invgaussMLE: Maximum Likelihood Parameter Estimation of an Inverse...
In STAR: Spike Train Analysis with R
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
Estimate inverse Gaussian model parameters by the maximum likelihood
method using possibly censored data. Two different parameterizations
of the inverse Gaussian distribution can be used.
Usage
1
2
3
invgaussMLE(yi, ni = numeric(length(yi)) + 1,
si = numeric(length(yi)) + 1,
parameterization = "sigma2")
Arguments
yi
vector of (possibly binned) observations or a
spikeTrain object.
ni
vector of counts for each value of yi; default: numeric(length(yi))+1.
si
vector of counts of uncensored observations for each
value of yi; default: numeric(length(yi))+1.
parameterization
parameterization used, "sigma2"
(default) of "boundary".
Details
The 2 different parameterizations of the inverse Gaussian distribution
are discussed in the manual of dinvgauss.
In the absence of censored data the ML estimates are available in
closed form (Lindsey, 2004, p 212) together with the Hessian matrix at
the MLE. In presence of censored data an initial guess for the
parameters is obtained using the uncensored data before maximizing the
likelihood function to the full data set using optim
with the BFGS method. ML
estimation is always performed with the "sigma2"
parameterization. Parameters and variance-covariance matrix are
transformed at the end if the "boundary" parameterization is
requested.
In order to ensure good behavior of the numerical optimization
routines, optimization is performed on the log of the parameters
(mu and sigma2).
Standard errors are obtained from the inverse of the observed
information matrix at the MLE. They are transformed to go from the log
scale used by the optimization routine, when the latter is used (ie,
for censored data) to the parameterization requested.
Value
A list of class durationFit with the following components:
estimate
the estimated parameters, a named vector.
se
the standard errors, a named vector.
logLik
the log likelihood at maximum.
r
a function returning the log of the relative likelihood function.
mll
a function returning the opposite of the log likelihood
function using the log of the parameters.
call
the matched call.
Note
The returned standard errors (component se) are valid in the asymptotic limit. You
should plot contours using function r in the returned list and
check that the contours are reasonably close to ellipses.
Author(s)
Christophe Pouzat christophe.pouzat@gmail.com
References
Lindsey, J.K. (2004) Introduction to Applied Statistics: A
Modelling Approach. OUP.
See Also
dinvgauss,lnormMLE,gammaMLE,weibullMLE,llogisMLE,rexpMLE.
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## Simulate sample of size 100 from an inverse Gaussian
## distribution
set.seed(1102006,"Mersenne-Twister")
sampleSize <- 100
mu.true <- 0.075
sigma2.true <- 3
sampleSize <- 100
sampIG <- rinvgauss(sampleSize,mu=mu.true,sigma2=sigma2.true)
## Make a maximum likelihood fit
sampIGmleIG <- invgaussMLE(sampIG)
## Compare estimates with actual values
rbind(est = coef(sampIGmleIG),se = sampIGmleIG$se,true = c(mu.true,sigma2.true))
## In the absence of censoring the MLE of the inverse Gaussian is available in a
## closed form together with its variance (ie, the observed information matrix)
## we can check that we did not screw up at that stage by comparing the observed
## information matrix obtained numerically with the analytical one. To do that we
## use the MINUS log likelihood function returned by invgaussMLE to get a numerical
## estimate
detailedFit <- optim(par=as.vector(log(sampIGmleIG$estimate)),
fn=sampIGmleIG$mll,
method="BFGS",
hessian=TRUE)
