tensor_predictors/sim/sim-tsir.R

library(tensorPredictors)
suppressPackageStartupMessages(library(Rdimtools))

# Source utility function used in most simulations (extracted for convenience)
setwd("~/Work/tensorPredictors/sim/")
source("./sim_utils.R")

# Data set sample size in every simulation
sample.size <- 500L
# Nr. of per simulation replications
reps <- 10L
# number of observation/response axes (order of the tensors)
orders <- c(2L, 3L, 4L)
# auto correlation coefficient for the mode-wise auto scatter matrices `Omegas`
rhos <- seq(0, 0.8, by = 0.2)


setwd("~/Work/tensorPredictors/sim/")
base.name <- format(Sys.time(), "sim-tsir-%Y%m%dT%H%M")
# data sampling routine
sample.data <- function(sample.size, betas, Omegas) {
    dimF <- mapply(ncol, betas)

    # responce is a standard normal variable
    y <- sort(rnorm(sample.size))
    y.pow <- Reduce(function(a, b) outer(a, b, `+`), Map(seq, 0L, len = dimF))
    F <- t(outer(y, as.vector(y.pow), `^`)) / as.vector(factorial(y.pow))
    dim(F) <- c(dimF, sample.size)

    matplot(mat(F, length(dim(F))), type = "l")
    abline(h = 0, lty = "dashed")

    matplot(y, scale(mat(F, length(dim(F))), scale = FALSE), type = "l")
    abline(h = 0, lty = "dashed")


    # sample predictors from tensor normal X | Y = y (last axis is sample axis)
    sample.axis <- length(betas) + 1L
    Sigmas <- Map(solve, Omegas)
    mu_y <- mlm(F, Map(`%*%`, Sigmas, betas))
    X <- mu_y + rtensornorm(sample.size, 0, Sigmas, sample.axis)

    # Make `y` into a `Y` tensor with variable axis all of dim 1
    Y <- array(y, dim = c(rep(1L, length(dimF)), sample.size))

    list(X = X, F = F, Y = Y, sample.axis = sample.axis)
}

# Create a CSV logger to write simulation results to
log.file <- paste(base.name, "csv", sep = ".")
logger <- CSV.logger(
    file.name = log.file,
    header = c("rho", "order", "sample.size", "rep", "beta.version", outer(
        "dist.subspace", c("gmlm", "gmlm.1d", "tsir", "sir"),
        paste, sep = "."
    ))
)

for (order in orders) {
    # setup problem dimensions given the tensor order
    dimX <- rep(2L, order)
    dimF <- rep(2L, order)

    # as well as the projections onto rank 1 matrices
    proj.betas <- Map(.projRank, rep(1L, order))

    for (rho in rhos) {
        # Scatter matrices (Omegas) given as autoregression structure `AR(rho)`
        Omegas <- Map(function(p) rho^abs(outer(1:p, 1:p, `-`)), dimX)

        # Version 1 betas (reductions)
        beta.version <- 1L
        betas <- Map(function(nr, nc) {
            `[<-`(matrix(0, nr, nc), 1, , 1)
        }, dimX, dimF)
        B.true <- Reduce(kronecker, rev(betas))
        # `(B.min %*% vec(X)) | Y = y` proportional to `(1 + y)^order`)
        B.min.true <- B.true[, 1L, drop = FALSE]

        # Version 1: repeated simulations
        for (rep in seq_len(reps)) {
            # Sample training data
            c(X, F, Y, sample.axis) %<-% sample.data(sample.size, betas, Omegas)

            # Fit models to provided data
            fit.gmlm  <- gmlm_tensor_normal(X, F, sample.axis = sample.axis, proj.betas = proj.betas)
            fit.gmlm.y <- gmlm_tensor_normal(X, Y, sample.axis = sample.axis)
            fit.tsir  <- TSIR(X, drop(Y), d = rep(1L, order), sample.axis = sample.axis)
            fit.sir   <- do.sir(mat(X, sample.axis), drop(Y), ndim = 1L)

            # Extract minimal reduction matrices from fitted models
            B.gmlm  <- qr.Q(qr(Reduce(kronecker, rev(fit.gmlm$betas))))[, 1L, drop = FALSE]
            B.gmlm.y <- Reduce(kronecker, rev(fit.gmlm.y$betas))
            B.tsir  <- Reduce(kronecker, rev(fit.tsir))
            B.sir   <- fit.sir$projection

            # Compute estimation to true minimal `B` distance
            dist.subspace.gmlm   <- dist.subspace(B.min.true, B.gmlm,   normalize = TRUE)
            dist.subspace.gmlm.y <- dist.subspace(B.min.true, B.gmlm.y, normalize = TRUE)
            dist.subspace.tsir   <- dist.subspace(B.min.true, B.tsir,   normalize = TRUE)
            dist.subspace.sir    <- dist.subspace(B.min.true, B.sir,    normalize = TRUE)

            # Write to simulation log file (CSV file)
            logger()

            # and print progress
            cat(sprintf("order %d, rho %.2f, version %d: rep: %d/%d\n",
                order, rho, beta.version, rep, reps))
        }


        # Version 2 betas (reductions)
        beta.version <- 2L
        betas <- Map(function(nr, nc) {
            tcrossprod((-1)^seq_len(nr), (-1)^seq_len(nc))
        }, dimX, dimF)
        B.true <- Reduce(kronecker, rev(betas))
        # `(B.min %*% vec(X)) | Y = y` proportional to `(1 - y)^order`)
        B.min.true <- B.true[, 1L, drop = FALSE]

