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In hysteresis: Tools for Modeling Rate-Dependent Hysteretic Processes and Ellipses
Hysteresis
An R package for fitting rate-dependent hysteretic loops
Hysteresis loops occur when an output variable can have multiple possible values at one input value depending on the history of the system and the direction of change in the input. This package contains functions to simulate, fit, and obtain parameter values along with their standard errors (Yang and Parkhurst) from hysteresis loops of the form
$$x_{t}=b_{x}cos(2t\pi/T+\phi)+c_{x}+e_{x,t}$$
$$y_{t}=b_{y}cos(2t\pi/T+\phi)^{n}+Rsin(2t\pi/T+\phi)^{m}+c_{y}+e_{y,t}$$
where e is a random error term. These generalized transcendental equations (Lapshin) form a hysteresis loop for a given frequency or period and set of time points t=1,2...n.
The plot below uses the function mloop which simulates hysteresis loops to show the effects of choosing various odd values for n and m.
library(knitr)
library(hysteresis)
par(mfrow=c(3,3),mai=c(0,0.2,0.2,0),ann=FALSE,xaxt="n",yaxt="n",oma=c(0,0,3,0))
for (i in c(1,3,15)){
for (j in c(1,3,15)){
obj<-mloop(m=i,n=j,n.points=100,period=99)
plot(floop(obj$x,obj$y,m=i,n=j,period=99),xlim=c(-0.8,0.8),ylim=c(-0.8,0.8))
if (i==1) title(paste("n=",j,sep=""))
if (j==1) title(ylab=paste("m=",i,sep=""),line=0,cex.sub=2)
}
}
title("Hysteresis Loops for Odd Values of m and n",outer=TRUE)
It is also possible to use even values for n.
par(mfrow=c(3,3),mai=c(0,0.2,0.2,0),ann=FALSE,xaxt="n",yaxt="n",oma=c(0,0,3,0))
for (i in c(1,3,15)){
for (j in c(2,4,16)){
obj<-mloop(m=i,n=j,n.points=100,period=99)
plot(floop(obj$x,obj$y,m=i,n=j,period=99),xlim=c(-0.8,0.8),ylim=c(-0.8,0.8))
if (i==1) title(paste("n=",j,sep=""))
if (j==2) title(ylab=paste("m=",i,sep=""),line=0,cex.sub=2)
}
}
title("Hysteresis Loops for Odd Values of m and Even Values of n",outer=TRUE)
A special case is when n=1 and m=1, this makes the hysteresis loop an ellipse. The centroid of the hysteresis loop is given by cx and cy as shown in the plot below of ellipses.
obj<-mloop(cx=0,cy=0,n.points=100,period=99)
obj2<-mloop(cx=1.5,cy=0,n.points=100,period=99)
obj3<-mloop(cx=0,cy=1.5,n.points=100,period=99)
plot(obj$x,obj$y,type="l",xlim=c(-2,3),ylim=c(-2,3),xlab="x",ylab="y",col="#6600CC",main="Centroid Given by cx and cy")
points(0,0,pch=19,col="#6600CC")
text(x=0,y=0.15,"(cx=0,cy=0)",col="#6600CC")
lines(obj2$x,obj2$y,col="#00FF66")
points(1.5,0,pch=19,col="#00FF66")
text(x=1.5,y=0.15,"(cx=1.5,cy=0)",col="#00FF66")
lines(obj3$x,obj3$y,col="#FF6600")
points(0,1.5,pch=19,col="#FF6600")
text(x=0,y=1.65,"(cx=0,cy=1.5)",col="#FF6600")
The saturation points describe where the direction of the input changes sign. The distances from the central point to the saturation points are given by b.x and b.y (saturation points at ($c_{x} \pm b_{x},c_{y} \pm b_{y}$))
for (i in c(1,2,4)){
obj<-mloop(b.x=i,n.points=100,period=99)
plot(obj$x,obj$y,xlim=c(-5,10),ylim=c(-1.4,1.4),type="l",main=paste("b.x=",i,sep=""),xlab="x",ylab="y")
points(i,0.8,pch=19)
legend(i,1,legend=c("Saturation Point","x=cx+b.x","y=cy+b.y"),bty="n")
}
for (i in c(0.8,2,4)){
obj<-mloop(b.y=i,n.points=100,period=99)
plot(obj$x,obj$y,xlim=c(-1,2),ylim=c(-5,5),type="l",main=paste("b.y=",i,sep=""),xlab="x",ylab="y")
points(0.6,i,pch=19)
legend(0.6,i,legend=c("Saturation Point","x=cx+b.x","y=cy+b.y"),bty="n")
}
Retention, or the output split point R, is the vertical distance from center to upper loop trajectory.
