Agenda

  • Classification: Basic Concepts
  • Decision Tree Induction
  • Bayes Classification Methods
  • Rule-Based Classification
  • Model Evaluation and Selection
  • Techniques to Improve Classification Accuracy: Ensemble Methods
  • Summary


Supervised vs. Unsupervised Learning

  • Supervised learning (classification)
    • Supervision: The training data (observations, measurements, etc.) are accompanied by labels indicating the class of the observations
    • New data is classified based on the training set
  • Unsupervised learning (clustering)
    • The class labels of training data is unknown
    • Given a set of measurements, observations, etc. with the aim of establishing the existence of classes or clusters in the data


  • Classification
    • predicts categorical class labels (discrete or nominal)
    • classifies data (constructs a model) based on the training set and the values (class labels) in a classifying attribute and uses it in classifying new data
  • Numeric Prediction
    • models continuous-valued functions, i.e., predicts unknown or missing values
  • Typical applications
    • Credit/loan approval:
    • Medical diagnosis: if a tumor is cancerous or benign
    • Fraud detection: if a transaction is fraudulent
    • Web page categorization: which category it is

Prediction Problems: Classification vs. Numeric Prediction



Classification—A Two-Step Process

  • Model construction: describing a set of predetermined classes
    • Each tuple/sample is assumed to belong to a predefined class, as determined by the class label attribute
    • The set of tuples used for model construction is training set
    • The model is represented as classification rules, decision trees, or mathematical formulae
  • Model usage: for classifying future or unknown objects
    • Estimate accuracy of the model
      • The known label of test sample is compared with the classified result from the model
      • Accuracy rate is the percentage of test set samples that are correctly classified by the model
      • Test set is independent of training set (otherwise overfitting)
    • If the accuracy is acceptable, use the model to classify new data
  • Note: If the test set is used to select models, it is called validation (test) set


Process (1): Model Construction



Process (2): Using the Model in Prediction



Decision Tree Induction: An Example



Algorithm for Decision Tree Induction

  • Basic algorithm (a greedy algorithm)
    • Tree is constructed in a top-down recursive divide-and-conquer manner
    • At start, all the training examples are at the root
    • Attributes are categorical (if continuous-valued, they are discretized in advance)
    • Examples are partitioned recursively based on selected attributes
    • Test attributes are selected on the basis of a heuristic or statistical measure (e.g., information gain)
  • Conditions for stopping partitioning
    • All samples for a given node belong to the same class
    • There are no remaining attributes for further partitioning – majority voting is employed for classifying the leaf
    • There are no samples left


Brief Review of Entropy

  • Entropy (Information Theory)
  • A measure of uncertainity associated with a random variable
  • Calculation: for a discrete random variable Y taking m distinct values {y1,...,ym},
    \[H(Y)=-\sum_{i=1}^{m}p_{i}log(p_{i}), p_{i}=P(Y=y_{i})\]
  • Interpretation:
    • Higher entropy=>higher uncertainty
    • Lower entropy=>lower uncertainty
  • Conditional entropy
    \[H(Y|X)=\sum_{x}p(x)H(Y|X=x)\]



Attribute Selection Measure: Information Gain (ID3/C4.5)

  • Attribute Selection Measure: Information Gain (ID3/C4.5)
  • Select the attribute with the highest information gain
  • Let pi be the probability that an arbitrary tuple in D belongs to class Ci, estimated by

\[ \frac{ |C_{i,D}|} {|D|}\]

  • Expected information (entropy) needed to classify a tuple in D:

\[Info(D)=-\sum_{i=1}^{m} p_{i}log_{2}(p_{i})\]

  • Information needed (after using A to split D into v partitions) to classify D:

\[Info_{A}(D)=-\sum_{j=1}^{v} \frac{|D_{j}|}{D}\times Info (D_{j})\]

  • Information gained by branching on attribute A

\[Gain(A)=Info(D)-Info_{A}(D)\]



Attribute Selection: Information Gain



Computing Information-Gain for Continuous-Valued Attributes

  • Let attribute A be a continuous-valued attribute
  • Must determine the best split point for A
    • Sort the value A in increasing order
    • Typically, the midpoint between each pair of adjacent values is considered as a possible split point
      • (ai+ai+1)/2 is the midpoint between the values of ai and ai+1
    • The point with the minimum expected information requirement for A is selected as the split-point for A
  • Split:
    • D1 is the set of tuples in D satisfying A ≤ split-point, and D2 is the set of tuples in D satisfying A > split-point


