,
Juan Aparicio [aut] ,
Víctor España [aut] | Title: | A Boosting Approach to Data Envelopment Analysis |
|---|---|
| Description: | Includes functions to estimate production frontiers and make ideal output predictions in the Data Envelopment Analysis (DEA) context using both standard models from DEA and Free Disposal Hull (FDH) and boosting techniques. In particular, EATBoosting (Guillen et al., 2023 <doi:10.1016/j.eswa.2022.119134>) and MARSBoosting. Moreover, the package includes code for estimating several technical efficiency measures using different models such as the input and output-oriented radial measures, the input and output-oriented Russell measures, the Directional Distance Function (DDF), the Weighted Additive Measure (WAM) and the Slacks-Based Measure (SBM). |
| Authors: | Maria D. Guillen [cre, aut] ,
Juan Aparicio [aut] ,
Víctor España [aut] |
| Maintainer: | Maria D. Guillen <[email protected]> |
| License: | AGPL (>= 3) |
| Version: | 0.1.0 |
| Built: | 2024-11-04 19:54:46 UTC |
| Source: | https://github.com/mariaguilleng/boostingdea |
This function adds the best pair of basis functions to the model
AddBF(data, x, y, ForwardModel, knots_list, Kp, minspan, Le, linpreds, err_min)AddBF(data, x, y, ForwardModel, knots_list, Kp, minspan, Le, linpreds, err_min)
data |
data |
x |
Column input indexes in |
y |
Column output indexes in |
ForwardModel |
|
knots_list |
|
Kp |
Maximum degree of interaction allowed. |
minspan |
|
Le |
|
linpreds |
|
err_min |
Minimum error in the split. |
A list containing the matrix of basis functions (B), a
list of basis functions (BF), a list of selected knots
(knots_list) and the minimum error (err_min).
The dataset consists of 31 banks operating in Taiwan.
data(banks)data(banks)
banks is a dataframe with 31 banks (rows) and 6 variables
(outputs) named Financial.funds (deposits and borrowed funds in
millions of TWD), Labor (number of employees),
Physical.capital (net amount of fixed assets in millions of TWD),
Finalcial.investments (financial assets, securities, and equity
investments in millions of TWD), Loans (loans and discounts in millions
of TWD) and Revenue (interests from financial investments and loans).
The dataset has been extracted from the “Condition and Performance of Domestic Banks” published by the Central Bank of China (Taiwan) and the Taiwan Economic Journal (TEJ) for the year 2010. The “Condition and Performance of Domestic Banks” was downloaded from http://www.cbc.gov.tw/ct.asp?xItem=1062&ctNode=535&mp=2
Juo, J. C., Fu, T. T., Yu, M. M., & Lin, Y. H. (2015). Profit-oriented productivity change. Omega, 57, 176-187.
This function predicts the expected output through a DEA model.
BBC_in( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )BBC_in( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
matrix with the the predicted score
This function predicts the expected output through a DEA model.
BBC_out( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )BBC_out( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
matrix with the the predicted score
This function computes the root mean squared error (RMSE) for a set of EATBoost models built with a grid of given hyperparameters.
bestEATBoost( training, test, x, y, num.iterations, learning.rate, num.leaves, verbose = TRUE )bestEATBoost( training, test, x, y, num.iterations, learning.rate, num.leaves, verbose = TRUE )
training |
Training |
test |
Test |
x |
Column input indexes in |
y |
Column output indexes in |
num.iterations |
Maximum number of iterations the algorithm will perform |
learning.rate |
Learning rate that control overfitting of the algorithm. Value must be in (0,1] |
num.leaves |
Maximum number of terminal leaves in each tree at each iteration |
verbose |
Controls the verbosity. |
A data.frame with the sets of hyperparameters and the root
mean squared error (RMSE) and mean square error (MSE) associated for each
model.
This funcion computes the root mean squared error (RMSE) for a set of MARSBoost models built with a grid of given hyperparameters.
bestMARSBoost( training, test, x, y, num.iterations, learning.rate, num.terms, verbose = TRUE )bestMARSBoost( training, test, x, y, num.iterations, learning.rate, num.terms, verbose = TRUE )
training |
Training |
test |
Test |
x |
Column input indexes in |
y |
Column output indexes in |
num.iterations |
Maximum number of iterations the algorithm will perform |
learning.rate |
Learning rate that control overfitting of the algorithm. Value must be in (0,1] |
num.terms |
Maximum number of reflected pairs created by the forward algorithm of MARS. |
verbose |
Controls the verbosity. |
A data.frame with the sets of hyperparameters and the root
mean squared error (RMSE) associated for each model.
