Composite Indicators - Compind
Description
Compind package contains functions to enhance several approaches to the Composite Indicators (CIs) methods, focusing, in particular, on the normalisation and weighting-aggregation steps.
Author(s)
Francesco Vidoli, Elisa Fusco Maintainer: Francesco Vidoli <fvidoli@gmail.com>
References
Daraio, C., Simar, L. (2005) "Introducing environmental variables in nonparametric frontier models: a probabilistic approach", Journal of productivity analysis, 24(1), 93-121.
Fusco E. (2015) "Enhancing non compensatory composite indicators: A directional proposal", European Journal of Operational Research, 242(2), 620-630.
Fusco E. (2023) "Potential improvements approach in composite indicators construction: the Multi-directional Benefit of the Doubt model", Socio-Economic Planning Sciences, vol. 85, 101447
Fusco, E., Liborio, M.P., Rabiei-Dastjerdi, H., Vidoli, F., Brunsdon, C. and Ekel, P.I. (2023), Harnessing Spatial Heterogeneity in Composite Indicators through the Ordered Geographically Weighted Averaging (OGWA) Operator. Geographical Analysis. https://doi.org/10.1111/gean.12384
R. Lahdelma, P. Salminen (2001) "SMAA-2: Stochastic multicriteria acceptability analysis for group decision making", Operations Research, 49(3), pp. 444-454
OECD (2008) "Handbook on constructing composite indicators: methodology and user guide".
Mazziotta C., Mazziotta M., Pareto A., Vidoli F. (2010) "La sintesi di indicatori territoriali di dotazione infrastrutturale: metodi di costruzione e procedure di ponderazione a confronto", Rivista di Economia e Statistica del territorio, n.1.
Melyn W. and Moesen W.W. (1991) "Towards a synthetic indicator of macroeconomic performance: unequal weighting when limited information is available", Public Economic research Paper 17, CES, KU Leuven.
Van Puyenbroeck T. and Rogge N. (2017) "Geometric mean quantity index numbers with Benefit-of-the-Doubt weights", European Journal of Operational Research, 256(3), 1004-1014.
Rogge N., de Jaeger S. and Lavigne C. (2017) "Waste Performance of NUTS 2-regions in the EU: A Conditional Directional Distance Benefit-of-the-Doubt Model", Ecological Economics, vol.139, pp. 19-32.
Simar L., Vanhems A. (2012) "Probabilistic characterization of directional distances and their robust versions", Journal of Econometrics, 166(2), 342-354.
UNESCO (1974)"Social indicators: problems of definition and of selection", Paris.
Vidoli F., Fusco E., Mazziotta C. (2015) "Non-compensability in composite indicators: a robust directional frontier method", Social Indicators Research, 122(3), 635-652.
Vidoli F., Mazziotta C. (2013) "Robust weighted composite indicators by means of frontier methods with an application to European infrastructure endowment", Statistica Applicata, Italian Journal of Applied Statistics.
Zanella A., Camanho A.S. and Dias T.G. (2015) "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, vol. 245(2), pp. 517-530.
Better Life Index 2017 indicators
Description
Data related to BLI Edition 2017 (OECD, 2017) for all 38 OECD and non-OECD countries (Data extracted on: 1902円2020円).
For more info, please see https://data-explorer.oecd.org.
Usage
data(BLI_2017)Format
BLI_2017 is a dataset with 38 observations and 12 indicators.
- country
OECD and non-OECD countries.
- housing
Housing.
- income
Income and wealth.
- jobs
Jobs and earnings.
- community
Community engagement.
- education
Education.
- environment
Environment quality.
- civic
Civic engagement.
- health
Health.
- satisfaction
Life satisfaction.
- safety
Personal security (safety).
- worklife
Work-Life balance.
Author(s)
Fusco E.
Examples
data(BLI_2017)Europe 2020 indicators
Description
Europe 2020, a strategy for jobs and smart, sustainable and inclusive growth, is based on five EU headline targets which are currently measured by eight headline indicators, Headline indicators, Eurostat, year 1990-2012 (Last update: 21/11/2013).
For more info, please see https://ec.europa.eu/eurostat/en/web/products-statistics-in-focus/-/KS-SF-12-039.
Usage
data(EU_2020)Format
EU_2020 is a dataset with 30 observations and 12 indicators (190 indicator per year).
- geo
EU-Member States including EU (28 countries) and EU (27 countries) row.
- employXXXX
Employment rate - age group 20-64, year XXXX (1992-2012).
- perc_GDPXXXX
Gross domestic expenditure on R&D (GERD), year XXXX (1990-2012).
- gas_emissXXXX
Greenhouse gas emissions - base year 1990, year XXXX (1990-2011).
- share_renXXXX
Share of renewable energy in gross final energy consumption, year XXXX (2004-2011).
- prim_enerXXXX
Primary energy consumption, year XXXX (1990-2011).
- final_energyXXXX
Final energy consumption, year XXXX (1990-2011).
- final_energyXXXX
Early leavers from education and training - Perc. of the population aged 18-24 with at most lower secondary education and not in further education or training, year XXXX (1992-2012).
- tertiaryXXXX
Tertiary educational attainment - age group 30-34, year XXXX (2000-2012).
- risk_povertyXXXX
People at risk of poverty or social exclusion - 1000 persons Perc. of total population, year XXXX (2004-2012).
- low_workXXXX
People living in households with very low work intensity - 1000 persons Perc. of total population, year XXXX (2004-2012).
- risk_povertyXXXX
People at risk of poverty after social transfers - 1000 persons Perc. of total population, year XXXX (2003-2012).
- deprivedXXXX
Severely materially deprived people - 1000 persons Perc. of total population, year XXXX (2003-2012).
Author(s)
Vidoli F.
Examples
data(EU_2020)EU NUTS1 Transportation data
Description
Eurostat regional transport statistics (reg_tran) data, year 2012.
Usage
data(EU_NUTS1)Format
EU_NUTS1 is a dataset with 34 observations and two indicators describing transportation infrastructure endowment of the main (in terms of population and GDP) European NUTS1 regions: France, Germany, Italy, Spain (United Kingdom has been omitted, due to lack of data concerning railways).
- roads
Calculated as (2 * Motorways - Kilometres per 1000 km2 + Other roads - Kilometres per 1000 km2 )/3
- trains
Calculated as (2 *Railway lines double+Electrified railway lines)/3
Author(s)
Vidoli F.
References
Vidoli F., Mazziotta C., "Robust weighted composite indicators by means of frontier methods with an application to European infrastructure endowment", Statistica Applicata, Italian Journal of Applied Statistics, 2013.
Examples
data(EU_NUTS1)Multivariate mixed bandwidth selection for exogenous variables
Description
A function for the selection of optimal multivariate mixed bandwidths for the kernel density estimation of continuous and discrete exogenous variables.
Usage
bandwidth_CI(x, indic_col, ngood, nbad, Q=NULL, Q_ord=NULL)Arguments
x
A data frame containing simple indicators.
indic_col
Simple indicators column number.
ngood
The number of desirable outputs; it has to be greater than 0.
nbad
The number of undesirable outputs; it has to be greater than 0.
Q
A matrix containing continuous exogenous variables.
