gam.fit4.r 56 KB
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## (c) Simon N. Wood (2013-2015). Provided under GPL 2.
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## Routines for gam estimation beyond exponential family.

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dDeta <- function(y,mu,wt,theta,fam,deriv=0) {
## What is available directly from the family are derivatives of the 
## deviance and link w.r.t. mu. This routine converts these to the
## required derivatives of the deviance w.r.t. eta.
## deriv is the order of derivative of the smoothing parameter score 
## required.
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## This version is based on ratios of derivatives of links rather 
## than raw derivatives of links. g2g = g''/g'^2, g3g = g'''/g'^3 etc 
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   r <- fam$Dd(y, mu, theta, wt, level=deriv)  
   d <- list(Deta=0,Dth=0,Dth2=0,Deta2=0,EDeta2=0,Detath=0,
             Deta3=0,Deta2th=0,Detath2=0,
             Deta4=0,Deta3th=0,Deta2th2=0)
   if (fam$link=="identity") { ## don't waste time on transformation
      d$Deta <- r$Dmu;d$Deta2 <- r$Dmu2
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      d$EDeta2 <- r$EDmu2;d$Deta.Deta2 <- r$Dmu/r$Dmu2
      d$Deta.EDeta2 <- r$Dmu/r$EDmu2
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      if (deriv>0) {
        d$Dth <- r$Dth; d$Detath <- r$Dmuth
        d$Deta3 <- r$Dmu3; d$Deta2th <- r$Dmu2th
      }
      if (deriv>1) {
        d$Deta4 <- r$Dmu4; d$Dth2 <- r$Dth2; d$Detath2 <- r$Dmuth2
        d$Deta2th2 <- r$Dmu2th2; d$Deta3th <- r$Dmu3th
      }
      return(d)
   }

   ig1 <- fam$mu.eta(fam$linkfun(mu)) 
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   ig12 <- ig1^2
   
   g2g <- fam$g2g(mu)

##   ig12 <- ig1^2;ig13 <- ig12 * ig1
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   d$Deta <- r$Dmu * ig1
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   d$Deta2 <- r$Dmu2*ig12 - r$Dmu*g2g*ig1
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   d$EDeta2 <- r$EDmu2*ig12
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   d$Deta.Deta2 <- r$Dmu/(r$Dmu2*ig1 - r$Dmu*g2g)
   d$Deta.EDeta2 <- r$Dmu/(r$EDmu2*ig1)
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   if (deriv>0) {
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      ig13 <- ig12 * ig1
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      d$Dth <- r$Dth 
      d$Detath <- r$Dmuth * ig1
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      g3g <- fam$g3g(mu)
      d$Deta3 <- r$Dmu3*ig13 - 3*r$Dmu2 * g2g * ig12 + r$Dmu * (3*g2g^2 - g3g)*ig1
      d$Deta2th <- r$Dmu2th*ig12 - r$Dmuth*g2g*ig1
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   }
   if (deriv>1) {
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     g4g <- fam$g4g(mu)
     d$Deta4 <- ig12^2*r$Dmu4 - 6*r$Dmu3*ig13*g2g + r$Dmu2*(15*g2g^2-4*g3g)*ig12 - 
                       r$Dmu*(15*g2g^3-10*g2g*g3g  +g4g)*ig1
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     d$Dth2 <- r$Dth2
     d$Detath2 <- r$Dmuth2 * ig1 
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     d$Deta2th2 <- ig12*r$Dmu2th2 - r$Dmuth2*g2g*ig1
     d$Deta3th <-  ig13*r$Dmu3th - 3 *r$Dmu2th*g2g*ig12 + r$Dmuth*(3*g2g^2-g3g)*ig1
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   }
   d
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} ## dDeta
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fetad.test <- function(y,mu,wt,theta,fam,eps = 1e-7,plot=TRUE) {
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## test family derivatives w.r.t. eta
  
  dd <- dDeta(y,mu,wt,theta,fam,deriv=2)
  dev <- fam$dev.resids(y, mu, wt,theta)
  mu1 <- fam$linkinv(fam$linkfun(mu)+eps)
  dev1 <- fam$dev.resids(y,mu1, wt,theta)
  Deta.fd <- (dev1-dev)/eps
  cat("Deta: rdiff = ",range(dd$Deta-Deta.fd)," cor = ",cor(dd$Deta,Deta.fd),"\n")
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  plot(dd$Deta,Deta.fd);abline(0,1)
  nt <- length(theta)
  for (i in 1:nt) {
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    th1 <- theta;th1[i] <- th1[i] + eps
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    dev1 <- fam$dev.resids(y, mu, wt,th1)   
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    Dth.fd <- (dev1-dev)/eps
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    um <- if (nt>1) dd$Dth[,i] else dd$Dth
    cat("Dth[",i,"]: rdiff = ",range(um-Dth.fd)," cor = ",cor(um,Dth.fd),"\n")
    plot(um,Dth.fd);abline(0,1)
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  }
  ## second order up...
  dd1 <- dDeta(y,mu1,wt,theta,fam,deriv=2)
  Deta2.fd <- (dd1$Deta - dd$Deta)/eps
  cat("Deta2: rdiff = ",range(dd$Deta2-Deta2.fd)," cor = ",cor(dd$Deta2,Deta2.fd),"\n")
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  plot(dd$Deta2,Deta2.fd);abline(0,1)
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  Deta3.fd <- (dd1$Deta2 - dd$Deta2)/eps
  cat("Deta3: rdiff = ",range(dd$Deta3-Deta3.fd)," cor = ",cor(dd$Deta3,Deta3.fd),"\n")
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  plot(dd$Deta3,Deta3.fd);abline(0,1)
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  Deta4.fd <- (dd1$Deta3 - dd$Deta3)/eps
  cat("Deta4: rdiff = ",range(dd$Deta4-Deta4.fd)," cor = ",cor(dd$Deta4,Deta4.fd),"\n")
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  plot(dd$Deta4,Deta4.fd);abline(0,1)
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  ## and now the higher derivs wrt theta...
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  ind <- 1:nt
  for (i in 1:nt) {
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    th1 <- theta;th1[i] <- th1[i] + eps
    dd1 <- dDeta(y,mu,wt,th1,fam,deriv=2)
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    Detath.fd <- (dd1$Deta - dd$Deta)/eps 
    um <- if (nt>1) dd$Detath[,i] else dd$Detath
    cat("Detath[",i,"]: rdiff = ",range(um-Detath.fd)," cor = ",cor(um,Detath.fd),"\n")
    plot(um,Detath.fd);abline(0,1)
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    Deta2th.fd <- (dd1$Deta2 - dd$Deta2)/eps
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    um <- if (nt>1) dd$Deta2th[,i] else dd$Deta2th
    cat("Deta2th[",i,"]: rdiff = ",range(um-Deta2th.fd)," cor = ",cor(um,Deta2th.fd),"\n") 
    plot(um,Deta2th.fd);abline(0,1)
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    Deta3th.fd <- (dd1$Deta3 - dd$Deta3)/eps
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    um <- if (nt>1) dd$Deta3th[,i] else dd$Deta3th
    cat("Deta3th[",i,"]: rdiff = ",range(um-Deta3th.fd)," cor = ",cor(um,Deta3th.fd),"\n")
    plot(um,Deta3th.fd);abline(0,1)
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    ## now the 3 second derivative w.r.t. theta terms
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    Dth2.fd <- (dd1$Dth - dd$Dth)/eps
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    um <- if (nt>1) dd$Dth2[,ind] else dd$Dth2
    er <- if (nt>1) Dth2.fd[,i:nt] else Dth2.fd
    cat("Dth2[",i,",]: rdiff = ",range(um-er)," cor = ",cor(as.numeric(um),as.numeric(er)),"\n")
    plot(um,er);abline(0,1)
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    Detath2.fd <- (dd1$Detath - dd$Detath)/eps
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    um <- if (nt>1) dd$Detath2[,ind] else dd$Detath2
    er <- if (nt>1) Detath2.fd[,i:nt] else Detath2.fd
    cat("Detath2[",i,",]: rdiff = ",range(um-er)," cor = ",cor(as.numeric(um),as.numeric(er)),"\n")
    ## cat("Detath2[",i,",]: rdiff = ",range(dd$Detath2-Detath2.fd)," cor = ",cor(dd$Detath2,Detath2.fd),"\n")
    plot(um,er);abline(0,1)
 