This function is used to simulate the data in a single output scenario.
CobbDouglas(N, nX)CobbDouglas(N, nX)
N |
Sample size. |
nX |
Number of inputs. Possible values: |
data.frame with the simulated data.
This function denotes if a node dominates another one or if there is no Pareto-dominance relationship.
comparePareto(t1, t2)comparePareto(t1, t2)
t1 |
A first node. |
t2 |
A second node. |
-1 if t1 dominates t2, 1 if t2 dominates t1 and 0 if there are no Pareto-dominance relationships.
This function generates two new basis functions from a variable and a knot.
CreateBF(data, xi, knt, B, p)CreateBF(data, xi, knt, B, p)
data |
|
xi |
|
knt |
Knot for creating the new basis function(s). |
B |
|
p |
|
Matrix of basis function (B) updated with the new basis
functions.
This function generates two new cubic basis functions from a variable and a knot previously created during MARS algorithm.
CreateCubicBF(data, xi, knt, B, side)CreateCubicBF(data, xi, knt, B, side)
data |
|
xi |
Variable index of the new basis function(s). |
knt |
Knots for creating the new basis function(s). |
B |
Matrix of basis functions. |
side |
Side of the basis function. |
Matrix of basis functions updated with the new basis functions.
This function predicts the expected output through a DEA model.
DDF( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE, direction.vector )DDF( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE, direction.vector )
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
direction.vector |
Direction vector. Valid values are: |
matrix with the the predicted score
This function estimates a production frontier satisfying Data Envelope Analysis axioms using the radial output measure.
This function saves information about the DEA model.
DEA(data, x, y) DEA_object(data, x, y, pred, score)DEA(data, x, y) DEA_object(data, x, y, pred, score)
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
pred |
Output predictions using the BBC radial output measure |
score |
Efficiency score using the BBC radial output measure |
A DEA object.
A DEA object.
This function creates a deep Efficiency Analysis Tree and a set of possible prunings by the weakest-link pruning procedure.
deepEAT(data, x, y, numStop = 5, max.leaves)deepEAT(data, x, y, numStop = 5, max.leaves)
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
numStop |
Minimum number of observations in a node for a split to be attempted. |
max.leaves |
Maximum number of leaf nodes. |
A list containing each possible pruning for the deep tree and its associated alpha value.
This function estimates a stepped production frontier through regression trees.
EAT(data, x, y, numStop = 5, max.leaves, na.rm = TRUE)EAT(data, x, y, numStop = 5, max.leaves, na.rm = TRUE)
data |
|
x |
Column input indexes in data. |
y |
Column output indexes in data. |
numStop |
Minimum number of observations in a node for a split to be attempted. |
max.leaves |
Maximum number of leaf nodes. |
na.rm |
|
The EAT function generates a regression tree model based on CART under a new approach that guarantees obtaining a stepped production frontier that fulfills the property of free disposability. This frontier shares the aforementioned aspects with the FDH frontier but enhances some of its disadvantages such as the overfitting problem or the underestimation of technical inefficiency.
An EAT object containing:
data
df: data frame containing the variables in the model.
x: input indexes in data.
y: output indexes in data.
input_names: input variable names.
output_names: output variable names.
row_names: rownames in data.
control
fold: fold hyperparameter value.
numStop: numStop hyperparameter value.
max.leaves: max.leaves hyperparameter value.
max.depth: max.depth hyperparameter value.
na.rm: na.rm hyperparameter value.
tree: list structure containing the EAT nodes.
nodes_df: data frame containing the following information for each node.
id: node index.
SL: left child node index.
N: number of observations at the node.
Proportion: proportion of observations at the node.
the output predictions.
R: the error at the node.
index: observation indexes at the node.
model
nodes: total number of nodes at the tree.
leaf_nodes: number of leaf nodes at the tree.
a: lower bound of the nodes.
y: output predictions.
This function saves information about the Efficiency Analysis Trees model.
EAT_object(data, x, y, rownames, numStop, max.leaves, na.rm, tree)EAT_object(data, x, y, rownames, numStop, max.leaves, na.rm, tree)
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
rownames |
|
numStop |
Minimum number of observations in a node for a split to be attempted. |
max.leaves |
Depth of the tree. |
na.rm |
|
tree |
|
An EAT object.