Q_ord
A matrix containing discrete exogenous variables.
Details
Author thanks Nicky Rogge for his help and for making available the original code of the bandwidth function.
Value
bandwidth
A matrix containing the optimal bandwidths for the exogenous variables indicate in Q and Q_ord.
ci_method
"bandwidth_CI
Author(s)
Fusco E., Rogge N.
Examples
data(EU_2020)
indic <- c("employ_2011", "gasemiss_2011","deprived_2011")
dat <- EU_2020[-c(10,18),indic]
Q_GDP <- EU_2020[-c(10,18),"percGDP_2011"]
# Conditional robust BoD Constrained VWR
band = bandwidth_CI(dat, ngood=1, nbad=2, Q = Q_GDP)
Adjusted Mazziotta-Pareto Index (AMPI) method
Description
Adjusted Mazziotta-Pareto Index (AMPI) is a non-compensatory composite index that allows to take into account the time dimension, too. The calculation part is similat to the MPI framework, but the standardization part make the scores obtained over the years comparable.
Usage
ci_ampi(x, indic_col, gp, time, polarity, penalty = "NEG")Arguments
x
A data.frame containing simple indicators in a Long Data Format.
indic_col
Simple indicators column number.
gp
Goalposts; to facilitate the interpretation of results, the goalposts can be chosen so that 100 represents a reference value (e.g., the average in a given year).
time
The time variable (mandatory); if the analysis is carried out over a single year, it is necessary to create a constant variable (i.e. dataframe@year <- 2014).
polarity
Polarity vector: "POS" = positive, "NEG" = negative. The polarity of a individual indicator is the sign of the relationship between the indicator and the phenomenon to be measured (e.g., in a well-being index, "GDP per capita" has 'positive' polarity and "Unemployment rate" has 'negative' polarity).
penalty
Penalty direction; Use "NEG" (default) in case of 'increasing' or 'positive' composite index (e.g., well-being index)), "POS" in case of 'decreasing' or 'negative' composite index (e.g., poverty index).
Details
Author thanks Leonardo Alaimo for their help and for making available the original code of the AMPI function. Federico Roscioli for his integrations to the original code.
Value
An object of class "CI". This is a list containing the following elements:
ci_ampi_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="ampi".
ci_penalty
Matrix containing penalties only.
ci_norm
List containing only the normalised indicators for each year.
Author(s)
Fusco E., Alaimo L., Giovagnoli C., Patelli L., F. Roscioli
References
Mazziotta, M., Pareto, A. (2013) "A Non-compensatory Composite Index for Measuring Well-being over Time", Cogito. Multidisciplinary Research Journal Vol. V, no. 4, pp. 93-104
Mazziotta, M., Pareto, A. (2016)."On a Generalized Non-compensatory Composite Index for Measuring Socio-economic Phenomena", Cogito. Social Indicators Research, Vol. 127, no. 3, pp. 983-1003
See Also
Examples
data(EU_2020)
data_test = EU_2020[,c("employ_2010","employ_2011","finalenergy_2010","finalenergy_2011")]
EU_2020_long<-reshape(data_test,
varying=c("employ_2010","employ_2011","finalenergy_2010","finalenergy_2011"),
direction="long",
idvar="geo",
sep="_")
CI <- ci_ampi(EU_2020_long,
indic_col=c(2:3),
gp=c(50, 100),
time=EU_2020_long[,1],
polarity= c("POS", "POS"),
penalty="POS")
CI$ci_ampi_est
CI$ci_penalty
CI$ci_norm
Benefit of the Doubt approach (BoD)
Description
Benefit of the Doubt approach (BoD) is the application of Data Envelopment Analysis (DEA) to the field of composite indicators. It was originally proposed by Melyn and Moesen (1991) to evaluate macroeconomic performance.
Usage
ci_bod(x,indic_col)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
Value
An object of class "CI". This is a list containing the following elements:
ci_bod_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="bod".
ci_bod_weights
Raw weights assigned to the simple indicators (Dual values - prices - in the dual DEA formulation).
Author(s)
Vidoli F.
References
OECD (2008) "Handbook on constructing composite indicators: methodology and user guide".
Melyn W. and Moesen W.W. (1991) "Towards a synthetic indicator of macroeconomic performance: unequal weighting when limited information is available", Public Economic research Paper 17, CES, KU Leuven.
Witte, K. D., Rogge, N. (2009) "Accounting for exogenous influences in a benevolent performance evaluation of teachers". Tech. rept. Working Paper Series ces0913, Katholieke Universiteit Leuven, Centrum voor Economische Studien.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_bod(Indic)
# validating BoD score
w = CI$ci_bod_weights
Indic[,1]*w[,1] + Indic[,2]*w[,2]
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),polarity = c("POS","POS"), method=2)
CI = ci_bod(data_norm$ci_norm,c(1:2))
Constrained Benefit of the Doubt approach (BoD)
Description
The constrained Benefit of the Doubt function lets to introduce additional constraints to the weight variation in the optimization procedure so that all the weights obtained are greater than a lower value (low_w) and less than an upper value (up_w).
Usage
ci_bod_constr(x,indic_col,up_w,low_w)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
up_w
Importance weights upper bound.
low_w
Importance weights lower bound.
Value
An object of class "CI". This is a list containing the following elements:
ci_bod_constr_est
Constrained composite indicator estimated values.
ci_method
Method used; for this function ci_method="bod_constrained".
ci_bod_constr_weights
Raw constrained weights assigned to the simple indicators.
Author(s)
Rogge N., Vidoli F.
References
Van Puyenbroeck T. and Rogge N. (2017) "Geometric mean quantity index numbers with Benefit-of-the-Doubt weights", European Journal of Operational Research, Volume 256, Issue 3, Pages 1004 - 1014.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_bod_constr(Indic,up_w=1,low_w=0.05)
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),polarity = c("POS","POS"), method=2)
CI = ci_bod_constr(data_norm$ci_norm,c(1:2),up_w=1,low_w=0.05)
Constrained Benefit of the Doubt approach (BoD) in presence of undesirable indicators
Description
The constrained Benefit of the Doubt function introduces additional constraints to the weight variation in the optimization procedure (Constrained Virtual Weights Restriction) allowing to restrict the importance attached to a single indicator expressed in percentage terms, ranging between a lower and an upper bound (VWR); this function, furthermore, allows to calculate the composite indicator simultaneously in presence of undesirable (bad) and desirable (good) indicators allowing to impose a preference structure (ordVWR).
Usage
ci_bod_constr_bad(x, indic_col, ngood=1, nbad=1, low_w=0, pref=NULL)Arguments
x
A data.frame containing simple indicators; the order is important: first columns must contain the desirable indicators, while second ones the undesirable indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
ngood
The number of desirable outputs; it has to be greater than 0.
nbad
The number of undesirable outputs; it has to be greater than 0.
low_w
Importance weights lower bound.
pref
The preference vector among indicators; For example if Indic1 is the most important, Indic2,Indic3 are more important than Indic4 and no preference judgment on Indic5 (= not included in the vector), the pref vector can be written as: c("Indic1", "Indic2","Indic3","Indic4")
Value
An object of class "CI". This is a list containing the following elements:
ci_bod_constr_bad_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="bod_constr_bad".
ci_bod_constr_bad_weights
Raw weights assigned to each simple indicator.
ci_bod_constr_bad_target
Indicator target values.
