Getting started with deaviz • deaviz

deaviz turns the numbers behind a Data Envelopment Analysis (DEA), whether the inputs/outputs profiles or the computational outcome of various DEA models, into plots: efficiency distributions, input/output relationships, an efficient frontier representation, projection biplots, benchmarking networks, cross-efficiency maps, and multi-period trajectories for panel data. This vignette walks through the typical workflow on two bundled datasets.

The workflow has three steps that map onto the package’s naming convention:

wrap your data in a dea_data() object (inputs, outputs, DMU labels);
compute_*() the quantities you need (efficiency, cross-efficiency, weights, self-organising maps);
plot_*() the result.

library(deaviz)

Most plots compute efficiency scores internally, which relies on the Benchmarking package; a few embeddings/layouts use smacof, igraph/graphlayouts, or kohonen. These are all Suggests: install them to reproduce every figure below. (Where a suggested package is missing, the corresponding chunk is simply skipped, so this vignette always builds.)

The data object

deaviz ships chinese_cities, a classic cross-sectional DEA benchmark of 35 cities with three inputs and three outputs (Sueyoshi, 1992). dea_data() records which columns are inputs, which are outputs, and which identifies the DMU. Columns can be given by name or by position.

So the dea_data object can be defined either by the input and output variable names as follows:

d <- dea_data(
  chinese_cities,
  inputs  = c("industrial_labour_force", "working_funds", "investments"),
  outputs = c("gross_industrial_output", "profit_and_tax", "retail_sales"),
  id      = "DMU"
)
d
#> <dea_data>
#>   DMUs    : 35
#>   Inputs  : 3 (industrial_labour_force, working_funds, investments)
#>   Outputs : 3 (gross_industrial_output, profit_and_tax, retail_sales)

or equivalently by the location of them:

d <- dea_data(
  chinese_cities,
  inputs  = 2:4,
  outputs = 5:7,
  id      = "DMU"
)

Everything downstream takes this d object. If your input/output columns are prefixed i_ / o_, dea_data() detects them automatically and you can skip the inputs/outputs arguments.

Efficiency scores

compute_efficiency() returns radial efficiency scores (plus peer weights and multiplier weights). Returns-to-scale and orientation are arguments.

eff <- compute_efficiency(x = d, rts = "vrs", orientation = "in")
head(round(eff$eff, 3))
#> [1] 1.000 0.944 0.805 0.779 0.875 0.686

Distributions: the shape of efficiency

Start by looking at the spread. plot_efficiency_distributions() shows the distribution of efficiency scores, while plot_io_distributions() shows the raw input/output variables.

plot_efficiency_distributions(d, rts = "vrs", title = "Chinese Cities Efficiency Scores", subtitle = "Variable Return To Scale")

plot_io_distributions(d, type = "box", x_angle = 30)

Note the x_angle = 30: the input/output names are long, so tilting the x-axis tick labels keeps them readable. Every plot whose x-axis carries variable or DMU names accepts x_angle.

plot_io_efficients() counts how many DMUs are efficient versus inefficient.

plot_io_efficients(d, rts = "vrs", transparency = 1)

Input/output relationships and the frontier

plot_io_scatter() lays out every input-against-output pair if no vector of inputs and/or outputs is assigned to vars = and if a vector provided, then the scatterplots will be limited to the pair-wise combination of the the variables. Regardless, the visual marks are coloured by efficiency.


plot_io_scatter(d, vars = c("industrial_labour_force", "gross_industrial_output", "retail_sales") , color = "vrs")

It is possible to plot scatters against the efficiency scores of the DMUs as well:


plot_io_scatter(d, vars = c("industrial_labour_force", "gross_industrial_output", "retail_sales"), efficiency = "vrs" , color = "vrs")

The only frontier visualization plot is available via plot_io_costa_frontier() and it collapses all inputs and outputs onto a single aggregated frontier (Bana e Costa et al., 2016).

plot_io_costa_frontier(d)

The textbook frontier

For teaching it is often clearest to take a single input against a single output and draw the efficient frontier the way textbooks do. plot_io_frontier() does exactly that: pick one input and one output, choose the returns-to-scale, and it shades the production-possibility set and traces the frontier, colouring each unit by its efficiency in that two-variable sub-model.

plot_io_frontier(d, input = "working_funds", output = "retail_sales",
                 rts = "vrs")

Turn on peers_network to see which efficient peers an inefficient city is benchmarked against; the arrows are weighted by the envelopment ( $\lambda$ ) weights.

plot_io_frontier(d, input = "working_funds", output = "retail_sales",
                 rts = "vrs", peers_network = TRUE, labels = "Chengdu")

Projection biplots

Two ways to project the multidimensional input/output space onto a readable plane. plot_io_pca_biplot() uses PCA, drawing the input/output loading vectors; plot_io_mds() uses metric (ratio) multidimensional scaling via the smacof majorization algorithm (de Leeuw & Mair, 2009). The graphical use of such projections for DEA follows Adler & Raveh (2008).