for (i in c(1,2,4)){
obj<-mloop(retention=i,n.points=100,period=99)
plot(obj$x,obj$y,xlim=c(-1,1),ylim=c(-5,5),type="l",main=paste("retention=",i,sep=""),xlab="x",ylab="y")
segments(0,0,0,i)
text(0.3,0.5,"Retention")
}
Finally the phase.angle, $\phi$, changes the location of points along the loop, but does not change the form of the loop itself. When phase.angle is zero, the loop starting point is also the saturation point.
obj<-mloop(retention=0.5,n.points=100,period=99)
plot(obj$x,obj$y,type="l",xlab="x",ylab="y",main="Starting Points for Different Values of phase.angle",xlim=c(-0.6,0.8))
for (i in c(0,90,180,260)){
obj2<-mloop(phase.angle=i,retention=0.5,n.points=1,period=99)
points(obj2$x,obj2$y,pch=19,col="gold",cex=2)
points(obj2$x,obj2$y,col="gold",cex=4)
text(obj2$x+.08,obj2$y,round(i,2))
}
Fitting Ellipses
The Process
Hysteresis contains one method for fitting hysteresis loops given any n and m in the function floop.
In the special case of an ellipse where n=1 and m=1, four methods are available in the function fel.
The two-step simple harmonic regression (harmonic2) method, the default, generally produces estimates that are less
biased and have lower variances than those produced by the other methods. Since the focus is on rate-dependent hysteresis,
knowledge of time for the observations is required (or if unknown, times may be assumed to be equally spaced). On the
other hand, if the objective is solely to fit an ellipse, observation times are not needed for the other three methods.
set.seed(24)
ellipse1 <- mel(method=2,retention=0.4,b.x=0.6,b.y=0.8,cx=0,cy=0,sd.x=0.1,sd.y=0.1,phase.angle=0,period=24,n.points=24)
#The function **mel** can be used as an alternative to **mloop** for simulating ellipses, and it is useful because it offers four different ellipse parameterizations.
model <- fel(ellipse1$x,ellipse1$y,method="harmonic2",period=24,times="equal")
#period=24 and times="equal" are used to say that 24 equally spaced points make up an ellipse.
model
In addition to the fundamental values of the model, fel also calculates a wide variety of derived parameters. Definitions for these parameters can be found using help(loop.parameters).
model$Estimates
A wide variety of functions have S3 methods for objects of class ellipsefit produced by fel.
The most important of these is summary.ellipsefit which can be used to bootstrap and summarize models
produced by fel.
summary(model,N=10000,studentize=TRUE)
Another important S3 method is for the function plot.
plot(model,main="2-step Simple Harmonic Regression Ellipse Example")
In addition, S3 methods exist for fitted, print, and residuals.
Comparison of Ellipse Estimation Methods
The two most useful ellipse estimation methods implemented by fel are the 'harmonic2' and 'direct' methods. The 'direct' method (Flusser and Halir) fits an ellipse without requiring time information and is more stable than the other two methods in fel, 'lm' and 'nls', which are based on linear least squares and ellipse-specific nonlinear regression respectively. The 'direct' method does not yet have delta method standard errors available.
modeldirect <- fel(ellipse1$x,ellipse1$y,method="direct",period=24,times="equal")
summodel<-summary(modeldirect,N=10000,studentize=TRUE)
summodel
plot(modeldirect,main="Direct Ellipse Example")
The 'direct' method uses different fundamental parameters than the 'harmonic2' method. However summary results for b.x, b.y, and retention are still available from the matrix of values produced by summary.ellipsefit.