Author(s)
Fusco E., Rogge N.
References
Rogge N., de Jaeger S. and Lavigne C. (2017) "Waste Performance of NUTS 2-regions in the EU: A Conditional Directional Distance Benefit-of-the-Doubt Model", Ecological Economics, vol.139, pp. 19-32.
Zanella A., Camanho A.S. and Dias T.G. (2015) "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, vol. 245(2), pp. 517-530.
See Also
Examples
data(EU_2020)
indic <- c("employ_2011", "percGDP_2011", "gasemiss_2011","deprived_2011")
dat <- EU_2020[-c(10,18),indic]
# BoD Constrained VWR
CI_BoD_C = ci_bod_constr_bad(dat, ngood=2, nbad=2, low_w=0.05, pref=NULL)
CI_BoD_C$ci_bod_constr_bad_est
# BoD Constrained ordVWR
importance <- c("gasemiss_2011","percGDP_2011","employ_2011")
CI_BoD_C = ci_bod_constr_bad(dat, ngood=2, nbad=2, low_w=0.05, pref=importance)
CI_BoD_C$ci_bod_constr_bad_est
Directional Benefit of the Doubt (D-BoD) model
Description
Directional Benefit of the Doubt (D-BoD) model enhance non-compensatory property by introducing directional penalties in a standard BoD model in order to consider the preference structure among simple indicators.
Usage
ci_bod_dir(x, indic_col, dir)Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
dir
Main direction. For example you can set the average rates of substitution.
Value
An object of class "CI". This is a list containing the following elements:
ci_bod_dir_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="bod_dir".
Author(s)
Vidoli F., Fusco E.
References
Fusco E. (2015) "Enhancing non compensatory composite indicators: A directional proposal", European Journal of Operational Research, 242(2), 620-630.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_bod_dir(Indic,dir=c(1,1))
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),polarity = c("POS","POS"), method=2)
CI = ci_bod_dir(data_norm$ci_norm,c(1:2),dir=c(1,0.5))
Multi-directional Benefit of the Doubt approach (MDBoD)
Description
Multi-directional Benefit of the Doubt (MDBoD) allows to introduce the non-compensability among simple indicators in a standard BOD in an objective manner: the preference structure, i.e., the direction, is determined directly from the data and is specific for each unit.
Usage
ci_bod_mdir(x,indic_col)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
Value
An object of class "CI". This is a list containing the following elements:
ci_bod_mdir_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="bod".
ci_bod_mdir_spec
Simple indicators specific scores.
ci_bod_mdir_dir
Directions for each simple indicator and unit.
Author(s)
Fusco E.
References
Fusco E. (2023) "Potential improvements approach in composite indicators construction: the Multi-directional Benefit of the Doubt model", Socio-Economic Planning Sciences, vol. 85, 101447
See Also
Examples
data(BLI_2017)
CI <- ci_bod_mdir(BLI_2017,c(2:12))
Variance weighted Benefit of the Doubt approach (BoD variance weighted)
Description
Variance weighted Benefit of the Doubt approach (BoD variance weighted) is a particular form of BoD method with additional information in the optimization problem. In particular it has been added weight constraints (in form of an Assurance region type I (AR I)) endogenously determined in order to take into account the ratio of the vertical variability of each simple indicator relative to one another.
Usage
ci_bod_var_w(x,indic_col,boot_rep = 5000)Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
boot_rep
The number of bootstrap replicates (default=5000) for the estimates of the nonparametric bootstrap (first order normal approximation) confidence intervals for the variances of the simple indicators.
Details
For more informations about the estimation of the confidence interval for the variances, please see function boot.ci, package boot.
Value
An object of class "CI". This is a list containing the following elements:
ci_bod_var_w_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="bod_var_w".
Author(s)
Vidoli F.
References
Vidoli F., Mazziotta C. (2013) "Robust weighted composite indicators by means of frontier methods with an application to European infrastructure endowment", Statistica Applicata, Italian Journal of Applied Statistics.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_bod_var_w(Indic)
Weighting method based on Factor Analysis
Description
Factor analysis groups together collinear simple indicators to estimate a composite indicator that captures as much as possible of the information common to individual indicators.
Usage
ci_factor(x,indic_col,method="ONE",dim)
Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
method
If method = "ONE" (default) the composite indicator estimated values are equal to first component scores; if method = "ALL" the composite indicator estimated values are equal to component score multiplied by its proportion variance; if method = "CH" it can be choose the number of the component to take into account.
dim
Number of chosen component (if method = "CH", default is 3).
Value
An object of class "CI". This is a list containing the following elements:
ci_factor_est
Composite indicator estimated values.
loadings_fact
Variance explained by principal factors (in percentage terms).
ci_method
Method used; for this function ci_method="factor".
Author(s)
Vidoli F.
References
OECD (2008) "Handbook on constructing composite indicators: methodology and user guide".
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_factor(Indic)
data(EU_NUTS1)
CI = ci_factor(EU_NUTS1,c(2:3), method="ALL")
data(EU_2020)
data_norm = normalise_ci(EU_2020,c(47:51),polarity = c("POS","POS","POS","POS","POS"), method=2)
CI3 = ci_factor(data_norm$ci_norm,c(1:5),method="CH", dim=3)
Weighting method based on Factor analysis of mixed data (FAMD)
Description
Factor analysis of mixed data (FAMD) can be seen as a principal component method dedicated to analyze a data set containing both quantitative and qualitative variables making possible to compute composite indicators taking into account continous, dummy, or factor variables
Usage
ci_factor_mixed(x,indic_col,method="ONE",dim)
Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
method
If method = "ONE" (default) the composite indicator estimated values are equal to first component scores; if method = "ALL" the composite indicator estimated values are equal to component score multiplied by its proportion variance; if method = "CH" it can be choose the number of the component to take into account.
dim
Number of chosen component (if method = "CH", default is 3).
Value
An object of class "CI". This is a list containing the following elements:
ci_factor_est
Composite indicator estimated values.
loadings_fact
Variance explained by principal factors (in percentage terms).
ci_method
Method used; for this function ci_method="factor_mixed".
Author(s)
Luis Carlos Castillo Tellez
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
i3 <- seq(0, 1, len = 100)
i3 = as.factor(ifelse(i3>0.5,1,0))
Indic = data.frame(i1, i2, i3)
CI = ci_factor_mixed(Indic,c(1:3))
CI2 = ci_factor_mixed(Indic,c(1:3), method="ALL")
CI3 = ci_factor_mixed(Indic,c(1:3), method="CH", dim=2)
Weighting method based on generalized mean
Description
Generalized means are a family of functions for aggregating sets of numbers (it include as special cases the Pythagorean means, arithmetic, geometric, and harmonic means). The generalized mean is also known as power mean or Holder mean.
Usage
ci_generalized_mean(x, indic_col, p, na.rm=TRUE)Arguments
x
A data.frame containing simple indicators.
indic_col
Simple indicators column number.
p
Exponent p (real number).
na.rm
Remove NA values before processing; default is TRUE.
Value
An object of class "CI". This is a list containing the following elements:
ci_generalized_mean_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="generalized_mean".
