# Normalizations Comparisons

Load libraries

    library(ggplot2)
    library(SurprisalAnalysis)

    ##

    ##

    ##

    library(data.table)
    library(pheatmap)
    library(latex2exp)

Let us test the normalization methods here pseudocount vs log1p

    #' Plot Bland Altman and density plots of the difference between the two zero handling
    #' methods
    #'
    #' @param aa the gene expression matrix
    #' @param pc the pseudocount
    #'
    #' @return two plots, the Bland Altman plot and the density plot
    plot.norm.diffs <- function(X, pc){

      mat_logpc <- log(X + pc)
      mat_log1p <- log1p(X)

      avg_vals  <- (mat_logpc + mat_log1p) / 2
      diff_vals <- mat_logpc - mat_log1p

      df_ba <- data.frame(
        Avg  = as.vector(avg_vals),
        Diff = as.vector(diff_vals)
      )

      p_ba <- ggplot(df_ba, aes(x = Avg, y = Diff)) +
        geom_point(size = 0.8, color = "darkred") +
        geom_hline(yintercept = 0, linetype = 2, color = "navy") +
        labs(
          title = "Bland–Altman Plot",
          subtitle = "log(x+1e-6) vs log1p(x)",
          x = "Average of two transforms",
          y = "Difference (log(x+1e-6) - log1p(x))"
        ) +
        theme(
          panel.grid.major = element_blank(),
          panel.grid.minor = element_blank(),
          panel.background = element_blank(),
          axis.line = element_line(colour = "black"),
          plot.title = element_text(hjust = 0.5, size = 20),
          axis.text = element_text(size = 15),
          text = element_text(size = 18)
        )

      df_diff <- data.frame(Diff = as.vector(diff_vals))

      p_density <- ggplot(df_diff, aes(x = Diff)) +
        geom_density(aes(y = after_stat(scaled)), fill = "steelblue", alpha = 0.5) +
        geom_vline(xintercept = 0, linetype = 2, color = "navy") +
        labs(
          title = "Distribution of Differences",
          x = "Difference (log(x+1e-6) - log1p(x))",
          y = "Density"
        ) +
        theme(
          panel.grid.major = element_blank(),
          panel.grid.minor = element_blank(),
          panel.background = element_blank(),
          axis.line = element_line(colour = "black"),
          plot.title = element_text(hjust = 0.5, size = 20),
          axis.text = element_text(size = 15),
          text = element_text(size = 18)
        )

      return(p_ba + p_density)
    }

    #pseudocount
    pc <- 1e-6

    df <- read.table(system.file("extdata", "GSE60450_Lactation-GenewiseCounts.txt", package = "SurprisalAnalysis"),header = TRUE, sep = "\t", check.names = FALSE)

    df[,3:ncol(df)]->testing.case.1

    df$EntrezGeneID->rownames(testing.case.1)

    as.matrix(testing.case.1)->testing.case.1

    stopifnot(all(c("EntrezGeneID","Length") %in% names(df)))

    gene_ids <- df$EntrezGeneID
    len_bp   <- df$Length
    counts   <- as.matrix(df[ , -(1:2)])

    keep <- is.finite(len_bp) & (len_bp > 0)
    counts <- counts[keep, , drop = FALSE]
    len_bp <- len_bp[keep]
    gene_ids <- gene_ids[keep]
    rownames(counts) <- gene_ids

    len_kb <- len_bp / 1000

    #FPKM
    #FPKM_ij = counts_ij / (len_kb_i * libsize_j_in_millions)
    libsize        <- colSums(counts, na.rm = TRUE)
    libsize_mill   <- libsize / 1e6

    fpkm <- sweep(counts, 1, len_kb, "/")
    fpkm <- sweep(fpkm, 2, libsize_mill, "/")

