Contents

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.

library("MASS") 
data("crabs") 
dim(crabs)
## [1] 200   8
crabs[1:4,] 
##   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
table(crabs$sex)
## 
##   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)
summary(m1)
## 
## 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”.

library(MLInterfaces)
fcrabs = crabs[crabs$sex == "F", ] 
ml1 = MLearn( sp~RW, fcrabs,glmI.logistic(thresh=.5), c(1:30, 51:80), family=binomial) 
ml1
## 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
confuMat(ml1)
##      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.

set.seed(123) 
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)
confuMat(sml1)
##      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)
confuMat(smx1)
##      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"))
\label{fig:figdopa}Pairs plot of the 5 quantitative features of the crabs data.Points are colored by species.

Figure 2: Pairs plot of the 5 quantitative features of the crabs data.Points are colored by species

2.5.2 Principal components; biplot

Principal components analysis transforms the multivariate data X into a new coordinate system. If the original variables are X1, . . . , Xp, then the variables in the new representation are denoted PC1, . . . , PCp. These new variables have the properties that PC1 is the linear combination of the X1, . . . , Xp having maximal variance, PC2 is the variance-maximizing linear combination of residuals of X after projecting into the hyperplane normal to PC1, and so on. If most of the variation in Xn×p can be captured in a low dimensional linear subspace of the space spanned by the columns of X, then the scatterplots of the first few principal components give a good representation of the structure in the data.

Formally, we can compute the PC using the singular value decomposition of X, in which X = U DV2, where Un×p and Vp×p are orthonormal, and D is a diagonal matrix of \(p\) nonnegative singular values. The principal components transformation is XV = UD, and if D is structured so that Dii ≥ Djj whenever i > j, then column i of XV is PCi. Note also that Dii = √n − 1 sd(PCi).

pc1 = prcomp( crabs[,-c(1:3)] )
pairs(pc1$x, col=ifelse(crabs$sp=="B", "blue", "orange"))
\label{fig:figdopa}Pairs plot of the crabs data in principal component coordinates

Figure 3: Pairs plot of the crabs data in principal component coordinates

The plot is shown in Figure 3.

The biplot, Figure 4, shows the data in PC space and also shows the relative contributions of the original variables in composing the transformation.

biplot(pc1, choices=2:3, col=c("#80808080", "red"))
\label{fig:figdopa} Biplot of the principal component analysis of the crabs data.

Figure 4: Biplot of the principal component analysis of the crabs data

2.5.3 Clustering

A familiar technique for displaying multivariate data in high-throughput biology is called the heatmap. In this display, samples are clustered as columns, and features as rows. The clustering technique used by default is R hclust. This procedure builds a clustering tree for the data as follows. Distances are computed between each pair of feature vectors for all N observations. The two closest pair is joined and regarded as a new object, so there are N-1 objects (clusters) at this point. This process is repeated until 1 cluster is formed; the clustering tree shows the process by which clusters are created via this agglomeration process.

The most crucial choice when applying this method is the initial choice of the distance metric between the features.

Once clusters are being formed, there are several ways to measure distances between them, based on the initial between-feature distances. Single-linkage clustering takes the distance between two clusters to be the shortest distance between any two members of the different clusters; average linkage averages all the distances between members; complete-linkage uses hte maximum distance between any two members of the different clusters. Other methods are also available in hclust.

Figure 5 shows cluster trees for samples and features. The default color choice is not great, thus we specify own using the col argument. A tiled display at the top, defined via the argument ColSideColors shows the species codes for the samples. An important choice to be made when calling heatmap is the value of the argument scale, whose default setting is to scale the rows, but not the columns.

X = data.matrix(crabs[,-c(1:3)])
heatmap(t(X), ColSideColors=ifelse(crabs$sp=="O", "orange", "blue"), col =
colorRampPalette(c("blue", "white", "red"))(255))
\label{fig:figdopa}Heatmap plot of the crabs data, including dendrograms representing hierarchical clustering of the rows and columns.