        # Version 2: repeated simulations (identical to Version 1)
        for (rep in seq_len(reps)) {
            # Sample training data
            c(X, F, Y, sample.axis) %<-% sample.data(sample.size, betas, Omegas)

            # Fit models to provided data
            fit.gmlm  <- gmlm_tensor_normal(X, F, sample.axis = sample.axis, proj.betas = proj.betas)
            fit.gmlm.y <- gmlm_tensor_normal(X, Y, sample.axis = sample.axis)
            fit.tsir  <- TSIR(X, drop(Y), d = rep(1L, order), sample.axis = sample.axis)
            fit.sir   <- do.sir(mat(X, sample.axis), drop(Y), ndim = 1L)

            # Extract minimal reduction matrices from fitted models
            B.gmlm  <- qr.Q(qr(Reduce(kronecker, rev(fit.gmlm$betas))))[, 1L, drop = FALSE]
            B.gmlm.y <- Reduce(kronecker, rev(fit.gmlm.y$betas))
            B.tsir  <- Reduce(kronecker, rev(fit.tsir))
            B.sir   <- fit.sir$projection

            # Compute estimation to true minimal `B` distance
            dist.subspace.gmlm   <- dist.subspace(B.min.true, B.gmlm,   normalize = TRUE)
            dist.subspace.gmlm.y <- dist.subspace(B.min.true, B.gmlm.y, normalize = TRUE)
            dist.subspace.tsir   <- dist.subspace(B.min.true, B.tsir,   normalize = TRUE)
            dist.subspace.sir    <- dist.subspace(B.min.true, B.sir,    normalize = TRUE)

            # Write to simulation log file (CSV file)
            logger()

            # and print progress
            cat(sprintf("order %d, rho %.2f, version %d: rep: %d/%d\n",
                order, rho, beta.version, rep, reps))
        }
    }
}

### read simulation results and generate plots
if (!interactive()) { pdf(file = paste(base.name, "pdf", sep = ".")) }

# Read simulation results back from log file
sim <- read.csv(log.file)

# Source utility function used in most simulations (extracted for convenience)
source("./sim_utils.R")

# aggregate results
grouping <- sim[c("rho", "order", "beta.version")]
sim.dist <- sim[startsWith(names(sim), "dist")]
aggr <- aggregate(sim.dist, grouping, mean)

# reset (possible lost) config
orders <- sort(unique(aggr$order))
rhos <- sort(unique(aggr$rho))

# new grouping for the aggregates
layout(rbind(
    matrix(seq_len(2 * length(orders)), ncol = 2),
    2 * length(orders) + 1
), heights = c(rep(6L, length(orders)), 1L))

col.methods <- c(
    gmlm   = "#000000",
    gmlm.y = "#FF0000",
    tsir   = "#009E73",
    sir    = "#999999"
)

for (group in split(aggr, aggr[c("order", "beta.version")])) {
    order <- group$order[[1]]
    beta.version <- group$beta.version[[1]]

    rho <- group$rho

    plot(range(rho), 0:1, main = sprintf("V%d, order %d", beta.version, order),
        type = "n", bty = "n", axes = FALSE, xlab = expression(rho), ylab = "Subspace Distance")
    axis(1, at = rho)
    axis(2, at = seq(0, 1, by = 0.2))
    with(group, {
        lines(rho, dist.subspace.gmlm,   col = col.methods["gmlm"],   lwd = 3, type = "b", pch = 16)
        lines(rho, dist.subspace.gmlm.y, col = col.methods["gmlm.y"], lwd = 3, type = "b", pch = 16)
        lines(rho, dist.subspace.tsir,   col = col.methods["tsir"],   lwd = 2, type = "b", pch = 16)
        lines(rho, dist.subspace.sir,    col = col.methods["sir"],    lwd = 2, type = "b", pch = 16)
    })
    if (order == 3L && beta.version == 2L) {
        abline(v = 0.5, lty = "dotted", col = "black")
        abline(h = group$dist.subspace.tsir[which(rho == 0.5)],
            lty = "dotted", col = "black")
    }
}
methods <- c("GMLM", "GMLM.y", "TSIR", "SIR")
restor.par <- par(
    fig = c(0, 1, 0, 1),
    oma = c(0, 0, 0, 0),
    mar = c(1, 0, 0, 0),
    new = TRUE
)
plot(0, type = "n", bty = "n", axes = FALSE, xlab = "", ylab = "")
legend("bottom", col = col.methods[tolower(methods)], legend = methods,
    horiz = TRUE, lty = 1, bty = "n", lwd = c(3, 2, 2), pch = 16)
par(restor.par)



### Version 1
# 2'nd order
# 1 + 2 y + y^2
# (1 + y)^2

# 3'rd order
# 1 + 3 y + 3 y^2 + y^4
# (1 + y)^3

# 4'th order
# 1 + 4 y + 6 y^2 + 4 y^3 + y^4
# (1 + y)^4


### Version 2
# 2'nd order
# 1 - 2 y + y^2
# (1 - y)^2 = 1 - 2 y + y^2

# 3'rd order
# 1 - 3 y + 3 y^2 - y^3
# -(y - 1)^3

# 4'th order
# 1 - 4 y + 6 y^2 - 4 y^3 + y^4
# (y - 1)^4