03/11/98 machine learning

1 CSE573 Autumn 1997

03/11/98 Machine Learning

• Administrative– Finish this topic– The rest of the time is yours– Final exam Tuesday, Mar. 17, 2:30-4:20 p.m., here– Additional help

• today after class (me), Thursday 2-4 (SteveW), Friday after class (me), Saturday 10-12 (me)

• Last time– general introduction to Machine Learning

• Machine Learning often isn’t what we mean by the word

– “he learned the difference between right and wrong”

– varieties of Machine Learning• reinforcement learning (learn plans/policies)

• explanation-based learning (learn control rules to speed up problem solving)

• inductive (concept) learning (learn a general description from examples)

– decision tree learning is an interesting special case

• This time– finish the decision tree construction algorithm


Example

Day Outlook Temp Humidity Wind Play?D1 Sunny Hot High Weak NoD2 Sunny Hot High Strong NoD3 Overcast Hot High Weak YesD4 Rain Mild High Weak YesD5 Rain Cool Normal Weak YesD6 Rain Cool Normal Strong NoD7 Overcast Cool Normal Strong YesD8 Sunny Mild High Weak NoD9 Sunny Cool Normal Weak YesD10 Rain Mild Normal Weak YesD11 Sunny Mild Normal Strong YesD12 Overcast Mild Normal Weak YesD13 Overcast Hot Normal Weak YesD14 Rain Mild High Strong No


Basic Algorithm

• Recall, a node in the tree represents a conjunction of attribute values. We will try to build “the shortest possible” tree that classifies all the training examples correctly. In the algorithm we also store the list of attributes we have not used so far for classification.

• Initialization: tree {} attributes {all attributes} examples {all training examples}

• Recursion:– Choose a new attribute A with possible values {ai}

– For each ai, add a subtree formed by recursively building the tree with

• the current node as root

• all attributes except A

• all examples where A=ai


Basic Algorithm (cont.)

• Termination (working on a single node):

– If all examples have the same classification, then this combination of attribute values is sufficient to classify all (training) examples. Return the unanimous classification.

– If examples is empty, then there are no examples with this combination of attribute values. Associate some “guess” with this combination.

– If attributes is empty, then the training data is not sufficient to discriminate. Return some “guess” based on the remaining examples.


What Makes a Good Attribute for Splitting?

D1,D2, ...,D14

D1 D2 D14...

DAYD1,D2, ...,D14

ALL

D1,D2, ...,D14

D1,D2, ...,D14

HUMIDITY

D1, S2, D3, D4, D8, D14

D5, D6, D7D9, D10, D11D12, D13

= high = normal

D1,D2, ...,D14

OUTLOOK

D1, D3, D8, D9, D11

D3, D7, D12, D13

= sunny = overcast = rain

D4, D5, D6, D10, D14

= D1= D2

= D14

= TRUE


How to choose the next attribute

• What is our goal in building the tree in the first place?– Maximize accuracy over the entire data set

– Minimize expected number of tests to classify an example in the training set

• (In both cases this can argue for building the shortest tree.)

• We can’t really do the first looking only at the training set: we can only build a tree accurate for our subset and assume the characteristics of the full data set are the same.

• To minimize the expected number of tests– the best test would be one where each branch has all positive or all negative

instances

– the worst test would be one where the proportion of positive to negative instances is the same in every branch

• knowledge of A would provide no information about the example’s ultimate classification


The Entropy (Disorder) of a Collection

• Suppose S is a collection containing positive and negative examples of the target concept:– Entropy(S) – (p+ log2 p+ + p- log2 p

-)

– where p+ is the fraction of examples that are positive and p- is the fraction of examples that are negative

• Good features– minimum of 0 where p+ = 0 and where p- = 0

– maximum of 1 where p+ = p- = 0.5

• Interpretation: how far away are we from having a leaf node in the tree?

• The best attribute would reduce the entropy in the child collections as quickly as possible.


Entropy and Information Gain

• The best attribute is one that maximizes the expected decrease in entropy– if entropy decreases to 0, the tree need not be expanded

further

– if entropy does not decrease at all, the attribute was useless

• Gain is defined to be– Gain(S, A) = Entropy(S) – v values(A) p{A=v} Entropy(S{A=v})

– where p{A=v} is the proportion of S where A=v, and

– S{A=v} is the collection taken by selecting those elements of S where

A=v


Expected Information Gain Calculation

[10+,15-]E(2/5) = 0.97

[8+,2-]E(8/10) = 0.72

[1+,11-]E(1/12) = 0.43

[1+,2-]E(1/3) = 0.92

(10) (12) (3)

