MPerformanceE function

Modified Performance Equation

MPerformanceE is used to calculate $y$ values at given $x$ values using the modified performance equation or one of its simplified versions. UTF-8

MPerformanceE(P, x, simpver = 1)

Arguments

• P: the parameters of the modified performance equation or one of its simplified versions.
• x: the given $x$ values.
• simpver: an optional argument to use the simplified version of the modified performance equation.

Details

When simpver = NULL, the modified performance equation is selected:

$$\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = c\left(1-e^{-K_{1}\left(x-x_{\mathrm{min}}\right)}\right)^{a}\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right)^{b};$$ $$\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

Here, $x$ and $y$ represent the independent and dependent variables, respectively; and $c$, $K_{1}$, $K_{2}$, $x_{\mathrm{min}}$, $x_{\mathrm{max}}$, $a$, and $b$ are constants to be estimated, where $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ represents the lower and upper intersections between the curve and the $x$-axis. $y$ is defined as 0 when $x < x_{\mathrm{min}}$ or $x > x_{\mathrm{max}}$. There are seven elements in P, representing the values of $c$, $K_{1}$, $K_{2}$, $x_{\mathrm{min}}$, $x_{\mathrm{max}}$, $a$, and $b$, respectively.

$\quad$ When simpver = 1, the simplified version 1 is selected:

$$\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = c\left(1-e^{-K_{1}x}\right)^{a}\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right)^{b};$$ $$\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are six elements in P, representing the values of $c$, $K_{1}$, $K_{2}$, $x_{\mathrm{max}}$, $a$, and $b$ respectively.

$\quad$ When simpver = 2, the simplified version 2 is selected:

$$\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = c\left(1-e^{-K_{1}\left(x-x_{\mathrm{min}}\right)}\right)\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right);$$ $$\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are five elements in P representing the values of $c$, $K_{1}$, $K_{2}$, $x_{\mathrm{min}}$, and $x_{\mathrm{max}}$, respectively.

$\quad$ When simpver = 3, the simplified version 3 is selected:

$$\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = c\left(1-e^{-K_{1}x}\right)\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right);$$ $$\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are four elements in P representing the values of $c$, $K_{1}$, $K_{2}$, and $x_{\mathrm{max}}$, respectively.

$\quad$ When simpver = 4, the simplified version 4 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 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.

Returns

The $y$ values predicted by the modified performance equation or one of its simplified versions.

Note

We have added two parameters $a$ and $b$ in the original performance equation (i.e., simpver = 2) to increase the flexibility for data fitting. The cases of simpver = 4 and simpver = 5 are used to describe the rotated and right-shifted Lorenz curve (see Lian et al. [2023] for details).

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")

Shi, P., Ge, F., Sun, Y., Chen, C. (2011) A simple model for describing the effect of temperature on insect developmental rate. Journal of Asia-Pacific Entomology

14, 15$-$20. tools:::Rd_expr_doi("10.1016/j.aspen.2010.11.008")

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")

Author(s)

Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .

See Also

areaovate, curveovate, fitLorenz, fitovate, fitsigmoid, MbetaE, MBriereE, MLRFE, sigmoid

Examples

x4   <- seq(0, 40, len=2000)
Par4 <- c(0.117, 0.090, 0.255, 5, 35, 1, 1)
y4   <- MPerformanceE(P=Par4, x=x4, simpver=NULL)

dev.new()
plot( x4, y4, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

graphics.off()