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I was working on data science harvard homework problem. It is a two class classification problem in which they plot the decision boundary for random forest, svm and decision tree. The problem has 2 features.

The question they asked was:

there is a tradeoff between the bias and the variance of a classifier. We want to choose a model that generalizes well to unseen data. With a high-variance classifier we run the risk of overfitting to noisy or unrepresentative training data. In contrast, classifier with a high bias typically produce simpler models that tend to underfit the training data, failing to capture important regularities.

Discuss the differences in the above decision surfaces in terms of their complexity and sensitivity to the training data. How do these properties relate to bias and variance?

In the solution part they have written the following:

Solution: The decision surfaces for the decision tree and random forest are very complex (wiggly contours with complex shapes). The decision tree is by far the most sensitive, showing only extreme classification probabilities that are heavily influenced by single points (see red stripes that seem to be drawn just to encapsulate observed "red" points). The random forest shows lower sensitivity, with isolated points having much less extreme classification probabilities. The SVM is the least sensitive, since it has a very smooth decision boundary.

The complexity of the classifier corresponds to lower bias, since it can be more "true" to the training data, but this also makes the classifier more sensitive to random variations in the data, and thus increases variance.

I couldnt understand what they are saying in the solution. can some one please explain me what is being explained in the figure and in the solution part??

The figures of the plot is shown below.

enter image description here enter image description here enter image description here

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    $\begingroup$ I think the author likes the "island" in the svm plot beacuse it is a shape with low entropy compared to the rf or the cart. $\endgroup$ – EngrStudent Dec 26 '16 at 16:10
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The images you present are the same as those here: link.

The following is some code, translated to R with some adjustments, to work through this. The RF selected (2 trees) is not acceptable. This is not apples-to-apples, so any of the authors' assertions about "entropy" can be mis-informative.

First we get the data:

#reproducibility
set.seed(136526) #I like to use question number as random seed

#libraries
library(data.table)      #to read the url
library(randomForest)    #to have randomForests 
library(miscTools)       #column medians

#main program

#get data
wine_df = fread("https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality-red.csv")

#conver to frame
wine_df <- as.data.frame(wine_df)

#parse data
Y <- (wine_df[,12])
X <- wine_df[,-12]

Next we find the right size of random forest for it.

max_trees <- 100     #same range
N_retest <- 35      #fair sample size

err <- matrix(0,max_trees,N_retest)  #initialize for the loop

for (i in 1:max_trees){

     for (j in 1:N_retest){

          #fit random forest with "i" number of trees
          my_rf <- randomForest(x = X, y = Y, ntree = i)

          #pop out sum of squared residuals divided by n
          temp <- mean(my_rf$mse)
          err[i,j] <- temp
     }
}

Now we can look at how many elements should be in the ensemble:

#make friendly for boxplot
err_frame <- as.data.frame(t(err))
names(err_frame) <- as.character(1:max_trees)

#central tendency
my_meds <- colMedians((err_frame))

#normalized slope of central tendency
est <- smooth.spline(x = 1:max_trees,y = my_meds,spar = 0.7)
pred <- predict(est)
my_sl <- c(diff(pred$y)/diff(pred$x))
my_sl <- (0.7-0.4)*(my_sl-min(my_sl))/(max(my_sl)-min(my_sl))+0.4

#make boxplot
boxplot(err_frame, 
        main = "MSE vs. number of trees", 
        xlab = "number of trees", 
        ylab = "forest mean MSE", xlim= c(0,75))

#draw central tendency (red)
lines(est, col="red", lwd=2)

#draw slope
lines(pred$x,c(0.4,my_sl),col="green")
points(pred$x,c(0.4,my_sl),col="green", pch=16)

grid()
legend(x = 60,y = 0.6,c("bxp","fit","slope"), 
       col = c("black","Red","Green"), 
       lty = c(NA, 1,1), 
       pch = c(22,-1,20),
       pt.cex = c(1.2,1,1) )

And it gives us this, which I then manually draw blue and black lines on in a version of midangle-skree heuristic to get a "decent" ensemble size of 30. It is two tangent lines from the slope: one at highest slope, one at right end of domain. We make a ray from intersection of those tangent lines to the slope-line along the mid-angle. The next highest point after the intersection informs tree-count.

enter image description here

Now that we have a decent random forest we can look at errors. First we compute the error.

# make "final" model
my_rf_fin <- randomForest(x = X, y = (Y), ntree = 30)

#predict on it
pred_fin <- predict(my_rf_fin)

#compute error
fit_err <- pred_fin - Y

The first plots to start with are basic EDA plots including the 4-plot of error.

#EDA on error
par(mfrow = n2mfrow(4) )

#run seq
plot(fit_err, type="l")
grid()

#lag plot
plot(fit_err[2:length(fit_err)],fit_err[1:(length(fit_err)-1)] )
abline(a = 0,b=1, col="Green", lwd=2)
grid()

#histogram
hist(fit_err,breaks = 128, main = "")
grid()

#normal quantile
qqnorm(fit_err, main = "")
grid()

par(mfrow = c(1,1))

Which yields:

enter image description here

The error is reasonably well behaved. It is narrow tailed. There is a non-Gaussian set of samples on the right side of the lag plot. The central part of the distribution looks triangular. It isn't Gaussian, but it wasn't expected to be. This is a discrete level output modeled as continuous.

Here is a variability plot of actual vs. predicted, and of error vs. predicted.

enter image description here

If systematically over-predicts the poorest class as better than rated, and under-predicts the highest class as poorer than rated.

This random forest is less poorly constructed, and likely is a healthier function approximator.

Next steps: make the boundary plot like yours on the first 2 principle components.

Notes on the code:

  • I'm not a big scikit.learn guy, so I am going to misunderstand parts of what they are doing. Standard disclaimers apply.
  • Two trees in an ensemble is a contradiction in terms like "one man
    army". The random forest is no "one man army" because it would be
    CART as a non-weak learner. The author did a disservice to an
    ensemble learner by selecting 2 elements as the ensemble size. The
    big joy of a random forest is you can add ensemble elements. Never
    (ever) accept a random forest smaller than 20 trees. Double-check
    any forest smaller than 50 trees.
  • The author has no split between training/validation or test. They
    use all the data to fit the learners. A better way is to split into those groups then determine the ensemble parameters, then make the model with the combined train/valid data. I don't see that here.
  • Author does not specify whether the "y" is discretized or continuous. This means the RF might be living in regression instead of classification.
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