fitAM: Fitting the 2-D Profiles of Apical Meristems
In biogeom: Biological Geometries
fitAM R Documentation
Fitting the 2-D Profiles of Apical Meristems
Description
fitAM is used to estimate the parameters of any non-linear equation
the user defines for describing the 2-D profiles of shoot or root apical meristems.
Usage
fitAM(expr, x, y, ini.val, method = "Nelder-Mead", control = list(),
lower = -Inf, upper = Inf, par.list = FALSE, stand.fig = TRUE, fig.opt = FALSE,
angle = NULL, xlim = NULL, ylim = NULL, unit = NULL, main = NULL)
Arguments
expr
the non-linear equation for describing the 2-D profiles of shoot or root apical meristems.
x
the x coordinates of an apical meristem (AM)'s profile.
y
the y coordinates of an AM's profile.
ini.val
the list of initial values for the model parameters.
method
an optional argument to select an optimization method.
control
the list of control parameters for using the optim function in package stats.
lower
the lower bounds on the variables for the L-BFGS-B algorithm.
upper
the upper bounds on the variables for the L-BFGS-B algorithm.
par.list
an optional argument to show the list of parameters on the screen.
stand.fig
an optional argument to draw the observed and predicted profiles of an AM at the standard state
(i.e., the origin for indicating the curve predicted by expr is located at (0, 0),
and the x-axis of the predicted curve at this state is aligned with the x-axis).
fig.opt
an optional argument to draw the observed and predicted profiles of an AM
at arbitrary angle between the rotated x-axis and the actual x-axis.
angle
the angle between the rotated x-axis and the actual x-axis, which can be defined by the user.
xlim
the range of the x-axis over which to plot the predicted AM's profile curve.
ylim
the range of the y-axis over which to plot the predicted AM's profile curve.
unit
the unit of the x-axis and the y-axis when showing the predicted AM's profile curve.
main
the main title of the figure.
Details
Here, the rotated x-axis means that for a scanned (photographed) AM the x-axis of the AM's profile
curve at the standard state predicted by expr might deviate from the actual horizontal axis.
The Nelder-Mead algorithm (Nelder and Mead, 1965) and the optimization method
(referred to as L-BFGS-B) proposed by Byrd et al. (1995) in which each variable can be given
a lower and/or upper bound can be selected to carry out the optimization of minimizing
the residual sum of squares (RSS) between the observed and predicted y values.
The optim function in package stats was used to carry out
the Nelder-Mead algorithm and the L-BFGS-B algorithm.
When angle = NULL, the observed AM's profile curve will be shown at its initial angle in the scanned image;
when angle is a numerical value (e.g., \pi/4) defined by the user,
it indicates that the x-axis at the standard state
is rotated by the amount (\pi/4) counterclockwise from the actutal x-axis.
Value
par
the estimates of the model parameters.
r.sq
the coefficient of determination between the observed and predicted y values
on the AM's profile curve.
RSS
the residual sum of squares between the observed and predicted y values
on the AM's profile curve.
x.stand.obs
the observed x coordinates of the points
on the AM's profile curve at the standard state.
x.stand.pred
the predicted x coordinates of the points
on the AM's profile curve at the standard state.
y.stand.obs
the observed y coordinates of the points
on the AM's profile curve at the standard state.
y.stand.pred
the predicted y coordinates of the points
on the AM's profile curve at the standard state.
x.obs
the observed x coordinates of the points on the AM's profile curve
at the transferred polar angles as defined by the user.
x.pred
the predicted x coordinates of the points on the AM's profile curve
at the transferred polar angles as defined by the user.
y.obs
the observed y coordinates of the points on the AM's profile curve
at the transferred polar angles as defined by the user.
y.pred
the predicted y coordinates of the points on the AM's profile curve
at the transferred polar angles as defined by the user.
Note
In the arguments, expr can be flexibly defined by the user, but it requires taking the form as
myfun <- function(P, x){...}, where P represents the parameter vector, and x
is the independent variable. In the outputs, the data length of the predicted values is the same as that of
the observations.
