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Decision Trees with Minimal Test Costs (ICML 2004, Banff, Canada) Charles X. Ling, Univ of Western Ontario, Canada Qiang Yang, HK UST, Hong Kong Etc. Goal When test examples contain missing values Decide what to do? Do a test to obtain a value

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decision trees with minimal test costs icml 2004 banff canada

Decision Trees with Minimal Test Costs(ICML 2004, Banff, Canada)

Charles X. Ling, Univ of Western Ontario, Canada

Qiang Yang, HK UST, Hong Kong

Etc.

slide2
Goal
  • When test examples contain missing values
    • Decide what to do?
      • Do a test to obtain a value
      • Skip the attribute and continue with the rest of the decision making
    • Ultimate decision: minimize total cost
      • Total cost=mis-classification cost + test cost
assumptions
Assumptions
  • Mis-classification cost, for one case
    • FP=false positive
    • FN=false negative
  • Test cost
    • = C(i) for attribute A_i, regardless of values
  • Examples in weather domain
expected results
Expected results
  • Let there be a total of P+N training cases
    • If P*FN+N*FP < C*(P+N)
      • Then?
    • IF P*FN+N*FP >> C*(P+N)
      • Then?
leaf labeling
Leaf Labeling
  • How to label a node?
    • P=positive examples
    • N=negative examples
    • P*FN=
    • N*FP=
    • Label the node as + if …
splitting criteria on attribute a
Splitting Criteria on Attribute A
  • Total Cost=?
    • Say two branches: 1 and 2
    • (P1, N1) on branch 1, and (P2,N2) on 2
    • Test cost=
    • Training data may contain missing values on this attribute
      • Ratio = P0/N0
    • Total=? (page 3)
properties
Properties
  • The relative difference of misclassification cost and test cost  tree!
  • Attributes with small test cost is always chosen first
  • When test cost is increased, the attribute goes down the tree
test strategies
Test Strategies
  • OST=optimal sequential test

Always test when value is missing, following the order given by tree

  • Stops whenever meeting missing value

Uses the ratio of P0/N0 to make decision at node

experiments
Experiments
  • Comparison with C4.5
costs in machine learning
Costs in Machine Learning
  • Most inductive learning algorithms: minimizing classification errors
    • Different types of misclassification have different costs, e.g. FP and FN
  • In this talk:
    • Test costs should also be considered
    • Cost sensitive learning considers a variety of costs; see survey by Peter Turney (2000)
applications
Applications
  • Medical Practice
    • Doctors may ask a patient to go through a number of tests (e.g., Blood tests, X-rays)
    • Which of these new tests will bring about higher value?
  • Biological Experimental Design
    • When testing a new drug, new tests are costly
    • which experiments to perform?
previous work
Previous Work
  • Many previous works consider the two types of cost separately – an obvious oversight
  • (Turney 1995): ICET, uses genetic algorithm to build trees to minimize the total cost
  • (Zubek and Dieterrich 2002): a Markov Decision Process (MDP), searches in a state space for optimal policies
  • (Greiner et al. 2002): PAC learning
an example of the problem
An Example of The Problem

Training: with ?, cannot obtain values

Goal 1: build a tree that minimizes the total cost

Test: with many ?, may obtain values at a cost

Goal 2: obtain test values at a cost to minimize the total cost

example medical diagnosis

blood test

pressure

essay

?

?

?

temperature

cardiogram

?

39oc

Example – Medical Diagnosis

Is the patient healthy?

Which test should be taken first?

Which test to perform next?

Concern: cost the patient as little as possible while maintaining low mis-diagnosis risk

what are total costs
What are Total Costs?
  • Assumption: binary classes, costs: FP and FN
  • Goal: minimize total cost
    • Total cost = misclassification cost + test cost
  • Previous Work
    • Information Gain as a attribute selection criterion
  • In this work, need a new attribute selection criterion
    • Total cost = Sum_i { Probability(i)*Cost(i)}
attribute selection criterion c4 5
Attribute Selection Criterion: C4.5

Select the attribute with minimal total cost

    • C4.5: minimal entropy
    • If growing a tree has a smaller total cost
      • then choose an attribute with minimal total cost
      • else stop and form a leaf
  • How to label a leaf node?
    • Label leaf according to minimal total cost
    • If (P×FN  N×FP) then class = positiveelse class = negative
    • Example: {P, P, P, N, N, N, N} FN=$10, FP=$1
      • Information Gain+Majority class: Predict N, 3 mistakes.
      • Total cost: if we predict P, 4*FP=4*1=$4;
        • If we predict N, 3*FN = $30. Conclusion: Predict P.
minimal cost summary
Minimal cost: summary
  • Attribute selection criterion: minimal total cost(Ctotal = Cmc + Ctest) instead of minimal entropy in C4.5
  • If growing a tree has a smaller total cost, then choose an attribute with minimal total cost. Otherwise, stop and form a leaf.
  • Label leaf also according to minimal total cost:
    • Suppose the leaf have P positive examples and N negative examples
    • FP denotes the cost of a false positive example and FN false negative
    • If (P×FN N×FP)THEN label = positive ELSE label = negative
a tree building example
A Tree Building Example

