MBriereE {biogeom}

Modified Briere Equation

Description

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

Usage

MBriereE(P, x, simpver = 1)

Arguments

P

the parameters of the modified Brière equation or one of its simplified versions.

x

the given x values.

simpver

an optional argument to use the simplified version of the modified Brière equation.

Details

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

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = a\left|x(x-x_{\mathrm{min}})(x_{\mathrm{max}}-x)^{1/m}\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; and a, m, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta 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 five elements in P, representing the values of a, m, 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 = a\left|x^{2}(x_{\mathrm{max}}-x)^{1/m}\right|^{\delta};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P, representing the values of a, m, 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 = ax(x-x_{\mathrm{min}})(x_{\mathrm{max}}-x)^{1/m};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P representing the values of a, m, 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 = ax^{2}(x_{\mathrm{max}}-x)^{1/m};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

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

Value

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

Note

We have added a parameter \delta in the original Brière 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

Brière, J.-F., Pracros, P, Le Roux, A.-Y., Pierre, J.-S. (1999) A novel rate model of temperature-dependent development for arthropods. Environmental Entomology 28, 22-29. doi:10.1093/ee/28.1.22

Cao, L., Shi, P., Li, L., Chen, G. (2019) A new flexible sigmoidal growth model. Symmetry 11, 204. doi:10.3390/sym11020204

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

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, MbetaE, MLRFE, MPerformanceE, sigmoid

Examples

x2   <- seq(-5, 15, len=2000)
Par2 <- c(0.01, 3, 0, 10, 1)
y2   <- MBriereE(P=Par2, x=x2, simpver=NULL)

dev.new()
plot( x2, y2, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

graphics.off()