fitLorenz is used to estimate the parameters of the rotated and right-shifted Lorenz curve
using version 4 or 5 of MPerformanceE, or the Lorenz equations including
SarabiaE, SCSE, and SHE.
fitLorenz(expr, z, ini.val, simpver = 4,
method = "Nelder-Mead", control = list(),
lower = -Inf, upper = Inf,
par.list = FALSE, fig.opt = FALSE, np = 2000,
xlab=NULL, ylab=NULL, main = NULL, subdivisions = 100L,
rel.tol = .Machine$double.eps^0.25,
abs.tol = rel.tol, stop.on.error = TRUE,
keep.xy = FALSE, aux = NULL, par.limit = TRUE)
the cumulative proportion of the number of an entity of interest, i.e., the number of households of a city, the number of leaves of a plant.
the cumulative proportion of the size of an entity of interest.
the \(x\) coordinates of the rotated and right-shifted y1 versus x1.
the \(y\) coordinates of the rotated and right-shifted y1 versus x1.
the estimates of the model parameters.
the coefficient of determination between the observed and predicted \(y\) values.
the residual sum of squares between the observed and predicted \(y\) values.
the number of data points used in the data fitting.
the \(x\)-coordinate of the maximum value point on the rotated and right-shifted Lorenz curve.
the \(y\)-coordinate of the maximum value point on the rotated and right-shifted Lorenz curve.
the Lorenz asymmetry coefficient associated with the rotated and right-shifted Lorenz curve, which equals \(x_c/\sqrt{2}\).
the calculated Gini coefficient.
version 4 or 5 of MPerformanceE, or the Lorenz equations including
SarabiaE, SCSE, and SHE.
the observations of size distribution (i.e., the household income distribution, the leaf size distribution).
the initial values of the model parameters.
an optional argument to use version 4 or 5 of MPerformanceE.
an optional argument to select an optimization method.
the list of control parameters for using the optim function in package stats.
the lower bounds on the variables for the L-BFGS-B algorithm.
the upper bounds on the variables for the L-BFGS-B algorithm.
the option to show the list of parameters on the screen.
an optional argument to draw the original and rotated Lorenz curves.
the number of data points to draw the predicted original and rotated Lorenz curves.
the label of the \(x\)-axis when showing the original Lorenz curve.
the label of the \(y\)-axis when showing the original Lorenz curve.
the main title of the figure.
please see the arguments for the integrate function in package stats.
please see the arguments for the integrate function in package stats.
please see the arguments for the integrate function in package stats.
please see the arguments for the integrate function in package stats.
please see the arguments for the integrate function in package stats.
please see the arguments for the integrate function in package stats.
an optional argument to limit the numerical ranges of model parameters of the three Lorenz
equations including SarabiaE, SCSE, and SHE.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
Here, ini.val only includes the initial values of the model parameters as a list.
The Nelder-Mead algorithm (Nelder and Mead, 1965) and the optimization method
(referred to as L-BFGS-B) proposed by Byrd et al. (1995) in which each variable can be given
a lower and/or upper bound can be selected to carry out the optimization of minimizing
the residual sum of squares (RSS) between the observed and predicted \(y\) values.
The optim function in package stats was used to carry out
the Nelder-Mead algorithm and the L-BFGS-B algorithm. When the user chooses
the L-BFGS-B algorithm, par.limit should be set to be FALSE.
Here, versions 4 and 5 of MPerformanceE and the Lorenz equations including
SarabiaE, SCSE, and SHE can be used to fit the rotated
and right-shifted Lorenz curve.
\(\quad\) When simpver = 4, the simplified version 4 of MPerformanceE is selected:
$$\mbox{if } x \in{\left(0, \ \sqrt{2}\right)},$$
$$y = c\left(1-e^{-K_{1}x}\right)^{a}\left(1-e^{K_{2}\left(x-\sqrt{2}\right)}\right)^{b};$$
$$\mbox{if } x \notin{\left(0, \ \sqrt{2}\right)},$$
$$y = 0.$$
There are five elements in P, representing
the values of \(c\), \(K_{1}\), \(K_{2}\), \(a\), and \(b\), respectively.
\(\quad\) When simpver = 5, the simplified version 5 of MPerformanceE is selected:
$$\mbox{if } x \in{\left(0, \ \sqrt{2}\right)},$$
$$y = c\left(1-e^{-K_{1}x}\right)\left(1-e^{K_{2}\left(x-\sqrt{2}\right)}\right);$$
$$\mbox{if } x \notin{\left(0, \ \sqrt{2}\right)},$$
$$y = 0.$$
There are three elements in P, representing
the values of \(c\), \(K_{1}\), and \(K_{2}\), respectively.
\(\quad\) For the Lorenz functions, the user can define any formulae that follow the below form:
Lorenz.fun <- function(P, x){...}, where P is the parameter vector, x is the preditor that
ranges between 0 and 1 representing the cumulative proportion of the number of individuals in a statistical
unit, and Lorenz.fun is the name of a Lorenz function defined by the user, which also ranges between 0 and 1
representing the cumulative
proportion of the income or size in a statistical unit. This package provides three representative Lorenz
functions: SarabiaE, SCSE, and SHE.
