Introduction
FuzzyPovertyR is a package for estimating fuzzy poverty
indexes. Broadly speaking, a fuzzy poverty index is an index that ranges
in the set \(Q=[0,1]\) (Alkire et al. (2015); Silber (2023)). A fuzzy indexes assigns to each
statistical unit a value in this interval according to a given
membership function (mf) \(\mu(x_i)\in
Q\) where \(x\) is a poverty
predicate. The higher the value of \(\mu\), the more the individual is regarded
as “poor” with respect to the poverty predicate \(x\). In socio-economic surveys \(x\) may be the equivalised disposable
income, or expenditure. However, in principle \(x\) may be a generic poverty predicate that
the researcher needs to analyse, for example this could be a variable
that relates to access to transports, services and other facilities.
Below we distinguish between monetary and supplementary poverty
indexes (or measures). A fuzzy monetary poverty measure is calculated
over a numeric vector of length \(n\) (the available sample size). A
supplementary poverty index is calculated on a data.frame
of items of a questionnaire that relates to other dimensions of poverty
other than monetary.
The dataset coming from the package is loaded in the environment
with
Fuzzy Monetary poverty measures
The package lets the user choose among different membership functions
trough the argument fm of the fm_construct
function. The membership function available are:
fm="verma" (see Cheli and Lemmi
(1995))
\[
\mu_i=\left(1-F_X(x_i)\right)^{\alpha-1} =
\left(\frac{\sum_{j=i+1} w_j|x_j> x_i}{\sum_{j\ge 2} w_j|x_j>
x_1}\right)^{\alpha-1}
\]
where \(F_X\) is the empirical
cumulative distribution function of \(X\) calculated for the \(i-th\) individual and \(w_i\) is the sampling weight of statistical
unit \(i\). The parameter \(\alpha\ge 2\) is chosen so that the average
over the function equals the head count ratio (i.e. the proportion of
units whose equivalised disposable income falls below the poverty line)
of the whole population.
fm="verma1999" (see Betti and
Verma (1999))
\[
\mu_i=(1-L_X(x_i))^{\alpha-1} =
\left(\frac{\sum_{j=i+1} w_jx_j|x_j> x_i}{\sum_{j\ge 2}
w_jx_j|x_j> x_1}\right)^{\alpha-1}
\]
where \(L_X\) is the Lorenz Curve of
\(X\) calculated for the \(i-th\) individual and \(w_i\) is the sampling weight of statistical
unit \(i\). Again, the parameter \(\alpha\ge 2\) is chosen once again so that
the average over the function equals the head count ratio of the whole
population or its estimate.
fm="verma" (see Betti and Verma
(2008))
\[
\begin{split}
\mu_i&=\left(1-F_X(x_i)\right)^{\alpha-1}\left(1-L_X(x_i)\right)=\\
&=\left(\frac{\sum_{j=i+1} w_j|x_j> x_i}{\sum_{j\ge 2}
w_j|x_j> x_1}\right)^{\alpha-1} \left(\frac{\sum_{j=i+1}
w_jx_j|x_j> x_i}{\sum_{j\ge 2} w_jx_j|x_j> x_1}\right)
\end{split}
\]
Again, the parameter \(\alpha\ge 2\)
is chosen once again so that the average over the function equals the
head count ratio of the whole population or its estimate.
fm="chakravarty" (see Chakravarty (2006))
\[
\mu_i = \begin{cases}
1 & x_i = 0\\
\frac{z - x_i}{z} & 0 \le x_i < z\\
0 & x_i \ge z
\end{cases}
\] where \(z\) is a threshold to
be chosen by the researcher.
fm="cerioli" (see Cerioli and
Zani (1990)) $$ _i=
\[
\mu_i = \begin{cases}
1, \quad 0<x_i<z_1\\
\frac{z_2-x_i}{z_2-z_1}, \quad z_1\le x_i<z_2\\
0 \quad x_i\ge z_2
\end{cases}
\]
$$ the values \(z_1\) and \(z_2\) have to be chosen by the
researcher.
fm="ZBM" (see Zedini and Belhadj
(2015) and Belhadj and Matoussi
(2010))
\[
\mu_i = \begin{cases}
1 & a \le x_i < b\\
\frac{-x_i}{c-b} + \frac{c}{c-b} & b \le x_i < c\\
0 & x_i < a \cup x_i
\ge c\\
\end{cases}
\] where \(a,c,b\) are
percentiles estimated with via the bootstrap technique.
