library(boostingDEA)
set.seed(1234)This vignette intends to explain the main functions of the
boostingDEA package. In it, techniques from the field of
machine learning are incorporated into solving problems
in the production theory context. Specifically, two
adaptations of the Gradient
Boosting technique are introduced: EATBoost and
MARSBoost. Gradient boosting is one of the variants of
ensemble methods where multiple weak models are created and combined to
get better performance as a whole. As a consequence, Gradient Boosting
gives a prediction model in the form of an ensemble of weak prediction
models. Specifically, at each step, a new weak model is trained to
predict the “error” of the current strong model. In this package, whilst
EATBoost uses an adaptation of regression trees known
as Efficiency
Analysis Trees as weak model, MARSBoost uses an
adaptation of Multivariate
Adaptive Regression Spline.
As previously said, we are dealing with a production theory context. Let us consider \(n\) Decision Making Units (DMUs) to be evaluated. \(DMU_i\) consumes \(\textbf{x}_i = (x_{1i}, ...,x_{mi}) \in R^{m}_{+}\) amount of inputs for the production of \(\textbf{y}_i = (y_{1i}, ...,y_{si}) \in R^{s}_{+}\) amount of outputs. The relative efficiency of each DMU in the sample is assessed regarding the so-called production possibility set or technology, which is the set of technically feasible combinations of \((\textbf{x, y})\). It is defined in general terms as:
\[\begin{equation} \Psi = \{(\textbf{x, y}) \in R^{m+s}_{+}: \textbf{x} \text{ can produce } \textbf{y}\} \end{equation}\]
Monotonicity (free disposability) of inputs and outputs is assumed, meaning that if \((\textbf{x, y}) \in \Psi\), then \((\textbf{x', y'}) \in \Psi\), as soon as \(\textbf{x'} \geq \textbf{x}\) and \(\textbf{y'} \leq \textbf{y}\). Often convexity of \(\Psi\) is also assumed. The efficient frontier of \(\Psi\) may be defined as \(\partial(\boldsymbol{\Psi}) := \{(\boldsymbol{x,y}) \in \boldsymbol{\Psi}: \boldsymbol{\hat{x}} < \boldsymbol{x}, \boldsymbol{\hat{y}} > \boldsymbol{y} \Rightarrow (\boldsymbol{\hat{x},\hat{y}}) \notin \boldsymbol{\Psi} \}\).
banks’ databaseThe banks database is included as a data object in the
boostingDEA library and is employed to exemplify the
package functions. The data corresponds to 31 Taiwanese banks for the
year 2010. The dataset was first obtained by Juo
et. al, 2015 from the “Condition and Performance of Domestic Banks”
published by the Central Bank of China (Taiwan) and the Taiwan Economic
Journal for the year 2010.
The following variables are collected for all banks:
Inputs :
Outputs :
Revenue can be interpreted as a combination of the
Financial.investments and Loans variables, and
can be used as the target variable for a mono-output scenario, while
Financial.investments and Loans for a
multi-output scenario.
data(banks)
banks
#> Financial.funds Labor Physical.capital
#> Export-Import Bank 25019 202 505
#> Bank of Taiwan 3171493 7951 76576
#> Taipei Fubon Bank 1222499 6434 12082
#> Bank of Kaohsiung 189169 914 2237
#> Land Bank 1846028 5732 22634
#> Cooperative Bank 2220071 8835 33638
#> First Bank 1602733 7048 22843
#> Hua Nan Bank 1595039 7126 24907
#> Chang Hwa Bank 1267731 6428 23778
#> Mega Bank 1589474 5033 13568
#> Cathay United Bank 1349708 6062 25285
#> The Shanghai Bank 577127 2265 10190
#> Union Bank 295386 2975 8051
#> Far Eastern Bank 344499 2378 2855
#> E. Sun Bank 936612 4583 14195
#> Cosmos Bank 109728 1685 6162
#> Taishin Bank 741883 6236 17278
#> Ta Chong Bank 322836 3215 3208
#> Jih Sun Bank 182228 1529 4238
#> Entie Bank 298330 1945 2114
#> China Trust Bank 1335080 9538 32436
#> Sunny Bank 213037 1790 9116
#> Bank of Panhsin 143268 1270 7416
#> Taiwan Business Bank 1035800 5010 14185
#> Taichung Bank 313451 1829 3243
#> China Development 77242 567 1219
#> Hwatai Bank 105657 856 1667
#> Cota Bank 108685 1078 1113
#> Industrial Bank of Taiwan 76383 282 2605