This function estimates a production frontier satisfying some classical production theory axioms, such as monotonicity and determinictiness, which is based upon the adaptation of the machine learning technique known as Gradient Tree Boosting
This function saves information about the EATBoost model
EATBoost(data, x, y, num.iterations, num.leaves, learning.rate) EATBoost_object( data, x, y, num.iterations, num.leaves, learning.rate, EAT.models, f0, prediction )EATBoost(data, x, y, num.iterations, num.leaves, learning.rate) EATBoost_object( data, x, y, num.iterations, num.leaves, learning.rate, EAT.models, f0, prediction )
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
num.iterations |
Maximum number of iterations the algorithm will perform |
num.leaves |
Maximum number of terminal leaves in each tree at each iteration. |
learning.rate |
Learning rate that control overfitting of the algorithm. Value must be in (0,1] |
EAT.models |
List of the EAT models created in each iterations |
f0 |
Initial predictions of the model (they correspond to maximum value of each output variable) |
prediction |
Final predictions of the original data |
A EATBoost object.
A EATBoost object.
Calculates the efficiency score corresponding to the given model using the given measure
efficiency( model, measure = "rad.out", data, x, y, heuristic = TRUE, direction.vector = NULL, weights = NULL )efficiency( model, measure = "rad.out", data, x, y, heuristic = TRUE, direction.vector = NULL, weights = NULL )
model |
Model object for which efficiency score is computed. Valid classes
are: |
measure |
Efficiency measure used. Valid measures are: |
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
heuristic |
Only used if |
direction.vector |
Only used when |
weights |
Only used when |
matrix with the the predicted score
This function predicts the expected output through a DEA model.
ERG(data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE)ERG(data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE)
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
matrix with the the predicted score
This function solves a Quadratic Programming Problem to obtain a set of coefficients.
EstimCoeffsForward(B, y)EstimCoeffsForward(B, y)
B |
|
y |
Output |
vector with the coefficients estimated.
This function gets the estimation of the response variable and updates Pareto-coordinates and the observation index for both new nodes.
estimEAT(data, leaves, t, xi, s, y)estimEAT(data, leaves, t, xi, s, y)
data |
Data to be used. |
leaves |
List structure with leaf nodes or pending expansion nodes. |
t |
Node which is being split. |
xi |
Variable index that produces the split. |
s |
Value of xi variable that produces the split. |
y |
Column output indexes in data. |
Left and right children nodes.
This function estimates a production frontier satisfying Free Disposal HUll axioms using the radial output measure.
This function saves information about the FDH model.
FDH(data, x, y) FDH_object(data, x, y, pred, score)FDH(data, x, y) FDH_object(data, x, y, pred, score)
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
pred |
Output predictions using the BBC radial output measure |
score |
Efficiency score using the BBC radial output measure |
A FDH object.
A FDH object.
EATBoost leaves supportsCalculates the inferior corner of the leaves supports of a
EATBoost model.
get.a.EATBoost(EATBoost_model)get.a.EATBoost(EATBoost_model)
EATBoost_model |
Model from class |
data.frame with the leave supports
EATBoost
Calculates the inferior corner of the support of all leave nodes
of every tree created in the EATBoost model
get.a.trees(EATBoost_model)get.a.trees(EATBoost_model)
EATBoost_model |
Model from class |
list of matrix. The length of the list is equal to
the num.iterations of the EATBoost_model. Each matrix
corresponds to a tree, where the number of columns is the number of input
variables and the number of rows to the number of leaves
EATBoost
Calculates the superior corner of the support of all leave nodes
of every tree created in the EATBoost model
get.b.trees(EATBoost_model)get.b.trees(EATBoost_model)
EATBoost_model |
Model from class |
list of matrix. The length of the list is equal to
the num.iterations of the EATBoost_model. Each matrix
corresponds to a tree, where the number of columns is the number of input
variables and the number of rows to the number of leaves
Calculates the intersection between two leave nodes from
different trees of a EATBoost model.
get.intersection.a(comb_a_actual, comb_b_actual)get.intersection.a(comb_a_actual, comb_b_actual)
comb_a_actual |
Inferior corner of first leave support |
comb_b_actual |
Superior corner of first leave support |
vector with the intersection. NULL if intersection
is not valid.
