Markov mobility indices

Scalar summaries of how much mixing a transition matrix implies — larger values mean more mobility (a faster-churning distribution), values near zero mean a near-immobile system. Several classical indices are provided:

prais

Prais index \((k - \mathrm{tr}(P)) / (k - 1)\) — average probability of leaving a class.

determinant

\(1 - |\det(P)|\).

L2

\(1 - |\lambda_2|\), one minus the second-largest eigenvalue modulus (the mixing rate).

shorrock1

\((k - k \sum_i \pi_i p_{ii}) / (k - 1)\), weighted by an initial distribution ini.

shorrock2

\(\sum_i \sum_j \pi_i p_{ij} |i - j| / (k - 1)\), penalising longer jumps.

Usage

mobility(x, measure = "all", ini = NULL)

Arguments

x

A row-stochastic matrix, or an object of class sddr_markov.

measure

One of "prais", "determinant", "L2", "shorrock1", "shorrock2", or "all" (default) to return every index.

ini

Optional initial distribution (length k) for the Shorrock indices; defaults to uniform.

Value

A named numeric vector of the requested index or indices.

References

Prais, S. J. (1955). Measuring social mobility. JRSS A, 118(1), 56-66. Shorrocks, A. F. (1978). The measurement of mobility. Econometrica, 46(5), 1013-1024. doi:10.2307/1911433

Examples

P <- matrix(c(0.7, 0.2, 0.1,
              0.2, 0.6, 0.2,
              0.1, 0.3, 0.6), nrow = 3, byrow = TRUE)
mobility(P)
#>       prais determinant          L2   shorrock1   shorrock2 
#>   0.5500000   0.8100000   0.4381966   0.5500000   0.2166667 
mobility(P, "prais")
#> prais 
#>  0.55