    #TPM
    # TPM_ij = (counts_ij / len_kb_i) / sum_i(counts_ij / len_kb_i) * 1e6
    rpk  <- sweep(counts, 1, len_kb, "/")
    scale <- colSums(rpk, na.rm = TRUE)
    tpm <- sweep(rpk, 2, scale, "/") * 1e6

    testing.case.2 <- fpkm
    testing.case.3 <- tpm

    plot.norm.diffs(testing.case.1, pc)

    plot.norm.diffs(testing.case.2, pc)

    plot.norm.diffs(testing.case.3, pc)

    baseline <- data.frame(raw = surprisal_analysis(testing.case.1)[[1]][,1],
               fpkm = surprisal_analysis(testing.case.2)[[1]][,1],
               tpm = surprisal_analysis(testing.case.3)[[1]][,1])

    baseline.pseudo <- ggplot(baseline)+
      geom_line(aes(x=1:nrow(baseline), y=raw, colour="Raw"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=raw, colour="Raw"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=fpkm, color = "FPKM Normalized"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=fpkm, colour = "FPKM Normalized"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), size=1,)+
      geom_point(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), shape=8, stroke=1)+
      theme(
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.background = element_blank(),
        axis.line = element_line(colour = "black"),
        plot.title = element_text(hjust = 0.5, size = 20),
        axis.text = element_text(size = 15),
        text = element_text(size = 18)
      )+
      scale_color_manual(values=c("Raw"="#9ECAD6", "FPKM Normalized" = "#4A9782","TPM Normalized"="#004030" ))+
      labs(x="Sample", title=TeX("$\\lambda_0$"), y="Value",colour="Normalization")

    baseline.pseudo

    lambda_1 <- data.frame(raw = surprisal_analysis(testing.case.1)[[1]][,2],
                           fpkm = surprisal_analysis(testing.case.2)[[1]][,2],
                           tpm = surprisal_analysis(testing.case.3)[[1]][,2])

    lambda_1.pseudo <- ggplot(lambda_1)+
      geom_line(aes(x=1:nrow(baseline), y=raw, colour="Raw"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=raw, colour="Raw"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=fpkm, color = "FPKM Normalized"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=fpkm, colour = "FPKM Normalized"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), size=1,)+
      geom_point(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), shape=8, stroke=1)+
      theme(
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.background = element_blank(),
        axis.line = element_line(colour = "black"),
        plot.title = element_text(hjust = 0.5, size = 20),
        axis.text = element_text(size = 15),
        text = element_text(size = 18)
      )+
      scale_color_manual(values=c("Raw"="#9ECAD6", "FPKM Normalized" = "#4A9782","TPM Normalized"="#004030" ))+
      labs(x="Sample", title=TeX("$\\lambda_1$"), y="Value",colour="Normalization")

    lambda_1.pseudo

    lambda_2 <- data.frame(raw = surprisal_analysis(testing.case.1)[[1]][,3],
                           fpkm = surprisal_analysis(testing.case.2)[[1]][,3],
                           tpm = surprisal_analysis(testing.case.3)[[1]][,3])

    lambda_2.pseudo <-   ggplot(lambda_2)+
      geom_line(aes(x=1:nrow(baseline), y=raw, colour="Raw"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=raw, colour="Raw"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=fpkm, color = "FPKM Normalized"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=fpkm, colour = "FPKM Normalized"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), size=1,)+
      geom_point(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), shape=8, stroke=1)+
      theme(
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.background = element_blank(),
        axis.line = element_line(colour = "black"),
        plot.title = element_text(hjust = 0.5, size = 20),
        axis.text = element_text(size = 15),
        text = element_text(size = 18)
      )+
      scale_color_manual(values=c("Raw"="#9ECAD6", "FPKM Normalized" = "#4A9782","TPM Normalized"="#004030" ))+
      labs(x="Sample", title=TeX("$\\lambda_2$"), y="Value",colour="Normalization")

