MbetaE {biogeom}

Modified Beta Equation

Description

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

Usage

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.

Value

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

Note

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

Author(s)

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

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. 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. doi:10.1111/nyas.14862

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()