This function evaluates a node and checks if it fulfills the conditions to be a final node.
isFinalNode(obs, data, numStop)isFinalNode(obs, data, numStop)
obs |
Observation in the evaluated node. |
data |
Data with predictive variable. |
numStop |
Minimum number of observations in a node to be split. |
True if the node is a final node and false in any other case.
Create an adapted version of Multivariate Adaptive Regression Splines (MARS) model to estimate a production frontier satisfying some classical production theory axioms, such as monotonicity and concavity.
MARSAdapted( data, x, y, nterms, Kp = 1, d = 2, err_red = 0.01, minspan = 0, endspan = 0, linpreds = FALSE, na.rm = TRUE )MARSAdapted( data, x, y, nterms, Kp = 1, d = 2, err_red = 0.01, minspan = 0, endspan = 0, linpreds = FALSE, na.rm = TRUE )
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
nterms |
Maximum number of reflected pairs created by the forward algorithm of MARS. |
Kp |
Maximum degree of interaction allowed. Default is |
d |
Generalized Cross Validation (GCV) penalty per knot. Default is
|
err_red |
Minimum reduced error rate for the addition of two new basis
functions. Default is |
minspan |
Minimum number of observations between knots. When
|
endspan |
Minimum number of observations before the first and after the
final knot. When |
linpreds |
|
na.rm |
|
An AdaptedMARS object.
This function saves information about the adapted Multivariate Adaptive Frontier Splines model.
MARSAdapted_object( data, x, y, rownames, nterms, Kp, d, err_red, minspan, endspan, na.rm, MARS.Forward, MARS.Forward.Smooth )MARSAdapted_object( data, x, y, rownames, nterms, Kp, d, err_red, minspan, endspan, na.rm, MARS.Forward, MARS.Forward.Smooth )
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
rownames |
|
nterms |
Maximum number of terms created by the forward algorithm . |
Kp |
Maximum degree of interaction allowed. Default is |
d |
Generalized Cross Validation (GCV) penalty per knot. Default is
|
err_red |
Minimum reduced error rate for the addition of two new basis
functions. Default is |
minspan |
Minimum number of observations between knots. When
|
endspan |
Minimum number of observations before the first and after the
final knot. When |
na.rm |
|
MARS.Forward |
The Multivariate Adaptive Frontier Splines model after applying the forward algorithm without the smoothing procedures |
MARS.Forward.Smooth |
The Multivariate Adaptive Frontier Splines model after applying the forward algorithm after applying the smoothing procedure |
A MARSAdapted object.
This function smoothes the Forward MARS predictor.
MARSAdaptedSmooth(data, nX, knots, y)MARSAdaptedSmooth(data, nX, knots, y)
data |
|
nX |
number of inputs in |
knots |
|
y |
output indexes in |
List containing the set of knots from backward (knots),
the new cubic knots (cubic_knots) and the set of coefficients
(alpha).
This function estimates a production frontier satisfying some classical production theory axioms, such as monotonicity and concavity, which is based upon the adaptation of the machine learning technique known as LS-boosting using adapted Multivariate Adaptive Regression Splines (MARS) as base learners.
This function saves information about the LS-Boosted Multivariate Adaptive Frontier Splines model.
MARSBoost(data, x, y, num.iterations, num.terms, learning.rate) MARSBoost_object( data, x, y, num.iterations, learning.rate, num.terms, MARS.models, f0, prediction, prediction.smooth )MARSBoost(data, x, y, num.iterations, num.terms, learning.rate) MARSBoost_object( data, x, y, num.iterations, learning.rate, num.terms, MARS.models, f0, prediction, prediction.smooth )
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
num.iterations |
Maximum number of iterations the algorithm will perform |
num.terms |
Maximum number of reflected pairs created by the forward algorithm of MARS. |
learning.rate |
Learning rate that control overfitting of the algorithm. Value must be in (0,1] |
MARS.models |
List of the adapted forward MARS models created in each iterations |
f0 |
Initial predictions of the model (they correspond to maximum value of each output variable) |
prediction |
Final predictions of the original data without applying the smoothing procedure |
prediction.smooth |
Final predictions of the original data after applying the smoothing procedure |
A MARSBoost object.
A MARSBoost object.
This function computes the mean squared error between two numeric vectors.
mse(y, yPred)mse(y, yPred)
y |
Vector of actual data. |
yPred |
Vector of predicted values. |
Mean Squared Error.