fm="belhadj2015" (see Besma
(2015))
\[
\mu_i = \begin{cases} 1 & x_i <
z_1\\ \mu^1 = 1-\frac{1}{2}\left(\frac{x_i-z_1}{z_1}\right)^b
& z_1 \le x_i < z\\ \mu^2 = 1-\frac{1}{2}\left(\frac{z_2 -
x_i}{z_2}\right)^b & z \le x_i < z_2\\ 0 & x_i
\ge z_2 \end{cases}
\] where \(z^*\) is the flex
point of \(\mu_i\), \(z_1\) and \(z_2\) have to be chosen by the researcher,
and \(b\ge 1\) is a shape parameter
ruling the degree of convexity of the function. In particular, when
\(b=1\) the trend is linear.
fm="belhadj2011" (see Belhadj
(2011))
\[
\mu_i = \begin{cases}
1 &
0 < x_i < z_{min} \\
\frac{-x_i}{z_{\max} - z_{\min}} + \frac{-z_{\max}}{z_{\max} -
z_{\min}} & z_{min} \le x_i < z_{max}\\
0 &
x_i \ge z_{max}
\end{cases}
\] where \(z_{min}\) and \(z_{MAX}\) have to be chosen by the
researcher
For each of the functions below the breakdown argument
can be specified in case the user’s want to obtain estimates for given
sub-domains.
Example using fm=verma, fm=verma1999 and
fm=ZBM
The computation of a fuzzy poverty index that uses the
fm="verma"argument goes trough the following steps:
- Estimation of the Head Count Ratio (HCR). The package
FuzzyPovertyR provides the function HCR to
estimate the Head Count Ratio from data. It outputs a list of three
elements: a classification of units into being poor or not poor, the
poverty line, and the value itself.
hcr = HCR(predicate = eusilc$eq_income, weight = eusilc$DB090, p = 0.5, q = 0.6)$HCR # add poverty threshold
if needed, the package has a built-in function
eq_predicate to calculate the equivalised disposable income
using some equivalence scales.
- Construction of the Fuzzy Monetary measure.
#> [1] "verma"
#> [1] "FuzzyMonetary"
#> Fuzzy monetary results:
#>
#> Summary of verma membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.004 0.036 0.194 0.528 1
#>
#> Estimate(s):
#>
#> [1] 0.246
#>
#> Parameter(s):
#>
#> alpha
#> 3.787
When alpha = NULL (the default) this function solves a
non-linear equation finding the value \(\alpha\) in interval that
equates the expected value of the poverty measure to the Head Count
Ratio calculated above (see #eq-betti2006). This can be avoided by
specifying a numeric value of \(\alpha\).
verma = fm_construct(predicate = eusilc$eq_income, fm = "verma", weight = eusilc$DB090, ID = NULL, interval = c(1,10), alpha = 2)
The result of fm_construct using fm="verma"
is a list containing
- a
data.frame of individuals’ membership functions
sorted in descending order (i.e. from most poor to least poor)
head(verma$results)
#> ID predicate weight mu
#> 1 44 3.225806 465.3585 1.0000000
#> 2 450 6.666667 1010.5060 0.9946732
#> 3 372 12.903226 304.7067 0.9930663
#> 4 490 17.857143 1177.4120 0.9868551
#> 5 245 20.000000 853.0255 0.9823546
#> 6 130 42.424242 1141.3410 0.9763241
- The estimated FM measure (note that this equals the HCR if
breakdown=NULL). However, one can obtain estimates for
sub-domains using the breakdown argument as follows
verma.break = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,10), alpha = NULL, breakdown = eusilc$db040,
fm="verma")
summary(verma.break)
#> Fuzzy monetary results:
#>
#> Summary of verma membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.004 0.036 0.194 0.528 1
#>
#> Estimate(s):
#>
#> a b c d e f g h i j k l m
#> 0.189 0.165 0.166 0.268 0.217 0.178 0.363 0.294 0.186 0.155 0.157 0.272 0.327
#> n o p q r s t u v w x y z
#> 0.247 0.237 0.255 0.356 0.163 0.375 0.076 0.509 0.242 0.206 0.271 0.313 0.213
#>
#> Parameter(s):
#>
#> alpha
#> 3.787
verma.break$estimate
#> a b c d e f g
#> 0.18866339 0.16538025 0.16562162 0.26821706 0.21695323 0.17829490 0.36323070
#> h i j k l m n
#> 0.29418642 0.18600937 0.15484057 0.15722022 0.27162839 0.32717227 0.24665062
#> o p q r s t u
#> 0.23685247 0.25459850 0.35638295 0.16349880 0.37543084 0.07613034 0.50897058
#> v w x y z
#> 0.24174548 0.20595260 0.27131691 0.31315140 0.21313753
alpha = verma$parameters$alpha
alpha
#> [1] 2
With almost identical procedures is also possible to compute the
index for fm="verma1999" and fm="TFR".