Gain Ratio for Attribute Selection (C4.5)

  • Information gain measure is biased towards attributes with a large number of values
  • C4.5 (a successor of ID3) uses gain ratio to overcome the problem (normalization to information gain)

\[SplitInfo_{A}(D)=-\sum_{j=1}^{v} \frac{|D_{j}|}{|D|}\times log_{2}(\frac{|D_{j}|}{|D|})\]
\[GainRatio(A) = \frac{Gain(A)}{SplitInfo(A)}\]

  • Ex.
    • gain_ratio(income) = 0.029/1.557 = 0.019
  • The attribute with the maximum gain ratio is selected as the splitting attribute


Gini Index (CART, IBM IntelligentMiner)

  • If a data set D contains examples from n classes, gini index, gini(D) is defined as

\[gini(D)=1-\sum_{j=1}^{n}p^{2}j\]

  • where pj is the relative frequency of class j in D
  • If a data set D is split on A into two subsets D1 and D2, the gini index gini(D) is defined as

\[gini_{A}(D)=\frac{|D_{1}|}{|D|}gini(D_{1})+\frac{|D_{2}|}{|D|} gini(D_{1})\]

  • Reduction in Impurity:

\[\Delta gini(A)=gini(D)-gini_{A}(D)\]

  • The attribute provides the smallest ginisplit(D) (or the largest reduction in impurity) is chosen to split the node (need to enumerate all the possible splitting points for each attribute)


Computation of Gini Index

  • Ex. D has 9 tuples in buys_computer = “yes” and 5 in “no”

\[gini(D)=1-\left ( \frac{9}{14} \right )^{2}-\left ( \frac{5}{14} \right )^{2}=0.459\]

  • Suppose the attribute income partitions D into 10 in D1: {low, medium} and 4 in D2

\[gini_{income\epsilon \left \{ low,medium \right \}}(D)=\left ( \frac{10}{14} \right )Gini(D_{1})+\left ( \frac{4}{14} \right )Gini(D_{2})\]

\[\frac{10}{14}\left (1-\left ( \frac{7}{10} \right )^{2}-\left ( \frac{3}{10} \right )^{2}  \right )+\frac{4}{14}\left (1-\left ( \frac{2}{4} \right )^{2}-\left ( \frac{2}{4} \right )^{2}  \right )\]

\[=0.443=gini_{income\epsilon \left \{ high \right \}}(D)\]

    • Gini{low,high} is 0.458; Gini{medium,high} is 0.450. Thus, split on the {low,medium} (and {high}) since it has the lowest Gini index
  • All attributes are assumed continuous-valued
  • May need other tools, e.g., clustering, to get the possible split values
  • Can be modified for categorical attributes


Comparing Attribute Selection Measures

  • The three measures, in general, return good results but
    • Information gain:
      • biased towards multivalued attributes
    • Gain ratio:
      • tends to prefer unbalanced splits in which one partition is much smaller than the others
    • Gini index:
      • biased to multivalued attributes
      • has difficulty when # of classes is large
      • tends to favor tests that result in equal-sized partitions and purity in both partitions


Other Attribute Selection Measures

  • CHAID: a popular decision tree algorithm, measure based on χ2 test for independence
  • C-SEP: performs better than info. gain and gini index in certain cases
  • G-statistic: has a close approximation to χ^2 distribution
  • MDL (Minimal Description Length) principle (i.e., the simplest solution is preferred):
    • The best tree as the one that requires the fewest # of bits to both (1) encode the tree, and (2) encode the exceptions to the tree
  • Multivariate splits (partition based on multiple variable combinations)
    • CART: finds multivariate splits based on a linear comb. of attrs.
  • Which attribute selection measure is the best?
    • Most give good results, none is significantly superior than others


Overfitting and Tree Pruning

  • Overfitting: An induced tree may overfit the training data
    • Too many branches, some may reflect anomalies due to noise or outliers
    • Poor accuracy for unseen samples
  • Two approaches to avoid overfitting
    • Prepruning: Halt tree construction early ̵ do not split a node if this would result in the goodness measure falling below a threshold
      • Difficult to choose an appropriate threshold
    • Postpruning: Remove branches from a “fully grown” tree—get a sequence of progressively pruned trees
      • Use a set of data different from the training data to decide which is the “best pruned tree”