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    Deta2th2.fd <- (dd1$Deta2th - dd$Deta2th)/eps
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    um <- if (nt>1) dd$Deta2th2[,ind] else dd$Deta2th2
    er <- if (nt>1) Deta2th2.fd[,i:nt] else Deta2th2.fd
    cat("Deta2th2[",i,",]: rdiff = ",range(um-er)," cor = ",cor(as.numeric(um),as.numeric(er)),"\n")
    ## cat("Deta2th2[",i,",]: rdiff = ",range(dd$Deta2th2-Deta2th2.fd)," cor = ",cor(dd$Deta2th2,Deta2th2.fd),"\n") 
    ind <- max(ind)+1:(nt-i) 
    plot(um,er);abline(0,1)
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  }
} ## fetad.test

fmud.test <- function(y,mu,wt,theta,fam,eps = 1e-7) {
## test family deviance derivatives w.r.t. mu
  dd <- fam$Dd(y, mu, theta, wt, level=2) 
  dev <- fam$dev.resids(y, mu, wt,theta)
  dev1 <- fam$dev.resids(y, mu+eps, wt,theta)
  Dmu.fd <- (dev1-dev)/eps
  cat("Dmu: rdiff = ",range(dd$Dmu-Dmu.fd)," cor = ",cor(dd$Dmu,Dmu.fd),"\n")
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  nt <- length(theta)
  for (i in 1:nt) {
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    th1 <- theta;th1[i] <- th1[i] + eps
    dev1 <- fam$dev.resids(y, mu, wt,th1)
    Dth.fd <- (dev1-dev)/eps
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    um <- if (nt>1) dd$Dth[,i] else dd$Dth
    cat("Dth[",i,"]: rdiff = ",range(um-Dth.fd)," cor = ",cor(um,Dth.fd),"\n")
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  }
  ## second order up...
  dd1 <- fam$Dd(y, mu+eps, theta, wt, level=2)
  Dmu2.fd <- (dd1$Dmu - dd$Dmu)/eps
  cat("Dmu2: rdiff = ",range(dd$Dmu2-Dmu2.fd)," cor = ",cor(dd$Dmu2,Dmu2.fd),"\n")
  Dmu3.fd <- (dd1$Dmu2 - dd$Dmu2)/eps
  cat("Dmu3: rdiff = ",range(dd$Dmu3-Dmu3.fd)," cor = ",cor(dd$Dmu3,Dmu3.fd),"\n")
  Dmu4.fd <- (dd1$Dmu3 - dd$Dmu3)/eps
  cat("Dmu4: rdiff = ",range(dd$Dmu4-Dmu4.fd)," cor = ",cor(dd$Dmu4,Dmu4.fd),"\n")
  ## and now the higher derivs wrt theta 
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  ind <- 1:nt
  for (i in 1:nt) {
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    th1 <- theta;th1[i] <- th1[i] + eps
    dd1 <- fam$Dd(y, mu, th1, wt, level=2)
    Dmuth.fd <- (dd1$Dmu - dd$Dmu)/eps
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    um <- if (nt>1) dd$Dmuth[,i] else dd$Dmuth
    cat("Dmuth[",i,"]: rdiff = ",range(um-Dmuth.fd)," cor = ",cor(um,Dmuth.fd),"\n")
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    Dmu2th.fd <- (dd1$Dmu2 - dd$Dmu2)/eps
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    um <- if (nt>1) dd$Dmu2th[,i] else dd$Dmu2th
    cat("Dmu2th[",i,"]: rdiff = ",range(um-Dmu2th.fd)," cor = ",cor(um,Dmu2th.fd),"\n")
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    Dmu3th.fd <- (dd1$Dmu3 - dd$Dmu3)/eps
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    um <- if (nt>1) dd$Dmu3th[,i] else dd$Dmu3th
    cat("Dmu3th[",i,"]: rdiff = ",range(um-Dmu3th.fd)," cor = ",cor(um,Dmu3th.fd),"\n")

    ## now the 3 second derivative w.r.t. theta terms...

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    Dth2.fd <- (dd1$Dth - dd$Dth)/eps
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    um <- if (nt>1) dd$Dth2[,ind] else dd$Dth2
    er <- if (nt>1) Dth2.fd[,i:nt] else Dth2.fd
    cat("Dth2[",i,",]: rdiff = ",range(um-er)," cor = ",cor(as.numeric(um),as.numeric(er)),"\n")

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    Dmuth2.fd <- (dd1$Dmuth - dd$Dmuth)/eps
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    um <- if (nt>1) dd$Dmuth2[,ind] else dd$Dmuth2
    er <- if (nt>1) Dmuth2.fd[,i:nt] else Dmuth2.fd
    cat("Dmuth2[",i,",]: rdiff = ",range(um-er)," cor = ",cor(as.numeric(um),as.numeric(er)),"\n")
 
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    Dmu2th2.fd <- (dd1$Dmu2th - dd$Dmu2th)/eps
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    um <- if (nt>1) dd$Dmu2th2[,ind] else dd$Dmu2th2
    er <- if (nt>1) Dmu2th2.fd[,i:nt] else Dmu2th2.fd
    cat("Dmu2th2[",i,",]: rdiff = ",range(um-er)," cor = ",cor(as.numeric(um),as.numeric(er)),"\n")
    ind <- max(ind)+1:(nt-i)
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  }
}



gam.fit4 <- function(x, y, sp, Eb,UrS=list(),
            weights = rep(1, nobs), start = NULL, etastart = NULL, 
            mustart = NULL, offset = rep(0, nobs),U1=diag(ncol(x)), Mp=-1, family = gaussian(), 
            control = gam.control(), deriv=2,
            scale=1,scoreType="REML",null.coef=rep(0,ncol(x)),...) {
## Routine for fitting GAMs beyond exponential family.
## Inputs as gam.fit3 except that family is of class "extended.family", while
## sp contains the vector of extended family parameters, followed by the log smoothing parameters,
## followed by the log scale parameter if scale < 0

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  ## some families have second derivative of deviance, and hence iterative weights
  ## very close to zero for some data. This can lead to poorly scaled sqrt(w)z
  ## and it is better to base everything on wz...
  if (is.null(family$use.wz)) family$use.wz <- FALSE

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  if (family$n.theta>0) { ## there are extra parameters to estimate
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    ind <- 1:family$n.theta
    theta <- sp[ind] ## parameters of the family
    family$putTheta(theta)
    sp <- sp[-ind]   ## log smoothing parameters
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  } else theta <- family$getTheta() ## fixed value

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  ## penalized <- if (length(UrS)>0) TRUE else FALSE
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  if (scale>0) scale.known <- TRUE else {
    ## unknown scale parameter, trial value supplied as 
    ## final element of sp. 
    scale.known <- FALSE
    nsp <- length(sp)
    scale <- exp(sp[nsp])
    sp <- sp[-nsp]
  }
  
  x <- as.matrix(x)  
  nSp <- length(sp) 
  rank.tol <- .Machine$double.eps*100 ## tolerance to use for rank deficiency
  q <- ncol(x)
  n <- nobs <- nrow(x)  
  
  xnames <- dimnames(x)[[2]]
  ynames <- if (is.matrix(y)) rownames(y) else names(y)
  ## Now a stable re-parameterization is needed....

  if (length(UrS)) {
      rp <- gam.reparam(UrS,sp,deriv)
      T <- diag(q)
      T[1:ncol(rp$Qs),1:ncol(rp$Qs)] <- rp$Qs
      T <- U1%*%T ## new params b'=T'b old params
    
      null.coef <- t(T)%*%null.coef  
     
      if (!is.null(start)) start <- t(T)%*%start

      ## form x%*%T in parallel 
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      x <- .Call(C_mgcv_pmmult2,x,T,0,0,control$nthreads)
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      rS <- list()
      for (i in 1:length(UrS)) {
        rS[[i]] <- rbind(rp$rS[[i]],matrix(0,Mp,ncol(rp$rS[[i]])))
      } ## square roots of penalty matrices in current parameterization
      Eb <- Eb%*%T ## balanced penalty matrix
      rows.E <- q-Mp
      Sr <- cbind(rp$E,matrix(0,nrow(rp$E),Mp))
      St <- rbind(cbind(rp$S,matrix(0,nrow(rp$S),Mp)),matrix(0,Mp,q))
  } else { 
      T <- diag(q); 
      St <- matrix(0,q,q) 
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      rSncol <- rows.E <- Eb <- Sr <- 0   
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      rS <- list(0)
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      rp <- list(det=0,det1 = 0,det2 = 0,fixed.penalty=FALSE)
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  }

  ## re-parameterization complete. Initialization....

  nvars <- ncol(x)
  if (nvars==0) stop("emtpy models not available")
  if (is.null(weights)) weights <- rep.int(1, nobs)
  if (is.null(offset)) offset <- rep.int(0, nobs)

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  linkinv <- family$linkinv
  valideta <- family$valideta
  validmu <- family$validmu
  dev.resids <- family$dev.resids

  ## need an initial `null deviance' to test for initial divergence...
  ## if (!is.null(start)) null.coef <- start - can be on edge of feasible - not good
  null.eta <- as.numeric(x%*%null.coef + as.numeric(offset))

