Nothing
#
# Copyright 2007-2019 by the individuals mentioned in the source code history
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#------------------------------------------------------------------------------
# Author: Michael D. Hunter
# Date: 2015-04-08
# Filename: StateSpaceContinuous.R
# Purpose: Create and test three example of continuous time state space models.
# Example 1 is an undamped linear oscillator.
# Example 2 is a damped linear oscillator with dynamic noise.
# Example 3 is a general dynamic model with multiple indicators per factor.
# All examples are fit to simulated data. Their parameter estimates are
# compared to their true generating values. This last example was also fit
# with a completely R-based implementation of the same algorithm that is now
# implemented in the C++ backend. The C++ code reaches the same solution
# as the R code, and is about 70x faster on the same machine.
#------------------------------------------------------------------------------
#------------------------------------------------------------------------------
# Load packages
require(mvtnorm)
require(Matrix)
require(OpenMx)
#------------------------------------------------------------------------------
# Example 1
# Undamped linear oscillator, i.e. a noisy sine wave.
# Measurement error, but no dynamic error, single indicator.
# This example works great.
#--------------------------------------
# Data Generation
set.seed(405)
tlen <- 200
t <- seq(1.2, 50, length.out=tlen)
freqParam <- .5
initialCond <- matrix(c(2.5, 0))
x <- initialCond[1,1]*cos(freqParam*t)
plot(t, x, type='l')
measVar <- 1.5
y <- cbind(obs=x+rnorm(tlen, sd=sqrt(measVar)), tim=t)
plot(t, y[,1], type='l')
#--------------------------------------
# Model Specification
#Note: the bounds are here only to keep CSOLNP from
# stepping too far off a cliff. With the bounds in
# place, CSOLNP finds the right solution. Without
# the bounds, CSOLNP goes crazy.
cdim <- list('obs', c('ksi', 'ksiDot'))
amat <- mxMatrix('Full', 2, 2, c(F, T, F, T), c(0, -.1, 1, -.2), name='A', lbound=-10)
bmat <- mxMatrix('Zero', 2, 1, name='B')
cmat <- mxMatrix('Full', 1, 2, FALSE, c(1, 0), name='C', dimnames=cdim)
dmat <- mxMatrix('Zero', 1, 1, name='D')
qmat <- mxMatrix('Zero', 2, 2, name='Q')
rmat <- mxMatrix('Diag', 1, 1, TRUE, .4, name='R', lbound=1e-6)
xmat <- mxMatrix('Full', 2, 1, TRUE, c(0, 0), name='x0', lbound=-10, ubound=10)
pmat <- mxMatrix('Diag', 2, 2, FALSE, 1, name='P0')
umat <- mxMatrix('Zero', 1, 1, name='u')
tmat <- mxMatrix('Full', 1, 1, name='time', labels='data.tim')
osc <- mxModel("LinearOscillator",
amat, bmat, cmat, dmat, qmat, rmat, xmat, pmat, umat, tmat,
mxExpectationStateSpace('A', 'B', 'C', 'D', 'Q', 'R', 'x0', 'P0', 'u', t='time', scores=TRUE),
mxFitFunctionML(),
mxData(y, 'raw'))
#osc <- mxOption(osc, 'Calculate Hessian', 'No')
#osc <- mxOption(osc, 'Standard Errors', 'No')
#osc <- mxOption(osc, 'Major iterations', 0)
oscr <- mxRun(osc)
#--------------------------------------
# Results Examination
summary(oscr)
(ssFreqParam <- mxEval(sqrt(-A[2,1]), oscr))
freqParam
omxCheckWithinPercentError(ssFreqParam, freqParam, 1)
(ssMeasVar <- mxEval(R, oscr))
measVar
omxCheckWithinPercentError(ssMeasVar, measVar, 10)
dampingParam <- 0
(ssDampingParam <- mxEval(-A[2,2], oscr))
dampingParam
omxCheckCloseEnough(ssDampingParam, dampingParam, .01)
rms <- function(x, y){sqrt(mean((x-y)^2))}
(ssInitialCond <- mxEval(x0, oscr))
initialCond
omxCheckTrue(rms(ssInitialCond, initialCond) < .1)
#------------------------------------------------------------------------------
# Example 1 continued
# Data generation and re-fitting
new_data <- mxGenerateData(oscr, nrow(y))
new_data$tim <- y[, 'tim']
nosc <- mxModel(osc, mxData(new_data, 'raw'))
noscr <- mxRun(nosc)
# Parameters look close
cbind(coef(oscr), coef(noscr))
se <- summary(oscr)$parameters[,6]