Author(s)
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be,
Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
References
Byrd, R.H., Lu, P., Nocedal, J., Zhu, C. (1995) A limited memory algorithm for bound constrained
optimization. SIAM Journal on Scientific Computing 16, 1190-1208. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/0916069")}
Shi, P., Chen, L., Quinn, B.K., Yu, K., Miao, Q., Guo, X., Lian, M., Gielis, J., Niklas, K.J. (2023)
A simple way to calculate the volume and surface area of avian eggs.
Annals of the New York Academy of Sciences 1524, 118-131. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/nyas.15000")}
See Also
PlanCoor, SAMs, SurfaceAreaAM, VolumeAM
Examples
#### See Shi et al. (2025) for details #########################################
# Shi, P., Liu, X., Gielis, J., Beirinckx, B., Niklas, K.J. (2025)
# Comparison of six non-linear equations in describing the 2-D
# profiles of apical meristems. American Journal of Botany (under review).
################################################################################
data(SAMs)
uni.sam <- sort( unique(SAMs$Genus) )
ind <- 2
Data <- SAMs[SAMs$Genus==uni.sam[ind], ]
x0 <- Data$x
y0 <- Data$y
Res1 <- adjdata(x0, y0, ub.np=200, times=1.2, len.pro=1/20)
X <- Res1$x
Y <- Res1$y
dev.new()
plot( X, Y, pch=1, cex.lab=1.5, cex.axis=1.5,
xlab=expression(paste(italic(x), " (unitless)", sep="")),
ylab=expression(paste(italic(y), " (unitless)", sep="")) )
x1 <- X
y1 <- Y
if(TRUE){
# The code results in an upward-opening profile curve with
# all y-values less than zero, facilitating data fitting by
# providing suitable initial values for model parameters.
y1 <- min(Y)-Y
x1 <- X-mean(X)
# To normalize the profile coordinates,
# and make x-coordinates range between -1 and 1
x2 <- x1/max(abs(x1))
y2 <- y1/max(abs(x1))
rm(x1)
rm(y1)
x1 <- x2
y1 <- y2
}
#### Fitting the catenary equation ###############################################
myfun1 <- function(P, x){
a <- P[1]
return( a * cosh(x/a) )
}
x0.ini <- 0
y0.ini <- -min(y1)
theta.ini <- 0
a.ini <- c( 0.75*abs(min(y1)), abs(min(y1)), 1.25*abs(min(y1)) )
ini.val1 <- list(x0.ini, y0.ini, theta.ini, a.ini )
Res1 <- fitAM( myfun1, x=x1, y=y1, ini.val=ini.val1,
control=list(reltol=1e-20, maxit=20000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Catenary equation" )
###################################################################################
#### Fitting the parabolic equation ###############################################
myfun2 <- function(P, x){
alpha1 <- P[1]
alpha2 <- P[2]
alpha1*x + alpha2*x^2
}
x0.ini <- 0
y0.ini <- -abs(min(y1))
theta.ini <- 0
beta1.ini <- 1e-3
beta2.ini <- c(1, 5, 10, 15)
ini.val2 <- list(x0.ini, y0.ini, theta.ini, beta1.ini, beta2.ini)
Res2 <- fitAM( myfun2, x=x1, y=y1, ini.val=ini.val2,
control=list(factr=1e-12, maxit=20000),
method="L-BFGS-B", lower=c(-Inf, -Inf, -pi/2/4, -Inf, 0),
upper=c(Inf, Inf, pi/2/4, Inf, Inf),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Parabolic equation" )
###################################################################################
#### Fitting the hybrid catenary-parabolic equation ###############################
myfun3 <- function(P, x){
alpha <- P[1]
beta <- P[2]
gamma <- P[3]
alpha * cosh(beta * x) + gamma * x^2 - alpha
}
x0.ini <- 0
y0.ini <- -abs(min(y1))
theta.ini <- 0
alpha.ini <- c(0.1, 0.5, 1, 2.5, 5)
beta.ini <- 1/abs(min(y1))
gamma.ini <- 1
ini.val3 <- list(x0.ini, y0.ini, theta.ini, alpha.ini, beta.ini, gamma.ini)
Res3 <- fitAM( myfun3, x=x1, y=y1, ini.val=ini.val3,
control=list(reltol=1e-30, maxit=50000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Hybrid catenary-parabolic equation" )
###################################################################################
