"fldafit" <- 
function(data, q = 5, h = 1, p = 1, tol = 0.001, maxit = 50, pert = 
	0.10000000000000001, grid = seq(0.01, 1, length = 100))
{
	# This is the main function to implement the FLDA procedure.
	initfit <- fldainit(data = data, pert = pert, grid = grid, h = h, p = p,
		q = q)
	# Initialize the parameters
	parameters <- initfit$parameters
	vars <- initfit$vars
	S <- initfit$S
	FullS <- initfit$FullS
	like.old <- 0
	like.new <- 1
	ind <- 1
	# Main loop. Iterates between M and E steps and stops when the
	# loglikelihood  has converged.
	while(abs(like.old - like.new)/like.new > tol & (ind <= maxit)) {
		parameters <- Mstep(parameters, data, vars, S, tol)
		vars <- Estep(parameters, data, vars, S)
		like.old <- like.new
		like.new <- calcobslike(parameters, data, S)
		print(paste("Iteration ", ind, ", LogLikelihood =", round(
			like.new, 3)))
		ind <- ind + 1
	}
	# Enforce parameter constraint (7)
	constrainedfit <- constraints(data, parameters, vars, FullS)
	parameters <- constrainedfit$parameters
	vars <- constrainedfit$vars
	list(parameters = parameters, vars = vars, S = S, FullS = FullS, like
		 = like.new)
}

"fldainit" <- 
function(data, pert = 0, grid = seq(0.01, 1, length = 100), h = 1, p = 2, q = 5
	)
{
	# 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,  ]
	K <- length(table(data$class))
	N <- length(table(data$curve))
	points <- matrix(unlist(tapply(1:length(data$curve), data$curve, 
		function(x, S, y, pert, q)
	{
		Sij <- S[x,  ]
		yij <- y[x]
		solve(t(Sij) %*% Sij + pert * diag(q)) %*% t(Sij) %*% yij
	}
	, S, data$y, pert, q)), N, q, byrow = T)
	classmean <- matrix(unlist(tapply(1:N, data$class, function(x, points)
	{
		apply(points[x,  ], 2, mean)
	}
	, points)), K, q, byrow = T)
	# Initialize lambda.zero, Lambda and alpha as defined in paper.
	lambda.zero <- apply(classmean, 2, mean)
	Lambda <- as.matrix(svd(scale(classmean, scale = F))$v[, 1:h])
	alpha <- matrix(apply(scale(classmean, scale = F), 1, function(x, 
		Lambda)
	{
		t(Lambda) %*% x
	}
	, Lambda), K, h, byrow = T)
	centerpoints <- t(t(points) - lambda.zero - (Lambda %*% t(alpha[data$
		class,  ])))
	# Initialize Theta (from Section 5.1), gamma and sigma
	# squared. Note the parameter called sigma is actually sigma squared.
	Theta <- as.matrix(svd(centerpoints)$v[, 1:p])
	gamma <- matrix(apply(centerpoints, 1, function(x, Theta)
	{
		t(Theta) %*% x
	}
	, Theta), N, p, byrow = T)
	coefs <- Lambda %*% t(alpha[data$class,  ]) + Theta %*% t(gamma) + 
		lambda.zero
	allcoefs <- coefs[, data$curve]
	sigma <- mean(unlist(tapply(1:length(data$curve), data$curve, function(
		x, S, allcoefs, y)
	{
		Sij <- S[x,  ]
		coefij <- allcoefs[, x[1]]
		yij <- y[x]
		(yij - Sij %*% coefij)^2
	}
	, S, allcoefs, data$y)))
	# Initialize D and gprod. gprod is the expected value of gamma %*% t(gamma).
	D <- apply(gamma^2, 2, mean)
	if(length(D) == 1)
		D <- as.matrix(D)
	gprod <- array(unlist(tapply(1:length(data$curve), data$curve, 
		function(x, S, D, Theta, sigma, gamma, data)
	{
		Sij <- S[x,  ]
		Cgamma <- solve(diag(1/D) + t(Theta) %*% t(Sij) %*% Sij %*% 
			Theta/sigma)
		gammaij <- gamma[data$curve[x[1]],  ]
		gammaij %*% t(gammaij) + Cgamma
	}
	, S, D, Theta, sigma, gamma, data)), c(p, p, N))
	list(S = S, FullS = FullS, parameters = list(lambda.zero = lambda.zero,
		Lambda = Lambda, alpha = alpha, Theta = Theta, sigma = sigma,
		D = D), vars = list(gamma = gamma, gprod = gprod))
}