#> Bank SinoPac 936418 4670 8721
#> Shin Kong Bank 428995 3146 7123
#> Finalcial.investments Loans Revenue
#> Export-Import Bank 3125 81996 1056
#> Bank of Taiwan 904580 2091100 41007
#> Taipei Fubon Bank 392491 866282 19402
#> Bank of Kaohsiung 16740 163054 2957
#> Land Bank 227086 1706964 31506
#> Cooperative Bank 502569 1799753 35510
#> First Bank 507630 1260072 27084
#> Hua Nan Bank 393020 1256618 25668
#> Chang Hwa Bank 257105 1060005 21638
#> Mega Bank 344886 1332661 29489
#> Cathay United Bank 162490 891448 21904
#> The Shanghai Bank 229432 392894 9215
#> Union Bank 13865 191209 8708
#> Far Eastern Bank 290831 239958 6616
#> E. Sun Bank 357497 602776 16911
#> Cosmos Bank 1681 70222 5251
#> Taishin Bank 203118 539549 16319
#> Ta Chong Bank 109832 262868 7191
#> Jih Sun Bank 26690 129907 3219
#> Entie Bank 31500 197734 5410
#> China Trust Bank 506711 949894 29185
#> Sunny Bank 5532 173909 4128
#> Bank of Panhsin 2272 107411 2824
#> Taiwan Business Bank 157870 935304 18056
#> Taichung Bank 14528 247600 5770
#> China Development 114442 72326 1873
#> Hwatai Bank 3121 88922 2284
#> Cota Bank 3709 85595 2283
#> Industrial Bank of Taiwan 32293 66947 1577
#> Bank SinoPac 267021 700600 17922
#> Shin Kong Bank 17372 328574 8226Data envelopment analysis (DEA) is the standard nonparametric method for the estimation of production frontiers. In this context, the technology is calculated under assumptions of free disposability, convexity, deterministicness and minimal extrapolation.
The radial input DEA model can be computed using the
DEA(data,x,y) function. Furthermore, in the case of the
mono-output scenario, the ideal output value for the DMU in order to be
efficient is also calculated.
x <- 1:3
y <- 6
DEA_model <- DEA(banks,x,y)
pred_DEA <- predict(DEA_model, banks, x, y)
pred_DEA
#> Revenue_pred
#> 1 1056.000
#> 2 41007.000
#> 3 23610.840
#> 4 4267.654
#> 5 31506.000
#> 6 35510.000
#> 7 30296.522
#> 8 30266.830
#> 9 25833.058
#> 10 29489.000
#> 11 26756.522
#> 12 11868.767
#> 13 8708.000
#> 14 6741.728
#> 15 19558.692
#> 16 5251.000
#> 17 17265.890
#> 18 7191.000
#> 19 5284.268
#> 20 5410.000
#> 21 29185.000
#> 22 6663.290
#> 23 4733.154
#> 24 21162.123
#> 25 6669.596
#> 26 2194.405
#> 27 2868.251
#> 28 2557.894
#> 29 1577.000
#> 30 18187.708
#> 31 10258.301Similarly, FDH introduced also estimates production frontiers, but it is based upon only two axioms: free disposability and deterministicness. Therefore, it can be considered as the skeleton of DEA, since the convex hull of the DEA coincides with the DEA’s frontier.
In the same fashion, the radial input FDH model can be computed in R
using the FDH(data,x,y) function, where the ideal output in
the case of the mono-output case is calculated as well.
x <- 1:3
y <- 6
FDH_model <- FDH(banks,x,y)
pred_FDH <- predict(FDH_model, banks, x, y)
pred_FDH
#> Revenue_pred
#> 1 1056
#> 2 41007
#> 3 19402
#> 4 2957
#> 5 31506
#> 6 35510
#> 7 29489
#> 8 29489
#> 9 21638
#> 10 29489
#> 11 21904
#> 12 9215
#> 13 8708
#> 14 6616
#> 15 16911
#> 16 5251
#> 17 16319
#> 18 7191
#> 19 3219
#> 20 5410
#> 21 29185
#> 22 5251
#> 23 2824
#> 24 18056
#> 25 5770
#> 26 1873
#> 27 2284
#> 28 2283
#> 29 1577
#> 30 17922
#> 31 8226The EATBoost algorithm is an adaptation of the Gradient Tree Boosting algorithm to estimate production technologies. However, unlike the standard Gradient Tree Boosting algorithm which uses regression trees as base learners, the EATBoost technique uses EAT trees. Further modifications are also made to satisfy the required theoretical conditions. In particular, the algorithm was modified to deal with the axiom of free disposability in inputs and outputs and to provide estimates that envelop the data cloud from above. These two same postulates are also key in the definition of the standard FDH estimator of a technology. Therefore, this new approach shares similarities with the FDH methodology, but with the advantage that it avoids the typical problem of overfitting.
