"fitfclust" <-
function(x=NULL,curve=NULL,timeindex=NULL,data=NULL, q = 5, h = 1, p = 5,
  K = 2, tol = 0.001, maxit = 20, pert =  
           0.01, grid = seq(0, 1, length = 100), hard = F, plot= F,trace=F)
{
  # This is the main function to implement the FClust procedure.
  library(splines)
  if (is.null(data))
  data <- list(x=x,curve=curve,timeindex=timeindex)
  initfit <- fclustinit(data = data, pert = pert, grid = grid, h = h,
                        p = p, q = q, K = K)
  # Initialize the parameters
  parameters <- initfit$parameters
  vars <- initfit$vars
  S <- initfit$S
  FullS <- initfit$FullS
  sigma.old <- 0
  sigma.new <- 1
  ind <- 1
  # Main loop. Iterates between M and E steps and stops when
  # sigma  has converged.
  while(abs(sigma.old - sigma.new)/sigma.new > tol & (ind <= maxit)) {
    parameters <- fclustMstep(parameters, data, vars, S, tol, p, hard)
    vars <- fclustEstep(parameters, data, vars, S, hard)
    sigma.old <- sigma.new
    sigma.new <- parameters$sigma[1]
    if (trace)
      print(paste("Iteration", ind,": Sigma = ",sigma.new))
  # Plot cluster mean curves.  
    if(plot){
      cluster.mean <- matrix(0,K,dim(FullS)[1])
      for(k in 1:K)
        cluster.mean[k,] <- FullS %*% (parameters$lambda.zero + parameters$Lambda %*% parameters$alpha[k,  ])
      plot(grid,grid,ylim=range(cluster.mean),type='n',ylab="Cluster Mean")
      for(k in 1:K)
        lines(grid,cluster.mean[k,], col = 4, lwd = 2)
    }
    ind <- ind + 1
  }
  # Enforce parameter constraint (7)
  cfit <- fclustconst(data, parameters, vars, FullS)
  list(data=data,parameters = cfit$parameters, vars = cfit$vars, FullS = FullS,grid=grid)
}

"fclustinit" <-
function(data, pert = 0, grid = seq(0.01, 1, length = 100), h = 1, p = 2, q = 5,K = K){
  S <- FullS <- NULL
  # This function initializes all the parameters.
  # Produce spline basis matrix
    FullS <- cbind(1, ns(grid, df = (q - 1)))
    FullS <- svd(FullS)$u
    S <- FullS[data$timeindex,  ]
  N <- length(table(data$curve))
  # Compute initial estimate of basis coefficients.
  points <- matrix(0,N,sum(q))
  for (i in 1:N){
    Si <- S[data$curve==i,]
    xi <- data$x[data$curve==i]
    points[i,] <- solve(t(Si) %*% Si + pert * diag(q)) %*% t(Si) %*%xi}
  # Use k-means to get initial cluster memberships from points.
  if(K > 1)
    class <- kmeans(points, K, 10)$cluster
  else class <- rep(1, N)
  # Initial estimates for the posterior probs of cluster membership.
  piigivej <- matrix(0, N, K)
  piigivej[col(piigivej) == class] <- 1
  # Calculate coefficeints for cluster means.
  classmean <- matrix(0,K,q)
  for (k in 1:K)
    classmean[k,] <- apply(points[class==k,],2,mean)
  # Initialize lambda.zero, Lambda and alpha as defined in paper.
  lambda.zero <- apply(classmean, 2, mean)
  Lambda <- as.matrix(svd(scale(classmean, scale = F))$v[, 1:h])
  alpha <- scale(classmean, scale = F)%*%Lambda
  # Calculate estimates for gamma and gprod.
  gamma <- t(t(points) - lambda.zero - (Lambda %*% t(alpha[class,  ])))
  gprod <- NULL
  for(i in 1:N)
    gprod <- cbind(gprod, (gamma[i,  ]) %*% t(gamma[i,  ]))
  gamma <- array(gamma[, rep(1:sum(q), rep(K, sum(q)))], c(N, K, sum(q)))
  gcov <- matrix(0, sum(q), N * sum(q))
  list(S = S, FullS = FullS, parameters = list(lambda.zero = lambda.zero,
    Lambda = Lambda, alpha = alpha), vars = list(gamma = gamma,
      gprod = gprod, gcov = gcov, piigivej = piigivej))
}

"fclustMstep" <-
function(parameters, data, vars, S, tol, p, hard)