Let’s have a look at the PCA biplot:

plot_io_pca_biplot(d, rts = "vrs")

The vectors are the dataset’s inputs and outputs and they show the direction towards which the value of the corresponding input or output increases in the 2d space.

In contrast, we can use an MDS algorithm to reduce the dimensionality of the dataset and represent the DMUs visually in a 2d plot.

plot_io_mds(d)

What to do with overcrowded plots? One solution offered by the deaviz package is to make the plot interactive so one can zoom into the plot and hover over the visual marks to get information about them.

plot_io_mds(d, interactive = TRUE)

Benchmarking networks

For inefficient DMUs, DEA identifies the efficient peers they are benchmarked against. plot_io_lambda_network() draws those peer relationships weighted by the envelopment ( $\lambda$ ) weights, laid out with Sammon mapping (Sammon, 1969) as in Porembski et al. (2005); plot_io_peer_network() lays out who is a peer to whom, therefore the edges are directed from the inefficient units to their targets.

plot_io_lambda_network(d, rts = "vrs")

plot_io_peer_network(d, rts = "vrs")

It is sometimes important in the networks to focus on and highlight a specific DMU and deaviz package addresses that need via the labels = argument:

plot_io_peer_network(d,layout = "fr",labels = "Xian", rts = "vrs")

Pay attention that the default layout of the network is changed by providing a pre-defined value to the layout = argument.

Cross-efficiency

Cross-efficiency scores every DMU using every other DMU’s optimal weights (Doyle & Green, 1994). compute_cross_efficiency() builds the matrix, which plot_cem_heatmap() displays. plot_cem_unfolding() unfolds the same matrix into a map of who rates whom favourably (Ashkiani & Mar-Molinero, 2017), and plot_cem_weights_heatmap() shows the underlying weight profiles.

cem <- compute_cross_efficiency(d)

plot_cem_heatmap(cem, x_angle = 90)

plot_cem_unfolding(cem)

plot_cem_weights_heatmap(d, x_angle = 30)

What if we want to highlight a specific DMU? Same as before, we can use the labels= argument:

plot_cem_weights_heatmap(d, x_angle = 30, labels = "Xian")

Profile plots

plot_io_radar() and plot_io_parcoo() show each DMU’s full input/output profile, as a radar polygon or a parallel-coordinates line.

plot_io_radar(d, efficiency = "vrs")

plot_io_parcoo(d, efficiency = "vrs", x_angle = 30)

Both accept a variables argument to show a subset of the profile: "all" (the default), "inputs", "outputs", or a vector of variable names or positions.

plot_io_parcoo(d, variables = "inputs", efficiency = "vrs", x_angle = 30)

Highlighting a single DMU: the focus view

This feature is so useful that it demands a dedicated section to explain it in further details. Pass one DMU name to labels and deaviz puts it centre stage: it is ringed and labelled, while everything else fades back. For the network plots the focus keeps the chosen DMU’s sub-network; for the panel biplot it keeps that DMU’s trajectory.

plot_io_pca_biplot(d, rts = "vrs", labels = "Beijing")

The amount of fade is tunable through the same fade argument: TRUE (default) uses a sensible level, FALSE turns it off, and a number sets the alpha of the faded marks directly (larger keeps them more visible).

plot_io_parcoo(d, efficiency = "vrs", labels = "Beijing", fade = 0.4)

Other labels modes are "all" (label everyone), "id" (number each marker), and "max.overlaps" (label as many as fit without collision).

plot_cem_unfolding(cem, labels = "id")

Self-organising maps

compute_som() trains a self-organising map (Kohonen, 2001) on the input/output profiles, via the kohonen package (Wehrens & Kruisselbrink, 2018); plot_io_som() colours the map by mean efficiency per node.

som <- compute_som(d)
plot_io_som(som)

Multi-period data: trajectories

For panel data, plot_panel_io_biplot() places every DMU-period on a shared PCA biplot and joins each DMU’s points into a trajectory over time. The bundled taiwanese_banks dataset is a balanced panel of 22 commercial banks over 2009-2011 (Kao & Liu, 2014). Inputs and outputs can be given by position.

plot_panel_io_biplot(
  taiwanese_banks, id = "DMU", period = "Year",
  inputs = 3:5, outputs = 6:8, labels = "Cathay"
)

Here the focus view keeps only Cathay’s three-year path lit while the other banks recede, and the loading vectors are spread apart so their labels stay legible.