Figure 5: Heatmap plot of the crabs data, including dendrograms representing hierarchical clustering of the rows and columns

Typically clustering is done in the absence of labels – it is an example of unsupervised machine learning. We can ask whether the clustering provided is a ‘good’ one using the measurement of a quantity called the silhouette. This is defined in R documentation as follows:

     For each observation i, the _silhouette width_ s(i) is defined as
     follows: 
      Put a(i) = average dissimilarity between i and all other points
     of the cluster to which i belongs (if i is the _only_ observation
     in its cluster, s(i) := 0 without further calculations). For all
     _other_ clusters C, put d(i,C) = average dissimilarity of i to all
     observations of C.  The smallest of these d(i,C) is b(i) := min_C
     d(i,C), and can be seen as the dissimilarity between i and its
     "neighbor" cluster, i.e., the nearest one to which it does _not_
     belong. Finally, 

             s(i) := ( b(i) - a(i) ) / max( a(i), b(i) ).

We can compute the silhouette for any partition of a dataset, and can use the hierarchical clustering result to define a partition as follows:

cl = hclust(dist(X)) 
tr = cutree(cl,2)
table(tr)
## tr
##   1   2 
## 105  95
library(cluster) 
sil = silhouette( tr, dist(X) )
plot(sil)

  • Question 8. In the preceding, we have used default dist, and default clustering algorithm for the heatmap. Investigate the impact of altering the choice of distance and clustering method on the clustering performance, both in relation to capacity to recover groups defined by species and in relation to the silhouette distribution.

  • Question 9. The PCA shows that the data configuration in PC2 and PC3 is at least bifurcated. Apply hierarchical and K-means clustering to the two-dimensional data in this subspace, and compare results with respect to capturing the species \(\times\) gender labels, and with respect to silhouette values. For example, load the exprs slot of crES [see just below for the definition of this structure] with the PCA reexpression of the features, call the result pcrES, and then:

    > ff = kmeansB(pcrES[2:3,], k=4)
> table(ff@clustIndices, crES$spsex)

2.6 Supervised learning

In this section we will examine procedures for polychotomous prediction. We want to be able to use the measurements to predict both species and sex of the crab. Again we would like to use the MLInterfaces infrastructure, so an ExpressionSet container will be useful.

feat2 = t(data.matrix(crabs[, -c(1:3)])) 
pd2 =new("AnnotatedDataFrame", crabs[,1:2]) 
crES = new("ExpressionSet",exprs=feat2, phenoData=pd2)
crES$spsex = paste(crES$sp,crES$sex, sep=":")
table(crES$spsex)
## 
## B:F B:M O:F O:M 
##  50  50  50  50

We will permute the samples so that simple selections for training set indices are random samples.

set.seed(1234) 
crES = crES[ , sample(1:200, size=200, replace=FALSE)]

2.6.1 RPART

A classic procedure is recursive partitioning.

library(rpart)
tr1 = MLearn(spsex~., crES, rpartI, 1:140)
## [1] "spsex"
tr1
## MLInterfaces classification output container
## The call was:
## MLearn(formula = spsex ~ ., data = crES, .method = rpartI, trainInd = 1:140)
## Predicted outcome distribution for test set:
## 
## B:F B:M O:F O:M 
##  15  20  12  13 
## Summary of scores on test set (use testScores() method for details):
##       B:F       B:M       O:F       O:M 
## 0.1847013 0.3631000 0.2205201 0.2316786
confuMat(tr1) 
##      predicted
## given B:F B:M O:F O:M
##   B:F  11   2   5   0
##   B:M   4  11   0   4
##   O:F   0   3   7   3
##   O:M   0   4   0   6

The actual tree is

plot(RObject(tr1))
text(RObject(tr1))

This procedure includes a diagnostic tool called the cost-complexity plot:

plotcp(RObject(tr1))

2.6.2 Random forests

A generalization of recursive partitioning is obtained by creating a collection of trees by bootstrap-sampling cases and randomly sampling from features available for splitting at nodes.

