implemented kpir_approx,

wip: kpir_sim

simulations/kpir_sim.R

@@ -20,9 +20,8 @@ sim <- function(X, Fy, shape, alpha.true, beta.true, max.iter = 500L) {
             )
 
            cat(sprintf(
-                "%3d | l = %-12.4f - dist = %-.4e - alpha(%d, %d) = %-.4e - beta(%d, %d) = %-.4e\n",
+                "%s(%3d) | l = %-12.4f - dist = %-.4e - alpha(%d, %d) = %-.4e - beta(%d, %d) = %-.4e\n",
-                iter, loss,
+                name, iter, loss, dist,
-                dist,
                 nrow(alpha), ncol(alpha), dist.alpha,
                 nrow(beta),  ncol(beta),  dist.beta
             ))
@@ -30,12 +29,11 @@ sim <- function(X, Fy, shape, alpha.true, beta.true, max.iter = 500L) {
     }
 
     # Initialize logger history targets
-    hist.base <- hist.new <- hist.momentum <- # hist.kron <-
+    hist.base <- hist.new <- hist.momentum <- hist.approx <- # hist.kron <-
         data.frame(iter = seq(0L, max.iter),
             loss = NA, dist = NA, dist.alpha = NA, dist.beta = NA,
             norm.alpha = NA, norm.beta = NA
         )
-    hist.kron <- NULL   # TODO: fit kron version
 
     # Base (old)
     kpir.base(X, Fy, shape, max.iter = max.iter, logger = logger("base"))
@@ -49,14 +47,56 @@ sim <- function(X, Fy, shape, alpha.true, beta.true, max.iter = 500L) {
     # # Residual Covariance Kronecker product assumpton version
     # kpir.kron(X, Fy, shape, max.iter = max.iter, logger = logger("kron"))
 
+    # Approximated MLE with Nesterov Momentum
+    kpir.approx(X, Fy, shape, max.iter = max.iter, logger = logger("approx"))
+
     # Add method tags
-    hist.base$type     <- factor("base")
+    hist.base$method     <- factor("base")
-    hist.new$type      <- factor("new")
+    hist.new$method      <- factor("new")
-    hist.momentum$type <- factor("momentum")
+    hist.momentum$method <- factor("momentum")
-    # hist.kron$type     <- factor("kron")
+    # hist.kron$method     <- factor("kron")
+    hist.approx$method   <- factor("approx")
 
     # Combine results and return
-    rbind(hist.base, hist.new, hist.momentum, hist.kron)
+    rbind(hist.base, hist.new, hist.momentum, hist.approx) #, hist.kron
+}
+
+## Plot helper functions
+plot.hist <- function(hist, response, ...) {
+    ggplot(hist, aes(x = iter, color = method, group = interaction(method, repetition))) +
+        geom_line(aes_(y = as.name(response)), na.rm = TRUE) +
+        geom_point(data = with(sub <- subset(hist, !is.na(as.symbol(response))),
+            aggregate(sub, list(method, repetition), tail, 1)
+        ), aes_(y = as.name(response))) +
+        labs(...) +
+        theme(legend.position = "bottom")
+}
+plot.stats <- function(hist, response, ..., title = "Stats") {
+    ggplot(hist, aes_(x = quote(iter), y = as.name(response),
+                      color = quote(method), group = quote(method))) +
+        geom_ribbon(aes(color = NULL, fill = method), alpha = 0.2,
+                    stat = "summary", fun.min = "min", fun.max = "max", na.rm = TRUE) +
+        geom_ribbon(aes(color = NULL, fill = method), alpha = 0.4,
+                    stat = "summary", na.rm = TRUE,
+                    fun.min = function(y) quantile(y, 0.25),
+                    fun.max = function(y) quantile(y, 0.75)) +
+        geom_line(stat = "summary", fun = "mean", na.rm = TRUE) +
+        labs(title = title, ...) +
+        theme(legend.position = "bottom")
+}
+plot.mean <- function(hist, response, ..., title = "Mean") {
+    ggplot(hist, aes_(x = quote(iter), y = as.name(response),
+                      color = quote(method), group = quote(method))) +
+        geom_line(stat = "summary", fun = "mean", na.rm = TRUE) +
+        labs(title = title, ...) +
+        theme(legend.position = "bottom")
+}
+plot.median <- function(hist, response, ..., title = "Median") {
+    ggplot(hist, aes_(x = quote(iter), y = as.name(response),
+                      color = quote(method), group = quote(method))) +
+        geom_line(stat = "summary", fun = "median", na.rm = TRUE) +
+        labs(title = title, ...) +
+        theme(legend.position = "bottom")
 }
 
 ## Generate some test data / DEBUG
@@ -68,7 +108,10 @@ r <- sample(1:15, 1) # 5
 print(c(n, p, q, k, r))
 
 hist <- NULL
-for (rep in 1:20) {
+reps <- 20
+for (rep in 1:reps) {
+    cat(sprintf("%4d / %d simulation rep. started\n", rep, reps))
+
     alpha.true <- alpha <- matrix(rnorm(q * r), q, r)
     beta.true <- beta <- matrix(rnorm(p * k), p, k)
     y <- rnorm(n)
@@ -87,122 +130,481 @@ for (rep in 1:20) {
     hist <- rbind(hist, hist.sim)
 }
 
+# Save simulation results
+datetime <- format(Sys.time(), "%Y%m%dT%H%M")
+saveRDS(hist, file = sprintf("AR_%s.rds", datetime))
 
