1. Model assessment and cross-validation - overview
Bias-variance decomposition and tradeoff
Analytical validation methods (AIC, BIC)
Resampling methods (cross-validation, bootstrap methods)
Data Mining and statistical learning 2008

2. Bias, variance and model complexity
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3. Bias, variance decomposition, and tradeoff
Expected prediction error
INSERT formula 7.8
We would like to minimize Err!
Increasing the complexity of models increases variance and decreases bias
Example: Smoothing based on nearest neighbours.
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4. Data Mining and statistical learning 2008
Loss function, training error, and test error - quantitative response (regression)
Loss functions (examples)
Training error:
Test error:
where the expectation is taken over the joint distribution of (X, Y)
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5. Data Mining and statistical learning 2008
Loss function, training error, and test error - qualitative response (classification)
Model: G=[1, 2, …, K], pk(X)=Pr(G=k| X).
Decision rule:
Examples of loss functions:
0-1 Loss
INSERT form. 7.4
Cross-entropy loss (log-likelihood)
INSERT form. 7.5
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6. Training error, and test error - qualitative response (classification)
0-1-loss: misclassification rate
Cross-entropy loss:
Test error
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7. Data Mining and statistical learning 2008
Model selection
Assume that the given model fα(x) is dependent on some tuning parameter (model complexity parameter) α
Examples of α:
No. predictors (multiple regression)
Degree of a polynomial (polynomial regression)
Penalty factor (smoothing splines, ridge regression)
Window width (kernels)
The aim of model selection: To find a model having minimum test error
Data Mining and statistical learning 2008

8. Model selection and assessment
Estimate the performance of different models in order to choose the best one.
Model assessment
Having chosen the final model, estimate the test error
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9. Model selection and assessment in data-rich applications
Training set (to produce a fit, appr. 50%)
Validation set (for model selection, 25 %)
Test set (for model assessment, 25%)
Example (splines)
Fit the training set using models with smoothing factors λ1, …, λn
Using fitted splines f1*(x),…fn*(x), estimate the prediction error using the validation set and choose the model #i producing minimal error.
Estimate the generalization error using the test set and model #i
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10. Model selection and assessment in applications with insufficient data
Analytical expressions to select and assess models
Cp (correction for the number of inputs or basis functions)
AIC (Akaike’s information criterion)
BIC (Bayesian information criterion)
Resampling
Cross-validation
The bootstrap
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11. Analytical validation methods
Background:
A model typically overfits the data.
The prediction error will on average be higher than the training error
Terminology:
The difference between the average training and prediction error is called optimism
Basic idea:
Find an analytical expression for the optimism.
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12. Data Mining and statistical learning 2008
Optimism
Training error rate:
In-sample error:
Optimism:
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13. Analytical expressions for the optimism
For squared error, 0-1 loss, and some other loss functions
2. For linear models with d inputs or basis functions
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14. Data Mining and statistical learning 2008
Cp scores
Basic idea:
when d parameters are fitted under squared loss.
Compute
and choose the model with smallest Cp score
Properties:
Penalization inreases with increasing model complexity (d)
and decreases as the training sample size increases
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15. Akaike’s information criterion
When the likelihood is maximized it holds asymptotically that
where
Given a tuning parameter , we set
where d() is the effective number of parameters.
For Gaussian models, AIC is equivalent to Cp
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16. Effective number of parameters (d())
For linear smoothers:
Examples:
Simple linear regression (#exact param), ridge regression
Smoothing splines
Kernel smoothers
Define the effective number of parameters as
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17. Bayesian information criterion
Based on Bayesian theory we set
For Gaussian models
Properties:
BIC =AIC if ”2” is substituted for log(N)
Since log(N)>1 for N>7, BIC penalizes complex models more heavily than AIC
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18. Data Mining and statistical learning 2008
Features of AIC and BIC
For large models (assymptotical property)
BIC chooses the right model (if it is present among alternatives)
AIC chooses too complex models
For small models
BIC chooses too simple models
AIC is OK
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19. Data Mining and statistical learning 2008
Resampling methods
Cross-validation
K-fold cross-validation (rough scheme, show picture):
Divide data-set in K roughly equally-sized subsets
Remove subset #i and fit the model using remaining data.
Predict the function values for subset #i using the fitted model.
Repeat steps 2-3 for different i
CV= squared difference between observed values and predicted values (another function is possible)
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20. Data Mining and statistical learning 2008
Resampling methods
Cross-validation
Note: if K=N then method is leave-one-out cross-validation.
K-fold cross-validation:
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21. Model selection using cross-validation
Having a model depending on a tuning (complexity) parameter, choose the one with smallest CV:
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22. Data Mining and statistical learning 2008
Prediction error and cross-validation curve estimated from a single training set
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23. Generalized cross-validation
Basic idea: approximate the leave-one-out CV to make it faster Used for linear smoothers:
Note: In smoothing problems, GCV is similiar to AIC
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24. Data Mining and statistical learning 2008
The bootstrap
Why do we need the bootstrap?
To estimate the uncertainty of parameter estimates
Example:
Having a sample X1, …, Xn from an unknown distribution
we estimate its mean (expectation) by computing the sample mean
How to find uncertainty of the sample mean?
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25. Implementation of the crude nonparametric bootstrap
Sample with replacement form the observed data to obtain a bootstrap sample
Repeat step 1 B times
Compute parameter estimates using the bootstrap samples
Compute the variance of estimates from step 3
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26. Data Mining and statistical learning 2008
Bootstrap replicates
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27. Benefits and drawbacks of the nonparametric bootstrap
The uncertainty of an estimate can be obtained without any information about the underlying distribution
It is not applicable to small data sets
If the distribution is known (except for the level of some parameters) the bootstarp may be slightly less efficient than conventional parametric methods
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28. The bootstrap and estimation of prediction errors
Fit the model to bootstrap samples (role=training) and examine how well it predicts the training set (role=prediction)
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29. An improved bootstrap estimator of the prediction error
The leave-one-out bootstrap is similar to two-fold CV, and produces biased estimates of the expected squared prediction error.
Solution:
.632-estimator(pulls bias down)
Data Mining and statistical learning 2008