Enhancements to Basic Decision Tree Induction

  • Allow for continuous-valued attributes
    • Dynamically define new discrete-valued attributes that partition the continuous attribute value into a discrete set of intervals
  • Handle missing attribute values
    • Assign the most common value of the attribute
    • Assign probability to each of the possible values
  • Attribute construction
    • Create new attributes based on existing ones that are sparsely represented
    • This reduces fragmentation, repetition, and replication


Classification in Large Databases

  • Classification—a classical problem extensively studied by statisticians and machine learning researchers
  • Scalability: Classifying data sets with millions of examples and hundreds of attributes with reasonable speed
  • Why is decision tree induction popular?
    • relatively faster learning speed (than other classification methods)
    • convertible to simple and easy to understand classification rules
    • can use SQL queries for accessing databases
    • comparable classification accuracy with other methods
  • RainForest (VLDB’98 — Gehrke, Ramakrishnan & Ganti)
    • Builds an AVC-list (attribute, value, class label)


Scalability Framework for RainForest

  • Separates the scalability aspects from the criteria that determine the quality of the tree
  • Builds an AVC-list: AVC (Attribute, Value, Class_label)
  • AVC-set (of an attribute X )
    • Projection of training dataset onto the attribute X and class label where counts of individual class label are aggregated
  • AVC-group (of a node n )
    • Set of AVC-sets of all predictor attributes at the node n


Rainforest: Training Set and Its AVC Sets



BOAT (Bootstrapped Optimistic Algorithm for Tree Construction)

  • Use a statistical technique called bootstrapping to create several smaller samples (subsets), each fits in memory
  • Each subset is used to create a tree, resulting in several trees
  • These trees are examined and used to construct a new tree T’
    • It turns out that T’ is very close to the tree that would be generated using the whole data set together
  • Adv: requires only two scans of DB, an incremental alg.


Presentation of Classification Results



Visualization of a Decision Tree in SGI/MineSet 3.0



Interactive Visual Mining by Perception-Based Classification (PBC)



Bayesian Classification: Why?

  • A statistical classifier: performs probabilistic prediction, i.e., predicts class membership probabilities
  • Foundation: Based on Bayes’ Theorem.
  • Performance: A simple Bayesian classifier, naïve Bayesian classifier, has comparable performance with decision tree and selected neural network classifiers
  • Incremental: Each training example can incrementally increase/decrease the probability that a hypothesis is correct — prior knowledge can be combined with observed data
  • Standard: Even when Bayesian methods are computationally intractable, they can provide a standard of optimal decision making against which other methods can be measured


Bayes’ Theorem: Basics

  • Total probability Theorem:

\[P(B)=\sum_{i=1}^{M}P(B|A_{i})P(A_{i})\]

  • Bayes’ Theorem:

\[P(H|X)=\frac{P(X|H)P(H)}{P(X)}=P(X|H)\times P(H)/P(X)\]

    • Let X be a data sample (“evidence”): class label is unknown
    • Let H be a hypothesis that X belongs to class C
    • Classification is to determine P(H|X), (i.e., posteriori probability): the probability that the hypothesis holds given the observed data sample X
    • P(H) (prior probability): the initial probability
      • E.g., X will buy computer, regardless of age, income, …
    • P(X): probability that sample data is observed
    • P(X|H) (likelihood): the probability of observing the sample X, given that the hypothesis holds
      • E.g., Given that X will buy computer, the prob. that X is 31..40, medium income


Prediction Based on Bayes’ Theorem

  • Given training data X, posteriori probability of a hypothesis H, P(H|X), follows the Bayes’ theorem
    \[P(H|X)=\frac{P(X|H)P(H)}{P(X)}=P(X|H)\times P(H)/P(X)\]
  • Informally, this can be viewed as
    • posteriori = likelihood x prior/evidence
  • Predicts X belongs to Ci iff the probability P(Ci|X) is the highest among all the P(Ck|X) for all the k classes
  • Practical difficulty: It requires initial knowledge of many probabilities, involving significant computational cost


Classification Is to Derive the Maximum Posteriori

  • Let D be a training set of tuples and their associated class labels, and each tuple is represented by an n-D attribute vector X = (x1, x2, …, xn)
  • Suppose there are m classes C1, C2, …, Cm.
  • Classification is to derive the maximum posteriori, i.e., the maximal P(Ci|X)
  • This can be derived from Bayes’ theorem

\[P(C_{i}|X)=\frac{P(X|C_{i})P(C_{i})}{P(X)}\]