-saveRDS(hist, file = "AR.rds")
+# for GGPlot2, as factors for grouping
 hist$repetition <- factor(hist$repetition)
 
-ggplot(hist, aes(x = iter, color = type, group = interaction(type, repetition))) +
+plot.hist(hist, "loss")
-    geom_line(aes(y = loss)) +
+dev.print(png, file = sprintf("sim01_loss_%s.png", datetime), width = 768, height = 768, res = 125)
-    geom_point(data = with(sub <- subset(hist, !is.na(loss)),
+plot.stats(hist, "loss")
-        aggregate(sub, list(type, repetition), tail, 1)
+dev.print(png, file = sprintf("sim01_loss_stats_%s.png", datetime), width = 768, height = 768, res = 125)
-    ), aes(y = loss)) +
-    labs(
-        title = bquote(paste("Optimization Objective: negative log-likelihood ",
-            l(hat(alpha), hat(beta)))),
-        subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
-            "20 repetitions, ", n == .(n), ", ",
-            p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
-        x = "nr. of iterations",
-        y = expression(l(hat(alpha), hat(beta))),
-        color = "method"
-    ) +
-    theme(legend.position = "bottom")
 
-dev.print(png, file = "sim01_loss.png", width = 768, height = 768, res = 125)
+plot.hist(hist, "dist")
+dev.print(png, file = sprintf("sim01_dist_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "dist")
+dev.print(png, file = sprintf("sim01_dist_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist(hist, "dist.alpha")
+dev.print(png, file = sprintf("sim01_dist_alpha_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "dist.alpha")
+dev.print(png, file = sprintf("sim01_dist_alpha_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist(hist, "dist.beta")
+dev.print(png, file = sprintf("sim01_dist_beta_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "dist.beta")
+dev.print(png, file = sprintf("sim01_dist_beta_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist(hist, "norm.alpha")
+dev.print(png, file = sprintf("sim01_norm_alpha_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "norm.alpha")
+dev.print(png, file = sprintf("sim01_norm_alpha_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist(hist, "norm.beta")
+dev.print(png, file = sprintf("sim01_norm_beta_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "norm.beta")
+dev.print(png, file = sprintf("sim01_norm_beta_stats_%s.png", datetime), width = 768, height = 768, res = 125)
 
 
-ggplot(hist, aes(x = iter, color = type, group = interaction(type, repetition))) +
-    geom_line(aes(y = dist)) +
-    geom_point(data = with(sub <- subset(hist, !is.na(dist)),
-        aggregate(sub, list(type, repetition), tail, 1)
-    ), aes(y = dist)) +
-    labs(
-        title = bquote(paste("Distance of estimate ", hat(B), " to true ", B == alpha %*% beta)),
-        subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
-            "20 repetitions, ", n == .(n), ", ",
-            p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
-        x = "nr. of iterations",
-        y = expression(abs(B * B^T - hat(B) * hat(B)^T)),
-        color = "method"
-    ) +
-    theme(legend.position = "bottom")
-
-dev.print(png, file = "sim01_dist.png", width = 768, height = 768, res = 125)
 
 
-ggplot(hist, aes(x = iter, color = type, group = interaction(type, repetition))) +
-    geom_line(aes(y = dist.alpha)) +
-    geom_point(data = with(sub <- subset(hist, !is.na(dist.alpha)),
-        aggregate(sub, list(type, repetition), tail, 1)
-    ), aes(y = dist.alpha)) +
-    labs(
-        title = bquote(paste("Distance of estimate ", hat(alpha), " to true ", alpha)),
-        subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
-            "20 repetitions, ", n == .(n), ", ",
-            p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
-        x = "nr. of iterations",
-        y = expression(abs(alpha * alpha^T - hat(alpha) * hat(alpha)^T)),
-        color = "method"
-    ) +
-    theme(legend.position = "bottom")
 
