Supervised Learning, Classification, Discrimination

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Supervised Learning, Classification,
Discrimination
SLIDES RECYCLED FROM ppt slides by Darlene
Goldstein http//statwww.epfl.ch/davison/teaching/
Microarrays/
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Gene expression data

• Data on G genes for n samples

mRNA samples
sample1 sample2 sample3 sample4 sample5 1
0.46 0.30 0.80 1.51 0.90 ... 2 -0.10 0.49
0.24 0.06 0.46 ... 3 0.15 0.74 0.04 0.10
0.20 ... 4 -0.45 -1.03 -0.79 -0.56 -0.32 ... 5 -0.
06 1.06 1.35 1.09 -1.09 ...
Genes
Gene expression level of gene i in mRNA sample j
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Machine learning tasks

• Task assign objects to classes (groups) on the
  basis of measurements made on the objects
• Unsupervised classes unknown, want to discover
  them from the data (cluster analysis)
• Supervised classes are predefined, want to use
  a (training or learning) set of labeled objects
  to form a classifier for classification of future
  observations

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Discrimination

• Objects (e.g. arrays) are to be classified as
  belonging to one of a number of predefined
  classes 1, 2, , K
• Each object associated with a class label (or
  response) Y ? 1, 2, , K and a feature vector
  (vector of predictor variables) of G
  measurements X (X1, , XG)
• Aim predict Y from X.

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Example Tumor Classification

• Reliable and precise classification essential for
  successful cancer treatment
• Current methods for classifying human
  malignancies rely on a variety of morphological,
  clinical and molecular variables
• Uncertainties in diagnosis remain likely that
  existing classes are heterogeneous
• Characterize molecular variations among tumors by
  monitoring gene expression (microarray)
• Hope that microarrays will lead to more reliable
  tumor classification (and therefore more
  appropriate treatments and better outcomes)

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Tumor Classification Using Gene Expression Data

• Three main types of statistical problems
  associated with tumor classification
• Identification of new/unknown tumor classes using
  gene expression profiles (unsupervised learning
  clustering)
• Classification of malignancies into known classes
  (supervised learning discrimination)
• Identification of marker genes that
  characterize the different tumor classes (feature
  or variable selection).

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Classifiers

• A predictor or classifier partitions the space of
  gene expression profiles into K disjoint subsets,
  A1, ..., AK, such that for a sample with
  expression profile X(X1, ...,XG) ? Ak the
  predicted class is k
• Classifiers are built from a learning set (LS)
• L (X1, Y1), ..., (Xn,Yn)
• Classifier C built from a learning set L
• C( . ,L) X ? 1,2, ... ,K
• Predicted class for observation X
• C(X,L) k if X is in Ak

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Decision Theory (I)

• Can view classification as statistical decision
  theory must decide which of the classes an
  object belongs to
• Use the observed feature vector X to aid in
  decision making
• Denote population proportion of objects of class
  k as pk p(Y k)
• Assume objects in class k have feature vectors
  with density pk(X) p(XY k)

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Decision Theory (II)

• One criterion for assessing classifier quality is
  the misclassification rate,
• p(C(X)?Y)
• A loss function L(i,j) quantifies the loss
  incurred by erroneously classifying a member of
  class i as class j
• The risk function R(C) for a classifier is the
  expected (average) loss
• R(C) EL(Y,C(X))

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Decision Theory (III)

• Typically L(i,i) 0
• In many cases can assume symmetric loss with
  L(i,j) 1 for i ? j (so that different types of
  errors are equivalent)
• In this case, the risk is simply the
  misclassification probability
• There are some important examples, such as in
  diagnosis, where the loss function is not
  symmetric

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Maximum likelihood discriminant rule

• A maximum likelihood estimator (MLE) chooses the
  parameter value that makes the chance of the
  observations the highest
• For known class conditional densities pk(X), the
  maximum likelihood (ML) discriminant rule
  predicts the class of an observation X by
• C(X) argmaxk pk(X)

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Fisher Linear Discriminant Analysis

• First applied in 1935 by M. Barnard at the
  suggestion of R. A. Fisher (1936), Fisher linear
  discriminant analysis (FLDA)
• finds linear combinations of the gene expression
  profiles XX1,...,XG with large ratios of
  between-groups to within-groups sums of squares -
  discriminant variables
• predicts the class of an observation X by the
  class whose mean vector is closest to X in terms
  of the discriminant variables