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  #old.pdev <- sum(dev.resids(y, linkinv(null.eta), weights,theta)) + t(null.coef)%*%St%*%null.coef 

  #if (!is.null(start)) { ## check it's at least better than null.coef
  #  pdev <- sum(dev.resids(y, linkinv(x%*%start+as.numeric(offset)), weights,theta)) + t(start)%*%St%*%start
  #  if (pdev>old.pdev) start <- mustart <- etastart <- NULL
  #}
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  ## call the families initialization code...

  if (is.null(mustart)) {
    eval(family$initialize)
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    mukeep <- NULL
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  } else {
    mukeep <- mustart
    eval(family$initialize)
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    #mustart <- mukeep
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  }
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  old.pdev <- sum(dev.resids(y, linkinv(null.eta), weights,theta)) + t(null.coef)%*%St%*%null.coef 

  if (!is.null(start)) { ## check it's at least better than null.coef
    pdev <- sum(dev.resids(y, linkinv(x%*%start+as.numeric(offset)), weights,theta)) + t(start)%*%St%*%start
    if (pdev>old.pdev) start <- mukeep <- etastart <- NULL
  }

  if (!is.null(mukeep)) mustart <- mukeep

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  ## and now finalize initialization of mu and eta...

  eta <- if (!is.null(etastart)) etastart
         else if (!is.null(start)) 
              if (length(start) != nvars) 
                  stop("Length of start should equal ", nvars, 
                  " and correspond to initial coefs for ", deparse(xnames))
              else {
                  coefold <- start
                  etaold <- offset + as.vector(if (NCOL(x) == 1) 
                  x * start
                  else x %*% start)
              }
              else family$linkfun(mustart)
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   mu <- linkinv(eta);etaold <- eta
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   coefold <- null.coef
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   conv <-  boundary <- FALSE
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   dd <- dDeta(y,mu,weights,theta,family,0) ## derivatives of deviance w.r.t. eta
   w <- dd$Deta2 * .5;
   wz <- w*(eta-offset) - .5*dd$Deta
   z <- (eta-offset) - dd$Deta.Deta2
   good <- is.finite(z)&is.finite(w)

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   for (iter in 1:control$maxit) { ## start of main fitting iteration 
      if (control$trace) cat(iter," ")
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    #  dd <- dDeta(y,mu,weights,theta,family,0) ## derivatives of deviance w.r.t. eta
    #  w <- dd$Deta2 * .5;
    #  wz <- w*(eta-offset) - .5*dd$Deta
    #  z <- (eta-offset) - dd$Deta.Deta2
    #  good <- is.finite(z)&is.finite(w)
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      if (control$trace&sum(!good)>0) cat("\n",sum(!good)," not good\n")
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      if (sum(!good)) {
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        use.wy <- TRUE
        good <- is.finite(w)&is.finite(wz)
        z[!is.finite(z)] <- 0 ## avoid NaN in .C call - unused anyway
      } else use.wy <- family$use.wz
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      oo <- .C(C_pls_fit1,   
               y=as.double(z[good]),X=as.double(x[good,]),w=as.double(w[good]),wy = as.double(wz[good]),
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                     E=as.double(Sr),Es=as.double(Eb),n=as.integer(sum(good)),
                     q=as.integer(ncol(x)),rE=as.integer(rows.E),eta=as.double(z),
                     penalty=as.double(1),rank.tol=as.double(rank.tol),
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                     nt=as.integer(control$nthreads),use.wy=as.integer(use.wy))
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      posdef <- oo$n >= 0
      if (!posdef) { ## then problem is indefinite - switch to +ve weights for this step
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        if (control$trace) cat("**using positive weights\n")
        # problem is that Fisher can be very poor for zeroes  

        ## index weights that are finite and positive 
        good <- is.finite(dd$Deta2)
        good[good] <- dd$Deta2[good]>0 
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        w[!good] <- 0
        wz <- w*(eta-offset) - .5*dd$Deta
        z <- (eta-offset) - dd$Deta.Deta2
        good <- is.finite(z)&is.finite(w) 
        if (sum(!good)) {
          use.wy <- TRUE
          good <- is.finite(w)&is.finite(wz)
          z[!is.finite(z)] <- 0 ## avoid NaN in .C call - unused anyway
        } else use.wy <- family$use.wz
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        oo <- .C(C_pls_fit1, ##C_pls_fit1,
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                  y=as.double(z[good]),X=as.double(x[good,]),w=as.double(w[good]),wy = as.double(wz[good]),
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                     E=as.double(Sr),Es=as.double(Eb),n=as.integer(sum(good)),
                     q=as.integer(ncol(x)),rE=as.integer(rows.E),eta=as.double(z),
                     penalty=as.double(1),rank.tol=as.double(rank.tol),
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                     nt=as.integer(control$nthreads),use.wy=as.integer(use.wy))
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      }
      start <- oo$y[1:ncol(x)] ## current coefficient estimates
      penalty <- oo$penalty ## size of penalty

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      eta <- drop(x%*%start) ## the linear predictor (less offset)
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      if (any(!is.finite(start))) { ## test for breakdown
          conv <- FALSE
          warning("Non-finite coefficients at iteration ", 
                  iter)
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          return(list(REML=NA)) ## return immediately signalling failure
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      }        
     
      mu <- linkinv(eta <- eta + offset)
      dev <- sum(dev.resids(y, mu, weights,theta)) 

      ## now step halve under non-finite deviance...
      if (!is.finite(dev)) {
         if (is.null(coefold)) {
            if (is.null(null.coef)) 
              stop("no valid set of coefficients has been found:please supply starting values", 
                    call. = FALSE)
            ## Try to find feasible coefficients from the null.coef and null.eta
            coefold <- null.coef
            etaold <- null.eta
         }
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         #warning("Step size truncated due to divergence", 
         #            call. = FALSE)
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         ii <- 1
         while (!is.finite(dev)) {
               if (ii > control$maxit) 
                    stop("inner loop 1; can't correct step size")
               ii <- ii + 1
               start <- (start + coefold)/2
               eta <- (eta + etaold)/2               
               mu <- linkinv(eta)
               dev <- sum(dev.resids(y, mu, weights,theta))
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         }
         boundary <- TRUE
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         penalty <- t(start)%*%St%*%start ## reset penalty too
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         if (control$trace) 
                  cat("Step halved: new deviance =", dev, "\n")
      } ## end of infinite deviance correction

      ## now step halve if mu or eta are out of bounds... 
      if (!(valideta(eta) && validmu(mu))) {
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         #warning("Step size truncated: out of bounds", 
         #         call. = FALSE)
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         ii <- 1
         while (!(valideta(eta) && validmu(mu))) {
                  if (ii > control$maxit) 
                    stop("inner loop 2; can't correct step size")
                  ii <- ii + 1
                  start <- (start + coefold)/2
                  eta <- (eta + etaold)/2 
                  mu <- linkinv(eta)
         }
         boundary <- TRUE
         dev <- sum(dev.resids(y, mu, weights))
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         penalty <- t(start)%*%St%*%start ## need to reset penalty too
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         if (control$trace) 
                  cat("Step halved: new deviance =", dev, "\n")
      } ## end of invalid mu/eta handling

      ## now check for divergence of penalized deviance....
  
      pdev <- dev + penalty  ## the penalized deviance 
      if (control$trace) cat("penalized deviance =", pdev, "\n")
     
      div.thresh <- 10*(.1+abs(old.pdev))*.Machine$double.eps^.5

      if (pdev-old.pdev>div.thresh) { ## solution diverging
         ii <- 1 ## step halving counter
         if (iter==1) { ## immediate divergence, need to shrink towards zero 
               etaold <- null.eta; coefold <- null.coef
         }
         while (pdev -old.pdev > div.thresh)  { ## step halve until pdev <= old.pdev
           if (ii > 100) 
              stop("inner loop 3; can't correct step size")
           ii <- ii + 1
           start <- (start + coefold)/2 
           eta <- (eta + etaold)/2               
           mu <- linkinv(eta)
           dev <- sum(dev.resids(y, mu, weights,theta))
       
           pdev <- dev + t(start)%*%St%*%start ## the penalized deviance
           if (control$trace) 
                  cat("Step halved: new penalized deviance =", pdev, "\n")
        }
     } ## end of pdev divergence

