ICS 278: Data Mining Lectures 7 and 8: Classification Algorithms

1
ICS 278 Data MiningLectures 7 and 8
Classification Algorithms

• Padhraic Smyth
• Department of Information and Computer Science
• University of California, Irvine

2
Notation

• Variables X, Y.. with values x, y (lower case)
• Vectors indicated by X
• Components of X indicated by Xj with values xj
• Matrix data set D with n rows and p columns
• jth column contains values for variable Xj
• ith row contains a vector of measurements on
  object i, indicated by x(i)
• The jth measurement value for the ith object is
  xj(i)
• Unknown parameter for a model q
• Can also use other Greek letters, like a, b, d, g
  ew
• Vector of parameters q

3
Classification

• Predictive modeling predict Y given X
• Y is real-valued gt regression
• Y is categorical gt classification
• Classification
• Many applications speech recognition, document
  classification, OCR, loan approval, face
  recognition, etc

4
Classification v. Regression

• Similar in many ways
• both learn a mapping from X to Y
• Both sensitive to dimensionality of X
• Generalization to new data is important in both
• Test error versus model complexity
• Many models can be used for either classification
  or regression, e.g.,
• Trees, neural networks
• Most important differences
• Categorical Y versus real-valued Y
• Different score functions
• E.g., classification error versus squared error

5
Decision Region Terminlogy
6
Probabilistic view of Classification

• Notation let there be K classes c1,..cK
• Class marginals p(ck) probability of class k
• Class-conditional probabilities p(
  x ck ) probability of x given ck , k 1,K
• Posterior class probabilities (by Bayes rule)
  p( ck x ) p( x ck ) p(ck) /
  p(x) , k 1,K
• where p(x)
  S p( x cj ) p(cj)
• In theory this is all we need.in practice
  this may not be best approach.

7
Example of Probabilistic Classification
p( x c1 )
p( x c2 )
8
Example of Probabilistic Classification
p( x c1 )
p( x c2 )
1
p( c1 x )
0.5
0
9
Example of Probabilistic Classification
p( x c1 )
p( x c2 )
1
p( c1 x )
0.5
0
10
Decision Regions and Bayes Error Rate
p( x c1 )
p( x c2 )
Class c2
Class c1
Class c2
Class c1
Class c2
Optimal decision regions regions where 1 class
is more likely Optimal decision regions ?
optimal decision boundaries
11
Decision Regions and Bayes Error Rate
p( x c1 )
p( x c2 )
Class c2
Class c1
Class c2
Class c1
Class c2
Optimal decision regions regions where 1 class
is more likely Optimal decision regions ?
optimal decision boundaries Bayes error rate
fraction of examples misclassified by optimal
classifier
shaded area above (see equation
10.3 in text)
12
Procedure for optimal Bayes classifier

• For each class learn a model p( x ck )
• E.g., each class is multivariate Gaussian with
  its own mean and covariance
• Use Bayes rule to obtain p( ck x )
• gt this yields the optimal decision
  regions/boundaries
• gt use these decision regions/boundaries for
  classification
• Correct in theory. but practical problems
  include
• How do we model p( x ck ) ?
• Even if we know the model for p( x ck ),
  modeling a distribution or density will be very
  difficult in high dimensions (e.g., p 100)
• Alternative approach model the decision
  boundaries directly

13
Three types of classifiers

• Generative (or class-conditional) classifiers
• Learn models for p( x ck ), use Bayes rule to
  find decision boundaries
• Examples naïve Bayes models, Gaussian
  classifiers
• Regression (or posterior class probabilities)
• Learn a model for p( ck x ) directly
• Example logistic regression (see lecture 5/6),
  neural networks
• Discriminative classifiers
• No probabilities
• Learn the decision boundaries directly
• Examples
• Linear boundaries perceptrons, linear SVMs
• Piecewise linear boundaries decision trees,
  nearest-neighbor classifiers
• Non-linear boundaries non-linear SVMs
• Note one can usually post-fit class
  probability estimates p( ck x ) to a
  discriminative classifier