## We should not forget that the "mll" function uses the log of the parameters while
## the "se" component of sampIGmleIG list is expressed on the linear scale we must therefore
## transform one into the other as follows (Kalbfleisch, 1985, p71):
## if x = exp(u) and y = exp(v) and if we have the information matrix in term of
## u and v (that's the hessian component of list detailedFit above), we create matrix:
## du/dx du/dy
## Q =
## dv/dx dv/dy
## and we get I in term of x and y by the following matrix product:
## I(x,y) <- t(Q) %*% I(u,v) %*% Q
## In the present case:
## du/dx = 1/exp(u), du/dy = 0, dv/dx = 0, dv/dy = 1/exp(v)
## Therefore:
Q <- diag(1/exp(detailedFit$par))
numericalI <- t(Q) %*% detailedFit$hessian %*% Q
seComp <- rbind(sampIGmleIG$se, sqrt(diag(solve(numericalI))))
colnames(seComp) <- c("mu","sigma2")
rownames(seComp) <- c("analytical", "numerical")
seComp
## We can check the relative differences between the 2
apply(seComp,2,function(x) abs(diff(x))/x[1])
## Not run:
## Estimate the log relative likelihood on a grid to plot contours
Mu <- seq(coef(sampIGmleIG)[1]-4*sampIGmleIG$se[1],
coef(sampIGmleIG)[1]+4*sampIGmleIG$se[1],
sampIGmleIG$se[1]/10)
Sigma2 <- seq(coef(sampIGmleIG)[2]-4*sampIGmleIG$se[2],
coef(sampIGmleIG)[2]+4*sampIGmleIG$se[2],
sampIGmleIG$se[2]/10)
sampIGmleIGcontour <- sapply(Mu, function(mu) sapply(Sigma2, function(s2) sampIGmleIG$r(mu,s2)))
## plot contours using a linear scale for the parameters
## draw four contours corresponding to the following likelihood ratios:
## 0.5, 0.1, Chi2 with 2 df and p values of 0.95 and 0.99
X11(width=12,height=6)
layout(matrix(1:2,ncol=2))
contour(Mu,Sigma2,t(sampIGmleIGcontour),
levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
labels=c("log(0.5)",
"log(0.1)",
"-1/2*P(Chi2=0.95)",
"-1/2*P(Chi2=0.99)"),
xlab=expression(mu),ylab=expression(sigma^2),
main="Log Relative Likelihood Contours"
)
points(coef(sampIGmleIG)[1],coef(sampIGmleIG)[2],pch=3)
points(mu.true,sigma2.true,pch=16,col=2)
## The contours are not really symmetrical about the MLE we can try to
## replot them using a log scale for the parameters to see if that improves
## the situation
contour(log(Mu),log(Sigma2),t(sampIGmleIGcontour),
levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
labels="",
xlab=expression(log(mu)),ylab=expression(log(sigma^2)),
main="Log Relative Likelihood Contours",
sub="log scale for the parameters")
points(log(coef(sampIGmleIG)[1]),log(coef(sampIGmleIG)[2]),pch=3)
points(log(mu.true),log(sigma2.true),pch=16,col=2)
## Even with the log scale the contours are not ellipsoidal, so let us compute profiles
## For that we are going to use the returned MINUS log likelihood function
logMuProfFct <- function(logMu,...) {
myOpt <- optimise(function(x) sampIGmleIG$mll(c(logMu,x))+logLik(sampIGmleIG),...)
as.vector(unlist(myOpt[c("objective","minimum")]))
}
logMuProfCI <- function(logMu,
CI,
a=logS2Seq[1],
b=logS2Seq[length(logS2Seq)]) logMuProfFct(logMu,c(a,b))[1] - qchisq(CI,1)/2
logS2ProfFct <- function(logS2,...) {
myOpt <- optimise(function(x) sampIGmleIG$mll(c(x,logS2))+logLik(sampIGmleIG),...)
as.vector(unlist(myOpt[c("objective","minimum")]))
}
logS2ProfCI <- function(logS2, CI,
a=logMuSeq[1],
b=logMuSeq[length(logMuSeq)]) logS2ProfFct(logS2,c(a,b))[1] - qchisq(CI,1)/2
## We compute profiles (on the log scale) eploxing +/- 3 times
## the se about the MLE
logMuSE <- sqrt(diag(solve(detailedFit$hessian)))[1]
logMuSeq <- seq(log(coef(sampIGmleIG)[1])-3*logMuSE,
log(coef(sampIGmleIG)[1])+3*logMuSE,
logMuSE/10)
logS2SE <- sqrt(diag(solve(detailedFit$hessian)))[2]
logS2Seq <- seq(log(coef(sampIGmleIG)[2])-3*logS2SE,
log(coef(sampIGmleIG)[2])+3*logS2SE,
logS2SE/10)
logMuProf <- sapply(logMuSeq,logMuProfFct,
lower=logS2Seq[1],
upper=logS2Seq[length(logS2Seq)])
## Get 95
logMuCI95 <- c(uniroot(logMuProfCI,
interval=c(logMuSeq[1],log(coef(sampIGmleIG)[1])),
CI=0.95)$root,
uniroot(logMuProfCI,
interval=c(log(coef(sampIGmleIG)[1]),logMuSeq[length(logMuSeq)]),
CI=0.95)$root
)
logMuCI99 <- c(uniroot(logMuProfCI,
interval=c(logMuSeq[1],log(coef(sampIGmleIG)[1])),
