1. UNIT3. Classification: Basic Concepts
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

2. Classification vs. Numeric Prediction
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

3. 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

4. Learning Step: Model Construction
Classification
Algorithms
Training
Data
Classifier
(Model)
IF rank = ‘professor’
OR years > 6
THEN tenured = ‘yes’

5. Classification step: Estimating accuracy of model
Classifier
Testing
Data
Unseen Data
(Jeff, Professor, 4)
Tenured?

6. 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

7. Decision Tree Induction: An Example
Training data set: Buys_computer
The data set follows an example of Quinlan’s ID3 (Playing Tennis)
Resulting tree:
age?
overcast
student?
credit rating?
<=30
>40 no
yes
31..40
fair
excellent

8. 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 |Ci, D|/|D|
Expected information (entropy) needed to classify a tuple in D:
Information needed (after using A to split D into v partitions) to classify D:
Information gained by branching on attribute A
I : the expected information needed to classify a given sample
E (entropy) : expected information based on the partitioning into subsets by A

9. Attribute Selection: Information Gain
Class P: buys_computer = “yes”
Class N: buys_computer = “no”
means “age <=30” has 5 out of 14 samples, with 2 yes’es and 3 no’s. Hence
Similarly,

10. 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)
GainRatio(A) = 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

11. Gini Index (CART, IBM IntelligentMiner)
If a data set D contains examples from n classes, gini index, gini(D) is defined as
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
Reduction in Impurity:
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)

12. Computation of Gini Index
Ex. D has 9 tuples in buys_computer = “yes” and 5 in “no”
Suppose the attribute income partitions D into 10 in D1: {low, medium} and 4 in D2
Gini{low,high} is 0.458; Gini{medium,high} is 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

13. 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”

14. Bayesian Classification: Why?
A statistical classifier: performs probabilistic prediction, i.e., predicts class membership probabilities. (the probability that given tuple belongs to a particular class)
Foundation: Based on Bayes’ Theorem. There are two types of Bayesian classification methods.
(i) Naïve Bayesian classifier : A simple Bayesian classifier, has comparable performance with decision tree and selected neural network classifiers. This classifier assume that the effect of an attribute value on a given class is independent of the values of other attributes. (class conditional independence)
(ii) Bayesian belief networks: These are graphical models, which allows the representation of dependencies among subsets of attributes.

15. Bayes’ Theorem: Basics
Bayes’ Theorem proposed by Thomas Bayes.
Bayes’ Theorem:
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 , medium income

16. Prediction Based on Bayes’ Theorem
Given training data X, posteriori probability of a hypothesis H, P(H|X), follows the Bayes’ theorem
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

17. Naïve Bayesian Classification: 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
Since P(X) is constant for all classes, only
needs to be maximized

18. Naïve Bayesian Classification
A simplified assumption: attributes are conditionally independent (i.e., no dependence relation between attributes):
This greatly reduces the computation cost: Only counts the class distribution

19. Naïve Bayes Classifier: Training Dataset
C1:buys_computer = ‘yes’
C2:buys_computer = ‘no’
Data to be classified:
X = (age <=30,
Income = medium,
Student = yes
Credit_rating = Fair)

20. 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”) = x x x = 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”) =0.643*0.044 = 0.028
P(X|buys_computer = “no”) * P(buys_computer = “no”) = 0.007
Therefore, X belongs to class (“buys_computer = yes”)

21. Naïve Bayes Classifier – Pros & Cons
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

22. Bayesian Belief Networks
Bayesian belief networks (also known as Bayesian networks, probabilistic networks): These provide graphical model of causal relationship and they allow class conditional dependencies between subsets of variables.
A belief network is defined by 2 components (i) A directed acyclic graph (ii) A set of conditional probability table.
Each node in directed acyclic graph represents a random variable. Each arc represents a probabilistic dependence. If an arc is drawn from node Y to node Z, then Y is parent of Z and Z is descendent to Y.
Each variable is conditionally independent of its non-descendents in the graph, given its parents. 22 22