summodel$values
The four plots below illustrate a comparison of the four methods for fitting an ellipse to simulated data. The data
points are based on the simulated red ellipse; the fitted ellipse is in black.
set.seed(13)
par(mfrow=c(2,2))
halfellipse <- mel(method=2,cx=20,cy=25,retention=1.2,b.x=14,b.y=0.8,sd.x=1,sd.y=0.2,period=24,n.points=16,phase.angle=pi/2)
halftrueellipse <- mel(method=2,cx=20,cy=25,retention=1.2,b.x=14,b.y=0.8,phase.angle=0,period=99,n.points=100)
harmodel<-fel(halfellipse$x,halfellipse$y,method="harmonic2",period=24,times="equal")
dirmodel<-fel(halfellipse$x,halfellipse$y,method="direct",period=24,times="equal")
lmmodel<-fel(halfellipse$x,halfellipse$y,method="lm",period=24,times="equal")
nlsmodel<-fel(halfellipse$x,halfellipse$y,method="nls",period=24,times="equal",control=c(n.iter=500))
plot(harmodel,main="Harmonic2 Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
plot(dirmodel,main="Direct Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
plot(lmmodel,main="Linear Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
plot(nlsmodel,main="Non-Linear Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
Geometric Method
The geometric ellipse method used here is based on the work of Gander, Golub and Strebel, and the code used is an
R translation of the Matlab code they provided. This method directly minimizes the Euclidean distances from the
ellipse through Gauss-Newton minimization. Standard errors are obtainable through either the Delta Method
and bootstrapping, however the use of bootstrapping through summary.ellipsefit is discouraged on these ellipses
as the geometric method is extremely computationally expensive. The "geometric" method works best when sd.x=sd.y,
and it is important to check for false convergence. One way to check for false convergence is to examine the plot
after fitting the data.
set.seed(101)
ellip <- mel(rote.deg=45,semi.major=5,semi.minor=3,n.points=13,sd.x=0.4,sd.y=0.4)
true.ellip <- mel(rote.deg=45,semi.major=5,semi.minor=3,n.points=100,period=100)
ellip.geometric <- fel(ellip$x,ellip$y,method="geometric")
ellip.geometric$values
plot(ellip.geometric,main="Geometric Model")
lines(true.ellip$x,true.ellip$y,col="red")
Bootstrapping Fitted Ellipses
The function summary.ellipsefit bootstraps the x and y residuals of a fitted ellipse separately to produce
standard errors and less biased estimates of ellipse parameters. These residuals are easy to obtain using the
'harmonic2' model which gives fitted points when fitting the ellipse, but somewhat more difficult to obtain from the
other methods which do not use time as a variable in fitting the model and therefore cannot place observations on the
ellipse. The function fel, therefore, gives two methods for producing x and y residuals using these methods. If
times="unknown", fitted values are taken to be the points on the ellipse closest to their realized values. If
times="equal" or a numeric vector and the period of the ellipse is known, then the distances between points on the
ellipse are taken as given and only the starting point of the ellipse is chosen to minimize the sum of squared distances
between fitted and realized values. If times are available, it is always better to give them, as the residuals given
by times='unknown' will lead to standard errors for ellipse parameters that are biased downwards. If times really are
unknown, a good alternative is to use the delta standard errors from the function delta.error which is currently
available for every method except the direct.
In addition, residuals can be studentized within the summary.ellipsefit function by keeping studentize=TRUE, which is the default. Simulations suggest that studentization improves 95% bootstrap coverage intervals for all four methods.
The value N gives the number of bootstrap replicates, its default is 1000 which may be low in some situations (Efron 1979).
In each replication, residuals are resampled with replacement and added to the original fitted values produced by fel. The simulated ellipse is then refit using the original method and parameter estimates are obtained. The standard deviations of these estimates are then used to give parameter standard errors, and less biased parameter estimates are obtained by subtracting the estimated bias produced by the method, mean(bootstrap estimates) - (original estimate), from the original estimate.
Note, if reproducible results are desired use set.seed() command.