CI=0.99)$root,
uniroot(logMuProfCI,
interval=c(log(coef(sampIGmleIG)[1]),logMuSeq[length(logMuSeq)]),
CI=0.99)$root
)
logS2Prof <- sapply(logS2Seq,logS2ProfFct,
lower=logMuSeq[1],
upper=logMuSeq[length(logMuSeq)])
## Get 95
logS2CI95 <- c(uniroot(logS2ProfCI,
interval=c(logS2Seq[1],log(coef(sampIGmleIG)[2])),
CI=0.95)$root,
uniroot(logS2ProfCI,
interval=c(log(coef(sampIGmleIG)[2]),logS2Seq[length(logS2Seq)]),
CI=0.95)$root
)
logS2CI99 <- c(uniroot(logS2ProfCI,
interval=c(logS2Seq[1],log(coef(sampIGmleIG)[2])),
CI=0.99)$root,
uniroot(logS2ProfCI,
interval=c(log(coef(sampIGmleIG)[2]),logS2Seq[length(logS2Seq)]),
CI=0.99)$root
)
## Add profiles to the previous plot
lines(logMuSeq,logMuProf[2,],col=2,lty=2)
lines(logS2Prof[2,],logS2Seq,col=2,lty=2)
## We can now check the deviations of the (profiled) deviances
## from the asymptotic parabolic curves
X11()
layout(matrix(1:4,nrow=2))
oldpar <- par(mar=c(4,4,2,1))
logMuSeqOffset <- logMuSeq-log(coef(sampIGmleIG)[1])
logMuVar <- diag(solve(detailedFit$hessian))[1]
plot(logMuSeq,2*logMuProf[1,],type="l",xlab=expression(log(mu)),ylab="Deviance")
lines(logMuSeq,logMuSeqOffset^2/logMuVar,col=2)
points(log(coef(sampIGmleIG)[1]),0,pch=3)
abline(h=0)
abline(h=qchisq(0.95,1),lty=2)
abline(h=qchisq(0.99,1),lty=2)
lines(rep(logMuCI95[1],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logMuCI95[2],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logMuCI99[1],2),c(0,qchisq(0.99,1)),lty=2)
lines(rep(logMuCI99[2],2),c(0,qchisq(0.99,1)),lty=2)
## We can also "linearize" this last graph
plot(logMuSeq,
sqrt(2*logMuProf[1,])*sign(logMuSeqOffset),
type="l",
xlab=expression(log(mu)),
ylab=expression(paste("signed ",sqrt(Deviance)))
)
lines(logMuSeq,
sqrt(logMuSeqOffset^2/logMuVar)*sign(logMuSeqOffset),
col=2)
points(log(coef(sampIGmleIG)[1]),0,pch=3)
logS2SeqOffset <- logS2Seq-log(coef(sampIGmleIG)[2])
logS2Var <- diag(solve(detailedFit$hessian))[2]
plot(logS2Seq,2*logS2Prof[1,],type="l",xlab=expression(log(sigma^2)),ylab="Deviance")
lines(logS2Seq,logS2SeqOffset^2/logS2Var,col=2)
points(log(coef(sampIGmleIG)[2]),0,pch=3)
abline(h=0)
abline(h=qchisq(0.95,1),lty=2)
abline(h=qchisq(0.99,1),lty=2)
lines(rep(logS2CI95[1],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logS2CI95[2],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logS2CI99[1],2),c(0,qchisq(0.99,1)),lty=2)
lines(rep(logS2CI99[2],2),c(0,qchisq(0.99,1)),lty=2)
## We can also "linearize" this last graph
plot(logS2Seq,
sqrt(2*logS2Prof[1,])*sign(logS2SeqOffset),
type="l",
xlab=expression(log(sigma^2)),
ylab=expression(paste("signed ",sqrt(Deviance)))
)
lines(logS2Seq,
sqrt(logS2SeqOffset^2/logS2Var)*sign(logS2SeqOffset),
col=2)
points(log(coef(sampIGmleIG)[2]),0,pch=3)
par(oldpar)
## make a parametric boostrap to check the distribution of the deviance
nbReplicate <- 1000 #10000
sampleSize <- 100
system.time(
devianceIG100 <- lapply(1:nbReplicate,
function(idx) {
if ((idx
sampIG <- rinvgauss(sampleSize,mu=mu.true,sigma2=sigma2.true)
sampIGmleIG <- invgaussMLE(sampIG)
Deviance <- -2*sampIGmleIG$r(mu.true,sigma2.true)
logPara <- log(coef(sampIGmleIG))
logParaSE <- sampIGmleIG$se/coef(sampIGmleIG)
intervalMu <- function(n) c(-n,n)*logParaSE[1]+logPara[1]
intervalS2 <- function(n) c(-n,n)*logParaSE[2]+logPara[2]
logMuProfFct <- function(logMu,...) {
optimise(function(x)
sampIGmleIG$mll(c(logMu,x))+logLik(sampIGmleIG),...)$objective
}
logMuProfCI <- function(logMu,
CI,
a=intervalS2(4)[1],
b=intervalS2(4)[2])
logMuProfFct(logMu,c(a,b)) - qchisq(CI,1)/2
logS2ProfFct <- function(logS2,...) {
optimise(function(x)
sampIGmleIG$mll(c(x,logS2))+logLik(sampIGmleIG),...)$objective
}
logS2ProfCI <- function(logS2, CI,
a=intervalMu(4)[1],
b=intervalMu(4)[2])
logS2ProfFct(logS2,c(a,b)) - qchisq(CI,1)/2
factor <- 4
while((logMuProfCI(intervalMu(factor)[2],0.99) *
logMuProfCI(logPara[1],0.99) >= 0) ||
