MPerformanceE {biogeom}

Modified Performance Equation

Description

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

Usage

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.

Value

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).

Author(s)

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

References

Huey, R.B., Stevenson, R.D. (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. American Zoologist 19, 357-366. 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. 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. 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. doi:10.1111/nyas.14862

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()