-dev.print(png, file = "sim01_dist_alpha.png", width = 768, height = 768, res = 125)
+n <- 200    # Sample Size
+p <- 11 # sample(1:15, 1)
+q <- 3  # sample(1:15, 1)
+k <- 7  # sample(1:15, 1)
+r <- 5  # sample(1:15, 1)
+print(c(n, p, q, k, r))
+
+hist <- NULL
+reps <- 20
+
+Delta.1 <- sqrt(0.5)^abs(outer(seq_len(q), seq_len(q), `-`))
+Delta.2 <- sqrt(0.5)^abs(outer(seq_len(p), seq_len(p), `-`))
+Delta <- kronecker(Delta.1, Delta.2)
+for (rep in 1:reps) {
+    cat(sprintf("%4d / %d simulation rep. started\n", rep, reps))
+
+    alpha.true <- alpha <- matrix(rnorm(q * r), q, r)
+    beta.true <- beta <- matrix(rnorm(p * k), p, k)
+    y <- rnorm(n)
+    Fy <- do.call(cbind, Map(function(slope, offset) {
+            sin(slope * y + offset)
+        },
+        head(rep(seq(1, ceiling(0.5 * k * r)), each = 2), k * r),
+        head(rep(c(0, pi / 2), ceiling(0.5 * k * r)), k * r)
+    ))
+    X <- tcrossprod(Fy, kronecker(alpha, beta)) + CVarE:::rmvnorm(n, sigma = Delta)
+
+    hist.sim <- sim(X, Fy, shape = c(p, q, k, r), alpha.true, beta.true)
+    hist.sim$repetition <- rep
+
+    hist <- rbind(hist, hist.sim)
+}
+
+
+# Save simulation results
+datetime <- format(Sys.time(), "%Y%m%dT%H%M")
+saveRDS(hist, file = sprintf("sim02_%s.rds", datetime))
+
+# for GGPlot2, as factors for grouping
+hist$repetition <- factor(hist$repetition)
+
+plot.hist(hist, "loss")
+dev.print(png, file = sprintf("sim02_loss_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "loss")
+dev.print(png, file = sprintf("sim02_loss_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist(hist, "dist")
+dev.print(png, file = sprintf("sim02_dist_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "dist")
+dev.print(png, file = sprintf("sim02_dist_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.mean(hist, "dist")
+plot.median(hist, "dist")
+
+plot.hist(hist, "dist.alpha")
+dev.print(png, file = sprintf("sim02_dist_alpha_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "dist.alpha")
+dev.print(png, file = sprintf("sim02_dist_alpha_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.mean(hist, "dist.alpha")
+plot.median(hist, "dist.alpha")
+
+plot.hist(hist, "dist.beta")
+dev.print(png, file = sprintf("sim02_dist_beta_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "dist.beta")
+dev.print(png, file = sprintf("sim02_dist_beta_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.mean(hist, "dist.beta")
+plot.median(hist, "dist.beta")
+
+plot.hist(hist, "norm.alpha")
+dev.print(png, file = sprintf("sim02_norm_alpha_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "norm.alpha")
+dev.print(png, file = sprintf("sim02_norm_alpha_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist(hist, "norm.beta")
+dev.print(png, file = sprintf("sim02_norm_beta_%s.png", datetime), width = 768, height = 768, res = 125)
+plot.stats(hist, "norm.beta")
+dev.print(png, file = sprintf("sim02_norm_beta_stats_%s.png", datetime), width = 768, height = 768, res = 125)
+
+plot.hist2 <- function(hist, response, type = "all", ...) {
+    # Extract final results from history
+    sub <- na.omit(hist[c("iter", response, "method", "repetition")])
+    sub <- aggregate(sub, list(sub$method, sub$repetition), tail, 1)
+
+    # Setup ggplot
+    p <- ggplot(hist, aes_(x = quote(iter),
+                           y = as.name(response),
+                           color = quote(method),
+                           group = quote(interaction(method, repetition))))
+    # Add requested layers
+    if (type == "all") {
+        p <- p + geom_line(na.rm = TRUE)
+        p <- p + geom_point(data = sub)
+    } else if (type == "mean") {
+        p <- p + geom_line(alpha = 0.5, na.rm = TRUE, linetype = "dotted")
+        p <- p + geom_point(data = sub, alpha = 0.5)
+        p <- p + geom_line(aes(group = method),
+                           stat = "summary", fun = "mean", na.rm = TRUE)
+    } else if (type == "median") {
+        p <- p + geom_line(alpha = 0.5, na.rm = TRUE, linetype = "dotted")
+        p <- p + geom_point(data = sub, alpha = 0.5)
+        p <- p + geom_line(aes(group = method),
+                           stat = "summary", fun = "median", na.rm = TRUE)
+    }
+    # return with theme and annotations 
+    p + labs(...) + theme(legend.position = "bottom")
+}
+
+plot.hist2(hist, "dist.alpha", "all", title = "all") + coord_trans(x = "log1p")
+plot.hist2(hist, "dist.alpha", "mean", title = "mean") + coord_trans(x = "log1p")
+plot.hist2(hist, "dist.alpha", "median", title = "median") + coord_trans(x = "log1p")
+
+
+################################################################################
+###                                   EEG                                    ###
+################################################################################
+
+suppressPackageStartupMessages({
+    library(pROC)
+})
+
+# acc: Accuracy. P(Yhat = Y). Estimated as: (TP+TN)/(P+N).
+acc <- function(y_true, y_pred) mean(round(y_pred) == y_true)
+# err: Error rate. P(Yhat != Y). Estimated as: (FP+FN)/(P+N).
+err <- function(y_true, y_pred) mean(round(y_pred) != y_true)
+# fpr: False positive rate. P(Yhat = + | Y = -). aliases: Fallout.
+fpr <- function(y_true, y_pred) mean((round(y_pred) == 1)[y_true == 0])
+# tpr: True positive rate.  P(Yhat = + | Y = +). aliases: Sensitivity, Recall.
+tpr <- function(y_true, y_pred) mean((round(y_pred) == 1)[y_true == 1])
+# fnr: False negative rate. P(Yhat = - | Y = +). aliases: Miss.
+fnr <- function(y_true, y_pred) mean((round(y_pred) == 0)[y_true == 1])
+# tnr: True negative rate.  P(Yhat = - | Y = -).
+tnr <- function(y_true, y_pred) mean((round(y_pred) == 0)[y_true == 0])
+
+# Load EEG dataset
+dataset <- readRDS('eeg_analysis/eeg_data.rds')
+
+eeg_cross_validation <- function(nrFolds = 10L) {
+    # Set dimenional parameters.
+    n <- nrow(dataset)  # sample size (nr. of people)
+    p <- 64L            # nr. of predictors (count of sensorce)