#> [1] "verma1999"
#> [1] "FuzzyMonetary"
#> Fuzzy monetary results:
#>
#> Summary of verma1999 membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0 0.006 0.131 0.591 1
#>
#> Estimate(s):
#>
#> [1] 0.246
#>
#> Parameter(s):
#>
#> alpha
#> 12.067
#> [1] "TFR"
#> [1] "FuzzyMonetary"
#> Fuzzy monetary results:
#>
#> Summary of TFR membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.005 0.042 0.199 0.52 1
#>
#> Estimate(s):
#>
#> [1] 0.246
#>
#> Parameter(s):
#>
#> alpha
#> 3.062
Example using fm=belhadj2015 and
fm=cerioli
The construction of a fuzzy index using the membership function as
Besma (2015) or Cerioli and Zani (1990) is obtained by
specifying fm="belhadj2015" or fm="cerioli".
Let us begin by fm="belhadj2015". For this mf the arguments
z1, z2 and b need user’s
specification. The value z that correspond to the flex
points of the mf or to the point where the two mf touch together is
calculated by the function.
The parameter b \((>=1)\) rules the shape of the
membership functions (set b=1 for linearity)
z1 = 20000; z2 = 70000; b = 2
belhadj = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "belhadj2015", z1 = z1, z2 = z2, b = b)
#> Fuzzy monetary results:
#>
#> Summary of belhadj2015 membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.991 1 1 1 1
#>
#> Estimate(s):
#>
#> [1] 0.939
#>
#> Parameter(s):
#>
#> z1 z2 z b
#> 20000.00 70000.00 47547.17 2.00
Using fm="cerioli". Again we have to set the values of
z1, z2 as follows:
z1 = 10000; z2 = 70000
cerioli = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "cerioli", z1 = z1, z2 = z2)
#> Fuzzy monetary results:
#>
#> Summary of cerioli membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.788 0.935 1 1 1
#>
#> Estimate(s):
#>
#> [1] 0.895
#>
#> Parameter(s):
#>
#> z1 z2
#> 10000 70000
Example using fm=chakravarty and
fm=belhadj2011
Chakravarty (2006) fuzzy index is
obtained setting fm = "chakravarty". The argument
z needs user’s specification as follows:
z = 60000
chakravarty = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "chakravarty", z = z)
#> Fuzzy monetary results:
#>
#> Summary of chakravarty membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.622 0.768 0.84 0.903 1
#>
#> Estimate(s):
#>
#> [1] 0.761
#>
#> Parameter(s):
#>
#> z
#> 60000
again is is possible to specify the breakdown argument
to obtain estimates at sub-domains.
chakravarty.break = fm_construct(predicate = eusilc$eq_income, eusilc$DB090, fm = "chakravarty", z = z, breakdown = eusilc$db040)
knitr::kable(data.frame(verma.break$estimate, chakravarty.break$estimate), col.names = c("Verma", "Chakravarty"), digits = 4)
| a |
0.1887 |
0.7041 |
| b |
0.1654 |
0.7340 |
| c |
0.1656 |
0.6114 |
| d |
0.2682 |
0.7740 |
| e |
0.2170 |
0.7886 |
| f |
0.1783 |
0.6871 |
| g |
0.3632 |
0.8167 |
| h |
0.2942 |
0.8128 |
| i |
0.1860 |
0.7439 |
| j |
0.1548 |
0.7600 |
| k |
0.1572 |
0.7827 |
| l |
0.2716 |
0.8143 |
| m |
0.3272 |
0.7918 |
| n |
0.2467 |
0.7797 |
| o |
0.2369 |
0.7523 |
| p |
0.2546 |
0.7596 |
| q |
0.3564 |
0.8127 |
| r |
0.1635 |
0.7804 |
| s |
0.3754 |
0.8489 |
| t |
0.0761 |
0.7434 |
| u |
0.5090 |
0.8221 |
| v |
0.2417 |
0.7170 |
| w |
0.2060 |
0.6673 |
| x |
0.2713 |
0.8170 |
| y |
0.3132 |
0.7286 |
| z |
0.2131 |
0.7966 |
Instead, for fm = "belhadj2011" the values of \(z_{min}\) and \(z_{MAX}\) need user’s specification as
follows:
zmin = 5000; zmax = 60000
belhadj2011 = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "belhadj2011", z_min = zmin, z_max = zmax)
#> Fuzzy monetary results:
#>
#> Summary of belhadj2011 membership function:
#>
#> Quantiles:
#>
#> 0% 20% 40% 60% 80% 100%
#> 0 0.678 0.838 0.916 0.985 1
#>
#> Estimate(s):
#>
#> [1] 0.823
#>
#> Parameter(s):
#>
#> z_min z_max
#> 5000 60000
Example using fm=ZBM
As previously showed to compute the index related to this mf is
necessary to compute three parameters \(a, b,
c\) which are computed with bootstrap techniques (Zedini and Belhadj (2015)). Moreover the
definition of the index require the knowledge of the household size. The
index is obtained as follows:
ZBM = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "ZBM", hh.size = eusilc$ncomp)
Fuzzy supplementary
The package include also a multidimensional fuzzy poverty index known
as Fuzzy Supplementary (FS) index (see Betti and
Verma (2008) and Betti, Gagliardi, and
Verma (2018)). This index is defined with multiple steps. The
package has an ad-hoc function for each step (excluding the third one).