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Gaussian ML Discriminant Rules

• For multivariate Gaussian (normal) class
  densities XY k N(?k, ?k), the ML classifier
  is
• C(X) argmink (X - ?k) ?k-1 (X - ?k) log ?k
  
• In general, this is a quadratic rule (Quadratic
  discriminant analysis, or QDA)
• In practice, population mean vectors ?k and
  covariance matrices ?k are estimated by
  corresponding sample quantities

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Gaussian ML Discriminant Rules

• When all class densities have the same covariance
  matrix, ?k ????the discriminant rule is linear
  (Linear discriminant analysis, or LDA FLDA for k
  2)
• C(X) argmink (X - ?k) ?-1 (X - ?k)
• When all class densities have the same diagonal
  covariance matrix ?diag(?12 ?G2), the
  discriminant rule is again linear (Diagonal
  linear discriminant analysis, or DLDA)

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Nearest Neighbor Classification

• Based on a measure of distance between
  observations (e.g. Euclidean distance or one
  minus correlation)
• k-nearest neighbor rule (Fix and Hodges (1951))
  classifies an observation X as follows
• find the k observations in the learning set
  closest to X
• predict the class of X by majority vote, i.e.,
  choose the class that is most common among those
  k observations.
• The number of neighbors k can be chosen by
  cross-validation (more on this later)

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How to construct a tree predictor

• BINARY RECURSIVE PARTITIONING
• Binary split parent node into two child nodes
• Recursive each child node can be treated as
  parent node
• Partitioning data set is partitioned into
  mutually exclusive subsets in each split

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Tree construction
High 17 Low 83
Is BP lt 91?
No
Yes
High 12 Low 88
High 70 Low 30
Is age lt 62.5?
Classified as high risk!
No
Yes
High 2 Low 98
High 23 Low 77
Classified as low risk!
Is ST present?
Yes
No
High 11 Low 89
High 50 Low 50
Classified as low risk!
Classified as high risk!
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Classification Trees

• Partition the feature space into a set of
  rectangles, then fit a simple model in each one
• Binary tree structured classifiers are
  constructed by repeated splits of subsets (nodes)
  of the measurement space X into two descendant
  subsets (starting with X itself)
• Each terminal subset is assigned a class label
  the resulting partition of X corresponds to the
  classifier
• RPART function in R

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Classification Tree

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Three Aspects of Tree Construction

• Split Selection Rule
• Split-stopping Rule
• Class assignment Rule
• Different approaches to these three issues
  (e.g. CART Classification And Regression Trees,
  Breiman et al. (1984) C4.5 and C5.0, Quinlan
  (1993)).

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Three Rules (CART)

• Splitting At each node, choose split maximizing
  decrease in impurity (e.g. Gini index, entropy,
  misclassification error)
• Split-stopping Grow large tree, prune to obtain
  a sequence of subtrees, then use cross-validation
  to identify the subtree with lowest
  misclassification rate
• Class assignment For each terminal node, choose
  the class minimizing the resubstitution estimate
  of misclassification probability, given that a
  case falls into this node

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Other Classifiers Include

• Support vector machines (SVMs)
• Neural networks
• Random forest predictors
• HUNDREDS more

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Feature selection and missing data

• Feature selection
• Automatic with trees
• For DA, NN need preliminary selection
• Need to account for selection when assessing
  performance
• Missing data
• Automatic imputation with trees
• Otherwise, impute (or ignore)

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Performance Assessment-error rate- test set
error- learning set error (aka resubstitution
error)-cross-validation
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Performance assessment (I)

• Resubstitution estimation error rate on the
  learning set
• Problem downward bias
• Test set estimation divide cases in learning
  set into two sets, L1 and L2 classifier built
  using L1, error rate computed for L2. L1 and L2
  must be iid.
• Problem reduced effective sample size

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Performance assessment (II)

• V-fold cross-validation (CV) estimation Cases
  in learning set randomly divided into V subsets
  of (nearly) equal size. Build classifiers
  leaving one set out test set error rates
  computed on left out set and averaged.
• Bias-variance tradeoff smaller V can give
  larger bias but smaller variance
• Out-of-bag estimation only used when dealing
  with bagged predictors

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Performance assessment (III)

• Common error to do feature selection using all of
  the data, then CV only for model building and
  classification
• However, usually features are unknown and the
  intended inference includes feature selection.
  Then, CV estimates as above tend to be downward
  biased.
• Features should be selected only from the
  learning set used to build the model (and not the
  entire learning set)