# Check that the old and new data estimates (data generated by sine+noise vs exact discrete)
# all with confidence intervals of one another.
withinCI <- (coef(noscr) < (coef(oscr) + 1.5*se)) & (coef(noscr) > (coef(oscr) - 1.5*se))
omxCheckEquals(withinCI, rep(TRUE, length(se)))
#------------------------------------------------------------------------------
# Compare frontend and backend scores
fstart <- Sys.time()
ks <- mxKalmanScores(noscr)
fstop <- Sys.time()
bstart <- Sys.time()
ksb <- mxKalmanScores(noscr, frontend=FALSE)
bstop <- Sys.time()
# Compare time to compute
fstop-fstart
bstop-bstart # 70x faster in backend
# Compare values
omxCheckTrue(all.equal(ks, ksb[1:8]))
# values equal to roughly 1.5e-8 #.Machine$double.eps ^ 0.5
cor(cbind(ks$xPredicted, noscr$expectation$xPredicted))
cor(cbind(ks$xUpdated, noscr$expectation$xUpdated))
cor(cbind(ks$xSmoothed, noscr$expectation$xSmoothed))
omxCheckCloseEnough( rms(ks$xPredicted, noscr$expectation$xPredicted), 0, 1e-10)
omxCheckCloseEnough( rms(ks$xUpdated, noscr$expectation$xUpdated), 0, 1e-10)
omxCheckCloseEnough( rms(ks$xSmoothed, noscr$expectation$xSmoothed), 0, 1e-10)
omxCheckCloseEnough( rms(t(apply(ks$PPredicted, 3, vech)), noscr$expectation$PPredicted), 0, 1e-10)
omxCheckCloseEnough( rms(t(apply(ks$PUpdated, 3, vech)), noscr$expectation$PUpdated), 0, 1e-10)
omxCheckCloseEnough( rms(t(apply(ks$PSmoothed, 3, vech)), noscr$expectation$PSmoothed), 0, 1e-10)
plot(new_data$tim, new_data$obs, type='l')
lines(new_data$tim, ks$xSmoothed[-1, 1], col='blue')
#------------------------------------------------------------------------------
# Example 2
# Damped linear oscillator example
# There is measurement noise and there are unmeasured dynamic disturbances.
# These disturbances are called dynamic noise.
# There is a single indicator.
# TODO Figure out what combination of variables can really be estimated for
# this kind of model. It is clear that measurement noise and SOME dynamic
# noise can be estimated, but not all.
#--------------------------------------
# Data Generation
xdim <- 2
udim <- 1
ydim <- 1
tdim <- 200
set.seed(315)
tA <- matrix(c(0, -.3, 1, -.7), xdim, xdim)
tB <- matrix(c(0), xdim, udim)
tC <- matrix(c(1, 0), ydim, xdim)
tD <- matrix(c(0), ydim, udim)
tQ <- matrix(c(0), xdim, xdim); diag(tQ) <- c(0, 2.2)
tR <- matrix(c(0), ydim, ydim); diag(tR) <- c(1.5)
x0 <- matrix(c(0, 1), xdim, 1)
P0 <- diag(c(1), xdim)
tdx <- matrix(0, xdim, tdim+1)
tx <- matrix(0, xdim, tdim+1)
tu <- matrix(0, udim, tdim)
ty <- matrix(0, ydim, tdim)
tT <- matrix((0:tdim)/4, nrow=1, ncol=tdim+1)
tI <- diag(1, nrow=xdim)
tx[,1] <- x0
for(i in 2:(tdim+1)){
q <- t(rmvnorm(1, rep(0, xdim), tQ))
tdx[,i] <- tA %*% tx[,i-1] + tB %*% tu[,i-1] + q
expA <- as.matrix(expm(tA * (tT[,i]-tT[,i-1])))
intA <- solve(tA) %*% (expA - tI)
eF <- as.matrix( expm( rbind(cbind(-tA, tQ), cbind(matrix(0, xdim, xdim), t(tA))) * (tT[,i]-tT[,i-1])))
