Description
Efficiency Analysis Trees.
Description
Functions are provided to determine production frontiers and technical efficiency measures through non-parametric techniques based upon regression trees. The package includes code for estimating radial input, output, directional and additive measures, plotting graphical representations of the scores and the production frontiers by means of trees, and determining rankings of importance of input variables in the analysis. Additionally, an adaptation of Random Forest by a set of individual Efficiency Analysis Trees for estimating technical efficiency is also included. More details in: <doi:10.1016/j.eswa.2020.113783>.
README.md
eat
The EAT algorithm performs a regression tree based on CART methodology under a new approach that guarantees obtaining a frontier as estimator that fulfills the property of free disposability. This new technique has been baptized as Efficiency Analysis Trees. Some of its main functions are:
To create homogeneous groups of DMUs in terms of their inputs and to know for each of these groups, what is the maximum expected output.
To know which DMUs exercise best practices and which of them do not obtain a performance according to their resources level.
To know what variables are more relevant in obtaining efficient levels of output.
Installation
You can install the released version of eat from CRAN with:
install.packages("eat")
And the development version from GitHub with:
devtools::install_github("MiriamEsteve/EAT")
Example
library(eat)
data("PISAindex")
- EAT model with 1 input (
NBMC) and 1 output (S_PISA)
single_model <- EAT(data = PISAindex,
x = 15, # input
y = 3) # output
#> [conflicted] Will prefer dplyr::filter over any other package
- Print an
EATobject
print(single_model)
#> [1] y: [ 551 ] || R: 11507.5 n(t): 72
#>
#> | [2] PFC < 77.2 --> y: [ 478 ] || R: 2324.47 n(t): 34
#>
#> | | [4] PFC < 65.45 --> y: [ 428 ] <*> || R: 390.17 n(t): 16
#>
#> | | [5] PFC >= 65.45 --> y: [ 478 ] <*> || R: 637.08 n(t): 18
#>
#> | [3] PFC >= 77.2 --> y: [ 551 ] <*> || R: 2452.83 n(t): 38
#>
#> <*> is a leaf node
- Summary of an
EATobject
summary(single_model)
#>
#> Formula: S_PISA ~ PFC
#>
#> # ========================== #
#> # Summary for leaf nodes #
#> # ========================== #
#>
#> id n(t) % S_PISA R(t)
#> 3 38 53 551 2452.83
#> 4 16 22 428 390.17
#> 5 18 25 478 637.08
#>
#> # ========================== #
#> # Tree #
#> # ========================== #
#>
#> Interior nodes: 2
#> Leaf nodes: 3
#> Total nodes: 5
#>
#> R(T): 3480.08
#> numStop: 5
#> fold: 5
#> max.depth:
#> max.leaves:
#>
#> # ========================== #
#> # Primary & surrogate splits #
#> # ========================== #
#>
#> Node 1 --> {2,3} || PFC --> {R: 4777.31, s: 77.2}
#>
#> Node 2 --> {4,5} || PFC --> {R: 1027.25, s: 65.45}
- Number of leaf nodes of an
EATobject
size(single_model)
#> The number of leaf nodes of the EAT model is: 3
- Frontier levels of output for an
EATobject
frontier.levels(single_model)
#> The frontier levels of the outputs at the leaf nodes are:
#> S_PISA
#> 1 551
#> 2 428
#> 3 478
- Descriptive analysis for an
EATobject
descriptiveEAT <- descrEAT(single_model)
descriptiveEAT
#> Node n(t) % mean var sd min Q1 median Q3 max RMSE
#> 1 1 72 100 455.06 2334.59 48.32 336 416.75 466.0 495.25 551 107.27
#> 2 2 34 47 416.88 1223.02 34.97 336 397.25 415.5 435.75 478 70.16
#> 3 3 38 53 489.21 851.95 29.19 419 478.00 494.0 504.50 551 68.17
#> 4 4 16 22 394.62 684.65 26.17 336 381.50 398.0 414.00 428 41.90
#> 5 5 18 25 436.67 889.29 29.82 386 415.25 433.5 468.00 478 50.48
- Plot the frontier
frontier(object = single_model,
FDH = TRUE,
observed.data = TRUE,
rwn = TRUE)
#> Warning: ggrepel: 8 unlabeled data points (too many overlaps). Consider
#> increasing max.overlaps
- EAT model with 13 inputs and 3 outputs
multioutput <- EAT(data = PISAindex,
x = 6:18,
y = 3:5)
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> Warning in preProcess(data = data, x = x, y = y, numStop = numStop, fold = fold, : Rows with NA values have been omitted .