Note
The generalized mean with the exponent p can be espressed as:
M_p(I_1,\dots,I_n) = \left( \frac{1}{n} \sum_{i=1}^n I_i^p \right)^{\frac{1}{p}}
Particular case are:
p=-\infty: minimum, p=-1: harmonic mean, p=0: geometric mean,
p=1: arithmetic mean, p=2: root-mean-square and p=\infty: maximum.
Author(s)
Vidoli F.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_generalized_mean(Indic, p=-1) # harmonic mean
data(EU_NUTS1)
CI = ci_generalized_mean(EU_NUTS1,c(2:3),p=2) # geometric mean
Intertemporal analysis for geometric mean quantity index numbers
Description
Intertemporal analysis for geometric mean quantity index numbers with Benefit-of-the-Doubt weights - see function ci_bod_constr .
Usage
ci_geom_bod_intertemp(x0,x1,indic_col,up_w,low_w,bench)Arguments
x0
A data.frame containing simple indicators - time 0
x1
A data.frame containing simple indicators - time 1
indic_col
A numeric list indicating the positions of the simple indicators.
up_w
Weights upper bound.
low_w
Weights lower bound.
bench
Row number of the benchmark unit
Value
An object of class "CI". This is a list containing the following elements:
ci_geom_bod_intertemp_est
A matrix containing the Overall Change (period t1 vs t0), the Change Effect (period t1 vs t0), the Benchmark Effect (period t1 vs t0) and Weight Effect (period t1 vs t0).
ci_method
Method used; for this function ci_method="Intertemporal_effects_Geometric_BoD".
Author(s)
Rogge N., Vidoli F.
References
Van Puyenbroeck T. and Rogge N. (2017) "Geometric mean quantity index numbers with Benefit-of-the-Doubt weights", European Journal of Operational Research, Volume 256, Issue 3, Pages 1004 - 1014
See Also
Examples
i1_t1 <- seq(0.3, 0.5, len = 100)
i2_t1 <- seq(0.3, 1, len = 100)
Indic_t1 = data.frame(i1_t1, i2_t1)
i1_t0 <- i1_t1 - rnorm (100, 0.2, 0.03)
i2_t0 <- i2_t1 - rnorm (100, 0.2, 0.03)
Indic_t0 = data.frame(i1_t0, i2_t0)
intertemp = ci_geom_bod_intertemp(Indic_t0,Indic_t1,c(1:2),up_w=0.95,low_w=0.05,1)
intertemp
Generalized geometric mean quantity index numbers
Description
This function use the geometric mean to aggregate the single indicators. Two weighting criteria has been implemented: EQUAL: equal weighting and BOD: Benefit-of-the-Doubt weights following the Puyenbroeck and Rogge (2017) approach.
Usage
ci_geom_gen(x,indic_col,meth,up_w,low_w,bench)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
meth
"EQUAL" = Equal weighting set, "BOD" = Benefit-of-the-Doubt weighting set.
up_w
if meth="BOD"; upper bound of the weighting set.
low_w
if meth="BOD"; lower bound of the weighting set.
bench
Row number of the benchmark unit used to normalize the data.frame x.
Value
An object of class "CI". This is a list containing the following elements:
If meth = "EQUAL":
ci_mean_geom_est
: Composite indicator estimated values.
ci_method
: Method used; for this function ci_method="mean_geom".
If meth = "BOD":
ci_geom_bod_est
: Constrained composite indicator estimated values.
ci_geom_bod_weights
: Raw constrained weights assigned to the simple indicators.
ci_method
: Method used; for this function ci_method="geometric_bod".
Author(s)
Rogge N., Vidoli F.
References
Van Puyenbroeck T. and Rogge N. (2017) "Geometric mean quantity index numbers with Benefit-of-the-Doubt weights", European Journal of Operational Research, Volume 256, Issue 3, Pages 1004 - 1014
See Also
Examples
i1 <- seq(0.3, 1, len = 100) - rnorm (100, 0.1, 0.03)
i2 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.1, 0.03)
i3 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.1, 0.03)
Indic = data.frame(i1, i2,i3)
geom1 = ci_geom_gen(Indic,c(1:3),meth = "EQUAL")
geom1$ci_mean_geom_est
geom1$ci_method
geom2 = ci_geom_gen(Indic,c(1:3),meth = "BOD",0.7,0.3,100)
geom2$ci_geom_bod_est
geom2$ci_geom_bod_weights
Mean-Min Function
Description
The Mean-Min Function (MMF) is an intermediate case between arithmetic mean, according to which no unbalance is penalized, and min function, according to which the penalization is maximum. It depends on two parameters that are respectively related to the intensity of penalization of unbalance (\alpha) and intensity of complementarity (\beta) among indicators.
Usage
ci_mean_min(x, indic_col, alpha, beta)Arguments
x
A data.frame containing simple indicators.
indic_col
Simple indicators column number.
alpha
The intensity of penalisation of unbalance among indicators, 0 \le \alpha \le 1
beta
The intensity of complementarity among indicators, \beta \ge 0
Value
An object of class "CI". This is a list containing the following elements:
ci_mean_min_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="mean_min".
Author(s)
Vidoli F.
References
Casadio Tarabusi, E., & Guarini, G. (2013) "An unbalance adjustment method for development indicators", Social indicators research, 112(1), 19-45.
See Also
Examples
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),c("NEG","POS"),method=2)
CI = ci_mean_min(data_norm$ci_norm, alpha=0.5, beta=1)
Mazziotta-Pareto Index (MPI) method
Description
Mazziotta-Pareto Index (MPI) is a non-linear composite index method which transforms a set of individual indicators in standardized variables and summarizes them using an arithmetic mean adjusted by a "penalty" coefficient related to the variability of each unit (method of the coefficient of variation penalty).
Usage
ci_mpi(x, indic_col, penalty="POS")Arguments
x
A data.frame containing simple indicators.
indic_col
Simple indicators column number.
penalty
Penalty direction; Use "POS" (default) in case of 'increasing' or 'positive' composite index (e.g., well-being index)), "NEG" in case of 'decreasing' or 'negative' composite index (e.g., poverty index).
Value
An object of class "CI". This is a list containing the following elements:
ci_mpi_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="mpi".
Author(s)
Vidoli F., Scaccabarozzi A.
References
De Muro P., Mazziotta M., Pareto A. (2011), "Composite Indices of Development and Poverty: An Application to MDGs", Social Indicators Research, Volume 104, Number 1, pp. 1-18.
See Also
Examples
data(EU_NUTS1)
# Please, pay attention. MPI can be calculated only with this standardization method:
# method=1, z.mean=100 and z.std=10
# For more info, please see references.
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),c("NEG","POS"),method=1,z.mean=100, z.std=10)
CI = ci_mpi(data_norm$ci_norm, penalty="NEG")
Ordered Geographically Weighted Average (OWA)
Description
The Ordered Geographically Weighted Averaging (OWA) operator is an extension of the multi-criteria decision aggregation method called OWA (Yager, 1988) that accounts for spatial heterogeneity.