\(\quad\) Here, the Gini coefficient (GC) is calculated as follows when MPerformanceE
is selected:
$$\mbox{GC} = 2\int_{0}^{\sqrt{2}}y\,dx,$$
where \(x\) and \(y\) are the independent and dependent variables
in version 4 or 5 of MPerformanceE, respectively.
\(\quad\) However, the Gini coefficient (GC) is calculated as follows when a Lorenz function, e.g.,
SCSE, is selected:
$$\mbox{GC} = 2\int_{0}^{1}y\,dx,$$
where \(x\) and \(y\) are the independent and dependent variables
in the Lorenz function, respectively.
\(\quad\) For SarabiaE and SHE, there are explicit formulae for GC
(Sarabia, 1997; Sitthiyot and Holasut, 2023).
\(\quad\) In addition, the function provides the Lorenz asymmetry coefficient (LAC) based on
the rotated and right-shifted Lorenz curve (RRLC), which is used to examine whether the RRLC is
skewed or symmetrical (Chen et al., 2025). The LAC takes the form:
$$\mbox{LAC} = \frac{x_c}{2},$$
where \(x_c\) represents the \(x\)-coordinate of the maximum value point on the RRLC.
When LAC > 0.5, the RRLC is left-skewed; when LAC < 0.5, the RRLC is right-skewed;
when LAC = 0.5, the RRLC is bilaterally symmetrical about the vertical line \(x = x_c\).
The three cases of the LAC numerical values correspond to three size distribution patterns:
(i) inequality driven by abundant large individuals, (ii) inequality dominated by a few large individuals,
and (iii) parity in contributions between small and large individuals (Chen et al., 2025).
Byrd, R.H., Lu, P., Nocedal, J., Zhu, C. (1995) A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16, 1190\(-\)1208. tools:::Rd_expr_doi("10.1137/0916069")
Chen, Y., Jiang, F., Damgaard, C.F., Shi, P., Weiner, J. (2025) Re-expression of the Lorenz asymmetry coefficient on the rotated and right-shifted Lorenz curve of leaf area distributions. Plants 14, 1345. tools:::Rd_expr_doi("10.3390/plants14091345")
Huey, R.B., Stevenson, R.D. (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. American Zoologist 19, 357\(-\)366. tools:::Rd_expr_doi("10.1093/icb/19.1.357")
Lian, M., Shi, P., Zhang, L., Yao, W., Gielis, J., Niklas, K.J. (2023) A generalized performance equation and its application in measuring the Gini index of leaf size inequality. Trees \(-\) Structure and Function 37, 1555\(-\)1565. tools:::Rd_expr_doi("10.1007/s00468-023-02448-8")
Lorenz, M.O. (1905) Methods of measuring the concentration of wealth. Journal of the American Statistical Association 9(70), 209\(-\)219. tools:::Rd_expr_doi("10.2307/2276207")
Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308\(-\)313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308")
Sarabia, J.-M. (1997) A hierarchy of Lorenz curves based on the generalized Tukey's lambda distribution. Econometric Reviews 16, 305\(-\)320. tools:::Rd_expr_doi("10.1080/07474939708800389")
Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516, 123\(-\)134. tools:::Rd_expr_doi("10.1111/nyas.14862")
Sitthiyot, T., Holasut, K. (2023) A universal model for the Lorenz curve with novel applications for datasets containing zeros and/or exhibiting extreme inequality. Scientific Reports 13, 4729. tools:::Rd_expr_doi("10.1038/s41598-023-31827-x")
LeafSizeDist, MPerformanceE, SarabiaE,
SCSE, SHE
data(LeafSizeDist)
CulmNumber <- c()
for(i in 1:length(LeafSizeDist$Code)){
temp <- as.numeric( strsplit(LeafSizeDist$Code[i], "-", fixed=TRUE)[[1]][1] )
CulmNumber <- c(CulmNumber, temp)
}
uni.CN <- sort( unique(CulmNumber) )
ind <- CulmNumber==uni.CN[1]
A0 <- LeafSizeDist$A[ind]
ini.val1 <- list(0.5, 0.1, c(0.01, 0.1, 1, 5, 10), 1, 1)
ini.val2 <- list(0.5, 0.1, c(0.01, 0.1, 1, 5, 10))
resu1 <- fitLorenz(MPerformanceE, z=A0, ini.val=ini.val1, simpver=4, fig.opt=TRUE)
resu2 <- fitLorenz(MPerformanceE, z=A0, ini.val=ini.val2, simpver=5, fig.opt=TRUE)
resu1$par
resu2$par
ini.val3 <- list(0.9, c(10, 50, 100, 500), 1, 0)
resu3 <- fitLorenz( SarabiaE, z=A0, ini.val=ini.val3, par.limit=TRUE,
fig.opt=TRUE, control=list(reltol=1e-20, maxit=10000) )
resu3$par
resu3$GC
graphics.off()
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