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     ## get new weights and pseudodata (needed now for grad testing)...
     dd <- dDeta(y,mu,weights,theta,family,0) ## derivatives of deviance w.r.t. eta
     w <- dd$Deta2 * .5;
     wz <- w*(eta-offset) - .5*dd$Deta
     z <- (eta-offset) - dd$Deta.Deta2
     good <- is.finite(z)&is.finite(w) 
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     ## convergence testing...
477
     if (posdef && abs(pdev - old.pdev)/(0.1 + abs(pdev)) < control$epsilon) {
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       ## Need to check coefs converged adequately, to ensure implicit differentiation
       ## ok. Testing coefs unchanged is problematic under rank deficiency (not guaranteed to
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       ## drop same parameter every iteration!)
481
       grad <- 2 * t(x[good,])%*%((w[good]*(x%*%start)[good]-wz[good]))+ 2*St%*%start 
482
       if (max(abs(grad)) > control$epsilon*max(abs(start+coefold))/2) {
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         old.pdev <- pdev  ## not converged quite enough
         coef <- coefold <- start
         etaold <- eta 
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         ##muold <- mu
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       } else { ## converged
         conv <- TRUE
         coef <- start
         break 
       }
     } else { ## not converged
       old.pdev <- pdev
       coef <- coefold <- start
       etaold <- eta 
     }
   } ## end of main loop
   
   ## so at this stage the model has been fully estimated
   coef <- as.numeric(T %*% coef)
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   ## now obtain derivatives, if these are needed...
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   check.derivs <- FALSE
   while (check.derivs) { ## debugging code to check derivatives
     eps <- 1e-7
     fmud.test(y,mu,weights,theta,family,eps = eps)
     fetad.test(y,mu,weights,theta,family,eps = eps)
   }   

510
   dd <- dDeta(y,mu,weights,theta,family,deriv)
511
   w <- dd$Deta2 * .5
512
   z <- (eta-offset) - dd$Deta.Deta2 ## - .5 * dd$Deta[good] / w
513
   wf <- pmax(0,dd$EDeta2 * .5) ## Fisher type weights 
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   wz <- w*(eta-offset) - 0.5*dd$Deta ## Wz finite when w==0
  
   gdi.type <- if (any(abs(w)<.Machine$double.xmin*1e20)||any(!is.finite(z))) 1 else 0   
   good <- is.finite(wz)&is.finite(w)   
      
   residuals <- z - (eta - offset)
   residuals[!is.finite(residuals)] <- NA 
   z[!is.finite(z)] <- 0 ## avoid passing NA etc to C code  
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   ntot <- length(theta) + length(sp)
   rSncol <- unlist(lapply(UrS,ncol))
525
   ## Now drop any elements of dd that have been dropped in fitting...
526 527
   if (sum(!good)>0) { ## drop !good from fields of dd, weights and pseudodata
     z <- z[good]; w <- w[good]; wz <- wz[good]; wf <- wf[good]
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     dd$Deta <- dd$Deta[good];dd$Deta2 <- dd$Deta2[good] 
     dd$EDeta2 <- dd$EDeta2[good]
     if (deriv>0) dd$Deta3 <- dd$Deta3[good]
     if (deriv>1) dd$Deta4 <- dd$Deta4[good]
     if (length(theta)>1) {
         if (deriv>0) {  
         dd$Dth <- dd$Dth[good,]; 
         dd$Detath <- dd$Detath[good,]; dd$Deta2th <- dd$Deta2th[good,]
         if (deriv>1) {  
           dd$Detath2 <- dd$Detath2[good,]; dd$Deta3th <- dd$Deta3th[good,]
           dd$Deta2th2 <- dd$Deta2th2[good,];dd$Dth2 <- dd$Dth2[good,]
         }
       }
     } else {
       if (deriv>0) { 
         dd$Dth <- dd$Dth[good]; 
         dd$Detath <- dd$Detath[good]; dd$Deta2th <- dd$Deta2th[good]
         if (deriv>1) {
           dd$Detath2 <- dd$Detath2[good]; dd$Deta3th <- dd$Deta3th[good]
           dd$Deta2th2 <- dd$Deta2th2[good]; dd$Dth2 <- dd$Dth2[good]
         }
       } 
     }
   }
552

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   oo <- .C(C_gdi2,
            X=as.double(x[good,]),E=as.double(Sr),Es=as.double(Eb),rS=as.double(unlist(rS)),
            U1 = as.double(U1),sp=as.double(exp(sp)),theta=as.double(theta),
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            z=as.double(z),w=as.double(w),wz=as.double(wz),wf=as.double(wf),Dth=as.double(dd$Dth),
            Det=as.double(dd$Deta),
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            Det2=as.double(dd$Deta2),Dth2=as.double(dd$Dth2),Det.th=as.double(dd$Detath),
            Det2.th=as.double(dd$Deta2th),Det3=as.double(dd$Deta3),Det.th2 = as.double(dd$Detath2),
            Det4 = as.double(dd$Deta4),Det3.th=as.double(dd$Deta3th), Deta2.th2=as.double(dd$Deta2th2),
561
            beta=as.double(coef),b1=as.double(rep(0,ntot*ncol(x))),w1=as.double(rep(0,ntot*length(z))),
562
            D1=as.double(rep(0,ntot)),D2=as.double(rep(0,ntot^2)),
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            P=as.double(0),P1=as.double(rep(0,ntot)),P2 = as.double(rep(0,ntot^2)),
            ldet=as.double(1-2*(scoreType=="ML")),ldet1 = as.double(rep(0,ntot)), 
            ldet2 = as.double(rep(0,ntot^2)),
            rV=as.double(rep(0,ncol(x)^2)),
            rank.tol=as.double(.Machine$double.eps^.75),rank.est=as.integer(0),
	    n=as.integer(sum(good)),q=as.integer(ncol(x)),M=as.integer(nSp),
            n.theta=as.integer(length(theta)), Mp=as.integer(Mp),Enrow=as.integer(rows.E),
            rSncol=as.integer(rSncol),deriv=as.integer(deriv),
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	    fixed.penalty = as.integer(rp$fixed.penalty),nt=as.integer(control$nthreads),
572
            type=as.integer(gdi.type),dVkk=as.double(rep(0,nSp^2)))
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## test code used to ensure type 0 and type 1 produce identical results, when both should work. 
#   oot <- .C(C_gdi2,
#            X=as.double(x[good,]),E=as.double(Sr),Es=as.double(Eb),rS=as.double(unlist(rS)),
#            U1 = as.double(U1),sp=as.double(exp(sp)),theta=as.double(theta),
#            z=as.double(z),w=as.double(w),wz=as.double(wz),wf=as.double(wf),Dth=as.double(dd$Dth),
#            Det=as.double(dd$Deta),
#            Det2=as.double(dd$Deta2),Dth2=as.double(dd$Dth2),Det.th=as.double(dd$Detath),
#            Det2.th=as.double(dd$Deta2th),Det3=as.double(dd$Deta3),Det.th2 = as.double(dd$Detath2),
#            Det4 = as.double(dd$Deta4),Det3.th=as.double(dd$Deta3th), Deta2.th2=as.double(dd$Deta2th2),
#            beta=as.double(coef),b1=as.double(rep(0,ntot*ncol(x))),w1=rep(0,ntot*length(z)),
#            D1=as.double(rep(0,ntot)),D2=as.double(rep(0,ntot^2)),
#            P=as.double(0),P1=as.double(rep(0,ntot)),P2 = as.double(rep(0,ntot^2)),
#            ldet=as.double(1-2*(scoreType=="ML")),ldet1 = as.double(rep(0,ntot)), 
#            ldet2 = as.double(rep(0,ntot^2)),
#            rV=as.double(rep(0,ncol(x)^2)),
#            rank.tol=as.double(.Machine$double.eps^.75),rank.est=as.integer(0),
#	    n=as.integer(sum(good)),q=as.integer(ncol(x)),M=as.integer(nSp),
#            n.theta=as.integer(length(theta)), Mp=as.integer(Mp),Enrow=as.integer(rows.E),
#            rSncol=as.integer(rSncol),deriv=as.integer(deriv),
#	    fixed.penalty = as.integer(rp$fixed.penalty),nt=as.integer(control$nthreads),
#            type=as.integer(1))
594 595
   rV <- matrix(oo$rV,ncol(x),ncol(x)) ## rV%*%t(rV)*scale gives covariance matrix 
   rV <- T %*% rV   
596 597
   ## derivatives of coefs w.r.t. sps etc...
   db.drho <- if (deriv) T %*% matrix(oo$b1,ncol(x),ntot) else NULL 
598
   dw.drho <- if (deriv) matrix(oo$w1,length(z),ntot) else NULL
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   Kmat <- matrix(0,nrow(x),ncol(x)) 
   Kmat[good,] <- oo$X                    ## rV%*%t(K)%*%(sqrt(wf)*X) = F; diag(F) is edf array 

