MLRFE {biogeom}R Documentation

Modified Lobry-Rosso-Flandrois (LRF) Equation

Description

MLRFE is used to calculate yy values at given xx 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 xx 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:

\mboxifx(xmin, xmin+xmax2),\mbox{if } x \in{\left(x_{\mathrm{min}}, \ \frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}\right)},

y=yopt{(xxmin)(xxmax)2(xmaxxopt)[(xmaxxopt)(xxopt)(xminxopt)(xopt+xmax2x)]}δ;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};

\mboxifx[xmin+xmax2, xmax),\mbox{if } x \in{\left[\frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y=yopt{(xxmax)(xxmin)2(xoptxmin)[(xoptxmin)(xxopt)(xoptxmax)(xopt+xmin2x)]}δ;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};

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

y=0.y = 0.

Here, xx and yy represent the independent and dependent variables, respectively; yopty_{\mathrm{opt}}, xoptx_{\mathrm{opt}}, xminx_{\mathrm{min}}, xmaxx_{\mathrm{max}}, and δ\delta are constants to be estimated; yopty_{\mathrm{opt}} represents the maximum yy, and xoptx_{\mathrm{opt}} is the xx value associated with the maximum yy (i.e., yopty_{\mathrm{opt}}); and xminx_{\mathrm{min}} and xmaxx_{\mathrm{max}} represents the lower and upper intersections between the curve and the xx-axis. There are five elements in P, representing the values of yopty_{\mathrm{opt}}, xoptx_{\mathrm{opt}}, xminx_{\mathrm{min}}, xmaxx_{\mathrm{max}}, and δ\delta, respectively.

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

\mboxifx(0, xmax2),\mbox{if } x \in{\left(0, \ \frac{x_{\mathrm{max}}}{2}\right)},

y=yopt{x(xxmax)2(xmaxxopt)[(xmaxxopt)(xxopt)+xopt(xopt+xmax2x)]}δ;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};

\mboxifx[xmax2, xmax),\mbox{if } x \in{\left[\frac{x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y=yopt{(xxmax)x2xopt[xopt(xxopt)(xoptxmax)(xopt2x)]}δ;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};

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

y=0.y = 0.

There are four elements in P, representing the values of yopty_{\mathrm{opt}}, xoptx_{\mathrm{opt}}, xmaxx_{\mathrm{max}}, and δ\delta, respectively.

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

\mboxifx(xmin, xmin+xmax2),\mbox{if } x \in{\left(x_{\mathrm{min}}, \ \frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}\right)},

y=yopt(xxmin)(xxmax)2(xmaxxopt)[(xmaxxopt)(xxopt)(xminxopt)(xopt+xmax2x)];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]};

\mboxifx[xmin+xmax2, xmax),\mbox{if } x \in{\left[\frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y=yopt(xxmax)(xxmin)2(xoptxmin)[(xoptxmin)(xxopt)(xoptxmax)(xopt+xmin2x)];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]};

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

y=0.y = 0.

There are four elements in P, representing the values of yopty_{\mathrm{opt}}, xoptx_{\mathrm{opt}}, xminx_{\mathrm{min}}, and xmaxx_{\mathrm{max}}, respectively.

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

\mboxifx(0, xmax2),\mbox{if } x \in{\left(0, \ \frac{x_{\mathrm{max}}}{2}\right)},

y=yoptx(xxmax)2(xmaxxopt)[(xmaxxopt)(xxopt)+xopt(xopt+xmax2x)];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]};

\mboxifx[xmax2, xmax),\mbox{if } x \in{\left[\frac{x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y=yopt(xxmax)x2xopt[xopt(xxopt)(xoptxmax)(xopt2x)];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]};

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

y=0.y = 0.

There are three elements in P, representing the values of yopty_{\mathrm{opt}}, xoptx_{\mathrm{opt}}, and xmaxx_{\mathrm{max}}, respectively.

Value

The yy values predicted by the modified LRF equation or one of its simplified versions.

Note

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

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

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, 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()

[Package biogeom version 1.4.3 Index]