Spatial Modeling with Spatially Varying Coefficient Processes

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1 Spatial Modeling with Spatially Varying Coefficient Processes Alan E. Gelfand, Hyon-Jung Kim, C. F. Sirmans, Sudipto Banerjee Presenters: Halley Brantley and Chris Krut September 28, 2015

2 Introduction Want to model selling price of single family homes in Baton Rouge. Covariates Age See Figure 1. Pg. 392 Square feet of liv Square feet of ot # of bathrooms

3 Linear Model Y (s) = X(s)β + W (s) + ɛ β = (β 0, β 1, β 2, β 3, β 4 ) X(s) = (1, Age, LivingArea, OtherArea, Bathrooms) ɛ N(0, τ 2 ) independent error term Spatial Components 1. s i is the location of observation i 2. Spatial random effect W (s) N(0, σ 2 H) 3. (H) ij = ρ(s i s j φ) 4. cov(y (s i ), Y (s j ) β, τ 2, φ) = σ 2 ρ(s i s j φ)

4 Challenge: βs may not be constant across space Possibility 1: divide area into similar blocks. Must assume coefficient is constant on specified area. Arbitrary scale of resolution Can t interpolate the surface Possibility 2: Assume β is a polynomial or spline function of lat/lon. Arbitrary specification of polynomial could be too inflexible. Splines are more flexible, but need to specify number and location of knots.

5 Gaussian Process Definition Y (s) is a Gaussian process with mean function µ(s) and covariance function cov(y (s), Y (s )) = H(s, s ) if for every subset of locations s 1,..., s n the vector Ỹ = (Y (s 1),..., Y (s n )) T Ỹ MV N n ( µ, H) (1) where µ = (µ(s 1 ),..., µ(s n )) T and H is a matrix such that {H} ij = H(s i, s j ; φ). To be a valid covariance function H(s, s : φ) must be positive semidefinite in the sense that it generates covariance matrices H which are positive semi definite v T Hv 0[1, Pg. 80].

6 Spatial Intercept Model: Process Model Y (s) = β 0 + x(s)β 1 + β 0 (s) + ε(s) E(ε(s)) = 0 cov(ε(s), ε(s )) = τ 2 I(s = s ) β 0 (s) Gaussian Process(0,H(s, s ; φ, σ 2 )) H(s, s ; φ, σ 2 ) = σ 2 ρ(s s ; φ). β 0 (s) = β 0 + β 0 (s) Assign Prior distributions to remaining parameters (β 0, β 1, σ 2 0, τ 2, φ 0 )

7 Spatial Intercept Model: Discretized Model Observe Y (s), x(s), over set of locations s 1,..., s n. Y (s i ) = β 0 + x(s i )β 1 + β 0 (s i ) + ε(s i ) ε(s i ) N(0, τ 2 ) (Y (s i ) β 0, β 1, β 0 (s i )) N(β 0 + x(s i )β 1 + β 0 (s i ), τ 2 ) β 0 = (β 0 (s 1 ),..., β 0 (s n )) β 0 MV N(0, σ 2 0H 0 (φ 0 )) Assign Prior distributions to remaining parameters (β 0, β 1, σ 2 0, τ 2, φ 0 )

8 Spatial Slope Model: Process Model Y (s) = β 0 + x(s)β 1 + x(s)β 1 (s) + ε(s) E(ε(s)) = 0 cov(ε(s), ε(s )) = τ 2 I(s = s ) β 0 (s) Gaussian Process(0,H(s, s ; φ, σ 2 )) H(s, s ; φ, σ 2 ) = σ 2 ρ(s s ; φ)). β 1 (s) = β 1 + β 1 (s) Assign Prior distributions to remaining parameters (β 0, β 1, σ 2 0, τ 2, φ 0 ) The process Y (s) ends up being heterogeneous and non-stationary. var(y (s) β0, β 1, τ 2, σ 2 1, φ 1 ) = x 2 (s)σ τ 2 cov(y (s), Y (s ) β 0, β 1, τ 2, σ 2 1, φ 1 ) = x(s)x(s )σ 2 1ρ 1 (s s ; φ 1 )