   D2 <- matrix(oo$D2,ntot,ntot); ldet2 <- matrix(oo$ldet2,ntot,ntot)
   bSb2 <- matrix(oo$P2,ntot,ntot)
   ## compute the REML score...
   ls <- family$ls(y,weights,n,theta,scale)
   nt <- length(theta)
   lsth1 <- ls$lsth1[1:nt];
   lsth2 <- as.matrix(ls$lsth2)[1:nt,1:nt] ## exclude any derivs w.r.t log scale here
   REML <- (dev+oo$P)/(2*scale) - ls$ls + (oo$ldet - rp$det)/2 - 
           as.numeric(scoreType=="REML") * Mp * log(2*pi*scale)/2
   REML1 <- REML2 <- NULL
   if (deriv) {
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     det1 <- oo$ldet1
     if (nSp) {
       ind <- 1:nSp + length(theta)
       det1[ind] <- det1[ind] - rp$det1
     }
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     REML1 <- (oo$D1+oo$P1)/(2*scale) - c(lsth1,rep(0,length(sp))) + (det1)/2
     if (deriv>1) {
       ls2 <- D2*0;ls2[1:nt,1:nt] <- lsth2 
621
       if (nSp) ldet2[ind,ind] <- ldet2[ind,ind] - rp$det2
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       REML2 <- (D2+bSb2)/(2*scale) - ls2 + ldet2/2
     }
   } 

   if (!scale.known&&deriv) { ## need derivatives wrt log scale, too 
      Dp <- dev + oo$P
      dlr.dlphi <- -Dp/(2 *scale) - ls$lsth1[nt+1] - Mp/2
      d2lr.d2lphi <- Dp/(2*scale) - ls$lsth2[nt+1,nt+1] 
      d2lr.dspphi <- -(oo$D1+oo$P1)/(2*scale) 
      d2lr.dspphi[1:nt] <- d2lr.dspphi[1:nt] - ls$lsth2[nt+1,1:nt]
      REML1 <- c(REML1,dlr.dlphi)
      if (deriv==2) {
              REML2 <- rbind(REML2,as.numeric(d2lr.dspphi))
              REML2 <- cbind(REML2,c(as.numeric(d2lr.dspphi),d2lr.d2lphi))
      }
   }
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   nth <- length(theta)
   if (deriv>0&&family$n.theta==0&&nth>0) { ## need to drop derivs for fixed theta
     REML1 <- REML1[-(1:nth)]
     if (deriv>1) REML2 <- REML2[-(1:nth),-(1:nth)]
     db.drho <- db.drho[,-(1:nth),drop=FALSE]
   }  
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   names(coef) <- xnames
   names(residuals) <- ynames
648
   wtdmu <- sum(weights * mu)/sum(weights) ## changed from y
649
   nulldev <- sum(dev.resids(y, rep(wtdmu,length(y)), weights))
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   n.ok <- nobs - sum(weights == 0)
   nulldf <- n.ok
652
   ww <- wt <- rep.int(0, nobs)
653
   wt[good] <- wf 
654
   ww[good] <- w
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   if (deriv && nrow(dw.drho)!=nrow(x)) {
      w1 <- dw.drho
      dw.drho <- matrix(0,nrow(x),ncol(w1))
      dw.drho[good,] <- w1
   }
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   aic.model <- family$aic(y, mu, theta, weights, dev) # note: incomplete 2*edf needs to be added
 

   list(coefficients = coef,residuals=residuals,fitted.values = mu,
        family=family, linear.predictors = eta,deviance=dev,
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        null.deviance=nulldev,iter=iter,
        weights=wt, ## note that these are Fisher type weights 
667
        prior.weights=weights,
668
        working.weights = ww, ## working weights
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        df.null = nulldf, y = y, converged = conv,
        boundary = boundary,
        REML=REML,REML1=REML1,REML2=REML2,
672
        rV=rV,db.drho=db.drho,dw.drho=dw.drho,
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        scale.est=scale,reml.scale=scale,
        aic=aic.model,
        rank=oo$rank.est,
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        K=Kmat,control=control,
        dVkk = matrix(oo$dVkk,nSp,nSp)
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        #,D1=oo$D1,D2=D2,
        #ldet=oo$ldet,ldet1=oo$ldet1,ldet2=ldet2,
        #bSb=oo$P,bSb1=oo$P1,bSb2=bSb2,
        #ls=ls$ls,ls1=ls$lsth1,ls2=ls$lsth2
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       )
 
} ## gam.fit4



gam.fit5 <- function(x,y,lsp,Sl,weights=NULL,offset=NULL,deriv=2,family,
                     control=gam.control(),Mp=-1,start=NULL){
690
## NOTE: offset handling - needs to be passed to ll code
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## fit models by general penalized likelihood method, 
## given doubly extended family in family. lsp is log smoothing parameters
## Stabilization strategy:
## 1. Sl.repara
## 2. Hessian diagonally pre-conditioned if +ve diagonal elements
##    (otherwise indefinite anyway)
## 3. Newton fit with perturbation of any indefinite hessian
## 4. At convergence test fundamental rank on balanced version of 
##    penalized Hessian. Drop unidentifiable parameters and 
##    continue iteration to adjust others.
## 5. All remaining computations in reduced space.
##    
## Idea is that rank detection takes care of structural co-linearity,
## while preconditioning and step 1 take care of extreme smoothing parameters
## related problems. 

707
  penalized <- if (length(Sl)>0) TRUE else FALSE
708 709 710

  nSp <- length(lsp)
  q <- ncol(x)
711
  nobs <- length(y)
712 713 714
  
  if (penalized) {
    Eb <- attr(Sl,"E") ## balanced penalty sqrt
715
 
716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732
    ## the stability reparameterization + log|S|_+ and derivs... 
    rp <- ldetS(Sl,rho=lsp,fixed=rep(FALSE,length(lsp)),np=q,root=TRUE) 
    x <- Sl.repara(rp$rp,x) ## apply re-parameterization to x
    Eb <- Sl.repara(rp$rp,Eb) ## root balanced penalty 
    St <- crossprod(rp$E) ## total penalty matrix
    E <- rp$E ## root total penalty
    attr(E,"use.unscaled") <- TRUE ## signal initialization code that E not to be further scaled   
    if (!is.null(start)) start  <- Sl.repara(rp$rp,start) ## re-para start
    ## NOTE: it can be that other attributes need re-parameterization here
    ##       this should be done in 'family$initialize' - see mvn for an example. 

  } else { ## unpenalized so no derivatives required
    deriv <- 0 
    rp <- list(ldetS=0,rp=list())
    St <- matrix(0,q,q)
    E <- matrix(0,0,q) ## can be needed by initialization code
  }
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  ## now call initialization code, but make sure that any 
  ## supplied 'start' vector is not overwritten...
  start0 <- start
  
  ## Assumption here is that the initialization code is fine with
  ##  re-parameterized x...

  eval(family$initialize) 
   
  if (!is.null(start0)) start <- start0 
  coef <- as.numeric(start)

  if (is.null(weights)) weights <- rep.int(1, nobs)
  if (is.null(offset)) offset <- rep.int(0, nobs)
 

749
  ## get log likelihood, grad and Hessian (w.r.t. coefs - not s.p.s) ...
750
  llf <- family$ll
751
  ll <- llf(y,x,coef,weights,family,offset=offset,deriv=1) 
752
  ll0 <- ll$l - (t(coef)%*%St%*%coef)/2
753 754 755 756
  rank.checked <- FALSE ## not yet checked the intrinsic rank of problem 
  rank <- q;drop <- NULL
  eigen.fix <- FALSE
  converged <- FALSE
757 758 759
  check.deriv <- FALSE; eps <- 1e-5 
  drop <- NULL;bdrop <- rep(FALSE,q) ## by default nothing dropped
  perturbed <- 0 ## counter for number of times perturbation tried on possible saddle
760 761
  for (iter in 1:(2*control$maxit)) { ## main iteration
    ## get Newton step... 
762 763 764 765
    if (check.deriv) {
      fdg <- ll$lb*0; fdh <- ll$lbb*0
      for (k in 1:length(coef)) {
        coef1 <- coef;coef1[k] <- coef[k] + eps
766
        ll.fd <- llf(y,x,coef1,weights,family,offset=offset,deriv=1)
767 768 769 770
        fdg[k] <- (ll.fd$l-ll$l)/eps
        fdh[,k] <- (ll.fd$lb-ll$lb)/eps
      }
    }
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    grad <- ll$lb - St%*%coef 
    Hp <- -ll$lbb+St
    D <- diag(Hp)
    indefinite <- FALSE
    if (sum(D <= 0)) { ## Hessian indefinite, for sure
      D <- rep(1,ncol(Hp))
      if (eigen.fix) {
778 779
        eh <- eigen(Hp,symmetric=TRUE);
        ev <- abs(eh$values)
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        Hp <- eh$vectors%*%(ev*t(eh$vectors))
      } else {
        Ib <- diag(rank)*abs(min(D))
        Ip <- diag(rank)*abs(max(D)*.Machine$double.eps^.5)
        Hp <- Hp  + Ip + Ib
      }
      indefinite <- TRUE
    } else { ## Hessian could be +ve def in which case Choleski is cheap!
      D <- D^-.5 ## diagonal pre-conditioner
      Hp <- D*t(D*Hp) ## pre-condition Hp
      Ip <- diag(rank)*.Machine$double.eps^.5   
    }
    L <- suppressWarnings(chol(Hp,pivot=TRUE))
    mult <- 1
    while (attr(L,"rank") < rank) { ## rank deficient - add ridge penalty 
      if (eigen.fix) {
        eh <- eigen(Hp,symmetric=TRUE);ev <- eh$values
        thresh <- max(min(ev[ev>0]),max(ev)*1e-6)*mult
        mult <- mult*10
        ev[ev<thresh] <- thresh
        Hp <- eh$vectors%*%(ev*t(eh$vectors)) 
        L <- suppressWarnings(chol(Hp,pivot=TRUE))
      } else {
        L <- suppressWarnings(chol(Hp+Ip,pivot=TRUE))
        Ip <- Ip * 100 ## increase regularization penalty
      }
      indefinite <- TRUE
    }