+    t <- 256L           # nr. of time points (measurements)
+
+    # Extract dimension names from X.
+    nNames <- dataset$PersonID
+    tNames <- as.character(seq(t))
+    pNames <- unlist(strsplit(colnames(dataset)[2 + t * seq(p)], '_'))[c(T, F)]
+
+    # Split into X-y.
+    X <- as.matrix(dataset[, -(1:2)])
+    y <- dataset$Case_Control
+    # Reshape X as 3D tenros of shape (n, t, p) aka. samples, timesteps, predictors.
+    # (Each of the n rows in X iterate over the time bevore switching sensorce.)
+    dim(X) <- c(n, t, p)
+    dimnames(X) <- list(nNames, tNames, pNames)
+
+    # Setup Cross-Validation result
+    CV <- data.frame(
+        fold = (seq_len(n) %% nrFolds) + 1L,
+        y_true = y,
+        y_pred = NA
+    )
+
+    # 
+
+}
+
+#' @param ppc Number of "p"redictor "p"rincipal "c"omponents.
+#' @param tpc Number of "t"ime "p"rincipal "c"omponents.
+egg_analysis_reduced <- function(methods, ppc, tpc) {
+    # Set dimenional parameters.
+    n <- nrow(dataset)  # sample size (nr. of people)
+    p <- 64L            # nr. of predictors (count of sensorce)
+    t <- 256L           # nr. of time points (measurements)
+
+    # Extract dimension names from X.
+    nNames <- dataset$PersonID
+    tNames <- as.character(seq(t))
+    pNames <- unlist(strsplit(colnames(dataset)[2 + t * seq(p)], '_'))[c(T, F)]
+
+    # Split into X-y.
+    X <- as.matrix(dataset[, -(1:2)])
+    y <- dataset$Case_Control
+    # Reshape X as 3D tenros of shape (n, t, p) aka. samples, timesteps, predictors.
+    # (Each of the n rows in X iterate over the time bevore switching sensorce.)
+    X <- array(X, dim = c(n, t, p),
+                dimnames = list(nNames, tNames, pNames))
+    # Reorder axis to (p, t, n) = (predictors, timesteps, samples).
+    X <- aperm(X, c(3, 2, 1))
+
+    # Compute Mean of X.
+    X_mean <- apply(X, c(1, 2), mean)
+    X_center <- X - as.vector(X_mean)
+
+    # Compute "left" and "right" cov-matrices.
+    Sigma_t <- matrix(apply(apply(X_center, 3,  crossprod), 1, mean), t, t)
+    Sigma_p <- matrix(apply(apply(X_center, 3, tcrossprod), 1, mean), p, p)
+    # Get "left", "right" principal components.
+    V_p <- svd(Sigma_p, ppc, 0L)$u
+    V_t <- svd(Sigma_t, tpc, 0L)$u
+
+    # Reduce dimension.
+    X_reduced <- apply(X_center, 3, function(x) crossprod(V_p, x %*% V_t))
+    dim(X_reduced) <- c(ppc, tpc, n)
+
+    # Vectorize to shape of (predictors * timesteps, samples) and transpose to
+    # (samples, predictors * timesteps).
+    X_vec <- t(matrix(X_reduced, ppc * tpc, n))
+
+    loo.cv <- expand.grid(method = names(methods), fold = 1:n)
+    loo.cv$y_true <- y[loo.cv$fold]
+    loo.cv$y_pred <- NA
+
+    # Performe LOO cross-validation for each method.
+    for (i in 1L:n) {
+        # Print progress.
+        cat(sprintf("\rCross-Validation (p-PC: %d, t-PC: %d): %4d/%d",
+                    ppc, tpc, i, n))
+        # Leave Out the i-th element.
+        X_train <- X_vec[-i, ]
+        X_test <- X_vec[i, ]
+        y_train <- y[-i]
+        # Center y.
+        y_train <- scale(y_train, center = TRUE, scale = FALSE)
+
+        # For each method.
+        for (method.name in names(methods)) {
+            method <- methods[[method.name]]
+            # Compute reduction using current method under common API.
+            sdr <- method(X_train, y_train, ppc, tpc)
+            B <- kronecker(sdr$alpha, sdr$beta)
+            # Fit a linear model (which ensures a common sdr direction if possible).
+            model <- glm(y ~ x, family = binomial(link = "logit"),
+                        data = data.frame(y = y[-i], x = X_train %*% B))
+            # Predict out of sample and store in LOO CV data.frame.
+            y_pred <- predict(model, data.frame(x = X_test %*% B), type = "response")
+            loo.cv[loo.cv$method == method.name & loo.cv$fold == i, 'y_pred'] <- y_pred
+        }
+    }
+
+    for (method.name in names(methods)) {
+        labels <- loo.cv[loo.cv$method == method.name, 'y_true']
+        predictions <- loo.cv[loo.cv$method == method.name, 'y_pred']
+        ROC <- roc(unlist(labels), unlist(predictions), quiet = TRUE)
+        # Combined accuracy, error, ...
+        cat("\nMethod: ", method.name, "\n",
+            "acc: ", acc(unlist(labels), unlist(predictions)), "\n",
+            "err: ", err(unlist(labels), unlist(predictions)), "\n",
+            "fpr: ", fpr(unlist(labels), unlist(predictions)), "\n",
+            "tpr: ", tpr(unlist(labels), unlist(predictions)), "\n",
+            "fnr: ", fnr(unlist(labels), unlist(predictions)), "\n",
+            "tnr: ", tnr(unlist(labels), unlist(predictions)), "\n",
+            "auc: ", ROC$auc, "\n",
+            "auc sd: ", sqrt(var(ROC)), "\n",
+            sep = '')
+    }
+
+    loo.cv
+}
+
+methods <- list(
+    KPIR_LS  = function(...) tensor_predictor(..., method = "KPIR_LS"),
+    KPIR_MLE = function(...) tensor_predictor(..., method = "KPIR_MLE"),
+    KPFC1    = function(...) tensor_predictor(..., method = "KPFC1"),
+    KPFC2    = function(...) tensor_predictor(..., method = "KPFC2"),
+    LSIR     = LSIR
+)
+
+#                 ppc, tpc
+#              ------------
+params <- list( c(  4,   3)
+              , c( 15,  15)
+              , c( 30,  20)
+)
+
+for (param in params) {
+    c(ppc, tpc) %<-% param
+    sim <- egg_analysis_reduced(methods, ppc, tpc)
+
+    attr(sim, 'param') <- c(ppc = ppc, tpc = tpc)
+
+    saveRDS(sim, file = sprintf('eeg_analysis_reduced_%d_%d.rds', ppc, tpc))
+}
+
+
+
+# plot.hist(hist, "loss",
+#     title = bquote(paste("Optimization Objective: negative log-likelihood ",
+#         l(hat(alpha), hat(beta)))),
+#     subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#         "20 repetitions, ", n == .(n), ", ",
+#         p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#     x = "nr. of iterations",
+#     y = expression(l(hat(alpha), hat(beta)))
+# )
