MLRFE is used to calculate \(y\) values at given \(x\) values
using the modified LRF equation or one of its simplified versions.
MLRFE(P, x, simpver = 1)The \(y\) values predicted by the modified LRF equation or one of its simplified versions.
the parameters of the modified LRF equation or one of its simplified versions.
the given \(x\) values.
an optional argument to use the simplified version of the modified LRF equation.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
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.
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. 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. tools:::Rd_expr_doi("10.1111/nyas.14862")
areaovate, curveovate, fitovate, fitsigmoid,
MbetaE, MBriereE, MPerformanceE, sigmoid
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()
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