9 Spatial Slope Model: Discretized Model Observe Y (s), x(s), over set of locations s 1,..., s n. Y (s i ) = β 0 + x(s i )β 1 + x(s i )β 1 (s i ) + ε(s) ε(s i ) N(0, τ 2 ) (Y (s i ) β 0, β 1, β 1 (s i )) N(β 0 + x(s i )β 1 + x(s i )β 1 (s i ), τ 2 ) β 1 = (β 1 (s 1 ),..., β 1 (s n )) β 1 MV N(0, σ 2 1H 1 (φ 1 )) Assign Prior distributions to remaining parameters (β 0, β 1, σ 2 0, τ 2, φ 0 )

10 Spatial Varying Coefficients Model Y (s) = β 0 + x(s)β 1 + β 0 (s) + x(s)β 1 (s) + ε(s) Y (s i ) = β 0 + x(s i )β 1 + β 0 (s i ) + x(s i )β 1 (s i ) + ε(s i ) Vector of spatial random effects at observed locations: β = (β 0 (s 1 ), β 1 (s 1 ),..., β 0 (s n ), β 1 (s n )) T β MV N(0, H(φ) T ) ( ) T11 T Positive definite matrix: T = 12 T 21 T 22 cov(β 0 (s i ), β 0 (s j )) = ρ(s i s j : φ)t 11 cov(β 1 (s i ), β 1 (s j )) = ρ(s i s j : φ)t 22 cov(β 0 (s i ), β 1 (s j )) = ρ(s i s j : φ)t 12

11 Multiple Regression with Spatially Varying coefficients Y (s) = X T (s) β(s) + ε(s) Y (s) = X T (s)µ β + X T (s)β(s) + ε(s) X T (s) = (1, x 1 (s),..., x p (s)) β(s) = µ β + β(s) µ β = (β 0, β 1,..., β p ) β(s) = (β 0 (s), β 1 (s),..., β p (s))

12 Multiple Regression with Spatially Varying coefficients y = X(1 µ β ) + Xβ + ε y = (Y (s 1 ),..., Y (s n )) T X(s 1 ) X(s 2 ) X =... X(s) µ β = (β 0, β 1,..., β p ) T β = (β 0 (s 1 ),..., β p (s 1 ),..., β 0 (s n ),..., β p (s n )) T β MV N(0, H(φ) T ) T is a p + 1 p + 1 positive definite matrix. cov(β i (s), β j (s )) = ρ(s s : φ)t i+1j+1

13 Marginalized Likelihood L(µ β, T, τ 2, φ y) = f(y X, µ β, β, T, φ, τ 2 )π(β)dβ L(µ β, T, τ 2, φ y) = (X(H(φ) T )X T + τ 2 I) 1/2 exp( 1 2 (y X(1 µ β)) T (X(H(φ) T )X T + τ 2 I) 1 (y X(1 µ β )))

14 Slice Sampling Want to sample from posterior f(µ β, T, τ 2, φ y) Introduce an auxiliary variable U unif(0, L(µ β, T, τ 2, φ y)) and use the fact f(µ β, T, τ 2, φ, U y) = I(U < L(µ β, T, τ 2, φ y))π(µ β, T, τ 2, φ) Sample U from unif(0, L(µ β, T, τ 2, φ y)). Sample µ β, T, τ 2, φ from π(µ β, T, τ 2, φ) subject to I(U < L(µ β, T, τ 2, φ y)).