+# plot.stats(hist, "loss",
+#     title = bquote(paste("Optimization Objective: negative log-likelihood ",
+#         l(hat(alpha), hat(beta)))),
+#     subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#         "20 repetitions, ", n == .(n), ", ",
+#         p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#     x = "nr. of iterations",
+#     y = expression(l(hat(alpha), hat(beta)))
+# )
+
+
+# dev.print(png, file = sprintf("sim01_loss_stat_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
+
+
+# ggplot(hist, aes(x = iter, color = method, group = interaction(method, repetition))) +
+#     geom_line(aes(y = dist)) +
+#     geom_point(data = with(sub <- subset(hist, !is.na(dist)),
+#         aggregate(sub, list(method, repetition), tail, 1)
+#     ), aes(y = dist)) +
+#     labs(
+#         title = bquote(paste("Distance of estimate ", hat(B), " to true ", B == alpha %*% beta)),
+#         subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#             "20 repetitions, ", n == .(n), ", ",
+#             p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#         x = "nr. of iterations",
+#         y = expression(abs(B * B^T - hat(B) * hat(B)^T)),
+#         color = "method"
+#     ) +
+#     theme(legend.position = "bottom")
+
+# dev.print(png, file = sprintf("sim01_dist_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
+
+# ggplot(hist, aes(x = iter, y = dist, color = method, group = method)) +
+#     geom_ribbon(aes(color = NULL, fill = method), alpha = 0.2,
+#                 stat = "summary", fun.min = "min", fun.max = "max", na.rm = TRUE) +
+#     geom_ribbon(aes(color = NULL, fill = method), alpha = 0.4,
+#                 stat = "summary", fun.min = function(y) quantile(y, 0.25),
+#                                   fun.max = function(y) quantile(y, 0.75), na.rm = TRUE) +
+#     geom_line(stat = "summary", fun = "mean", na.rm = TRUE) +
+#     labs(
+#         title = bquote(paste("Distance of estimate ", hat(B), " to true ", B == alpha %*% beta)),
+#         subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#             "20 repetitions, ", n == .(n), ", ",
+#             p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#         x = "nr. of iterations",
+#         y = expression(abs(B * B^T - hat(B) * hat(B)^T)),
+#         color = "method"
+#     ) +
+#     theme(legend.position = "bottom")
+
+# dev.print(png, file = sprintf("sim01_dist_stat_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
+
+
+# ggplot(hist, aes(x = iter, color = method, group = interaction(method, repetition))) +
+#     geom_line(aes(y = dist.alpha)) +
+#     geom_point(data = with(sub <- subset(hist, !is.na(dist.alpha)),
+#         aggregate(sub, list(method, repetition), tail, 1)
+#     ), aes(y = dist.alpha)) +
+#     labs(
+#         title = bquote(paste("Distance of estimate ", hat(alpha), " to true ", alpha)),
+#         subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#             "20 repetitions, ", n == .(n), ", ",
+#             p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#         x = "nr. of iterations",
+#         y = expression(abs(alpha * alpha^T - hat(alpha) * hat(alpha)^T)),
+#         color = "method"
+#     ) +
+#     theme(legend.position = "bottom")
+
+# dev.print(png, file = sprintf("sim01_dist_alpha_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
+
+
+# ggplot(hist, aes(x = iter, color = method, group = interaction(method, repetition))) +
+#     geom_line(aes(y = dist.beta)) +
+#     geom_point(data = with(sub <- subset(hist, !is.na(dist.beta)),
+#         aggregate(sub, list(method, repetition), tail, 1)
+#     ), aes(y = dist.beta)) +
+#     labs(
+#         title = bquote(paste("Distance of estimate ", hat(beta), " to true ", beta)),
+#         subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#             "20 repetitions, ", n == .(n), ", ",
+#             p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#         x = "nr. of iterations",
+#         y = expression(abs(beta * beta^T - hat(beta) * hat(beta)^T)),
+#         color = "method"
+#     ) +
+#     theme(legend.position = "bottom")
+
+# dev.print(png, file = sprintf("sim01_dist_beta_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
+
+
+# ggplot(hist, aes(x = iter, color = method, group = interaction(method, repetition))) +
+#     geom_line(aes(y = norm.alpha)) +
+#     geom_point(data = with(sub <- subset(hist, !is.na(norm.alpha)),
+#         aggregate(sub, list(method, repetition), tail, 1)
+#     ), aes(y = norm.alpha)) +
+#     labs(
+#         title = expression(paste("Norm of ", hat(alpha))),
+#         subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#             "20 repetitions, ", n == .(n), ", ",
+#             p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#         x = "nr. of iterations",
+#         y = expression(abs(hat(alpha))[F]),
+#         color = "method"
+#     ) +
+#     theme(legend.position = "bottom")
+
+# dev.print(png, file = sprintf("sim01_norm_alpha_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
+
+# ggplot(hist, aes(x = iter, color = method, group = interaction(method, repetition))) +
+#     geom_line(aes(y = norm.beta)) +
+#     geom_point(data = with(sub <- subset(hist, !is.na(norm.beta)),
+#         aggregate(sub, list(method, repetition), tail, 1)
+#     ), aes(y = norm.beta)) +
+#     labs(
+#         title = expression(paste("Norm of ", hat(beta))),
+#         subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
+#             "20 repetitions, ", n == .(n), ", ",
+#             p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
+#         x = "nr. of iterations",
+#         y = expression(abs(hat(beta))[F]),
+#         color = "method"
+#     ) +
+#     theme(legend.position = "bottom")
+
+# dev.print(png, file = sprintf("sim01_norm_beta_%s.png", datetime),
+#           width = 768, height = 768, res = 125)
 
