-#' Optimize parameters
-#'
-#' Optimize the parameters of a mixture of logistic regressions model, possibly using
-#' \code{mu <- computeMu(...)} as a partial starting point.
+#' Wrapper function for OptimParams class
#'
#' @param K Number of populations.
#' @param link The link type, 'logit' or 'probit'.
-#' @param optargs a list with optional arguments:
-#' \itemize{
-#' \item 'M' : list of moments of order 1,2,3: will be computed if not provided.
-#' \item 'X,Y' : input/output, mandatory if moments not given
-#' \item 'exact': use exact formulas when available?
-#' \item weights Weights on moments when minimizing sum of squares
-#' }
+#' @param X Data matrix of covariables
+#' @param Y Output as a binary vector
#'
#' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
#' outputs the list of optimized parameters
#' \item β: regression matrix, size dxK
#' \item b: intercepts, size K
#' }
-#' x0 is a vector containing respectively the K-1 first elements of p, then β by
-#' columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
+#' θ0 is a vector containing respectively the K-1 first elements of p, then β by
+#' columns, and finally b: \code{θ0 = c(p[1:(K-1)],as.double(β),b)}.
#'
#' @seealso \code{multiRun} to estimate statistics based on β, and
#' \code{generateSampleIO} for I/O random generation.
#' # Optimize parameters from estimated μ
#' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
#' μ = computeMu(io$X, io$Y, list(K=2))
-#' M <- computeMoments(io$X, io$Y)
-#' o <- optimParams(2, "logit", list(M=M))
-#' x0 <- c(1/2, as.double(μ), c(0,0))
-#' par0 <- o$run(x0)
+#' o <- optimParams(io$X, io$Y, 2, "logit")
+#' θ0 <- list(p=1/2, β=μ, b=c(0,0))
+#' par0 <- o$run(θ0)
#' # Compare with another starting point
-#' x1 <- c(1/2, 2*as.double(μ), c(0,0))
-#' par1 <- o$run(x1)
+#' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
+#' par1 <- o$run(θ1)
#' o$f( o$linArgs(par0) )
#' o$f( o$linArgs(par1) )
#' @export
-optimParams = function(K, link=c("logit","probit"), optargs=list())
+optimParams = function(X, Y, K, link=c("logit","probit"))
{
# Check arguments
+ if (!is.matrix(X) || any(is.na(X)))
+ stop("X: numeric matrix, no NAs")
+ if (!is.numeric(Y) || any(is.na(Y)) || any(Y!=0 | Y!=1))
+ stop("Y: binary vector with 0 and 1 only")
link <- match.arg(link)
- if (!is.list(optargs))
- stop("optargs: list")
- if (!is.numeric(K) || K < 2)
- stop("K: integer >= 2")
-
- M <- optargs$M
- if (is.null(M))
- {
- if (is.null(optargs$X) || is.null(optargs$Y))
- stop("If moments are not provided, X and Y are required")
- M <- computeMoments(optargs$X,optargs$Y)
- }
+ if (!is.numeric(K) || K!=floor(K) || K < 2)
+ stop("K: integer >= 2")
# Build and return optimization algorithm object
- methods::new("OptimParams", "li"=link, "M1"=as.double(M[[1]]),
- "M2"=as.double(M[[2]]), "M3"=as.double(M[[3]]), "K"=as.integer(K))
+ methods::new("OptimParams", "li"=link, "X"=X,
+ "Y"=as.integer(Y), "K"=as.integer(K))
}
-# Encapsulated optimization for p (proportions), β and b (regression parameters)
-#
-# @field li Link, 'logit' or 'probit'
-# @field M1 Estimated first-order moment
-# @field M2 Estimated second-order moment (flattened)
-# @field M3 Estimated third-order moment (flattened)
-# @field K Number of populations
-# @field d Number of dimensions
-#
+#' Encapsulated optimization for p (proportions), β and b (regression parameters)
+#'
+#' Optimize the parameters of a mixture of logistic regressions model, possibly using
+#' \code{mu <- computeMu(...)} as a partial starting point.