(logMuProfCI(intervalMu(factor)[1],0.99) *
logMuProfCI(logPara[1],0.99) >= 0)
) factor <- factor+1
##browser()
logMuCI95 <- c(uniroot(logMuProfCI,
interval=c(intervalMu(factor)[1],logPara[1]),
CI=0.95)$root,
uniroot(logMuProfCI,
interval=c(logPara[1],intervalMu(factor)[2]),
CI=0.95)$root
)
logMuCI99 <- c(uniroot(logMuProfCI,
interval=c(intervalMu(factor)[1],logPara[1]),
CI=0.99)$root,
uniroot(logMuProfCI,
interval=c(logPara[1],intervalMu(factor)[2]),
CI=0.99)$root
)
factor <- 4
while((logS2ProfCI(intervalS2(factor)[2],0.99) *
logS2ProfCI(logPara[2],0.99) >= 0) ||
(logS2ProfCI(intervalS2(factor)[1],0.99) *
logS2ProfCI(logPara[2],0.99) >= 0)
) factor <- factor+1
logS2CI95 <- c(uniroot(logS2ProfCI,
interval=c(intervalS2(factor)[1],logPara[2]),
CI=0.95)$root,
uniroot(logS2ProfCI,
interval=c(logPara[2],intervalS2(factor)[2]),
CI=0.95)$root
)
logS2CI99 <- c(uniroot(logS2ProfCI,
interval=c(intervalS2(factor)[1],logPara[2]),
CI=0.99)$root,
uniroot(logS2ProfCI,
interval=c(logPara[2],intervalS2(factor)[2]),
CI=0.99)$root
)
list(deviance=Deviance,
logMuCI95=logMuCI95,
logMuNorm95=qnorm(c(0.025,0.975),logPara[1],logParaSE[1]),
logMuCI99=logMuCI99,
logMuNorm99=qnorm(c(0.005,0.995),logPara[1],logParaSE[1]),
logS2CI95=logS2CI95,
logS2Norm95=qnorm(c(0.025,0.975),logPara[2],logParaSE[2]),
logS2CI99=logS2CI99,
logS2Norm99=qnorm(c(0.005,0.995),logPara[2],logParaSE[2]))
}
)
)[3]
## Find out how many times the true parameters was within the computed CIs
nLogMuCI95 <- sum(sapply(devianceIG100,
function(l) l$logMuCI95[1] <= log(mu.true) &&
log(mu.true)<= l$logMuCI95[2]
)
)
nLogMuNorm95 <- sum(sapply(devianceIG100,
function(l) l$logMuNorm95[1] <= log(mu.true) &&
log(mu.true)<= l$logMuNorm95[2]
)
)
nLogMuCI99 <- sum(sapply(devianceIG100,
function(l) l$logMuCI99[1] <= log(mu.true) &&
log(mu.true)<= l$logMuCI99[2]
)
)
nLogMuNorm99 <- sum(sapply(devianceIG100,
function(l) l$logMuNorm99[1] <= log(mu.true) &&
log(mu.true)<= l$logMuNorm99[2]
)
)
## Check if these counts are compatible with the nominal CIs
c(prof95Mu=nLogMuCI95,norm95Mu=nLogMuNorm95)
qbinom(c(0.005,0.995),nbReplicate,0.95)
c(prof95Mu=nLogMuCI99,norm95Mu=nLogMuNorm99)
qbinom(c(0.005,0.995),nbReplicate,0.99)
nLogS2CI95 <- sum(sapply(devianceIG100,
function(l) l$logS2CI95[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2CI95[2]
)
)
nLogS2Norm95 <- sum(sapply(devianceIG100,
function(l) l$logS2Norm95[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2Norm95[2]
)
)
nLogS2CI99 <- sum(sapply(devianceIG100,
function(l) l$logS2CI99[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2CI99[2]
)
)
nLogS2Norm99 <- sum(sapply(devianceIG100,
function(l) l$logS2Norm99[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2Norm99[2]
)
)
## Check if these counts are compatible with the nominal CIs
c(prof95S2=nLogS2CI95,norm95S2=nLogS2Norm95)
qbinom(c(0.005,0.995),nbReplicate,0.95)
c(prof95S2=nLogS2CI99,norm95S2=nLogS2Norm99)
qbinom(c(0.005,0.995),nbReplicate,0.99)
## Get 95 and 99% confidence intervals for the QQ plot
ci <- sapply(1:nbReplicate,
function(idx) qchisq(qbeta(c(0.005,0.025,0.975,0.995),
idx,
nbReplicate-idx+1),
df=2)
)
## make QQ plot
X <- qchisq(ppoints(nbReplicate),df=2)
Y <- sort(sapply(devianceIG100,function(l) l$deviance))
X11()
plot(X,Y,type="n",
xlab=expression(paste(chi[2]^2," quantiles")),
ylab="MC quantiles",
main="Deviance with true parameters after ML fit of IG data",
sub=paste("sample size:", sampleSize,"MC replicates:", nbReplicate)
)
abline(a=0,b=1)
lines(X,ci[1,],lty=2)
lines(X,ci[2,],lty=2)
lines(X,ci[3,],lty=2)
lines(X,ci[4,],lty=2)
lines(X,Y,col=2)
## End(Not run)
STAR documentation built on May 2, 2019, 11:44 a.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
Estimate inverse Gaussian model parameters by the maximum likelihood method using possibly censored data. Two different parameterizations of the inverse Gaussian distribution can be used.