S =

Gain(S,A) = 0.97 - (10/25 .72 + 12/25 .43 + 3/25 .92) = 0.97 - .60

= 0.37


Example

Day Outlook Temp Humidity Wind Play?D1 Sunny Hot High Weak NoD2 Sunny Hot High Strong NoD3 Overcast Hot High Weak YesD4 Rain Mild High Weak YesD5 Rain Cool Normal Weak YesD6 Rain Cool Normal Strong NoD7 Overcast Cool Normal Strong YesD8 Sunny Mild High Weak NoD9 Sunny Cool Normal Weak YesD10 Rain Mild Normal Weak YesD11 Sunny Mild Normal Strong YesD12 Overcast Mild Normal Weak YesD13 Overcast Hot Normal Weak YesD14 Rain Mild High Strong No

S: [9+, 5-]E(9/14) = 0.940


Choosing the First Attribute

Humidity

S: [9+, 5-]E = 0.940

High Low

S: [3+, 4-]E = 0.985

S: [6+, 1-]E = 0.592

Wind

S: [9+, 5-]E = 0.940

High Low

S: [6+, 2-]E = 0.811

S: [3+, 3-]E = 1.000

Gain(S, Humidity) = .940 - (7/14).985 - (7/14) .592

= .151

Gain(S, Wind) = .940 - (8/14).811 - (6/14)1.00= .048

Gain(S, Outlook) = .246Gain(S, Temperature) = .029


After the First Iteration

Outlook

Sunny Rain

? Yes ?

Overcast

D1, D2, …, D149+ 5-

D1, D2, D8, D9, D11[3+, 2-]E=.970 D3, D7, D12, D13

[4+, 0-]

D4, D5, D6, D10, D14[3+, 2-]

Gain(Ssunny, Humidity) = .970Gain(Ssunny, Temp) = .570Gain(Ssunny, Wind) = .019


Final Tree

Outlook

Sunny Rain

Humidity Yes Wind

Overcast

No Yes No Yes

High Low Strong Weak


Some Additional Technical Problems

• Noise in the data– Not much you can do about it

• Overfitting– What’s good for the training set may not be good for the

full data set

• Missing values– Attribute values omitted in training set cases or in

subsequent (untagged) cases to be classified


Data Overfitting

• Overfitting, definition– Given a set of trees T, a tree t T is said to overfit the training data if

there is some alternative tree t’, such that t has better accuracy than t’ over the training examples, but t’ has better accuracy than t over the entire set of examples

• The decision not to stop until attributes or examples are exhausted is somewhat arbitrary– you could always stop and take the majority decision, and the tree would

be shorter as a result!

• The standard stopping rule provides 100% accuracy on the training set, but not necessarily on the test set– if there is noise in the training data

– if the training data is too small to give good coverage• likely to be spurious correlation


Overfitting (continued)

• How to avoid overfitting– stop growing the tree before it perfectly classifies the training

data– allow overfitting, but post-prune the tree

• Training and validation sets– training set is used to build the tree– a separate validation set is used to evaluate the accuracy over

subsequent data, and to evaluate the impact of pruning• validation set is unlikely to exhibit the same noise and spurious

correlation

– rule of thumb: 2/3 to the training set, 1/3 to the validation set


Reduced Error Pruning

• Pruning a node consists of removing all subtrees, making it a leaf, and assigning it the most common classification of the associated training examples.

• Prune nodes iteratively and greedily: next remove the node that most improves accuracy over the validation set– but never remove a node that decreases accuracy

• A good method if you have lots of cases


Overfitting (continued)

• How to avoid overfitting– stop growing the tree before it perfectly classifies the training

data

– allow overfitting, but post-prune the tree

• Training and validation sets– training set is used to form the learned hypothesis

– validation set used to evaluate the accuracy over subsequent data, and to evaluate the impact of pruning

– justification: validation set is unlikely to exhibit the same noise and spurious correlation

– rule of thumb: 2/3 to the training set, 1/3 to the validation set


Missing Attribute Values

• Situations– missing attribute value(s) in the training set

– missing value(s) in the validation or subsequent tests

• Quick and dirty methods– assign it the same value most common for other training examples at

the same node

– assign it the same value most common for other training examples at the same node that have the same classification

• “Fractional” method– assign a probability to each value of A based on observed frequencies

– create “fractional cases” with these probabilities

– weight information gain with each case’s fraction


Example: Fractional Values

D1, D2, D8, D9, D11[3+, 2-]E=.970

Day Outlook Temp Humidity Wind Play?D1 xx Hot High Weak NoD2 xx Hot High Strong NoD8 xx Mild High ??? NoD9 xx Cool Normal Weak YesD11 xx Mild Normal Strong Yes

D2(1.0), D8(0.5), D11(1.0) [1.5+,1.5-]

E=1D1(1.0), D8(0.5), D9(1.0)

[1.5+,1.5-]E=1

wind=weak wind=strong


Decision Tree Learning

• The problem: given a data set, produce the shortest-depth decision tree that accurately classifies the data

• The (heuristic): build the tree greedily on the basis of expected entropy loss

• Common problems

– the training set is not a good surrogate for the full data set• noise

• spurious correlations

– thus the optimal tree for the test set may not be accurate for the full data set (overfitting)

– missing values in training set or subsequent cases

03/11/98 machine learning

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