{
  # This function implements the M step of the EM algorithm.
  K <- dim(parameters$alpha)[1]
  alpha <- parameters$alpha
  Lambda <- parameters$Lambda
  gamma <- vars$gamma
  gcov <- vars$gcov
  curve <- data$curve
  piigivej <- vars$piigivej
  N <- dim(gamma)[1]
  K <- dim(alpha)[1]
  h <- dim(alpha)[2]
  n <- length(curve)
  q <- dim(S)[2]
  sigma.old <- 2
  sigma <- 1
  # Compute pi.
  if(hard)
    parameters$pi <- rep(1/K, K)
  else parameters$pi <- apply(vars$piigivej, 2, mean)
  # Compute rank p estimate of Gamma
  ind <- matrix(rep(c(rep(c(1, rep(0, q - 1)), N), 0), q)[1:(N*q^2)], N * q, q)
  gsvd <- svd(vars$gprod %*% ind/N)
  gsvd$d[ - (1:p)] <- 0
  parameters$Gamma <- gsvd$u %*% diag(gsvd$d) %*% t(gsvd$u)
  # This loop iteratively calculates alpha and then Lambda and stops
  # when they have converged.
  loopnumber <- 1
  while((abs(sigma.old[1] - sigma[1])/sigma[1] > tol) & (loopnumber <10)){
    sigma.old <- sigma
  # Calculate lambda.zero.
    gamma.pi <- diag(S %*% t(apply(gamma * as.vector(piigivej),
                                   c(1, 3), sum)[curve,  ]))
    alpha.pi <- t(matrix(apply(alpha, 2, function(x, piigivej,K)
                               {rep(1, K) %*% (piigivej * x)}
                               , t(piigivej), K), N, h)[curve,  ])
    lambda.zero <- solve(t(S) %*% S) %*% t(S) %*% (data$x - diag(
                S %*% Lambda %*% alpha.pi) - gamma.pi)
    x <- data$x - S %*% lambda.zero
    # Calculate alpha.
    for(i in 1.:K) {
      S.Lam <- S %*% Lambda
      S.Lam.pi <- S.Lam * piigivej[curve, i]
      if(sum(piigivej[, i]) > 10^(-4))
        alpha[i,  ] <- solve(t(S.Lam.pi) %*% S.Lam) %*%
          t(S.Lam.pi) %*% (x - diag(S %*% t(gamma[curve, i,  ])))
      else print("Warning: empty cluster")