The EATBoost function receives as arguments the data
(data) containing the study variables, the indexes of the
predictor variables or inputs (x) and the indexes of the
predicted variables or outputs (y). Moreover, the
num.iterations, the learning.rate and
num.leaves are hyperparameters for the model and are
compulsory.
num.iterations: The maximum number of iterations the
algorithm will perform.
learning.rate: Learning rate. It controls the
overfitting of the algorithm. Value must be in \((0,1]\).
num.leaves: The maximum number of terminal leaves in
each tree at each iteration
The function returns an EATBoost object.
x <- 1:3
y <- 4:5
EATBoost_model <- EATBoost(banks, x, y,
num.iterations = 4,
num.leaves = 4,
learning.rate = 0.6)To try to find the best hyperparameters, we can resort to a grid of
parameters values tested through training and
test samples in a user specified proportion. In the
package, this can be done through the function
bestEATBoost. This function instead of receiving as
arguments a single value for each hyperparameter, receives a
vector, and evaluates each possible combination in the grid
through Mean Square Error (MSE) and Root Mean Square Error (RMSE).
Finally, it returns a data.frame with each possible
combination ordered by RMSE.
N <- nrow(banks)
x <- 1:3
y <- 4:5
selected <- sample(1:N, N * 0.8) # Training indexes
training <- banks[selected, ] # Training set
test <- banks[- selected, ] # Test set
grid_EATBoost <- bestEATBoost(training, test, x, y,
num.iterations = c(5,6,7),
learning.rate = c(0.4, 0.5, 0.6),
num.leaves = c(6,7,8),
verbose = FALSE)
head(grid_EATBoost)
#> num.iterations learning.rate num.leaves RMSE MSE
#> 1 5 0.6 7 175893.4 30938480461
#> 2 5 0.6 6 176614.1 31192532114
#> 3 5 0.5 8 177724.9 31586138341
#> 4 5 0.5 6 177761.0 31598963379
#> 5 5 0.5 7 177897.8 31647611711
#> 6 6 0.6 7 179046.0 32057468193EATboost_model_tuned <- EATBoost(banks, x, y,
num.iterations = grid_EATBoost[1, "num.iterations"],
learning.rate = grid_EATBoost[1, "learning.rate"],
num.leaves = grid_EATBoost[1, "num.leaves"])
pred_EATBoost <- predict(EATboost_model_tuned, banks, x)
pred_EATBoost
#> Finalcial.investments_pred Loans_pred
#> 1 12364.87 102624.5
#> 2 904580.00 2091100.0
#> 3 691348.93 1549138.7
#> 4 122533.01 193094.5
#> 5 822648.85 1882858.7
#> 6 822648.85 1882858.7
#> 7 691348.93 1549138.7
#> 8 691348.93 1549138.7
#> 9 691348.93 1549138.7
#> 10 691348.93 1549138.7
#> 11 691348.93 1549138.7
#> 12 294033.05 539098.2
#> 13 126639.74 223212.2
#> 14 298139.78 534092.6
#> 15 363099.13 714838.7
#> 16 122533.01 109424.3
#> 17 338139.38 649284.0
#> 18 152306.78 281589.1
#> 19 122533.01 152398.3
#> 20 122533.01 218082.2
#> 21 691348.93 1549138.7
#> 22 122533.01 196109.7
#> 23 122533.01 134015.3
#> 24 691348.93 1549138.7
#> 25 129866.45 267298.2
#> 26 122533.01 109424.3
#> 27 122533.01 109424.3
#> 28 50061.72 106765.5
#> 29 78804.67 104620.7
#> 30 363099.13 714838.7
#> 31 298139.78 544228.3The MARSBoost algorithm is an adaptation of the LS-Boosting algorithm to estimate production technologies. In this case, the base learner used in the algorithm is an adaptation of the MARS model. MARS essentially builds flexible models by fitting piecewise linear regressions; that is, the non-linearity of a model is approximated through the use of separate regression slopes in distinct intervals of the predictor variable space. The combinations of these models, which do not have a continuous first derivative, led to sharp trends. For this reason, a smoothing procedure can be applied. Thus, the estimator obtained without the smoothing procedure presents similarities with the one obtained by DEA, while the estimate in the second stage resembles more well-known (smoothed) functional forms typical of production theory; like Cobb-Douglas, CES or Translog.