#### Fitting the performance equation #############################################
myfun4 <- function(P, x){
c <- P[1]
K1 <- P[2]
K2 <- P[3]
xmin <- 0
xmax <- P[4]
# x[x > xmax] <- xmax
# x[x < xmin] <- xmin
-c * ( 1-exp(-K1*(x-xmin)) )*( 1-exp(K2*(x-xmax)) )
}
x0.ini <- min(x1)
y0.ini <- 0
theta.ini <- -pi/12
c.ini <- abs(min(y1))
K1.ini <- 1
K2.ini <- 1
xmax.ini <- max(x1)*2
ini.val4 <- list(x0.ini, y0.ini, theta.ini, c.ini, K1.ini, K2.ini, xmax.ini)
Res4 <- fitAM( myfun4, x=x1, y=y1, ini.val=ini.val4, method="L-BFGS-B",
lower=c(-Inf, -Inf, -pi/2/4, 0, 0, 0, 0),
upper=c(Inf, Inf, pi/2/4, Inf, Inf, Inf, Inf),
control=list(factr=1e-12, maxit=20000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Performance equation" )
###################################################################################
#### Fitting the superparabolic equation equation #################################
myfun5 <- function(P, x){
beta1 <- P[1]
beta2 <- P[2]
beta1 * abs(x)^beta2
}
x0.ini <- 0
y0.ini <- -max( y1 )
theta.ini <- 0
beta1.ini <- max(x1)/2
beta2.ini <- c(1.5, 2, 2.5)
ini.val5 <- list(x0.ini, y0.ini, theta.ini, beta1.ini, beta2.ini)
Res5 <- fitAM( myfun5, x=x1, y=y1, ini.val=ini.val5,
control=list(reltol=1e-20, maxit=20000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Superparabolic equation" )
###################################################################################
#### Fitting the superellipse equation equation ###################################
myfun6 <- function(P, x){
A <- P[1]
B <- P[2]
n <- P[3]
-B*(1-abs(x/A)^n)*(1/n)
}
x0.ini <- 0
y0.ini <- 0
theta.ini <- c(0, pi/2)
A.ini <- max(x1)
B.ini <- -min(y1)
n.ini <- c(1, 2, 3)
ini.val6 <- list(x0.ini, y0.ini, theta.ini, A.ini, B.ini, n.ini)
Res6 <- fitAM( myfun6, x=x1, y=y1, ini.val=ini.val6,
control=list(reltol=1e-20, maxit=20000),
par.list=FALSE, stand.fig = FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Superellipse equation" )
###################################################################################
graphics.off()
biogeom documentation built on Aug. 24, 2025, 5:08 p.m.
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| fitAM | R Documentation |
Fitting the 2-D Profiles of Apical Meristems
Description
fitAM is used to estimate the parameters of any non-linear equation
the user defines for describing the 2-D profiles of shoot or root apical meristems.
Usage
fitAM(expr, x, y, ini.val, method = "Nelder-Mead", control = list(),
lower = -Inf, upper = Inf, par.list = FALSE, stand.fig = TRUE, fig.opt = FALSE,
angle = NULL, xlim = NULL, ylim = NULL, unit = NULL, main = NULL)
Arguments
expr |
the non-linear equation for describing the 2-D profiles of shoot or root apical meristems. |
x |
the |
y |
the |
ini.val |
the list of initial values for the model parameters. |
method |
an optional argument to select an optimization method. |
control |
the list of control parameters for using the |
lower |
the lower bounds on the variables for the |
upper |
the upper bounds on the variables for the |
par.list |
an optional argument to show the list of parameters on the screen. |
stand.fig |
an optional argument to draw the observed and predicted profiles of an AM at the standard state
(i.e., the origin for indicating the curve predicted by |
fig.opt |
an optional argument to draw the observed and predicted profiles of an AM
at arbitrary angle between the rotated |
angle |
the angle between the rotated |
xlim |
the range of the |
ylim |
the range of the |
unit |
the unit of the |
main |
the main title of the figure. |
Details
Here, the rotated x-axis means that for a scanned (photographed) AM the x-axis of the AM's profile
curve at the standard state predicted by expr might deviate from the actual horizontal axis.