tQd <- expA %*% eF[1:xdim, (xdim+1):(2*xdim)]
# Whiten the continuous-time error then transform it to the continuous time error cov
s <- svd(tQ)
dinv <- s$d
dinv[s$d > 0] <- 1/s$d[s$d > 0]
W <- diag(sqrt(dinv)) %*% t(s$v)
qd <- t(chol(tQd)) %*% W %*% q
tx[,i] <- expA %*% tx[, i-1] + intA %*% tB %*% tu[,i-1] + qd
#ty[,i-1] <- tC %*% tx[,i-1] + tD %*% tu[,i-1] + t(rmvnorm(1, rep(0, ydim), tR))
ty[,i-1] <- tC %*% tx[,i] + tD %*% tu[,i-1] + t(rmvnorm(1, rep(0, ydim), tR))
}
#plot(tx[1,], type='l')
rownames(ty) <- paste('y', 1:ydim, sep='')
rownames(tx) <- paste('x', 1:xdim, sep='')
plot(tT[,-1], ty[1,], type='l')
plot(tT[1,], tx[1,], type='l')
#--------------------------------------
# Model Specification
cdim <- list(rownames(ty), c('ksi', 'ksiDot'))
amat <- mxMatrix('Full', xdim, xdim, c(F, T, F, T), c(0, -.1, 1, -.2), name='A')
bmat <- mxMatrix('Zero', xdim, udim, name='B')
cmat <- mxMatrix('Full', ydim, xdim, FALSE, c(1, 0), name='C', dimnames=cdim)
dmat <- mxMatrix('Zero', udim, udim, name='D')
qmat <- mxMatrix('Diag', xdim, xdim, c(F,T), c(0, 1), name='Q', lbound=1e-6)
rmat <- mxMatrix('Diag', ydim, ydim, TRUE, 1.5, name='R')
xmat <- mxMatrix('Full', xdim, 1, c(T,F), c(0, 1), name='x0')
pmat <- mxMatrix('Diag', xdim, xdim, FALSE, 1, name='P0')
umat <- mxMatrix('Zero', udim, 1, name='u')
tmat <- mxMatrix('Full', 1, 1, name='time', labels='data.times')
dosc <- mxModel("DampedLinearOscillator",
amat, bmat, cmat, dmat, qmat, rmat, xmat, pmat, umat, tmat,
mxExpectationStateSpaceContinuousTime('A', 'B', 'C', 'D', 'Q', 'R', 'x0', 'P0', 'u', t='time'),
mxFitFunctionML(),
mxData(cbind(t(ty), times=tT[,-1]), 'raw'))
#dosc <- mxOption(dosc, 'Calculate Hessian', 'No')
#dosc <- mxOption(dosc, 'Standard Errors', 'No')
#dosc <- mxOption(dosc, 'Major iterations', 0)
doscr <- mxRun(dosc)
#--------------------------------------
# Results Examination
summary(doscr)
mxEval(A, doscr)
tA
omxCheckTrue(rms(mxEval(A, doscr)[2,1], tA[2,1]) < .07)
mxEval(R, doscr)
tR
omxCheckTrue(rms(mxEval(R, doscr), tR) < .2)
mxEval(Q, doscr)
tQ
omxCheckTrue(rms(mxEval(Q, doscr)[2,2], tQ[2,2]) < .7)
mxEval(x0, doscr) #poorly estimated, likely due to dynamic error
x0
omxCheckTrue(rms(mxEval(x0, doscr)[1,1], x0[1,1]) < .7)
estParam <- coef(doscr)
truParam <- c(tA[2,], tQ[2,2], tR, x0[1,1])
estSE <- summary(doscr)$parameters[,6]
ex3Tvals <- (estParam - truParam) / estSE
omxCheckTrue(all(abs(ex3Tvals) < 0.6))
s <- solve(kronecker(diag(xdim), mxEval(A, doscr)) + kronecker(mxEval(A, doscr), diag(xdim)))
(asym_lvar <- matrix(-s %*% matrix(mxEval(Q, doscr)), xdim, xdim))
(obs_lvar <- var(t(tx)))
omxCheckTrue(rms(vech(asym_lvar), vech(obs_lvar)) < .7)
(asym_ovar <- mxEval(C, doscr) %*% asym_lvar %*% t(mxEval(C, doscr)) + mxEval(R, doscr))
(obs_ovar <- var(t(ty)))
omxCheckTrue(rms(vech(asym_ovar), vech(obs_ovar)) < 1.1)