- Ranking of importance of variables for EAT
rankingEAT(object = multioutput,
barplot = TRUE,
threshold = 70,
digits = 2)
#> $scores
#> Importance
#> AAE 100.00
#> WS 98.45
#> S 84.51
#> NBMC 83.37
#> HW 83.31
#> ABK 67.97
#> GDP_PPP 65.37
#> AIC 64.89
#> EQ 57.11
#> PR 57.05
#> I 57.05
#> PS 45.41
#> PFC 31.67
#>
#> $barplot
- Plot an EAT model
plotEAT(object = multioutput)
- Tuning an EAT model
n <- nrow(PISAindex) # Observations in the dataset
t_index <- sample(1:n, n * 0.7) # Training indexes
training <- PISAindex[t_index, ] # Training set
test <- PISAindex[-t_index, ] # Test set
bestEAT(training = training,
test = test,
x = 6:18,
y = 3:5,
numStop = c(5, 7, 10),
fold = c(5, 7))
#> Warning in preProcess(test, x, y, na.rm = na.rm): Rows with NA values have been omitted .
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> numStop fold RMSE leaves
#> 1 7 5 66.94 10
#> 2 7 7 66.94 10
#> 3 5 7 71.87 8
#> 4 5 5 84.60 7
#> 5 10 5 85.06 5
#> 6 10 7 85.06 5
- Efficiency scores EAT
single_model <- EAT(data = PISAindex,
x = 15, # input
y = 3) # output
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
scores_EAT <- efficiencyEAT(data = PISAindex,
x = 15,
y = 3,
object = single_model,
scores_model = "BCC.OUT",
digits = 3,
FDH = TRUE)
#> EAT_BCC_OUT FDH_BCC_OUT
#> SGP 1.000 1.000
#> JPN 1.042 1.000
#> KOR 1.062 1.000
#> EST 1.040 1.000
#> NLD 1.095 1.095
#> POL 1.078 1.000
#> CHE 1.113 1.113
#> CAN 1.064 1.064
#> DNK 1.118 1.118
#> SVN 1.087 1.024
#> BEL 1.104 1.062
#> FIN 1.056 1.056
#> SWE 1.104 1.104
#> GBR 1.091 1.091
#> NOR 1.124 1.124
#> DEU 1.095 1.095
#> IRL 1.111 1.069
#> AUT 1.124 1.082
#> CZE 1.109 1.044
#> LVA 1.131 1.066
#> FRA 1.118 1.075
#> ISL 1.160 1.116
#> NZL 1.085 1.043
#> PRT 1.120 1.055
#> AUS 1.095 1.054
#> RUS 1.000 1.000
#> ITA 1.021 1.021
#> SVK 1.187 1.037
#> LUX 1.155 1.155
#> HUN 1.146 1.000
#> LTU 1.143 1.060
#> ESP 1.141 1.075
#> USA 1.098 1.056
#> BLR 1.015 1.015
#> MLT 1.193 1.106
#> HRV 1.006 1.006
#> ISR 1.193 1.106
#> TUR 1.021 1.000
#> UKR 1.019 1.000
#> CYP 1.255 1.182
#> GRC 1.058 1.058
#> SRB 1.086 1.000
#> MYS 1.091 1.068
#> ALB 1.026 1.000
#> BGR 1.127 1.127
#> ARE 1.270 1.177
#> MNE 1.152 1.128
#> ROU 1.122 1.122
#> KAZ 1.204 1.179
#> MDA 1.000 1.000
#> AZE 1.075 1.048
#> THA 1.005 1.005
#> URY 1.293 1.200
#> CHL 1.241 1.169
#> QAT 1.315 1.239
#> MEX 1.021 1.021
#> BIH 1.075 1.048
#> CRI 1.149 1.149
#> JOR 1.114 1.093
#> PER 1.059 1.032
#> GEO 1.117 1.089
#> MKD 1.036 1.036
#> LBN 1.115 1.115
#> COL 1.036 1.036
#> BRA 1.183 1.158
#> ARG 1.183 1.158
#> IDN 1.081 1.081
#> SAU 1.238 1.215
#> MAR 1.135 1.135
#> PAN 1.173 1.173
#> PHL 1.199 1.168
#> DOM 1.274 1.241
#>
#> Model Mean Std. Dev. Min Q1 Median Q3 Max
#> EAT 1.114 0.074 1 1.061 1.110 1.110 1.315
#> FDH 1.081 0.065 1 1.030 1.069 1.069 1.241
- Efficiency scores CEAT
scores_CEAT <- efficiencyCEAT(data = PISAindex,
x = 15,
y = 3,
object = single_model,
scores_model = "BCC.INP",
digits = 3,
