting Bayesian models
MCMC algorithms for fitting Bayesian models p. 1/1 MCMC algorithms for fitting Bayesian models Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota MCMC algorithms for fitting Bayesian models
More informationBayesian spatial quantile regression
Brian J. Reich and Montserrat Fuentes North Carolina State University and David B. Dunson Duke University E-mail:reich@stat.ncsu.edu Tropospheric ozone Tropospheric ozone has been linked with several adverse
More informationBayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota,
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department
More informationPROBABILITY DISTRIBUTIONS. J. Elder CSE 6390/PSYC 6225 Computational Modeling of Visual Perception
PROBABILITY DISTRIBUTIONS Credits 2 These slides were sourced and/or modified from: Christopher Bishop, Microsoft UK Parametric Distributions 3 Basic building blocks: Need to determine given Representation:
More informationHierarchical Modeling for non-gaussian Spatial Data
Hierarchical Modeling for non-gaussian Spatial Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department
More informationMotivation Scale Mixutres of Normals Finite Gaussian Mixtures Skew-Normal Models. Mixture Models. Econ 690. Purdue University
Econ 690 Purdue University In virtually all of the previous lectures, our models have made use of normality assumptions. From a computational point of view, the reason for this assumption is clear: combined
More informationHierarchical Linear Models
Hierarchical Linear Models Statistics 220 Spring 2005 Copyright c 2005 by Mark E. Irwin The linear regression model Hierarchical Linear Models y N(Xβ, Σ y ) β σ 2 p(β σ 2 ) σ 2 p(σ 2 ) can be extended
More informationCBMS Lecture 1. Alan E. Gelfand Duke University
CBMS Lecture 1 Alan E. Gelfand Duke University Introduction to spatial data and models Researchers in diverse areas such as climatology, ecology, environmental exposure, public health, and real estate
More informationBasics of Point-Referenced Data Models
Basics of Point-Referenced Data Models Basic tool is a spatial process, {Y (s), s D}, where D R r Chapter 2: Basics of Point-Referenced Data Models p. 1/45 Basics of Point-Referenced Data Models Basic
More informationBayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Andrew O. Finley 1 and Sudipto Banerjee 2 1 Department of Forestry & Department of Geography, Michigan
More informationPractical considerations for survival models
Including historical data in the analysis of clinical trials using the modified power prior Practical considerations for survival models David Dejardin 1 2, Joost van Rosmalen 3 and Emmanuel Lesaffre 1
More informationRonald Christensen. University of New Mexico. Albuquerque, New Mexico. Wesley Johnson. University of California, Irvine. Irvine, California
Texts in Statistical Science Bayesian Ideas and Data Analysis An Introduction for Scientists and Statisticians Ronald Christensen University of New Mexico Albuquerque, New Mexico Wesley Johnson University
More informationHierarchical models. Dr. Jarad Niemi. August 31, Iowa State University. Jarad Niemi (Iowa State) Hierarchical models August 31, / 31
Hierarchical models Dr. Jarad Niemi Iowa State University August 31, 2017 Jarad Niemi (Iowa State) Hierarchical models August 31, 2017 1 / 31 Normal hierarchical model Let Y ig N(θ g, σ 2 ) for i = 1,...,
More informationHierarchical Nearest-Neighbor Gaussian Process Models for Large Geo-statistical Datasets
Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geo-statistical Datasets Abhirup Datta 1 Sudipto Banerjee 1 Andrew O. Finley 2 Alan E. Gelfand 3 1 University of Minnesota, Minneapolis,
More informationFrailty Modeling for clustered survival data: a simulation study
Frailty Modeling for clustered survival data: a simulation study IAA Oslo 2015 Souad ROMDHANE LaREMFiQ - IHEC University of Sousse (Tunisia) souad_romdhane@yahoo.fr Lotfi BELKACEM LaREMFiQ - IHEC University
More informationSpatial Misalignment
Spatial Misalignment Jamie Monogan University of Georgia Spring 2013 Jamie Monogan (UGA) Spatial Misalignment Spring 2013 1 / 28 Objectives By the end of today s meeting, participants should be able to:
More informationBayesian Inference on Joint Mixture Models for Survival-Longitudinal Data with Multiple Features. Yangxin Huang
Bayesian Inference on Joint Mixture Models for Survival-Longitudinal Data with Multiple Features Yangxin Huang Department of Epidemiology and Biostatistics, COPH, USF, Tampa, FL yhuang@health.usf.edu January
More informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 31st January 2006 Part VI Session 6: Filtering and Time to Event Data Session 6: Filtering and
More informationCross-covariance Functions for Tangent Vector Fields on the Sphere
Cross-covariance Functions for Tangent Vector Fields on the Sphere Minjie Fan 1 Tomoko Matsuo 2 1 Department of Statistics University of California, Davis 2 Cooperative Institute for Research in Environmental
More informationBayesian linear regression
Bayesian linear regression Linear regression is the basis of most statistical modeling. The model is Y i = X T i β + ε i, where Y i is the continuous response X i = (X i1,..., X ip ) T is the corresponding
More informationPart 7: Hierarchical Modeling
Part 7: Hierarchical Modeling!1 Nested data It is common for data to be nested: i.e., observations on subjects are organized by a hierarchy Such data are often called hierarchical or multilevel For example,
More informationModeling Real Estate Data using Quantile Regression
Modeling Real Estate Data using Semiparametric Quantile Regression Department of Statistics University of Innsbruck September 9th, 2011 Overview 1 Application: 2 3 4 Hedonic regression data for house prices
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference Sudipto Banerjee and Andrew O. Finley 2 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department
More informationLocal Likelihood Bayesian Cluster Modeling for small area health data. Andrew Lawson Arnold School of Public Health University of South Carolina
Local Likelihood Bayesian Cluster Modeling for small area health data Andrew Lawson Arnold School of Public Health University of South Carolina Local Likelihood Bayesian Cluster Modelling for Small Area
More informationLongitudinal breast density as a marker of breast cancer risk
Longitudinal breast density as a marker of breast cancer risk C. Armero (1), M. Rué (2), A. Forte (1), C. Forné (2), H. Perpiñán (1), M. Baré (3), and G. Gómez (4) (1) BIOstatnet and Universitat de València,
More informationReliability Monitoring Using Log Gaussian Process Regression
COPYRIGHT 013, M. Modarres Reliability Monitoring Using Log Gaussian Process Regression Martin Wayne Mohammad Modarres PSA 013 Center for Risk and Reliability University of Maryland Department of Mechanical
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Researchers in diverse areas such as climatology, ecology, environmental health, and real estate marketing are increasingly faced with the task of analyzing data
More informationMarkov Chain Monte Carlo (MCMC)
Markov Chain Monte Carlo (MCMC Dependent Sampling Suppose we wish to sample from a density π, and we can evaluate π as a function but have no means to directly generate a sample. Rejection sampling can
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 7 Approximate
More informationLecture 5 Models and methods for recurrent event data
Lecture 5 Models and methods for recurrent event data Recurrent and multiple events are commonly encountered in longitudinal studies. In this chapter we consider ordered recurrent and multiple events.
More informationPattern Recognition and Machine Learning. Bishop Chapter 2: Probability Distributions
Pattern Recognition and Machine Learning Chapter 2: Probability Distributions Cécile Amblard Alex Kläser Jakob Verbeek October 11, 27 Probability Distributions: General Density Estimation: given a finite
More informationSpatial Smoothing in Stan: Conditional Auto-Regressive Models
Spatial Smoothing in Stan: Conditional Auto-Regressive Models Charles DiMaggio, PhD, NYU School of Medicine Stephen J. Mooney, PhD, University of Washington Mitzi Morris, Columbia University Dan Simpson,
More informationBayesian Inference for Regression Parameters
Bayesian Inference for Regression Parameters 1 Bayesian inference for simple linear regression parameters follows the usual pattern for all Bayesian analyses: 1. Form a prior distribution over all unknown
More informationMaking rating curves - the Bayesian approach
Making rating curves - the Bayesian approach Rating curves what is wanted? A best estimate of the relationship between stage and discharge at a given place in a river. The relationship should be on the
More informationSampling bias in logistic models
Sampling bias in logistic models Department of Statistics University of Chicago University of Wisconsin Oct 24, 2007 www.stat.uchicago.edu/~pmcc/reports/bias.pdf Outline Conventional regression models
More informationBayesian inference & process convolution models Dave Higdon, Statistical Sciences Group, LANL
