Raster regression

Connor Donegan

April 27, 2023

This vignette provides a tutorial for fitting spatial regression models to raster data using geostan. The term “raster” is used here to refer to any regularly spaced set of observations such that the data can be represented spatially by a rectangular grid. Remotely sensed imagery is a common form of raster data.

geostan can be used for spatial regression with fairly large raster data layers, although the functionality of these models will often be limited to the estimation of regression coefficients and spatial autocorrelation parameters. Limited experience thus far finds that geostan’s spatial autoregressive models can be fit to raster layers with two hundred thousand observations using a laptop computer and fewer than ten minutes of sampling time.

0.1 Demonstration

Start by loading some necessary R packages.

library(geostan)
library(sf)
set.seed(1127)

We will create a small raster data layer for the purpose of illustration.

row <- 40
col <- 30
c(N <- row * col)
## [1] 1200
sfc = st_sfc(st_polygon(list(rbind(c(0,0), c(col,0), c(col,row), c(0,0)))))
grid <- st_make_grid(sfc, cellsize = 1, square = TRUE)
grid <- st_as_sf(grid)
W <- shape2mat(grid, style = "W", queen = FALSE)
grid$z <- sim_sar(w = W, rho = 0.9)
grid$y <- -0.5 * grid$z + sim_sar(w = W, rho = .9, sigma = .3)
plot(grid[,'z'])

The following R code will fit a spatial autoregressive model to these data:

fit <- stan_sar(y ~ z, data = grid, C = W)

The stan_sar function will take the spatial weights matrix W and pass it through a function called prep_sar_data which will calculate the eigenvalues of the spatial weights matrix using base::eigen, as required for computational reasons. This step is prohibitive for large data sets (e.g., \(N = 100,000\)).

The following code would normally be used to fit a conditional autoregressive (CAR) model:

C <- shape2mat(grid, style = "B", queen = FALSE)
car_list <- prep_car_data(C, "WCAR")
fit <- stan_car(y ~ z, data = grid, car_parts = car_list)

Here, the prep_car_data function calculates the eigenvalues of the spatial weights matrix using base::eigen, which is not feasible for large N.

The prep_sar_data2 and prep_car_data2 functions are designed for raster layers. As input, they require the dimensions of the grid (number of rows and number of columns). The eigenvalues are produced very quickly using Equation 5 from Griffith (2000). The methods have certain restrictions. First, this is only applicable to raster layers—regularly spaced, rectangular grids of observations. Second, to define which observations are adjacent to one another, the “rook” criteria is used (spatially, only observations that share an edge are defined as neighbors to one another). Third, the spatial adjacency matrix will be row-standardized. This is standard (and required) for SAR models, and it corresponds to the “WCAR” specification of the CAR model (see Donegan 2022).

The following code will fit a SAR model to our grid data, and is suitable for much larger raster layers:

