PEAXAI

PEAXAI (Probabilistic Efficiency Analysis Using Explainable Artificial Intelligence) is an R package for estimating technical efficiency through a probabilistic framework that integrates Data Envelopment Analysis (DEA), machine learning classifiers, and explainable artificial intelligence (XAI) tools.

The package uses DEA-derived efficiency labels to train supervised machine learning models that estimate the probability that each decision-making unit (DMU) is efficient. This probability-based approach allows users to move beyond a purely deterministic efficient/inefficient classification and to analyse efficiency under uncertainty.

In addition, PEAXAI provides post-hoc interpretability tools to explain the estimated efficiency probabilities. These tools include global and local feature importance, Shapley value explanations, permutation importance, sensitivity analysis, counterfactual improvement recommendations, efficiency rankings, and peer identification based on user-defined probability thresholds.

The package is designed for researchers and practitioners interested in productivity and efficiency analysis, benchmarking, policy evaluation, operational research, and explainable machine learning applications.

Please check the vignette for more details.

Metadata Table

Installation

You can install the released version of the package from CRAN with:

install.packages("PEAXAI")

And the development version from GitHub with:

install.packages("devtools")
devtools::install_github("rgonzalezmoyano/PEAXAI")

Dependencies

PEAXAI depends on R and imports the following packages:

Benchmarking
caret
deaR
dplyr
kernelshap
iml
isotone
lime
np
PRROC
pROC
rminer
rms
stats

The following packages are suggested for examples, documentation and vignettes:

ggplot2
knitr
rmarkdown

Main functions

The main functions exported by PEAXAI are:

Basic example

The following example illustrates the main workflow of the package using the example dataset included in PEAXAI.

library(PEAXAI)

data("firms", package = "PEAXAI")

# Select firms from the Autonomous Community of Valencia
data <- subset(firms, autonomous_community == "Comunidad Valenciana")
data <- data[, -ncol(data)]

# Define inputs and output
x <- c(1:4)
y <- c(5)

# Define returns to scale
RTS <- "vrs"

# Define machine learning method
methods <- list(
  "nnet" = list(
    tuneGrid = expand.grid(
      size = c(1, 5, 10, 20),
      decay = 10^seq(-5, -1, by = 1)
    ),
    preProcess = c("center", "scale"),
    entropy = TRUE,
    skip = TRUE,
    maxit = 1000,
    MaxNWts = 100000,
    trace = FALSE,
    weights = NULL
  )
)

# Cross-validation settings
trControl <- list(
  method = "cv",
  number = 5
)

# Metrics used to select the best model
metric_priority <- c("Balanced_Accuracy", "F1", "ROC_AUC")

# Define imbalance treatment
imbalance_rate <- seq(0.05, 0.4, 0.05)

# Set seed for reproducibility
seed <- 1

# Fit PEAXAI model
models <- PEAXAI_fitting(
  data = data,
  x = x,
  y = y,
  RTS = RTS,
  imbalance_rate = imbalance_rate,
  methods = methods,
  trControl = trControl,
  metric_priority = metric_priority,
  hold_out = NULL,
  verbose = TRUE,
  seed = seed
)

# Extract final model
final_model <- models[["best_model_fit"]][["nnet"]]

# Estimate efficiency probabilities
probabilities <- PEAXAI_predict(
  data = data,
  x = x,
  y = y,
  final_model = final_model
)

head(probabilities)

Explainable AI analysis

PEAXAI allows users to compute global and local explanations for the estimated efficiency probabilities.