14
Which type of classifier is appropriate?

• Lets look at the score functions
• c(i) true class, c(x(i) q) class predicted
  by the classifier
• Class-mismatch loss functions
• S(q) 1/n Si Cost c(i), c(x(i) q)
  
• where cost(i, j) cost of misclassifying
  true class i as predicted class j
• e.g., cost(i,j) 0 if ij, 1 otherwise
  (misclassification error or 0-1 loss)
• and more generally cost(i,j) is a matrix of K
  x K losses (e.g., surgery, spam email, etc)
• Class-probability loss functions
• S(q) 1/n Si log p(c(i) x(i)
  q ) (log probability score)
• or S(q) 1/n Si c(i) p(c(i)
  x(i) q ) 2 (Brier score)

15
Example classifying spam email

• 0-1 loss function
• Appropriate if we just want to maximize accuracy
• Asymmetric cost matrix
• Appropriate if missing non-spam emails is more
  costly than failing to detect spam emails
• Probability loss
• Appropriate if we wanted to rank all emails by
  p(spam email features), e.g., to allow the
  user to look at emails via a ranked list.
• In general dont solve a harder problem than you
  need to, or dont model aspects of the problem
  you dont need to (e.g., modeling p(xc)) -
  Vapnik, 1996.

16
Classes of classifiers

• Class-conditional/probabilistic, based on p( x
  ck ),
• Naïve Bayes (simple, but often effective in high
  dimensions)
• Parametric generative models, e.g., Gaussian (can
  be effective in low-dimensional problems leads
  to quadratic boundaries in general)
• Regression-based, p( ck x ) directly
• Logistic regression simple, linear in odds
  space
• Neural network non-linear extension of logistic,
  can be difficult to work with
• Discriminative models, focus on locating optimal
  decision boundaries
• Linear discriminants, perceptrons simple,
  sometimes effective
• Support vector machines generalization of linear
  discriminants, can be quite effective,
  computational complexity is an issue
• Nearest neighbor simple, can scale poorly in
  high dimensions
• Decision trees swiss army knife, often
  effective in high dimensionis

17
Naïve Bayes Classifiers

• Generative probabilistic model with conditional
  independence assumption on p( x ck ), i.e.
  p( x ck ) P p( xj
  ck )
• Typically used with nominal variables
• Real-valued variables discretized to create
  nominal versions
• (alternative is to model each p( xj ck ) with a
  parametric model less widely used)
• Comments
• Simple to train (just estimate conditional
  probabilities for each feature-class pair)
• Often works surprisingly well in practice
• e.g., state of the art for text-classification,
  basis of many widely used spam filters
• Feature selection can be helpful, e.g.,
  information gain
• Note that even if CI assumptions are not met, it
  may still be able to approximate the optimal
  decision boundaries (seems to happen in practice)
• However. on most problems can usually be beaten
  with a more complex model (plus more work)

18
Announcements

• Homework 2 now online on the Web page
• Due next Thursday in class
• Homework 1 still being graded
• Projects
• Interim report due 2 weeks from now (more details
  later)
• More traffic data now online
• Locations of VDS stations now known (contact Ram
  Hariharan)
• Schedule
• Today more on classification
• Next clustering, pattern-finding, dimension
  reduction
• After that specific topics such as text, Web,
  credit scoring, etc

19
Link between Logistic Regression and Naïve Bayes
Naïve Bayes
Logistic Regression
20
Linear Discriminant Classifiers

• Linear Discriminant Analysis (LDA)
• Earliest known classifier (1936, R.A. Fisher)
• See section 10.4 for math details
• Find a projection onto a vector such that means
  for each class (2 classes) are separated as much
  as possible (with variances taken into account
  appropriately)
• Reduces to a special case of parametric Gaussian
  classifier in certain situations
• Many subsequent variations on this basic theme
  (e.g., regularized LDA)
• Other linear discriminants
• Decision boundary (p-1) dimensional hyperplane
  in p dimensions
• Perceptron learning algorithms (pre-dated neural
  networks)
• Simple error correction based learning
  algorithms
• SVMs use a sophisticated margin idea for
  selecting the hyperplane