The Nelder-Mead algorithm (Nelder and Mead, 1965) and the optimization method
(referred to as L-BFGS-B) proposed by Byrd et al. (1995) in which each variable can be given
a lower and/or upper bound can be selected to carry out the optimization of minimizing
the residual sum of squares (RSS) between the observed and predicted y values.
The optim function in package stats was used to carry out
the Nelder-Mead algorithm and the L-BFGS-B algorithm.
When angle = NULL, the observed AM's profile curve will be shown at its initial angle in the scanned image;
when angle is a numerical value (e.g., \pi/4) defined by the user,
it indicates that the x-axis at the standard state
is rotated by the amount (\pi/4) counterclockwise from the actutal x-axis.
Value
par |
the estimates of the model parameters. |
r.sq |
the coefficient of determination between the observed and predicted |
RSS |
the residual sum of squares between the observed and predicted |
x.stand.obs |
the observed |
x.stand.pred |
the predicted |
y.stand.obs |
the observed |
y.stand.pred |
the predicted |
x.obs |
the observed |
x.pred |
the predicted |
y.obs |
the observed |
y.pred |
the predicted |
Note
In the arguments, expr can be flexibly defined by the user, but it requires taking the form as
myfun <- function(P, x){...}, where P represents the parameter vector, and x
is the independent variable. In the outputs, the data length of the predicted values is the same as that of
the observations.
Author(s)
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
References
Byrd, R.H., Lu, P., Nocedal, J., Zhu, C. (1995) A limited memory algorithm for bound constrained
optimization. SIAM Journal on Scientific Computing 16, 1190-1208. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/0916069")}
Shi, P., Chen, L., Quinn, B.K., Yu, K., Miao, Q., Guo, X., Lian, M., Gielis, J., Niklas, K.J. (2023)
A simple way to calculate the volume and surface area of avian eggs.
Annals of the New York Academy of Sciences 1524, 118-131. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/nyas.15000")}
See Also
PlanCoor, SAMs, SurfaceAreaAM, VolumeAM
Examples
#### See Shi et al. (2025) for details #########################################
# Shi, P., Liu, X., Gielis, J., Beirinckx, B., Niklas, K.J. (2025)
# Comparison of six non-linear equations in describing the 2-D
# profiles of apical meristems. American Journal of Botany (under review).
################################################################################
data(SAMs)
uni.sam <- sort( unique(SAMs$Genus) )
ind <- 2
Data <- SAMs[SAMs$Genus==uni.sam[ind], ]
x0 <- Data$x
y0 <- Data$y
Res1 <- adjdata(x0, y0, ub.np=200, times=1.2, len.pro=1/20)
X <- Res1$x
Y <- Res1$y
dev.new()
plot( X, Y, pch=1, cex.lab=1.5, cex.axis=1.5,
xlab=expression(paste(italic(x), " (unitless)", sep="")),
ylab=expression(paste(italic(y), " (unitless)", sep="")) )