Comparison of Bootstrapped Ellipses
The fitted black ellipses from above are then bootstrapped to reduce bias.
par(mfrow=c(2,2))
harsummodel<-summary(harmodel,N=1000,studentize=TRUE)
dirsummodel<-summary(dirmodel,N=1000,studentize=TRUE)
lmsummodel<-summary(lmmodel,N=1000,studentize=TRUE)
nlssummodel<-summary(nlsmodel,N=1000,studentize=TRUE)
plot(harsummodel,main="Bootstrapped Harmonic2 Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
plot(dirsummodel,main="Bootstrapped Direct Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
plot(lmsummodel,main="Bootstrapped Lm Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
plot(nlssummodel,main="Bootstrapped Nls Model",xlim=c(5,34),ylim=c(23.4,26.9))
lines(halftrueellipse$x,halftrueellipse$y,col="red")
Fitting Multiple Ellipses Simultaneously
The argument subjects in the function fel can be used to fit multiple ellipses, which share the same period,
at one time. In this case fel produces an object of class ellipsefitlist instead of ellipsefit, and methods for
objects of class ellipsefitlist exist for the functions summary, plot, and print. Ellipses are separated
by levels given by the argument subjects, which can be either a vector or a list of vectors treated as factors. Below
is an example of fitting multiple ellipses using the subjects option.
data(EllipseData)
models <- fel(EllipseData$X,EllipseData$Y,method="harmonic2",subjects=EllipseData$subjects,subset=EllipseData$repeated==1)
models
summodels<-summary(models)
summodels
plot(summodels,main="Fitting Multiple Ellipses Simultaneously")
To output summary results to excel readable file at current directory
## write.table(models$Estimates,"file_name.txt") and
## write.table(summodels$Boot.Estimates,"file_name.txt")
Fitting Hysteresis Loops
The function floop can be used to fit hysteresis loops with specific values of n and m as arguments to floop. Below is an example of a
hysteresis loop with n=5, m=3.
loop <- mloop(n=5, m=3,sd.x=0.02,sd.y=0.02)
fitloop <- floop(loop$x,loop$y,n=5, m=3,period=24,times="equal")
fitloop$Estimates
plot(fitloop,main="Fitted Hysteresis Loop")
summary(fitloop)
Acknowledgments
NIFA MRF 25-008/W-2173 Impacts of Stress Factors on Performance, Health, and Well Being of Farm Animals
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Hysteresis
An R package for fitting rate-dependent hysteretic loops
Hysteresis loops occur when an output variable can have multiple possible values at one input value depending on the history of the system and the direction of change in the input. This package contains functions to simulate, fit, and obtain parameter values along with their standard errors (Yang and Parkhurst) from hysteresis loops of the form $$x_{t}=b_{x}cos(2t\pi/T+\phi)+c_{x}+e_{x,t}$$ $$y_{t}=b_{y}cos(2t\pi/T+\phi)^{n}+Rsin(2t\pi/T+\phi)^{m}+c_{y}+e_{y,t}$$
where e is a random error term. These generalized transcendental equations (Lapshin) form a hysteresis loop for a given frequency or period and set of time points t=1,2...n.
The plot below uses the function mloop which simulates hysteresis loops to show the effects of choosing various odd values for n and m.
library(knitr) library(hysteresis) par(mfrow=c(3,3),mai=c(0,0.2,0.2,0),ann=FALSE,xaxt="n",yaxt="n",oma=c(0,0,3,0)) for (i in c(1,3,15)){ for (j in c(1,3,15)){ obj<-mloop(m=i,n=j,n.points=100,period=99) plot(floop(obj$x,obj$y,m=i,n=j,period=99),xlim=c(-0.8,0.8),ylim=c(-0.8,0.8)) if (i==1) title(paste("n=",j,sep="")) if (j==1) title(ylab=paste("m=",i,sep=""),line=0,cex.sub=2) } } title("Hysteresis Loops for Odd Values of m and n",outer=TRUE)
It is also possible to use even values for n.