    lambda_2.pseudo

    baseline <- data.frame(raw = surprisal_analysis(testing.case.1, "log1p")[[1]][,1],
                           fpkm = surprisal_analysis(testing.case.2, "log1p")[[1]][,1],
                           tpm = surprisal_analysis(testing.case.3, "log1p")[[1]][,1])

    baseline.log1p <-  ggplot(baseline)+
      geom_line(aes(x=1:nrow(baseline), y=raw, colour="Raw"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=raw, colour="Raw"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=fpkm, color = "FPKM Normalized"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=fpkm, colour = "FPKM Normalized"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), size=1,)+
      geom_point(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), shape=8, stroke=1)+
      theme(
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.background = element_blank(),
        axis.line = element_line(colour = "black"),
        plot.title = element_text(hjust = 0.5, size = 20),
        axis.text = element_text(size = 15),
        text = element_text(size = 18)
      )+
      scale_color_manual(values=c("Raw"="#9ECAD6", "FPKM Normalized" = "#4A9782","TPM Normalized"="#004030" ))+
      labs(x="Sample", title=TeX("$\\lambda_0$"), y="Value",colour="Normalization")

    baseline.log1p

    lambda_1 <- data.frame(raw = surprisal_analysis(testing.case.1, "log1p")[[1]][,2],
                           fpkm = surprisal_analysis(testing.case.2, "log1p")[[1]][,2],
                           tpm = surprisal_analysis(testing.case.3, "log1p")[[1]][,2])

    lambda_1.log1p <- ggplot(lambda_1)+
      geom_line(aes(x=1:nrow(baseline), y=raw, colour="Raw"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=raw, colour="Raw"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=fpkm, color = "FPKM Normalized"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=fpkm, colour = "FPKM Normalized"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), size=1,)+
      geom_point(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), shape=8, stroke=1)+
      theme(
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.background = element_blank(),
        axis.line = element_line(colour = "black"),
        plot.title = element_text(hjust = 0.5, size = 20),
        axis.text = element_text(size = 15),
        text = element_text(size = 18)
      )+
      scale_color_manual(values=c("Raw"="#9ECAD6", "FPKM Normalized" = "#4A9782","TPM Normalized"="#004030" ))+
      labs(x="Sample", title=TeX("$\\lambda_1$"), y="Value",colour="Normalization")

    lambda_1.log1p

    lambda_2 <- data.frame(raw = surprisal_analysis(testing.case.1, "log1p")[[1]][,3],
                           fpkm = surprisal_analysis(testing.case.2, "log1p")[[1]][,3],
                           tpm = surprisal_analysis(testing.case.3, "log1p")[[1]][,3])

    lambda_2.log1p <- ggplot(lambda_2)+
      geom_line(aes(x=1:nrow(baseline), y=raw, colour="Raw"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=raw, colour="Raw"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=fpkm, color = "FPKM Normalized"), size=1)+
      geom_point(aes(x=1:nrow(baseline), y=fpkm, colour = "FPKM Normalized"), shape=8, stroke=1)+
      geom_line(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), size=1,)+
      geom_point(aes(x=1:nrow(baseline), y=tpm, colour="TPM Normalized"), shape=8, stroke=1)+
      theme(
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.background = element_blank(),
        axis.line = element_line(colour = "black"),
        plot.title = element_text(hjust = 0.5, size = 20),
        axis.text = element_text(size = 15),
        text = element_text(size = 18)
      )+
      scale_color_manual(values=c("Raw"="#9ECAD6", "FPKM Normalized" = "#4A9782","TPM Normalized"="#004030" ))+
      labs(x="Sample", title=TeX("$\\lambda_2$"), y="Value",colour="Normalization")

    lambda_2.log1p

    #(baseline.pseudo+baseline.log1p)/(lambda.1.pseudo+lambda.1.log1p)/(lambda_2.pseudo+lambda_2.log1p)

    running.test.case.1<-data.frame(Sample = as.numeric(rownames(testing.case.1)), testing.case.1)