1 Bayesian inference & process convolution models Dave Higdon, Statistical Sciences Group, LANL 2 MOVING AVERAGE SPATIAL MODELS Kernel basis representation for spatial processes z(s) Define m basis functions
More informationspbayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models
spbayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models Andrew O. Finley 1, Sudipto Banerjee 2, and Bradley P. Carlin 2 1 Michigan State University, Departments
More informationDisease mapping with Gaussian processes
EUROHEIS2 Kuopio, Finland 17-18 August 2010 Aki Vehtari (former Helsinki University of Technology) Department of Biomedical Engineering and Computational Science (BECS) Acknowledgments Researchers - Jarno
More informationUsing Estimating Equations for Spatially Correlated A
Using Estimating Equations for Spatially Correlated Areal Data December 8, 2009 Introduction GEEs Spatial Estimating Equations Implementation Simulation Conclusion Typical Problem Assess the relationship
More informationHierarchical Modelling for non-gaussian Spatial Data
Hierarchical Modelling for non-gaussian Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Spatial Generalized Linear Models Often data
More informationA Fully Nonparametric Modeling Approach to. BNP Binary Regression
A Fully Nonparametric Modeling Approach to Binary Regression Maria Department of Applied Mathematics and Statistics University of California, Santa Cruz SBIES, April 27-28, 2012 Outline 1 2 3 Simulation
More informationBayesian Multivariate Logistic Regression
Bayesian Multivariate Logistic Regression Sean M. O Brien and David B. Dunson Biostatistics Branch National Institute of Environmental Health Sciences Research Triangle Park, NC 1 Goals Brief review of
More informationMetropolis-Hastings Algorithm
Strength of the Gibbs sampler Metropolis-Hastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to
More informationPhysician Performance Assessment / Spatial Inference of Pollutant Concentrations
Physician Performance Assessment / Spatial Inference of Pollutant Concentrations Dawn Woodard Operations Research & Information Engineering Cornell University Johns Hopkins Dept. of Biostatistics, April
More informationMarkov Random Fields
Markov Random Fields 1. Markov property The Markov property of a stochastic sequence {X n } n 0 implies that for all n 1, X n is independent of (X k : k / {n 1, n, n + 1}), given (X n 1, X n+1 ). Another
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry
More informationBayesian Inference in a Normal Population
Bayesian Inference in a Normal Population September 17, 2008 Gill Chapter 3. Sections 1-4, 7-8 Bayesian Inference in a Normal Population p.1/18 Normal Model IID observations Y = (Y 1,Y 2,...Y n ) Y i N(µ,σ
More informationPart 8: GLMs and Hierarchical LMs and GLMs
Part 8: GLMs and Hierarchical LMs and GLMs 1 Example: Song sparrow reproductive success Arcese et al., (1992) provide data on a sample from a population of 52 female song sparrows studied over the course
More informationHierarchical Modeling and Analysis for Spatial Data
Hierarchical Modeling and Analysis for Spatial Data Bradley P. Carlin, Sudipto Banerjee, and Alan E. Gelfand brad@biostat.umn.edu, sudiptob@biostat.umn.edu, and alan@stat.duke.edu University of Minnesota
More informationPoint process with spatio-temporal heterogeneity
Point process with spatio-temporal heterogeneity Jony Arrais Pinto Jr Universidade Federal Fluminense Universidade Federal do Rio de Janeiro PASI June 24, 2014 * - Joint work with Dani Gamerman and Marina
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS Parametric Distributions Basic building blocks: Need to determine given Representation: or? Recall Curve Fitting Binary Variables
More informationGeneralized common spatial factor model
Biostatistics (2003), 4, 4,pp. 569 582 Printed in Great Britain Generalized common spatial factor model FUJUN WANG Eli Lilly and Company, Indianapolis, IN 46285, USA MELANIE M. WALL Division of Biostatistics,
More informationGAUSSIAN PROCESS REGRESSION
GAUSSIAN PROCESS REGRESSION CSE 515T Spring 2015 1. BACKGROUND The kernel trick again... The Kernel Trick Consider again the linear regression model: y(x) = φ(x) w + ε, with prior p(w) = N (w; 0, Σ). The
More informationBayesian Hierarchical Models
Bayesian Hierarchical Models Gavin Shaddick, Millie Green, Matthew Thomas University of Bath 6 th - 9 th December 2016 1/ 34 APPLICATIONS OF BAYESIAN HIERARCHICAL MODELS 2/ 34 OUTLINE Spatial epidemiology
More information