Spatial Markov chain for distribution dynamics

Estimates a spatial Markov chain (Rey, 2001): a set of class-transition probability matrices, each conditioned on the spatial context of a unit at the start of the period. Where markov() asks "given a unit's own class, where does it move next?", spatial_markov() asks "given a unit's own class and the class of its spatial neighbourhood, where does it move next?". Comparing the conditional matrices with the pooled (aspatial) matrix reveals whether, and how, spatial context shapes distribution dynamics.

The neighbourhood is summarised by the spatial lag: a (row-standardised) weighted average of neighbours' values. Supply either a spatial weights matrix via weights (the lag is computed for you, per period), or a precomputed spatial-lag column via lag (keeping the estimator independent of any particular spatial-data stack).

Usage

spatial_markov(
  data,
  id,
  time,
  value,
  weights = NULL,
  lag = NULL,
  k = 5,
  m = k,
  breaks = NULL,
  lag_breaks = NULL,
  fixed = TRUE,
  row_standardize = TRUE
)

Arguments

data

A data frame in long format with one row per unit and period.

id

Name (character scalar) of the column identifying the unit.

time

Name (character scalar) of the column giving the period. Periods are compared in sorted order within each unit.

value

Name (character scalar) of the numeric column to analyse.

weights

Optional spatial weights: a square numeric matrix whose row and column names match the unit ids, or an spdep listw or nb object (e.g. built from an sf layer with spdep::poly2nb() and spdep::nb2listw()). Used to compute the spatial lag of value within each period. Requires a balanced panel. Exactly one of weights or lag must be supplied.

lag

Optional name (character scalar) of a column giving a precomputed spatial lag of value.

k

Integer number of classes to discretise value into. Default 5 (quintiles). Ignored if breaks is supplied.

m

Integer number of spatial-lag classes to condition on. Defaults to k.

breaks

Optional numeric vector of interior class boundaries. If supplied these fixed cut points are used instead of data-driven quantiles, giving length(breaks) + 1 classes.

lag_breaks

Optional numeric vector of interior class boundaries for the spatial lag (the lag analogue of breaks). If supplied, these fixed cut points are used instead of data-driven quantiles. Supplying both breaks and lag_breaks makes the classification fully explicit, which is how exact agreement with giddy (cutoffs / lag_cutoffs) is obtained.

fixed

Logical. If TRUE (default) a single set of quantile breaks is estimated from the pooled distribution and applied to every period (absolute mobility). If FALSE, breaks are re-estimated within each period (relative mobility). Ignored when breaks is supplied.

row_standardize

Logical; row-standardise weights before computing the lag (default TRUE). Ignored when lag is supplied.

Value

An object of class sddr_spatial_markov: a list with elements

matrices

Length-m list of k by k conditional transition probability matrices, one per spatial-lag class.

transitions

Length-m list of conditional count matrices.

pooled

The k by k pooled (aspatial) transition matrix; this equals the matrix from markov() with the same class definition.

pooled_transitions

Pooled count matrix.

steady_states

Length-m list of conditional ergodic distributions (NA where a conditional matrix has unvisited classes).

classes, lag_classes

k and m.

n, n_transitions

Units contributing, and total transitions.

data

Input augmented with lagval, value state and lag state.

References

Rey, S. J. (2001). Spatial empirics for economic growth and convergence. Geographical Analysis, 33(3), 195-214. doi:10.1111/j.1538-4632.2001.tb00444.x

See also

markov()

Examples

set.seed(1)
n <- 12; periods <- 8
# Ring contiguity: each unit neighbours the next and previous on a circle.
W <- matrix(0, n, n, dimnames = list(1:n, 1:n))
for (i in 1:n) {
  W[i, i %% n + 1] <- 1
  W[i, (i - 2) %% n + 1] <- 1
}
df <- do.call(rbind, lapply(seq_len(periods), function(p)
  data.frame(id = 1:n, time = p, value = rnorm(n))))
sm <- spatial_markov(df, "id", "time", "value", weights = W, k = 3)
sm
#> <sddr> spatial Markov chain
#>   units: 12 | transitions: 84 | value classes: 3 | lag classes: 3 | breaks: fixed
#> 
#> Pooled transition matrix (aspatial):
#>       1     2     3
#> 1 0.267 0.367 0.367
#> 2 0.500 0.214 0.286
#> 3 0.231 0.423 0.346
#> 
#> Conditional on spatial-lag class 1 (31 transitions):
#>       1     2     3
#> 1 0.308 0.154 0.538
#> 2 0.700 0.200 0.100
#> 3 0.375 0.250 0.375
#> 
#> Conditional on spatial-lag class 2 (29 transitions):
#>       1     2     3
#> 1 0.333 0.556 0.111
#> 2 0.100 0.300 0.600
#> 3 0.100 0.600 0.300
#> 
#> Conditional on spatial-lag class 3 (24 transitions):
#>       1     2     3
#> 1 0.125 0.500 0.375
#> 2 0.750 0.125 0.125
#> 3 0.250 0.375 0.375