23. Bayesian Belief Networks
The below given figure1 is simple belief networks for four random variables
Nodes: random variables
Links: dependency
X and Y are the parents of Z, and Y is the parent of P
No dependency between Z and P
Has no loops/cycles Y Z P X
The below given figure2 is simple belief networks for six boolean variables. The arc in this figure represents causal knowledge. (Ex:- A person having lung cancer is influenced by his family history / person is smoker)

24. Bayesian Belief Network: An Example
Family
History (FH)
Smoker (S)
CPT: Conditional Probability Table for variable LungCancer: LC
~LC
(FH, S)
(FH, ~S)
(~FH, S)
(~FH, ~S)
0.8
0.2
0.5
0.7
0.3
0.1
0.9
LungCancer
(LC)
Emphysema
shows the conditional probability for each possible combination of its parents
Derivation of the probability of a particular combination of values of X, from CPT:
PositiveXRay
Dyspnea
Bayesian Belief Network

25. Using IF-THEN Rules for Classification
Rule based classifier make use of a set of IF-THEN rules for classification. We can express a rule of the form,
r : (condition) -> y
LHS - called as rule antecedent (or) precondition
RHS - called as rule consequent
Rule set can be expressed as,
R = { r1, r2, r3, …….., rn }
Simply, it represent the knowledge in the form of IF-THEN rules
Ex:- R: IF age = youth AND student = yes THEN buys_computer = yes
(or)
R: ((age = youth) ^ (student = yes)) -> buys_computer = yes
If the condition holds true for given tuple , then then antecedent is satisfied.

26. Using IF-THEN Rules for Classification
Assessment of a rule: Using coverage and accuracy we can asses the rule.
ncovers = No.of tuples covered by R
ncorrect = No. of tuples correctly classified by R
coverage(R) = ncovers /|D| /* D: training data set */
accuracy(R) = ncorrect / coverage
Ex:- R : (age > 40) ^ (income = medium) -> buys_computer=yes
ncovers = (the tuples covered by R)
ncorrect = (the tuples correctly classified by R)
coverage(R) = 3/14 =
accuracy(R) = 2/0.214 = 9.34
Easy to interpret
Easy to generate
Can classify new tuples rapidly
Highly expensive

27. 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
age?
student?
credit rating?
<=30
>40 no
yes
31..40
fair
excellent
Example: Rule extraction from 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

28. Classification by Backpropagation
Back propagation is a neural network learning algorithm. Learning is the process of modifying the weights in order to produce a network that performs some function.
Started by psychologists and neurobiologists to develop and test computational analogues of neurons
A neural network is a set of connected input/output units where each connection has a weight associated with it
During the learning phase, the network learns by adjusting the weights so as to be able to predict the correct class label of the input tuples
Many neural network Algorithms existed. One of the popular algorithm is back propagation and it performs learning on multilayer feed-forward networks.

29. Neural Network as a Classifier
Weakness
Long training time
Require a number of parameters typically best determined empirically, e.g., the network topology or “structure.”
Poor interpretability: Difficult to interpret the symbolic meaning behind the learned weights and of “hidden units” in the network
Strength
High tolerance to noisy data
Ability to classify untrained patterns
Well-suited for continuous-valued inputs and outputs
Successful on an array of real-world data, e.g., hand-written letters
Algorithms are inherently parallel
Techniques have recently been developed for the extraction of rules from trained neural networks

30. A Multi-Layer Feed-Forward Neural Network
Output vector
Output layer
Hidden layer
MK: Note – different notation than used in book. Will have to standardize notation.
wij
Input layer
Input vector: X

31. How A Multi-Layer Neural Network Works
Each layer is made up of units. The inputs to the network correspond to the attributes measured for each training tuple.
Inputs are fed simultaneously into the units making up the input layer
They are then weighted and fed simultaneously to a hidden layer
The number of hidden layers is arbitrary, although usually only one
The weighted outputs of the last hidden layer are input to units making up the output layer, which emits the network's prediction
The units of input layer called as i/p units and output layer called as o/p units and hidden layer units called as neurodes.
The network is feed-forward: None of the weights cycles back to an input unit or to an output unit of a previous layer

