Package 'psdea'

Data Envelopment Analysis with User-Defined and Learned Reference Sets

Description

Slacks-based data envelopment analysis for panel data in which the reference set is an explicit argument. A weight matrix, supplied by the user or learned from the data, selects the units that may enter each decision-making unit's frontier. The package reports the unrestricted and the restricted score, the ratio of the two, and an exact decomposition of the shortfall into the part the restricted technology can remove and the part it cannot.

Details

The main function is psdea_panel. Weight matrices are built with psdea_spw or learned with psdea_learn; scores can be inspected with psdea_moran. Every linear program is solved by a two-phase revised simplex implemented in R (psdea_lp), so the package has no compiled code and no dependencies beyond base R.

References

Tone, K. (2001) A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research 130, 498–509.

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Provincial panel of China, 2003–2024

Description

A panel of 30 Chinese provinces with the inputs, output and undesirable output used in the article. Energy, carbon dioxide and gross domestic product for 2023 and 2024 are provisional; the columns whose names begin with flag_ record whether a value is observed, official or nowcast.

Format

A data frame with 660 rows and 22 variables. The variables used in the examples are code (1–30, matching the weight matrices), prov, year, energy (10\,000 tonnes of coal equivalent), capital (100 million yuan at constant 2003 prices), gdp_real (same units) and co2 (million tonnes).

Source

Provincial statistical yearbooks and the CEADs emission inventory.

Examples

data(psdea_china_energy)
str(psdea_china_energy[1:5, c("prov", "year", "energy", "gdp_real")])

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Learn the reference set from the data

Description

Reconstructs each unit's bundle as a sparse non-negative combination of the other units' bundles by penalised least squares, solved with cyclic coordinate descent. The support of the solution is the peer set.

Usage

psdea_learn(Z, lambda = 0.05, maxit = 500L, tol = 1e-4)

psdea_learn_data(data, vars, lambda = 0.05, by = NULL)

Arguments

Z	matrix with units in the columns and standardised inputs and outputs in the rows.
lambda	penalty on the sum of the weights. Larger values give sparser peer sets.
maxit	maximum number of coordinate-descent sweeps.
tol	a weight is treated as zero below this value when counting peers.
data	a data frame.
vars	names of the variables forming the bundle.
by	optional name of the period variable; the matrix is then built for each period and averaged.

Details

The problem is a penalised non-negative reconstruction, not a convex-hull representation: the weights are not constrained to sum to one and normalising them afterwards does not change the optimisation problem. The peer set is chosen on the same observations that produce the scores, so the matrix is a selection rule, not an estimate of a technology club.

Value

A row-normalised matrix of weights, with attributes n_peers and iters.

Examples

set.seed(1)
Z <- matrix(runif(40, 1, 2), 4, 10)
W <- psdea_learn(Z, lambda = 0.05)
attr(W, "n_peers")

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Two-phase revised simplex

Description

Solves $\min c'x$ subject to $Ax=b$ and $x\ge 0$ with a two-phase revised simplex and Bland's rule for pricing.

Usage

psdea_lp(A, b, c, maxit = 20000L, tol = 1e-10)

Arguments

A	constraint matrix, $m \times n$.
b	right-hand side, length $m$.
c	objective coefficients, length $n$.
maxit	maximum number of pivots.
tol	tolerance used in the pricing and ratio tests.

Value

A list with x (the solution), obj, it (pivots) and status: 0 optimal, 1 infeasible, 2 iteration limit reached.

Examples

A <- matrix(c(1, 2, -1, 0,
              3, 1,  0, -1), 2, 4, byrow = TRUE)
psdea_lp(A, c(4, 6), c(1, 1, 0, 0))$obj   # 2.8

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Spatial autocorrelation diagnostics

Description

Moran's $I$ and local indicators of spatial association for a vector of scores, with permutation inference. These are diagnostics: $I$ is the slope of the spatial lag on the score, not a share of variance.

Usage

psdea_moran(x, W, nperm = 499L, seed = 1L)

psdea_lisa(x, W, nperm = 499L, seed = 1L)

Arguments

x	numeric vector of scores, one per unit.
W	square weight matrix.
nperm	number of permutations.
seed	seed for the permutations.

Value

psdea_moran returns a named vector with the observed statistic and its p-value; psdea_lisa returns a data frame with the local statistic and its conditional p-value for each unit.