  • Since P(X) is constant for all classes, only 

\[P(C_{i}|X)=P(X|C_{i})P(C_{i})\]

needs to be maximized



Naïve Bayes Classifier

  • A simplified assumption: attributes are conditionally independent (i.e., no dependence relation between attributes):

\[P(X|C_{i})=\prod_{k=1}^{n}P(x_{k}|C_{i})=P(x_{1}|C_{i})\times P(x_{2}|C_{i})\times ... \times P(x_{n}|C_{i})\]

  • This greatly reduces the computation cost: Only counts the class distribution
  • If Ak is categorical, P(xk|Ci) is the # of tuples in Ci having value xk for Ak divided by |Ci, D| (# of tuples of Ci in D)
  • If Ak is continous-valued, P(xk|Ci) is usually computed based on Gaussian distribution with a mean μ and standard deviation σ

\[g(x,\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma ^{2}}}\]

    • and P(xk|Ci) is 

\[P(X|C_{i})=g(x_{k},\mu_{C_{i}},\sigma_{C_{i}})\]



Naïve Bayes Classifier: Training Dataset



Naïve Bayes Classifier: An Example

  • P(Ci): P(buys_computer = “yes”) = 9/14 = 0.643
    P(buys_computer =“no”) = 5/14= 0.357
  • Compute P(X|Ci) for each class
    P(age = “<=30” | buys_computer = “yes”) = 2/9 = 0.222
    P(age = “<= 30” | buys_computer = “no”) = 3/5 = 0.6
    P(income = “medium” | buys_computer = “yes”) = 4/9 = 0.444
    P(income = “medium” | buys_computer = “no”) = 2/5 = 0.4
    P(student = “yes” | buys_computer = “yes) = 6/9 = 0.667
    P(student = “yes” | buys_computer = “no”) = 1/5 = 0.2
    P(credit_rating = “fair” | buys_computer = “yes”) = 6/9 = 0.667
    P(credit_rating = “fair” | buys_computer = “no”) = 2/5 = 0.4
  • X = (age <= 30 , income = medium, student = yes, credit_rating = fair)
    P(X|Ci) : P(X|buys_computer = “yes”) = 0.222 x 0.444 x 0.667 x 0.667 = 0.044
    P(X|buys_computer = “no”) = 0.6 x 0.4 x 0.2 x 0.4 = 0.019
    P(X|Ci)*P(Ci) : P(X|buys_computer = “yes”) * P(buys_computer = “yes”) = 0.028
    P(X|buys_computer = “no”) * P(buys_computer = “no”) = 0.007
    Therefore, X belongs to class (“buys_computer = yes”)


Avoiding the Zero-Probability Problem

  • Naïve Bayesian prediction requires each conditional prob. be non-zero. Otherwise, the predicted prob. will be zero

\[P(X|C_{i})=\prod_{k=1}^{n}P(x_{k}|C_{i})\]

  • Ex. Suppose a dataset with 1000 tuples, income=low (0), income= medium (990), and income = high (10)
  • Use Laplacian correction (or Laplacian estimator)
    • Adding 1 to each case
      • Prob(income = low) = 1/1003
      • Prob(income = medium) = 991/1003
      • Prob(income = high) = 11/1003
    • The “corrected” prob. estimates are close to their “uncorrected” counterparts


Naïve Bayes Classifier: Comments

  • Advantages
    • Easy to implement
    • Good results obtained in most of the cases
  • Disadvantages
    • Assumption: class conditional independence, therefore loss of accuracy
    • Practically, dependencies exist among variables
      • E.g., hospitals: patients: Profile: age, family history, etc.
        • Symptoms: fever, cough etc., Disease: lung cancer, diabetes, etc.
      • Dependencies among these cannot be modeled by Naïve Bayes Classifier
  • How to deal with these dependencies? Bayesian Belief Networks 


Using IF-THEN Rules for Classification

  • Represent the knowledge in the form of IF-THEN rules
    • R: IF age = youth AND student = yes THEN buys_computer = yes
    • Rule antecedent/precondition vs. rule consequent
  • Assessment of a rule: coverage and accuracy
    • ncovers = # of tuples covered by R
    • ncorrect = # of tuples correctly classified by R
    • coverage(R) = ncovers /|D| /* D: training data set */
    • accuracy(R) = ncorrect / ncovers
  • If more than one rule are triggered, need conflict resolution
    • Size ordering: assign the highest priority to the triggering rules that has the “toughest” requirement (i.e., with the most attribute tests)
    • Class-based ordering: decreasing order of prevalence or misclassification cost per class
    • Rule-based ordering (decision list): rules are organized into one long priority list, according to some measure of rule quality or by experts