The steps are:
Step 1 - Identification
Step 2 - Transformation
Step 3 - Factor analysis to identify dimensions of
poverty
Step 4 - Calculation of weights
Step 5 - Calculation of scores in dimensions
Step 6 - Calculation of the \(\alpha\) parameter
Step 7 - Construction of the FS measure for each
dimension
Step 1 - Identification
This step is pretty simple. The user has to select the columns of the
data that correspond to the items that he/she has decided to keep in the
analysis.
Step 1 is done with the following selection
# eusilc = na.omit(eusilc)
step1 = eusilc[,4:23]
If the data in the dataset are not “ordered” in the right way,
i.e. the highest values represents the highest deprivation it is
possible to invert them using the following function:
#Create a dataframe in which the variable X is not ordered in the right way:
data=data.frame("X"=rep(c(1,2,3,4),20), "Y"=rep(c(7,8,9,1),20))
#Crete vec_order a vector of length n with TRUE or FALSE. True if the order of the variable is to be inverted, False otherwise
vec_order=c(TRUE,FALSE)
head(fs_order(data=data, vec_order))
#> X Y
#> 1 4 7
#> 2 3 8
#> 3 2 9
#> 4 1 1
#> 5 4 7
#> 6 3 8
Step 3 - Factor Analysis
This fuzzy supplementary measure use factor analysis to undercover
latent dimension in the data. There are multiple approaches to get
factor analysis in R which we do not cover in this vignette, however the
user can check for example the lavaan package. Anyways,
factor analysis is not mandatory and the user may wish to undertake a
different approach to undercover a latent structure in the data. Indeed,
it is possible to skip factor analysis or to use a personal assignment
of columns in dimensions.
Regardless of the chosen method, to go trough Step 3 the user has to
specify a numeric vector of the same length of the number of items
selected in Step 1 that assigns each column to a given dimension.
dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5)
Steps 4 and 5 - Calculation of weights and measures within
dimensions
In this step, the weights to be assigned to each item, belonging to a
given dimension \(h\), are determined
separately within each dimension. Such weighting procedure takes into
account of two different aspects: the dispersion of the deprivation
indicator and its correlation with other deprivation indicators in the
given dimension. The weight of item \(j\) belonging to dimension \(h\) is taken as
\[
\omega_{hj}=\omega^a_{hj}\times \omega^b_{hj}, \quad h=1,\dots,H \quad
\text{and}\quad j=1,\dots,k_h
\]
\(\omega_{hj}^a\) is taken as
proportional to the coefficient of variation of the complement to one of
the positive score for the variable concerned
\[
\omega^a_{hj}\propto \frac{\sigma_{s_{hj}}}{1-\bar{s}_{hj}}
\]
where \(\sigma_{s_{hj}}\) is the
standard deviation of the deprivation score \(s\) for item \(j\) in dimension \(h\) and \(\bar
s_{hj}\) its sample mean. The second factor is
\[
\omega^b_{hj}\propto\Biggl(\frac{1}{1+\sum_{j=1}^{k_h}\rho_{e_{hjhj^{\ast}}}|\rho_{e_{hjhj^{\ast}}}<r^\ast_{e_{hj}}}\Biggr)\times
\Biggl(\frac{1}{1+\sum_{j=1}^{k_h}\rho_{e_{hjhj^{\ast}}}|\rho_{e_{hjhj^{\ast}}}\ge
r^\ast_{e_{hj}}}\Biggr)
\]
where \(\rho_{e_{hjhj^{\ast}}}\) is
the kendall’s correlation coefficient between deprivation indicators
corresponding to items j and \(j^{\ast}\) in the \(h\)-dimension and \(r^\ast_{e_{hj}}\) is a critical value.