The MARSBoost function works similarly to the
EATBoost one. It receives as arguments the data
(data) containing the study variables, the indexes of the
predictor variables or inputs (x), the indexes of the
predicted variables or outputs (y) and a set of
hyperparameters:
num.iterations: The maximum number of iterations the
algorithm will perform.
learning.rate: Learning rate. It controls the
overfitting of the algorithm. Value must be in \((0,1]\).
num.terms: The maximum number of reflected pairs in
each model at each iteration
The function returns an MARSBoost object and can be only
used in mono-ouput scenarios.
x <- 1:3
y <- 6
MARSBoost_model <- MARSBoost(banks, x, y,
num.iterations = 4,
learning.rate = 0.6,
num.terms = 4)In this case, to find the best hyperparameters, we can resort to the
bestMARSBoost function. Here, we can create a grid of
hyperparameters to find the optimal value for
num.iterations, learning.rate and
num.terms.
N <- nrow(banks)
x <- 1:3
y <- 6
selected <- sample(1:N, N * 0.8) # Training indexes
training <- banks[selected, ] # Training set
test <- banks[- selected, ] # Test set
grid_MARSBoost <- bestMARSBoost(training, test, x, y,
num.iterations = c(5,6,7),
learning.rate = c(0.4, 0.5, 0.6),
num.terms = c(6,7,8),
verbose = FALSE)
head(grid_MARSBoost)
#> num.iterations learning.rate num.terms RMSE MSE
#> 1 7 0.6 6 2115.814 4476669
#> 2 7 0.6 7 2115.814 4476669
#> 3 7 0.6 8 2115.814 4476669
#> 4 6 0.6 6 2162.278 4675447
#> 5 6 0.6 7 2162.278 4675447
#> 6 6 0.6 8 2162.278 4675447MARSBoost_model_tuned <- MARSBoost(banks, x, y,
num.iterations = grid_MARSBoost[1, "num.iterations"],
learning.rate = grid_MARSBoost[1, "learning.rate"],
num.terms = grid_MARSBoost[1, "num.terms"])
pred_MARSBoost <- predict(MARSBoost_model_tuned, banks, x)
pred_MARSBoost
#> Revenue_pred
#> 1 1121.456
#> 2 44258.838
#> 3 25826.754
#> 4 5023.932
#> 5 31877.200
#> 6 35519.006
#> 7 30324.119
#> 8 30377.385
#> 9 27174.489
#> 10 29507.871
#> 11 28484.046
#> 12 13802.012
#> 13 8845.106
#> 14 8594.639
#> 15 20638.981
#> 16 5309.583
#> 17 17591.556
#> 18 8431.431
#> 19 5922.635
#> 20 7511.006
#> 21 29204.369
#> 22 7280.988
#> 23 5654.287
#> 24 22442.894
#> 25 8082.041
#> 26 2510.994
#> 27 3350.320
#> 28 3408.779
#> 29 2681.269
#> 30 20388.933
#> 31 11175.128Technical inefficiency is defined as the distance from a point that belongs to \(\Psi\) to the production frontier \(\partial(\Psi)\). For a point located inside \(\Psi\), it is evident that there are many possible paths to the frontier, each associated with a different technical inefficiency measure.