Usage
1 2 3 | invgaussMLE(yi, ni = numeric(length(yi)) + 1,
si = numeric(length(yi)) + 1,
parameterization = "sigma2")
|
Arguments
yi |
vector of (possibly binned) observations or a
|
ni |
vector of counts for each value of |
si |
vector of counts of uncensored observations for each
value of |
parameterization |
parameterization used, |
Details
The 2 different parameterizations of the inverse Gaussian distribution
are discussed in the manual of dinvgauss.
In the absence of censored data the ML estimates are available in
closed form (Lindsey, 2004, p 212) together with the Hessian matrix at
the MLE. In presence of censored data an initial guess for the
parameters is obtained using the uncensored data before maximizing the
likelihood function to the full data set using optim
with the BFGS method. ML
estimation is always performed with the "sigma2"
parameterization. Parameters and variance-covariance matrix are
transformed at the end if the "boundary" parameterization is
requested.
In order to ensure good behavior of the numerical optimization
routines, optimization is performed on the log of the parameters
(mu and sigma2).
Standard errors are obtained from the inverse of the observed information matrix at the MLE. They are transformed to go from the log scale used by the optimization routine, when the latter is used (ie, for censored data) to the parameterization requested.
Value
A list of class durationFit with the following components:
estimate |
the estimated parameters, a named vector. |
se |
the standard errors, a named vector. |
logLik |
the log likelihood at maximum. |
r |
a function returning the log of the relative likelihood function. |
mll |
a function returning the opposite of the log likelihood function using the log of the parameters. |
call |
the matched call. |
Note
The returned standard errors (component se) are valid in the asymptotic limit. You
should plot contours using function r in the returned list and
check that the contours are reasonably close to ellipses.
Author(s)
Christophe Pouzat christophe.pouzat@gmail.com
References
Lindsey, J.K. (2004) Introduction to Applied Statistics: A Modelling Approach. OUP.
See Also
dinvgauss,lnormMLE,gammaMLE,weibullMLE,llogisMLE,rexpMLE.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 | ## Simulate sample of size 100 from an inverse Gaussian
## distribution
set.seed(1102006,"Mersenne-Twister")
sampleSize <- 100
mu.true <- 0.075
sigma2.true <- 3
sampleSize <- 100
sampIG <- rinvgauss(sampleSize,mu=mu.true,sigma2=sigma2.true)
## Make a maximum likelihood fit
sampIGmleIG <- invgaussMLE(sampIG)
## Compare estimates with actual values
rbind(est = coef(sampIGmleIG),se = sampIGmleIG$se,true = c(mu.true,sigma2.true))
## In the absence of censoring the MLE of the inverse Gaussian is available in a
## closed form together with its variance (ie, the observed information matrix)
## we can check that we did not screw up at that stage by comparing the observed
## information matrix obtained numerically with the analytical one. To do that we
## use the MINUS log likelihood function returned by invgaussMLE to get a numerical
## estimate
detailedFit <- optim(par=as.vector(log(sampIGmleIG$estimate)),
fn=sampIGmleIG$mll,
method="BFGS",
hessian=TRUE)
## We should not forget that the "mll" function uses the log of the parameters while
## the "se" component of sampIGmleIG list is expressed on the linear scale we must therefore
## transform one into the other as follows (Kalbfleisch, 1985, p71):
## if x = exp(u) and y = exp(v) and if we have the information matrix in term of
## u and v (that's the hessian component of list detailedFit above), we create matrix:
## du/dx du/dy
## Q =
## dv/dx dv/dy
## and we get I in term of x and y by the following matrix product:
## I(x,y) <- t(Q) %*% I(u,v) %*% Q
## In the present case:
## du/dx = 1/exp(u), du/dy = 0, dv/dx = 0, dv/dy = 1/exp(v)
## Therefore:
Q <- diag(1/exp(detailedFit$par))