    running.test.case.2<-data.frame(Sample = as.numeric(rownames(testing.case.2)), testing.case.2)

    running.test.case.3<-data.frame(Sample = as.numeric(rownames(testing.case.3)), testing.case.3)

    testing.case.2.log1p <- GO_analysis_surprisal_analysis(surprisal_analysis(running.test.case.2, "log1p")[[2]], 0.85, 2, key_type = "ENTREZID", flip = FALSE, species.db.str =  "org.Mm.eg.db", top_GO_terms=15)

    testing.case.2.pseudo <- GO_analysis_surprisal_analysis(surprisal_analysis(running.test.case.2, "pseudocount")[[2]], 0.85, 2, key_type = "ENTREZID", flip = FALSE, species.db.str =  "org.Mm.eg.db", top_GO_terms=15)

    #ggplot(testing.case.2.pseudo, aes(x = reorder(Description, -Count), y = Count, fill = p.adjust)) +
      #geom_col() +
      #coord_flip() +
      #scale_fill_gradient(low = "steelblue", high = "red") +
      #labs(
      #  title = "Top 15 Enriched GO Terms",
      #  x = "GO Term",
      #  y = "Gene Count",
      #  fill = "Adj. p-value"
      #) +
      #theme_minimal(base_size = 14)

    #Jaccard similarity

    extreme_pct_lambda <- function(sa_mat, lambda_col = "lambda_1", p = 0.05) {
      stopifnot(lambda_col %in% colnames(sa_mat))
      v <- sa_mat[, lambda_col]
      n <- length(v)
      k <- max(1, ceiling(p * n))

      top_ids <- names(sort(v, decreasing = TRUE))[seq_len(k)]
      bot_ids <- names(sort(v, decreasing = FALSE))[seq_len(k)]

      return(sort(unique(c(top_ids, bot_ids))))
    }

    sa1_log1p <- surprisal_analysis(running.test.case.1, "log1p")[[2]]
    sa1_pc    <- surprisal_analysis(running.test.case.1, "pseudocount")[[2]]

    sa2_log1p <- surprisal_analysis(running.test.case.2, "log1p")[[2]]
    sa2_pc    <- surprisal_analysis(running.test.case.2, "pseudocount")[[2]]

    sa3_log1p <- surprisal_analysis(running.test.case.3, "log1p")[[2]]
    sa3_pc    <- surprisal_analysis(running.test.case.3, "pseudocount")[[2]]

    set1_log1p <- extreme_pct_lambda(sa1_log1p, "lambda_1", 0.05)
    set1_pc    <- extreme_pct_lambda(sa1_pc,    "lambda_1", 0.05)

    set2_log1p <- extreme_pct_lambda(sa2_log1p, "lambda_1", 0.05)
    set2_pc    <- extreme_pct_lambda(sa2_pc,    "lambda_1", 0.05)

    set3_log1p <- extreme_pct_lambda(sa3_log1p, "lambda_1", 0.05)
    set3_pc    <- extreme_pct_lambda(sa3_pc,    "lambda_1", 0.05)

    sets <- list(
      `Raw-log1p`        = set1_log1p,
      `Raw-pseudocount`  = set1_pc,
      `FPKM-log1p`       = set2_log1p,
      `FPKM-pseudocount` = set2_pc,
      `TPM-log1p`        = set3_log1p,
      `TPM-pseudocount`  = set3_pc
    )

    pairwise_matrix <- function(sets, fun) {
      n <- length(sets)
      M <- matrix(0, n, n, dimnames = list(names(sets), names(sets)))
      for (i in seq_len(n)) for (j in seq_len(n)) M[i, j] <- fun(sets[[i]], sets[[j]])
      M
    }

    overlap_fun <- function(a, b) length(intersect(a, b))
    jaccard_fun <- function(a, b) {
      inter <- length(intersect(a, b)); uni <- length(union(a, b))
      if (uni == 0) return(NA_real_) else inter / uni
    }