 
-ggplot(hist, aes(x = iter, color = type, group = interaction(type, repetition))) +
-    geom_line(aes(y = dist.beta)) +
-    geom_point(data = with(sub <- subset(hist, !is.na(dist.beta)),
-        aggregate(sub, list(type, repetition), tail, 1)
-    ), aes(y = dist.beta)) +
-    labs(
-        title = bquote(paste("Distance of estimate ", hat(beta), " to true ", beta)),
-        subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
-            "20 repetitions, ", n == .(n), ", ",
-            p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
-        x = "nr. of iterations",
-        y = expression(abs(beta * beta^T - hat(beta) * hat(beta)^T)),
-        color = "method"
-    ) +
-    theme(legend.position = "bottom")
 
-dev.print(png, file = "sim01_dist_beta.png", width = 768, height = 768, res = 125)
 
 
-ggplot(hist, aes(x = iter, color = type, group = interaction(type, repetition))) +
-    geom_line(aes(y = norm.alpha)) +
-    geom_point(data = with(sub <- subset(hist, !is.na(norm.alpha)),
-        aggregate(sub, list(type, repetition), tail, 1)
-    ), aes(y = norm.alpha)) +
-    labs(
-        title = expression(paste("Norm of ", hat(alpha))),
-        subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
-            "20 repetitions, ", n == .(n), ", ",
-            p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
-        x = "nr. of iterations",
-        y = expression(abs(hat(alpha))[F]),
-        color = "method"
-    ) +
-    theme(legend.position = "bottom")
 
-dev.print(png, file = "sim01_norm_alpha.png", width = 768, height = 768, res = 125)
 
-ggplot(hist, aes(x = iter, color = type, group = interaction(type, repetition))) +
-    geom_line(aes(y = norm.beta)) +
-    geom_point(data = with(sub <- subset(hist, !is.na(norm.beta)),
-        aggregate(sub, list(type, repetition), tail, 1)
-    ), aes(y = norm.beta)) +
-    labs(
-        title = expression(paste("Norm of ", hat(beta))),
-        subtitle = bquote(paste(Delta[i][j] == 0.25, " * ", 0.5^abs(i - j), ", ",
-            "20 repetitions, ", n == .(n), ", ",
-            p == .(p), ", ", q == .(q), ", ", k == .(k), ", ", r == .(r))),
-        x = "nr. of iterations",
-        y = expression(abs(hat(beta))[F]),
-        color = "method"
-    ) +
-    theme(legend.position = "bottom")
 
-dev.print(png, file = "sim01_norm_beta.png", width = 768, height = 768, res = 125)
 
 
 
@@ -983,3 +1385,82 @@ dev.print(png, file = "sim01_norm_beta.png", width = 768, height = 768, res = 12
 