    piv <- attr(L,"pivot")
    ipiv <- piv;ipiv[piv] <- 1:ncol(L)
    step <- D*(backsolve(L,forwardsolve(t(L),(D*grad)[piv]))[ipiv])

    c.norm <- sum(coef^2)
    if (c.norm>0) { ## limit step length to .1 of coef length
      s.norm <- sqrt(sum(step^2))
      c.norm <- sqrt(c.norm)
      if (s.norm > .1*c.norm) step <- step*0.1*c.norm/s.norm
    }
    ## try the Newton step...
    coef1 <- coef + step 
821
    ll <- llf(y,x,coef1,weights,family,offset=offset,deriv=1) 
822
    ll1 <- ll$l - (t(coef1)%*%St%*%coef1)/2
823
    khalf <- 0;fac <- 2
824
    while ((!is.finite(ll1)||ll1 < ll0) && khalf < 25) { ## step halve until it succeeds...
825
      step <- step/fac;coef1 <- coef + step
826
      ll <- llf(y,x,coef1,weights,family,offset=offset,deriv=0)
827 828
      ll1 <- ll$l - (t(coef1)%*%St%*%coef1)/2
      if (ll1>=ll0) {
829
        ll <- llf(y,x,coef1,weights,family,offset=offset,deriv=1)
830 831
      } else { ## abort if step has made no difference
        if (max(abs(coef1-coef))==0) khalf <- 100
832 833
      }
      khalf <- khalf + 1
834
      if (khalf>5) fac <- 5
835
    } ## end step halve
836
 
837
    if (!is.finite(ll1) || ll1 < ll0) { ## switch to steepest descent... 
838 839 840 841
      step <- -.5*drop(grad)*mean(abs(coef))/mean(abs(grad))
      khalf <- 0
    }

842
    while ((!is.finite(ll1)||ll1 < ll0) && khalf < 25) { ## step cut until it succeeds...
843
      step <- step/10;coef1 <- coef + step
844
      ll <- llf(y,x,coef1,weights,family,offset=offset,deriv=0)
845 846
      ll1 <- ll$l - (t(coef1)%*%St%*%coef1)/2
      if (ll1>=ll0) {
847
        ll <- llf(y,x,coef1,weights,family,offset=offset,deriv=1)
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      } else { ## abort if step has made no difference
        if (max(abs(coef1-coef))==0) khalf <- 100
      }
      khalf <- khalf + 1
    }
853

854
    if ((is.finite(ll1)&&ll1 >= ll0)||iter==control$maxit) { ## step ok. Accept and test
855
      coef <- coef + step
856 857 858 859 860 861 862 863 864 865
      ## convergence test...
      ok <- (iter==control$maxit||(abs(ll1-ll0) < control$epsilon*abs(ll0) 
          && max(abs(grad)) < .Machine$double.eps^.5*abs(ll0))) 
      if (ok) { ## appears to have converged
        if (indefinite) { ## not a well defined maximum
          if (perturbed==5) stop("indefinite penalized likelihood in gam.fit5 ")
          if (iter<4||rank.checked) {
            perturbed <- perturbed + 1
            coef <- coef*(1+(runif(length(coef))*.02-.01)*perturbed) + 
                    (runif(length(coef)) - 0.5 ) * mean(abs(coef))*1e-5*perturbed 
866
            ll <- llf(y,x,coef,weights,family,offset=offset,deriv=1) 
867
            ll0 <- ll$l - (t(coef)%*%St%*%coef)/2
868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885
          } else {        
            rank.checked <- TRUE
            if (penalized) {
              Sb <- crossprod(Eb) ## balanced penalty
              Hb <- -ll$lbb/norm(ll$lbb,"F")+Sb/norm(Sb,"F") ## balanced penalized hessian
            } else Hb <- -ll$lbb/norm(ll$lbb,"F")
            ## apply pre-conditioning, otherwise badly scaled problems can result in
            ## wrong coefs being dropped...
            D <- abs(diag(Hb))
            D[D<1e-50] <- 1;D <- D^-.5
            Hb <- t(D*Hb)*D
            qrh <- qr(Hb,LAPACK=TRUE)
            rank <- Rrank(qr.R(qrh))
            if (rank < q) { ## rank deficient. need to drop and continue to adjust other params
              drop <- sort(qrh$pivot[(rank+1):q]) ## set these params to zero 
              bdrop <- 1:q %in% drop ## TRUE FALSE version
              ## now drop the parameters and recompute ll0...
              lpi <- attr(x,"lpi")
886 887
              xat <- attributes(x)
              xat$dim <- xat$dimnames <- NULL
888 889 890 891
              coef <- coef[-drop]
              St <- St[-drop,-drop]
              x <- x[,-drop] ## dropping columns from model matrix
              if (!is.null(lpi)) { ## need to adjust column indexes as well
892 893
                ii <- (1:q)[!bdrop];ij <- rep(NA,q)
                ij[ii] <- 1:length(ii) ## col i of old model matrix is col ij[i] of new 
894
               
895
                for (i in 1:length(lpi)) {
896
                  lpi[[i]] <- ij[lpi[[i]][!(lpi[[i]]%in%drop)]] # drop and shuffle up
897 898
                }
              } ## lpi adjustment done
899
              if (length(xat)>0) for (i in 1:length(xat)) attr(x,names(xat)[i]) <- xat[[i]]
900
              attr(x,"lpi") <- lpi
901
              attr(x,"drop") <- drop ## useful if family has precomputed something from x
902
              ll <- llf(y,x,coef,weights,family,offset=offset,deriv=1) 
903 904
              ll0 <- ll$l - (t(coef)%*%St%*%coef)/2
            } 
905
          }
906 907 908 909 910 911

        } else { ## not indefinite really converged
          converged <- TRUE
          break
        }
      } else ll0 <- ll1 ## step ok but not converged yet
912 913
    } else { ## step failed.
      converged  <- FALSE
914
      if (is.null(drop)) bdrop <- rep(FALSE,q)
915 916 917 918
      warning(paste("step failed: max abs grad =",max(abs(grad))))
      break
    }
  } ## end of main fitting iteration
919 920

  ## at this stage the Hessian (of pen lik. w.r.t. coefs) should be +ve definite,
921
  ## so that the pivoted Choleski factor should exist...
922 923
  if (iter == 2*control$maxit&&converged==FALSE) 
    warning(gettextf("iteration limit reached: max abs grad = %g",max(abs(grad))))
924 925 926 927 928 929 930

  ldetHp <- 2*sum(log(diag(L))) - 2 * sum(log(D)) ## log |Hp|

  if (!is.null(drop)) { ## create full version of coef with zeros for unidentifiable 
    fcoef <- rep(0,length(bdrop));fcoef[!bdrop] <- coef
  } else fcoef <- coef

931
  dVkk <- d1l <- d2l <- d1bSb <- d2bSb <- d1b <- d2b <- d1ldetH <- d2ldetH <- d1b <- d2b <- NULL
932 933

  if (deriv>0) {  ## Implicit differentiation for derivs...
934

935 936 937 938 939
    m <- nSp
    d1b <- matrix(0,rank,m)
    Sib <- Sl.termMult(rp$Sl,fcoef,full=TRUE) ## list of penalties times coefs
    if (nSp) for (i in 1:m) d1b[,i] <- 
       -D*(backsolve(L,forwardsolve(t(L),(D*Sib[[i]][!bdrop])[piv]))[ipiv])
940
  
941 942 943
    ## obtain the curvature check matrix...
    dVkk <- crossprod(L[,ipiv]%*%(d1b/D))

944 945 946 947
    if (!is.null(drop)) { ## create full version of d1b with zeros for unidentifiable 
      fd1b <-  matrix(0,q,m)
      fd1b[!bdrop,] <- d1b
    } else fd1b <- d1b
948

949 950
    ## Now call the family again to get first derivative of Hessian w.r.t
    ## smoothing parameters, in list d1H...
951

952 953
    ll <- llf(y,x,coef,weights,family,offset=offset,deriv=3,d1b=d1b)
    # d1l <- colSums(ll$lb*d1b) # cancels
954
    
955

956
    if (deriv>1) { ## Implicit differentiation for the second derivatives is now possible...
957

958 959 960 961 962 963 964 965
      d2b <- matrix(0,rank,m*(m+1)/2)
      k <- 0
      for (i in 1:m) for (j in i:m) {
        k <- k + 1
        v <- -ll$d1H[[i]]%*%d1b[,j] + Sl.mult(rp$Sl,fd1b[,j],i)[!bdrop] + Sl.mult(rp$Sl,fd1b[,i],j)[!bdrop]
        d2b[,k] <- -D*(backsolve(L,forwardsolve(t(L),(D*v)[piv]))[ipiv])
        if (i==j) d2b[,k] <- d2b[,k] + d1b[,i]
      } 
966
  
967 968
      ## Now call family for last time to get trHid2H the tr(H^{-1} d^2 H / drho_i drho_j)...