    }
    # Calculate Lambda given alpha. This is done by iterating
    # through each column of Lambda holding the other columns fixed.
    for(m in 1:h) {
      pi.alphasq <- apply(t(piigivej) * (alpha^2)[, m], 2,sum)[curve]
      pi.alpha <- apply(t(piigivej) * alpha[, m], 2, sum)[curve]
      S.Lambda <- t(S %*% Lambda)
      if(h != 1) {
        temp <- NULL
        for(i in 1:K) {
          temp <- cbind(temp, as.vector(rep(1, h - 1) %*% matrix((S.Lambda *
                 alpha[i,  ])[ - m,  ], h - 1,dim(S)[1])) * alpha[i, m])
        }
        otherLambda <- (temp * piigivej[curve,  ])%*%rep(1, K)
      }
      else otherLambda <- 0
      gamma.pi.alpha <- apply(gamma * as.vector(piigivej) *
              rep(alpha[, m], rep(N, K)), c(1, 3), sum)[curve,  ]
      Lambda[, m] <- solve(t(S * pi.alphasq) %*% S) %*% t(S) %*%
      (x * pi.alpha - otherLambda - (S *gamma.pi.alpha) %*% rep(1, sum(q)))
    }
    # Calculate sigma 
      ind <- matrix(rep(c(rep(F, q), rep(T, N * q)),N)
    [1:(N * N * q)], N, N * q, byrow = T)[rep(1:N, table(curve)),]
      mat1 <- matrix(rep(S, N), n, N * q)
      mat3 <- t(mat1)
      mat3[t(ind)] <- 0
      ind2 <- matrix(rep(c(rep(F, q), rep(T, N * q)),
      N)[1:(N * N * q)], N, N * q, byrow = T)[rep(1:N, rep(q, N)),]
      mat2 <- matrix(rep(t(gcov), N), N * q, N * q,byrow = T)
      mat2[ind2] <- 0
      econtrib <- 0
      for(i2 in 1:K) {
        vect1 <- x - S %*% Lambda %*% alpha[
             i2,  ] - (S * gamma[curve, i2,  ]) %*% rep(1, q)
        econtrib <- econtrib + t(piigivej[curve,i2] * vect1) %*% vect1
      }
      sigma <- as.vector((econtrib + sum(diag(mat1 %*% mat2 %*% mat3)))/n)
    loopnumber <- loopnumber + 1
  }
  parameters$lambda.zero <- as.vector(lambda.zero)
  parameters$alpha <- alpha
  parameters$Lambda <- Lambda
  parameters$sigma <- sigma
  parameters
}

"fclustEstep" <-
function(parameters, data, vars, S, hard)
{
  # This function performs the E step of the EM algorithm by
  # calculating the expected values of gamma and gamma %*% t(gamma)
  # given the current parameter estimates.
  par <- parameters
  N <- dim(vars$gamma)[1]
  K <- dim(vars$gamma)[2]
  q <- dim(vars$gamma)[3]
  Gamma <- par$Gamma
  Lambda.alpha <- par$lambda.zero + par$Lambda %*% t(par$alpha)
  vars$gprod <- vars$gcov <- NULL
  for(j in 1:N) {
    # Calculate expected value of gamma.
    Sj <- S[data$curve == j,  ]
    nj <- sum(data$curve == j)
    invvar <- diag(1/rep(par$sigma, nj))
    Cgamma <- Gamma - Gamma %*% t(Sj) %*% solve(diag(nj) + Sj %*%
              Gamma %*% t(Sj) %*% invvar) %*% invvar %*% Sj %*% Gamma
    centx <- data$x[data$curve == j] - Sj %*% Lambda.alpha
    vars$gamma[j,  ,  ] <- t(Cgamma %*% t(Sj) %*% invvar %*% centx)
    # Calculate pi i given j.
    covx <- Sj %*% par$Gamma %*% t(Sj) + solve(invvar)
    d <- exp( - diag(t(centx) %*% solve(covx) %*% centx)/2) * par$pi
    vars$piigivej[j,  ] <- d/sum(d)
    if(hard) {
      m <- order( - d)[1]
      vars$piigivej[j,  ] <- 0
      vars$piigivej[j, m] <- 1
    }
    # Calculate expected value of gamma %*% t(gamma).
    vars$gprod <- cbind(vars$gprod, t(matrix(vars$gamma[j,  ,  ],
                K, q)) %*% (matrix(vars$gamma[j,  ,  ], K, q) * vars$
                piigivej[j,  ]) + Cgamma)
    vars$gcov <- cbind(vars$gcov, Cgamma)
  }
  vars
}


"fclustconst" <-
function(data, parameters, vars, S)