15 Sampling β(s) 1. We can obtain the Marginal Posterior f( β, y) = f( β µ β, T, φ, τ 2, y) f(µ β, T, φ, τ 2 y) 2. Full Conditional is of a known form f( β µ β, T, φ, τ 2, y) = N(Bb, B) B = (X T X/τ 2 + H 1 (φ) T 1 ) 1 b = X T y/τ 2 + (H 1 (φ) T 1 )(1 µ β ) 3. We can generate from f( β, y) using the samples from f(µ β, T, φ, τ 2 y) and the full conditional distribution in 2.

16 Sampling β(s n+1 ) for a new location s n+1 1. We can obtain the Marginal Posterior f( β(s n+1 ), y) = f( β(s n+1 ) β, µ β, T, φ, τ 2 )f( β, µ β, T, φ, τ 2 y) 2. Conditional distribution has a known form f( β(s n+1 ) β, µ β, T, φ, τ 2 ) = N(µ, σ) µ = µ β + (h T new(φ)h 1 (φ) I)( β 1 n 1 µ β ) σ = (I h T new(φ)h 1 (φ)h new (φ))t h new (φ) = (ρ(s n+1 s 1 ; φ),..., ρ(s n+1 s 1 ; φ)) T 3. Similar to before we can use posterior samples to generate β(s n+1 )

17 Predict y(s n+1 ) from predictive posterior 1. The Predictive Posterior is given by f(y(s n+1 ) y) = f(y(s n+1 ) β(s n+1 ), τ 2 ) f( β(s n+1 ) β, µ β, T, φ) f( β, µ β, T, φ, τ 2 y) 2. Can sample from f(y(s n+1 ) y) by iteratively generating from f( β, µ β, T, φ, τ 2 y), f( β(s n+1 ) β, µ β, T, φ), f(y(s n+1 ) β(s n+1 ), τ 2 ).

18 Application to housing prices Compared 4 different models: 1. Allow all βs to vary spatially (multivariate). 2. Allow 3 of the βs to vary spatially. 3. Allow 2 of the βs to vary spatially. 4. Allow all βs to vary spatially (independently). Model comparison was based on the goodness-of-fit and penalty for model complexity. D = (Y obs (s, t) µ(s, t)) 2 + k σ 2 (s, t) (s,t) (s,t)

19 Spatial Results See Table 3, Pg. 392

20 Spatial Results See Figure 4, Pg. 393

21 Space-Time Regression with Varying Coefficients Y (s) = X T (s, t) β(s, t) + ε(s, t) β(s, t) = µ β + β(s, t) 1. β(s, t) = β(s) 2. β(s, t) = β(s) + α(t) 2.a α k (t) = I(t = 1)α k I(t = M)α km 2.b α(t) = (α 0 (t),..., α P (t)) T cov(α(t), α(t )) = ρ (2) (t t ; γ)t 3. β(s, t) = β t (s)(independent processes nested in t) (cov(β t (s), β t (s )) = ρ(s s ; φ (t) )T t 4. Separable covariance cov(β(s, t), β(s, t )) = ρ 1 (s s ; φ) ρ 2 (t t : γ)t

22 Spatio-Temporal Results Model prices over 4 years. Model 1: β(s, t) = β(s) Model 2: β(s, t) = β(s) + α(t) Model 2a: α(t) as 4 iid time dummies Model 2b: α(t) as multivariate temporal process Model 3: β(s, t) = β (t) (s)(independent process at each t) Model 3a: common µ b across t. Model 3b: µ (t) b Model 4: Separable spatio-temporal model.

23 See Table 6 Pg. 395

24 See Table 7 Pg. 395

25 Generalized Linear Model f(y(s i ) θ(s i )) = h(y(s i )) exp(θ(s i )y(s i ) b(θ(s i ))) L( β y) ( = exp y(si )X T (s i ) β(s i ) b(x T (s i ) β(s ) i )) θ(s i ) = X T (s i ) β(s i ) β N(1 n 1 µ β, H(φ) T ) Constructing a satisfactory sampler is quite challenging. More work needs to be done on model fitting side.

26 Carl Edward Rasmussen. Gaussian processes for machine learning

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