Aggregation over a group of items in a particular dimension is given by
a weighted mean taken over the items in that dimension \(s_{hi}=\sum w_{hj} s_{hj,i}/w_{hj}\), where
\(w_{hj}\) is the sampling weight of
the \(j\)-th deprivation item in the
\(h\)-th dimension. An overall score
for the \(i\)-th individual is
calculated as the un-weighted mean:
\[
s_i=\frac{\sum_h s_{hi}}{H}
\] Those two steps are implemented in the package with the
function fs_weight. It is necessary to define a value \(\rho\) which is a critical value to be used
for calculation of weights in the Kendall correlation matrix. If NULL,
i.e. not defined, rho is set equal to the point of largest gap between
the ordered set of correlation values encountered (see Betti and Verma,
2008).
#> [1] "FuzzySupplementary"
#> # A tibble: 20 × 5
#> Dimension Item w_a w_b w
#> <dbl> <chr> <dbl> <dbl> <dbl>
#> 1 1 HS040 0.804 0.483 0.388
#> 2 1 HS050 0.157 0.688 0.108
#> 3 1 HS060 0.928 0.508 0.471
#> 4 1 HS070 0.0807 0.504 0.0407
#> 5 2 HS080 0.111 0.611 0.0677
#> 6 2 HS090 0.597 0.586 0.350
#> 7 2 HS100 0.157 0.478 0.0751
#> 8 2 HS110 0.419 0.578 0.242
#> 9 2 HS120 0.759 0.883 0.670
#> 10 3 HS160 4.36 0.807 3.52
#> 11 3 HS170 2.69 0.630 1.69
#> 12 3 HS180 4.03 0.686 2.77
#> 13 3 HS190 3.26 1.32 4.29
#> 14 4 HH010 0.986 0.929 0.916
#> 15 4 HH020 0.427 0.900 0.384
#> 16 4 HH040 2.61 0.933 2.44
#> 17 4 HH050 0.220 0.980 0.215
#> 18 5 HH081 0.105 0.553 0.0583
#> 19 5 HH091 0.101 0.590 0.0593
#> 20 5 HX040 0.624 0.903 0.563
The output is a longitudinal data frame that contains the weights
\(w_a, w_b, w = w_a\times w_b\) , the
deprivation score \(s_{hi}\) for unit
\(i\) and dimension \(j\), and the overall score \(s_i\) (the average over dimensions).
knitr::kable(head(steps4_5$steps4_5), digits = 4, caption = "Results from Steps 4 and 5.")
Results from Steps 4 and 5.
| 1 |
HS040 |
0 |
1 |
0.8038 |
0.4828 |
0.3881 |
0.1476 |
0.4536 |
| 2 |
HS040 |
0 |
1 |
0.8038 |
0.4828 |
0.3881 |
0.0404 |
0.2703 |
| 3 |
HS040 |
0 |
1 |
0.8038 |
0.4828 |
0.3881 |
0.6149 |
0.5334 |
| 4 |
HS040 |
1 |
1 |
0.8038 |
0.4828 |
0.3881 |
1.0000 |
0.4972 |
| 5 |
HS040 |
1 |
1 |
0.8038 |
0.4828 |
0.3881 |
0.5327 |
0.4558 |
| 6 |
HS040 |
0 |
1 |
0.8038 |
0.4828 |
0.3881 |
0.1476 |
0.2527 |
Step 6 - Calculation of the parameter \(\alpha\)
This step is equivalent to that discussed in the Fuzzy Monetary
section when fm="verma", in-fact the mf is defined as:
\[
\mu_i=(1-F_S(s_i)^{\alpha-1}(1-L_S(s_i)).
\] The function fs_equate is used to find the value
of \(\alpha\) such that the expected
value of the mf equals the HCR.
alpha = fs_equate(steps4_5 = steps4_5, weight = eusilc$DB090, HCR = hcr, interval = c(1,10))
#> trying with alpha: 1 Expected Value: 0.5559
#> trying with alpha: 10 Expected Value: 0.0926
#> trying with alpha: 7.0204 Expected Value: 0.1285
#> trying with alpha: 4.0102 Expected Value: 0.2098
#> trying with alpha: 2.951 Expected Value: 0.2691
#> trying with alpha: 3.364 Expected Value: 0.2424
#> trying with alpha: 3.3089 Expected Value: 0.2457
#> trying with alpha: 3.3038 Expected Value: 0.246
#> trying with alpha: 3.3039 Expected Value: 0.246
#> trying with alpha: 3.3038 Expected Value: 0.246
#> Done.
(alternatively a user’s defined specification of the
alpha argument can be used as well.)
Step 7 - Construction of the Fuzzy Supplementary measure.
