MPerformanceE is used to calculate \(y\) values at given \(x\) values using
the modified performance equation or one of its simplified versions.
MPerformanceE(P, x, simpver = 1)The \(y\) values predicted by the modified performance equation or one of its simplified versions.
the parameters of the modified performance equation or one of its simplified versions.
the given \(x\) values.
an optional argument to use the simplified version of the modified performance equation.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
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.
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")
areaovate, curveovate, fitLorenz,
fitovate, fitsigmoid, MbetaE,
MBriereE, MLRFE, sigmoid
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()
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