hysteresis (version 2.7.2)

mel: Simulate (Make) an Ellipse

Description

Produces an ellipse based on 1 of 4 possible formulations: 1-Eigenvalues, 2-Hysteresis Coefs, 3-Amplitudes and 4-Algebraic Coefs.

Usage

mel(method=1,seed=NULL,...)
mel1(cx=32,cy=39,rote.deg=2,semi.major=7,semi.minor=0.23,
phase.angle=0,n.points=24,period=24,sd.x=0,sd.y=0)
mel2(cx=32,cy=39,b.x=6.99,b.y=0.244,retention=0.23,
phase.angle=0,n.points=24,period=24,sd.x=0,sd.y=0)
mel3(cx=32,cy=39,ampx=6.99,ampy=0.335,lag=2.888,phase.angle=0,
n.points=24,period=24,sd.x=0,sd.y=0)
mel4(x2=0.002293,xy=-.06960,y2=0.9976,x=2.567,y=-75.58,int=1432.7,
phase.angle=0,n.points=24,period=24,sd.x=0,sd.y=0)

Value

mel returns an object of class ellipsemake.

values

the nine fundamental parameters (cx,cy,rote.deg,semi.major,semi.minor,b.x,b.y,a,phase.angle) of which only four or five are used along with the four derived parameters (area, lag, retention, coercion).

method

the method used.

x

the input x.

y

the output y.

Arguments

method

selects which of the functions mel1, mel2, mel3, mel4 to use to describe the ellipse.

seed

integer, the starting seed.

...

arguments to the functions mel1, mel2, mel3, mel4 described below.

cx

Center of input x.

cy

Center of output y.

phase.angle

defines the starting point of the ellipse. Does not change ellipse shape.

rote.deg

Theta, angle of rotation. In degrees. Only used if method=1.

semi.major

Half length of major axis. Only used if method=1.

semi.minor

Half length of minor axis. Only used if method=1.

b.x

Saturation point x coordinate. Only used if method=2.

b.y

Saturation point y coordinate. Only used if method=2.

retention

another ellipse parameter used if method=2. split point, representing vertical distance from center to upper loop trajectory. It is the intersection of the loop and the output axis characterizing the distortion in the response at the average input challenge.

ampx

The range of the ellipse input values divided by 2. Only used if method=3.

ampy

The range of the ellipse output values divided by 2. Only used if method=3.

lag

The number of points between the location where the input reaches its maximum value and where the output reaches its maximum value. Lag is therefore dependent on the value chosen for period. Only used if method=3.

x2

Coefficient on x^2 in the equation found in details. Only used if method=4.

xy

Coefficient on xy in the equation found in details. Only used if method=4.

y2

Coefficient on y^2 in the equation found in details. Only used if method=4.

x

Coefficient on x in the equation found in details. Only used if method=4.

y

Coefficient on y in the equation found in details. Only used if method=4.

int

Coefficient on the intercept in the equation found in details. Only used if method=4.

n.points

Number of points on ellipse. Equally spaced around circumference of ellipse/period.

period

Number of points required to make a full loop around the ellipse.

sd.x

optional number specifying a normally distributed standard deviation for x.

sd.y

optional number specifying a normally distributed standard deviation for y.

Author

Spencer Maynes, Fan Yang, and Anne Parkhurst.

Details

All of the four methods can be used to specify a series of points that make up an ellipse. The function mel uses parameters to form an ellipse and find derived variables such as area, lag, retention, and coercion. Optionally, normally distributed random variation can be introduced in both the x and y directions. The first method is useful alongside the nls, lm and direct fitting methods, while the second is comparable to the harmonic2 ellipse fitting method. The third method for mel is included because it is the easiest to interpret. Finally the fourth method uses the equation 0=a0+a1*x^2+a2*xy+a3*y^2+a4*x+a5*y to form an ellipse. The "a" parameters here are marked as int, x2, xy, y2, x and y in the function itself.

References

Yang, F. and A. Parkhurst, Efficient Estimation of Elliptical Hysteresis. (submitted)

See Also

fel for fitting observations that form an ellipse and creating an ellipsefit object, plot.ellipsefitfor plotting an ellipsefit object. summary.ellipsefit for summarizing an ellipsefit object, and plot.ellipsesummary for plotting an ellipsesummary object.

Examples

ellipseA <- mel(method=3,cx=35, cy=39, ampx=7, ampy=2, lag=3, sd.x=0.2,sd.y=0.04)
 ellipseA.fit <- fel(ellipseA$x,ellipseA$y)
  plot(ellipseA.fit,xlab="Input",ylab="Output",main="Simulated Ellipse",
  putNumber=TRUE)
 boot.ellipseA.fit <- fel(ellipseA$x,ellipseA$y, boot=TRUE, seed=231)
  plot(boot.ellipseA.fit,xlab="Input",ylab="Output",
  main="Bootstrapped Ellipse",values="ellipse.all")

ellipse.eig <- mel(semi.major=7,semi.minor=4,rote.deg=30)
ellip.eigen.fit <- fel(ellipse.eig$x,ellipse.eig$y)
ellip.eigen.fit$Estimates
plot(ellip.eigen.fit,main="Ellipse from Eigenvalue Parameters",
show=c("semi.major","semi.minor","rote.deg"),values="ellipse")