numericalI <- t(Q) %*% detailedFit$hessian %*% Q
seComp <- rbind(sampIGmleIG$se, sqrt(diag(solve(numericalI))))
colnames(seComp) <- c("mu","sigma2")
rownames(seComp) <- c("analytical", "numerical")
seComp
## We can check the relative differences between the 2
apply(seComp,2,function(x) abs(diff(x))/x[1])
## Not run:
## Estimate the log relative likelihood on a grid to plot contours
Mu <- seq(coef(sampIGmleIG)[1]-4*sampIGmleIG$se[1],
coef(sampIGmleIG)[1]+4*sampIGmleIG$se[1],
sampIGmleIG$se[1]/10)
Sigma2 <- seq(coef(sampIGmleIG)[2]-4*sampIGmleIG$se[2],
coef(sampIGmleIG)[2]+4*sampIGmleIG$se[2],
sampIGmleIG$se[2]/10)
sampIGmleIGcontour <- sapply(Mu, function(mu) sapply(Sigma2, function(s2) sampIGmleIG$r(mu,s2)))
## plot contours using a linear scale for the parameters
## draw four contours corresponding to the following likelihood ratios:
## 0.5, 0.1, Chi2 with 2 df and p values of 0.95 and 0.99
X11(width=12,height=6)
layout(matrix(1:2,ncol=2))
contour(Mu,Sigma2,t(sampIGmleIGcontour),
levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
labels=c("log(0.5)",
"log(0.1)",
"-1/2*P(Chi2=0.95)",
"-1/2*P(Chi2=0.99)"),
xlab=expression(mu),ylab=expression(sigma^2),
main="Log Relative Likelihood Contours"
)
points(coef(sampIGmleIG)[1],coef(sampIGmleIG)[2],pch=3)
points(mu.true,sigma2.true,pch=16,col=2)
## The contours are not really symmetrical about the MLE we can try to
## replot them using a log scale for the parameters to see if that improves
## the situation
contour(log(Mu),log(Sigma2),t(sampIGmleIGcontour),
levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
labels="",
xlab=expression(log(mu)),ylab=expression(log(sigma^2)),
main="Log Relative Likelihood Contours",
sub="log scale for the parameters")
points(log(coef(sampIGmleIG)[1]),log(coef(sampIGmleIG)[2]),pch=3)
points(log(mu.true),log(sigma2.true),pch=16,col=2)
## Even with the log scale the contours are not ellipsoidal, so let us compute profiles
## For that we are going to use the returned MINUS log likelihood function
logMuProfFct <- function(logMu,...) {
myOpt <- optimise(function(x) sampIGmleIG$mll(c(logMu,x))+logLik(sampIGmleIG),...)
as.vector(unlist(myOpt[c("objective","minimum")]))
}
logMuProfCI <- function(logMu,
CI,
a=logS2Seq[1],
b=logS2Seq[length(logS2Seq)]) logMuProfFct(logMu,c(a,b))[1] - qchisq(CI,1)/2
logS2ProfFct <- function(logS2,...) {
myOpt <- optimise(function(x) sampIGmleIG$mll(c(x,logS2))+logLik(sampIGmleIG),...)
as.vector(unlist(myOpt[c("objective","minimum")]))
}
logS2ProfCI <- function(logS2, CI,
a=logMuSeq[1],
b=logMuSeq[length(logMuSeq)]) logS2ProfFct(logS2,c(a,b))[1] - qchisq(CI,1)/2
## We compute profiles (on the log scale) eploxing +/- 3 times
## the se about the MLE
logMuSE <- sqrt(diag(solve(detailedFit$hessian)))[1]
logMuSeq <- seq(log(coef(sampIGmleIG)[1])-3*logMuSE,
log(coef(sampIGmleIG)[1])+3*logMuSE,
logMuSE/10)
logS2SE <- sqrt(diag(solve(detailedFit$hessian)))[2]
logS2Seq <- seq(log(coef(sampIGmleIG)[2])-3*logS2SE,
log(coef(sampIGmleIG)[2])+3*logS2SE,
logS2SE/10)
logMuProf <- sapply(logMuSeq,logMuProfFct,
lower=logS2Seq[1],
upper=logS2Seq[length(logS2Seq)])
## Get 95
logMuCI95 <- c(uniroot(logMuProfCI,
interval=c(logMuSeq[1],log(coef(sampIGmleIG)[1])),
CI=0.95)$root,
uniroot(logMuProfCI,
interval=c(log(coef(sampIGmleIG)[1]),logMuSeq[length(logMuSeq)]),
CI=0.95)$root
)
logMuCI99 <- c(uniroot(logMuProfCI,
interval=c(logMuSeq[1],log(coef(sampIGmleIG)[1])),
CI=0.99)$root,
uniroot(logMuProfCI,