21
Nearest Neighbor Classifiers

• kNN select the k nearest neighbors to x from the
  training data and select the majority class from
  these neighbors
• k is a parameter
• Small k noisier estimates, Large k smoother
  estimates
• Best value of k often chosen by cross-validation
• Comments
• Virtually assumption free
• Interesting theoretical properties
  Bayes error lt error(kNN) lt 2 x Bayes error
  (asymptotically)
• Disadvantages
• Can scale poorly with dimensionality sensitive
  to distance metric
• Requires fast lookup at run-time to do
  classification with large n
• Does not provide any interpretable model

22
Local Decision Boundaries
Boundary? Points that are equidistant between
points of class 1 and 2 Note locally the
boundary is (1) linear (because of Euclidean
distance) (2) halfway between the 2 class
points (3) at right angles to connector
1
2
Feature 2
1
2
2
?
1
Feature 1
23
Finding the Decision Boundaries
1
2
Feature 2
1
2
2
?
1
Feature 1
24
Finding the Decision Boundaries
1
2
Feature 2
1
2
2
?
1
Feature 1
25
Finding the Decision Boundaries
1
2
Feature 2
1
2
2
?
1
Feature 1
26
Overall Boundary Piecewise Linear
Decision Region for Class 1
Decision Region for Class 2
1
2
Feature 2
1
2
2
?
1
Feature 1
27
Decision Tree Classifiers

• Widely used in practice
• Can handle both real-valued and nominal inputs
  (unusual)
• Good with high-dimensional data
• similar algorithms as used in constructing
  regression trees
• historically, developed both in statistics and
  computer science
• Statistics
• Breiman, Friedman, Olshen and Stone, CART, 1984
• Computer science
• Quinlan, ID3, C4.5 (1980s-1990s)

28
Decision Tree Example
Debt
Income
29
Decision Tree Example
Debt
Income gt t1
??
Income
t1
30
Decision Tree Example
Debt
Income gt t1
t2
Debt gt t2
Income
t1
??
31
Decision Tree Example
Debt
Income gt t1
t2
Debt gt t2
Income
t1
t3
Income gt t3
32
Decision Tree Example
Debt
Income gt t1
t2
Debt gt t2
Income
t1
t3
Income gt t3
Note tree boundaries are piecewise linear and
axis-parallel
33
Decision Trees are not stable
Moving just one example slightly may lead to
quite different trees and space partition! Lack
of stability against small perturbation of data.
Figure from Duda, Hart Stork, Chap. 8
34
Decision Tree Pseudocode
node tree-design (Data X,C) For i 1 to
d quality_variable(i) quality_score(Xi,
C) end node X_split, Threshold for
maxquality_variable Data_right, Data_left
split(Data, X_split, Threshold) if node
leaf? return(node) else node_right
tree-design(Data_right) node_left
tree-design(Data_left) end end
35
Binary split selection criteria

• Q(t) N1Q1(t) N2Q2(t), where t is the
  threshold
• Let p1k be the proportion of class k points in
  region 1
• Error criterion for a branch
• Q1(t) 1 - p1k
• Gini index Q1(t) Sk p1k (1 -
  p1k)
• Cross-entropy Q1(t) Sk p1k
  log p1k
• Cross-entropy and Gini work better in general
• Tend to give higher rank to splits with more
  extreme class distributions
• Consider (300,100) (100,300) split versus
  (400,0) (200 200)

36
Computational Complexity for a Binary Tree

• At the root node, for each of p variables
• Sort all values, compute quality for each split
• O(pN log N) time for real-valued or ordinal
  variables
• Subsequent internal node operations each take
  O(N log N)
• e.g., balanced tree of depth K requires
• pN log N 2(pN/2 log N/2) 4(pN/4 log N/4)
  . 2K(pN/2K log N/2K)
• pN(logN log(N/2) log(N/4) log N/2K)
• This assumes data are in main memory
• If data are on disk then repeated access of
  subsets at different nodes may be very slow
  (impossible to pre-index)

