LISA Markov chain

A Markov chain over the four quadrants of the Moran scatterplot (Rey, 2001). Each unit, in each period, is placed in a quadrant according to the sign of its value (relative to the cross-sectional mean) and the sign of its spatial lag:

HH

high value, high neighbourhood (quadrant 1).

LH

low value, high neighbourhood (quadrant 2).

LL

low value, low neighbourhood (quadrant 3).

HL

high value, low neighbourhood (quadrant 4).

The resulting 4 by 4 transition matrix describes how units move between local spatial-association types over time — capturing the co-evolution of a unit and its neighbourhood.

Usage

lisa_markov(data, id, time, value, weights, row_standardize = TRUE)

Arguments

data

A data frame in long format with one row per unit and period. The panel must be balanced (every unit present in every period).

id, time, value

Character scalars naming the unit, period, and numeric value columns.

weights

Spatial weights: a square numeric matrix whose row and column names match the unit ids, or an spdep listw or nb object.

row_standardize

Logical; row-standardise weights before computing the spatial lag (default TRUE).

Value

An object of class sddr_markov: a 4 by 4 transition matrix over the quadrants HH, LH, LL, HL, plus the per-unit-per-period quadrants matrix.

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

spatial_markov(), markov()

Examples

set.seed(1)
n <- 9; periods <- 8
W <- matrix(0, n, n, dimnames = list(1:n, 1:n))
for (i in 1:n) {                       # ring contiguity
  W[i, i %% n + 1] <- 1
  W[i, (i - 2) %% n + 1] <- 1
}
df <- data.frame(id = rep(1:n, times = periods),
                 time = rep(1:periods, each = n),
                 value = rnorm(n * periods))
lisa_markov(df, "id", "time", "value", weights = W)
#> <sddr> LISA Markov chain
#>   units: 9 | transitions: 63 | classes: 4 | breaks: fixed
#> 
#> Transition probability matrix (rows = from, cols = to):
#>       HH    LH    LL    HL
#> HH 0.316 0.368 0.263 0.053
#> LH 0.471 0.118 0.235 0.176
#> LL 0.308 0.154 0.308 0.231
#> HL 0.143 0.357 0.143 0.357
#> 
#> Ergodic (steady-state) distribution:
#>    HH    LH    LL    HL 
#> 0.321 0.251 0.245 0.183