set.seed(124) 
library(randomForest)
crES$spsex = factor(crES$spsex) # needed 3/2020 as fails with 'do regression?' error 
rf1 = MLearn(spsex~., crES, randomForestI, 1:140 )
## [1] "spsex"
rf1
## MLInterfaces classification output container
## The call was:
## MLearn(formula = spsex ~ ., data = crES, .method = randomForestI, 
##     trainInd = 1:140)
## Predicted outcome distribution for test set:
## 
## B:F B:M O:F O:M 
##  14  21  12  13 
## Summary of scores on test set (use testScores() method for details):
##       B:F       B:M       O:F       O:M 
## 0.2479667 0.3208333 0.2184333 0.2127667
cm = confuMat(rf1)
cm 
##      predicted
## given B:F B:M O:F O:M
##   B:F  13   2   2   1
##   B:M   1  14   1   3
##   O:F   0   2   9   2
##   O:M   0   3   0   7

The single split error rate is estimated at 28%.

  • Question 10. What is the out-of-bag error rate for rf1? Obtain a cross-validated estimate of misclassification error using randomForest with an xvalSpec().

2.6.3 Linear discriminants

ld1 = MLearn(spsex~., crES, ldaI, 1:140 ) 
## [1] "spsex"
ld1
## MLInterfaces classification output container
## The call was:
## MLearn(formula = spsex ~ ., data = crES, .method = ldaI, trainInd = 1:140)
## Predicted outcome distribution for test set:
## 
## B:F B:M O:F O:M 
##  19  18  12  11
confuMat(ld1)
##      predicted
## given B:F B:M O:F O:M
##   B:F  17   1   0   0
##   B:M   2  17   0   0
##   O:F   0   0  12   1
##   O:M   0   0   0  10
xvld = MLearn( spsex~., crES, ldaI, xvalSpec("LOG", 5,balKfold.xvspec(5)))
## [1] "spsex"
confuMat(xvld)
##      predicted
## given B:F B:M O:F O:M
##   B:F  49   1   0   0
##   B:M   5  45   0   0
##   O:F   0   0  46   4
##   O:M   0   0   0  50
  • Question 11. Use the balKfold function to generate an index set for partitions that is balanced with respect to class distribution. Check the balance and repeat the cross validation.