969
      llr <- llf(y,x,coef,weights,family,offset=offset,deriv=4,d1b=d1b,d2b=d2b,
970 971 972 973
                       Hp=Hp,rank=rank,fh = L,D=D)

      ## Now compute Hessian of log lik w.r.t. log sps using chain rule
       
974 975
      # d2la <- colSums(ll$lb*d2b) # cancels
      # k <- 0
976 977
      d2l <- matrix(0,m,m)
      for (i in 1:m) for (j in i:m) {
978 979 980
        # k <- k + 1
        d2l[j,i] <- d2l[i,j] <- # d2la[k] + # cancels
	                        t(d1b[,i])%*%ll$lbb%*%d1b[,j] 
981 982 983
      }
    } ## if (deriv > 1)
  } ## if (deriv > 0)
984 985

  ## Compute the derivatives of log|H+S|... 
986 987 988 989 990 991 992 993 994
  if (deriv > 0) {
    d1ldetH <- rep(0,m)
    d1Hp <- list()
    for (i in 1:m) {
      A <- -ll$d1H[[i]] + Sl.mult(rp$Sl,diag(q),i)[!bdrop,!bdrop]
      d1Hp[[i]] <- D*(backsolve(L,forwardsolve(t(L),(D*A)[piv,]))[ipiv,])  
      d1ldetH[i] <- sum(diag(d1Hp[[i]]))
    }
  } ## if (deriv > 0)
995

996 997 998 999 1000 1001 1002
  if (deriv > 1) {
    d2ldetH <- matrix(0,m,m)
    k <- 0
    for (i in 1:m) for (j in i:m) {
      k <- k + 1
      d2ldetH[i,j] <- -sum(d1Hp[[i]]*t(d1Hp[[j]])) - llr$trHid2H[k] 
      if (i==j) { ## need to add term relating to smoothing penalty
1003 1004 1005 1006 1007 1008 1009 1010
        #A <- t(Sl.mult(rp$Sl,diag(q),i,full=FALSE))
        #bind <- rowSums(abs(A))!=0 ## FIX: abs 3/3/16 
        #ind <- which(bind)
        #bind <- bind[!bdrop]
        #A <- A[!bdrop,!bdrop[ind]]
        A <- Sl.mult(rp$Sl,diag(q),i,full=TRUE)[!bdrop,!bdrop]
        bind <- rowSums(abs(A))!=0 ## row/cols of non-zero block
        A <- A[,bind] ## drop the zero columns  
1011 1012 1013 1014 1015
        A <- D*(backsolve(L,forwardsolve(t(L),(D*A)[piv,]))[ipiv,])
        d2ldetH[i,j] <- d2ldetH[i,j] + sum(diag(A[bind,]))
      } else d2ldetH[j,i] <- d2ldetH[i,j]
    }
  } ## if (deriv > 1)
1016 1017 1018

  ## Compute derivs of b'Sb...

1019
  if (deriv>0) {
1020
    # Sb <- St%*%coef
1021 1022 1023 1024
    Skb <- Sl.termMult(rp$Sl,fcoef,full=TRUE)
    d1bSb <- rep(0,m)
    for (i in 1:m) { 
      Skb[[i]] <- Skb[[i]][!bdrop]
1025 1026
      d1bSb[i] <- # 2*sum(d1b[,i]*Sb) + # cancels
                  sum(coef*Skb[[i]])
1027
    }
1028 1029
  }
 
1030 1031
  if (deriv>1) {
    d2bSb <- matrix(0,m,m)
1032
    # k <- 0
1033 1034 1035
    for (i in 1:m) {
      Sd1b <- St%*%d1b[,i] 
      for (j in i:m) {
1036 1037
         k <- k + 1
         d2bSb[j,i] <- d2bSb[i,j] <- 2*sum( # d2b[,k]*Sb + # cancels 
1038
         d1b[,i]*Skb[[j]] + d1b[,j]*Skb[[i]] + d1b[,j]*Sd1b)
1039 1040
      }
      d2bSb[i,i] <-  d2bSb[i,i] + sum(coef*Skb[[i]]) 
1041 1042 1043 1044
    }
  }

  ## get grad and Hessian of REML score...
1045
  REML <- -as.numeric(ll$l - drop(t(coef)%*%St%*%coef)/2 + rp$ldetS/2  - ldetHp/2  + Mp*log(2*pi)/2)
1046
 
1047 1048 1049
  REML1 <- if (deriv<1) NULL else -as.numeric( # d1l # cancels
                                   - d1bSb/2 + rp$ldet1/2  - d1ldetH/2 ) 

1050 1051 1052 1053 1054
  if (control$trace) {
    cat("\niter =",iter,"  ll =",ll$l,"  REML =",REML,"  bSb =",t(coef)%*%St%*%coef/2,"\n")
    cat("log|S| =",rp$ldetS,"  log|H+S| =",ldetHp,"  n.drop =",length(drop),"\n")
    if (!is.null(REML1)) cat("REML1 =",REML1,"\n")
  }
1055
  REML2 <- if (deriv<2) NULL else -( d2l - d2bSb/2 + rp$ldet2/2  - d2ldetH/2 ) 
1056
 ## bSb <- t(coef)%*%St%*%coef
1057 1058
  lpi <- attr(x,"lpi")
  if (is.null(lpi)) { 
1059
    linear.predictors <- if (is.null(offset)) as.numeric(x%*%coef) else as.numeric(x%*%coef+offset)
1060 1061 1062
    fitted.values <- family$linkinv(linear.predictors) 
  } else {
    fitted.values <- linear.predictors <- matrix(0,nrow(x),length(lpi))
1063
    if (!is.null(offset)) offset[[length(lpi)+1]] <- 0
1064 1065
    for (j in 1:length(lpi)) {
      linear.predictors[,j] <- as.numeric(x[,lpi[[j]],drop=FALSE] %*% coef[lpi[[j]]])
1066
      if (!is.null(offset[[j]])) linear.predictors[,j] <-  linear.predictors[,j] + offset[[j]]
1067 1068 1069 1070 1071
      fitted.values[,j] <- family$linfo[[j]]$linkinv( linear.predictors[,j]) 
    }
  }
  coef <- Sl.repara(rp$rp,fcoef,inverse=TRUE) ## undo re-parameterization of coef 
 
1072
  if (!is.null(drop)&&!is.null(d1b)) { ## create full version of d1b with zeros for unidentifiable 
1073 1074 1075
    db.drho <- matrix(0,length(bdrop),ncol(d1b));db.drho[!bdrop,] <- d1b
  } else db.drho <- d1b
  ## and undo re-para...
1076
  if (!is.null(d1b)) db.drho <- t(Sl.repara(rp$rp,t(db.drho),inverse=TRUE,both.sides=FALSE)) 
1077 1078 1079

  ret <- list(coefficients=coef,family=family,y=y,prior.weights=weights,
       fitted.values=fitted.values, linear.predictors=linear.predictors,
1080
       scale.est=1, ### NOTE: needed by newton, but what is sensible here? 
1081 1082
       REML= REML,REML1= REML1,REML2=REML2,
       rank=rank,aic = -2*ll$l, ## 2*edf needs to be added
1083 1084
       ##deviance = -2*ll$l,
       l= ll$l,## l1 =d1l,l2 =d2l,
1085 1086 1087 1088 1089
       lbb = ll$lbb, ## Hessian of log likelihood
       L=L, ## chol factor of pre-conditioned penalized hessian
       bdrop=bdrop, ## logical index of dropped parameters
       D=D, ## diagonal preconditioning matrix
       St=St, ## total penalty matrix
1090 1091 1092
       rp = rp$rp,
       db.drho = db.drho, ## derivative of penalty coefs w.r.t. log sps.
       #bSb = bSb, bSb1 =  d1bSb,bSb2 =  d2bSb,
1093
       S1=rp$ldet1,
1094 1095 1096
       #S=rp$ldetS,S1=rp$ldet1,S2=rp$ldet2,
       #Hp=ldetHp,Hp1=d1ldetH,Hp2=d2ldetH,
       #b2 = d2b)
1097
       niter=iter,H = ll$lbb,dH = ll$d1H,dVkk=dVkk)#,d2H=llr$d2H)
1098 1099 1100 1101 1102
    ## debugging code to allow components of 2nd deriv of hessian w.r.t. sp.s 
    ## to be passed to deriv.check.... 
    #if (!is.null(ll$ghost1)&&!is.null(ll$ghost2)) { 
    #  ret$ghost1 <- ll$ghost1; ret$ghost2 <- ret$ghost2
    #} 
1103
    ret
1104 1105
} ## end of gam.fit5