+#'
+#' @field li Link function, 'logit' or 'probit'
+#' @field X Data matrix of covariables
+#' @field Y Output as a binary vector
+#' @field K Number of populations
+#' @field d Number of dimensions
+#' @field W Weights matrix (iteratively refined)
+#'
setRefClass(
Class = "OptimParams",
fields = list(
# Inputs
- li = "character", #link 'logit' or 'probit'
- M1 = "numeric", #order-1 moment (vector size d)
- M2 = "numeric", #M2 easier to process as a vector
- M3 = "numeric", #M3 easier to process as a vector
+ li = "character", #link function
+ X = "matrix",
+ Y = "numeric",
+ Mhat = "numeric", #vector of empirical moments
# Dimensions
K = "integer",
+ n = "integer",
d = "integer",
# Weights matrix (generalized least square)
W = "matrix"
methods = list(
initialize = function(...)
{
- "Check args and initialize K, d"
+ "Check args and initialize K, d, W"
- callSuper(...)
- if (!hasArg("li") || !hasArg("M1") || !hasArg("M2") || !hasArg("M3")
- || !hasArg("K"))
- {
+ callSuper(...)
+ if (!hasArg("X") || !hasArg("Y") || !hasArg("K") || !hasArg("li"))
stop("Missing arguments")
- }
+ # Precompute empirical moments
+ M <- computeMoments(optargs$X,optargs$Y)
+ M1 <- as.double(M[[1]])
+ M2 <- as.double(M[[2]])
+ M3 <- as.double(M[[3]])
+ Mhat <<- matrix(c(M1,M2,M3), ncol=1)
+
+ n <<- nrow(X)
d <<- length(M1)
W <<- diag(d+d^2+d^3) #initialize at W = Identity
},
- expArgs = function(x)
+ expArgs = function(v)
{
- "Expand individual arguments from vector x"
+ "Expand individual arguments from vector v into a list"
list(
# p: dimension K-1, need to be completed
- "p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
- "β" = matrix(x[K:(K+d*K-1)], ncol=K),
- "b" = x[(K+d*K):(K+(d+1)*K-1)])
+ "p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
+ "β" = matrix(v[K:(K+d*K-1)], ncol=K),
+ "b" = v[(K+d*K):(K+(d+1)*K-1)])
},
- linArgs = function(o)
+ linArgs = function(L)
{
- " Linearize vectors+matrices into a vector x"
+ "Linearize vectors+matrices from list L into a vector"
- c(o$p[1:(K-1)], as.double(o$β), o$b)
+ c(L$p[1:(K-1)], as.double(L$β), L$b)
},
- f = function(x)
- {
- "Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"
-
- P <- expArgs(x)
- p <- P$p
- β <- P$β
+ computeW = function(θ)
+ {
+ dim <- d + d^2 + d^3
+ W <<- solve( matrix( .C("Compute_Omega",
+ X=as.double(X), Y=as.double(Y), M=as.double(M(θ)),
+ pn=as.integer(n), pd=as.integer(d),
+ W=as.double(W), PACKAGE="morpheus")$W, nrow=dim, ncol=dim) )
+ NULL #avoid returning W
+ },
+
+ M <- function(θ)
+ {
+ "Vector of moments, of size d+d^2+d^3"
+
+ p <- θ$p
+ β <- θ$β
λ <- sqrt(colSums(β^2))
- b <- P$b
+ b <- θ$b
# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
β2 <- apply(β, 2, function(col) col %o% col)
β3 <- apply(β, 2, function(col) col %o% col %o% col)
- return(
- sum( ( β %*% (p * .G(li,1,λ,b)) - M1 )^2 ) +
- sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) +
- sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) )
- },
+ matrix(c(
+ β %*% (p * .G(li,1,λ,b)),
+ β2 %*% (p * .G(li,2,λ,b)),
+ β3 %*% (p * .G(li,3,λ,b))), ncol=1)
+ },
+
+ f = function(θ)
+ {
+ "Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"
- grad_f = function(x)
+ A <- M(θ) - Mhat
+ t(A) %*% W %*% A
+ },
+
+ grad_f = function(θ)
{
"Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
- P <- expArgs(x)
- p <- P$p
- β <- P$β
+ -2 * t(grad_M(θ)) %*% getW(θ) %*% (Mhat - M(θ))
+ }
+
+ grad_M = function(θ)
+ {
+ "Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
+
+ L <- expArgs(θ)
+ p <- L$p
+ β <- L$β
λ <- sqrt(colSums(β^2))
μ <- sweep(β, 2, λ, '/')
- b <- P$b
+ b <- L$b
+
+ res <- matrix(nrow=nrow(W), ncol=0)
# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
β2 <- apply(β, 2, function(col) col %o% col)
G4 = .G(li,4,λ,b)
G5 = .G(li,5,λ,b)
- # (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
- F1 = as.double( 2 * ( β %*% (p * G1) - M1 ) )
- F2 = as.double( 2 * ( β2 %*% (p * G2) - M2 ) )
- F3 = as.double( 2 * ( β3 %*% (p * G3) - M3 ) )
-
+ # Gradient on p: K-1 columns, dim rows
km1 = 1:(K-1)
- grad <- #gradient on p
- t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
- t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
- t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ) %*% F3
+ res <- cbind(res, rbind(
+ t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ),
+ t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ),
+ t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] )))
- grad_β <- matrix(nrow=d, ncol=K)
+ # TODO: understand derivatives order and match the one in optim init param
for (i in 1:d)
{
# i determines the derivated matrix dβ[2,3]
dβ3_right[block,] <- dβ3_right[block,] + β2
dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
- grad_β[i,] <- t(dβ) %*% F1 + t(dβ2) %*% F2 + t(dβ3) %*% F3
+ res <- cbind(res, rbind(t(dβ), t(dβ2), t(dβ3)))
}
- grad <- c(grad, as.double(grad_β))
- grad = c(grad, #gradient on b
- t( sweep(β, 2, p * G2, '*') ) %*% F1 +
- t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
- t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
+ # Gradient on b
+ res <- cbind(res, rbind(
+ t( sweep(β, 2, p * G2, '*') ),
+ t( sweep(β2, 2, p * G3, '*') ),
+ t( sweep(β3, 2, p * G4, '*') )))
- grad
+ res
},
- run = function(x0)
+ run = function(θ0)
{
- "Run optimization from x0 with solver..."
-
- if (!is.list(x0))
- stop("x0: list")
- if (is.null(x0$β))
- stop("At least x0$β must be provided")
- if (!is.matrix(x0$β) || any(is.na(x0$β)) || ncol(x0$β) != K)
- stop("x0$β: matrix, no NA, ncol == K")
- if (is.null(x0$p))
- x0$p = rep(1/K, K-1)
- else if (length(x0$p) != K-1 || sum(x0$p) > 1)
- stop("x0$p should contain positive integers and sum to < 1")
+ "Run optimization from θ0 with solver..."
