# LOL Cross-Validation

#### Eric Bridgeford

#### 2020-06-25

# Cross Validation to Determine Error of Embedding Alg composed with Classifier

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

Here, we look at how to see the generalizability of a given model in the form of the cross-validated error. We simulate data with `n=200` and `d=30`:

    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")

We arbitrarily select LOL as our algorithm, and look at the leave-one-out (loo) cross-validated error with the LDA classifier. We project the resulting model to `3` dimensions and visualize the first `2`:

    result <- lol.xval.eval(X, Y, r, alg = lol.project.lol, alg.return="A",
                            classifier=MASS::lda, classifier.return="class", k='loo')

    data <- data.frame(x1=result$model$Xr[,1], x2=result$model$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(sprintf("Projected Data using LOL, L=%.2f", result$lhat))

# Cross Validation to Determine Optimal Embedding Dimensions of Embedding Alg composed with Classifier

    result <- lol.xval.optimal_dimselect(X, Y, rs=c(5, 10, 15), alg = lol.project.lol, alg.return="A",
                            classifier=MASS::lda, classifier.return="class", k='loo')

    data <- data.frame(x1=result$model$Xr[,1], x2=result$model$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(sprintf("Projected Data using LOL, L=%.2f", result$optimal.lhat))

    ggplot(result$foldmeans.data, aes(x=r, y=lhat)) +
      geom_line() +
      xlab("Embedding Dimensions, r") +
      ylab("Misclassification Rate, L") +
      ggtitle("Impact on Misclassification Rate of Embedding Dimension")

    print(sprintf("optimal dimension: %d", result$optimal.r))

    ## [1] "optimal dimension: 5"

    print(sprintf("Misclassification rate at rhat = %d: %.2f", result$optimal.r, result$optimal.lhat))

    ## [1] "Misclassification rate at rhat = 5: 0.07"
    ## [2] "Misclassification rate at rhat = 5: 0.07"