2.6.4 Neural net

nn1 = MLearn(spsex~., crES, nnetI, 1:140, size=3, decay=.1)
## [1] "spsex"
## # weights:  34
## initial  value 210.549773 
## iter  10 value 192.102775
## iter  20 value 147.838019
## iter  30 value 122.446145
## iter  40 value 42.634063
## iter  50 value 33.156533
## iter  60 value 32.631597
## iter  70 value 32.587472
## iter  80 value 32.584864
## final  value 32.584303 
## converged
nn1 
## MLInterfaces classification output container
## The call was:
## MLearn(formula = spsex ~ ., data = crES, .method = nnetI, trainInd = 1:140, 
##     size = 3, decay = 0.1)
## Predicted outcome distribution for test set:
## 
## B:F B:M O:F O:M 
##  18  19  12  11 
## Summary of scores on test set (use testScores() method for details):
##       B:F       B:M       O:F       O:M 
## 0.2948541 0.3078294 0.2091575 0.1881590
RObject(nn1) 
## a 5-3-4 network with 34 weights
## inputs: FL RW CL CW BD 
## output(s): spsex 
## options were - softmax modelling  decay=0.1
confuMat(nn1)
##      predicted
## given B:F B:M O:F O:M
##   B:F  17   1   0   0
##   B:M   1  18   0   0
##   O:F   0   0  12   1
##   O:M   0   0   0  10
xvnnBAD = MLearn( spsex~., crES, nnetI, xvalSpec("LOG", 5, function(data, clab,iternum) which( rep(1:5,each=40) == iternum ) ), size=3,decay=.1 )
## [1] "spsex"
## # weights:  34
## initial  value 59.944660 
## iter  10 value 52.383742
## iter  20 value 52.328922
## iter  30 value 36.162023
## iter  40 value 18.424810
## iter  50 value 16.742739
## iter  60 value 16.666492
## iter  70 value 16.531701
## iter  80 value 16.348753
## iter  90 value 16.341980
## iter 100 value 16.341716
## final  value 16.341716 
## stopped after 100 iterations
## # weights:  34
## initial  value 60.559984 
## iter  10 value 55.332627
## iter  20 value 54.471313
## iter  30 value 37.129183
## iter  40 value 18.743630
## iter  50 value 18.343335
## iter  60 value 18.341766
## iter  70 value 18.340791
## final  value 18.340126 
## converged
## # weights:  34
## initial  value 58.388107 
## iter  10 value 50.025755
## iter  20 value 34.567868
## iter  30 value 25.393417
## iter  40 value 19.119780
## iter  50 value 18.479760
## iter  60 value 18.457200
## iter  70 value 18.455017
## iter  70 value 18.455017
## final  value 18.455017 
## converged
## # weights:  34
## initial  value 56.312404 
## iter  10 value 54.231832
## iter  20 value 38.578144
## iter  30 value 21.749309
## iter  40 value 19.433643
## iter  50 value 18.175293
## iter  60 value 18.027653
## iter  70 value 18.022584
## iter  80 value 18.021740
## final  value 18.021733 
## converged
## # weights:  34
## initial  value 66.222486 
## iter  10 value 51.932804
## iter  20 value 42.369662
## iter  30 value 36.429562
## iter  40 value 23.026644
## iter  50 value 17.763455
## iter  60 value 17.626915
## iter  70 value 17.609194
## iter  80 value 17.608039
## iter  90 value 17.607812
## final  value 17.607805 
## converged
xvnnGOOD = MLearn( spsex~., crES, nnetI, xvalSpec("LOG", 5,balKfold.xvspec(5) ), size=3, decay=.1 )
## [1] "spsex"
## # weights:  34
## initial  value 236.655971 
## iter  10 value 221.577281
## iter  20 value 193.401577
## iter  30 value 109.642510
## iter  40 value 87.837208
## iter  50 value 53.337845
## iter  60 value 40.015235
## iter  70 value 39.529689
## iter  80 value 39.508922
## iter  90 value 39.485591
## iter 100 value 39.205868
## final  value 39.205868 
## stopped after 100 iterations
## # weights:  34
## initial  value 278.638252 
## iter  10 value 221.728674
## iter  20 value 180.982892
## iter  30 value 156.111919
## iter  40 value 123.000370
## iter  50 value 55.242191
## iter  60 value 34.086648
## iter  70 value 32.937901
## iter  80 value 32.792157
## iter  90 value 32.765436
## final  value 32.763993 
## converged
## # weights:  34
## initial  value 235.827834 
## iter  10 value 185.584490
## iter  20 value 137.388611
## iter  30 value 126.312306
## iter  40 value 83.167721
## iter  50 value 39.525854
## iter  60 value 34.327764
## iter  70 value 33.226308
## iter  80 value 32.262021
## iter  90 value 32.208223
## final  value 32.208198 
## converged
## # weights:  34
## initial  value 246.161602 
## iter  10 value 221.701021
## iter  20 value 152.075319
## iter  30 value 125.184824
## iter  40 value 122.328179
## iter  50 value 108.784116
## iter  60 value 69.806462
## iter  70 value 59.034110
## iter  80 value 46.754185
## iter  90 value 36.519916
## iter 100 value 35.531868
## final  value 35.531868 
## stopped after 100 iterations
## # weights:  34
## initial  value 235.690522 
## iter  10 value 221.074899
## iter  20 value 142.146905
## iter  30 value 122.632468
## iter  40 value 51.504304
## iter  50 value 35.382372
## iter  60 value 33.506367
## iter  70 value 32.601047
## iter  80 value 32.447009
## iter  90 value 32.407399
## final  value 32.405065 
## converged
confuMat(xvnnBAD)
##      predicted
## given B:F B:M O:F O:M
##   B:F  49   1   0   0
##   B:M   5  45   0   0
##   O:F   0   0  46   4
##   O:M   0   0   3  47
confuMat(xvnnGOOD)
##      predicted
## given B:F B:M O:F O:M
##   B:F  49   1   0   0
##   B:M   4  46   0   0
##   O:F   0   0  49   1
##   O:M   0   0   0  50