37
Splitting on a nominal attribute

• Nominal attribute with m values
• e.g., the name of a state or a city in marketing
  data
• 2m-1 possible subsets gt exhaustive search is
  O(2m-1)
• For small m, a simple approach is to branch on
  specific values
• But for large m this may not work well
• Neat trick for the 2-class problem
• For each predictor value calculate the proportion
  of class 1s
• Order the m values according to these proportions
• Now treat as an ordinal variable and select the
  best split (linear in m)
• This gives the optimal split for the Gini index,
  among all possible 2m-1 splits (Breiman et al,
  1984).

38
How to Choose the Right-Sized Tree?
Predictive Error
Error on Test Data
Error on Training Data
Size of Decision Tree
Ideal Range for Tree Size
39
Choosing a Good Tree for Prediction

• General idea
• grow a large tree
• prune it back to create a family of subtrees
• weakest link pruning
• score the subtrees and pick the best one
• Massive data sizes (e.g., n 100k data points)
• use training data set to fit a set of trees
• use a validation data set to score the subtrees
• Smaller data sizes (e.g., n 1k or less)
• use cross-validation
• use explicit penalty terms (e.g., Bayesian
  methods)

40
Example Spam Email Classification

• Data Set (from the UCI Machine Learning Archive)
• 4601 email messages from 1999
• Manually labelled as spam (60), non-spam (40)
• 54 features percentage of words matching a
  specific word/character
• Business, address, internet, free, george, !, ,
  etc
• Average/longest/sum lengths of uninterrupted
  sequences of CAPS
• Error Rates (Hastie, Tibshirani, Friedman, 2001)
• Training 3056 emails, Testing 1536 emails
• Decision tree 8.7
• Logistic regression error 7.6
• Naïve Bayes 10 (typically)

41
(No Transcript)
42
(No Transcript)
43
Treating Missing Data in Trees

• Missing values are common in practice
• Approaches to handing missing values
• During training
• Ignore rows with missing values (inefficient)
• During testing
• Send the example being classified down both
  branches and average predictions
• Replace missing values with an imputed value
  (can be suboptimal)
• Other approaches
• Treat missing as a unique value (useful if
  missing values are correlated with the class)
• Surrogate splits method
• Search for and store surrogate variables/splits
  during training

44
Other Issues with Classification Trees

• Why use binary splits?
• Multiway splits can be used, but cause
  fragmentation
• Linear combination splits?
• can produces small improvements
• optimization is much more difficult (need weights
  and split point)
• Trees are much less interpretable
• Model instability
• A small change in the data can lead to a
  completely different tree
• Model averaging techniques (like bagging) can be
  useful
• Tree bias
• Poor at approximating non-axis-parallel
  boundaries
• Producing rule sets from tree models (e.g., c5.0)

45
Why Trees are widely used in Practice

• Can handle high dimensional data
• builds a model using 1 dimension at time
• Can handle any type of input variables
• categorical, real-valued, etc
• most other methods require data of a single type
  (e.g., only real-valued)
• Trees are (somewhat) interpretable
• domain expert can read off the trees logic
• Tree algorithms are relatively easy to code and
  test

46
Limitations of Trees

• Representational Bias
• classification piecewise linear boundaries,
  parallel to axes
• regression piecewise constant surfaces
• High Variance
• trees can be unstable as a function of the
  sample
• e.g., small change in the data -gt completely
  different tree
• causes two problems
• 1. High variance contributes to prediction error
• 2. High variance reduces interpretability
• Trees are good candidates for model combining
• Often used with boosting and bagging
• Trees do not scale well to massive data sets
  (e.g., N in millions)
• repeated random access of subsets of the data