par(mfrow=c(3,3),mai=c(0,0.2,0.2,0),ann=FALSE,xaxt="n",yaxt="n",oma=c(0,0,3,0)) for (i in c(1,3,15)){ for (j in c(2,4,16)){ obj<-mloop(m=i,n=j,n.points=100,period=99) plot(floop(obj$x,obj$y,m=i,n=j,period=99),xlim=c(-0.8,0.8),ylim=c(-0.8,0.8)) if (i==1) title(paste("n=",j,sep="")) if (j==2) title(ylab=paste("m=",i,sep=""),line=0,cex.sub=2) } } title("Hysteresis Loops for Odd Values of m and Even Values of n",outer=TRUE)
A special case is when n=1 and m=1, this makes the hysteresis loop an ellipse. The centroid of the hysteresis loop is given by cx and cy as shown in the plot below of ellipses.
obj<-mloop(cx=0,cy=0,n.points=100,period=99) obj2<-mloop(cx=1.5,cy=0,n.points=100,period=99) obj3<-mloop(cx=0,cy=1.5,n.points=100,period=99) plot(obj$x,obj$y,type="l",xlim=c(-2,3),ylim=c(-2,3),xlab="x",ylab="y",col="#6600CC",main="Centroid Given by cx and cy") points(0,0,pch=19,col="#6600CC") text(x=0,y=0.15,"(cx=0,cy=0)",col="#6600CC") lines(obj2$x,obj2$y,col="#00FF66") points(1.5,0,pch=19,col="#00FF66") text(x=1.5,y=0.15,"(cx=1.5,cy=0)",col="#00FF66") lines(obj3$x,obj3$y,col="#FF6600") points(0,1.5,pch=19,col="#FF6600") text(x=0,y=1.65,"(cx=0,cy=1.5)",col="#FF6600")
The saturation points describe where the direction of the input changes sign. The distances from the central point to the saturation points are given by b.x and b.y (saturation points at ($c_{x} \pm b_{x},c_{y} \pm b_{y}$))
for (i in c(1,2,4)){ obj<-mloop(b.x=i,n.points=100,period=99) plot(obj$x,obj$y,xlim=c(-5,10),ylim=c(-1.4,1.4),type="l",main=paste("b.x=",i,sep=""),xlab="x",ylab="y") points(i,0.8,pch=19) legend(i,1,legend=c("Saturation Point","x=cx+b.x","y=cy+b.y"),bty="n") }
for (i in c(0.8,2,4)){ obj<-mloop(b.y=i,n.points=100,period=99) plot(obj$x,obj$y,xlim=c(-1,2),ylim=c(-5,5),type="l",main=paste("b.y=",i,sep=""),xlab="x",ylab="y") points(0.6,i,pch=19) legend(0.6,i,legend=c("Saturation Point","x=cx+b.x","y=cy+b.y"),bty="n") }
Retention, or the output split point R, is the vertical distance from center to upper loop trajectory.
for (i in c(1,2,4)){ obj<-mloop(retention=i,n.points=100,period=99) plot(obj$x,obj$y,xlim=c(-1,1),ylim=c(-5,5),type="l",main=paste("retention=",i,sep=""),xlab="x",ylab="y") segments(0,0,0,i) text(0.3,0.5,"Retention") }
Finally the phase.angle, $\phi$, changes the location of points along the loop, but does not change the form of the loop itself. When phase.angle is zero, the loop starting point is also the saturation point.
obj<-mloop(retention=0.5,n.points=100,period=99) plot(obj$x,obj$y,type="l",xlab="x",ylab="y",main="Starting Points for Different Values of phase.angle",xlim=c(-0.6,0.8)) for (i in c(0,90,180,260)){ obj2<-mloop(phase.angle=i,retention=0.5,n.points=1,period=99) points(obj2$x,obj2$y,pch=19,col="gold",cex=2) points(obj2$x,obj2$y,col="gold",cex=4) text(obj2$x+.08,obj2$y,round(i,2)) }
Fitting Ellipses
The Process
Hysteresis contains one method for fitting hysteresis loops given any n and m in the function floop. In the special case of an ellipse where n=1 and m=1, four methods are available in the function fel. The two-step simple harmonic regression (harmonic2) method, the default, generally produces estimates that are less biased and have lower variances than those produced by the other methods. Since the focus is on rate-dependent hysteresis, knowledge of time for the observations is required (or if unknown, times may be assumed to be equally spaced). On the other hand, if the objective is solely to fit an ellipse, observation times are not needed for the other three methods.