2.6.5 SVM

sv1 = MLearn(spsex~., crES, svmI, 1:140) 
## [1] "spsex"
sv1 
## MLInterfaces classification output container
## The call was:
## MLearn(formula = spsex ~ ., data = crES, .method = svmI, trainInd = 1:140)
## Predicted outcome distribution for test set:
## 
## B:F B:M O:F O:M 
##  23  16  12   9 
## Summary of scores on test set (use testScores() method for details):
##       B:M       B:F       O:M       O:F 
## 0.2474892 0.3082105 0.1902935 0.2540068
RObject(sv1)
## 
## Call:
## svm(formula = formula, data = data, probability = probability)
## 
## 
## Parameters:
##    SVM-Type:  C-classification 
##  SVM-Kernel:  radial 
##        cost:  1 
## 
## Number of Support Vectors:  129
confuMat(sv1) 
##      predicted
## given B:F B:M O:F O:M
##   B:F  16   0   2   0
##   B:M   5  13   0   1
##   O:F   2   1  10   0
##   O:M   0   2   0   8
xvsv = MLearn( spsex~., crES,svmI, xvalSpec("LOG", 5, balKfold.xvspec(5)))
## [1] "spsex"
confuMat(xvsv) 
##      predicted
## given B:F B:M O:F O:M
##   B:F  45   0   5   0
##   B:M  12  34   0   4
##   O:F   9   0  40   1
##   O:M   4   0   0  46

3 Learning with expression arrays

Here we will concentrate on ALL: acute lymphocytic leukemia, B-cell type.

3.1 Phenotype reduction

We will identify expression patterns that discriminate individuals with BCR/ABL fusion in B-cell leukemia.

library("ALL")
data("ALL") 
bALL = ALL[, substr(ALL$BT,1,1) == "B"]
fus = bALL[, bALL$mol.biol %in% c("BCR/ABL", "NEG")]
fus$mol.biol = factor(fus$mol.biol)
fus 
## ExpressionSet (storageMode: lockedEnvironment)
## assayData: 12625 features, 79 samples 
##   element names: exprs 
## protocolData: none
## phenoData
##   sampleNames: 01005 01010 ... 84004 (79 total)
##   varLabels: cod diagnosis ... date last seen (21 total)
##   varMetadata: labelDescription
## featureData: none
## experimentData: use 'experimentData(object)'
##   pubMedIds: 14684422 16243790 
## Annotation: hgu95av2

3.2 Nonspecific filtering

We can nonspecifically filter to 300 genes (to save computing time) with largest measures of robust variation across all samples:

mads = apply(exprs(fus),1,mad) 
fusk = fus[ mads > sort(mads,decr=TRUE)[300], ] 
fcol =ifelse(fusk$mol.biol=="NEG", "green", "red")

3.3 Exploratory work

For exploratory data analysis, a heatmap is customary.

heatmap(exprs(fusk), ColSideColors=fcol)

Principal components and a biplot may be more revealing. How many principal components are likely to be important?

PCg = prcomp(t(exprs(fusk)))
plot(PCg)

pairs(PCg$x[,1:5],col=fcol,pch=19)

biplot(PCg)

  • Question 12. Modify the biplot so that instead of plotting sample ID, the symbol "O" is plotted for a NEG sample and "+" is plotted for a BCR/ABL sample.

  • Question 13. Consider the following code

    chkT = function (x, eset=fusk) {
t.test(exprs(eset)[x, eset$mol.b == "NEG"], exprs(eset)[x, eset$mol.b == 
    "BCR/ABL"]) }

    Use it in conjunction with the biplot to interpret expression patterns of genes that appear to be important in defining the PCs.