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Aggregating classifiers

• Breiman (1996, 1998) found that gains in accuracy
  could be obtained by aggregating predictors built
  from perturbed versions of the learning set the
  multiple versions of the predictor are aggregated
  by voting.
• Let C(., Lb) denote the classifier built from the
  bth perturbed learning set Lb, and let wb denote
  the weight given to predictions made by this
  classifier. The predicted class for an
  observation x is given by
• argmaxk ?b wbI(C(x,Lb)
  k)

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Bagging

• Bagging Bootstrap aggregating
• Nonparametric Bootstrap (standard bagging)
  perturbed learning sets drawn at random with
  replacement from the learning sets predictors
  built for each perturbed dataset and aggregated
  by plurality voting (wb 1)
• Parametric Bootstrap perturbed learning sets
  are multivariate Gaussian
• Convex pseudo-data (Breiman 1996)

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Aggregation By-products Out-of-bag estimation
of error rate

• Out-of-bag error rate estimate unbiased
• Use the left out cases from each bootstrap sample
  as a test set
• Classify these test set cases, and compare to the
  class labels of the learning set to get the
  out-of-bag estimate of the error rate

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Aggregation By-products Case-wise information

• Class probability estimates (votes) (0,1) the
  proportion of votes for the winning class
  gives a measure of prediction confidence
• Vote margins (1,1) the proportion of votes for
  the true class minus the maximum of the
  proportion of votes for each of the other
  classes can be used to detect mislabeled
  (learning set) cases

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Aggregation By-products Variable Importance
Statistics

• Measure of predictive power
• For each tree, randomly permute the values of the
  jth variable for the out-of-bag cases, use to get
  new classifications
• Several possible importance measures

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Aggregation By-products Intrinsic Case
Proximities

• Proportion of trees for which cases i and j are
  in the same terminal node
• Clustering
• Outlier detection
• 1/sum(squared proximities of cases in same class)

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Boosting

• Freund and Schapire (1997), Breiman (1998)
• Data resampled adaptively so that the weights in
  the resampling are increased for those cases most
  often misclassified
• Predictor aggregation done by weighted voting

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Comparison of classifiers

• Dudoit, Fridlyand, Speed (JASA, 2002)
• FLDA
• DLDA
• DQDA
• NN
• CART
• Bagging and boosting

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Comparison study datasets

• Leukemia Golub et al. (1999)
• n 72 samples, G 3,571 genes
• 3 classes (B-cell ALL, T-cell ALL, AML)
• Lymphoma Alizadeh et al. (2000)
• n 81 samples, G 4,682 genes
• 3 classes (B-CLL, FL, DLBCL)
• NCI 60 Ross et al. (2000)
• N 64 samples, p 5,244 genes
• 8 classes

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Leukemia data, 2 classes Test set error
rates150 LS/TS runs
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Leukemia data, 3 classes Test set error
rates150 LS/TS runs
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Lymphoma data, 3 classes Test set error rates
N150 LS/TS runs
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NCI 60 data Test set error rates150 LS/TS runs
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Results

• In the main comparison of Dudoit et al, NN and
  DLDA had the smallest error rates, FLDA had the
  highest
• For the lymphoma and leukemia datasets,
  increasing the number of genes to G200 didn't
  greatly affect the performance of the various
  classifiers there was an improvement for the NCI
  60 dataset.
• More careful selection of a small number of genes
  (10) improved the performance of FLDA dramatically

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Comparison study Discussion (I)

• Diagonal LDA ignoring correlation between
  genes helped here
• Unlike classification trees and nearest
  neighbors, LDA is unable to take into account
  gene interactions
• Although nearest neighbors are simple and
  intuitive classifiers, their main limitation is
  that they give very little insight into
  mechanisms underlying the class distinctions

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Comparison study Discussion (II)

• Classification trees are capable of handling and
  revealing interactions between variables
• Useful by-product of aggregated classifiers
  prediction votes, variable importance statistics
• Variable selection A crude criterion such as
  BSS/WSS may not identify the genes that
  discriminate between all the classes and may not
  reveal interactions between genes
• With larger training sets, expect improvement in
  performance of aggregated classifiers

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Acknowledgements

• Sandrine Dudoit
• Jane Fridlyand
• Yee Hwa (Jean) Yang
• Terry Speed