set.seed(24) ellipse1 <- mel(method=2,retention=0.4,b.x=0.6,b.y=0.8,cx=0,cy=0,sd.x=0.1,sd.y=0.1,phase.angle=0,period=24,n.points=24) #The function **mel** can be used as an alternative to **mloop** for simulating ellipses, and it is useful because it offers four different ellipse parameterizations. model <- fel(ellipse1$x,ellipse1$y,method="harmonic2",period=24,times="equal") #period=24 and times="equal" are used to say that 24 equally spaced points make up an ellipse. model
In addition to the fundamental values of the model, fel also calculates a wide variety of derived parameters. Definitions for these parameters can be found using help(loop.parameters).
model$Estimates
A wide variety of functions have S3 methods for objects of class ellipsefit produced by fel. The most important of these is summary.ellipsefit which can be used to bootstrap and summarize models produced by fel.
summary(model,N=10000,studentize=TRUE)
Another important S3 method is for the function plot.
plot(model,main="2-step Simple Harmonic Regression Ellipse Example")
In addition, S3 methods exist for fitted, print, and residuals.
Comparison of Ellipse Estimation Methods
The two most useful ellipse estimation methods implemented by fel are the 'harmonic2' and 'direct' methods. The 'direct' method (Flusser and Halir) fits an ellipse without requiring time information and is more stable than the other two methods in fel, 'lm' and 'nls', which are based on linear least squares and ellipse-specific nonlinear regression respectively. The 'direct' method does not yet have delta method standard errors available.
modeldirect <- fel(ellipse1$x,ellipse1$y,method="direct",period=24,times="equal") summodel<-summary(modeldirect,N=10000,studentize=TRUE) summodel plot(modeldirect,main="Direct Ellipse Example")
The 'direct' method uses different fundamental parameters than the 'harmonic2' method. However summary results for b.x, b.y, and retention are still available from the matrix of values produced by summary.ellipsefit.
summodel$values
The four plots below illustrate a comparison of the four methods for fitting an ellipse to simulated data. The data points are based on the simulated red ellipse; the fitted ellipse is in black.
set.seed(13) par(mfrow=c(2,2)) halfellipse <- mel(method=2,cx=20,cy=25,retention=1.2,b.x=14,b.y=0.8,sd.x=1,sd.y=0.2,period=24,n.points=16,phase.angle=pi/2) halftrueellipse <- mel(method=2,cx=20,cy=25,retention=1.2,b.x=14,b.y=0.8,phase.angle=0,period=99,n.points=100) harmodel<-fel(halfellipse$x,halfellipse$y,method="harmonic2",period=24,times="equal") dirmodel<-fel(halfellipse$x,halfellipse$y,method="direct",period=24,times="equal") lmmodel<-fel(halfellipse$x,halfellipse$y,method="lm",period=24,times="equal") nlsmodel<-fel(halfellipse$x,halfellipse$y,method="nls",period=24,times="equal",control=c(n.iter=500)) plot(harmodel,main="Harmonic2 Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red") plot(dirmodel,main="Direct Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red") plot(lmmodel,main="Linear Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red") plot(nlsmodel,main="Non-Linear Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red")
Geometric Method
The geometric ellipse method used here is based on the work of Gander, Golub and Strebel, and the code used is an R translation of the Matlab code they provided. This method directly minimizes the Euclidean distances from the ellipse through Gauss-Newton minimization. Standard errors are obtainable through either the Delta Method and bootstrapping, however the use of bootstrapping through summary.ellipsefit is discouraged on these ellipses as the geometric method is extremely computationally expensive. The "geometric" method works best when sd.x=sd.y, and it is important to check for false convergence. One way to check for false convergence is to examine the plot after fitting the data.