It is noteworthy that the PCA is computed over the pooled data.

For panel work it pays to build a dea_panel_data() object once – it validates the panel (unique DMU-period rows, balancedness) and remembers which columns are the id, the period, and the inputs/outputs, so the plot_panel_* functions can be called without repeating them:

pd <- dea_panel_data(taiwanese_banks, inputs = 3:5, outputs = 6:8,
                     id = "DMU", period = "Year")
pd
#> A DEA panel: 22 DMUs x 3 periods (balanced) 
#>   periods: 2009, 2010, 2011 
#>   inputs:  labour, physical_capital, purchased_funds 
#>   outputs: demand_deposits, short_term_loans, long_term_loans

plot_panel_io_parcoo() offers a simpler view of the same panel: the periods form the parallel axes and each DMU is one line, tracing either a single input/output variable or the per-period efficiency score (each period’s cross-section solved as its own DEA problem).

plot_panel_io_parcoo(pd, y = "vrs", labels = "Cathay")

With color_by_trend = TRUE each line is coloured by the DMU’s overall change (last period minus first), so the biggest risers and fallers stand out.

plot_panel_io_parcoo(pd, y = "demand_deposits", color_by_trend = TRUE)

Interactive plots

Many plots accept interactive = TRUE, returning a plotly widget with hover tooltips instead of a static ggplot. This needs the plotly package and is best viewed in an HTML context:

plot_io_pca_biplot(d, rts = "vrs", labels = "Beijing", interactive = TRUE)

References

Adler, N., & Raveh, A. (2008). Presenting DEA graphically. Omega, 36(5), 715–729.

Ashkiani, S. (2019). Four Essays on Data Visualization and Anomaly Detection of Data Envelopment Analysis Problems (PhD thesis). Universitat Autonoma de Barcelona. https://ddd.uab.cat/record/240333

Ashkiani, S., & Mar-Molinero, C. (2017). Visualization of cross-efficiency matrices using multidimensional unfolding. In Recent Applications of Data Envelopment Analysis.

Bana e Costa, C. A., Soares de Mello, J. C. C. B., & Angulo Meza, L. (2016). A new approach to the bi-dimensional representation of the DEA efficient frontier with multiple inputs and outputs. European Journal of Operational Research, 255(1), 175–186. https://doi.org/10.1016/j.ejor.2016.05.012

Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. https://doi.org/10.1016/0377-2217(78)90138-8

de Leeuw, J., & Mair, P. (2009). Multidimensional scaling using majorization: SMACOF in R. Journal of Statistical Software, 31(3), 1–30. https://doi.org/10.18637/jss.v031.i03

Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45(5), 567–578. https://doi.org/10.1057/jors.1994.84

Kao, C., & Liu, S.-T. (2014). Multi-period efficiency measurement in data envelopment analysis: The case of Taiwanese commercial banks. Omega, 47, 90–98. https://doi.org/10.1016/j.omega.2013.09.001

Kohonen, T. (2001). Self-Organizing Maps (3rd ed.). Springer.

Porembski, M., Breitenstein, K., & Alpar, P. (2005). Visualizing efficiency and reference relations in data envelopment analysis with an application to the branches of a German bank. Journal of Productivity Analysis, 23(2), 203–221. https://doi.org/10.1007/s11123-005-1328-5

Sammon, J. W. (1969). A nonlinear mapping for data structure analysis. IEEE Transactions on Computers, C-18(5), 401–409. https://doi.org/10.1109/T-C.1969.222678

Sueyoshi, T. (1992). Measuring the industrial performance of Chinese cities by data envelopment analysis. Socio-Economic Planning Sciences, 26(2), 75–88. https://doi.org/10.1016/0038-0121(92)90015-W

Wehrens, R., & Kruisselbrink, J. (2018). Flexible self-organizing maps in kohonen 3.0. Journal of Statistical Software, 87(7), 1–18. https://doi.org/10.18637/jss.v087.i07

Where to next

Every function has its own help page with a full argument list and examples (e.g. ?plot_panel_io_biplot). To cite the package, see citation("deaviz").