"Mstep" <- 
function(parameters, data, vars, S, tol)
{
	# This function implements the M step of the EM algorithm. This is
	# performed in two parts using step1 and step2.
	parameters <- step1(parameters, data, vars, S)
	parameters <- step2(parameters, data, vars, S, tol)
	parameters
}

"step1" <- 
function(parameters, data, vars, S)
{
	# This function calculates the variance terms sigma and D.
	lambda.zero <- parameters$lambda.zero
	Lambda <- parameters$Lambda
	alpha <- parameters$alpha
	Theta <- parameters$Theta
	gamma <- vars$gamma
	gprod <- vars$gprod
	sigma <- parameters$sigma
	N <- dim(gamma)[1]
	p <- dim(gamma)[2]
	gammasq <- apply(gprod, 3, diag)
	# Calculate D.
	if(is.matrix(gammasq)) D <- as.vector(apply(gammasq, 1, mean)) else D <-
			as.matrix(mean(gammasq))
	coefs <- Lambda %*% t(alpha[data$class,  ]) + Theta %*% t(gamma) + 
		lambda.zero
	allcoefs <- coefs[, data$curve]
	# Calculate sigma.
	parameters$sigma <- mean(unlist(tapply(1:length(data$curve), data$
		curve, function(x, S, allcoefs, y, D, Theta, sigma)
	{
		Sij <- S[x,  ]
		coefij <- allcoefs[, x[1]]
		yij <- y[x]
		(yij - Sij %*% coefij)^2 + diag(Sij %*% Theta %*% solve(diag(
			1/D) + t(Theta) %*% t(Sij) %*% Sij %*% Theta/sigma) %*%
			t(Theta) %*% t(Sij))
	}
	, S, allcoefs, data$y, D, Theta, sigma)))
	parameters$D <- D
	parameters
}

"step2" <- 
function(parameters, data, vars, S, tol)
{
	# This function calculates lambda.zero, alpha, Lambda, and
	# Theta. It iterates between step2a and step2b until the
	# parameters have converged. Step2a calculates
	# Theta and lambda.zero while step2b calculates Lambda and alpha.
	R.old <- 2
	R.new <- 1
	class <- data$class
	curve <- data$curve
	q <- dim(S)[2]
	while(abs(R.old - R.new)/R.new > tol) {
		parameters <- step2a(parameters, data, vars, S, tol)
		parameters <- step2b(parameters, data, vars, S, tol)
		R.old <- R.new
		par <- parameters
		predy <- (S * t(par$lambda.zero + par$Lambda %*% t(par$alpha[
			class[curve],  ]) + par$Theta %*% t(vars$gamma[curve,
			]))) %*% rep(1., q)
		R.new <- sum((data$y - predy)^2)
	}
	parameters
}

