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Machine Learning in Real World: CART. Outline. CART Overview and Gymtutor Tutorial Example Splitting Criteria Handling Missing Values Pruning Finding Optimal Tree. CART – Classification And Regression Tree. Developed 1974-1984 by 4 statistics professors

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outline
Outline
  • CART Overview and Gymtutor Tutorial Example
  • Splitting Criteria
  • Handling Missing Values
  • Pruning
    • Finding Optimal Tree
cart classification and regression tree
CART – Classification And Regression Tree
  • Developed 1974-1984 by 4 statistics professors
    • Leo Breiman (Berkeley), Jerry Friedman (Stanford), Charles Stone (Berkeley), Richard Olshen (Stanford)
  • Focused on accurate assessment when data is noisy
  • Currently distributed by Salford Systems
cart tutorial data gymtutor
CART Tutorial Data: Gymtutor

CART HELP, Sec 3 in CARTManual.pdf

  • ANYRAQT Racquet ball usage (binary indicator coded 0, 1)
  • ONAER Number of on-peak aerobics classes attended
  • NSUPPS Number of supplements purchased
  • OFFAER Number of off-peak aerobics classes attended
  • NFAMMEM Number of family members
  • TANNING Number of visits to tanning salon
  • ANYPOOL Pool usage (binary indicator coded 0, 1)
  • SMALLBUS Small business discount (binary indicator coded 0, 1)
  • FIT Fitness score
  • HOME Home ownership (binary indicator coded 0, 1)
  • PERSTRN Personal trainer (binary indicator coded 0, 1)
  • CLASSES Number of classes taken.
  • SEGMENT Member’s market segment (1, 2, 3) – target
view data
View data
  • CART Menu: View -> Data Info …
cart model setup
CART Model Setup
  • Target -- required
  • Predictors (default – all)
  • Categorical
    • ANYRAQT, ANYPOOL, SMALLBUS, HOME
    • Categorical: if field name ends in “$”, or from values
  • Testing
    • default – 10-fold cross-validation
key cart features
Key CART features
  • Automated field selection
    • handles any number of fields
      • automatically selects relevant fields
  • No data preprocessing needed
      • Does not require any kind of variable transforms
      • Impervious to outliers
  • Missing value tolerant
      • Moderate loss of accuracy due to missing values
cart key parts of tree structured data analysis
CART: Key Parts of Tree Structured Data Analysis
  • Tree growing
    • Splitting rules to generate tree
    • Stopping criteria: how far to grow?
    • Missing values: using surrogates
  • Tree pruning
    • Trimming off parts of the tree that don’t work
    • Ordering the nodes of a large tree by contribution to tree accuracy … which nodes come off first?
  • Optimal tree selection
    • Deciding on the best tree after growing and pruning
    • Balancing simplicity against accuracy
cart is a form of binary recursive partitioning
CART is a form of Binary Recursive Partitioning
  • Data is split into two partitions
    • Q: Does C4.5 always have binary partitions?
  • Partitions can also be split into sub-partitions
    • hence procedure is recursive
  • CART tree is generated by repeated partitioning of data set
    • parent gets two children
    • each child produces two grandchildren
    • four grandchildren produce 8 great grandchildren
splits always determined by questions with yes no answers
Splits always determined by questions with YES/NO answers
  • Is continuous variable X£c ?
  • Does categorical variable D take on levels i, j, or k?
    • is GENDER M or F ?
  • Standard split:
    • if answer to question is YES a case goes left; otherwise it goes right
    • this is the form of all primary splits
    • example : Is AGE  62.5?
  • More complex conditions possible:
    • Boolean combinations: AGE<=62 OR BP<=91
    • Linear combinations: .66*AGE - .75*BP< -40
searching all possible splits
Searching all Possible Splits
  • For any node CART will examine ALL possible splits.
    • CART allows search over a random sample if desired
  • Look at first variable in our data set AGE with minimum value 40
    • Test split Is AGE £ 40?
      • Will separate out the youngest persons to the left
      • Could be many cases if many people have the same AGE
  • Next increase the AGE threshold to the next youngest person
      • Is AGE £ 43?
      • This will direct additional cases to the left
  • Continue increasing the splitting threshold value by value
    • each value is tested for how good the split is . . . how effective it is in separating the classes from each other
  • Q: Consider splits between values of the same class?
split tables

X

Split Tables

Q: Where splits need to be evaluated?