The function efficiency calculates the efficiency score
corresponding to the given model using the given measure. A dataset
(data) and the corresponding indexes of input(s)
(x) and output(s) (y) must be entered. It is
recommended that the dataset with the DMUs whose efficiency is to be
calculated coincide with those used to estimate the frontier. The
possible argument of this function are:
model: Model object for which efficiency score is
computed. Valid objects are the ones returned from functions
DEA, FDH, EATBoost and
MARSBoost.
measure: Efficiency measure used. Valid values
are:
rad.out: Banker Charnes and Cooper output-oriented
radial modelrad.in: Banker Charnes and Cooper input-oriented radial
modelRussell.out: output-oriented Russell modelRussell.in: input-oriented Russell modelDDF: Directional Distance Function model. The
directional vector is specified in the argument
direction.vectorWAM: Weight Additive ModelsERG: Slacks-Based Measure, which is mathematically
equivalent to the Enhanced Russel Measureheuristic: Only used if the model is
EATBoost. This indicates whether the heuristic or the exact
approach is used. This heuristic approach might be needed due to the
extreme complexity at a computational level of the EATBoot exact
efficiency approach.
direction.vector: Only used when the
measure is DDF. Indicates the direction
vector. The valid values are:
dmu: \((x_0,
y_0)\)unit: unit vectormean: mean values of each variablex,y)weights: Only used when the measure is
WAM. Valid values are: MIP: Measure of
Inefficiency Proportions RAM: Range Adjusted Measure
BAM: Bounded Adjusted Measure
normalized: normalized weighted additive model
x,y)For this section, the previously created models are used.
Let’s first see an example using the standard DEA and FDH techniques. For both techniques, all measures can be calculated.
x <- 1:3
y <- 6
efficiency(DEA_model,
measure = "rad.in",
banks, x, y)
#> DEA.rad.in
#> Export-Import Bank 1.0000000
#> Bank of Taiwan 1.0000000
#> Taipei Fubon Bank 0.8109065
#> Bank of Kaohsiung 0.6587345
#> Land Bank 1.0000000
#> Cooperative Bank 1.0000000
#> First Bank 0.8568164
#> Hua Nan Bank 0.8092421
#> Chang Hwa Bank 0.8212723
#> Mega Bank 1.0000000
#> Cathay United Bank 0.8010874
#> The Shanghai Bank 0.7700111
#> Union Bank 1.0000000
#> Far Eastern Bank 0.9807311
#> E. Sun Bank 0.8459681
#> Cosmos Bank 1.0000000
#> Taishin Bank 0.9350929
#> Ta Chong Bank 1.0000000
#> Jih Sun Bank 0.5748014
#> Entie Bank 1.0000000
#> China Trust Bank 1.0000000
#> Sunny Bank 0.5996823
#> Bank of Panhsin 0.5664026
#> Taiwan Business Bank 0.8326728
#> Taichung Bank 0.8558921
#> China Development 0.8181399
#> Hwatai Bank 0.7606673
#> Cota Bank 0.8866054
#> Industrial Bank of Taiwan 1.0000000
#> Bank SinoPac 0.9842813
#> Shin Kong Bank 0.7816094efficiency(FDH_model,
measure = "WAM",
weights = "RAM",
banks, x, y)
#> FDH.WAM
#> Export-Import Bank 0.00000000
#> Bank of Taiwan 0.00000000
#> Taipei Fubon Bank 0.00000000
#> Bank of Kaohsiung 0.00000000
#> Land Bank 0.00000000
#> Cooperative Bank 0.00000000
#> First Bank 0.10054236
#> Hua Nan Bank 0.11766372
#> Chang Hwa Bank 0.00000000
#> Mega Bank 0.00000000
#> Cathay United Bank 0.00000000
#> The Shanghai Bank 0.00000000
#> Union Bank 0.00000000
#> Far Eastern Bank 0.00000000
#> E. Sun Bank 0.00000000
#> Cosmos Bank 0.00000000
#> Taishin Bank 0.00000000
#> Ta Chong Bank 0.00000000
#> Jih Sun Bank 0.00000000
#> Entie Bank 0.00000000
#> China Trust Bank 0.00000000
#> Sunny Bank 0.02775541
#> Bank of Panhsin 0.00000000
#> Taiwan Business Bank 0.00000000
#> Taichung Bank 0.00000000
#> China Development 0.00000000
#> Hwatai Bank 0.00000000
#> Cota Bank 0.00000000
#> Industrial Bank of Taiwan 0.00000000
#> Bank SinoPac 0.00000000
#> Shin Kong Bank 0.00000000In the case of the EATBoost algorithm, all measures can be calculated as well.
x <- 1:3
y <- 4:5
efficiency(EATboost_model_tuned,
measure = "Russell.out",
heuristic = FALSE,
banks, x, y)
#> Calculating EATBoost Russell.out efficiency measure. This migth take a while...