    overlap_mat <- pairwise_matrix(sets, overlap_fun)
    diag(overlap_mat) <- vapply(sets, length, integer(1))
    jaccard_mat <- pairwise_matrix(sets, jaccard_fun)

    pheatmap::pheatmap(
      jaccard_mat,
      main = TeX("Jaccard similarity (Top & Bottom 5% of $\\lambda_0$)"),
      display_numbers = TRUE, number_format = "%.2f",
      fontsize_row = 10, fontsize_col = 10, border_color = NA,
      color = colorRampPalette(c("#1B3C53", "#456882", "#91C8E4"))(100)
    )

    set1_log1p <- extreme_pct_lambda(sa1_log1p, "lambda_2", 0.05)
    set1_pc    <- extreme_pct_lambda(sa1_pc,    "lambda_2", 0.05)

    set2_log1p <- extreme_pct_lambda(sa2_log1p, "lambda_2", 0.05)
    set2_pc    <- extreme_pct_lambda(sa2_pc,    "lambda_2", 0.05)

    set3_log1p <- extreme_pct_lambda(sa3_log1p, "lambda_2", 0.05)
    set3_pc    <- extreme_pct_lambda(sa3_pc,    "lambda_2", 0.05)

    sets <- list(
      `Raw-log1p`        = set1_log1p,
      `Raw-pseudocount`  = set1_pc,
      `FPKM-log1p`       = set2_log1p,
      `FPKM-pseudocount` = set2_pc,
      `TPM-log1p`        = set3_log1p,
      `TPM-pseudocount`  = set3_pc
    )

    overlap_mat <- pairwise_matrix(sets, overlap_fun)
    diag(overlap_mat) <- vapply(sets, length, integer(1))
    jaccard_mat <- pairwise_matrix(sets, jaccard_fun)

    pheatmap::pheatmap(
      jaccard_mat,
      main = TeX("Jaccard similarity (Top & Bottom 5% of $\\lambda_1$)"),
      display_numbers = TRUE, number_format = "%.2f",
      fontsize_row = 10, fontsize_col = 10, border_color = NA,
      color = colorRampPalette(c("#1B3C53", "#456882", "#91C8E4"))(100)
    )

    set1_log1p <- extreme_pct_lambda(sa1_log1p, "lambda_3", 0.05)
    set1_pc    <- extreme_pct_lambda(sa1_pc,    "lambda_3", 0.05)

    set2_log1p <- extreme_pct_lambda(sa2_log1p, "lambda_3", 0.05)
    set2_pc    <- extreme_pct_lambda(sa2_pc,    "lambda_3", 0.05)

    set3_log1p <- extreme_pct_lambda(sa3_log1p, "lambda_3", 0.05)
    set3_pc    <- extreme_pct_lambda(sa3_pc,    "lambda_3", 0.05)

    sets <- list(
      `Raw-log1p`        = set1_log1p,
      `Raw-pseudocount`  = set1_pc,
      `FPKM-log1p`       = set2_log1p,
      `FPKM-pseudocount` = set2_pc,
      `TPM-log1p`        = set3_log1p,
      `TPM-pseudocount`  = set3_pc
    )

    overlap_mat <- pairwise_matrix(sets, overlap_fun)
    diag(overlap_mat) <- vapply(sets, length, integer(1))
    jaccard_mat <- pairwise_matrix(sets, jaccard_fun)

    pheatmap::pheatmap(
      jaccard_mat,
      main = TeX("Jaccard similarity (Top & Bottom 5% of $\\lambda_2$)"),
      display_numbers = TRUE, number_format = "%.2f",
      fontsize_row = 10, fontsize_col = 10, border_color = NA,
      color = colorRampPalette(c("#1B3C53", "#456882", "#91C8E4"))(100)
    )