{
  # This function enforces the constraint (7) from the paper on the
  # parameters. This means that the alphas can be interpreted as the
  # number of standard deviations that the groups are apart etc.
  par <- parameters
  A <- t(S) %*% solve(par$sigma * diag(dim(S)[1]) + S %*% par$Gamma %*%
                      t(S)) %*% S
  svdA <- svd(A)
  sqrtA <- diag(sqrt(svdA$d)) %*% t(svdA$u)
  negsqrtA <- svdA$u %*% diag(1/sqrt(svdA$d))
  finalsvd <- svd(sqrtA %*% par$Lambda)
  par$Lambda <- negsqrtA %*% finalsvd$u
  if(dim(par$Lambda)[2] > 1)
    par$alpha <- t(diag(finalsvd$d) %*% t(finalsvd$v) %*% t(par$alpha))
  else par$alpha <- t(finalsvd$d * t(finalsvd$v) %*% t(par$alpha))
  meanalpha <- apply(par$alpha, 2, mean)
  par$alpha <- t(t(par$alpha) - meanalpha)
  par$lambda.zero <- par$lambda.zero + par$Lambda %*% meanalpha
  list(parameters = par, vars = vars)
}

"nummax" <-
function(X)
{
	ind <- rep(1, dim(X)[1])
	m <- X[, 1]
	if(dim(X)[2] > 1)
		for(i in 2:dim(X)[2]) {
			test <- X[, i] > m
			ind[test] <- i
			m[test] <- X[test, i]
		}
	list(ind = ind, max = m)
}

"fclust.pred" <-
function(fit,data=NULL,reweight=F)
{
  # This function produces the alpha hats used to provide a low
  # dimensional pictorial respresentation of each curve. It also
  # produces a class prediction for each curve. It takes as
  # input the fit from fldafit (for predictions on the original data)
  # or the fit and a new data set (for predictions on new data).
  if (is.null(data))
    data <- fit$data
  FullS <- fit$FullS
  par <- fit$parameters
  curve <- data$curve
  time <- data$time
  N <- length(table(curve))
  h <- dim(par$alpha)[2]
  alpha.hat <- matrix(0, N, h)
  K <- dim(fit$par$alpha)[1]
  distance <- matrix(0, N, K)
  Calpha <- array(0, c(N, h, h))
  for(i in 1:N) {
    Sij <- FullS[time[curve == i],  ]
    xij <- data$x[curve == i]
    n <- length(xij)
    Sigma <- par$sigma * diag(n) + Sij %*% par$Gamma %*% t(Sij)
    # Calculate covariance for each alpha hat.
    InvCalpha <- t(par$Lambda) %*% t(Sij) %*% solve(Sigma) %*% Sij %*%
      par$Lambda
    Calpha[i,  ,  ] <- solve(InvCalpha)
    # Calculate each alpha hat.
    alpha.hat[i,  ] <- Calpha[i,  ,  ] %*% t(par$Lambda) %*% t(
       Sij) %*% solve(Sigma) %*% (xij - Sij %*% par$lambda.zero)
    # Calculate the matrix of distances, relative to the
    # appropriate metric of each curve from each class centroid. 
    for (k in 1:K){
      y <- as.vector(alpha.hat[i,])-fit$par$alpha[k,]
      distance[i,k] <- t(y)%*%InvCalpha %*%y}}
    # Calculate final class predictions for each curve.
  class.pred <- rep(1, N)
  log.pi <- log(fit$par$pi)
  if (!reweight)
    log.pi <- rep(0,K)
  probs <- t(exp(log.pi-t(distance)/2))
  probs <- probs/apply(probs,1,sum)
  m <- probs[,1]
  if(K != 1)
    for(k in 2:K) {
      test <- (probs[, k] > m)
      class.pred[test] <- k
      m[test] <- probs[test, k]
    }
  list(Calpha = Calpha, alpha.hat = alpha.hat, class.pred = class.pred,
       distance = distance, m = m,probs=probs)
}

"fclust.curvepred" <-
function(fit, data=NULL, index=NULL, tau = 0.95, tau1 = 0.975)
{
  if (is.null(data))
    data <- fit$data
  if (is.null(index))
    index <- 1:length(table(data$curve))
  tau2 <- tau/tau1
  sigma <- fit$par$sigma
  Gamma <- fit$par$Gamma