Examples

x <- c(1, 2, 3, 4)
W <- psdea_spw(c(1, 2, 3), c(2, 3, 4), 1:4)
psdea_moran(x, W, nperm = 99)

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Efficiency against an unrestricted and a restricted frontier

Description

Estimates a slacks-based measure with undesirable outputs for each period of a panel, once against all units and, when a weight matrix is supplied, once against the units selected by that matrix. Returns the decomposed shortfall.

Usage

psdea_panel(data, y, x, year, id = NULL, bad = NULL, W = NULL,
            rts = c("crs", "vrs"))

## S3 method for class 'psdea'
print(x, ...)

## S3 method for class 'psdea'
summary(object, ...)

Arguments

data	a data frame in long format, one row per unit and period.
y	name of the single desirable output.
x	for psdea_panel, a character vector with the names of the inputs; for the print method, the fitted object.
year	name of the period variable; one frontier is built per period.
id	name of the unit identifier. When NULL the row order is used and a variable .id is created.
bad	optional character vector with the names of undesirable outputs.
W	optional square matrix of order equal to the number of units in a period. A non-zero off-diagonal entry means that the column unit may enter the frontier of the row unit. The unit itself is always included.
rts	either "crs" or "vrs".
object	an object returned by psdea_panel.
...	further arguments, currently ignored.

Details

The shares are $(1-\rho^N)/(1-\rho^G)$ and $(\rho^N-\rho^G)/(1-\rho^G)$; they sum to one and are undefined on the frontier, where the denominator is zero. They are not equal to $1-\mathrm{TGR}$.

Value

A data frame of class psdea with the columns id, year, eff (unrestricted score), eff_loc (restricted score), tgr (their ratio), s_local and s_gap (the two shares of the shortfall, missing when the unrestricted score is one) and n_peers (the size of the reference set).

See Also

psdea_spw, psdea_learn

Examples

data(psdea_china_energy)
d <- subset(psdea_china_energy, year == 2015 & code <= 6)
W <- psdea_w_contiguity[1:6, 1:6]
fit <- psdea_panel(d, "gdp_real", c("capital", "energy"), "year",
                   id = "code", bad = "co2", W = W)
fit
fit$s_local + fit$s_gap

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Slacks-based measure for one cross-section

Description

Computes the slacks-based measure of Tone (2001) with undesirable outputs for every column of a cross-section, optionally against the restricted reference technology defined by a weight matrix.

Usage

psdea_sbm(X, Y, B = NULL, vrs = FALSE, W = NULL)

Arguments

X	matrix of inputs, rows are inputs and columns are units.
Y	matrix of desirable outputs, same layout.
B	matrix of undesirable outputs, or NULL.
vrs	logical; variable rather than constant returns to scale.
W	optional square weight matrix; see psdea_panel.

Value

A list with eff, and, when W is given, eff_loc, n_peers, tgr, s_local and s_gap.

References

Tone, K. (2004) Dealing with undesirable outputs in DEA: a slacks-based measure approach. GRIPS Research Report I-2003-0005.

Examples

X <- matrix(c(1, 1, 1), nrow = 1)
Y <- matrix(c(1, 2, 4), nrow = 1)
psdea_sbm(X, Y)$eff            # 0.25 0.50 1.00
W <- matrix(c(0, 1, 0,
              1, 0, 1,
              0, 1, 0), 3, 3, byrow = TRUE)
psdea_sbm(X, Y, W = W)$eff_loc # 0.50 1.00 1.00

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Build a weight matrix

Description

Builds a square weight matrix from a table of links, or reads one from a comma-separated file. The matrix is used only through its non-zero pattern.

Usage

psdea_spw(from, to, units, row_standardise = TRUE, directed = FALSE)

psdea_spw_read(file, units = NULL, row_standardise = TRUE)

Arguments

from, to	vectors of unit names or indices giving the links.
units	character vector with the unit names, in the order used by the data; the row and column names of the result follow this order.
row_standardise	logical; divide each row by its sum.
directed	logical; if FALSE the links are made symmetric.
file	path to a comma-separated file without header.

Value

A numeric matrix with row and column names units.

Examples

units <- c("A", "B", "C")
psdea_spw(c("A", "B"), c("B", "C"), units)

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Spatial weight matrices for the Chinese provinces

Description

Matrices of order 30 used in the article: land-border contiguity (65 links plus one for Hainan), four nearest neighbours, and inverse distance with a median cut-off. Rows and columns follow the province code.

Format

A 30 by 30 numeric matrix, row-standardised.

Source

Own construction from provincial boundaries and capital coordinates.

Examples

data(psdea_w_contiguity)
table(rowSums(psdea_w_contiguity > 0))

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