 # log(det(kronecker(D.1, D.2)))
 # p * log(det(D.1)) + q * log(det(D.2))
+
+
+
+     d <- ggplot(mtcars, aes(cyl, mpg)) + geom_point()
+     d + stat_summary(fun.data = "mean_cl_boot", colour = "red", size = 2)
+
+     # Orientation follows the discrete axis
+     ggplot(mtcars, aes(mpg, factor(cyl))) +
+       geom_point() +
+       stat_summary(fun.data = "mean_cl_boot", colour = "red", size = 2)
+     
+     # You can supply individual functions to summarise the value at
+     # each x:
+     d + stat_summary(fun = "median", colour = "red", size = 2, geom = "point")
+     d + stat_summary(fun = "mean", colour = "red", size = 2, geom = "point")
+     d + aes(colour = factor(vs)) + stat_summary(fun = mean, geom="line")
+     
+     d + stat_summary(fun = mean, fun.min = min, fun.max = max, colour = "red")
+     
+     d <- ggplot(diamonds, aes(cut))
+     d + geom_bar()
+     d + stat_summary(aes(y = price), fun = "mean", geom = "bar")
+     
+     # Orientation of stat_summary_bin is ambiguous and must be specified directly
+     ggplot(diamonds, aes(carat, price)) +
+       stat_summary_bin(fun = "mean", geom = "bar", orientation = 'y')
+     
+     
+     # Don't use ylim to zoom into a summary plot - this throws the
+     # data away
+     p <- ggplot(mtcars, aes(cyl, mpg)) +
+       stat_summary(fun = "mean", geom = "point")
+     p
+     p + ylim(15, 30)
+     # Instead use coord_cartesian
+     p + coord_cartesian(ylim = c(15, 30))
+     
+     # A set of useful summary functions is provided from the Hmisc package:
+     stat_sum_df <- function(fun, geom="crossbar", ...) {
+       stat_summary(fun.data = fun, colour = "red", geom = geom, width = 0.2, ...)
+     }
+     d <- ggplot(mtcars, aes(cyl, mpg)) + geom_point()
+     # The crossbar geom needs grouping to be specified when used with
+     # a continuous x axis.
+     d + stat_sum_df("mean_cl_boot", mapping = aes(group = cyl))
+     d + stat_sum_df("mean_sdl", mapping = aes(group = cyl))
+     d + stat_sum_df("mean_sdl", fun.args = list(mult = 1), mapping = aes(group = cyl))
+     d + stat_sum_df("median_hilow", mapping = aes(group = cyl))
+
+     # An example with highly skewed distributions:
+     if (require("ggplot2movies")) {
+     set.seed(596)
+     mov <- movies[sample(nrow(movies), 1000), ]
+      m2 <-
+        ggplot(mov, aes(x = factor(round(rating)), y = votes)) +
+        geom_point()
+      m2 <-
+        m2 +
+        stat_summary(
+          fun.data = "mean_cl_boot",
+          geom = "crossbar",
+          colour = "red", width = 0.3
+        ) +
+        xlab("rating")
+     m2
+     # Notice how the overplotting skews off visual perception of the mean
+     # supplementing the raw data with summary statistics is _very_ important
+     
+     # Next, we'll look at votes on a log scale.
+     
+     # Transforming the scale means the data are transformed
+     # first, after which statistics are computed:
+     m2 + scale_y_log10()
+     # Transforming the coordinate system occurs after the
+     # statistic has been computed. This means we're calculating the summary on the raw data
+     # and stretching the geoms onto the log scale.  Compare the widths of the
+     # standard errors.
+     m2 + coord_trans(y="log10")
+     }

tensorPredictors/R/kpir_approx.R

@@ -91,14 +91,8 @@ kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
     S.2 <- n^-1 * tcrossprod(mat(R, 2), mat(R %x_3% Delta.1.inv, 2))
 
     # Evaluate negative log-likelihood (2 pi term dropped)
-    loss <- -0.5 * n * (p * q * log(s) - p * log(det(Delta.1)) -
+    loss <- -0.5 * (n * (p * q * log(s) - p * log(det(Delta.1)) -
-        q * log(det(Delta.2)) - s * sum(S.1 * Delta.1.inv))
+        q * log(det(Delta.2))) - s * sum(S.1 * Delta.1.inv))
-
-    # Gradient "generating" tensor
-    G <- (sum(S.1 * Delta.1.inv) - p * q / s) * R
-    G <- G + R %x_2% ((diag(q, p, p) - s * (Delta.2.inv %*% S.2)) %*% Delta.2.inv)
-    G <- G + R %x_3% ((diag(p, q, q) - s * (Delta.1.inv %*% S.1)) %*% Delta.1.inv)
-    G <- G + s * (R %x_2% Delta.2.inv %x_3% Delta.1.inv)
 