  Lambda <- fit$par$Lambda
  alpha <- fit$par$alpha
  lambda.zero <- as.vector(fit$par$lambda.zero)
  S <- fit$FullS
  N <- length(index)
  upci <-lowci <- uppi <- lowpi <- gpred <- matrix(0,N,nrow(S))
  etapred <- matrix(0,N,ncol(S))
  ind <- 1
  Lambda.alpha <- lambda.zero + Lambda %*% t(alpha)
  for (i in index){
    y <- data$x[data$curve == i]
    Si <- S[data$time[data$curve == i],  ]
    ni <- dim(Si)[1]
    invvar <- diag(1/rep(sigma, ni))
    covx <- Si %*% Gamma %*% t(Si) + solve(invvar)
    centx <- data$x[data$curve == i] - Si %*% Lambda.alpha
    d <- exp( - diag(t(centx) %*% solve(covx) %*% centx)/2) * fit$par$pi
    pi <- d/sum(d)
    K <- length(pi)
    mu <- lambda.zero + Lambda %*% t(alpha * pi) %*% rep(1, K)
    cov <- (Gamma - Gamma %*% t(Si) %*% solve(diag(ni) + Si %*% Gamma %*%
                                              t(Si)/sigma) %*% Si %*% Gamma/sigma)/sigma
    etapred[ind,] <- mu + cov %*% t(Si) %*% (y - Si %*% mu)
    ord <- order( - pi)
    numb <- sum(cumsum(pi[ord]) <= tau1) + 1
    v <- diag(S %*% (cov * sigma) %*% t(S))
    pse <- sqrt(v + sigma)
    se <- sqrt(v)
    lower.p <- upper.p <- lower.c <- upper.c <- matrix(0, nrow(S), numb)
    for(j in 1:numb) {
      mu <- lambda.zero + Lambda %*% alpha[ord[j],  ]
      mean <- S %*% (mu + cov %*% t(Si) %*% (y - Si %*% mu))
      upper.p[, j] <- mean + qnorm(tau2) * pse
      lower.p[, j] <- mean - qnorm(tau2) * pse
      upper.c[, j] <- mean + qnorm(tau2) * se
      lower.c[, j] <- mean - qnorm(tau2) * se
    }
    upci[ind,] <- nummax(upper.c)$max
    lowci[ind,] <-  - nummax( - lower.c)$max
    uppi[ind,] <- nummax(upper.p)$max
    lowpi[ind,] <-  - nummax( - lower.p)$max
    gpred[ind,] <- as.vector(S %*%etapred[ind,])
    ind <- ind+1
  }
  meancurves <- S%*%Lambda.alpha
  list(etapred = etapred, gpred = gpred,  upci = upci,lowci = lowci,  uppi = uppi, lowpi = lowpi,index=index,grid=fit$grid,data=data,meancurves=meancurves)
}

"fclust.discrim" <-
function(fit,absvalue=F){
    S <- fit$FullS
    sigma <- fit$par$sigma
    n <- nrow(S)
    Gamma <- fit$par$Gamma
    Sigma <- S%*%Gamma%*%t(S)+sigma*diag(n)
    Lambda <- fit$par$Lambda
    discrim <- solve(Sigma)%*%S%*%Lambda
    if (absvalue)
      discrim <- abs(discrim)
    n <- ncol(discrim)
    nrows <- ceiling(sqrt(n))
    par(mfrow=c(nrows,nrows))
    for (i in 1:n){
    plot(fit$grid,discrim[,i],ylab=paste("Discriminant Function ",i),xlab="Time",type='n')
    lines(fit$grid,discrim[,i],lwd=3)
    abline(0,0)}}

"fclust.plotcurves" <-
function(object=NULL,fit=NULL,index=NULL,ci=T,pi=T,clustermean=F){
    if (is.null(object))
      object <- fclust.curvepred(fit)
    if (is.null(index))
      index <- 1:length(table(object$data$curve))
    r <- ceiling(sqrt(length(index)))
    par(mfrow=c(r,r))
    for (i in index){
      grid <- object$grid
      upci <- object$upci[i,]
      uppi <- object$uppi[i,]
      lowci <- object$lowci[i,]
      lowpi <- object$lowpi[i,]
      gpred <- object$gpred[i,]
      meancurves <- (object$mean)
      if (clustermean)
        yrange <- c(min(c(lowpi,meancurves)),max(c(uppi,meancurves)))
      else
        yrange <- c(min(lowpi),max(uppi))
      plot(grid,grid,ylim=yrange,ylab="Predictions",xlab="Time",type='n',
           main=paste("Curve ",i))
