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Additive Models and Trees

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Additive Models and Trees

Lecture Notes for CMPUT 466/551

Nilanjan Ray

Principal Source: Department of Statistics, CMU

- GAM: Generalized Additive Models
- CART: Classification and Regression Trees
- MARS: Multiple Adaptive Regression Splines

What is GAM?

The functions fjare smoothing functions in general, such as splines, kernel

functions, linear functions, and so on…

Each function could be different, e.g., f1 can be linear, f2 can be a natural

spline, etc.

Compare GAM with Linear Basis Expansions (Ch. 5 of [HTF])

Similarities? Dissimilarities?

Any similarity (in principle) with Naïve Bayes model?

- Non-parametric functions (linear smoother)
- Smoothing splines (Basis expansion)
- Simple k-nearest neighbor (raw moving average)
- Locally weighted average by using kernel weighting
- Local linear regression, local polynomial regression

- Linear functions
- Functions of more than one variables (interaction term)
- Example:

Backfitting algorithm

- Initialize:
- Cycle:j = 1,2,…, p,…,1,2,…, p,…, (m cycles)
Until the functions change less than a prespecified threshold

Computational Advantage?

Convergence?

How to choose fitting functions?

Model:

Additive Logistic Regression: Backfitting

Fitting logistic regression (P99)

Fitting additive logistic regression (P262)

1. where

1.

2.

2.

Iterate:

Iterate:

a.

a.

b.

b.

c.

Using weighted least squares to fit a linear model to zi with weights wi, give new estimates

c. Using weighted backfitting algorithm to fit an additive model to zi with weights wi, give new estimates

3. Continue step 2 until converge

3.Continue step 2 until converge

- Input variables (predictors):
- 48 quantitative variables: percentage of words in the email that match a given word. Examples include business, address, internet, etc.
- 6 quantitative variables: percentage of characters in the email that match a given character, such as ‘ch;’, ch(, etc.
- The average length of uninterrupted sequences of capital letters
- The length of the longest uninterrupted sequence of capital letters
- The sum of length of uninterrupted length of capital letters

- Output variable: SPAM (1) or Email (0)
- fj’s are taken as cubic smoothing splines

Sensitivity: Probability of predicting spam given true state is spam =

Specificity: Probability of predicting email given true state is email =

- Useful flexible extensions of linear models
- Backfitting algorithm is simple and modular
- Interpretability of the predictors (input variables) are not obscured
- Not suitable for very large data mining applications (why?)

- Overview
- Principle behind: Divide and conquer
- Partition the feature space into a set of rectangles
- For simplicity, use recursive binary partition

- Fit a simple model (e.g. constant) for each rectangle
- Classification and Regression Trees (CART)
- Regress Trees
- Classification Trees

- Popular in medical applications

- An example (in regression case):

- How to grow a tree?
- How large should we grow the tree?

- Partition the space into M regions: R1, R2, …, RM.

Note that this is still an additive model

- The best partition: to minimize the sum of squared error:
- Finding the global minimum is computationally infeasible
- Greedy algorithm: at each level choose variable j and value s as:
- The greedy algorithm makes the tree unstable
- The error made at the upper level will be propagated to the lower level

- Trade-off between bias and variance
- Very large tree: overfit (low bias, high variance)
- Small tree (low variance, high bias): might not capture the structure

- Strategies:
- 1: split only when we can decrease the error (usually short-sighted)
- 2: Cost-complexity pruning (preferred)

Penalty on the complexity/size of the tree

Cost: sum of squared errors

- Cost-complexity pruning:
- Pruning: collapsing some internal nodes
- Cost complexity:
- Choose best alpha: weakest link pruning (p.270, [HTF])
- Each time collapse an internal node which add smallest error
- Choose from this tree sequence the best one by cross-validation

- Classify the observations in node m to the major class in the node:
- Pmk is the proportion of observation of class k in node m

- Define impurity for a node:
- Misclassification error:
- Entropy:
- Gini index :

- Entropy and Gini are more sensitive
- To grow the tree: use Entropy or Gini
- To prune the tree: use Misclassification rate (or any other method)

Node impurity measures versus class proportion for 2-class problem

- Categorical Predictors
- Problem: Consider splits of sub tree t into tL and tR based on categorical predictor x which has q possible values: 2(q-1)-1 ways !
- Treat the categorical predictor as ordered by say proportion of class 1

- Linear Combination Splits
- Split the node based on
- Improve the predictive power
- Hurt interpretability

- Instability of Trees
- Inherited from the hierarchical nature
- Bagging (section 8.7 of [HTF]) can reduce the variance

Construct B number of trees from B bootstrap samples– bootstrap trees

is computed from the bth bootstrap sample

in this case a tree

Bagging reduces the variance of the original tree by aggregation

Majority vote

Average

- In multi-dimensional spline the basis functions grow exponentially– curse of dimensionality
- A partial remedy is a greedy forward search algorithm
- Create a simple basis-construction dictionary
- Construct basis functions on-the-fly
- Choose the best-fit basis function at each step

- 1-dim linear spline (t represents the knot)
- Basis collections C:
|C| = 2 * N * p

- Initialize basis set M with a constant function
- Form candidates (cross-product of M with set C)
- Add the best-fit basis pair (decrease residual error the most) into M
- Repeat from step 2 (until e.g. |M| >= threshold)

M (new)

M (old)

C

The final model M typically overfits the data

=>Need to reduce the model size (# of terms)

Backward deletion procedure

- Remove term which causes the smallest increase in residual error
- Compute
- Repeat step 1
Choose the model size with minimum GCV.

- M(.) measures effective # of parameters:
- r: # of linearly independent basis functions
- K: # of knots selected
- c = 3

- Piecewise linear reflected basis
- Allow operation on local region
- Fitting N reflected basis pairs takes O(N) instead of O(N^2)
- Left-part is zero, right-part differs by a constant

X[i-1]

X[i]

X[i+1]

X[i+2]

- Hierarchical model (reduce search computation)
- High-order term exists => some lower-order “footprints” exist

- Restriction: Each input appear at most once in a product:
e.g. (Xj - t1) * (Xj - t1) is not considered

- Set upper limit on order of interaction
- Upper limit of 1 => additive model

- MARS for classification
- Use multi-response Y (N*K indicator matrix)
- Masking problem may occur
- Better solution: “optimal scoring” (Chapter 12.5 of [HTF])

IF

- replace piecewise linear basis by step functions
- keep only the newly formed product terms in M (leaf nodes of a binary tree)
THEN

MARS forward procedure

= CART tree growing procedure