Usage
ci_ogwa(x, id, indic_col, atleastjp, coords,
kernel = "bisquare", adaptive = F, bw,
p = 2, theta = 0, longlat = F, dMat)Arguments
x
A data.frame containing score of the simple indicators.
id
Units' unique identifier.
indic_col
Simple indicators column number.
coords
A two-column matrix of latitude and longitude coordinates.
atleastjp
Fuzzy linguistic quantifier "At least j".
kernel
function chosen as follows: gaussian: wgt = exp(-.5*(vdist/bw)^2); exponential: wgt = exp(-vdist/bw); bisquare: wgt = (1-(vdist/bw)^2)^2 if vdist < bw, wgt=0 otherwise; tricube: wgt = (1-(vdist/bw)^3)^3 if vdist < bw, wgt=0 otherwise; boxcar: wgt=1 if dist < bw, wgt=0 otherwise.
adaptive
if TRUE calculate an adaptive kernel where the bandwidth (bw) corresponds to the number of nearest neighbours (i.e. adaptive distance); default is FALSE, where a fixed kernel is found (bandwidth is a fixed distance).
bw
bandwidth used in the weighting function.
p
the power of the Minkowski distance, default is 2, i.e. the Euclidean distance.
theta
an angle in radians to rotate the coordinate system, default is 0.
longlat
if TRUE, great circle distances will be calculated.
dMat
a pre-specified distance matrix, it can be calculated by the function gw.dist .
Value
An object of class "CI". This is a list containing the following elements:
CI_OGWA_n
Composite indicator estimated values for OGWA-.
CI_OGWA_p
Composite indicator estimated values for OGWA+.
wp
OGWA weights' vector "More than j".
wn
OGWA weights' vector "At least j".
ci_method
Method used; for this function ci_method="ogwa".
Author(s)
Fusco E., Liborio M.P.
References
Fusco, E., Liborio, M.P., Rabiei-Dastjerdi, H., Vidoli, F., Brunsdon, C. and Ekel, P.I. (2023), Harnessing Spatial Heterogeneity in Composite Indicators through the Ordered Geographically Weighted Averaging (OGWA) Operator. Geographical Analysis. https://doi.org/10.1111/gean.12384
See Also
Examples
data(data_HPI)
data_HPI_2019 = data_HPI[data_HPI$year==2019,]
Indic_name = c("Life_Expectancy","Ladder_of_life","Ecological_Footprint")
Indic_norm = normalise_ci(data_HPI_2019, Indic_name, c("POS","POS","NEG"),method=2)$ci_norm
Indic_norm = Indic_norm[Indic_norm$Life_Expectancy>0 &
Indic_norm$Ladder_of_life>0 &
Indic_norm$Ecological_Footprint >0,]
Indic_CI = data.frame(Indic_norm,
data_HPI_2019[rownames(Indic_norm),
c("lat","long","HPI","ISO","Country")])
atleast = 2
coord = Indic_CI[,c("lat","long")]
CI_ogwa_n = ci_ogwa(Indic_CI, id="ISO",
indic_col=c(1:3),
atleastjp=atleast,
coords=as.matrix(coord),
kernel = "gaussian",
adaptive=FALSE,
longlat=FALSE)$CI_OGWA_n
#CI_ogwa_p = ci_ogwa(Indic_CI, id="ISO",
# indic_col=c(1:3),
# atleastjp=atleast,
# coords=as.matrix(coord),
# kernel = "gaussian",
# adaptive=FALSE,
# longlat=FALSE)$CI_OGWA_p
Ordered Weighted Average (OWA)
Description
The Ordered Weighted Averaging (OWA) operator is a multi-criteria decision aggregation method that is structurally non-compensatory (Yager, 1988).
Usage
ci_owa(x, id, indic_col, atleastjp)Arguments
x
A data.frame containing score of the simple indicators.
id
Units' unique identifier.
indic_col
Simple indicators column number.
atleastjp
Fuzzy linguistic quantifier "At least j".
Value
An object of class "CI". This is a list containing the following elements:
CI_OWA_n
Composite indicator estimated values for OWA-.
CI_OWA_p
Composite indicator estimated values for OWA+.
wp
OWA weights' vector "More than j".
wn
OWA weights' vector "At least j".
ci_method
Method used; for this function ci_method="owa".
Author(s)
Fusco E., Liborio M.P.
References
Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on systems, Man, and Cybernetics, 18(1), 183-190.
See Also
Examples
data(data_HPI)
data_HPI = data_HPI[complete.cases(data_HPI),]
data_HPI_2019 = data_HPI[data_HPI$year==2019,]
Indic_name = c("Life_Expectancy","Ladder_of_life","Ecological_Footprint")
Indic_norm = data.frame("ISO"=data_HPI_2019$ISO,
normalise_ci(data_HPI_2019[, Indic_name],
c(1:3),
c("POS","POS","NEG"),
method=2)$ci_norm)
Indic_norm = Indic_norm[Indic_norm$Life_Expectancy>0 &
Indic_norm$Ladder_of_life>0 &
Indic_norm$Ecological_Footprint >0 ,]
atleast = 2
CI_owa_n = ci_owa(Indic_norm, id="ISO",
indic_col=c(2:4),
atleastjp=atleast)$CI_OWA_n
CI_owa_p = ci_owa(Indic_norm, id="ISO",
indic_col=c(2:4),
atleastjp=atleast)$CI_OWA_p
Robust Benefit of the Doubt approach (RBoD)
Description
Robust Benefit of the Doubt approach (RBoD) is the robust version of the BoD method. It is based on the concept of the expected minimum input function of order-m so "in place of looking for the lower boundary of the support of F, as was typically the case for the full-frontier (DEA or FDH), the order-m efficiency score can be viewed as the expectation of the maximal score, when compared to m units randomly drawn from the population of units presenting a greater level of simple indicators", Daraio and Simar (2005).
Usage
ci_rbod(x,indic_col,M,B)Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
M
The number of elements in each of the bootstrapped samples.
B
The number of bootstrap replicates.
Value
An object of class "CI". This is a list containing the following elements:
ci_rbod_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="rbod".
Author(s)
Vidoli F.
References
Daraio, C., Simar, L. "Introducing environmental variables in nonparametric frontier models: a probabilistic approach", Journal of productivity analysis, 2005, 24(1), 93 - 121.
Vidoli F., Mazziotta C., "Robust weighted composite indicators by means of frontier methods with an application to European infrastructure endowment", Statistica Applicata, Italian Journal of Applied Statistics, 2013.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_rbod(Indic,B=10)
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),polarity = c("POS","POS"), method=2)
CI = ci_rbod(data_norm$ci_norm,c(1:2),M=10,B=20)
Robust constrained Benefit of the Doubt approach (BoD) in presence of undesirable indicators
Description
The Robust constrained Benefit of the Doubt function introduces additional constraints to the weight variation in the optimization procedure (Constrained Virtual Weights Restriction) allowing to restrict the importance attached to a single indicator expressed in percentage terms, ranging between a lower and an upper bound (VWR); this function, furthermore, allows to calculate the composite indicator simultaneously in presence of undesirable (bad) and desirable (good) indicators allowing to impose a preference structure (ordVWR). This function is the robust version of the ci_bod_constr_bad: it is based on the concept of the expected minimum input function of order-m (Daraio and Simar, 2005) allowing to compare the unit under analysis against M peers by extracting B samples with replacement.