+
+ if (!is.list(θ0))
+ stop("θ0: list")
+ if (is.null(θ0$β))
+ stop("At least θ0$β must be provided")
+ if (!is.matrix(θ0$β) || any(is.na(θ0$β)) || ncol(θ0$β) != K)
+ stop("θ0$β: matrix, no NA, ncol == K")
+ if (is.null(θ0$p))
+ θ0$p = rep(1/K, K-1)
+ else if (length(θ0$p) != K-1 || sum(θ0$p) > 1)
+ stop("θ0$p should contain positive integers and sum to < 1")
# Next test = heuristic to detect missing b (when matrix is called "beta")
- if (is.null(x0$b) || all(x0$b == x0$β))
- x0$b = rep(0, K)
- else if (any(is.na(x0$b)))
- stop("x0$b cannot have missing values")
+ if (is.null(θ0$b) || all(θ0$b == θ0$β))
+ θ0$b = rep(0, K)
+ else if (any(is.na(θ0$b)))
+ stop("θ0$b cannot have missing values")
- op_res = constrOptim( linArgs(x0), .self$f, .self$grad_f,
+ op_res = constrOptim( linArgs(θ0), .self$f, .self$grad_f,
ui=cbind(
rbind( rep(-1,K-1), diag(K-1) ),
matrix(0, nrow=K, ncol=(d+1)*K) ),
ci=c(-1,rep(0,K-1)) )
+ # debug:
+ print(computeW(expArgs(op_res$par)))
+ # We get a first non-trivial estimation of W
+ # TODO: loop, this redefine f, so that we can call constrOptim again...
+ # Stopping condition? N iterations? Delta <= ε ?
+
expArgs(op_res$par)
}
)
# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
# = E[g^{(order)}(z)] with z~N(b,diag(λ))
+# by numerically evaluating the integral.
#
# @param link Link, 'logit' or 'probit'
# @param order Order of derivative
# NOTE: weird "integral divergent" error on inputs:
# link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
# Switch to pracma package for that (but it seems slow...)
-
- exactComp <- FALSE #TODO: global, or argument...
-
- if (exactComp && link == "probit")
- {
- # Use exact computations
- sapply( seq_along(λ), function(k) {
- .exactProbitIntegral(order, λ[k], b[k])
- })
- }
-
- else
- {
- # Numerical integration
- sapply( seq_along(λ), function(k) {
- res <- NULL
- tryCatch({
- # Fast code, may fail:
- res <- stats::integrate(
- function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
- lower=-Inf, upper=Inf )$value
- }, error = function(e) {
- # Robust slow code, no fails observed:
- sink("/dev/null") #pracma package has some useless printed outputs...
- res <- pracma::integral(
- function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
- xmin=-Inf, xmax=Inf, method="Kronrod")
- sink()
- })
- res
- })
- }
-}
-
-# TODO: check these computations (wrong atm)
-.exactProbitIntegral <- function(order, λ, b)
-{
- c1 = (1/sqrt(2*pi)) * exp( -.5 * b/((λ^2+1)^2) )
- if (order == 1)
- return (c1)
- c2 = b - λ^2 / (λ^2+1)
- if (order == 2)
- return (c1 * c2)
- if (order == 3)
- return (c1 * (λ^2 - 1 + c2^2))
- if (order == 4)
- return ( (c1*c2/((λ^2+1)^2)) * (-λ^4*((b+1)^2+1) -
- 2*λ^3 + λ^2*(2-2*b*(b-1)) + 6*λ + 3 - b^2) )
- if (order == 5) #only remaining case...
- return ( c1 * (3*λ^4+c2^4+6*c1^2*(λ^2-1) - 6*λ^2 + 6) )
+ sapply( seq_along(λ), function(k) {
+ res <- NULL
+ tryCatch({
+ # Fast code, may fail:
+ res <- stats::integrate(
+ function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
+ lower=-Inf, upper=Inf )$value
+ }, error = function(e) {
+ # Robust slow code, no fails observed:
+ sink("/dev/null") #pracma package has some useless printed outputs...
+ res <- pracma::integral(
+ function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
+ xmin=-Inf, xmax=Inf, method="Kronrod")
+ sink()
+ })
+ res
+ })
}
# Derivatives list: g^(k)(x) for links 'logit' and 'probit'
.deriv <- list(
"probit"=list(
# 'probit' derivatives list;
- # TODO: exact values for the integral E[g^(k)(λz+b)]
+ # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
function(x) exp(-x^2/2)/(sqrt(2*pi)) * -x, #g''
function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1), #g^(3)
projects / morpheus.git / blobdiff