set.seed(101) ellip <- mel(rote.deg=45,semi.major=5,semi.minor=3,n.points=13,sd.x=0.4,sd.y=0.4) true.ellip <- mel(rote.deg=45,semi.major=5,semi.minor=3,n.points=100,period=100) ellip.geometric <- fel(ellip$x,ellip$y,method="geometric") ellip.geometric$values plot(ellip.geometric,main="Geometric Model") lines(true.ellip$x,true.ellip$y,col="red")
Bootstrapping Fitted Ellipses
The function summary.ellipsefit bootstraps the x and y residuals of a fitted ellipse separately to produce standard errors and less biased estimates of ellipse parameters. These residuals are easy to obtain using the 'harmonic2' model which gives fitted points when fitting the ellipse, but somewhat more difficult to obtain from the other methods which do not use time as a variable in fitting the model and therefore cannot place observations on the ellipse. The function fel, therefore, gives two methods for producing x and y residuals using these methods. If times="unknown", fitted values are taken to be the points on the ellipse closest to their realized values. If times="equal" or a numeric vector and the period of the ellipse is known, then the distances between points on the ellipse are taken as given and only the starting point of the ellipse is chosen to minimize the sum of squared distances between fitted and realized values. If times are available, it is always better to give them, as the residuals given by times='unknown' will lead to standard errors for ellipse parameters that are biased downwards. If times really are unknown, a good alternative is to use the delta standard errors from the function delta.error which is currently available for every method except the direct.
In addition, residuals can be studentized within the summary.ellipsefit function by keeping studentize=TRUE, which is the default. Simulations suggest that studentization improves 95% bootstrap coverage intervals for all four methods.
The value N gives the number of bootstrap replicates, its default is 1000 which may be low in some situations (Efron 1979). In each replication, residuals are resampled with replacement and added to the original fitted values produced by fel. The simulated ellipse is then refit using the original method and parameter estimates are obtained. The standard deviations of these estimates are then used to give parameter standard errors, and less biased parameter estimates are obtained by subtracting the estimated bias produced by the method, mean(bootstrap estimates) - (original estimate), from the original estimate. Note, if reproducible results are desired use set.seed() command.
Comparison of Bootstrapped Ellipses
The fitted black ellipses from above are then bootstrapped to reduce bias.
par(mfrow=c(2,2)) harsummodel<-summary(harmodel,N=1000,studentize=TRUE) dirsummodel<-summary(dirmodel,N=1000,studentize=TRUE) lmsummodel<-summary(lmmodel,N=1000,studentize=TRUE) nlssummodel<-summary(nlsmodel,N=1000,studentize=TRUE) plot(harsummodel,main="Bootstrapped Harmonic2 Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red") plot(dirsummodel,main="Bootstrapped Direct Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red") plot(lmsummodel,main="Bootstrapped Lm Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red") plot(nlssummodel,main="Bootstrapped Nls Model",xlim=c(5,34),ylim=c(23.4,26.9)) lines(halftrueellipse$x,halftrueellipse$y,col="red")
Fitting Multiple Ellipses Simultaneously
The argument subjects in the function fel can be used to fit multiple ellipses, which share the same period, at one time. In this case fel produces an object of class ellipsefitlist instead of ellipsefit, and methods for objects of class ellipsefitlist exist for the functions summary, plot, and print. Ellipses are separated by levels given by the argument subjects, which can be either a vector or a list of vectors treated as factors. Below is an example of fitting multiple ellipses using the subjects option.
data(EllipseData) models <- fel(EllipseData$X,EllipseData$Y,method="harmonic2",subjects=EllipseData$subjects,subset=EllipseData$repeated==1) models summodels<-summary(models) summodels plot(summodels,main="Fitting Multiple Ellipses Simultaneously")
To output summary results to excel readable file at current directory
## write.table(models$Estimates,"file_name.txt") and ## write.table(summodels$Boot.Estimates,"file_name.txt")
Fitting Hysteresis Loops
The function floop can be used to fit hysteresis loops with specific values of n and m as arguments to floop. Below is an example of a hysteresis loop with n=5, m=3.
loop <- mloop(n=5, m=3,sd.x=0.02,sd.y=0.02) fitloop <- floop(loop$x,loop$y,n=5, m=3,period=24,times="equal") fitloop$Estimates plot(fitloop,main="Fitted Hysteresis Loop") summary(fitloop)
Acknowledgments
NIFA MRF 25-008/W-2173 Impacts of Stress Factors on Performance, Health, and Well Being of Farm Animals
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