Usage
ci_rbod_constr_bad(x, indic_col, ngood=1, nbad=1, low_w=0, pref=NULL, M, B) Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
ngood
The number of desirable outputs; it has to be greater than 0.
nbad
The number of undesirable outputs; it has to be greater than 0.
low_w
Importance weights lower bound.
pref
The preference vector among indicators; For example if Indic1 is the most important, Indic2,Indic3 are more important than Indic4 and no preference judgment on Indic5 (= not included in the vector), the pref vector can be written as: c("Indic1", "Indic2","Indic3","Indic4")
M
The number of elements in each of the bootstrapped samples.
B
The number of bootstrap replicates.
Value
An object of class "CI". This is a list containing the following elements:
ci_rbod_constr_bad_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="rbod_constr_bad".
ci_rbod_constr_bad_weights
Raw weights assigned to each simple indicator.
ci_rbod_constr_bad_target
Indicator target values.
Author(s)
Fusco E., Rogge N.
References
Rogge N., de Jaeger S. and Lavigne C. (2017) "Waste Performance of NUTS 2-regions in the EU: A Conditional Directional Distance Benefit-of-the-Doubt Model", Ecological Economics, vol.139, pp. 19-32.
Zanella A., Camanho A.S. and Dias T.G. (2015) "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, vol. 245(2), pp. 517-530.
See Also
ci_bod_constr , ci_bod_constr_bad
Examples
data(EU_2020)
indic <- c("employ_2011", "percGDP_2011", "gasemiss_2011","deprived_2011")
dat <- EU_2020[-c(10,18),indic]
# Robust BoD Constrained VWR
CI_BoD_C = ci_rbod_constr_bad(dat, ngood=2, nbad=2, low_w=0.05, pref=NULL, M=10, B=50)
CI_BoD_C$ci_rbod_constr_bad_est
# Robust BoD Constrained ordVWR
importance <- c("gasemiss_2011","percGDP_2011","employ_2011")
CI_BoD_C = ci_rbod_constr_bad(dat, ngood=2, nbad=2, low_w=0.05, pref=importance, M=10, B=50)
CI_BoD_C$ci_rbod_constr_bad_est
Conditional robust constrained Benefit of the Doubt approach (BoD) in presence of undesirable indicators
Description
The Conditional robust constrained Benefit of the Doubt function introduces additional constraints to the weight variation in the optimization procedure (Constrained Virtual Weights Restriction) allowing to restrict the importance attached to a single indicator expressed in percentage terms, ranging between a lower and an upper bound (VWR); this function, furthermore, allows to calculate the composite indicator simultaneously in presence of undesirable (bad) and desirable (good) indicators allowing to impose a preference structure (ordVWR). This function, in addition to being robust against outlier data (see ci_rbod_constr_bad function) allows to take into account external contextual continuous (Q) or/and ordinal (Q_ord) variables.
Usage
ci_rbod_constr_bad_Q(x, indic_col, ngood=1, nbad=1,
low_w=0, pref=NULL, M, B, Q=NULL, Q_ord=NULL, bandwidth)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
ngood
The number of desirable outputs; it has to be greater than 0.
nbad
The number of undesirable outputs; it has to be greater than 0.
low_w
Importance weights lower bound.
pref
The preference vector among indicators; For example if Indic1 is the most important, Indic2,Indic3 are more important than Indic4 and no preference judgment on Indic5 (= not included in the vector), the pref vector can be written as: c("Indic1", "Indic2","Indic3","Indic4")
M
The number of elements in each of the bootstrapped samples.
B
The number of bootstrap replicates.
Q
A matrix containing continuous exogenous variables.
Q_ord
A matrix containing discrete exogenous variables.
bandwidth
Multivariate mixed bandwidth for exogenous variables; it can be calculated by bandwidth_CI function.
Value
An object of class "CI". This is a list containing the following elements:
ci_rbod_constr_bad_Q_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="rbod_constr_bad_Q".
ci_rbod_constr_bad_Q_weights
Raw weights assigned to each simple indicator.
ci_rbod_constr_bad_Q_target
Indicator target values.
Author(s)
Fusco E., Rogge N.
References
Rogge N., de Jaeger S. and Lavigne C. (2017) "Waste Performance of NUTS 2-regions in the EU: A Conditional Directional Distance Benefit-of-the-Doubt Model", Ecological Economics, vol.139, pp. 19-32.
Zanella A., Camanho A.S. and Dias T.G. (2015) "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, vol. 245(2), pp. 517-530.
See Also
ci_rbod_constr_bad , ci_bod_constr_bad
Examples
data(EU_2020)
indic <- c("employ_2011", "gasemiss_2011","deprived_2011")
dat <- EU_2020[-c(10,18),indic]
Q_GDP <- EU_2020[-c(10,18),"percGDP_2011"]
# Conditional robust BoD Constrained VWR
band = bandwidth_CI(dat, ngood=1, nbad=2, Q = Q_GDP)
CI_BoD_C = ci_rbod_constr_bad_Q(dat,
ngood=1,
nbad=2,
low_w=0.05,
pref=NULL,
M=10,
B=50,
Q=Q_GDP,
bandwidth = band$bandwidth)
CI_BoD_C$ci_rbod_constr_bad_Q_est
# # Conditional robust BoD Constrained ordVWR
# import <- c("gasemiss_2011","employ_2011", "deprived_2011")
#
# CI_BoD_C2 = ci_rbod_constr_bad_Q(dat,
# ngood=1,
# nbad=2,
# low_w=0.05,
# pref=import,
# M=10,
# B=50,
# Q=Q_GDP,
# bandwidth = band$bandwidth)
# CI_BoD_C2$ci_rbod_constr_bad_Q_est
Directional Robust Benefit of the Doubt approach (D-RBoD)
Description
Directional Robust Benefit of the Doubt approach (D-RBoD) is the directional robust version of the BoD method.
Usage
ci_rbod_dir(x,indic_col,M,B,dir)Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
M
The number of elements in each of the bootstrapped samples.
B
The number of bootstap replicates.
dir
Main direction. For example you can set the average rates of substitution.
Value
An object of class "CI". This is a list containing the following elements:
ci_rbod_dir_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="rbod_dir".
Author(s)
Fusco E., Vidoli F.
References
Daraio C., Simar L., "Introducing environmental variables in nonparametric frontier models: a probabilistic approach", Journal of productivity analysis, 2005, 24(1), 93 121.
Simar L., Vanhems A., "Probabilistic characterization of directional distances and their robust versions", Journal of Econometrics, 2012, 166(2), 342 354.
Vidoli F., Fusco E., Mazziotta C., "Non-compensability in composite indicators: a robust directional frontier method", Social Indicators Research, Springer Netherlands.
See Also
Examples
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),polarity = c("POS","POS"), method=2)
CI = ci_rbod_dir(data_norm$ci_norm, c(1:2), M = 25, B = 50, c(1,0.1))
Robust multi-directional Benefit of the Doubt approach (MDRBoD)
Description
Robust Multi-directional Benefit of the Doubt (MDRBoD) allows to introduce the non-compensability among simple indicators in a standard Robust BOD in an objective manner: the preference structure, i.e., the direction, is determined directly from the data and is specific for each unit and these estimated values are calculated as the reference sample varies in order to smooth out the effect of outliers or out-of-range data.