Cmc = min(P×FN, N×FP)

Ctest = 0Ctotal = Cmc + Ctest

P:N

Attribute A with a test cost C

Consider attribute A for a potential splitting attribute

A = v1

A = v2

C’mc= min(P1×FN, N1×FP) + min(P2×FN, N2×FP)

C’test = (P1 + N1 + P2 + N2) × C

C’total = C’mc + C’test

P2:N2

P1:N1

  • If C’total < Ctotal, splitting on A would reduce the total cost Choose an attribute with the minimal total cost for splitting
  • If C’totalCtotal for all remaining attributes, no further sub-tree will be built, and the set will become a leaf.
missing values values
Missing values: ? values
  • First, how to handle ? values in training data
  • Previous work
    • built ? branch;
    • problematic
  • We will
    • deal with unknown values in the training set:
    • no branch for ? will be built,
    • examples are “gathered” inside the internal nodes
desirable properties

A1

A1

P

All test costs are 300

A6

A6

P

N

P

N

P

P

P

P

P

P

All test costs are 20

P

N

N

P

N

N

All test costs are 0

Desirable Properties

1. Effect of difference between misclassification costs and the test costs

slide21

A1

A2

A3

A4

A5

A6

# 1

20

20

20

20

20

20

# 2

200

20

100

100

200

200

# 3

200

100

100

100

20

200

A1

A2

A5

A6

A6

A1

N

P

P

P

P

P

P

A1

P

P

N

N

P

N

N

P

N

P

N

P

P

P

N

P

N

P

P

2. Prefer attribute with smaller test costs

slide22

A6

A6

A1

P

P

A1

N

P

P

A2

N

A6

A6

P

P

P

P

P

N

N

P

N

N

N

N

P

P

N

P

N

Cost of A1=20

Cost of A1=50

Cost of A1=80

3. If test cost increases, attribute tends to be “pushed” down and “falls out” of the tree

cost reduction
Cost Reduction

P:N

A

P1:N1

P2:N2

  • Equivalent to Information Gain
    • Min{P*FN, N*FP} – [cost(A) + Min{P1*FN,N1*FP} + Min{P2*FN, N2*FP}]
  • We choose the attribute with the largest such value as a splitting attribute
missing values in test cases
Missing values in test cases

A New patient arrives:

ost intuition
OST: Intuition
  • Follow the test-cost sensitive decision tree
  • When reaching a node, make a decision
    • If the node is a leaf node…
    • If the node is an internal node, and we have missing value, then obtain the value
  • Evaluation: calculate the total cost when the ground truth is known
four testing strategies

A1

A6

A6

P

P

P

P

P

N

N

P

N

N

A1

P

N

P

N

P

P

Four Testing Strategies
  • First: Optimal Sequential Test (OST)(Simple batch test: do all tests)
  • Second: No test will be performed, predict with internal node
  • Third: No test will be performed, predict with weighted sum of subtrees
  • Fourth: A new tree is built dynamically for each test case using only the known attributes
experiment settings
Experiment - settings
  • Five dataset, binary-class
  • 60/40 for training/testing, repeat 5 times
  • Unknown values for training/test examples are selected randomly by a specific probability
  • Also compare to C4.5 tree, using OST for testing
results with different of unknown

No test, internal

No test, lazy tree

C4.5 tree, OST

Results with different % of unknown
  • OST is best; M4 and C4.5 next; M3 is worst
  • OST not increase with more ?; others do overall

No test, distributed

results with different test costs

No test, internal

No test, lazy tree

C4.5 tree, OST

Results with different test costs
  • With large test costs, OST = M2 = M3 = M4
  • C4.5 is much worse (tree building is cost-insensitive)

No test, distributed

results with unbalanced class costs

No test, internal

No test, distributed

No test, lazy tree

C4.5 tree, OST

Results with unbalanced class costs
  • With large test costs, OST = M2 = M4
  • C4.5 is much worse (tree building is cost-insensitive)
  • M3 is worse than M2… (M3 is used in C4.5)
comparing ost c4 5 cross 6 datasets
Comparing OST/C4.5 cross 6 datasets
  • OST always outperforms C4.5
conclusions
Conclusions
  • New tree building algorithm for minimal costs
    • Desirable properties
    • Computationally efficient (similar to C4.5)
  • Test strategies (OST and batch) are very effective
  • Can solve many real-world diagnosis problems
future work
Future Work
  • More intelligent “Batch Test” methods
  • Consider cost of additional batch test
    • Optimal sequential batch testbatch 1 = (test1, test 2)batch 2 = (test 3, test 4, test 5), …
  • Other learning algorithms with minimal total cost
  • A wrapper that works for any “black box”
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