Rule Extraction from a Decision Tree

  • Rules are easier to understand than large trees
  • One rule is created for each path from the root to a leaf
  • Each attribute-value pair along a path forms a conjunction: the leaf holds the class prediction
  • Rules are mutually exclusive and exhaustive
  • Example: Rule extraction from our buys_computer decision-tree
    IF age = young AND student = no THEN buys_computer = no
    IF age = young AND student = yes THEN buys_computer = yes
    IF age = mid-age THEN buys_computer = yes
    IF age = old AND credit_rating = excellent THEN buys_computer = no
    IF age = old AND credit_rating = fair THEN buys_computer = yes



Rule Induction: Sequential Covering Method

  • Sequential covering algorithm: Extracts rules directly from training data
  • Typical sequential covering algorithms: FOIL, AQ, CN2, RIPPER
  • Rules are learned sequentially, each for a given class Ci will cover many tuples of Ci but none (or few) of the tuples of other classes
  • Steps:
    • Rules are learned one at a time
    • Each time a rule is learned, the tuples covered by the rules are removed
    • Repeat the process on the remaining tuples until termination condition, e.g., when no more training examples or when the quality of a rule returned is below a user-specified threshold
  • Comp. w. decision-tree induction: learning a set of rules simultaneously


Sequential Covering Algorithm

while (enough target tuples left)
    generate a rule
    remove positive target tuples satisfying this rule


Rule Generation

  • To generate a rule
    while(true)
           find the best predicate p
           if foil-gain(p) > threshold then add p to current rule
           else break



How to Learn-One-Rule?

  • Start with the most general rule possible: condition = empty
  • Adding new attributes by adopting a greedy depth-first strategy
    • Picks the one that most improves the rule quality
  • Rule-Quality measures: consider both coverage and accuracy
    • Foil-gain (in FOIL & RIPPER): assesses info_gain by extending condition

\[FOIL-Gain=pos{'} \times \left ( log_{2}\frac{pos'}{pos'+neg'}-log_{2}\frac{pos}{pos+neg} \right )\]

      • favors rules that have high accuracy and cover many positive tuples
  • Rule pruning based on an independent set of test tuples

\[FOIL-Prune(R)=\frac{pos-neg}{pos+neg} \right )\]

      • Pos/neg are # of positive/negative tuples covered by R.
      • If FOIL_Prune is higher for the pruned version of R, prune R


Model Evaluation and Selection

  • Evaluation metrics: How can we measure accuracy? Other metrics to consider?
  • Use validation test set of class-labeled tuples instead of training set when assessing accuracy
  • Methods for estimating a classifier’s accuracy:
    • Holdout method, random subsampling
    • Cross-validation
    • Bootstrap
  • Comparing classifiers:
    • Confidence intervals
    • Cost-benefit analysis and ROC Curves



Classifier Evaluation Metrics: Confusion Matrix

  • Given m classes, an entry, CM i,j in a confusion matrix indicates # of tuples in class i that were labeled by the classifier as class j
  • May have extra rows/columns to provide totals



Classifier Evaluation Metrics: Accuracy, Error Rate, Sensitivity and Specificity

  • Classifier Accuracy, or recognition rate: percentage of test set tuples that are correctly classified
    Accuracy = (TP + TN)/All
  • Error rate: 1 – accuracy, or
    Error rate = (FP + FN)/All
  • Class Imbalance Problem:
    • One class may be rare, e.g. fraud, or HIV-positive
    • Significant majority of the negative class and minority of the positive class
    • Sensitivity: True Positive recognition rate
      • Sensitivity = TP/P
    • Specificity: True Negative recognition rate
      • Specificity = TN/N



Classifier Evaluation Metrics: Precision and Recall, and F-measures

  • Precision: exactness – what % of tuples that the classifier labeled as positive are actually positive

\[precision=\frac{TP}{TP+FP}\]

  • Recall: completeness – what % of positive tuples did the classifier label as positive?

\[recall=\frac{TP}{TP+FN}\]

  • Perfect score is 1.0
  • Inverse relationship between precision & recall
  • F measure (F1 or F-score): harmonic mean of precision and recall,

\[F=\frac{2\times precision \times recall}{precision+recall}\]

  • : weighted measure of precision and recall
    • assigns ß times as much weight to recall as to precision
\[F_{\beta }=\frac{(1+\beta ^{2})\times precision \times recall}{\beta ^{2}\times precision+recall }\]


Classifier Evaluation Metrics: Example

 