Usage
ci_rbod_mdir(x,indic_col,M, B, interval)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
M
The number of elements in each of the bootstrapped samples.
B
The number of bootstrap replicates.
interval
Desired probability for Student distribution [see function qt()]; default = 0.05.
Value
An object of class "CI". This is a list containing the following elements:
ci_rbod_mdir_est
Composite indicator estimated values.
conf
lower_ci and upper_ci; Estimated confidence interval for the composite indicator estimated values.
ci_method
Method used; for this function ci_method="rbod_mdir".
ci_rbod_mdir_spec
Simple indicators specific scores.
ci_rbod_mdir_dir
Directions for each simple indicator and unit.
Author(s)
Vidoli F.
References
F. Vidoli, E. Fusco, G. Pignataro, C. Guccio (2024) "Multi-directional Robust Benefit of the Doubt model: An application to the measurement of the quality of acute care services in OECD countries", Socio-Economic Planning Sciences. https://doi.org/10.1016/j.seps.2024.101877
See Also
Examples
data(BLI_2017)
CI <- ci_rbod_mdir(BLI_2017,c(2:12), M=6,B=20, interval=0.05)
Spatial robust Benefit of the Doubt approach (Sp-RBoD)
Description
The Spatial robust Benefit of the Doubt approach (Sp-RBoD) method allows to take into account the spatial contextual condition into the robust Benefit of the Doubt method.
Usage
ci_rbod_spatial(x, indic_col, M=20, B=100, W) Arguments
x
A data.frame containing score of the simple indicators.
indic_col
Simple indicators column number.
M
The number of elements in each of the bootstrapped samples; default is 20.
B
The number of bootstrap replicates; default is 100.
W
The spatial weights matrix. A square non-negative matrix with no NAs representing spatial weights; may be a matrix of class "sparseMatrix" (spdep package)
Value
An object of class "CI". This is a list containing the following elements:
ci_rbod_spatial_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="rbod_spatial".
Author(s)
Fusco E., Vidoli F.
References
Fusco E., Vidoli F., Sahoo B.K. (2018) "Spatial heterogeneity in composite indicator: a methodological proposal", Omega, Vol. 77, pp. 1-14
See Also
Examples
data(EU_NUTS1)
coord = EU_NUTS1[,c("Long","Lat")]
k<-knearneigh(as.matrix(coord), k=5)
k_nb<-knn2nb(k)
W_mat <-nb2mat(k_nb,style="W",zero.policy=TRUE)
CI = ci_rbod_spatial(EU_NUTS1,c(2:3),M=10,B=20, W=W_mat)
Constrained stochastic multi-objective acceptability analysis (C-SMAA)
Description
Stochastic multiobjective acceptability analysis (SMAA) is a multicriteria decision support technique for multiple decision makers based on exploring the weight space. Inaccurate or uncertain input data can be represented as probability distributions. In SMAA the decision makers need not express their preferences explicitly or implicitly; instead the technique analyses what kind of valuations would make each alternative the preferred one. The method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative, a central weight vector representing the typical valuations resulting in that decision, and a confidence factor measuring whether the input data is accurate enough for making an informed decision. (R Lahdelma, J. Hokkanen and P. Salminen, 1998); this function, in particular, allows to restricts the range of allowable weights within the SMAA analysis.
Usage
ci_smaa_constr(x,indic_col,rep, label, low_w=NULL)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
rep
Number of samples.
label
A factor column useful to identify units.
low_w
Importance weights lower bound vector; default is NULL (for standard SMAA)
Details
Author thanks Giuliano Resce and Raffaele Lagravinese for their help and for making available the original code of the SMAA function.\
The lower bound vector must be set as a vector of the same size as the number of simple indicators; for example - in the presence of two indicators - if you want to constrain only one indicator, you must write: low_w = c (0,0.2).
Value
An object of class "CI". This is a list containing the following elements:
ci_smaa_constr_rank_freq
Frequence of the SMAA ranks based on the sampled alternatives' values. The rows represent the analysis units while the first column represents the number of times the unit was in first rank, the second one in second rank and so on.
ci_smaa_constr_average_rank
The average rank.
ci_smaa_constr_values
The alternative values based on a set of samples from the criteria values distribution and the samples set from the feasible weight space.
ci_method
Method used; for this function ci_method="smaa_const".
Author(s)
Vidoli F.
References
R. Lahdelma, P. Salminen (2001) "SMAA-2: Stochastic multicriteria acceptability analysis for group decision making", Operations Research, 49(3), pp. 444-454
S. Greco, A. Ishizaka, B. Matarazzo and G. Torrisi (2017) "Stochastic multi-attribute acceptability analysis (SMAA): an application to the ranking of Italian regions", Regional Studies
R. Lagravinese, P. Liberati and G. Resce (2017) "Exploring health outcomes by stochastic multi-objective acceptability analysis: an application to Italian regions", Working Papers. Collection B: Regional and sectoral economics, 1703, Universidade de Vigo, GEN - Governance and Economics research Network.
See Also
Examples
# ----- Define a function for plotting a matrix ----- #
myImagePlot <- function(x, ...){
min <- min(x)
max <- max(x)
yLabels <- rownames(x)
xLabels <- colnames(x)
title <-c()
# check for additional function arguments
if( length(list(...)) ){
Lst <- list(...)
if( !is.null(Lst$zlim) ){
min <- Lst$zlim[1]
max <- Lst$zlim[2]
}
if( !is.null(Lst$yLabels) ){
yLabels <- c(Lst$yLabels)
}
if( !is.null(Lst$xLabels) ){
xLabels <- c(Lst$xLabels)
}
if( !is.null(Lst$title) ){
title <- Lst$title
}
}
# check for null values
if( is.null(xLabels) ){
xLabels <- c(1:ncol(x))
}
if( is.null(yLabels) ){
yLabels <- c(1:nrow(x))
}
layout(matrix(data=c(1,2), nrow=1, ncol=2), widths=c(4,1), heights=c(1,1))
# Red and green range from 0 to 1 while Blue ranges from 1 to 0
ColorRamp <- rgb( seq(0,1,length=256), # Red
seq(0,1,length=256), # Green
seq(1,0,length=256)) # Blue
ColorLevels <- seq(min, max, length=length(ColorRamp))
# Reverse Y axis
reverse <- nrow(x) : 1
yLabels <- yLabels[reverse]
x <- x[reverse,]
# Data Map
par(mar = c(3,5,2.5,2))
image(1:length(xLabels), 1:length(yLabels), t(x), col=ColorRamp, xlab="",
ylab="", axes=FALSE, zlim=c(min,max))
if( !is.null(title) ){
title(main=title)
}
axis(BELOW<-1, at=1:length(xLabels), labels=xLabels, cex.axis=0.7)
axis(LEFT <-2, at=1:length(yLabels), labels=yLabels, las= HORIZONTAL<-1,
cex.axis=0.7)
# Color Scale
par(mar = c(3,2.5,2.5,2))
image(1, ColorLevels,
matrix(data=ColorLevels, ncol=length(ColorLevels),nrow=1),
col=ColorRamp,
xlab="",ylab="",
xaxt="n")
layout(1)
}
# ----- END plot function ----- #
data(EU_NUTS1)
# Standard SMAA
test <- ci_smaa_constr(EU_NUTS1,c(2,3), rep=200, label = EU_NUTS1[,1])
# source("http://www.phaget4.org/R/myImagePlot.R")
# myImagePlot(test$ci_smaa_constr_rank_freq)
test$ci_smaa_constr_average_rank
# Constrained SMAA
test2 <- ci_smaa_constr(EU_NUTS1,c(2,3), rep=200, label = EU_NUTS1[,1], low_w=c(0.2,0.2) )
# myImagePlot(test2$ci_smaa_constr_rank_freq)
test2$ci_smaa_constr_average_rank
Wroclaw Taxonomic Method
Description
Wroclaw taxonomy method (also known as the dendric method), originally developed at the University of Wroclaw, is based on the distance from a theoretical unit characterized by the best performance for all indicators considered; the composite indicator is therefore based on the sum of euclidean distances from the ideal unit and normalized by a measure of variability of these distance (mean + 2*std).