#>
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#> EATBoost.Russell.out
#> Export-Import Bank 2.604169
#> Bank of Taiwan 1.000000
#> Taipei Fubon Bank 1.774850
#> Bank of Kaohsiung 4.252005
#> Land Bank 2.362838
#> Cooperative Bank 1.341532
#> First Bank 1.295660
#> Hua Nan Bank 1.495926
#> Chang Hwa Bank 2.075210
#> Mega Bank 1.583506
#> Cathay United Bank 2.996247
#> The Shanghai Bank 1.326845
#> Union Bank 5.150572
#> Far Eastern Bank 1.625453
#> E. Sun Bank 1.100791
#> Cosmos Bank 37.225596
#> Taishin Bank 1.434063
#> Ta Chong Bank 1.228972
#> Jih Sun Bank 2.882052
#> Entie Bank 2.496422
#> China Trust Bank 1.497620
#> Sunny Bank 11.638757
#> Bank of Panhsin 27.589735
#> Taiwan Business Bank 3.017762
#> Taichung Bank 5.009301
#> China Development 1.291816
#> Hwatai Bank 20.245692
#> Cota Bank 7.372348
#> Industrial Bank of Taiwan 2.001521
#> Bank SinoPac 1.190069
#> Shin Kong Bank 9.409211However, due to the extreme complexity at a computational level of
the exact efficiency approach, the heuristic hyperparameter
can be specified to resort to the simpler less time-consuming heuristic
approach. In fact, heuristic is the default mode for
EATBoost.
efficiency(EATboost_model_tuned,
measure = "Russell.out",
banks, x, y,
heuristic = TRUE)
#> EATBoost.heu.Russell.out
#> Export-Import Bank 2.604169
#> Bank of Taiwan 1.000000
#> Taipei Fubon Bank 1.774850
#> Bank of Kaohsiung 4.252005
#> Land Bank 2.362838
#> Cooperative Bank 1.341532
#> First Bank 1.295660
#> Hua Nan Bank 1.495926
#> Chang Hwa Bank 2.075210
#> Mega Bank 1.583506
#> Cathay United Bank 2.996247
#> The Shanghai Bank 1.326845
#> Union Bank 5.150572
#> Far Eastern Bank 1.625453
#> E. Sun Bank 1.100791
#> Cosmos Bank 37.225596
#> Taishin Bank 1.434063
#> Ta Chong Bank 1.228972
#> Jih Sun Bank 2.882052
#> Entie Bank 2.496422
#> China Trust Bank 1.497620
#> Sunny Bank 11.638757
#> Bank of Panhsin 27.589735
#> Taiwan Business Bank 3.017762
#> Taichung Bank 5.009301
#> China Development 1.291816
#> Hwatai Bank 20.245692
#> Cota Bank 7.372348
#> Industrial Bank of Taiwan 2.001521
#> Bank SinoPac 1.190069
#> Shin Kong Bank 9.409211Finally, for the MARSBoost algorithm, only the radial output measure can be calculated.
efficiency(MARSBoost_model_tuned, "rad.out", banks, x, 6)
#> MARSBoost.rad.out
#> Export-Import Bank 1.061985
#> Bank of Taiwan 1.079300
#> Taipei Fubon Bank 1.331139
#> Bank of Kaohsiung 1.698996
#> Land Bank 1.011782
#> Cooperative Bank 1.000254
#> First Bank 1.119632
#> Hua Nan Bank 1.183473
#> Chang Hwa Bank 1.255869
#> Mega Bank 1.000640
#> Cathay United Bank 1.300404
#> The Shanghai Bank 1.497777
#> Union Bank 1.015745
#> Far Eastern Bank 1.299069
#> E. Sun Bank 1.220447
#> Cosmos Bank 1.011156
#> Taishin Bank 1.077980
#> Ta Chong Bank 1.172498
#> Jih Sun Bank 1.839899
#> Entie Bank 1.388356
#> China Trust Bank 1.000664
#> Sunny Bank 1.763805
#> Bank of Panhsin 2.002226
#> Taiwan Business Bank 1.242960
#> Taichung Bank 1.400700
#> China Development 1.340627
#> Hwatai Bank 1.466865
#> Cota Bank 1.493114
#> Industrial Bank of Taiwan 1.700234
#> Bank SinoPac 1.137648
#> Shin Kong Bank 1.358513