For example, global feature importance can be obtained as follows:

importance_method <- list(
  name = "SHAP",
  nsim = 200
)

relative_importance <- PEAXAI_global_importance(
  final_model = final_model,
  x = x,
  y = y,
  explain_data = data,
  reference_data = data,
  importance_method = importance_method
)

relative_importance

Counterfactual efficiency targets

Given a set of probability thresholds, PEAXAI can estimate the input-output changes required for a DMU to reach a desired probability of being efficient.

efficiency_thresholds <- seq(0.75, 0.95, 0.1)

directional_vector <- list(
  relative_importance = relative_importance,
  baseline = "mean"
)

targets <- PEAXAI_counterfactuals(
  data = data,
  x = x,
  y = y,
  final_model = final_model,
  efficiency_thresholds = efficiency_thresholds,
  directional_vector = directional_vector,
  n_expand = 0.25,
  n_grid = 50,
  max_y = 1,
  min_x = 1
)

head(targets[["0.85"]][["counterfactual_dataset"]])
head(targets[["0.85"]][["inefficiencies"]])

Efficiency rankings

Efficiency rankings can be computed either from the predicted probabilities or from attainable counterfactual improvements.

ranking_predicted <- PEAXAI_ranking(
  data = data,
  x = x,
  y = y,
  final_model = final_model,
  rank_basis = "predicted"
)

head(ranking_predicted)

If counterfactual targets have already been computed, an attainable-efficiency ranking can be obtained as follows:

ranking_attainable <- PEAXAI_ranking(
  data = data,
  x = x,
  y = y,
  final_model = final_model,
  efficiency_thresholds = efficiency_thresholds,
  targets = targets,
  rank_basis = "attainable"
)

head(ranking_attainable[["0.85"]])

Peer identification

PEAXAI also identifies benchmark peers for each DMU according to estimated efficiency probabilities and selected probability thresholds.

peers <- PEAXAI_peer(
  data = data,
  x = x,
  y = y,
  final_model = final_model,
  targets = targets,
  efficiency_thresholds = efficiency_thresholds,
  weighted = FALSE,
  relative_importance = relative_importance
)

head(peers)

Repository structure

The repository is organised as follows:

PEAXAI/
├── R/
│   └── R source files containing the main functions of the package
├── data/
│   └── Example datasets included in the package
├── man/
│   └── Function documentation generated with roxygen2
├── vignettes/
│   └── Package vignette with a complete worked example
├── DESCRIPTION
│   └── R package metadata, dependencies, authors and license information
├── NAMESPACE
│   └── Exported functions and imported methods
├── README.md
│   └── General description, installation instructions and usage examples
└── LICENCE.txt
    └── Full license text

Data

The package includes example data that can be loaded directly in R.

The vignette illustrates the use of PEAXAI with a dataset of Spanish food industry firms obtained from the SABI database for the year 2023. The dataset includes firms with more than 50 employees and contains financial and operational variables relevant for productivity and efficiency analysis.

The output variable used in the vignette is:

operating_income

The input variables used in the vignette are:

total_assets
employees
fixed_assets
personnel_expenses

The variable autonomous_community indicates the geographical location of each firm within one of Spain’s autonomous communities.

Documentation

After installing the package, the documentation of each function can be accessed in R using:

?PEAXAI_fitting
?PEAXAI_predict
?PEAXAI_global_importance
?PEAXAI_local_importance
?PEAXAI_counterfactuals
?PEAXAI_ranking
?PEAXAI_peer

The package vignette can be opened with:

browseVignettes("PEAXAI")

License

This software is distributed under the GPL-3 license.

Please see the LICENCE.txt file for the full license text.

Citation

If you use PEAXAI, please cite the associated manuscript:

@article{gonzalez_moyano_peaxai,
  title   = {Probability-based Technical Efficiency Analysis through Machine Learning},
  author  = {González Moyano, Ricardo and Aparicio, Juan and Zofío, José Luis and España, Víctor},
  journal = {In review},
  year    = {2025}
}

Contact

For questions, bug reports or suggestions, please use the GitHub Issues page:

https://github.com/rgonzalezmoyano/PEAXAI/issues

or contact the maintainer:

Ricardo González Moyano
ricardo.gonzalezm@umh.es