Transcript Data Mining Lecture 1 - University of California, Irvine

ICS 278: Data Mining
Lectures 7 and 8: Classification Algorithms
Padhraic Smyth
Department of Information and Computer Science
University of California, Irvine
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC 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 = q
– Can also use other Greek letters, like a, b, d, g ew
– Vector of parameters = q
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Region Terminlogy
TWO-CLASS DATA IN A TWO-DIMENSIONAL FEATURE SPACE
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Decision
Region 1
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Decision
Region 2
Feature 2
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0
Decision
Boundary
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Feature 1
Lecture 7: Classification
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Padhraic Smyth, UC Irvine
Probabilistic view of Classification
•
Notation: let there be K classes c1,…..cK
•
Class marginals: p(ck) = probability of class k
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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.
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Example of Probabilistic Classification
p( x | c2 )
Data Mining Lectures
Lecture 7: Classification
p( x | c1 )
Padhraic Smyth, UC Irvine
Example of Probabilistic Classification
p( x | c2 )
p( x | c1 )
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p( c1 | x )
0.5
0
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Example of Probabilistic Classification
p( x | c2 )
p( x | c1 )
1
p( c1 | x )
0.5
0
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Regions and Bayes Error Rate
p( x | c2 )
Class c2
Class c1
Class c2
p( x | c1 )
Class c2
Class c1
Optimal decision regions = regions where 1 class is more likely
Optimal decision regions  optimal decision boundaries
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Regions and Bayes Error Rate
p( x | c2 )
Class c2
Class c1
Class c2
p( x | c1 )
Class c2
Class c1
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)
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
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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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Three types of classifiers
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Generative (or class-conditional) classifiers:
– Learn models for p( x | ck ), use Bayes rule to find decision boundaries
– Examples: naïve Bayes models, Gaussian classifiers
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Regression (or posterior class probabilities):
– Learn a model for p( ck | x ) directly
– Example: logistic regression (see lecture 5/6), neural networks
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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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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 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(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
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Lecture 7: Classification
(Brier score)
Padhraic Smyth, UC Irvine
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.
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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 lowdimensional 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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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)
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Link between Logistic Regression and Naïve Bayes
Naïve Bayes
P(C | d )
P (C )
P( w | C )
log
 log
  log
P(C | d )
P (C ) wd
P( w | C )
Logistic Regression
P (C | d )
log
 a   bw  w
P (C | d )
wd
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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”
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
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Feature 2
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2
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Feature 1
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Finding the Decision Boundaries
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Feature 2
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Feature 1
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Finding the Decision Boundaries
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Feature 2
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Feature 1
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Finding the Decision Boundaries
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Feature 2
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Feature 1
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Overall Boundary = Piecewise Linear
Decision Region
for Class 1
Decision Region
for Class 2
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2
Feature 2
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2
?
2
1
Feature 1
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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)
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Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Tree Example
Debt
Income
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Lecture 7: Classification
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Decision Tree Example
Debt
Income > t1
??
t1
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Income
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Tree Example
Debt
Income > t1
t2
Debt > t2
t1
Income
??
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Tree Example
Debt
Income > t1
t2
Debt > t2
t3
t1
Income
Income > t3
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Tree Example
Debt
Income > t1
t2
Debt > t2
t3
Income
t1
Income > t3
Note: tree boundaries are piecewise
linear and axis-parallel
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Binary split selection criteria
•
Q(t) = N1Q1(t) + N2Q2(t), where t is the threshold
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Let p1k be the proportion of class k points in region 1
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Error criterion for a branch
Q1(t) = 1 - p1k*
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Gini index:
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Cross-entropy:
Q1(t) = Sk p1k log p1k
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Cross-entropy and Gini work better in general
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Q1(t) = Sk p1k (1 - p1k)
– Tend to give higher rank to splits with more extreme class distributions
– Consider [(300,100) (100,300)] split versus [(400,0) (200 200)]
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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)
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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:
–
–
–
–
Data Mining Lectures
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).
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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)
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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)
–
–
–
–
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Training: 3056 emails, Testing: 1536 emails
Decision tree = 8.7%
Logistic regression: error = 7.6%
Naïve Bayes = 10% (typically)
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
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Model instability
– A small change in the data can lead to a completely different tree
– Model averaging techniques (like bagging) can be useful
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Tree “bias”
– Poor at approximating non-axis-parallel boundaries
•
Producing rule sets from tree models (e.g., c5.0)
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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 realvalued)
• Trees are (somewhat) interpretable
– domain expert can “read off” the tree’s logic
• Tree algorithms are relatively easy to code and test
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
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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
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Lecture 7: Classification
Padhraic Smyth, UC Irvine
green = majority vote
purple = averaging
the probabilities
From Hastie, Tibshirani,
And Friedman, 2001
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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)
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
–
–
–
–
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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
Lecture 7: Classification
Padhraic Smyth, UC Irvine
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
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Decision Tree Classifiers
Classification
Task
Representation
Data Mining Lectures
Decision boundaries =
hierarchy of axis-parallel
Score Function
Cross-validated
error
Search/Optimization
Greedy search in
tree space
Data
Management
None specified
Models,
Parameters
Tree
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Naïve Bayes Classifier
Classification
Task
Representation
Conditional independence
probability model
Score Function
Likelihood
Search/Optimization
Closed form
probability estimates
Data
Management
None specified
Models,
Parameters
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Conditional
probability tables
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Padhraic Smyth, UC Irvine
Logistic Regression
Task
Representation
Classification
Log-odds(C) = linear
function of X’s
Score Function
Search/Optimization
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Log-likelihood
Iterative (Newton) method
Data
Management
None specified
Models,
Parameters
Logistic
weights
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Nearest Neighbor Classifier
Task
Classification
Representation
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Memory-based
Score Function
Cross-validated error
(for selecting k)
Search/Optimization
None
Data
Management
None specified
Models,
Parameters
None
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Support Vector Machines
Task
Representation
Hyperplanes
Score Function
“Margin”
Search/Optimization
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Classification
Convex optimization
(quadratic programming)
Data
Management
None specified
Models,
Parameters
None
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Software (same as for Regression)
•
MATLAB
•
R
•
Commercial tools
– 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
– General purpose statistical computing environment (successor to S)
– Free (!)
– Widely used by statisticians, has a huge library of functions and visualization
tools
– SAS, other statistical packages
– Data mining packages
– Often are not progammable: offer a fixed menu of items
Data Mining Lectures
Lecture 7: Classification
Padhraic Smyth, UC Irvine
Reading
•
For this class: Chapter 10:
•
Suggested background reading for further information:
– Covers both general concepts in classification and a broad range of classifiers
– 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