#------------------------------------------------------------------------------
# Example 3
# Full bore example of general dynamics with dynamic error,
# measurement error, and multiple indicators per latent variable.
#--------------------------------------
# Data generation
xdim <- 3
udim <- 2
ydim <- 9
tdim <- 500
set.seed(948)
tA <- matrix(c(-.4, 0, 0, 0, -.9, .1, 0, -.1, -.9), xdim, xdim)
tB <- matrix(c(0), xdim, udim)
tC <- matrix(c(runif(3, .4, 1), rep(0, ydim), runif(3, .4, 1), rep(0, ydim), runif(3, .4, 1)), ydim, xdim)
tD <- matrix(c(0), ydim, udim)
tQ <- matrix(c(0), xdim, xdim); diag(tQ) <- runif(xdim, 1, 2) # 0, 1
tR <- matrix(c(0), ydim, ydim); diag(tR) <- runif(ydim)
x0 <- matrix(c(rnorm(xdim)), xdim, 1)
P0 <- diag(c(runif(xdim)))
tdx <- matrix(0, xdim, tdim+1)
tx <- matrix(0, xdim, tdim+1)
tu <- matrix(0, udim, tdim)
ty <- matrix(0, ydim, tdim)
tT <- matrix(0:tdim, nrow=1, ncol=tdim+1)
tI <- diag(1, nrow=xdim)
tx[,1] <- x0
for(i in 2:(tdim+1)){
q <- t(rmvnorm(1, rep(0, xdim), tQ))
tdx[,i] <- tA %*% tx[,i-1] + tB %*% tu[,i-1] + q
expA <- as.matrix(expm(tA * (tT[,i]-tT[,i-1])))
intA <- solve(tA) %*% (expA - tI)
eF <- as.matrix( expm( rbind(cbind(-tA, tQ), cbind(matrix(0, xdim, xdim), t(tA))) * (tT[,i]-tT[,i-1])))
tQd <- expA %*% eF[1:xdim, (xdim+1):(2*xdim)]
# Whiten the continuous-time error then transform it to the continuous time error cov
s <- svd(tQ)
dinv <- s$d
dinv[s$d > 0] <- 1/s$d[s$d > 0]
W <- diag(sqrt(dinv)) %*% t(s$v)
qd <- t(chol(tQd)) %*% W %*% q
tx[,i] <- expA %*% tx[, i-1] + intA %*% tB %*% tu[,i-1] + qd
#ty[,i-1] <- tC %*% tx[,i-1] + tD %*% tu[,i-1] + t(rmvnorm(1, rep(0, ydim), tR))
ty[,i-1] <- tC %*% tx[,i] + tD %*% tu[,i-1] + t(rmvnorm(1, rep(0, ydim), tR))
}
#plot(tx[1,], type='l')
rownames(ty) <- paste('y', 1:ydim, sep='')
rownames(tx) <- paste('x', 1:xdim, sep='')
plot(ty[1,], type='l')
#--------------------------------------
# Model Specification
smod <- mxModel(
name='SSContinuous',
mxMatrix(name='A', values=tA, nrow=xdim, ncol=xdim, free=c(TRUE, FALSE, FALSE, FALSE, TRUE, TRUE, FALSE, FALSE, TRUE), labels=c('a', NA, NA, NA, 'b', 'c', NA, 'csym[1,1]', 'b'), ubound=c(0, NA, NA, NA, 0, NA, NA, NA, 0)),
mxAlgebra(name='csym', -c),
mxMatrix(name='B', values=0, nrow=xdim, ncol=udim, free=FALSE),
mxMatrix(name='C', values=tC, nrow=ydim, ncol=xdim, free=(tC!=0), dimnames=list(rownames(ty), rownames(tx))),