DEA = TRUE)
#> CEAT_BCC_INP DEA_BCC_INP
#> SGP 0.878 1.000
#> JPN 0.872 0.986
#> KOR 0.878 0.989
#> EST 0.857 0.969
#> NLD 0.736 0.824
#> POL 0.862 0.968
#> CHE 0.697 0.777
#> CAN 0.768 0.865
#> DNK 0.693 0.772
#> SVN 0.821 0.920
#> BEL 0.735 0.821
#> FIN 0.787 0.888
#> SWE 0.724 0.809
#> GBR 0.750 0.840
#> NOR 0.680 0.757
#> DEU 0.723 0.809
#> IRL 0.731 0.816
#> AUT 0.712 0.792
#> CZE 0.788 0.880
#> LVA 0.758 0.843
#> FRA 0.725 0.808
#> ISL 0.669 0.739
#> NZL 0.769 0.862
#> PRT 0.776 0.865
#> AUS 0.754 0.845
#> RUS 0.846 0.936
#> ITA 0.756 0.832
#> SVK 0.717 0.787
#> LUX 0.660 0.729
#> HUN 0.779 0.864
#> LTU 0.768 0.851
#> ESP 0.755 0.838
#> USA 0.756 0.846
#> BLR 0.750 0.826
#> MLT 0.703 0.771
#> HRV 0.792 0.875
#> ISR 0.703 0.771
#> TUR 0.866 0.953
#> UKR 0.831 0.916
#> CYP 0.628 0.678
#> GRC 0.754 0.822
#> SRB 0.767 0.829
#> MYS 0.734 0.792
#> ALB 1.000 1.000
#> BGR 0.661 0.691
#> ARE 0.616 0.663
#> MNE 0.698 0.698
#> ROU 0.662 0.701
#> KAZ 0.696 0.696
#> MDA 0.771 0.825
#> AZE 0.967 0.967
#> THA 0.744 0.787
#> URY 0.602 0.637
#> CHL 0.638 0.692
#> QAT 0.591 0.599
#> MEX 0.746 0.755
#> BIH 0.782 0.782
#> CRI 0.615 0.615
#> JOR 0.682 0.731
#> PER 0.795 0.795
#> GEO 0.798 0.798
#> MKD 0.768 0.768
#> LBN 0.735 0.735
#> COL 0.739 0.739
#> BRA 0.697 0.697
#> ARG 0.693 0.693
#> IDN 0.735 0.735
#> SAU 0.674 0.674
#> MAR 0.748 0.748
#> PAN 0.770 0.770
#> PHL 0.780 0.780
#> DOM 0.804 0.804
#>
#> Model Mean Std. Dev. Min Q1 Median Q3 Max
#> CEAT 0.749 0.077 0.591 0.698 0.749 0.749 1
#> DEA 0.805 0.094 0.599 0.739 0.801 0.801 1
- Efficiency jitter plot
efficiencyJitter(object = single_model,
df_scores = scores_EAT$EAT_BCC_OUT,
scores_model = "BCC.OUT",
lwb = 1.2)
- Efficiency density plot
efficiencyDensity(df_scores = scores_EAT[, 3:4],
model = c("EAT", "FDH"))
- RFEAT model
forest <- RFEAT(data = PISAindex,
x = 6:18, # input
y = 5, # output
numStop = 5,
m = 30,
s_mtry = "BRM",
na.rm = TRUE)
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
- Print a
RFEATobject
print(forest)
#>
#> Formula: M_PISA ~ NBMC + WS + S + PS + ABK + AIC + HW + EQ + PR + PFC + I + AAE + GDP_PPP
#>
#> # ========================== #
#> # Forest #
#> # ========================== #
#>
#> Error: 738.42
#> numStop: 5
#> No. of trees (m): 30
#> No. of inputs tried (s_mtry): BRM
- Plot the Out-of-Bag error for a forest made up of k trees
plotRFEAT(forest)
- RFEAT ranking
rankingRFEAT(object = forest, barplot = TRUE,
digits = 2)
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> $scores
#> Importance
#> PS 14.25
#> PR 14.08
#> AAE 13.97
#> EQ 11.86
#> S 10.80
#> HW 9.36
#> AIC 6.36
#> I 4.49
#> NBMC 3.26
#> WS -1.79
#> GDP_PPP -4.68
#> PFC -4.77
#> ABK -6.11
#>
#> $barplot
- Tuning a RFEAT model
bestRFEAT(training = training,
test = test,
x = 6:18,
y = 3:5,
numStop = c(5, 10),
m = c(30, 40),
s_mtry = c("BRM", "3"))
#> Warning in preProcess(test, x, y, na.rm = na.rm): Rows with NA values have been omitted .
