SHE {biogeom}

Sitthiyot-Holasut Equation

Description

SHE is used to calculate y values at given x values using the Sitthiyot-Holasut equation. The equation describes the y coordinates of the Lorenz curve.

Usage

SHE(P, x)

Arguments

P

the parameters of the Sitthiyot-Holasut equation.

x

the given x values ranging between 0 and 1.

Details

\mbox{if } x > \delta,

y = \left(1-\rho\right)\,\left[\left(\frac{2}{P+1}\right)\left(\frac{x-\delta}{1-\delta}\right)\right] + \rho\,\left[\left(1-\omega\right)\left(\frac{x-\delta}{1-\delta}\right)^{P}+\omega\,\left\{1-\left[1-\left(\frac{x-\delta}{1-\delta}\right)\right]^{\frac{1}{P}}\right\}\right];

\mbox{if } x \le \delta,

y = 0.

Here, x and y represent the independent and dependent variables, respectively; and \delta, \rho, \omega and P are constants to be estimated, where 0 \le \delta < 1, 0 \le \rho \le 1, 0 \le \omega \le 1, and P \ge 1. There are four elements in P, representing the values of \delta, \rho, \omega and P, respectively.

Value

The y values predicted by the Sitthiyot-Holasut equation.

Note

The numerical range of x should range between 0 and 1. When x < \delta, the x value is assigned to be \delta.

Author(s)

Peijian Shi [e-mail], Johan Gielis [e-mail], Brady K. Quinn [e-mail].

References

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. doi:10.1038/s41598-023-31827-x

See Also

fitLorenz, MPerformanceE, SarabiaE, SCSE

Examples

X1  <- seq(0, 1, len=2000)
Pa3 <- c(0, 1, 0.446, 1.739)
Y3  <- SHE(P=Pa3, x=X1)

dev.new()
plot( X1, Y3, cex.lab=1.5, cex.axis=1.5, type="l", asp=1, xaxs="i", 
      yaxs="i", xlim=c(0, 1), ylim=c(0, 1), 
      xlab="Cumulative proportion of the number of infructescences", 
      ylab="Cumulative proportion of the infructescence length" )

graphics.off()