# Principal Component Analysis (PCA)

#### Eric Bridgeford

#### 2020-06-25

    require(lolR)
    require(ggplot2)
    require(MASS)
    n=400
    d=30
    r=3

Data for this notebook will be `n=400` examples of `d=30` dimensions.

# PCA

## Stacked Cigar Simulation

We first visualize the first `2` dimensions:

    testdat <- lol.sims.cigar(n, d)
    X <- testdat$X
    Y <- testdat$Y

    data <- data.frame(x1=X[,1], x2=X[,2], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, y=x2, color=y)) +
      geom_point() +
      xlab("x1") +
      ylab("x2") +
      ggtitle("Simulated Data")

Projecting with PCA to `3` dimensions and visualizing the first `2`:

    result <- lol.project.pca(X, r)

    data <- data.frame(x1=result$Xr[,1], x2=result$Xr[,2], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, y=x2, color=y)) +
      geom_point() +
      xlab("x1") +
      ylab("x2") +
      ggtitle("Projected Data using PCA")

Projecting with LDA to `K-1=1` dimensions:

    liney <- MASS::lda(result$Xr, Y)
    result <- predict(liney, result$Xr)
    lhat <- 1 - sum(result$class == Y)/length(Y)

    data <- data.frame(x1=result$x[,1], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, fill=y)) +
      geom_density(adjust=1.5, alpha=0.6) +
      xlab("$x_1$") +
      ylab("Density") +
      ggtitle(sprintf("PCA-LDA, L = %.2f", lhat))

## Trunk Simulation

We visualize the first `2` dimensions:

    testdat <- lol.sims.rtrunk(n, d)
    X <- testdat$X
    Y <- testdat$Y

    data <- data.frame(x1=X[,1], x2=X[,2], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, y=x2, color=y)) +
      geom_point() +
      xlab("x1") +
      ylab("x2") +
      ggtitle("Simulated Data")

Projecting with PCA to `3` dimensions and visualizing the first `2`:

    result <- lol.project.pca(X, r)

    data <- data.frame(x1=result$Xr[,1], x2=result$Xr[,2], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, y=x2, color=y)) +
      geom_point() +
      xlab("x1") +
      ylab("x2") +
      ggtitle("Projected Data using PCA")

Projecting with LDA to `K-1=1` dimensions:

    liney <- MASS::lda(result$Xr, Y)
    result <- predict(liney, result$Xr)
    lhat <- 1 - sum(result$class == Y)/length(Y)

    data <- data.frame(x1=result$x[,1], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, fill=y)) +
      geom_density(adjust=1.5, alpha=0.6) +
      xlab("x1") +
      ylab("Density") +
      ggtitle(sprintf("PCA-LDA, L = %.2f", lhat))

## Rotated Trunk Simulation

We visualize the first `2` dimensions:

    testdat <- lol.sims.rtrunk(n, d, rotate=TRUE)
    X <- testdat$X
    Y <- testdat$Y

    data <- data.frame(x1=X[,1], x2=X[,2], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, y=x2, color=y)) +
      geom_point() +
      xlab("x1") +
      ylab("x2") +
      ggtitle("Simulated Data")

Projecting with PCA to `3` dimensions and visualizing the first `2`:

    result <- lol.project.pca(X, r)

    data <- data.frame(x1=result$Xr[,1], x2=result$Xr[,2], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, y=x2, color=y)) +
      geom_point() +
      xlab("x1") +
      ylab("x2") +
      ggtitle("Projected Data using PCA")

Projecting with LDA to `K-1=1` dimensions:

    liney <- MASS::lda(result$Xr, Y)
    result <- predict(liney, result$Xr)
    lhat <- 1 - sum(result$class == Y)/length(Y)

    data <- data.frame(x1=result$x[,1], y=Y)
    data$y <- factor(data$y)
    ggplot(data, aes(x=x1, fill=y)) +
      geom_density(adjust=1.5, alpha=0.6) +
      xlab("x1") +
      ylab("Density") +
      ggtitle(sprintf("PCA-LDA, L = %.2f", lhat))