1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190
efsud <- function(x,y,lsp,Sl,weights=NULL,offset=NULL,family,
                     control=gam.control(),Mp=-1,start=NULL) {
## Extended Fellner-Schall method
## tr(S^-S_j) is returned by ldetS as ldet1 - S1 from gam.fit5
## b'S_jb is computed as d1bSb in gam.fit5
## tr(V S_j) will need to be computed using Sl.termMult
##   Sl returned by ldetS and Vb computed as in gam.fit5.postproc.
  tol <- 1e-6
  lsp <- lsp + 2.5
  mult <- 1
  fit <- gam.fit5(x=x,y=y,lsp=lsp,Sl=Sl,weights=weights,offset=offset,deriv=0,family=family,
                     control=control,Mp=Mp,start=start)
  score.hist <- rep(0,200)
  for (iter in 1:200) {
    start <- fit$coefficients
    ## obtain Vb...
    ipiv <- piv <- attr(fit$L,"pivot")
    p <- length(piv)
    ipiv[piv] <- 1:p
    Vb <- crossprod(forwardsolve(t(fit$L),diag(fit$D,nrow=p)[piv,,drop=FALSE])[ipiv,,drop=FALSE])
    if (sum(fit$bdrop)) { ## some coefficients were dropped...
      q <- length(fit$bdrop)
      ibd <- !fit$bdrop
      Vtemp <- Vb; Vb <- matrix(0,q,q)
      Vb[ibd,ibd] <- Vtemp
    }
    Vb <- Sl.repara(fit$rp,Vb,inverse=TRUE)
    SVb <- Sl.termMult(Sl,Vb) ## this could be made more efficient
    trVS <- rep(0,length(SVb))
    for (i in 1:length(SVb)) {
      ind <- attr(SVb[[i]],"ind")
      trVS[i] <- sum(diag(SVb[[i]][,ind]))
    }
    Sb <- Sl.termMult(Sl,start,full=TRUE)
    bSb <- rep(0,length(Sb))
    for (i in 1:length(Sb)) {
      bSb[i] <- sum(start*Sb[[i]])
    }
    a <- pmax(0,fit$S1*exp(-lsp) - trVS)
    r <- a/pmax(0,bSb)
    r[a==0&bSb==0] <- 1
    r[!is.finite(r)] <- 1e6
    lsp1 <- pmin(lsp + log(r)*mult,12)
    old.reml <- fit$REML
    fit <- gam.fit5(x=x,y=y,lsp=lsp1,Sl=Sl,weights=weights,offset=offset,deriv=0,
                    family=family,control=control,Mp=Mp,start=start)
    ## some step length control...
   
    if (fit$REML<=old.reml) { ## improvement
      if (max(abs(log(r))<.05)) { ## consider step extension
        lsp2 <- pmin(lsp + log(r)*mult*2,12) ## try extending step...
        fit2 <- gam.fit5(x=x,y=y,lsp=lsp2,Sl=Sl,weights=weights,offset=offset,deriv=0,family=family,
                     control=control,Mp=Mp,start=start)
     
        if (fit2$REML < fit$REML) { ## improvement - accept extension
          fit <- fit2;lsp <- lsp2
	  mult <- mult * 2
        } else { ## accept old step
          lsp <- lsp1
        }
      } else lsp <- lsp1
    } else { ## no improvement 
      while (fit$REML > old.reml&&mult>1) { ## don't contract below 1 as update doesn't have to improve REML 
          mult <- mult/2 ## contract step
          lsp1 <- pmin(lsp + log(r)*mult,12)
	  fit <- gam.fit5(x=x,y=y,lsp=lsp1,Sl=Sl,weights=weights,offset=offset,deriv=0,family=family,
                        control=control,Mp=Mp,start=start)
      }
      lsp <- lsp1
      if (mult<1) mult <- 1
    }
    score.hist[iter] <- fit$REML
    if (iter==1) old.ll <- fit$l else {
      if (abs(old.ll-fit$l)<tol*abs(fit$l)) break
      old.ll <- fit$l
    }
  }
  fit$sp <- exp(lsp)
  fit$niter <- iter
  fit$outer.info <- list(iter = iter,score.hist=score.hist[1:iter])
  fit$outer.info$conv <- if (iter==200) "iteration limit reached" else "full convergence"
  fit
} ## efsud

gam.fit5.post.proc <- function(object,Sl,L,lsp0,S,off) {
1191 1192
## object is object returned by gam.fit5, Sl is penalty object, L maps working sp
## vector to full sp vector 
1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245
## Computes:
## R - unpivoted Choleski of estimated expected hessian of ll 
## Vb - the Bayesian cov matrix,
## Ve - "frequentist" alternative
## F - the EDF matrix
## edf = diag(F) and edf2 = diag(2F-FF)
## Main issue is that lbb and lbb + St must be +ve definite for
## F to make sense.
## NOTE: what comes in is in stabilizing parameterization from 
##       gam.fit5, and may have had parameters dropped. 
##       possibly initial reparam needs to be undone here as well
##       before formation of F....
  lbb <- -object$lbb ## Hessian of log likelihood in fit parameterization
  p <- ncol(lbb)
  ipiv <- piv <- attr(object$L,"pivot")
  ipiv[piv] <- 1:p
  ##  Vb0 <- crossprod(forwardsolve(t(object$L),diag(object$D,nrow=p)[piv,])[ipiv,])

  ## need to pre-condition lbb before testing rank...
  lbb <- object$D*t(object$D*lbb)
 
  R <- suppressWarnings(chol(lbb,pivot=TRUE)) 
  
  if (attr(R,"rank") < ncol(R)) { 
    ## The hessian of the -ve log likelihood is not +ve definite
    ## Find the "nearest" +ve semi-definite version and use that
    retry <- TRUE;tol <- 0
    eh <- eigen(lbb,symmetric=TRUE)
    mev <- max(eh$values);dtol <- 1e-7
    while (retry) {
      eh$values[eh$values<tol*mev] <- tol*mev
      R <- sqrt(eh$values)*t(eh$vectors)
      lbb <- crossprod(R)
      Hp <- lbb + object$D*t(object$D*object$St) ## pre-conditioned Hp
      ## Now try to invert it by Choleski with diagonal pre-cond,
      ## to get Vb
      object$L <- suppressWarnings(chol(Hp,pivot=TRUE))
      if (attr(object$L,"rank")==ncol(Hp)) {
        R <- t(t(R)/object$D) ## so R'R = lbb (original)
        retry <- FALSE
      } else { ##  failure: make more +ve def
        tol <- tol + dtol;dtol <- dtol*10
      }
    } ## retry  
  } else { ## hessian +ve def, so can simply use what comes from fit directly
    ipiv <- piv <- attr(R,"pivot")
    ipiv[piv] <- 1:p
    R <- t(t(R[,ipiv])/object$D) ## so now t(R)%*%R = lbb (original)
  } 
  ## DL'LD = penalized Hessian, which needs to be inverted
  ## to DiLiLi'Di = Vb, the Bayesian cov matrix...
  ipiv <- piv <- attr(object$L,"pivot")
  ipiv[piv] <- 1:p
1246
  Vb <- crossprod(forwardsolve(t(object$L),diag(object$D,nrow=p)[piv,,drop=FALSE])[ipiv,,drop=FALSE])
1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260

  ## Insert any zeroes required as a result of dropping 
  ## unidentifiable parameters...
  if (sum(object$bdrop)) { ## some coefficients were dropped...
    q <- length(object$bdrop)
    ibd <- !object$bdrop
    Vtemp <- Vb; Vb <- matrix(0,q,q)
    Vb[ibd,ibd] <- Vtemp
    Rtemp <- R; R <- matrix(0,q,q)
    R[ibd,ibd] <- Rtemp
    lbbt <- lbb;lbb <- matrix(0,q,q)
    lbb[ibd,ibd] <- lbbt
  }  

1261
  edge.correct <- FALSE 
1262
  ## compute the smoothing parameter uncertainty correction...
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