32. Defining a Network Topology
Decide the network topology: Specify # of units in the input layer, # of hidden layers (if > 1), # of units in each hidden layer, and # of units in the output layer
Normalize the input values for each attribute measured in the training tuples to [0.0—1.0]
One input unit per domain value, each initialized to 0
Output, if for classification and more than two classes, one output unit per class is used
Once a network has been trained and its accuracy is unacceptable, repeat the training process with a different network topology or a different set of initial weights

33. Backpropagation
Back propagation learns by Iteratively processing a set of training tuples & compare the network's prediction for each tuple with the actual known target value. (categorical/continuous values)
For each training tuple, the weights are modified to minimize the mean squared error between the network's prediction and the actual target value
Modifications are made in the “backwards” direction: from the output layer, through each hidden layer down to the first hidden layer, hence “backpropagation”
Steps
Initialize weights to small random numbers, associated with biases
Propagate the inputs forward (by applying activation function)
Backpropagate the error (by updating weights and biases)
Terminating condition (when error is very small, etc.)

34. Neuron: A Hidden/Output Layer Unitmk f
weighted
sum
Input
vector x
output y
Activation
function
weight
vector w å w0 w1 wn x0 x1 xn
bias
An n-dimensional input vector x is mapped into variable y by means of the scalar product and a nonlinear function mapping
The inputs to unit are outputs from the previous layer. They are multiplied by their corresponding weights to form a weighted sum, which is added to the bias associated with unit. Then a nonlinear activation function is applied to it. 34 34

35. SVM—History and Applications
Vapnik and colleagues (1992)—groundwork from Vapnik & Chervonenkis’ statistical learning theory in 1960s
Features: training can be slow but accuracy is high owing to their ability to model complex nonlinear decision boundaries (margin maximization)
Used for: classification and numeric prediction
Applications:
handwritten digit recognition, object recognition, speaker identification, benchmarking time-series prediction tests

36. SVM—Support Vector Machines
A relatively new classification method for both linear and nonlinear data
A support vector machine is an algorithm that works as follows. It uses a nonlinear mapping to transform the original training data into a higher dimension
With the new dimension, it searches for the linear optimal separating hyperplane (i.e., “decision boundary” separating the tuples of one class from another)
With an appropriate nonlinear mapping to a sufficiently high dimension, data from two classes can always be separated by a hyperplane
SVM finds this hyperplane using support vectors (“essential” training tuples) and margins (defined by the support vectors)

37. Support Vector Machines
SVM — When Data Is Linearly Separable
Let data D be (X1, y1), …, (X|D|, y|D|), where Xi is the set of training tuples associated with the class labels yi . Each yi can take one of two values, either +1 or -1 corresponding to the classes buys computer = yes and buys computer = no, respectively.
There are infinite lines (hyperplanes) separating the two classes but we want to find the best one (the one that minimizes classification error on unseen data). SVM searches for the hyperplane with the largest margin, i.e., maximum marginal hyperplane (MMH).
Ex:- consider an example based on two input attributes, A1 and A2, as shown in Figure. From the graph, we see that the 2-D data are linearly separable (or “linear,” ) bz a straight line can be drawn to separate all of the tuples of class +1 from all of the tuples of class -1.

38. SVM— Margins and Support Vectors
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39. SVM — Margins and Support Vectors
Large Margin
Small Margin
A separating hyperplane can be written as WX + b = 0;
Where W is a weight vector, namely, W = {w1, w2, …. , wn}, n is the number of attributes and b is a scalar, often referred to as a bias.

40. Support Vector Machines
Let us consider two input attributes, A1 and A2, as in below Figure. Training tuples are 2-D, e.g., X = (x1, x2), where x1 and x2 are the values of attributes A1 and A2, respectively, for X. If we think of b as an additional weight, w0, we can rewrite the above separating hyperplane as, w0+ w1 x1 + w2x2 = 0
Thus, any point that lies above the separating hyperplane satisfies, w0+ w1 x1 + w2x2 > 0
Similarly, any point that lies below the separating hyperplane satisfies, w0+ w1 x1 + w2x2 < 0
The weights can be adjusted so that the hyperplanes defining the “sides” of the margin can be written as
H1 : w0+ w1 x1 + w2x2 >= + 1 for yi = +1
H2 : w0+ w1 x1 + w2x2 <= -1 for yi = -1

41. Support Vector Machines
That is, any tuple that falls on or above H1 belongs to class +1, and any tuple that falls on or below H2 to class -1.
Any training tuples that fall on hyperplanes H1 or H2 (i.e., the “sides” defining the margin) satisfy above Equation are called support vectors.