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> numStop m s_mtry RMSE
#> 1 5 40 3 57.44
#> 2 5 40 BRM 57.72
#> 3 5 30 BRM 58.39
#> 4 5 30 3 59.13
#> 5 10 30 BRM 62.43
#> 6 10 40 BRM 63.18
#> 7 10 40 3 65.02
#> 8 10 30 3 68.43
- RFEAT scores
efficiencyRFEAT(data = PISAindex,
x = 6:18, # input
y = 5, # output
object = forest,
FDH = TRUE)
#> RFEAT_BCC_OUT FDH_BCC_OUT
#> SGP 0.936 1.000
#> JPN 1.024 1.000
#> KOR 1.004 1.000
#> EST 0.982 1.000
#> NLD 0.999 1.000
#> POL 0.982 1.000
#> CHE 1.026 1.002
#> CAN 1.010 1.008
#> DNK 1.017 1.014
#> SVN 1.004 1.000
#> BEL 1.006 1.000
#> FIN 1.021 1.018
#> SWE 1.029 1.028
#> GBR 1.019 1.000
#> NOR 1.039 1.030
#> DEU 1.028 1.032
#> IRL 1.031 1.032
#> AUT 1.024 1.034
#> CZE 1.014 1.000
#> LVA 0.997 1.000
#> FRA 1.037 1.000
#> ISL 1.070 1.042
#> NZL 1.048 1.045
#> PRT 0.995 1.000
#> AUS 1.054 1.051
#> RUS 0.963 1.000
#> ITA 1.011 1.000
#> SVK 0.997 1.000
#> LUX 1.053 1.000
#> HUN 1.008 1.000
#> LTU 1.020 1.000
#> ESP 1.030 1.000
#> USA 1.033 1.000
#> BLR 0.992 1.000
#> MLT 1.026 1.000
#> HRV 1.034 1.000
#> ISR 1.050 1.000
#> TUR 0.979 1.000
#> UKR 0.981 1.000
#> CYP 1.095 1.000
#> GRC 1.063 1.007
#> SRB 1.002 1.000
#> MYS 0.998 1.000
#> ALB 0.982 1.000
#> BGR 1.031 1.000
#> ARE 1.014 1.000
#> MNE 1.022 1.000
#> ROU 1.031 1.000
#> KAZ 1.018 1.000
#> MDA 1.003 1.000
#> AZE 0.972 1.000
#> THA 0.991 1.000
#> URY 1.045 1.000
#> CHL 1.097 1.005
#> QAT 1.070 1.000
#> MEX 1.005 1.000
#> BIH 1.042 1.000
#> CRI 1.085 1.000
#> JOR 1.046 1.000
#> PER 0.997 1.000
#> GEO 1.060 1.000
#> MKD 1.052 1.000
#> LBN 1.037 1.000
#> COL 1.055 1.000
#> BRA 1.074 1.000
#> ARG 1.157 1.000
#> IDN 1.034 1.000
#> SAU 1.100 1.000
#> MAR 1.031 1.000
#> PAN 1.116 1.000
#> PHL 1.049 1.000
#> DOM 1.145 1.000
#>
#> Model Mean Std. Dev. Min Q1 Median Q3 Max
#> RFEAT 1.029 0.039 0.936 1.003 1.026 1.026 1.157
#> FDH 1.005 0.012 1.000 1.000 1.000 1.000 1.051
- EAT and RFEAT predictions
input <- c(6, 7, 8, 12, 17)
output <- 3:5
EAT_model <- EAT(data = PISAindex, x = input, y = output)
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> Warning in preProcess(data = data, x = x, y = y, numStop = numStop, fold = fold, : Rows with NA values have been omitted .
RFEAT_model <- RFEAT(data = PISAindex, x = input, y = output)
#> [conflicted] Removing existing preference
#> [conflicted] Will prefer dplyr::filter over any other package
#> Warning in preProcess(data = data, x = x, y = y, numStop = numStop, na.rm = na.rm): Rows with NA values have been omitted .
# PREDICTIONS
predictions_EAT <- predict(object = EAT_model, newdata = PISAindex[, input])
predictions_RFEAT <- predict(object = RFEAT_model, newdata = PISAindex[, input])
Please, check the vignette for more details.