The parameter \(\alpha\) estimated
is used to calculate the fuzzy supplementary mf for each dimension of
deprivation separately as follows. The FS indicator for the \(h-th\) deprivation dimension for the \(i-th\) individual is defined as combination
of the \((1-F_{(S),hi})\) indicator and
the \((1-L_{(S),hi})\) indicator.
\[
\begin{split}
\mu_{hi}&=\biggl(1-F_{S_{h}}(s_{hi})\Biggr)^{\alpha-1}\biggl(1-L_{S_h}(s_{hi})\Biggr)=\\
&=\Biggl[\frac{\sum_{\gamma=i+1}w_{h\gamma}|s_{h\gamma}>s_{hi}}{\sum_{\gamma\ge
2}w_{h\gamma}|s_{h\gamma}>s_{h1}}\Biggr]^{\alpha-1}\Biggl[\frac{\sum_{\gamma=i+1}w_{h\gamma}s_{h\gamma}|s_{h\gamma}>s_{hi}}{\sum_{\gamma\ge
2}w_{h\gamma}s_{h\gamma}|s_{h\gamma}>s_{h1}}\Biggr]
\end{split}
\] The function fs_construct is used to compute the
FS for each dimension and overall as follows:
#> Fuzzy supplementary results:
#>
#> Summary of the membership functions:
#>
#> Quantiles:
#>
#> 0% 25% 50% 75% 100%
#> FS1 0 0.0000 0.1028 0.3146 1
#> FS2 0 0.0234 0.1761 0.4886 1
#> FS3 0 1.0000 1.0000 1.0000 1
#> FS4 0 0.0049 0.1082 0.1082 1
#> FS5 0 0.0000 0.0113 0.3381 1
#> Overall 0 0.0074 0.0975 0.3643 1
#>
#> Estimate(s):
#>
#> FS1 FS2 FS3 FS4 FS5 Overall
#> 0.161 0.224 0.829 0.200 0.191 0.246
#>
#> Parameter(s):
#>
#> Alpha:
#>
#> [1] 3.304
#> FS1 FS2 FS3 FS4 FS5 Overall
#> a 0.16017778 0.15721420 0.7504151 0.11843430 0.13690382 0.17803697
#> b 0.17203766 0.13353783 0.8603701 0.18226841 0.35306548 0.16150876
#> c 0.17790861 0.20949808 0.8747891 0.23864519 0.22568769 0.30225670
#> d 0.18430484 0.38429991 0.8634134 0.17047030 0.06218540 0.24525107
#> e 0.13157099 0.21602939 0.9432241 0.35550492 0.30772389 0.45379840
#> f 0.11635064 0.25468798 0.7003120 0.18457563 0.15552302 0.18651992
#> g 0.19789906 0.19517219 0.7868058 0.16063463 0.08637581 0.27243247
#> h 0.10378979 0.19398696 0.8526735 0.21615485 0.22137989 0.18955852
#> i 0.19944239 0.10733043 0.9561498 0.16923227 0.29106818 0.21228732
#> j 0.10950019 0.22043135 0.8774224 0.05680170 0.18878432 0.19766477
#> k 0.20243807 0.07183648 0.9791507 0.09036923 0.27431162 0.32986760
#> l 0.18773841 0.23105579 0.9185245 0.15254579 0.12587426 0.16756316
#> m 0.19459905 0.08661289 0.8812583 0.19715665 0.19712955 0.16909045
#> n 0.08403138 0.23751028 0.8101947 0.24143510 0.05576150 0.08021327
#> o 0.10717430 0.15153461 0.6907420 0.07810733 0.12316342 0.09665732
#> p 0.12462294 0.26074356 0.7678514 0.28953421 0.29440486 0.21144141
#> q 0.10731824 0.28134009 0.8083268 0.16137948 0.26594294 0.22523560
#> r 0.13292525 0.13075056 0.9532498 0.36621517 0.41196933 0.38421113
#> s 0.26416295 0.28100851 0.9387913 0.26507953 0.21010476 0.41753517
#> t 0.16740211 0.41950301 0.8725724 0.11691019 0.04229702 0.35450250
#> u 0.22746666 0.22160734 0.6666813 0.11469800 0.21659332 0.22168511
#> v 0.17997018 0.20148115 0.7284268 0.18935129 0.22035156 0.24724376
#> w 0.19114979 0.24834051 0.7252039 0.37794904 0.13056837 0.24049684
#> x 0.15863892 0.23342891 0.8680189 0.15527000 0.14885089 0.24134930
#> y 0.11294887 0.29472989 0.8436033 0.14557032 0.09870329 0.25374784
#> z 0.21039040 0.30499191 0.7309410 0.25553623 0.20458509 0.44289583
The output of the fs_construct function is a list
containing:
FS$estimate
#> FS1 FS2 FS3 FS4 FS5 Overall
#> a 0.16017778 0.15721420 0.7504151 0.11843430 0.13690382 0.17803697
#> b 0.17203766 0.13353783 0.8603701 0.18226841 0.35306548 0.16150876