"step2a" <- 
function(parameters, data, vars, S, tol)
{
	# This function calculates Theta and lambda.zero.
	y <- data$y
	alpha <- parameters$alpha
	Lambda <- parameters$Lambda
	Theta <- parameters$Theta
	gamma <- vars$gamma
	gprod <- vars$gprod
	curve <- data$curve
	class <- data$class
	q <- dim(S)[2]
	partial.centy <- y - (S * ((alpha %*% t(Lambda))[class[curve],  ] +
		(gamma %*% t(Theta))[curve,  ])) %*% rep(1, q)
	# Calculate lambda.zero
	lambda.zero <- solve(t(S) %*% S) %*% t(S) %*% partial.centy
	centy <- y - S %*% lambda.zero - (S * ((alpha %*% t(Lambda))[class[
		curve],  ])) %*% rep(1, q)
	R.old <- 2
	R.new <- 1
	p <- dim(gamma)[2.]
	# Calculate Theta. This is done by iterating through calculating each column
	# of Theta holding the other columns fixed.
	while(abs(R.old - R.new)/R.new > tol) {
		for(j in 1.:p) {
			ind <- rep(1., p)
			ind[j] <- 0.
			gammay <- gamma[curve, j] * centy - ((S %*% Theta) *
				t(matrix(gprod[j,  , curve], p, length(curve)))
				) %*% ind
			Sgammaj <- S * sqrt(gprod[j, j, curve])
			Theta[, j] <- solve(t(Sgammaj) %*% Sgammaj) %*% t(
				S) %*% gammay
		}
		R.old <- R.new
		R.new <- sum((centy - ((S %*% Theta) * gamma[curve,  ]) %*% rep(
			1., p))^2.)
	}
	parameters$Theta <- Theta
	parameters$lambda.zero <- as.vector(lambda.zero)
	parameters
}

"step2b" <- 
function(parameters, data, vars, S, tol)
{
	# This function calculates alpha and Lambda.
	y <- data$y
	alpha <- parameters$alpha
	Lambda <- parameters$Lambda
	lambda.zero <- parameters$lambda.zero
	Theta <- parameters$Theta
	gamma <- vars$gamma
	gprod <- vars$gprod
	curve <- data$curve
	class <- data$class
	alpha <- parameters$alpha
	K <- length(table(class))
	q <- dim(S)[2]
	h <- dim(alpha)[2]
	centy <- y - S %*% lambda.zero - (S * ((gamma %*% t(Theta))[curve,
		])) %*% rep(1, q)
	R.old <- 2
	R.new <- 1
	# This loop iteratively calculate alpha and then Lambda and stops
	# when they have converged.
	while(abs(R.old - R.new)/R.new > tol) {
		# Calculate alpha given Lambda.
		for(i in 1.:K) {
			Si.Lam <- S[class[curve] == i,  ] %*% Lambda
			yi <- centy[class[curve] == i]
			alpha[i,  ] <- solve(t(Si.Lam) %*% Si.Lam) %*% t(Si.Lam
				) %*% yi
		}
		R.old2 <- 1
		R.new2 <- 0
		# Calculate Lambda given alpha. This is done by iterating
		# through each column of Lambda holding the other columns fixed.
		while(abs(R.old2 - R.new2)/R.new2 > tol) {
			for(j in 1.:h) {
				ind <- rep(1., h)
				ind[j] <- 0.
				morecenty <- centy - ((S %*% Lambda) * alpha[
					class[curve],  ]) %*% ind
				Salpha <- S * alpha[class[curve], j]
				Lambda[, j] <- solve(t(Salpha) %*% Salpha) %*%
					t(Salpha) %*% morecenty
			}
			R.old2 <- R.new2
			R.new2 <- sum((centy - ((S %*% Lambda) * alpha[class[
				curve],  ]) %*% rep(1., h))^2)
		}
		R.old <- R.new
		R.new <- R.new2
	}
	parameters$alpha <- alpha
	parameters$Lambda <- Lambda
	parameters
}

"Estep" <- 
function(parameters, data, vars, S)
{
	# 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$gamma)[1]
	p <- dim(vars$gamma)[2]
	h <- dim(parameters$alpha)[2]
	# Calculate expected value of gamma.
	gamma <- matrix(unlist(tapply(1:length(data$curve), data$curve, 
		function(x, sigma, D, Theta, S, y, lambda.zero, Lambda, alpha,
		class, curve, h)
	{
		Sij <- S[x,  ]
		A <- Sij %*% Theta
		centy <- y[x] - Sij %*% lambda.zero - Sij %*% t(t(Lambda) *
			alpha[class[curve][x[1]],  ]) %*% rep(1, h)
		solve(sigma * diag(1/D) + t(A) %*% A) %*% t(A) %*% centy
	}
	, par$sigma, par$D, par$Theta, S, data$y, par$lambda.zero, par$Lambda,
		par$alpha, data$class, data$curve, h)), N, p, byrow = T)
	# Calculate expected value of gamma %*% t(gamma).
	gprod <- array(unlist(tapply(1:length(data$curve), data$curve, 
		function(x, S, D, Theta, sigma, gamma, curve)
	{
		Sij <- S[x,  ]
		Cgamma <- solve(diag(1/D) + t(Theta) %*% t(Sij) %*% Sij %*% 
			Theta/sigma)
		gammaij <- gamma[curve[x[1]],  ]
		gammaij %*% t(gammaij) + Cgamma
	}
	, S, par$D, par$Theta, par$sigma, gamma, data$curve)), c(p, p, N))
	vars$gamma <- gamma
	vars$gprod <- gprod
	vars
}