Usage
ci_wroclaw(x,indic_col)Arguments
x
A data.frame containing simple indicators.
indic_col
Simple indicators column number.
Details
Please pay attention that ci_wroclaw_est is the distance from the "ideal" unit; so, units with higher values for the simple indicators get lower values of composite indicator.
Value
An object of class "CI". This is a list containing the following elements:
ci_wroclaw_est
Composite indicator estimated values.
ci_method
Method used; for this function ci_method="wroclaw".
Author(s)
Vidoli F.
References
UNESCO, "Social indicators: problems of definition and of selection", Paris 1974.
Mazziotta C., Mazziotta M., Pareto A., Vidoli F., "La sintesi di indicatori territoriali di dotazione infrastrutturale: metodi di costruzione e procedure di ponderazione a confronto", Rivista di Economia e Statistica del territorio, n.1, 2010.
See Also
Examples
i1 <- seq(0.3, 0.5, len = 100) - rnorm (100, 0.2, 0.03)
i2 <- seq(0.3, 1, len = 100) - rnorm (100, 0.2, 0.03)
Indic = data.frame(i1, i2)
CI = ci_wroclaw(Indic)
data(EU_NUTS1)
CI = ci_wroclaw(EU_NUTS1,c(2:3))
data(EU_2020)
data_selez = EU_2020[,c(1,22,191)]
data_norm = normalise_ci(data_selez,c(2:3),c("POS","NEG"),method=3)
ci_wroclaw(data_norm$ci_norm,c(1:2))
Happy Planet Index 2017-2019 indicators
Description
Data related to Happy Planet Index for 151 countries and the period 2017-2019.
For more info, please see https://happyplanetindex.org.
Usage
data(data_HPI)Format
data_HPI is a dataset with 453 observations and 10 variables.
- Country
Country name
- ISO
ISO code
- year
Years 2017-2019
- Continent
Continent
- Population
Population (thousands)
- Life_Expectancy
Life Expectancy (years)
- Ladder_of_life
Ladder of life (Wellbeing) (0-10)
- Ecological_Footprint
Ecological Footprint (g ha)
- HPI
HPI
- GDP_per_capita
GDP per capita ($)
Author(s)
Fusco E.
References
Examples
data(data_HPI)Normalisation and polarity functions
Description
This function lets to normalise simple indicators according to the polarity of each one.
Usage
normalise_ci(x, indic_col, polarity, method=1, z.mean=0, z.std=1, ties.method ="average")
Arguments
x
A data frame containing simple indicators.
indic_col
Simple indicators column number.
method
Normalisation methods:
-
1 (default) = standardization or z-scores using the following formulation:
z_{ij}=z.mean \pm \frac{x_{ij}-M_{x_j}}{S_{x_j}}\cdot z.stdwhere
\pmdepends on polarity parameter and z.mean and z.std represent the shifting parameters. -
2 = Min-max method using the following formulation:
if polarity="POS":
\frac{x-min(x)}{max(x)-min(x)}if polarity="NEG":
\frac{max(x)-x}{max(x)-min(x)} -
3 = Ranking method. If polarity="POS" ranking is increasing, while if polarity="NEG" ranking is decreasing.
polarity
Polarity vector: "POS" = positive, "NEG" = negative. The polarity of a individual indicator is the sign of the relationship between the indicator and the phenomenon to be measured (e.g., in a well-being index, "GDP per capita" has 'positive' polarity and "Unemployment rate" has 'negative' polarity).
z.mean
If method=1, Average shifting parameter. Default is 0.
z.std
If method=1, Standard deviation expansion parameter. Default is 1.
ties.method
If method=3, A character string specifying how ties are treated, see rank for details. Default is "average".
Value
ci_norm
A data.frame containing normalised score of the choosen simple indicators.
norm_method
Normalisation method used.
Author(s)
Vidoli F.
References
OECD, "Handbook on constructing composite indicators: methodology and user guide", 2008, pag.30.
See Also
Examples
data(EU_NUTS1)
# Standard z-scores normalisation #
data_norm = normalise_ci(EU_NUTS1,c(2:3),c("NEG","POS"),method=1,z.mean=0, z.std=1)
summary(data_norm$ci_norm)
# Normalisation for MPI index #
data_norm = normalise_ci(EU_NUTS1,c(2:3),c("NEG","POS"),method=1,z.mean=100, z.std=10)
summary(data_norm$ci_norm)
data_norm = normalise_ci(EU_NUTS1,c(2:3),c("NEG","POS"),method=2)
summary(data_norm$ci_norm)
Visualize the percentage of outperforming units in robust methods
Description
This function generates a plot showing the proportion of outperforming units (i.e., units with a score greater than 1) across varying subset sizes (M) used in robust Benefit of the Doubt (BoD) models. It helps to analyze how the choice of subset size impacts the number of outperforming units identified by the model.
Usage
plot_M_robust_ci(x,indic_col,method,mvector,B,dir, interval)Arguments
x
A data.frame containing simple indicators.
indic_col
A numeric list indicating the positions of the simple indicators.
method
The method to be tested. Implemented methods are "rbod", "rbod", "rbod_mdir".
mvector
The vector of different elements in each of the bootstrapped samples to be tested.
B
The number of bootstrap replicates.
dir
Main direction. For example you can set the average rates of substitution.
interval
Desired probability for Student distribution [see function qt()]; default = 0.05.
Author(s)
Fusco E.
See Also
ci_rbod , ci_rbod_dir , ci_rbod_mdir
Examples
data(EU_NUTS1)
data_norm = normalise_ci(EU_NUTS1,c(2:3),polarity = c("POS","POS"), method=2)
plot_M_robust_ci(data_norm$ci_norm, c(1:2), method = "rbod",
mvector = seq(5,25,5), B = 10)
plot_M_robust_ci(data_norm$ci_norm, c(1:2), method = "rbod_dir",
mvector = seq(5,25,5), B = 10, dir=c(1,0.1))
data(BLI_2017)
plot_M_robust_ci(x = BLI_2017, indic_col = c(2:12), method = "rbod_mdir",
mvector = seq(5,10,1), B = 10, interval = 0.05)