mxMatrix(name='D', values=0, nrow=ydim, ncol=udim, free=FALSE),
# Note Factor error matrix is fixed! This is for model identification.
# I happen to fix the variances to their true values.
mxMatrix(name='Q', type='Diag', values=diag(tQ), nrow=xdim, ncol=xdim, free=FALSE),
mxMatrix(name='R', type='Diag', values=diag(tR), nrow=ydim, ncol=ydim, free=TRUE),
mxMatrix(name='x', values=x0, nrow=xdim, ncol=1, free=FALSE),
mxMatrix(name='P', values=P0, nrow=xdim, ncol=xdim, free=FALSE),
mxMatrix("Zero", udim, 1, name="u"),
mxMatrix("Full", 1, 1, name="time", labels="data.Time"),
mxData(observed=cbind(t(ty), Time=tT[,-1]), type='raw'),
mxExpectationSSCT(A='A', B='B', C='C', D='D', Q='Q', R='R', x0='x', P0='P', u='u', t='time'),
mxFitFunctionML()
)
# Uncomment for degugging
#smod <- mxOption(smod, 'Calculate Hessian', 'No')
#smod <- mxOption(smod, 'Standard Errors', 'No')
#smod <- mxOption(smod, 'Major iterations', 0)
srun <- mxRun(smod)
# takes about a minute on Squishy.
# The same model all in R takes 1.46 hours.
# Speed-up is approx 70x.
# Check likelihoods of initial parameters
# -2LL
#summary(srun)$Minus2LogLikelihood # when major iterations is 0
#2*dlmLL(y=t(ty), mod=mfun(tinit)) + 200*9*log(2*pi) # dlm gives back -LL - CONST, so adjust it. 200 is N, 9 is k
#--------------------------------------
# Results Examination
summary(srun)
mxEval(A, srun); tA
omxCheckTrue(rms(mxEval(A, srun)[tA!=0], tA[tA!=0]) < .02)
mxEval(C, srun)[tC!=0]; tC[tC!=0]
omxCheckTrue(rms(mxEval(C, srun)[tC!=0], tC[tC!=0]) < .1)
diag(mxEval(R, srun)); diag(tR)
omxCheckTrue(rms(diag(mxEval(R, srun)), diag(tR)) < .1)
restA <- matrix(c(
-0.3173239, 0.00000000, 0.00000000,
0.0000000, -1.09531962, -0.04313248,
0.0000000, 0.04313248, -1.09531962
), xdim, xdim, byrow=TRUE)
restC <- c(0.4789963, 0.6038891, 0.7602543, 0.9514004, 0.7272259,
1.0570274, 0.5431997, 1.0255853, 0.4885580)
restR <- c(0.3226227, 0.8767227, 1.1029966, 0.5607970, 0.1622717,
0.8304199, 0.7078884, 0.9453716, 0.9422717)
restAnew <- matrix(c(
-0.4381936, 0, 0,
0, -0.90029646, -0.08843642,
0, 0.08843642, -0.90029646), xdim, xdim, byrow=TRUE)
restCnew <- c(0.4779426, 0.7274788, 0.7311741, 0.9677682, 0.7325752,
1.0499232, 0.4789363, 1.0284996, 0.5794153)
restRnew <- c(0.3115990, 0.8528872, 1.1730305, 0.5813367, 0.1538722,
0.8300052, 0.7317202, 0.8460866, 0.9378043)
# At least 7x improvement in dynamics estimation
omxCheckTrue(rms(restA, tA) / rms(restAnew, tA) > 7)
# At most 30% decrement in factor loadings estimation
omxCheckTrue(rms(restCnew, tC[tC!=0]) / rms(restC, tC[tC!=0]) < 1.3)
# At most 20% decrement in residual variance estimation
omxCheckTrue(rms(restRnew, diag(tR)) / rms(restR, diag(tR)) < 1.2)
omxCheckCloseEnough(diag(mxEval(A, srun)), diag(restAnew), 0.05)
omxCheckCloseEnough(mxEval(C, srun)[tC!=0], restCnew, 0.01)
omxCheckCloseEnough(diag(mxEval(R, srun)), restRnew, 0.02)
s <- solve(kronecker(diag(xdim), mxEval(A, srun)) + kronecker(mxEval(A, srun), diag(xdim)))
(asym_lvar <- matrix(-s %*% matrix(mxEval(Q, srun)), xdim, xdim))
(obs_lvar <- var(t(tx)))
omxCheckTrue(rms(vech(asym_lvar), vech(obs_lvar)) < .05)
(asym_ovar <- mxEval(C, srun) %*% asym_lvar %*% t(mxEval(C, srun)) + mxEval(R, srun))
(obs_ovar <- var(t(ty)))
omxCheckTrue(rms(vech(asym_ovar), vech(obs_ovar)) < 0.05)
#------------------------------------------------------------------------------
# Done
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.