42. Associative Classification
Associative classification: Major steps
Mine data to find strong associations between frequent patterns (conjunctions of attribute-value pairs) and class labels
Association rules are generated in the form of
P1 ^ p2 … ^ pl  “Aclass = C” (conf, sup)
Organize the rules to form a rule-based classifier
Why effective?
It explores highly confident associations among multiple attributes and may overcome some constraints introduced by decision-tree induction, which considers only one attribute at a time
Associative classification has been found to be often more accurate than some traditional classification methods, such as C4.5 42 42

43. Typical Associative Classification Methods
CBA (Classification Based on Associations: Liu, Hsu & Ma, KDD’98)
Mine possible association rules in the form of
Cond-set (a set of attribute-value pairs)  class label
Build classifier: Organize rules according to decreasing precedence based on confidence and then support
CMAR (Classification based on Multiple Association Rules: Li, Han, Pei, ICDM’01)
Classification: Statistical analysis on multiple rules
CPAR (Classification based on Predictive Association Rules: Yin & Han, SDM’03)
Generation of predictive rules (FOIL-like analysis) but allow covered rules to retain with reduced weight
Prediction using best k rules
High efficiency, accuracy similar to CMAR 43

44. What Is Prediction?
(Numerical) prediction is similar to classification
construct a model
use model to predict continuous or ordered value for a given input
Prediction is different from classification
Classification refers to predict categorical class label
Prediction models continuous-valued functions
Major method for prediction: regression
model the relationship between one or more independent or predictor variables and a dependent or response variable
Regression analysis
Linear and multiple regression
Non-linear regression
Other regression methods: generalized linear model, Poisson regression, log-linear models, regression trees 44

45. Linear Regression
Linear regression: involves a response variable y and a single predictor variable x
y = w0 + w1 x
where w0 (y-intercept) and w1 (slope) are regression coefficients
Method of least squares: estimates the best-fitting straight line
Multiple linear regression: involves more than one predictor variable
Training data is of the form (X1, y1), (X2, y2),…, (X|D|, y|D|)
Ex. For 2-D data, we may have: y = w0 + w1 x1+ w2 x2
Solvable by extension of least square method or using SAS, S-Plus
Many nonlinear functions can be transformed into the above 45 45

46. Nonlinear Regression
Some nonlinear models can be modeled by a polynomial function
A polynomial regression model can be transformed into linear regression model. For example,
y = w0 + w1 x + w2 x2 + w3 x3
convertible to linear with new variables: x2 = x2, x3= x3
Other functions, such as power function, can also be transformed to linear model
Some models are intractable nonlinear (e.g., sum of exponential terms)
possible to obtain least square estimates through extensive calculation on more complex formulae 46 46

47. Classifier Accuracy Measures
classes
buy_computer = yes
buy_computer = no
total
recognition(%)
6954 46
7000
99.34
412
2588
3000
86.27
7366
2634
10000
95.52
Accuracy of a classifier M, acc(M): percentage of test set tuples that are correctly classified by the model M
Error rate (misclassification rate) of M = 1 – acc(M)
Given m classes, CMi,j, an entry in a confusion matrix, indicates # of tuples in class i that are labeled by the classifier as class j C1 C2
True positive
False negative
False positive
True negative
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48. Evaluating the Accuracy of a Classifier or Predictor
Holdout method and Random sampling
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
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49. Evaluating the Accuracy of a Classifier or Predictor
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 boostrap methods, and a common one is .632 boostrap
Suppose we are given a data set of d tuples. The data set 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 will end up in the bootstrap, and the remaining 36.8% will form the test set (since (1 – 1/d)d ≈ e-1 = 0.368)
Repeat the sampling procedue k times, overall accuracy of the model:
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