1 Overview

The term machine learning refers to a family of computational methods for analyzing multivariate datasets. Each data point has a vector of features in a shared feature space, and may have a class label from some fixed finite set.

Supervised learning refers to processes that help articulate rules that map feature vectors to class labels. The class labels are known and function as supervisory information to guide rule construction. Unsupervised learning refers to processes that discover structure in collections of feature vectors. Typically the structure consists of a grouping of objects into clusters.

This practical introduction to machine learning will begin with a survey of a low-dimensional dataset to fix concepts, and will then address problems coming from genomic data analysis, using RNA expression and chromatin state data.

Some basic points to consider at the start:

2 Getting acquainted with machine learning via the crabs data

2.1 Attaching and checking the data

The following steps bring the crabs data into scope and illustrate aspects of its structure.

## [1] 200   8
##   sp sex index  FL  RW   CL   CW  BD
## 1  B   M     1 8.1 6.7 16.1 19.0 7.0
## 2  B   M     2 8.8 7.7 18.1 20.8 7.4
## 3  B   M     3 9.2 7.8 19.0 22.4 7.7
## 4  B   M     4 9.6 7.9 20.1 23.1 8.2
##   F   M 
## 100 100
\label{fig:figbwplot} Boxplots of RW, the rear width in mm, stratified by species('B' or 'O' for blue or orange) and sex ('F' and 'M').

Figure 1: Boxplots of RW, the rear width in mm, stratified by species(‘B’ or ‘O’ for blue or orange) and sex (‘F’ and ‘M’)

The plot is shown in Figure 1.

We will regard these data as providing five quantitative features (FL, RW, CL, CW, BD)1 You may consult the manual page of {crabs} for an explanation of these abbreviations. and a pair of class labels (sex, sp=species). We may regard this as a four class problem, or as two two class problems.

2.2 A simple classifier derived by human reasoning

Our first problem does not involve any computations. If you want to write R code to solve the problem, do so, but use prose first.

  • Question 1. On the basis of the boxplots in Figure 1, comment on the prospects for predicting species on the basis of RW. State a rule for computing the predictions. Describe how to assess the performance of your rule.

2.3 Prediction via logistic regression

A simple approach to prediction involves logistic regression.

m1 = glm(sp~RW, data=crabs, family=binomial)
## Call:
## glm(formula = sp ~ RW, family = binomial, data = crabs)
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.44908    0.82210  -4.195 2.72e-05 ***
## RW           0.27080    0.06349   4.265 2.00e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Dispersion parameter for binomial family taken to be 1)
##     Null deviance: 277.26  on 199  degrees of freedom
## Residual deviance: 256.35  on 198  degrees of freedom
## AIC: 260.35
## Number of Fisher Scoring iterations: 4
  • Question 2. Write down the statistical model corresponding to the R expression above. How can we derive a classifier from this model?

  • Question 3. Perform the following computations. Discuss their interpretation. What are the estimated error rates of the two models? Is the second model, on the subset, better?

plot(predict(m1,type="response"), crabs$sp,)
table(predict(m1,type="response")>.5, crabs$sp)
m2 = update(m1, subset=(sex=="F"))
table(predict(m2,type="response")>.5, crabs$sp[crabs$sex=="F"])

2.4 The cross-validation concept

Cross-validation is a technique that is widely used for reducing bias in the estimation of predictive accuracy. If no precautions are taken, bias can be caused by overfitting a classification algorithm to a particular dataset; the algorithm learns the classification ‘’by heart’’, but performs poorly when asked to generalise it to new, unseen examples. Briefly, in cross-validation the dataset is deterministically partitioned into a series of training and test sets. The model is built for each training set and evaluated on the test set. The accuracy measures are averaged over this series of fits. Leave-one-out cross-validation consists of N fits, with N training sets of size N-1 and N test sets of size 1.

First let us use MLearn from the MLInterfaces package to fit a single logistic model. MLearn requires you to specify an index set for training. We use c(1:30, 51:80) to choose a training set of size 60, balanced between two species (because we know the ordering of records). This procedure also requires you to specify a probability threshold for classification. We use a typical default of 0.5. If the predicted probability of being “O” exceeds 0.5, we classify to “O”, otherwise to “B”.

fcrabs = crabs[crabs$sex == "F", ] 
ml1 = MLearn( sp~RW, fcrabs,glmI.logistic(thresh=.5), c(1:30, 51:80), family=binomial) 
## MLInterfaces classification output container
## The call was:
## MLearn(formula = sp ~ RW, data = fcrabs, .method = glmI.logistic(thresh = 0.5), 
##     trainInd = c(1:30, 51:80), family = binomial)
## Predicted outcome distribution for test set:
##  O 
## 40 
## Summary of scores on test set (use testScores() method for details):
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.7553  0.8861  0.9803  0.9355  0.9917  0.9997
##      predicted
## given  B  O
##     B  0 20
##     O  0 20
  • Question 4. What does the report on ml1 tell you about predictions with this model? Can you reconcile this with the results in model m2? [Hint – non-randomness of the selection of the training set is a problem.]

  • Question 5. Modify the MLearn call to obtain a predictor that is more successful on the test set.

Now we will illustrate cross-validation. First, we scramble the order of records in the ExpressionSet so that sequentially formed groups are approximately random samples.

sfcrabs = fcrabs[ sample(nrow(fcrabs)), ]

We invoke the MLearn method in two ways – first specifying a training index set, then specifying a five-fold cross-validation.

sml1 = MLearn( sp~RW, sfcrabs, glmI.logistic(thresh=.5),c(1:30, 51:80),family=binomial)
##      predicted
## given  B  O
##     B 15  6
##     O  8 11
smx1 = MLearn( sp~RW, sfcrabs, glmI.logistic(thresh=.5),xvalSpec("LOG", 5, function(data, clab, iternum) {which(rep(1:5, each=20) == iternum) }), family=binomial)
##      predicted
## given  B  O
##     B 36 14
##     O 14 36
  • Question 6. Define clearly the difference between models sml1 and smx1 and state the misclassification rate estimates associated with each model.

2.5 Exploratory multivariate analysis

2.5.1 Scatterplots

  • Question 7. Interpret the following code, whose result is shown in Figure 2. Modify it to depict the pairwise configurations with different colors for crab genders.
pairs(crabs[,-c(1:3)], col=ifelse(crabs$sp=="B", "blue", "orange"))