From: Benjamin Auder Date: Wed, 4 Dec 2019 15:07:11 +0000 (+0100) Subject: Prepare code for GLSMM implementation (generalized least squares) X-Git-Url: https://git.auder.net/doc/html/%24%7BgetWhatsApp%28link%29%7D?a=commitdiff_plain;h=4263503b0fbe45a6fa5b353c8405e30557b3dd70;p=morpheus.git Prepare code for GLSMM implementation (generalized least squares) --- diff --git a/pkg/R/optimParams.R b/pkg/R/optimParams.R index f62e75a..948167b 100644 --- a/pkg/R/optimParams.R +++ b/pkg/R/optimParams.R @@ -1,17 +1,9 @@ -#' 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$run(x0)} #' outputs the list of optimized parameters @@ -30,58 +22,57 @@ #' # Optimize parameters from estimated μ #' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit") #' μ = computeMu(io$X, io$Y, list(K=2)) -#' M <- computeMoments(io$X, io$Y) -#' o <- optimParams(2, "logit", list(M=M)) -#' x0 <- c(1/2, as.double(μ), c(0,0)) +#' o <- optimParams(io$X, io$Y, 2, "logit") +#' x0 <- list(p=1/2, β=μ, b=c(0,0)) #' par0 <- o$run(x0) #' # Compare with another starting point -#' x1 <- c(1/2, 2*as.double(μ), c(0,0)) +#' x1 <- list(p=1/2, β=2*μ, b=c(0,0)) #' par1 <- o$run(x1) #' o$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) + li = "character", #link function + X = "matrix", + Y = "numeric", + M1 = "numeric", M2 = "numeric", #M2 easier to process as a vector - M3 = "numeric", #M3 easier to process as a vector + M3 = "numeric", #same for M3 # Dimensions K = "integer", + n = "integer", d = "integer", # Weights matrix (generalized least square) W = "matrix" @@ -90,15 +81,19 @@ setRefClass( 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]]) + + n <<- nrow(X) d <<- length(M1) W <<- diag(d+d^2+d^3) #initialize at W = Identity }, @@ -121,31 +116,48 @@ setRefClass( c(o$p[1:(K-1)], as.double(o$β), o$b) }, - f = function(x) - { + getOmega = function(theta) + { + dim <- d + d^2 + d^3 + matrix( .C("Compute_Omega", + X=as.double(X), Y=as.double(Y), pn=as.integer(n), pd=as.integer(d), + p=as.double(theta$p), β=as.double(theta$β), b=as.double(theta$b), + W=as.double(W), PACKAGE="morpheus")$W, nrow=dim, ncol=dim) + }, + + f = function(theta) + { "Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)" - P <- expArgs(x) - p <- P$p - β <- P$β + p <- theta$p + β <- theta$β λ <- sqrt(colSums(β^2)) - b <- P$b + b <- theta$b # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1 β2 <- apply(β, 2, function(col) col %o% col) β3 <- apply(β, 2, function(col) col %o% col %o% col) - return( - sum( ( β %*% (p * .G(li,1,λ,b)) - M1 )^2 ) + - sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) + - sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) ) - }, + A <- matrix(c( + β %*% (p * .G(li,1,λ,b)) - M1, + β2 %*% (p * .G(li,2,λ,b)) - M2, + β3 %*% (p * .G(li,3,λ,b)) - M3), ncol=1) + t(A) %*% W %*% A + }, grad_f = function(x) { "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1" - P <- expArgs(x) + # TODO: formula -2 t(grad M(theta)) . W . (Mhat - M(theta)) + } + + grad_M = function(theta) + { + # TODO: adapt code below for grad of d+d^2+d^3 vector of moments, + # instead of grad (sum(Mhat-M(theta)^2)) --> should be easier + + P <- expArgs(x) p <- P$p β <- P$β λ <- sqrt(colSums(β^2)) @@ -213,6 +225,7 @@ setRefClass( grad }, + # TODO: rename x(0) into theta(0) --> θ run = function(x0) { "Run optimization from x0 with solver..." @@ -239,6 +252,10 @@ setRefClass( matrix(0, nrow=K, ncol=(d+1)*K) ), ci=c(-1,rep(0,K-1)) ) + # We get a first non-trivial estimation of W: getOmega(theta)^{-1} + # TODO: loop, this redefine f, so that we can call constrOptim again... + # Stopping condition? N iterations? Delta <= ε ? + expArgs(op_res$par) } ) @@ -246,6 +263,7 @@ setRefClass( # 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 @@ -257,56 +275,23 @@ setRefClass( # 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' @@ -314,7 +299,7 @@ setRefClass( .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) diff --git a/pkg/R/utils.R b/pkg/R/utils.R index 6d1c361..6ac9bec 100644 --- a/pkg/R/utils.R +++ b/pkg/R/utils.R @@ -74,6 +74,23 @@ normalize = function(X) computeMoments = function(X, Y) list( colMeans(Y * X), .Moments_M2(X,Y), .Moments_M3(X,Y) ) +# Computes the Omega matrix for generalized least square method +# +# @param X matrix of covariates (of size n*d) +# @param Y vector of responses (of size n) +# @param theta list with p, beta, b +# +# @return Matrix of size dimxdim where dim=d+d^2+d^3 +# +.Moments_M3 = function(X, Y) +{ + n = nrow(X) + d = ncol(X) + M3 = array(0,dim=c(d,d,d)) + array( .C("Moments_M3", X=as.double(X), Y=as.double(Y), pn=as.integer(n), + pd=as.integer(d), M3=as.double(M3), PACKAGE="morpheus")$M3, dim=c(d,d,d) ) +} + # Find the optimal assignment (permutation) between two sets (minimize cost) # # @param distances The distances matrix, in columns (distances[i,j] is distance between i diff --git a/pkg/src/functions.c b/pkg/src/functions.c index 0d632d3..42bb134 100644 --- a/pkg/src/functions.c +++ b/pkg/src/functions.c @@ -53,3 +53,10 @@ void Moments_M3(double* X, double* Y, int* pn, int* pd, double* M3) } } } + +void Compute_Omega(double* X, double* Y, int* pn, int* pd, double* W) +{ + // TODO: formula 1/N sum( t(g(Zi,theta)) g(Zi,theta) ) + // = 1/N sum( t( (XiYi-...) - theta[i] ) ( ... ) ) + // --> similar to Moments_M2 and M3 above +}