---
title: "Reference-set data envelopment analysis with psdea"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Reference-set data envelopment analysis with psdea}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

## The idea

A DEA score is a distance to a frontier, and the frontier is built from peers.
`psdea» makes the peer set an argument.  The weight matrix is used only through
its non-zero pattern: a non-zero off-diagonal entry means that the column unit
may enter the frontier of the row unit, and the unit itself is always included.

## A three-unit example

Three units use one unit of input and produce 1, 2 and 4 units of output.

```{r}
library(psdea)
X <- matrix(c(1, 1, 1), nrow = 1)
Y <- matrix(c(1, 2, 4), nrow = 1)
round(psdea_sbm(X, Y)$eff, 4)
```

The scores are the ratio to the best observed ratio, 4.  Restricting unit 1 to
units 1 and 2 lowers the best ratio to 2:

```{r}
W <- matrix(c(0, 1, 0,
              1, 0, 1,
              0, 1, 0), 3, 3, byrow = TRUE)
round(psdea_sbm(X, Y, W = W)$eff_loc, 4)
```

## A panel

```{r}
data(psdea_china_energy)
d <- subset(psdea_china_energy, year == 2015 & code <= 8)
W <- psdea_w_contiguity[1:8, 1:8]
fit <- psdea_panel(d, "gdp_real", c("capital", "energy"), "year",
                   id = "code", bad = "co2", W = W)
fit
```

The two shares decompose the shortfall, and they are missing on the frontier
because the denominator is zero there:

```{r}
fit[, c("eff", "eff_loc", "s_local", "s_gap")]
```

## Learning the peers

When the analyst has no prior on who the peers are, the matrix can be built by
sparse non-negative reconstruction of each unit's bundle:

```{r}
Wl <- psdea_learn_data(d, c("capital", "energy", "gdp_real"), lambda = 0.05, by = "year")
attr(Wl, "n_peers")
fit_l <- psdea_panel(d, "gdp_real", c("capital", "energy"), "year",
                     id = "code", bad = "co2", W = Wl)
mean(fit_l$tgr, na.rm = TRUE)
```