Actual Class\Predicted class

cancer = yes

cancer = no

Total

Recognition(%)

 

cancer = yes

90

210

300

30.00 ( sensitivity

 

cancer = no

140

9560

9700

98.56 ( specificity)

 

Total

230

9770

10000

96.40 ( accuracy )

Precision = 90/230 = 39.13% Recall = 90/300 = 30.00%



Evaluating Classifier Accuracy: Holdout & Cross-Validation Methods

  • Holdout method
    • Given data is randomly partitioned into two independent sets
      • Training set (e.g., 2/3) for model construction
      • Test set (e.g., 1/3) for accuracy estimation
    • Random sampling: a variation of holdout
      • Repeat holdout k times, accuracy = avg. of the accuracies obtained
  • Cross-validation (k-fold, where k = 10 is most popular)
    • Randomly partition the data into k mutually exclusive subsets, each approximately equal size
    • At i-th iteration, use Di as test set and others as training set
    • Leave-one-out: k folds where k = # of tuples, for small sized data
    • *Stratified cross-validation*: folds are stratified so that class dist. in each fold is approx. the same as that in the initial data



Evaluating Classifier Accuracy: Bootstrap

  • Bootstrap
    • Works well with small data sets
    • Samples the given training tuples uniformly with replacement
      • i.e., each time a tuple is selected, it is equally likely to be selected again and re-added to the training set
  • Several bootstrap methods, and a common one is .632 boostrap
    • A data set with d tuples is sampled d times, with replacement, resulting in a training set of d samples. The data tuples that did not make it into the training set end up forming the test set. About 63.2% of the original data end up in the bootstrap, and the remaining 36.8% form the test set (since (1 – 1/d)d ≈ e-1 = 0.368)
    • Repeat the sampling procedure k times, overall accuracy of the model:

\[Acc(M)=\frac{1}{k}\sum_{i=1}^{k}(0.632 \times Acc(M_{i})_{test-set}+0.368 \times Acc(M_{i})_{train-set})\]



Estimating Confidence Intervals: Classifier Models M1 vs. M2

  • Suppose we have 2 classifiers, M1 and M2, which one is better?
  • Use 10-fold cross-validation to obtain err'(M1) and err'(M2)
  • These mean error rates are just estimates of error on the true population of future data cases
  • What if the difference between the 2 error rates is just attributed to chance?
    • Use a test of statistical significance
    • Obtain confidence limits for our error estimates


Estimating Confidence Intervals: Null Hypothesis

  • Perform 10-fold cross-validation
  • Assume samples follow a t distribution with k–1 degrees of freedom (here, k=10)
  • Use t-test (or Student’s t-test)
  • Null Hypothesis: M1 & M2 are the same
  • If we can reject null hypothesis, then
    • we conclude that the difference between M1 & M2 is statistically significant
    • Chose model with lower error rate



Estimating Confidence Intervals: t-test

  • If only 1 test set available: pairwise comparison
    • For ith round of 10-fold cross-validation, the same cross partitioning is used to obtain err(M1)i and err(M2)i
    • Average over 10 rounds to get err'(M1) and err'(M2)
    • t-test computes t-statistic with k-1 degrees of freedom:

\[t=\frac{\bar{err}(M_{1})-\bar{err}(M_{2})}{\sqrt{var(M_{1}-M_{2})/k}}\]

\[var(M_{1}-M_{2})=\frac{1}{k}\sum_{i=1}^{k}\left [err(M_{1})_{i} - err(M_{2})_{i} -(\bar{err}(M_{1})-\bar{err}(M_{2}))\right ]^{2}\]

  • If two test sets available: use non-paired t-test

\[var(M_{1}-M_{2})=\sqrt{\frac{var(M_{1})}{k_{1}}+\frac{var(M_{2})}{k_{2}}}\]

where k1 & k2 are # of cross-validation samples used for M1 & M2, resp.