#> c 0.17790861 0.20949808 0.8747891 0.23864519 0.22568769 0.30225670
#> d 0.18430484 0.38429991 0.8634134 0.17047030 0.06218540 0.24525107
#> e 0.13157099 0.21602939 0.9432241 0.35550492 0.30772389 0.45379840
#> f 0.11635064 0.25468798 0.7003120 0.18457563 0.15552302 0.18651992
#> g 0.19789906 0.19517219 0.7868058 0.16063463 0.08637581 0.27243247
#> h 0.10378979 0.19398696 0.8526735 0.21615485 0.22137989 0.18955852
#> i 0.19944239 0.10733043 0.9561498 0.16923227 0.29106818 0.21228732
#> j 0.10950019 0.22043135 0.8774224 0.05680170 0.18878432 0.19766477
#> k 0.20243807 0.07183648 0.9791507 0.09036923 0.27431162 0.32986760
#> l 0.18773841 0.23105579 0.9185245 0.15254579 0.12587426 0.16756316
#> m 0.19459905 0.08661289 0.8812583 0.19715665 0.19712955 0.16909045
#> n 0.08403138 0.23751028 0.8101947 0.24143510 0.05576150 0.08021327
#> o 0.10717430 0.15153461 0.6907420 0.07810733 0.12316342 0.09665732
#> p 0.12462294 0.26074356 0.7678514 0.28953421 0.29440486 0.21144141
#> q 0.10731824 0.28134009 0.8083268 0.16137948 0.26594294 0.22523560
#> r 0.13292525 0.13075056 0.9532498 0.36621517 0.41196933 0.38421113
#> s 0.26416295 0.28100851 0.9387913 0.26507953 0.21010476 0.41753517
#> t 0.16740211 0.41950301 0.8725724 0.11691019 0.04229702 0.35450250
#> u 0.22746666 0.22160734 0.6666813 0.11469800 0.21659332 0.22168511
#> v 0.17997018 0.20148115 0.7284268 0.18935129 0.22035156 0.24724376
#> w 0.19114979 0.24834051 0.7252039 0.37794904 0.13056837 0.24049684
#> x 0.15863892 0.23342891 0.8680189 0.15527000 0.14885089 0.24134930
#> y 0.11294887 0.29472989 0.8436033 0.14557032 0.09870329 0.25374784
#> z 0.21039040 0.30499191 0.7309410 0.25553623 0.20458509 0.44289583
alpha the parameter \(\alpha\) estimated from data.
Again, it is possible to obtain results for sub-domains by specifying
the breakdown argument
| a |
0.1602 |
0.1572 |
0.7504 |
0.1184 |
0.1369 |
0.1780 |
| b |
0.1720 |
0.1335 |
0.8604 |
0.1823 |
0.3531 |
0.1615 |
| c |
0.1779 |
0.2095 |
0.8748 |
0.2386 |
0.2257 |
0.3023 |
| d |
0.1843 |
0.3843 |
0.8634 |
0.1705 |
0.0622 |
0.2453 |
| e |
0.1316 |
0.2160 |
0.9432 |
0.3555 |
0.3077 |
0.4538 |
| f |
0.1164 |
0.2547 |
0.7003 |
0.1846 |
0.1555 |
0.1865 |
| g |
0.1979 |
0.1952 |
0.7868 |
0.1606 |
0.0864 |
0.2724 |
| h |
0.1038 |
0.1940 |
0.8527 |
0.2162 |
0.2214 |
0.1896 |
| i |
0.1994 |
0.1073 |
0.9561 |
0.1692 |
0.2911 |
0.2123 |
| j |
0.1095 |
0.2204 |
0.8774 |
0.0568 |
0.1888 |
0.1977 |
| k |
0.2024 |
0.0718 |
0.9792 |
0.0904 |
0.2743 |
0.3299 |
| l |
0.1877 |
0.2311 |
0.9185 |
0.1525 |
0.1259 |
0.1676 |
| m |
0.1946 |
0.0866 |
0.8813 |
0.1972 |
0.1971 |
0.1691 |
| n |
0.0840 |
0.2375 |
0.8102 |
0.2414 |
0.0558 |
0.0802 |
| o |
0.1072 |
0.1515 |
0.6907 |
0.0781 |
0.1232 |
0.0967 |
| p |
0.1246 |
0.2607 |
0.7679 |
0.2895 |
0.2944 |
0.2114 |
| q |
0.1073 |
0.2813 |
0.8083 |
0.1614 |
0.2659 |
0.2252 |
| r |
0.1329 |
0.1308 |
0.9532 |
0.3662 |
0.4120 |
0.3842 |
| s |
0.2642 |
0.2810 |
0.9388 |
0.2651 |
0.2101 |
0.4175 |
| t |
0.1674 |
0.4195 |
0.8726 |
0.1169 |
0.0423 |
0.3545 |
| u |
0.2275 |
0.2216 |
0.6667 |
0.1147 |
0.2166 |
0.2217 |
| v |
0.1800 |
0.2015 |
0.7284 |
0.1894 |
0.2204 |
0.2472 |
| w |
0.1911 |
0.2483 |
0.7252 |
0.3779 |
0.1306 |
0.2405 |
| x |
0.1586 |
0.2334 |
0.8680 |
0.1553 |
0.1489 |
0.2413 |
| y |
0.1129 |
0.2947 |
0.8436 |
