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 Betti and Verma (2008))
\[ \begin{split} \mu_i&=\left(1-F^{(M)}_i\right)^{\alpha-1}\left(1-L^{(M)}_i\right)=\\ &=\left(\frac{\sum_j w_j|x_j> x_i}{\sum_j w_j|x_j> x_1}\right)^{\alpha-1} \left(\frac{\sum_j w_jx_j|x_j> x_i}{\sum_j w_jx_j|x_j> x_1}\right) \end{split} \]
where \(w_i\) is the sampling weight of unit \(i\).
fm="ZBM"(see Zedini and Belhadj (2015) and Belhadj and Matoussi (2010))
\[ \mu_Q(x_i) = \begin{cases} 1 & a \le x_i < b\\ \frac{-x_i}{c-b} + \frac{c}{c-b} & b \le x_i < c\\ 0 & x_i < a \cup x_i \ge c\\ \end{cases} \]
fm="belhadj"(see Besma (2015))
\[ \mu_Q(x_i) = \begin{cases} 1 & y_i < z_1\\ \mu^1 = 1-\frac{1}{2}\left(\frac{y_i-z_1}{z_1}\right)^b & z_1 \le y_i < z^{*}\\ \mu^2 = 1-\frac{1}{2}\left(\frac{z_2 - y_i}{z_2}\right)^b & z^{*} \le y_i < z_2\\ 0 & y_i \ge z_2 \end{cases} \]
\[ \mu_Q(x_i) = \begin{cases} 1 & 0 < x_i < z_{min} \\ \frac{-x_i}{z_{\max} - z_{\min}} + \frac{-z_{\max}}{z_{\max} - z_{\min}} & z_{min} \le x_i < z_{max}\\ 0 & x_i \ge z_max \end{cases} \]
fm="chakravarty"(see Chakravarty (2019))
\[ \mu_Q(x_i) = \begin{cases} 1 & x_i = 0\\ \frac{z'' - x_i}{z''} & 0 \le x_i < z''\\ 0 & x_i \ge z'' \end{cases} \]
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
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
FuzzyPovertyRprovides the functionHCRto 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)$HCR # add poverty thresholdif needed, the package has a built-in function
eq_predicate to calculate the equivalised disposable income
using some equivalence scales.
- Construction of the Fuzzy Monetary measure.
verma = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL,
HCR = hcr, interval = c(1,20), alpha = NULL, fm = "verma", verbose = FALSE)
verma$fm
#> [1] "verma"
class(verma)
#> [1] "FuzzyMonetary"
summary(verma)
#> 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
plot(verma)
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$eq_income, fm = "verma1999", weight = eusilc$DB090, ID = NULL, interval = c(1,10), alpha = 2)The result of fm_construct using fm="verma"
is a list containing
- a
data.frameof individuals’ membership functions sorted in descending order (i.e. from most poor to least poor)
head(verma$results)
#> ID predicate weight mu
#> 1 44 3.225806 465.3585 1.0000000
#> 2 450 6.666667 1010.5060 0.9999952
#> 3 372 12.903226 304.7067 0.9999924
#> 4 490 17.857143 1177.4120 0.9999774
#> 5 245 20.000000 853.0255 0.9999653
#> 6 130 42.424242 1141.3410 0.9999308- The estimated FM measure (note that this equals the HCR if
breakdown=NULL). However, one can obtain estimates for sub-domains using thebreakdownargument as follows
verma.break = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,10), alpha = NULL, breakdown = eusilc$db040)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
plot(verma.break)
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- The
alphaparameter.
Example using fm=belhadj2015
The construction of a fuzzy index using the membership function as
Besma (2015) is obtained by specifying
fm="belhadj". The arguments z1,
z2 and b need user’s specification (see
Formula above and Besma (2015)).
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 and returned as z_star.
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) summary(belhadj)
#> 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
plot(belhadj)
Example using fm=chakravarty
Chakravarty (2019) axiomatic fuzzy
index is obtained setting fm = "chakravarty". The argument
z needs user’s specification
z = 60000
chakravarty = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "chakravarty", z = z)summary(chakravarty)
#> 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
plot(chakravarty)
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)| Verma | Chakravarty | |
|---|---|---|
| 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 |
