MbetaE function

Modified Beta Equation

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

MbetaE(P, x, simpver = 1)

Arguments

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

Details

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

$$\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = y_{\mathrm{opt}}{ \left[\left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x-x_{\mathrm{min}}}{x_{\mathrm{opt}}-x_{\mathrm{min}}}\right)^{\frac{x_{\mathrm{opt}}-x_{\mathrm{min}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} \right] }^{\delta};$$ $$\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

Here, $x$ and $y$ represent the independent and dependent variables, respectively; $y_{\mathrm{opt}}$, $x_{\mathrm{opt}}$, $x_{\mathrm{min}}$, $x_{\mathrm{max}}$, and $\delta$ are constants to be estimated; $y_{\mathrm{opt}}$ represents the maximum $y$, and $x_{\mathrm{opt}}$ is the $x$ value associated with the maximum $y$ (i.e., $y_{\mathrm{opt}}$); and $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ represent 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 five elements in P, representing the values of $y_{\mathrm{opt}}$, $x_{\mathrm{opt}}$, $x_{\mathrm{min}}$, $x_{\mathrm{max}}$, and $\delta$, respectively.

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

$$\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = y_{\mathrm{opt}}{ \left[\left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x}{x_{\mathrm{opt}}}\right)^{\frac{x_{\mathrm{opt}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} \right] }^{\delta};$$ $$\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are four elements in P, representing the values of $y_{\mathrm{opt}}$, $x_{\mathrm{opt}}$, $x_{\mathrm{max}}$, and $\delta$, respectively.

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

$$\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = y_{\mathrm{opt}}{ \left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x-x_{\mathrm{min}}}{x_{\mathrm{opt}}-x_{\mathrm{min}}}\right)^{\frac{x_{\mathrm{opt}}-x_{\mathrm{min}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} };$$ $$\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are four elements in P, representing the values of $y_{\mathrm{opt}}$, $x_{\mathrm{opt}}$, $x_{\mathrm{min}}$, and $x_{\mathrm{max}}$, respectively.

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

$$\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = y_{\mathrm{opt}}{ \left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x}{x_{\mathrm{opt}}}\right)^{\frac{x_{\mathrm{opt}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} };$$ $$\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are three elements in P, representing the values of $y_{\mathrm{opt}}$, $x_{\mathrm{opt}}$, and $x_{\mathrm{max}}$, respectively.

Returns

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

References

Shi, P., Fan, M., Ratkowsky, D.A., Huang, J., Wu, H., Chen, L., Fang, S., Zhang, C. (2017) Comparison of two ontogenetic growth equations for animals and plants. Ecological Modelling 349, 1$-$10. tools:::Rd_expr_doi("10.1016/j.ecolmodel.2017.01.012")

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 .

Note

We have added a parameter $\delta$ in the original beta equation (i.e., simpver = 2) to increase the flexibility for data fitting.

See Also

areaovate, curveovate, fitovate, fitsigmoid, MBriereE, MLRFE, MPerformanceE, sigmoid

Examples

x1   <- seq(-5, 15, len=2000)
Par1 <- c(3, 3, 10, 2)
y1   <- MbetaE(P=Par1, x=x1, simpver=1)

dev.new()
plot( x1, y1,cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )
 
graphics.off()