x1 <- X
y1 <- Y
if(TRUE){
# The code results in an upward-opening profile curve with
# all y-values less than zero, facilitating data fitting by
# providing suitable initial values for model parameters.
y1 <- min(Y)-Y
x1 <- X-mean(X)
# To normalize the profile coordinates,
# and make x-coordinates range between -1 and 1
x2 <- x1/max(abs(x1))
y2 <- y1/max(abs(x1))
rm(x1)
rm(y1)
x1 <- x2
y1 <- y2
}
#### Fitting the catenary equation ###############################################
myfun1 <- function(P, x){
a <- P[1]
return( a * cosh(x/a) )
}
x0.ini <- 0
y0.ini <- -min(y1)
theta.ini <- 0
a.ini <- c( 0.75*abs(min(y1)), abs(min(y1)), 1.25*abs(min(y1)) )
ini.val1 <- list(x0.ini, y0.ini, theta.ini, a.ini )
Res1 <- fitAM( myfun1, x=x1, y=y1, ini.val=ini.val1,
control=list(reltol=1e-20, maxit=20000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Catenary equation" )
###################################################################################
#### Fitting the parabolic equation ###############################################
myfun2 <- function(P, x){
alpha1 <- P[1]
alpha2 <- P[2]
alpha1*x + alpha2*x^2
}
x0.ini <- 0
y0.ini <- -abs(min(y1))
theta.ini <- 0
beta1.ini <- 1e-3
beta2.ini <- c(1, 5, 10, 15)
ini.val2 <- list(x0.ini, y0.ini, theta.ini, beta1.ini, beta2.ini)
Res2 <- fitAM( myfun2, x=x1, y=y1, ini.val=ini.val2,
control=list(factr=1e-12, maxit=20000),
method="L-BFGS-B", lower=c(-Inf, -Inf, -pi/2/4, -Inf, 0),
upper=c(Inf, Inf, pi/2/4, Inf, Inf),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Parabolic equation" )
###################################################################################
#### Fitting the hybrid catenary-parabolic equation ###############################
myfun3 <- function(P, x){
alpha <- P[1]
beta <- P[2]
gamma <- P[3]
alpha * cosh(beta * x) + gamma * x^2 - alpha
}
x0.ini <- 0
y0.ini <- -abs(min(y1))
theta.ini <- 0
alpha.ini <- c(0.1, 0.5, 1, 2.5, 5)
beta.ini <- 1/abs(min(y1))
gamma.ini <- 1
ini.val3 <- list(x0.ini, y0.ini, theta.ini, alpha.ini, beta.ini, gamma.ini)
Res3 <- fitAM( myfun3, x=x1, y=y1, ini.val=ini.val3,
control=list(reltol=1e-30, maxit=50000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Hybrid catenary-parabolic equation" )
###################################################################################
#### Fitting the performance equation #############################################
myfun4 <- function(P, x){
c <- P[1]
K1 <- P[2]
K2 <- P[3]
xmin <- 0
xmax <- P[4]
# x[x > xmax] <- xmax
# x[x < xmin] <- xmin
-c * ( 1-exp(-K1*(x-xmin)) )*( 1-exp(K2*(x-xmax)) )
}
x0.ini <- min(x1)
y0.ini <- 0
theta.ini <- -pi/12
c.ini <- abs(min(y1))
K1.ini <- 1
K2.ini <- 1
xmax.ini <- max(x1)*2
ini.val4 <- list(x0.ini, y0.ini, theta.ini, c.ini, K1.ini, K2.ini, xmax.ini)
Res4 <- fitAM( myfun4, x=x1, y=y1, ini.val=ini.val4, method="L-BFGS-B",
lower=c(-Inf, -Inf, -pi/2/4, 0, 0, 0, 0),
upper=c(Inf, Inf, pi/2/4, Inf, Inf, Inf, Inf),
control=list(factr=1e-12, maxit=20000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Performance equation" )
###################################################################################
#### Fitting the superparabolic equation equation #################################
myfun5 <- function(P, x){
beta1 <- P[1]
beta2 <- P[2]
beta1 * abs(x)^beta2
}
x0.ini <- 0
y0.ini <- -max( y1 )
theta.ini <- 0
beta1.ini <- max(x1)/2
beta2.ini <- c(1.5, 2, 2.5)
ini.val5 <- list(x0.ini, y0.ini, theta.ini, beta1.ini, beta2.ini)
Res5 <- fitAM( myfun5, x=x1, y=y1, ini.val=ini.val5,
control=list(reltol=1e-20, maxit=20000),
par.list=FALSE, stand.fig=FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Superparabolic equation" )
###################################################################################
#### Fitting the superellipse equation equation ###################################
myfun6 <- function(P, x){
A <- P[1]
B <- P[2]
n <- P[3]
-B*(1-abs(x/A)^n)*(1/n)
}
x0.ini <- 0
y0.ini <- 0
theta.ini <- c(0, pi/2)
A.ini <- max(x1)
B.ini <- -min(y1)
n.ini <- c(1, 2, 3)
ini.val6 <- list(x0.ini, y0.ini, theta.ini, A.ini, B.ini, n.ini)
Res6 <- fitAM( myfun6, x=x1, y=y1, ini.val=ini.val6,
control=list(reltol=1e-20, maxit=20000),
par.list=FALSE, stand.fig = FALSE, fig.opt=TRUE, angle=NULL,
unit="unitless", main="Superellipse equation" )
###################################################################################
graphics.off()
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