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Topic 10 - Ensemble Methods

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Topic 10 - Ensemble Methods

Data Mining - Volinsky - 2011 - Columbia University

- Remember this picture?
- Always looking for balance between low complexity (‘good on average’ but bad for prediction) and high complexity (‘good for specific cases’ but might overfit)
- By combining many different models, ensembles make it easier to hit the ‘sweet spot’ of modelling.
- Best for models to draw from diverse, independent opinions
- Wisdom Of Crowds

Stest(q)

Strain(q)

Data Mining - Volinsky - 2011 - Columbia University

- Models are just models.
- Usually not true!
- The truth is often much more complex than any single model can capture.
- Combinations of simple models can be arbitrarily complex. (e.g. spam/robots models, neural nets, splines)

- Notion: An average of several measurements is often more accurate and stable than a single measurement
Accuracy: how well the model does for estimation and prediction

Stability: small changes in inputs have little effect on outputs

Data Mining - Volinsky - 2011 - Columbia University

- The ensemble predicts a target value as an average or a vote of the predictions (of several individual models)...
- Each model is fit independently of the others
- Final prediction is a combination of the independent predictions of all models

- For an continuous target, an ensemble averages predictions
- Usually weighted

- For a categorical target (classification), an ensemble may average the probabilities of the target values…or may use ‘voting’.
- Voting classifies a case into the class that was selected most by individual models

Data Mining - Volinsky - 2011 - Columbia University

- Voting example
- 5 independent classifiers
- 70% accuracy for each
- Use voting…
- What is the probability that the ensemble model is correct?
- Lets simulate it

- What about 100 examples?
- (not a realistic example, why?)

Data Mining - Volinsky - 2011 - Columbia University

- The beauty is that you can average together models of any kind!!!
- Don’t need fancy schemes – just average!
- But there are fancy schemes: each one has various ways of fitting many models to the same data, and use voting or averaging
- Stacking (Wolpert 92): fit many leave-1-out models
- Bagging (Breiman 96) build models on many permutations of original data
- Boosting (Freund & Shapire 96): iteratively re-model, using re-weighted data based on errors from previous models…
- Arcing (Breiman 98), Bumping (Tibshirani 97), Crumpling (Anderson & Elder 98) , Born-Again (Breiman 98):
- Bayesian Model Averaging - near to my heart…

- We’ll explore BMA, bagging and boosting…

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

- Idea: account for inherent variance of the model selection process
- Posterior Variance = Within-Model Variance + Between-Model Variance
- Data-driven model selection is risky: “Part of the evidence is spent specify the model” (Leamer, 1978)
- Model-based inferences can be over-precise

Data Mining - Volinsky - 2011 - Columbia University

- For some quantity of interest D: avg over all Models M, given the data D:
To calculate the first term properly, you need to integrate out model parameters q,

Where q is the MLE.

For the second term, note that

^

Data Mining - Volinsky - 2011 - Columbia University

- The approximations on the previous page allow you to calculate many posterior model probabilities quickly, and gives you the weights to use for averaging.
- But, how do you know which models to average over?
- Example, regression with p parameters
- Each subset of p is a ‘model’
- 2p possible models!

- Idea:

Data Mining - Volinsky - 2011 - Columbia University

- But how to find the best models without fitting all models?
- Solution: Leaps and Bounds algorithm can find the best model without fitting all models
- Goal: find the single best model for each model size

Don’t need to traverse this part of the tree since there is no way it can beat AB

Data Mining - Volinsky - 2011 - Columbia University

PMP = Posterior Model Probability

Best Models

Score on holdout data: BMA wins

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

- Different approach to model ensembles – mostly for classification
- Observed: when model predictions are not highly correlated, combining does well
- Big idea: can we fit models specifically to the “difficult” parts of the data?