This function calculates the Mean Square Error between the predicted value and the observations in a given node.
mse_tree(data, t, y)mse_tree(data, t, y)
data |
Data to be used. |
t |
A given node. |
y |
Column output indexes in data. |
Mean Square Error at a node.
This function finds the node where a register is located.
posIdNode(tree, idNode)posIdNode(tree, idNode)
tree |
A list containing EAT nodes. |
idNode |
Id of a specific node. |
Position of the node or -1 if it is not found.
This function predicts the expected output by a DEA
object.
## S3 method for class 'DEA' predict(object, newdata, x, y, ...)## S3 method for class 'DEA' predict(object, newdata, x, y, ...)
object |
A |
newdata |
|
x |
Inputs index. |
y |
Outputs index. |
... |
further arguments passed to or from other methods. |
data.frame with the predicted values. Valid measures are:
rad.out.
This function predicts the expected output by an EAT object.
## S3 method for class 'EAT' predict(object, newdata, x, ...)## S3 method for class 'EAT' predict(object, newdata, x, ...)
object |
An |
newdata |
|
x |
Inputs index. |
... |
further arguments passed to or from other methods. |
data.frame with the predicted values.
This function predicts the expected output by a EATBoost
object.
## S3 method for class 'EATBoost' predict(object, newdata, x, ...)## S3 method for class 'EATBoost' predict(object, newdata, x, ...)
object |
A |
newdata |
|
x |
Inputs index. |
... |
further arguments passed to or from other methods. |
data.frame with the predicted values.
This function predicts the expected output by a FDH
object.
## S3 method for class 'FDH' predict(object, newdata, x, y, ...)## S3 method for class 'FDH' predict(object, newdata, x, y, ...)
object |
A |
newdata |
|
x |
Inputs index. |
y |
Outputs index. |
... |
further arguments passed to or from other methods. |
data.frame with the predicted values. Valid measures are:
rad.out.
This function predicts the expected output by a MARS
object.
## S3 method for class 'MARSAdapted' predict(object, newdata, x, class = 1, ...)## S3 method for class 'MARSAdapted' predict(object, newdata, x, class = 1, ...)
object |
A |
newdata |
|
x |
Inputs index. |
class |
Model for prediction. |
... |
further arguments passed to or from other methods. |
data.frame with the predicted values.
This function predicts the expected output by a MARSBoost
object.
## S3 method for class 'MARSBoost' predict(object, newdata, x, class = 1, ...)## S3 method for class 'MARSBoost' predict(object, newdata, x, class = 1, ...)
object |
A |
newdata |
|
x |
Inputs index. |
class |
Model for prediction. |
... |
further arguments passed to or from other methods. |
data.frame with the predicted values.
This function predicts the expected value based on a set of inputs.
predictor(tree, register)predictor(tree, register)
tree |
|
register |
Set of independent values. |
The expected value of the dependent variable based on the given register.
This function arranges the data in the required format and displays error messages.
preProcess(data, x, y, na.rm = TRUE)preProcess(data, x, y, na.rm = TRUE)
data |
|
x |
Column input indexes in |
y |
Column output indexes in |
na.rm |
|
It returns a data.frame in the required format.
This function predicts the expected output through a DEA model.
Russell_in( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )Russell_in( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
matrix with the the predicted score
This function predicts the expected output through a DEA model.
Russell_out( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )Russell_out( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE )
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
matrix with the the predicted score
This function gets the variable and split value to be used in estimEAT, selects the best split and updates VarInfo, node indexes and leaves list.
split(data, tree, leaves, t, x, y, numStop)split(data, tree, leaves, t, x, y, numStop)
data |
Data to be used. |
tree |
List structure with the tree nodes. |
leaves |
List with leaf nodes or pending expansion nodes. |
t |
Node which is being split. |
x |
Column input indexes in data. |
y |
Column output indexes in data. |
numStop |
Minimum number of observations in a node to be split. |
Leaves and tree lists updated with the new child nodes.
This function predicts the expected output through a DEA model.
WAM( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE, weights )WAM( data, x, y, dataOriginal = data, xOriginal = x, yOriginal = y, FDH = FALSE, weights )
data |
|
x |
Vector. Column input indexes in data. |
y |
Vector. Column output indexes in data. |
dataOriginal |
|
xOriginal |
Vector. Column input indexes in original data. |
yOriginal |
Vector. Column output indexes in original data. |
FDH |
Binary decision variables |
weights |
Weights. Valid values are: |
matrix with the the predicted score