0.1456 |
0.0987 |
0.2537 |
| z |
0.2104 |
0.3050 |
0.7309 |
0.2555 |
0.2046 |
0.4429 |
The package contains also a function named
fs_construct_all which constructs the fuzzy supplementary
poverty measure based without step-by-step functions.
Variance Estimation
The variance of each Fuzzy Monetary measure can be estimated either
via Bootstrap (naive or calibrated) or Jackknife Repeated Replications.
We recommend the former each time the user has no knowledge of the
sampling design, while we recommend the Jackknife when there is full
information on the design and of the PSUs (see Betti, Gagliardi, and Verma (2018)). In the
following, for the unidimensional indices, we report only the examples
linked with fm=verma. For the other specification of the mf
the function is identical the only changes are linked with the
parameters required by the mf (see fm_construct).
Example using fm=verma
In case of fm="verma", we recommend the user to use the
value of alpha from obtained from the function
fm_construct. It is possible to specify different values of
the parameter (i.e. alpha=2). We do not recommend to leave
the argument alpha=NULL for the computation of
variance.
alpha = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = 0.12, interval = c(1,10), alpha = NULL)$alpha
boot.var = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", type = "bootstrap_naive", HCR = .12, alpha = alpha, verbose = F, R = 10)
# plot(boot.var)
fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", type = "jackknife", HCR = .12, alpha = 9, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)
#> $variance
#> [1] 5.324184e-06
#>
#> $type
#> [1] "jackknife"
#>
#> attr(,"class")
#> [1] "FuzzyMonetary"
which gives the bootstrap estimate or the jackknife estimate.
If there are multiple sub-domains or sub-populations that need
variance estimation, the user can specify the breakdown to the
breakdown argument of the function fm_var. For
example:
#> Variance of Fuzzy monetary results:
#>
#> Type of estimator:
#>
#> bootstrap_naive
#>
#> Estimate(s):
#>
#> a b c d e f g h
#> 0.004590 0.005659 0.007915 0.011578 0.005051 0.003710 0.008928 0.006941
#> i j k l m n o p
#> 0.002886 0.004883 0.007624 0.003945 0.010905 0.002366 0.006753 0.006203
#> q r s t u v w x
#> 0.015099 0.005297 0.004728 0.000368 0.007285 0.008584 0.007285 0.004070
#> y z
#> 0.009698 0.005751
#> Variance of Fuzzy monetary results:
#>
#> Type of estimator:
#>
#> jackknife
#>
#> Estimate(s):
#>
#> a b c d e f g h
#> 0.006067 0.009761 0.009576 0.012358 0.013549 0.003032 0.014162 0.024544
#> i j k l m n o p
#> 0.005101 0.006252 0.014782 0.003836 0.016164 0.002989 0.008660 0.007709
#> q r s t u v w x
#> 0.022870 0.008810 0.007936 0.000672 0.008556 0.010271 0.007848 0.007644
#> y z
#> 0.009668 0.005607
Fuzzy supplementary
variance = fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL,
dimensions = dimensions, breakdown = NULL, HCR = 0.12,
alpha = 2, rho = NULL, type = 'bootstrap_naive',
M = NULL, R = 50, verbose = F)
summary(variance)
The following uses the Jackknife
fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL, dimensions = dimensions,
stratum = eusilc$stratum, psu = eusilc$psu, verbose = F, f = .01,
breakdown = NULL, alpha = 3, rho = NULL, type = "jackknife", fixed = T)%>%summary()
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