      if (clustermean)
        for (k  in 1:ncol(meancurves))
          lines(grid,meancurves[,k],col=6,lty=2,lwd=2)
      if (ci){
        lines(grid,upci,col=3)
        lines(grid,lowci,col=3)}
      if (pi){
        lines(grid,uppi,col=4)
        lines(grid,lowpi,col=4)}
      lines(grid,gpred,col=2,lwd=2)
      lines(grid[object$data$time[object$data$curve==i]],object$data$x[object$data$curve==i],lwd=2)
      points(grid[object$data$time[object$data$curve==i]],object$data$x[object$data$curve==i],pch=19,cex=1.5)
    }
  }

"simdata" <-
structure(list(x = c(-0.227117840025744, -0.195701325120710, 
-0.123029609099774, -0.118619923345801, -0.110854003404887, -0.106908797785542, 
-0.107349467323043, -0.136947437978017, 0.147316597820139, 0.342213089281111, 
-0.374637137339911, -0.132569470675019, -0.106261313334034, -0.184045390440344, 
-0.176687325709244, -0.205370648502014, -0.207503074221494, -0.223960531903988, 
-0.179282627425149, 0.228839712560303, -0.406436079640189, -0.252485300162310, 
-0.130893822115844, -0.119483603887501, -0.157911812412119, -0.191173118294793, 
-0.186391707679505, -0.166101182172451, 0.017989520648983, 0.102677621538723, 
-0.329746791407803, -0.280156068523990, -0.194771696246444, -0.149493252880510, 
-0.171596928524117, -0.197948023475606, -0.217010552084032, -0.131802072904116, 
0.256737323621704, 0.358579862077912, -0.226652568726055, -0.194959419590792, 
-0.143850156023500, -0.100379181559697, -0.106534204920563, -0.182279005060978, 
-0.191699200147385, -0.209233185188917, -0.131366576347318, 0.168550341932226, 
-0.313790919034796, -0.116777642865946, -0.101128217858913, -0.104795564464629, 
-0.094204404109515, -0.106849583723510, -0.114156343156498, -0.113507166276034, 
-0.137597617774813, 0.340065783702146, -0.129486325727864, -0.130020108924852, 
-0.136359618769366, -0.162790169199692, -0.174869930544662, -0.193164071088511, 
-0.203685565671835, -0.093588403559253, 0.232631809906245, 0.465850115143711, 
-0.294711458343317, -0.0948725387147703, -0.110915079204378, 
-0.114290893388158, -0.160756593743237, -0.179658994991600, -0.191269772791207, 
-0.223781346510862, 0.203603299174188, 0.331811990712914, -0.467522085682574, 
-0.245553797585632, -0.158979981457772, -0.209709876424111, -0.220642628521095, 
-0.169252620230874, -0.0887299961627144, -0.0386754613677018, 
0.147942919391726, 0.414151222903763, -0.112767077728689, -0.124058844743678, 
-0.107685638343733, -0.128089652858022, -0.206697524949735, -0.152130234525033, 
-0.0748596280764934, -0.0450328357374213, 0.237789276941251, 
0.317903890876979, -0.325736649236121, -0.305421771375266, -0.136924941126976, 
-0.137501625477845, -0.148820202759715, -0.155879507713238, -0.196535108694717, 
-0.213591244573685, 0.185888155688001, 0.436176736410145, -0.238859561543531, 
-0.175022237353340, -0.150749556903914, -0.120811412581943, -0.0948573655562948, 
-0.114608637387264, -0.191172926480358, -0.179682834570726, 0.0754949932557839, 
0.119940540722110, -0.366315219441288, -0.178066636999900, -0.155689951246938, 
-0.107700926401963, -0.191011693833892, -0.229049245707688, -0.219123020434303, 
-0.129898949730155, 0.140321315344435, 0.478295571715595, -0.281431065900907, 
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