interval=c(log(coef(sampIGmleIG)[1]),logMuSeq[length(logMuSeq)]),
CI=0.99)$root
)
logS2Prof <- sapply(logS2Seq,logS2ProfFct,
lower=logMuSeq[1],
upper=logMuSeq[length(logMuSeq)])
## Get 95
logS2CI95 <- c(uniroot(logS2ProfCI,
interval=c(logS2Seq[1],log(coef(sampIGmleIG)[2])),
CI=0.95)$root,
uniroot(logS2ProfCI,
interval=c(log(coef(sampIGmleIG)[2]),logS2Seq[length(logS2Seq)]),
CI=0.95)$root
)
logS2CI99 <- c(uniroot(logS2ProfCI,
interval=c(logS2Seq[1],log(coef(sampIGmleIG)[2])),
CI=0.99)$root,
uniroot(logS2ProfCI,
interval=c(log(coef(sampIGmleIG)[2]),logS2Seq[length(logS2Seq)]),
CI=0.99)$root
)
## Add profiles to the previous plot
lines(logMuSeq,logMuProf[2,],col=2,lty=2)
lines(logS2Prof[2,],logS2Seq,col=2,lty=2)
## We can now check the deviations of the (profiled) deviances
## from the asymptotic parabolic curves
X11()
layout(matrix(1:4,nrow=2))
oldpar <- par(mar=c(4,4,2,1))
logMuSeqOffset <- logMuSeq-log(coef(sampIGmleIG)[1])
logMuVar <- diag(solve(detailedFit$hessian))[1]
plot(logMuSeq,2*logMuProf[1,],type="l",xlab=expression(log(mu)),ylab="Deviance")
lines(logMuSeq,logMuSeqOffset^2/logMuVar,col=2)
points(log(coef(sampIGmleIG)[1]),0,pch=3)
abline(h=0)
abline(h=qchisq(0.95,1),lty=2)
abline(h=qchisq(0.99,1),lty=2)
lines(rep(logMuCI95[1],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logMuCI95[2],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logMuCI99[1],2),c(0,qchisq(0.99,1)),lty=2)
lines(rep(logMuCI99[2],2),c(0,qchisq(0.99,1)),lty=2)
## We can also "linearize" this last graph
plot(logMuSeq,
sqrt(2*logMuProf[1,])*sign(logMuSeqOffset),
type="l",
xlab=expression(log(mu)),
ylab=expression(paste("signed ",sqrt(Deviance)))
)
lines(logMuSeq,
sqrt(logMuSeqOffset^2/logMuVar)*sign(logMuSeqOffset),
col=2)
points(log(coef(sampIGmleIG)[1]),0,pch=3)
logS2SeqOffset <- logS2Seq-log(coef(sampIGmleIG)[2])
logS2Var <- diag(solve(detailedFit$hessian))[2]
plot(logS2Seq,2*logS2Prof[1,],type="l",xlab=expression(log(sigma^2)),ylab="Deviance")
lines(logS2Seq,logS2SeqOffset^2/logS2Var,col=2)
points(log(coef(sampIGmleIG)[2]),0,pch=3)
abline(h=0)
abline(h=qchisq(0.95,1),lty=2)
abline(h=qchisq(0.99,1),lty=2)
lines(rep(logS2CI95[1],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logS2CI95[2],2),c(0,qchisq(0.95,1)),lty=2)
lines(rep(logS2CI99[1],2),c(0,qchisq(0.99,1)),lty=2)
lines(rep(logS2CI99[2],2),c(0,qchisq(0.99,1)),lty=2)
## We can also "linearize" this last graph
plot(logS2Seq,
sqrt(2*logS2Prof[1,])*sign(logS2SeqOffset),
type="l",
xlab=expression(log(sigma^2)),
ylab=expression(paste("signed ",sqrt(Deviance)))
)
lines(logS2Seq,
sqrt(logS2SeqOffset^2/logS2Var)*sign(logS2SeqOffset),
col=2)
points(log(coef(sampIGmleIG)[2]),0,pch=3)
par(oldpar)
## make a parametric boostrap to check the distribution of the deviance
nbReplicate <- 1000 #10000
sampleSize <- 100
system.time(
devianceIG100 <- lapply(1:nbReplicate,
function(idx) {
if ((idx
sampIG <- rinvgauss(sampleSize,mu=mu.true,sigma2=sigma2.true)
sampIGmleIG <- invgaussMLE(sampIG)
Deviance <- -2*sampIGmleIG$r(mu.true,sigma2.true)
logPara <- log(coef(sampIGmleIG))
logParaSE <- sampIGmleIG$se/coef(sampIGmleIG)
intervalMu <- function(n) c(-n,n)*logParaSE[1]+logPara[1]
intervalS2 <- function(n) c(-n,n)*logParaSE[2]+logPara[2]
logMuProfFct <- function(logMu,...) {
optimise(function(x)
sampIGmleIG$mll(c(logMu,x))+logLik(sampIGmleIG),...)$objective
}
logMuProfCI <- function(logMu,
CI,
a=intervalS2(4)[1],
b=intervalS2(4)[2])
logMuProfFct(logMu,c(a,b)) - qchisq(CI,1)/2
logS2ProfFct <- function(logS2,...) {
optimise(function(x)
sampIGmleIG$mll(c(x,logS2))+logLik(sampIGmleIG),...)$objective
}
logS2ProfCI <- function(logS2, CI,
a=intervalMu(4)[1],
b=intervalMu(4)[2])
logS2ProfFct(logS2,c(a,b)) - qchisq(CI,1)/2
factor <- 4