"calcobslike" <- 
function(parameters, data, S)
{
	#This function calculates the observed log likelihood (up to
	#constants) given current parameter estimates. 
	par <- parameters
	h <- dim(par$alpha)[2]
	like <- unlist(tapply(1:length(data$curve), data$curve, function(x,
		S, D, Theta, sigma, y, lambda.zero, alpha, class, curve, Lambda,
		h)
	{
		Sij <- S[x,  ]
		A <- Sij %*% Theta
		centy <- y[x] - Sij %*% lambda.zero - Sij %*% t(t(Lambda) *
			alpha[class[curve][x[1]],  ]) %*% rep(1, h)
		C <- sigma * diag(length(x)) + A %*% diag(D) %*% t(A)
		t(centy) %*% solve(C) %*% centy + sum(logb(eigen(C, symmetric
			 = T)$value))
	}
	, S, par$D, par$Theta, par$sigma, data$y, par$lambda.zero, par$alpha,
		data$class, data$curve, par$Lambda, h))
 - sum(like)/2
}

"constraints" <- 
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
	svdTheta <- svd(par$Theta)
	par$Theta <- svdTheta$u
	par$D <- par$D * svdTheta$d^2
	vars$gamma <- t(t(svdTheta$d * svdTheta$v) %*% t(vars$gamma))
	A <- t(S) %*% solve(par$sigma * diag(dim(S)[1]) + S %*% t(as.vector(
		par$D) * t(par$Theta)) %*% t(S %*% par$Theta)) %*% 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$alpha <- t(t(finalsvd$d * 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)
}

"fldapred" <- 
function(fit, data)
{
	# 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 and the original data (for training
	# error rates) or a new data set (for test predictions).
	par <- fit$parameters
	S <- fit$FullS[data$timeindex,  ]
	N <- length(data$class)
	h <- dim(par$alpha)[2]
	alpha.hat <- matrix(0, N, h)
	curve <- data$curve
	K <- dim(fit$par$alpha)[1]
	distance <- matrix(0, N, K)
	Calpha <- array(0, c(N, h, h))
	for(i in 1:N) {
		Sij <- S[curve == i,  ]
		yij <- data$y[curve == i]
		n <- length(yij)
		Sigma <- par$sigma * diag(n) + Sij %*% par$Theta %*% diag(
			par$D) %*% t(par$Theta) %*% t(Sij)
		# Calculate covariance for each alpha hat.
		InvCalpha <- t(par$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) %*% (yij - Sij %*% par$
			lambda.zero)
		# Calculate the matrix of distances, relative to the
		# appropriate metric of each curve from each class centroid. 
		distance[i,  ] <- apply(t(fit$par$alpha) - as.vector(alpha.hat[
			i,  ]), 2, function(x, InvCalpha)
		{
			t(x) %*% InvCalpha %*% x
		}
		, InvCalpha)
	}
	# Calculate final class predictions for each curve.
	class.pred <- rep(1, N)
	m <- distance[, 1]
	for(i in 2:K) {
		test <- (distance[, i] < m)
		class.pred[test] <- i
		m[test] <- distance[test, i]
	}
	list(Calpha = Calpha, alpha.hat = alpha.hat, class.pred = class.pred,
		distance = distance)
}