3.4 Classifier construction

3.4.1 Demonstrations

Diagonal LDA has a good reputation. Let’s try it first, followed by neural net and random forests. We will not attend to tuning the latter two, defaults or guesses for key parameters are used.

dld1 = MLearn( mol.biol~., fusk, dldaI, 1:40 )
## [1] "mol.biol"
dld1 
## MLInterfaces classification output container
## The call was:
## MLearn(formula = mol.biol ~ ., data = fusk, .method = dldaI, 
##     trainInd = 1:40)
## Predicted outcome distribution for test set:
## 
## BCR/ABL     NEG 
##      27      12
confuMat(dld1) 
##          predicted
## given     BCR/ABL NEG
##   BCR/ABL      15   1
##   NEG          12  11
nnALL = MLearn( mol.biol~., fusk, nnetI, 1:40, size=5, decay=.01, MaxNWts=2000 )
## [1] "mol.biol"
## # weights:  1506
## initial  value 32.454087 
## iter  10 value 27.625693
## iter  20 value 14.433671
## iter  30 value 11.781962
## iter  40 value 8.806493
## iter  50 value 6.903445
## iter  60 value 1.663919
## iter  70 value 1.362786
## iter  80 value 1.352153
## iter  90 value 1.230728
## iter 100 value 0.875583
## final  value 0.875583 
## stopped after 100 iterations
confuMat(nnALL)
##          predicted
## given     BCR/ABL NEG
##   BCR/ABL      14   2
##   NEG          10  13
rfALL = MLearn(
mol.biol~., fusk, randomForestI, 1:40 ) 
## [1] "mol.biol"
rfALL
## MLInterfaces classification output container
## The call was:
## MLearn(formula = mol.biol ~ ., data = fusk, .method = randomForestI, 
##     trainInd = 1:40)
## Predicted outcome distribution for test set:
## 
## BCR/ABL     NEG 
##      25      14 
## Summary of scores on test set (use testScores() method for details):
## BCR/ABL     NEG 
##   0.536   0.464
confuMat(rfALL)
##          predicted
## given     BCR/ABL NEG
##   BCR/ABL      15   1
##   NEG          10  13

None of these are extremely impressive, but the problem may just be very hard.

3.4.2 Gene set appraisal

  • Question 14. We can assess the predictive capacity of a set of genes by restricting the ExpressionSet to that set and using the best classifier appropriate to the problem. We can also assess the incremental effect of combining gene sets, relative to using them separately.

    One collection of gene sets that is straightforward to use and interpret is provided by the keggorthology package (see also GSEABase). Here’s how we can define the ExpressionSets for genes annotated by KEGG to Environmental (Genetic) Information Processing:

    library(keggorthology) 
data(KOgraph) 
adj(KOgraph,nodes(KOgraph)[1])
    ## $KO.Feb10root
## [1] "Metabolism"                          
## [2] "Genetic Information Processing"      
## [3] "Environmental Information Processing"
## [4] "Cellular Processes"                  
## [5] "Organismal Systems"                  
## [6] "Human Diseases"
    EIP = getKOprobes("Environmental Information Processing")
GIP = getKOprobes("Genetic Information Processing")
length(intersect(EIP, GIP)) 
    ## [1] 44
    EIPi = setdiff(EIP, GIP) 
GIP = setdiff(GIP, EIP) 
EIP = EIPi
Efusk = fusk[ featureNames(fusk)  %in% EIP, ]
Gfusk = fusk[ featureNames(fusk)  %in% EIP, ]

    Obtain and assess the predictive capacity of the genes annotated to "Cell Growth and Death".

  • Question 15. How many of the genes identified by RDA as important for discriminating fusion are annotated to Genetic Information Processing in the KEGG orthology?

