# Data-Fitting Function for the Rotated and Right-Shifted Lorenz Curve `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`. UTF-8 ```r fitLorenz(expr, z, ini.val, simpver = 4, control = list(), 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) ``` ## Arguments - `expr`: version 4 or 5 of `MPerformanceE`, or the Lorenz equations including `SarabiaE`, `SCSE`, and `SHE`. - `z`: the observations of size distribution (i.e., the household income distribution, the leaf size distribution). - `ini.val`: the initial values of the model parameters. - `simpver`: an optional argument to use version 4 or 5 of `MPerformanceE`. - `control`: the list of control parameters for using the `optim` function in package `stats`. - `par.list`: the option of showing the list of parameters on the screen. - `fig.opt`: an optional argument to draw the original and rotated Lorenz curves. - `np`: the number of data points to draw the predicted original and rotated Lorenz curves. - `xlab`: the label of the $x$-axis when showing the original Lorenz curve. - `ylab`: the label of the $y$-axis when showing the original Lorenz curve. - `main`: the main title of the figure. - `subdivisions`: please see the arguments for the `integrate` function in package `stats`. - `rel.tol`: please see the arguments for the `integrate` function in package `stats`. - `abs.tol`: please see the arguments for the `integrate` function in package `stats`. - `stop.on.error`: please see the arguments for the `integrate` function in package `stats`. - `keep.xy`: please see the arguments for the `integrate` function in package `stats`. - `aux`: please see the arguments for the `integrate` function in package `stats`. - `par.limit`: an optional argument to limit the numerical ranges of model parameters of the three Lorenz equations including `SarabiaE`, `SCSE`, and `SHE`. ## Details Here, `ini.val` only includes the initial values of the model parameters as a list. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used 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. 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 vector of parameter(s), `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). ## Returns - **x1**: 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. - **y1**: the cumulative proportion of the size of an entity of interest. - **x**: the $x$ coordinates of the rotated and right-shifted `y1` versus `x1`. - **y**: the $y$ coordinates of the rotated and right-shifted `y1` versus `x1`. - **par**: the estimates of the model parameters. - **r.sq**: the coefficient of determination between the observed and predicted $y$ values. - **RSS**: the residual sum of squares between the observed and predicted $y$ values. - **sample.size**: the number of data points used in the data fitting. - **GC**: the calculated Gini coefficient. ## Note When `MPerformanceE` is selected, the estimates of the model parameters denote those in `MPerformanceE` rather than being obtained by directly fitting the `y1` versus `x1` data; when a Lorenz function is selected, the estimates of the model parameters denote those in the Lorenz function. ## Author(s) Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca . ## References 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$-$1565. tools:::Rd_expr_doi("10.1007/s00468-023-02448-8") Lorenz, M.O. (1905) Methods of measuring the concentration of wealth. **Journal of the American Statistical Association** 9(70), 209$-$219. tools:::Rd_expr_doi("10.2307/2276207") Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. **Computer Journal** 7, 308$-$313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308") Sarabia, J.-M. (1997) A hierarchy of Lorenz curves based on the generalized Tukey's lambda distribution. **Econometric Reviews** 16, 305$-$320. tools:::Rd_expr_doi("10.1080/07474939708800389") Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. **Annals of the New York Academy of Sciences** 1516, 123$-$134. tools:::Rd_expr_doi("10.1111/nyas.14862") Sitthiyot, T., Holasut, K. (2023) A universal model for the Lorenz curve with novel applications for datasets containing zeros and/or exhibiting extreme inequality. **Scientific Reports** 13, 4729. tools:::Rd_expr_doi("10.1038/s41598-023-31827-x") ## See Also `LeafSizeDist`, `MPerformanceE`, `SarabiaE`, `SCSE`, `SHE` ## Examples ```r data(LeafSizeDist) CulmNumber <- c() for(i in 1:length(LeafSizeDist$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() ```

fitLorenz function