47
Evaluating Classification Results (in general)

• Summary statistics
• empirical estimate of score function on test
  data, eg., error rate
• More detailed breakdown
• E.g., confusion matrices
• Can be quite useful in detecting systematic
  errors
• Detection v. false-alarm plots (2 classes)
• Binary classifier with real-valued output for
  each example, where higher means more likely to
  be class 1
• For each possible threshold, calculate
• Detection rate fraction of class 1 detected
• False alarm rate fraction of class 2 detected
• Plot y (detection rate) versus x (false alarm
  rate)
• Also known as ROC, precision-recall,
  specificity/sensitivity

48
Bagging for Combining Classifiers

• Training data sets of size N
• Generate B bootstrap sampled data sets of size
  N
• Bootstrap sample sample with replacement
• e.g. B 100
• Build B models (e.g., trees), one for each
  bootstrap sample
• Intuition is that the bootstrapping perturbs
  the data enough to make the models more resistant
  to true variability
• For prediction, combine the predictions from the
  B models
• E.g., for classification p(c x) fraction of B
  models that predict c
• Plus generally improves accuracy on models such
  as trees
• Negative lose interpretability

49
green majority vote purple averaging the
probabilities
From Hastie, Tibshirani, And Friedman, 2001
50
Illustration of Boosting Color of points class
label Diameter of points weight at each
iteration Dashed line single stage classifier.
Green line combined, boosted classifier Dotted
blue in last two bagging (from G. Rätsch, Phd
thesis, 2001)
51
Support Vector Machines(will be discussed again
later)

• Support vector machines
• Use a different loss function, the margin
• Results in convex optimization problem, solvable
  by quadratic programming
• Decision boundary represented by examples in
  training data
• Linear version clever placement of the
  hyperplane
• Non-linear version kernel trick for
  high-dimensional problems
• Computational complexity can be O(N3) without
  speedups

52
Summary on Classifiers

• Simple models (but can be effective)
• Logistic regression
• Naïve Bayes
• K nearest-neighbors
• Decision trees
• Good for high-dimensional problems with different
  data types
• State of the art
• Support vector machines
• Boosted trees
• Many tradeoffs in interpretability, score
  functions, etc

53
Decision Tree Classifiers
Classification
Task
Decision boundaries hierarchy of axis-parallel
Representation
Cross-validated error
Score Function
Greedy search in tree space
Search/Optimization
Data Management
None specified
Models, Parameters
Tree
54
Naïve Bayes Classifier
Classification
Task
Conditional independence probability model
Representation
Score Function
Likelihood
Closed form probability estimates
Search/Optimization
Data Management
None specified
Models, Parameters
Conditional probability tables
55
Logistic Regression
Task
Classification
Log-odds(C) linear function of Xs
Representation
Score Function
Log-likelihood
Search/Optimization
Iterative (Newton) method
Data Management
None specified
Models, Parameters
Logistic weights
56
Nearest Neighbor Classifier
Task
Classification
Representation
Memory-based
Cross-validated error (for selecting k)
Score Function
Search/Optimization
None
Data Management
None specified
Models, Parameters
None
57
Support Vector Machines
Task
Classification
Representation
Hyperplanes
Score Function
Margin
Convex optimization (quadratic programming)
Search/Optimization
Data Management
None specified
Models, Parameters
None
58
Software (same as for Regression)

• MATLAB
• Many free toolboxes on the Web for regression
  and prediction
• e.g., see http//lib.stat.cmu.edu/matlab/ and
  in particular the CompStats toolbox
• R
• General purpose statistical computing environment
  (successor to S)
• Free (!)
• Widely used by statisticians, has a huge library
  of functions and visualization tools
• Commercial tools
• SAS, other statistical packages
• Data mining packages
• Often are not progammable offer a fixed menu of
  items

59
Reading

• For this class Chapter 10
• Covers both general concepts in classification
  and a broad range of classifiers
• Suggested background reading for further
  information
• Elements of Statistical Learning,
• T. Hastie, R. Tibshirani, and J. Friedman,
  Springer Verlag, 2001
• Learning from Kernels,
• B Schoelkopf and A. Smola, MIT Press, 2003.
• Classification Trees,
• Breiman, Friedman, Olshen, and Stone, Wadsworth
  Press, 1984.