| planarR {lazy.fa} | R Documentation |
Planar Rotation
planarR( func, A, initT, ..., method = "nlm", sim = 1, init = "pc", maxiter = 100, eps = 1e-08, epsd = 1e-06, stepmax = 1, SQUAREM = 3, nSQUAREM = 1, minalpha = -999, maxalpha = -0.5, always = 1, reset1 = 0, reset2 = 1, print = 0 )
func |
Criterion Function to be Minimized which must have |
A |
Factor Pattern Matrix to be rotated |
initT |
The initial orthogonal rotation matrix
to start planar rotation. |
... |
additional parameters to func such as
P, Delta, Q, and Lambda for oblique rotation with fixed R. |
method |
The method to be used for minimization |
sim |
= 0 to use ALS |
init |
= Initial rotation method: "pc", "varimax", "promax", or "random". |
maxiter |
max # of iterations |
eps |
convergence criterion for the relative improvement of criterion |
epsd |
convergence criterion for the max abs difference of thetas |
SQUAREM |
= 3 : See the help of iSQUAREM in lazy.accel package. |
nSQUAREM |
when to star iSQUAREM |
minalpha |
= -999 : See the help of iSQUAREM in lazy.accel package. |
maxalpha |
= -1 : See the help of iSQUAREM in lazy.accel package. |
always |
= 1 : See the help of iSQUAREM in lazy.accel package. |
reset1 |
= 0 : See the help of iSQUAREM in lazy.accel package. |
reset2 |
= 1 : See the help of iSQUAREM in lazy.accel package. |
print |
>= 1 to print the result |
A matrix will be rotated as
B=A %*% T
where
T = T_{21} T_{31} T_{32} ... T{ndim, ndim-1}
where T_{ij} is the ndim x ndim matrix defined as
T_{ij}[i,i]=cos(theta_{ij}); T_{ij}[j,j]=cos(theta_{ij});
T_{ij}[i,j]=sin(theta_{ij}); T_{ij}[j,i]=-sin(theta_{ij})
This reparametrizes the free elements of T to theta_{ij}, i > 2,
and we store them in thetas vector of length ndim*(ndim-1)/2.
func function must have the following parametes:
theta, i, j, Ta, A
If func has an additional set of parameters, they must be passed
through ... argument.
The minimization of func with respect to thetas will be performed
in two ways:
If sim=0, the ALS approach with inline SQUAREM algorithm will be used
where the conditional minimum of func with respect to each element of
thetas is sought using the method specified in the method parameter.
Available methods are: "nlm", "optimize", and "grid".
If sim=1 the func will be minimized with respect to all the elements of
thetas by R native "nlm" function.
When critOBPR is used to perform an Procrustes rotation,
sim=0 and method="grid" MUST be specified.
Note that for the oblique rotation with the presicribed factor correlation
matrix, R, the actual rotation matrix can be obtained as follows.
Let
R=Q %*% Delta %*% t(Q) and S=P %*% Lambda %*% t(P)
where R is the target factor correlation matrix and
S is the current factor correlation matrix associated with A,
Then, the rotated factor loadings which preserves R is given as
B=A%*%solve(t(W))
where
W=P %*%diag(1/sqrt(Lambda))%*%T%*%diag(sqrt(Delta))%*%t(Q)
or
B=A%*%invtW
where
invtW=P%*%diag(sqrt(Lambda))%*%T%*%diag(1/sqrt(Delta))%*%t(Q).
See the example below.
Note on the iSQUAREM:
Try always=1 with maxalpha=-1 first.
If it seems not working, use always=0 with maxalpha=1.
Changing reset1=1 and reset2=2 may help.
If all of the above fail, be patient and use SQUAREM=0.
A list of the following:
T, Ta, thetas, critval, sim, llll, SQUAREM, T0
T = The Orthogonal rotation matrix
Ta = The array of planar rotation matrices: i>j
T0 = The initial roation matrix
thetas = The vector of angles: i>j
critval = The value of criterion minimized
sim = The value of sim parameter
llll = The number of iterations required
SQUAREM = The value of SQUAREM parameter
# generate factor loadings
seed=1701
set.seed(seed)
n=30; ndim=5
A=gendatafa_A( n, ndim )$loadings
# Simple Orthomax rotation with kappa parameter:
# ... argument is kappa=1.
resp <- planarR( orthomax, A, NULL
, kappa=1
, maxiter=400, eps=1e-7, epsd=1e-7
, method="nlm", sim=1, init="pca" )
# print the result
T=resp$T
AT=A%*%T
Print(AT, fmt="7.3",fuzz=0.3)
Print(varimax(A,normalize=0)$loadings, fmt="7.3",fuzz=0.3)
# Target Factor Correlation
rho=0.2
R=matrix(rho,ndim,ndim); diag(R)=1
svd=svd(R); Q=svd$u; Delta=svd$d
# current factor dispersion is I
S=diag(ndim)
svd=svd(S); P=svd$u; Lambda=svd$d
# Oblique rotation with prescribed R:
# ... argument consists of P=P, Delta=Delta, Q=Q, Lambda=Lambda.
resp <- planarR( critOBR, A, NULL
, P=P, Delta=Delta, Q=Q, Lambda=Lambda
, maxiter=400, eps=1e-7, epsd=1e-7
, method="nlm", sim=1, init="pca" )
# recover the oblique rotation matrix from T.
T=resp$T
invtW=P %*% diag(sqrt(Lambda)) %*% T %*% diag(1/sqrt(Delta)) %*% t(Q)
W=solve(t(invtW))
AW=A%*%invtW
AW=reorder_fa(AW)$loadings
Print(AW, fmt="7.3",fuzz=0.3)
Print(t(W)%*%W, fmt="7.3")
# Oblique Procrustes rotation with prescribed factor correlation
seed=1701
set.seed(seed)
# Generate Target and loadings
n=50; ndim=6
temp=gendatafa_A( n, ndim )
Target=temp$loadings01
A=temp$loadings
Print(Target,A)
loc=( (1:ndim)[(1:ndim)%%2 == 0 | (1:ndim)%%3 == 0] )
Print(ndim,loc)
A[,loc]=-A[,loc]
# current factor dispersion is I
S=diag(ndim)
svd=svd(S); P=svd$u; Lambda=svd$d
# Target Factor Correlation
rho=0.2
R=matrix(rho,ndim,ndim); diag(R)=1
svd=svd(R); Q=svd$u; Delta=svd$d
resp <- planarR( critOBPR, A, NULL
, Target=Target, P=P, Delta=Delta, Q=Q, Lambda=Lambda
, sim=0, method="grid", print=2, SQUAREM=3
, maxiter=100, eps=1e-7, epsd=1e-7 )
T=resp$T
invtW=P %*% diag(sqrt(Lambda)) %*% T %*% diag(1/sqrt(Delta)) %*% t(Q)
W=solve(t(invtW))
AA=A%*%invtW
tWW=t(W)%*%W
Print(T,W)
Print(R,tWW)
Print(Target, AA,fmt="8.1 8.5", fuzz=0.3)