"testdata" <- 
list("class" = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3,
	3, 3, 3, 3, 3)
, "curve" = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3,
	3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5,
	5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8,
	8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10,
	10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11,
	12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13,
	13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15,
	15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17,
	17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18,
	18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20,
	20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22,
	22, 22, 22, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 23,
	23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25,
	25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27,
	27, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28,
	29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 30, 30,
	30, 30, 30)
, "Lambda" = matrix(c(0.24730773195061709, 0.51967921625471314, 0.38173517220668235, 
	-0.057218897873575779, -0.72095537582904312)
, nrow = 5, ncol = 1)
, "Theta" = matrix(c(0.61534580440637132, -0.58285320907745175, 1.6471693108723451, 
	-0.76056448836480961, -0.34441897588801434, 0.34604466163651526, 
	-0.082100399319948045
	, -1.062182378520208, 2.2067633452381563, 1.7770909113558511)
, nrow = 5, ncol = 2)
, "alpha" = matrix(c(1, 0, -1)
, nrow = 3, ncol = 1)
, "gamma" = matrix(c(1.0720954942669303, -0.67034488643394652, 2.2704109002018362, 
	0.32690661015543815, -0.73880090849605207, -0.63078230710211636, 
	-0.49323111713439866
	, 0.94615960180325809, -0.4035777883995188, 1.9405554825118283, 
	1.396620543010838, -0.370804162653463, -0.12306712790612784, 
	-0.72874784832688633, -2.0554536783033117, 0.60001896859387305, 
	-0.0047757844320472365, 1.079286319954841, -2.5057776373251377, 
	0.36146189986797495, 0.27441467344868081, -0.91206424637758077, 
	-1.1312094974699498, -0.57347983105777289, -0.12084701754879573, 
	-0.51729004392897715
	, 0.76688177010598613, -0.49688220043238068, 0.92704684096345336, 
	0.51527697844352371, 0.34774172672221942, -0.16437535453002708, 
	1.6525916536696372, 0.36610485379235252, 1.2187092337197449, 
	-0.78382738990284184, 1.1372262319201933, 0.3738434082185193, 
	-0.64962405039274729, 0.77116666873100181, 1.2882572885412773, 
	0.31828261213293196, 1.732230453356389, 0.024229957136957251, 
	0.085853029164094652, 0.31214870664122563, -0.70527635950352519, 
	1.2957451015342591, -0.29208168286150721, 0.96145577514805824, 
	-0.86285718481084739, -1.0411043047135922, -0.91743097616099711, 
	-1.2051053487884111
	, 0.19931335159537142, 0.46569923042786543, -2.4400655660013562, 
	-0.42069585727812747
	, -1.8032915443816269, -1.2648636631264898)
, nrow = 30, ncol = 2)
, "Y" = c(-0.28804195444736247, -0.060479806783910642, 0.090932120501865665, 
	0.10804255324516462, 0.060556986945978622, -0.008447092174873886, 
	-0.025641182397478725
	, -0.059409327285215995, -0.048411091069484616, -0.058277806575697831,
	-1.2175916550830526, -0.1545286380056124, 0.54489266536760816, 
	0.66669443715538268, 0.35794536517232284, -0.14213942831064486, 
	-0.5513919358953665, -0.60730361468145155, -0.1264976247137288, 
	0.74366217480661678, 0.47983595542597385, 0.0947325207102169, 
	-0.18134481662195956, -0.19681345667192401, -0.027174253885533357,
	0.16559534440729887, 0.35190441524225397, 0.30471805877102548, 
	-0.077464105137754175, -0.69999192446646541, -0.54724174630763156,
	0.057809819620243383, 0.45233216058760412, 0.55891438968018148, 
	0.38287605185056894, 0.06425231733667798, -0.25664050936261595, 
	-0.39553030996126315, -0.20411271009299142, 0.22226860445002158, 
	0.59969993343914041, -0.057028996633101764, -0.47771673930489028, 
	-0.56700000661983374
	, -0.33999721163310331, 0.065831738165180176, 0.48919324222542249,
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)