     # Call history callback (logger) before the first iteration
     if (is.function(logger)) {
@@ -125,12 +119,116 @@ kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
             alpha.moment <- alpha1 + ((a0 - 1) / a1) * (alpha1 - alpha0)
             beta.moment  <-  beta1 + ((a0 - 1) / a1) * ( beta1 -  beta0)
         }
+
+        # Extrapolated residuals
+        R <- X - (Fy %x_3% alpha.moment %x_2% beta.moment)
+
+        # Recompute Covariance Estimates and scaling factor
+        Delta.1 <- tcrossprod(mat(R, 3))
+        Delta.2 <- tcrossprod(mat(R, 2))
+        s <- sum(diag(Delta.1))
+
+        # Inverse Covariances
+        Delta.1.inv <- solve(Delta.1)
+        Delta.2.inv <- solve(Delta.2)
+
+        # cross dependent covariance estimates
+        S.1 <- n^-1 * tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3))
+        S.2 <- n^-1 * tcrossprod(mat(R, 2), mat(R %x_3% Delta.1.inv, 2))
+
+        # Gradient "generating" tensor
+        G <- (sum(S.1 * Delta.1.inv) - p * q / s) * R
+        G <- G + R %x_2% ((diag(q, p) - s * (Delta.2.inv %*% S.2)) %*% Delta.2.inv)
+        G <- G + R %x_3% ((diag(p, q) - s * (Delta.1.inv %*% S.1)) %*% Delta.1.inv)
+        G <- G + s * (R %x_2% Delta.2.inv %x_3% Delta.1.inv)
+
+        # Calculate Gradients
+        grad.alpha <- tcrossprod(mat(G, 3), mat(Fy %x_2% beta.moment, 3))
+        grad.beta  <- tcrossprod(mat(G, 2), mat(Fy %x_3% alpha.moment, 2))
+
+        # Backtracking line search (Armijo type)
+        # The `inner.prod` is used in the Armijo break condition but does not
+        # depend on the step size.
+        inner.prod <- sum(grad.alpha^2) + sum(grad.beta^2)
+
+        # backtracking loop
+        for (delta in step.size * 0.618034^seq.int(0L, len = max.line.iter)) {
+            # Update `alpha` and `beta` (note: add(+), the gradients are already
+            # pointing into the negative slope direction of the loss cause they are
+            # the gradients of the log-likelihood [NOT the negative log-likelihood])
+            alpha.temp <- alpha.moment + delta * grad.alpha
+            beta.temp  <- beta.moment  + delta * grad.beta
+
+            # Update Residuals, Covariances, ...
+            R <- X - (Fy %x_3% alpha.temp %x_2% beta.temp)
+            Delta.1 <- tcrossprod(mat(R, 3))
+            Delta.2 <- tcrossprod(mat(R, 2))
+            s <- sum(diag(Delta.1))
+            Delta.1.inv <- solve(Delta.1)
+            Delta.2.inv <- solve(Delta.2)
+            S.1 <- n^-1 * tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3))
+            # S.2 not needed
+
+            # Re-evaluate negative log-likelihood
+            loss.temp <- -0.5 * (n * (p * q * log(s) - p * log(det(Delta.1)) -
+                q * log(det(Delta.2))) - s * sum(S.1 * Delta.1.inv))
+
+            # Armijo line search break condition
+            if (loss.temp <= loss - 0.1 * delta * inner.prod) {
+                break
+            }
+        }
+
+        # Call logger (invoke history callback)
+        if (is.function(logger)) {
+            logger(iter, loss.temp, alpha.temp, beta.temp, Delta.1, Delta.2, delta)
+        }
+
+        # Enforce descent
+        if (loss.temp < loss) {
+            alpha0 <- alpha1
+            alpha1 <- alpha.temp
+            beta0 <- beta1
+            beta1 <- beta.temp
+
+            # check break conditions
+            if (mean(abs(alpha1)) + mean(abs(beta1)) < eps) {
+                break.reason <- "alpha, beta numerically zero"
+                break   # estimates are basically zero -> stop
+            }
+            if (inner.prod < eps * (p * q + r * k)) {
+                break.reason <- "mean squared gradient is smaller than epsilon"
+                break   # mean squared gradient is smaller than epsilon -> stop
+            }
+            if (abs(loss.temp - loss) < eps) {
+                break.reason <- "decrease is too small (slow)"
+                break   # decrease is too small (slow) -> stop
+            }
+
+            loss  <- loss.temp
+            no.nesterov <- FALSE    # always reset
+        } else if (!no.nesterov) {
+            no.nesterov <- TRUE     # retry without momentum
+            next
+        } else {
+            break.reason <- "failed even without momentum"
+            break       # failed even without momentum -> stop
+        }
+
+        # update momentum scaling
+        a0 <- a1
+        a1 <- nesterov.scaling(a1, iter)
+
+        # Set next iter starting step.size to line searched step size
+        # (while allowing it to encrease)
+        step.size <- 1.618034 * delta
+
     }
 
-    # Extrapolated residuals
+    list(
-    R <- X - (Fy %x_3% alpha.moment %x_2% beta.moment)
+        loss = loss,
-
+        alpha = alpha1, beta = beta1,
-    
+        Delta.1 = Delta.1, Delta.2 = Delta.2, tr.Delta = s,
-
+        break.reason = break.reason
-    list(loss = loss, alpha = alpha1, beta = beta1, Delta = Delta)
+    )
 }

tensorPredictors/R/kpir_kron.R

@@ -126,8 +126,8 @@ kpir.kron <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
         resid.trans <- resid %x_3% solve(Delta.1) %x_2% solve(Delta.2)
 
         # Calculate Gradients
-        grad.alpha <- tcrossprod(mat(resid.trans, 3), mat(Fy %x_2% beta, 3))
+        grad.alpha <- tcrossprod(mat(resid.trans, 3), mat(Fy %x_2% S.beta, 3))
-        grad.beta  <- tcrossprod(mat(resid.trans, 2), mat(Fy %x_3% alpha, 2))
+        grad.beta  <- tcrossprod(mat(resid.trans, 2), mat(Fy %x_3% S.alpha, 2))
 
         # Backtracking line search (Armijo type)
         # The `inner.prod` is used in the Armijo break condition but does not
@@ -161,7 +161,7 @@ kpir.kron <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
             }
         }
 
-        # Call logger (invoce history callback)
+        # Call logger (invoke history callback)
         if (is.function(logger)) {
             logger(iter, loss.temp, alpha.temp, beta.temp, Delta.1, Delta.2, delta)
         }
@@ -207,5 +207,10 @@ kpir.kron <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
 
     }
 
-    list(loss = loss, alpha = alpha1, beta = beta1, Delta = Delta, break.reason = break.reason)
+    list(
+        loss = loss,
+        alpha = alpha1, beta = beta1,
+        Delta.1 = Delta.1, Delta.2 = Delta.2,
+        break.reason = break.reason
+    )
 }

tensorPredictors/src/poi.c

@@ -80,15 +80,3 @@ extern SEXP FastPOI_C_sub(SEXP in_B, SEXP in_Delta,
     UNPROTECT(1);
     return out_Z;
 }
-
-/* List of registered routines (e.g. C entry points) */
-static const R_CallMethodDef CallEntries[] = {
-    {"FastPOI_C_sub", (DL_FUNC) &FastPOI_C_sub, 5},
-    {NULL, NULL, 0}
-};
-
-/* Restrict C entry points to registered routines. */
-void R_init_tensorPredictors(DllInfo *dll) {
-    R_registerRoutines(dll, NULL, CallEntries, NULL, NULL);
-    R_useDynamicSymbols(dll, FALSE);
-}