sar_list <- prep_sar_data2(row = row, col = col)
## Range of permissible rho values:  -1 1
fit <- stan_sar(y ~ z, 
      data = grid,
          centerx = TRUE,
          sar_parts = sar_list,
          iter = 500,
          chains = 4,
          slim = TRUE #,
          # cores = 4, # for multi-core processing
        )
## 
## *Setting prior parameters for intercept
## Distribution: normal
##   location scale
## 1    0.035     5
## 
## *Setting prior parameters for beta
## Distribution: normal
##   location scale
## 1        0     5
## 
## *Setting prior for SAR scale parameter (sar_scale)
## Distribution: student_t
##   df location scale
## 1 10        0     3
## 
## *Setting prior for SAR spatial autocorrelation parameter (sar_rho)
## Distribution: uniform
##   lower upper
## 1    -1     1
## 
## SAMPLING FOR MODEL 'foundation' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 0.000481 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 4.81 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 1: Iteration:   1 / 500 [  0%]  (Warmup)
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## Chain 1: 
## Chain 1:  Elapsed Time: 3.364 seconds (Warm-up)
## Chain 1:                0.787 seconds (Sampling)
## Chain 1:                4.151 seconds (Total)
## Chain 1: 
## 
## SAMPLING FOR MODEL 'foundation' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 0.000444 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 4.44 seconds.
## Chain 2: Adjust your expectations accordingly!
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## Chain 2: 
## Chain 2:  Elapsed Time: 1.456 seconds (Warm-up)
## Chain 2:                0.811 seconds (Sampling)
## Chain 2:                2.267 seconds (Total)
## Chain 2: 
## 
## SAMPLING FOR MODEL 'foundation' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 0.000454 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 4.54 seconds.
## Chain 3: Adjust your expectations accordingly!
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## Chain 3: 
## Chain 3:  Elapsed Time: 1.41 seconds (Warm-up)
## Chain 3:                0.7 seconds (Sampling)
## Chain 3:                2.11 seconds (Total)
## Chain 3: 
## 
## SAMPLING FOR MODEL 'foundation' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 0.000444 seconds
## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 4.44 seconds.
## Chain 4: Adjust your expectations accordingly!
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## Chain 4: 
## Chain 4:  Elapsed Time: 1.422 seconds (Warm-up)
## Chain 4:                0.687 seconds (Sampling)
## Chain 4:                2.109 seconds (Total)
## Chain 4:
print(fit)      
## Spatial Model Results 
## Formula: y ~ z
## Spatial method (outcome):  SAR 
## Likelihood function:  auto_gaussian 
## Link function:  identity 
## Residual Moran Coefficient:  NA 
## Observations:  1200 
## Data models (ME): none
## Inference for Stan model: foundation.
## 4 chains, each with iter=500; warmup=250; thin=1; 
## post-warmup draws per chain=250, total post-warmup draws=1000.
## 
##             mean se_mean    sd   2.5%    25%    50%    75%  97.5% n_eff  Rhat
## intercept  0.035   0.002 0.073 -0.104 -0.016  0.035  0.087  0.168  1009 1.000
## z         -0.510   0.000 0.009 -0.527 -0.516 -0.510 -0.503 -0.493  1350 0.998
## sar_rho    0.874   0.000 0.015  0.844  0.864  0.874  0.883  0.902   924 1.004
## sar_scale  0.312   0.000 0.006  0.299  0.307  0.311  0.316  0.324  1026 1.001
## 
## Samples were drawn using NUTS(diag_e) at Fri Mar  1 18:01:10 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

The user first creates the data list using prep_sar_data2 and then passes it to stan_sar using the sar_parts argument. Also, slim = TRUE is invoked to prevent the model from collecting N-length parameter vectors and quantities of interest (such as fitted values and log-likelihoods).

For large data sets and complex models, slim = TRUE can bring about computational improvements at the cost of losing some functionality (including the loss of convenience functions like sp_diag, me_diag, spatial, resid, and fitted). Many quantities of interest, such as fitted values and spatial trend terms, can still be calculated manually using the data and parameter estimates (intercept, coefficients, and spatial autocorrelation parameters).

The favorable MCMC diagnostics for this model (sufficiently large effective sample sizes n_eff, and Rhat values very near to 1), based on just 250 post-warmup iterations per chain with four MCMC chains, provides some indication as to how computationally efficient these spatial autoregressive models can be.

Also, note that Stan usually samples more efficiently when variables have been mean-centered. Using the centerx = TRUE argument in stan_sar (or any other model-fitting function in geostan) can be very helpful in this respect. Also note that the SAR models in geostan are (generally) no less computationally-efficient than the CAR models, and may even be slightly more efficient.

References

Donegan, Connor. 2022. “Building Spatial Conditional Autoregressive Models in the Stan Programming Language.” OSF Preprints. https://doi.org/10.31219/osf.io/3ey65.

Griffith, Daniel A. 2000. “Eigenfunction Properties and Approximations of Selected Incidence Matrices Employed in Spatial Analyses.” Linear Algebra and Its Applications 321 (1-3): 95–112.