4 Embedding features selection in cross-validation

We provide helper functions to conduct several kinds of feature selection in cross-validation, see help(fs.absT). Here we pick the top 30 features (ranked by absolute t statistic) for each cross-validation partition.

dldFS = MLearn( mol.biol~., fusk, dldaI, xvalSpec("LOG", 5, balKfold.xvspec(5), fs.absT(30) ))
## [1] "mol.biol"
dldFS 
## MLInterfaces classification output container
## The call was:
## MLearn(formula = mol.biol ~ ., data = fusk, .method = dldaI, 
##     trainInd = xvalSpec("LOG", 5, balKfold.xvspec(5), fs.absT(30)))
## Predicted outcome distribution for test set:
## 
## BCR/ABL     NEG 
##      42      37 
## history of feature selection in cross-validation available; use fsHistory()
confuMat(dld1)
##          predicted
## given     BCR/ABL NEG
##   BCR/ABL      15   1
##   NEG          12  11
confuMat(dldFS) 
##          predicted
## given     BCR/ABL NEG
##   BCR/ABL      34   3
##   NEG           8  34

5 Session information

sessionInfo() 
## R version 4.3.1 (2023-06-16)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 22.04.3 LTS
## 
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.18-bioc/R/lib/libRblas.so 
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB              LC_COLLATE=C              
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## time zone: America/New_York
## tzcode source: system (glibc)
## 
## attached base packages:
## [1] stats4    stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] keggorthology_2.54.0 hgu95av2.db_3.13.0   org.Hs.eg.db_3.18.0 
##  [4] graph_1.80.0         ALL_1.43.0           randomForest_4.7-1.1
##  [7] rpart_4.1.21         lattice_0.22-5       MASS_7.3-60         
## [10] gbm_2.1.8.1          MLInterfaces_1.82.0  cluster_2.1.4       
## [13] annotate_1.80.0      XML_3.99-0.14        AnnotationDbi_1.64.0
## [16] IRanges_2.36.0       S4Vectors_0.40.0     Biobase_2.62.0      
## [19] BiocGenerics_0.48.0  Rcpp_1.0.11          BiocStyle_2.30.0    
## 
## loaded via a namespace (and not attached):
##  [1] blob_1.2.4                  Biostrings_2.70.0          
##  [3] bitops_1.0-7                fastmap_1.1.1              
##  [5] RCurl_1.98-1.12             digest_0.6.33              
##  [7] survival_3.5-7              KEGGREST_1.42.0            
##  [9] gdata_3.0.0                 RSQLite_2.3.1              
## [11] genefilter_1.84.0           magrittr_2.0.3             
## [13] compiler_4.3.1              rlang_1.1.1                
## [15] sass_0.4.7                  tools_4.3.1                
## [17] yaml_2.3.7                  knitr_1.44                 
## [19] S4Arrays_1.2.0              bit_4.0.5                  
## [21] DelayedArray_0.28.0         abind_1.4-5                
## [23] nnet_7.3-19                 grid_4.3.1                 
## [25] xtable_1.8-4                e1071_1.7-13               
## [27] ada_2.0-5                   gtools_3.9.4               
## [29] SummarizedExperiment_1.32.0 cli_3.6.1                  
## [31] rmarkdown_2.25              crayon_1.5.2               
## [33] httr_1.4.7                  DBI_1.1.3                  
## [35] cachem_1.0.8                proxy_0.4-27               
## [37] zlibbioc_1.48.0             splines_4.3.1              
## [39] BiocManager_1.30.22         XVector_0.42.0             
## [41] matrixStats_1.0.0           vctrs_0.6.4                
## [43] Matrix_1.6-1.1              jsonlite_1.8.7             
## [45] bookdown_0.36               bit64_4.0.5                
## [47] magick_2.8.1                jquerylib_0.1.4            
## [49] codetools_0.2-19            GenomeInfoDb_1.38.0        
## [51] sfsmisc_1.1-16              GenomicRanges_1.54.0       
## [53] htmltools_0.5.6.1           GenomeInfoDbData_1.2.11    
## [55] R6_2.5.1                    evaluate_0.22              
## [57] png_0.1-8                   memoise_2.0.1              
## [59] bslib_0.5.1                 class_7.3-22               
## [61] SparseArray_1.2.0           xfun_0.40                  
## [63] MatrixGenerics_1.14.0       pkgconfig_2.0.3