Sorted by Blood Pressure

Sorted byAge

X

cart splitting criteria gini index
CART Splitting Criteria: Gini Index
  • If a data set T contains examples from n classes, gini index, gini(T) is defined as

where pj is the relative frequency of class j in T.

gini(T) is minimized if the classes in T are skewed.

  • Advanced: CART also has other splitting criteria
    • Twoing is recommended for multi-class
missing as a distinct splitter value
Missing as a distinct splitter value
  • CHAID treats missing as a distinct categorical value
    • e.g AGE is 25-44, 45-64, 65-95 or missing
    • method also adopted by C4.5
  • If missing is a distinct value then all cases with missing go the same way in the tree
  • Assumption: whatever the unknown value it is the same for all cases with missing value
  • Problem: can be more than one reason for a database field to be missing:
    • E.g. Income as a splitter wants to separate high from low
    • Levels most likely to be missing? High Income AND Low Income!
    • Don’t want to send both groups to same side of tree
cart treatment of missing primary splitters surrogates
CART Treatment of Missing Primary Splitters: Surrogates
  • CART uses a more refined method —a surrogate is used as a stand in for a missing primary field
    • surrogate should be a valid replacement for primary
  • Consider our example of INCOME
  • Other variables like Education or Occupation might work as good surrogates
    • Higher education people usually have higher incomes
    • People in high income occupations will usually (though not always) have higher incomes
  • Using surrogate means that missing on primary not all treated same way
  • Whether go left or right depends on surrogate value
    • thus record specific . . . some cases go left others go right
surrogates mimicking alternatives to primary splitters
Surrogates Mimicking Alternatives to Primary Splitters
  • A primary splitter is the best splitter of a node
  • A surrogate is a splitter that splits in a fashion similar to the primary
  • Surrogate — variable with near equivalent information
  • Why Useful?
    • If the primary is expensive or difficult to gather and the surrogate is not
      • Then consider using the surrogate instead
      • Loss in predictive accuracy might be slight
    • If primary splitter is MISSING then CART will use a surrogate
    • if top surrogate missing CART uses 2nd best surrogate etc
  • If all surrogates missing also CART uses majority rule
cart pruning method grow full tree then prune
CART Pruning Method: Grow Full Tree, Then Prune
  • You will never know when to stop . . . so don’t!
  • Instead . . . grow trees that are obviously too big
  • Largest tree grown is called “maximal” tree
  • Maximal tree could have hundreds or thousands of nodes
    • usually instruct CART to grow only moderately too big
    • rule of thumb: should grow trees about twice the size of the truly best tree
  • This becomes first stage in finding the best tree
  • Next we will have to get rid the parts of the overgrown tree that don’t work (not supported by test data)
tree pruning
Tree Pruning
  • Take a very large tree (“maximal” tree)
  • Tree may be radically over-fit
    • Tracks all the idiosyncrasies of THIS data set
    • Tracks patterns that may not be found in other data sets
    • At bottom of tree splits based on very few cases
    • Analogous to a regression with very large number of variables
  • PRUNE away branches from this large tree
    • But which branch to cut first?
  • CART determines a pruning sequence:
    • the exact order in which each node should be removed
    • pruning sequence determined for EVERY node
    • sequence determined all the way back to root node
order of pruning weakest link goes first
Order of Pruning: Weakest Link Goes First
  • Prune away "weakest link" — the nodes that add least to overall accuracy of the tree
    • contribution to overall tree a function of both increase in accuracy and size of node
    • accuracy gain is weighted by share of sample
    • small nodes tend to get removed before large ones
  • If several nodes have same contribution they all prune away simultaneously
    • Hence more than two terminal nodes could be cut off in one pruning
  • Sequence determined all the way back to root node
    • need to allow for possibility that entire tree is bad
    • if target variable is unpredictable we will want to prune back to root . . . the no model solution
pruning sequence example
Pruning Sequence Example

