MbetaE | R Documentation |
MbetaE
is used to calculate y
values at given x
values
using the modified beta equation or one of its simplified versions.
MbetaE(P, x, simpver = 1)
P |
the parameters of the modified beta equation or one of its simplified versions. |
x |
the given |
simpver |
an optional argument to use the simplified version of the modified beta equation. |
When simpver = NULL
, the modified beta equation is selected:
\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},
y = y_{\mathrm{opt}}{ \left[\left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x-x_{\mathrm{min}}}{x_{\mathrm{opt}}-x_{\mathrm{min}}}\right)^{\frac{x_{\mathrm{opt}}-x_{\mathrm{min}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} \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}}
represent 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 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, \ x_{\mathrm{max}}\right)},
y = y_{\mathrm{opt}}{ \left[\left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x}{x_{\mathrm{opt}}}\right)^{\frac{x_{\mathrm{opt}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} \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}}, \ x_{\mathrm{max}}\right)},
y = y_{\mathrm{opt}}{ \left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x-x_{\mathrm{min}}}{x_{\mathrm{opt}}-x_{\mathrm{min}}}\right)^{\frac{x_{\mathrm{opt}}-x_{\mathrm{min}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} };
\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, \ x_{\mathrm{max}}\right)},
y = y_{\mathrm{opt}}{ \left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x}{x_{\mathrm{opt}}}\right)^{\frac{x_{\mathrm{opt}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} };
\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.
The y
values predicted by the modified beta equation or one of its simplified versions.
We have added a parameter \delta
in the original beta equation (i.e., simpver = 2
) to increase the flexibility for data fitting.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
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")}
areaovate
, curveovate
, fitovate
, fitsigmoid
,
MBriereE
, MLRFE
, MPerformanceE
, sigmoid
x1 <- seq(-5, 15, len=2000)
Par1 <- c(3, 3, 10, 2)
y1 <- MbetaE(P=Par1, x=x1, simpver=1)
dev.new()
plot( x1, y1,cex.lab=1.5, cex.axis=1.5, type="l",
xlab=expression(italic(x)), ylab=expression(italic(y)) )
graphics.off()
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