data mining lectures lecture 7: classification padhraic smyth, uc irvine ics 278: data mining...

Data Mining Lectures Lecture 7: Classification Padhraic Smyth, UC Irvine

ICS 278: Data Mining

Lectures 7 and 8: Classification Algorithms

Padhraic SmythDepartment of Information and Computer Science

University of California, Irvine


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 = – Can also use other Greek letters, like ew

– Vector of parameters =


Classification

• Predictive modeling: predict Y given X– Y is real-valued => regression– Y is categorical => classification

• Classification– Many applications: speech recognition, document

classification, OCR, loan approval, face recognition, etc


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


Decision Region Terminlogy

2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

6

Feature 1

Fea

ture

2

TWO-CLASS DATA IN A TWO-DIMENSIONAL FEATURE SPACE

DecisionRegion 2

DecisionRegion 1

DecisionBoundary


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) = p( x | cj ) p(cj)

In theory this is all we need….in practice this may not be best approach.


Example of Probabilistic Classification

p( x | c1 )p( x | c2 )


Example of Probabilistic Classification

p( x | c1 )p( x | c2 )

p( c1 | x )1

0.5

0


Decision Regions and Bayes Error Rate

p( x | c1 )p( x | c2 )

Class c1 Class c2 Class c1Class c2Class c2

Optimal decision regions = regions where 1 class is more likely

Optimal decision regions optimal decision boundaries


Decision Regions and Bayes Error Rate

p( x | c1 )p( x | c2 )

Class c1 Class c2 Class c1Class c2Class 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)


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 )

=> this yields the optimal decision regions/boundaries => 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


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


Which type of classifier is appropriate?

• Lets look at the score functions:– c(i) = true class, c(x(i) ; ) = class predicted by the classifier

Class-mismatch loss functions:

S() = 1/n i Cost [c(i), c(x(i) ; ) ]

where cost(i, j) = cost of misclassifying true class i as predicted class j

e.g., cost(i,j) = 0 if i=j, = 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() = 1/n i log p(c(i) | x(i) ; ) (log probability score)

or S() = 1/n i [ c(i) – p(c(i) | x(i) ; ) ]2 (Brier score)


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: don’t solve a harder problem than you need to, or don’t model aspects of the problem you don’t need to (e.g., modeling p(x|c)) - Vapnik, 1996.


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


Naïve Bayes Classifiers

• Generative probabilistic model with conditional independence assumption on p( x | ck ), i.e. p( x | ck ) = 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)


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


Link between Logistic Regression and Naïve Bayes

( | ) ( ) ( | )log log log

( | ) ( ) ( | )w d

P C d P C P w C

P C d P C P w C

Naïve Bayes

Logistic Regression

( | )log

( | ) ww d

P C dw

P C d


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


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 < error(kNN) < 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”


Local Decision Boundaries

1

1

1

2

2

2

Feature 1

Feature 2

?

Boundary? Points that are equidistantbetween points of class 1 and 2Note: locally the boundary is(1) linear (because of Euclidean distance)(2) halfway between the 2 class points(3) at right angles to connector


Finding the Decision Boundaries

1

1

1

2

2

2

Feature 1

Feature 2

?


Overall Boundary = Piecewise Linear

1

1

1

2

2

2

Feature 1

Feature 2

?

Decision Region for Class 1

Decision Region for Class 2


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 (1980’s-1990’s)


Decision Tree Example

Income

Debt



t1 Income

DebtIncome > t1

??



t1

t2

Income

DebtIncome > t1

Debt > t2

??



t1t3

t2

Income

DebtIncome > t1

Debt > t2

Income > t3



t1t3

t2

Income

DebtIncome > t1

Debt > t2

Income > t3

Note: tree boundaries are piecewiselinear and axis-parallel


Figure from Duda, Hart & Stork, Chap. 8

Moving just one example slightly may lead to quite different trees and space partition!

Lack of stability against small perturbation of data.

Decision Trees are not stable


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 max{quality_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


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) = k p1k (1 - p1k)

• Cross-entropy: Q1(t) = k 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)]


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)


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 => 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 1’s – 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).


How to Choose the Right-Sized Tree?

PredictiveError

Size of Decision Tree

Error on Training Data

Error on Test Data

Ideal Rangefor Tree Size


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)


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)


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


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)


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 tree’s logic

• Tree algorithms are relatively easy to code and test


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 -> 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


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


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


green = majority votepurple = averaging the probabilities

From Hastie, Tibshirani,And Friedman, 2001


Illustration of Boosting:Color of points = class labelDiameter of points = weight at each iterationDashed line: single stage classifier. Green line: combined, boosted classifierDotted blue in last two: bagging (from G. Rätsch, Phd thesis, 2001)


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


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


Task

Decision Tree Classifiers

Representation

Score Function

Search/Optimization

Data Management

Models, Parameters

Classification

Decision boundaries = hierarchy of axis-parallel

Cross-validated error

Greedy search in tree space

None specified

Tree


Task

Naïve Bayes Classifier

Representation

Score Function

Search/Optimization

Data Management

Models, Parameters

Classification

Conditional independence probability model

Likelihood

Closed form probability estimates

None specified

Conditional probability tables


Task

Logistic Regression

Representation

Score Function

Search/Optimization

Data Management

Models, Parameters

Classification

Log-odds(C) = linear function of X’s

Log-likelihood

Iterative (Newton) method

None specified

Logistic weights


Task

Nearest Neighbor Classifier

Representation

Score Function

Search/Optimization

Data Management

Models, Parameters

Classification

Memory-based

Cross-validated error (for selecting k)

None

None specified

None


Task

Support Vector Machines

Representation

Score Function

Search/Optimization

Data Management

Models, Parameters

Classification

Hyperplanes

“Margin”

Convex optimization

(quadratic programming)

None specified

None


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


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.

data mining lectures lecture 7: classification padhraic smyth, uc irvine ics 278: data mining...

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