Estimating Confidence Intervals: Table for t-distribution

  • Symmetric
  • Significance level, e.g., sig = 0.05 or 5% means M1 & M2 are significantly different for 95% of population
  • Confidence limit, z = sig/2



Estimating Confidence Intervals: Statistical Significance

  • Are M1 & M2 significantly different?
    • Compute t. Select significance level (e.g. sig = 5%)
    • Consult table for t-distribution: Find t value corresponding to k-1 degrees of freedom (here, 9)
    • t-distribution is symmetric: typically upper % points of distribution shown → look up value for confidence limit z=sig/2 (here, 0.025)
    • If t > z or t < -z, then t value lies in rejection region:
      • Reject null hypothesis that mean error rates of M1 & M2 are same
      • Conclude: statistically significant difference between M1 & M2
    • Otherwise, conclude that any difference is chance



Model Selection: ROC Curves

 
  • ROC (Receiver Operating Characteristics) curves: for visual comparison of classification models
  • Originated from signal detection theory
  • Shows the trade-off between the true positive rate and the false positive rate
  • The area under the ROC curve is a measure of the accuracy of the model
  • Rank the test tuples in decreasing order: the one that is most likely to belong to the positive class appears at the top of the list
  • The closer to the diagonal line (i.e., the closer the area is to 0.5), the less accurate is the model
  • Vertical axis represents the true positive rate
  • Horizontal axis rep. the false positive rate

  • The plot also shows a diagonal line
  • A model with perfect accuracy will have an area of 1.0



Issues Affecting Model Selection

  • Accuracy
    • classifier accuracy: predicting class label
  • Speed
    • time to construct the model (training time)
    • time to use the model (classification/prediction time)
  • Robustness: handling noise and missing values
  • Scalability: efficiency in disk-resident databases
  • Interpretability
    • understanding and insight provided by the model
  • Other measures, e.g., goodness of rules, such as decision tree size or compactness of classification rules



Issues: Evaluating Classification Methods

  • Accuracy
    • classifier accuracy: predicting class label
    • predictor accuracy: guessing value of predicted attributes
  • Speed
    • time to construct the model (training time)
    • time to use the model (classification/prediction time)
  • Robustness: handling noise and missing values
  • Scalability: efficiency in disk-resident databases
  • Interpretability
    • understanding and insight provided by the model
  • Other measures, e.g., goodness of rules, such as decision tree size or compactness of classification rules


Predictor Error Measures

  • Measure predictor accuracy: measure how far off the predicted value is from the actual known value
  • Loss function : measures the error betw. yi and the predicted value yi’
    • Absolute error: | yi – yi’|
    • Squared error: (yi – yi’)2

  • Test error (generalization error): the average loss over the test set

    Mean absolute error:

    \[\frac{\sum_{i=1}^{d}|y_{i}-y_{i}^{'}|}{d}\]

    Mean squared error:

    \[\frac{\sum_{i=1}^{d}(y_{i}-y_{i}^{'})^{2}}{d}\]

    Relative absolute error:

    \[\frac{\sum_{i=1}^{d}|y_{i}-y_{i}^{'}|}{\sum_{i=1}^{d}|y_{i}-\bar{y}|}\]

    Relative squared error:

    \[\frac{\sum_{i=1}^{d}(y_{i}-y_{i}^{'})^{2}}{\sum_{i=1}^{d}(y_{i}-\bar{y})^{2}}\]

    The mean squared-error exaggerates the presence of outliers 

    Popularly use (square) root mean-square error, similarly, root relative squared error


Scalable Decision Tree Induction Methods

  • SLIQ (EDBT’96 — Mehta et al.)
    • Builds an index for each attribute and only class list and the current attribute list reside in memory
  • SPRINT (VLDB’96 — J. Shafer et al.)
    • Constructs an attribute list data structure
  • PUBLIC (VLDB’98 — Rastogi & Shim)
    • Integrates tree splitting and tree pruning: stop growing the tree earlier
  • RainForest (VLDB’98 — Gehrke, Ramakrishnan & Ganti)
    • Builds an AVC-list (attribute, value, class label)
  • BOAT (PODS’99 — Gehrke, Ganti, Ramakrishnan & Loh)
    • Uses bootstrapping to create several small samples


Data Cube-Based Decision-Tree Induction

  • Integration of generalization with decision-tree induction (Kamber et al.’97)
  • Classification at primitive concept levels
    • E.g., precise temperature, humidity, outlook, etc.
    • Low-level concepts, scattered classes, bushy classification-trees
    • Semantic interpretation problems
  • Cube-based multi-level classification
    • Relevance analysis at multi-levels
    • Information-gain analysis with dimension + level


Ensemble Methods: Increasing the Accuracy

  • Ensemble methods
    • Use a combination of models to increase accuracy
    • Combine a series of k learned models, M1, M2, …, Mk, with the aim of creating an improved model M*
  • Popular ensemble methods
    • Bagging: averaging the prediction over a collection of classifiers
    • Boosting: weighted vote with a collection of classifiers
    • Ensemble: combining a set of heterogeneous classifiers