24 Terminal Nodes

21 Terminal Nodes

18 Terminal Nodes

20 Terminal Nodes

now we test every tree in the pruning sequence
Now we test every tree in the pruning sequence
  • Take a test data set and drop it down the largest tree in the sequence and measure its predictive accuracy
    • how many cases right and how many wrong
    • measure accuracy overall and by class
  • Do same for 2nd largest tree, 3rd largest tree, etc
  • Performance of every tree in sequence is measured
  • Results reported in table and graph formats
  • Note that this critical stage is impossible to complete without test data
  • CART procedure requires test data to guide tree evaluation
training data vs test data error rates
Training Data Vs. Test Data Error Rates

No.

Terminal Nodes

  • Compare error rates measured by
    • learn data
    • large test set
  • Learn R(T) always decreases as tree grows (Q: Why?)
  • Test R(T) first declines then increases (Q: Why?)
  • Overfitting is the result tree of too much reliance on learn R(T)
  • Can lead to disasters when applied to new data

R(T)

Rts(T)

71 .00 .42

63 .00 .40

58 .03 .39

40 .10 .32

34 .12 .32

19 .20 .31

**10 .29 .30

9 .32 .34

7 .41 .47

6 .46 .54

5 .53 .61

2 .75 .82

1 .86 .91

why look at training data error rates or cost at all
Why look at training data error rates (or cost) at all?
  • First, provides a rough guide of how you are doing
    • Truth will typically be WORSE than training data measure
    • If tree performing poorly on training data error may not want to pursue further
    • Training data error rate more accurate for smaller trees
      • So reasonable guide for smaller trees
      • Poor guide for larger trees
  • At optimal tree training and test error rates should be similar
    • if not something is wrong
    • useful to compare not just overall error rate but also within node performance between training and test data
cart optimal tree
CART: Optimal Tree
  • Within a single CART run which tree is best?
  • Process of pruning the maximal tree can yield many sub-trees
  • Test data set or cross- validation measures the error rate of each tree
  • Current wisdom — select the tree with smallest error rate
  • Only drawback — minimum may not be precisely estimated
  • Typical error rate as a function of tree size has flat region
  • Minimum could be anywhere in this region
one se rule one standard error rule
One SE Rule -- One Standard Error Rule
  • Original monograph recommends NOT choosing minimum error tree because of possible instability of results from run to run
  • Instead suggest SMALLEST TREE within 1 SE of the minimum error tree
  • Tends to provide very stable results from run to run
  • Is possibly as accurate as minimum cost tree yet simpler
  • Current learning — one SERULE is good for small data sets
  • For large data sets one should pick most accurate tree
    • known as the zero SE rule
in what sense is the optimal tree best
In what sense is the optimal tree “best”?
  • Optimal tree has lowest or near lowest cost as determined by a test procedure
  • Tree should exhibit very similar accuracy when applied to new data
  • BUT Best Tree is NOT necessarily the one that happens to be most accurate on a single test database
    • trees somewhat larger or smaller than “optimal” may be preferred
  • Room for user judgment
    • judgment not about split variable or values
    • judgment as to how much of tree to keep
    • determined by story tree is telling
    • willingness to sacrifice a small amount of accuracy for simplicity
cart summary
CART Summary
  • CART Key Features
    • binary splits
    • gini index as splitting criteria
    • grow, then prune
    • surrogates for missing values
    • optimal tree – 1 SE rule
    • lots of nice graphics
decision tree summary
Decision Tree Summary
  • Decision Trees
    • splits – binary, multi-way
    • split criteria – entropy, gini, …
    • missing value treatment
    • pruning
    • rule extraction from trees
  • Both C4.5 and CART are robust tools
  • No method is always superior – experiment!

witten & eibe