Data Mining - Volinsky - 2011 - Columbia University

From HTF p. 339

Data Mining - Volinsky - 2011 - Columbia University

- Courtesy M. Littman

Data Mining - Volinsky - 2011 - Columbia University

- Courtesy M. Littman

Data Mining - Volinsky - 2011 - Columbia University

- Courtesy M. Littman

Data Mining - Volinsky - 2011 - Columbia University

- Fast algorithms - AdaBoost
- Flexible – can work with any classification algorithm
- Individual models don’t have to be good
- In fact, the method works best with bad models!
- (bad = slightly better than random guessing)
- Most common model – “boosted stumps”

Data Mining - Volinsky - 2011 - Columbia University

Boosting Example from HTF p. 302

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

Bagging = Boostrap aggregating

- Big Idea:
- To avoid overfitting of specific dataset, fit model to “bootstrapped” random sets of the data

- Bootstrap
- Random sample, with replacement, from the data set
- Size of sample = size of data
- X= (1,2,3,4,5,6,7,8,9,10)
- B1=(1,2,3,3,4,5,6,6,7,8)
- B2=(1,1,1,1,2,2,2,5,6,8)
- …

- Bootstrap sample have the same statistical properties as original data
- By creating similar datasets you can see how much stability there is in your data. If there is a lack of stability, averaging helps.

Data Mining - Volinsky - 2011 - Columbia University

- Training data sets of size N
- Generate B “bootstrap” sampled data sets of size N
- Build B models (e.g., trees), one for each bootstrap sample
- Intuition is that the bootstrapping “perturbs” the data enough to make the models more resistant to true variability
- Note: only ~62% of data included in any bootstrap sample
- Can use the rest as an out-of-sample estimate!

- For prediction, combine the predictions from the B models
- Voting or averaging based on“out-of-bag” sample
- Plus: generally improves accuracy on models such as trees
- Negative: lose interpretability

Data Mining - Volinsky - 2011 - Columbia University

HTF Bagging Example p 285

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

- Trees are great, but
- As we’ve seen, they are “unstable”
- Also, trees are sensitive to the primary split, which can lead the tree in inappropriate directions
- one way to see this: fit a tree on a random sample, or a bootstrapped sample of the data -

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

from G. Ridgeway, 2003

- Solution:
- random forests: an ensemble of decision trees
- Similar to bagging: inject randomness to overcome instability
- each tree is built on a random subset of the training data
- Boostrapped version of data

- at each split point, only a random subset of predictors are considered
- Use “out-of-bag” hold out sample to estimate size of each tree
- prediction is simply majority vote of the trees ( or mean prediction of the trees).

- Randomizing the variables used is the key
- Reduces correlation between models!

- Has the advantage of trees, with more robustness, and a smoother decision rule.

Data Mining - Volinsky - 2011 - Columbia University

HTF Example p 589

Data Mining - Volinsky - 2011 - Columbia University

Breiman, Leo (2001). "Random Forests". Machine Learning 45 (1), 5-32

Data Mining - Volinsky - 2011 - Columbia University

- Breiman’s original algorithm said: “to keep bias low, trees are to be grown to maximum depth”
- However, empirical evidence typically shows that “stumps” do best

Data Mining - Volinsky - 2011 - Columbia University

- Averaging models together has been shown to be effective for prediction
- Many weird names:
- See papers by Leo Breiman (e.g. “Bagging Predictors”, Arcing the Edge”, and “Random Forests” for more detail

- Key points
- Models average well if they are uncorrelated
- Can inject randomness to insure uncorrelated models
- Averaging small models better than large ones

- Also, can give more insight into variables than simple tree
- Variables that show up again and again must be good

Data Mining - Volinsky - 2011 - Columbia University

- Data: Wisconsin Breast Cancer
- Courtesy S. Urbanek

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

Data Mining - Volinsky - 2011 - Columbia University

- Random Forests from Leo Breiman himself
- Breiman, Leo (2001). "Random Forests". Machine Learning 45 (1), 5-32
- Hastie, Tibshirani, Friedman (HTF)
- Chapters 8,10,15,16

Data Mining - Volinsky - 2011 - Columbia University