MLRFE: Modified Lobry-Rosso-Flandrois (LRF) Equation

In biogeom: Biological Geometries

Modified Lobry-Rosso-Flandrois (LRF) Equation

Description

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

Usage

MLRFE(P, x, simpver = 1)

Arguments

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

Details

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

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ \frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}\right)},

y = y_{\mathrm{opt}}\left\{\frac{\left(x-x_{\mathrm{min}}\right)\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})-(x_{\mathrm{min}}-x_{\mathrm{opt}})(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]}\right\}^{\delta};

\mbox{if } x \in{\left[\frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}\left\{\frac{\left(x-x_{\mathrm{max}}\right)\left(x-x_{\mathrm{min}}\right)^{2}}{\left(x_{\mathrm{opt}}-x_{\mathrm{min}}\right)\left[(x_{\mathrm{opt}}-x_{\mathrm{min}})(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}+x_{\mathrm{min}}-2x)\right]}\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}} represents the lower and upper intersections between the curve and the x-axis. 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, \ \frac{x_{\mathrm{max}}}{2}\right)},

y = y_{\mathrm{opt}}\left\{\frac{x\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})+x_{\mathrm{opt}}(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]}\right\}^{\delta};

\mbox{if } x \in{\left[\frac{x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}\left\{\frac{\left(x-x_{\mathrm{max}}\right)x^{2}}{x_{\mathrm{opt}}\left[x_{\mathrm{opt}}(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}-2x)\right]}\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}}, \ \frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}\right)},

y = \frac{y_{\mathrm{opt}}\left(x-x_{\mathrm{min}}\right)\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})-(x_{\mathrm{min}}-x_{\mathrm{opt}})(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]};

\mbox{if } x \in{\left[\frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = \frac{y_{\mathrm{opt}}\left(x-x_{\mathrm{max}}\right)\left(x-x_{\mathrm{min}}\right)^{2}}{\left(x_{\mathrm{opt}}-x_{\mathrm{min}}\right)\left[(x_{\mathrm{opt}}-x_{\mathrm{min}})(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}+x_{\mathrm{min}}-2x)\right]};

\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, \ \frac{x_{\mathrm{max}}}{2}\right)},

y = \frac{y_{\mathrm{opt}}x\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})+x_{\mathrm{opt}}(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]};

\mbox{if } x \in{\left[\frac{x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = \frac{y_{\mathrm{opt}}\left(x-x_{\mathrm{max}}\right)x^{2}}{x_{\mathrm{opt}}\left[x_{\mathrm{opt}}(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}-2x)\right]};

\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 LRF equation or one of its simplified versions.

Note

We have added n parameter \delta in the original LRF 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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/nyas.14862")}

See Also

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

Examples

x3   <- seq(-5, 15, len=2000)
Par3 <- c(3, 3, 10, 2)
y3   <- MbetaE(P=Par3, x=x3, simpver=1)

dev.new()
plot( x3, y3, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )
 
graphics.off()

biogeom documentation built on Aug. 24, 2025, 5:08 p.m.