MLRFE {biogeom} | R Documentation |
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 |
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 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()