Bagging: Boostrap Aggregation

  • Analogy: Diagnosis based on multiple doctors’ majority vote
  • Training
    • Given a set D of d tuples, at each iteration i, a training set Di of d tuples is sampled with replacement from D (i.e., bootstrap)
    • A classifier model Mi is learned for each training set Di
  • Classification: classify an unknown sample X
    • Each classifier Mi returns its class prediction
    • The bagged classifier M* counts the votes and assigns the class with the most votes to X
  • Prediction: can be applied to the prediction of continuous values by taking the average value of each prediction for a given test tuple
  • Accuracy
    • Often significantly better than a single classifier derived from D
    • For noise data: not considerably worse, more robust
    • Proved improved accuracy in prediction



Boosting

  • Analogy: Consult several doctors, based on a combination of weighted diagnoses—weight assigned based on the previous diagnosis accuracy
  • How boosting works?
    • Weights are assigned to each training tuple
    • A series of k classifiers is iteratively learned
    • After a classifier Mi is learned, the weights are updated to allow the subsequent classifier, Mi+1, to pay more attention to the training tuples that were misclassified by Mi
    • The final M* combines the votes of each individual classifier, where the weight of each classifier's vote is a function of its accuracy
  • Boosting algorithm can be extended for numeric prediction
  • Comparing with bagging: Boosting tends to have greater accuracy, but it also risks overfitting the model to misclassified data



Adaboost (Freund and Schapire, 1997)

  • Given a set of d class-labeled tuples, (X1, y1), …, (Xd, yd)
  • Initially, all the weights of tuples are set the same (1/d)
  • Generate k classifiers in k rounds. At round i,
    • Tuples from D are sampled (with replacement) to form a training set Di of the same size
    • Each tuple’s chance of being selected is based on its weight
    • A classification model Mi is derived from Di
    • Its error rate is calculated using Di as a test set
    • If a tuple is misclassified, its weight is increased, o.w. it is decreased
  • Error rate: err(Xj) is the misclassification error of tuple Xj. Classifier Mi error rate is the sum of the weights of the misclassified tuples: 

\[error(M_{i})=\sum_{j}^{d}w_{j}\times err (X_{j})\]

  • The weight of classifier Mi’s vote is

\[log \frac{1-error(M_{i})}{error(M_{i})}\]



Random Forest (Breiman 2001)

  • Random Forest:
    • Each classifier in the ensemble is a decision tree classifier and is generated using a random selection of attributes at each node to determine the split
    • During classification, each tree votes and the most popular class is returned
  • Two Methods to construct Random Forest:
    • Forest-RI (random input selection): Randomly select, at each node, F attributes as candidates for the split at the node. The CART methodology is used to grow the trees to maximum size
    • Forest-RC (random linear combinations): Creates new attributes (or features) that are a linear combination of the existing attributes (reduces the correlation between individual classifiers)
  • Comparable in accuracy to Adaboost, but more robust to errors and outliers
  • Insensitive to the number of attributes selected for consideration at each split, and faster than bagging or boosting



Classification of Class-Imbalanced Data Sets

  • Class-imbalance problem: Rare positive example but numerous negative ones, e.g., medical diagnosis, fraud, oil-spill, fault, etc.
  • Traditional methods assume a balanced distribution of classes and equal error costs: not suitable for class-imbalanced data
  • Typical methods for imbalance data in 2-class classification:
    • Oversampling: re-sampling of data from positive class
    • Under-sampling: randomly eliminate tuples from negative class
    • Threshold-moving: moves the decision threshold, t, so that the rare class tuples are easier to classify, and hence, less chance of costly false negative errors
    • Ensemble techniques: Ensemble multiple classifiers introduced above
  • Still difficult for class imbalance problem on multiclass tasks



Summary

  • Classification is a form of data analysis that extracts models describing important data classes.
  • Effective and scalable methods have been developed for decision tree induction, Naive Bayesian classification, rule-based classification, and many other classification methods.
  • Evaluation metrics include: accuracy, sensitivity, specificity, precision, recall, F measure, and measure.
  • Stratified k-fold cross-validation is recommended for accuracy estimation. Bagging and boosting can be used to increase overall accuracy by learning and combining a series of individual models.
  • Significance tests and ROC curves are useful for model selection.
  • There have been numerous comparisons of the different classification methods; the matter remains a research topic
  • No single method has been found to be superior over all others for all data sets
  • Issues such as accuracy, training time, robustness, scalability, and interpretability must be considered and can involve trade-offs, further complicating the quest for an overall superior method



References

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