while((logMuProfCI(intervalMu(factor)[2],0.99) *
logMuProfCI(logPara[1],0.99) >= 0) ||
(logMuProfCI(intervalMu(factor)[1],0.99) *
logMuProfCI(logPara[1],0.99) >= 0)
) factor <- factor+1
##browser()
logMuCI95 <- c(uniroot(logMuProfCI,
interval=c(intervalMu(factor)[1],logPara[1]),
CI=0.95)$root,
uniroot(logMuProfCI,
interval=c(logPara[1],intervalMu(factor)[2]),
CI=0.95)$root
)
logMuCI99 <- c(uniroot(logMuProfCI,
interval=c(intervalMu(factor)[1],logPara[1]),
CI=0.99)$root,
uniroot(logMuProfCI,
interval=c(logPara[1],intervalMu(factor)[2]),
CI=0.99)$root
)
factor <- 4
while((logS2ProfCI(intervalS2(factor)[2],0.99) *
logS2ProfCI(logPara[2],0.99) >= 0) ||
(logS2ProfCI(intervalS2(factor)[1],0.99) *
logS2ProfCI(logPara[2],0.99) >= 0)
) factor <- factor+1
logS2CI95 <- c(uniroot(logS2ProfCI,
interval=c(intervalS2(factor)[1],logPara[2]),
CI=0.95)$root,
uniroot(logS2ProfCI,
interval=c(logPara[2],intervalS2(factor)[2]),
CI=0.95)$root
)
logS2CI99 <- c(uniroot(logS2ProfCI,
interval=c(intervalS2(factor)[1],logPara[2]),
CI=0.99)$root,
uniroot(logS2ProfCI,
interval=c(logPara[2],intervalS2(factor)[2]),
CI=0.99)$root
)
list(deviance=Deviance,
logMuCI95=logMuCI95,
logMuNorm95=qnorm(c(0.025,0.975),logPara[1],logParaSE[1]),
logMuCI99=logMuCI99,
logMuNorm99=qnorm(c(0.005,0.995),logPara[1],logParaSE[1]),
logS2CI95=logS2CI95,
logS2Norm95=qnorm(c(0.025,0.975),logPara[2],logParaSE[2]),
logS2CI99=logS2CI99,
logS2Norm99=qnorm(c(0.005,0.995),logPara[2],logParaSE[2]))
}
)
)[3]
## Find out how many times the true parameters was within the computed CIs
nLogMuCI95 <- sum(sapply(devianceIG100,
function(l) l$logMuCI95[1] <= log(mu.true) &&
log(mu.true)<= l$logMuCI95[2]
)
)
nLogMuNorm95 <- sum(sapply(devianceIG100,
function(l) l$logMuNorm95[1] <= log(mu.true) &&
log(mu.true)<= l$logMuNorm95[2]
)
)
nLogMuCI99 <- sum(sapply(devianceIG100,
function(l) l$logMuCI99[1] <= log(mu.true) &&
log(mu.true)<= l$logMuCI99[2]
)
)
nLogMuNorm99 <- sum(sapply(devianceIG100,
function(l) l$logMuNorm99[1] <= log(mu.true) &&
log(mu.true)<= l$logMuNorm99[2]
)
)
## Check if these counts are compatible with the nominal CIs
c(prof95Mu=nLogMuCI95,norm95Mu=nLogMuNorm95)
qbinom(c(0.005,0.995),nbReplicate,0.95)
c(prof95Mu=nLogMuCI99,norm95Mu=nLogMuNorm99)
qbinom(c(0.005,0.995),nbReplicate,0.99)
nLogS2CI95 <- sum(sapply(devianceIG100,
function(l) l$logS2CI95[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2CI95[2]
)
)
nLogS2Norm95 <- sum(sapply(devianceIG100,
function(l) l$logS2Norm95[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2Norm95[2]
)
)
nLogS2CI99 <- sum(sapply(devianceIG100,
function(l) l$logS2CI99[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2CI99[2]
)
)
nLogS2Norm99 <- sum(sapply(devianceIG100,
function(l) l$logS2Norm99[1] <= log(sigma2.true) &&
log(sigma2.true)<= l$logS2Norm99[2]
)
)
## Check if these counts are compatible with the nominal CIs
c(prof95S2=nLogS2CI95,norm95S2=nLogS2Norm95)
qbinom(c(0.005,0.995),nbReplicate,0.95)
c(prof95S2=nLogS2CI99,norm95S2=nLogS2Norm99)
qbinom(c(0.005,0.995),nbReplicate,0.99)
## Get 95 and 99% confidence intervals for the QQ plot
ci <- sapply(1:nbReplicate,
function(idx) qchisq(qbeta(c(0.005,0.025,0.975,0.995),
idx,
nbReplicate-idx+1),
df=2)
)
## make QQ plot
X <- qchisq(ppoints(nbReplicate),df=2)
Y <- sort(sapply(devianceIG100,function(l) l$deviance))
X11()
plot(X,Y,type="n",
xlab=expression(paste(chi[2]^2," quantiles")),
ylab="MC quantiles",
main="Deviance with true parameters after ML fit of IG data",
sub=paste("sample size:", sampleSize,"MC replicates:", nbReplicate)
)
abline(a=0,b=1)
lines(X,ci[1,],lty=2)
lines(X,ci[2,],lty=2)
lines(X,ci[3,],lty=2)
lines(X,ci[4,],lty=2)
lines(X,Y,col=2)
## End(Not run)
|
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