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## PowerPoint Slideshow about 'Lecture 12 – Model Assessment and Selection' - rudolpho-calvey

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Summary

- Bias, variance, model complexity
- Optimism of training error rate
- Estimates of in-sample prediction error, AIC
- Effective number of parameters
- The Bayesian approach and BIC
- Vapnik-Chernovekis dimension
- Cross-Validation
- Bootstrap method

Model Selection and Assessment

- Model selection:
- Estimating the performance of different models in order to chose the best
- Model assessment:
- Having chosen a final model, estimating its prediction error (generalization error) on new data
- If we were rich in data:

Train

Validation

Test

Bias-Variance Decomposition

- As we have seen before,
- The first term is the variance of the target around the true mean f(x0); the second term is the average by which our estimate is off from the true mean; the last term is variance of f^(x0)

* The more complex f, the lower the bias, but the higher the variance

Bias-Variance Decomposition (cont’d)

- For K-nearest neighbor
- For linear regression

Bias-Variance Decomposition (cont’d)

- For linear regression, where h(x0) is the vector of weights that produce fp(x0)=x0T(XTX)-1XTy and hence Var[(fp(x0)]=||h(x0)||22
- This variance changes with x0, but its average over the sample values xi is (p/N) 2

Example

- 50 observations and 20 predictors, uniformly distributed in the hypercube [0,1]20.
- Left: Y is 0 if X11/2 and apply k-NN
- Right: Y is 1 if j=110Xj is 5 and 0 otherwise

Prediction error

Squared bias

Variance

Optimism of Training Error

- The training error
- Is typically less than the true error
- In sample error
- Optimism
- For squared error, 0-1, and other losses, on can show in general

Optimism (cont’d)

- Thus, the amount by which the error under estimates the true error depends on how much yi affects its own prediction
- For linear model
- For additive model Y=f(X)+ and thus,

Optimism increases linearly with number of inputs or basis d,

decreases as training size increases

How to count for optimism?

- Estimate the optimism and add it to the training error, e.g., AIC, BIC, etc.
- Bootstrap and cross-validation, are direct estimates of this optimism error

Estimates of In-Sample Prediction Error

- General form of in-sample estimate is computed from
- Cp statistic: for an additive error model, when d parameters are fit under squared error loss,
- Using this criterion, adjust the training error by a factor proportional to the number of basis
- Akaike Information Criterion (AIC) is a similar but a more generally applicable estimate of Errin, when the log-likelihood loss function is used

Akaike Information Criterion (AIC)

- AIC relies on a relationship that holds asymptotically as N
- Pr(Y) is a family of densities for Y (contains the “true” density), “ hat” is the max likelihood estimate of , “loglik” is the maximized log-likelihood:

AIC (cont’d)

- For the Gaussian model, the AICCp
- For the logistic regression, using the binomial log-likelihood, we have
- AIC=-2/N. loglik + 2. d/N
- Choose the model that produces the smallest possible AIC
- What if we don’t know d?
- How about having tuning parameters?

AIC (cont’d)

- Given a set of models f(x) indexed by a tuning parameter , denote by err() and d() the training error and number of parameters
- The function AIC provides an estimate of the test error curve and we find the tuning parameter that maximizes it
- By choosing the best fitting model with d inputs, the effective number of parameters fit is more than d

The effective number of parameters

- Generalize num of parameters to regularization
- Effective num of parameters is: d(S) = trace(S)
- In sample error is:

The effective number of parameters

- Thus, for a regularized model:
- Hence
- and

The Bayesian Approach and BIC

- Bayesian information criterion (BIC)
- BIC/2 is also known as Schwartz criterion

BIC is proportional to AIC (Cp) with a factor 2 replaced by log (N).

BIC penalizes complex models more heavily, prefering Simpler models

BIC (cont’d)

- BIC is asymptotically consistent as a selection criteria: given a family of models, including the true one, the prob. of selecting the true one is 1 for N
- Suppose we have a set of candidate models Mm, m=1,..,M and corresponding model parameters m, and we wish to chose a best model
- Assuming a prior distribution Pr(m|Mm) for the parameters of each model Mm, compute the posterior probability of a given model!

BIC (cont’d)

- The posterior probability is
- Where Z represents the training data. To compare two models Mm and Ml, form the posterior odds
- If the posterior greater than one, chose m, otherwise l.

BIC (cont’d)

- Bayes factor: the rightmost term in posterior odds
- We need to approximate Pr(Z|Mm)
- A Laplace approximation to the integral gives
- ^m is the maximum likelihood estimate and dm is the number of free parameters of model Mm
- If the loss function is set as -2 log Pr(Z|Mm,^m), this is equivalent to the BIC criteria

BIC (cont’d)

- Thus, choosing the model with minimum BIC is equivalent to choosing the model with largest (approximate) posterior probability
- If we compute the BIC criterion for a set of M models, BICm, m=1,…,M, then the posterior of each model is estimates as

- Thus, we can estimate not only the best model, but also
- asses the relative merits of the models considered

Vapnik-Chernovenkis Dimension

- It is difficult to specify the number of parameters
- The Vapnik-Chernovenkis (VC) provides a general measure of complexity and associated bounds on optimism
- For a class of functions {f(x,)} indexed by a parameter vector , and xp.
- Assume f is in indicator function, either 0 or 1
- If =(0,1) and f is a linear indicator, I(0+1Tx>0), then it is reasonable to say complexity is p+1
- How about f(x, )=I(sin .x)?

VC Dimension (cont’d)

- The Vapnik-Chernovenkis dimension is a way of measuring the complexity of a class of functions by assessing how wiggly its members can be
- The VC dimension of the class {f(x,)} is defined to be the largest number of points (in some configuration) that can be shattered by members of {f(x,)}

VC Dimension (cont’d)

- A set of points is shattered by a class of functions if no matter how we assign a binary label to each point, a member of the class can perfectly separate them
- Example: VC dim of linear indicator function in 2D

VC Dimension (cont’d)

- Using the concepts of VC dimension, one can prove results about the optimism of training error when using a class of functions. E.g.
- If we fit N data points using a class of functions {f(x,)} having VC dimension h, then with probability at least 1- over training sets

For regression, a1=a2=1

Cherkassky and Mulier, 1998

VC Dimension (cont’d)

- The bounds suggest that the optimism increases with h and decreases with N in qualitative agreement with the AIC correction d/N
- The results of VC dimension bounds are stronger: they give a probabilistic upper bounds for all functions f(x,) and hence allow for searching over the class

VC Dimension (cont’d)

- Vapnik’s Structural Risk Minimization (SRM) is built around the described bounds
- SRM fits a nested sequence of models of increasing VC dimensions h1<h2<…, and then chooses the model with the smallest value of the upper bound
- Drawback is difficulty in computing VC dim
- A crude upper bound may not be adequate

Cross Validation (CV)

- The most widely used method
- Directly estimate the generalization error by applying the model to the test sample
- K-fold cross validation
- Use part of data to build a model, different part to test
- Do this for k=1,2,…,K and calculate the prediction error when predicting the kth part

CV (cont’d)

- :{1,…,N}{1,…,K} divides the data to groups
- Fitted function f^-(x), computed when removed
- CV estimate of prediction error is
- If K=N, is called leave-one-out CV
- Given a set of models f^-(x), the th model fit with the kth part removed. For this set of models we have

CV (cont’d)

- CV() should be minimized over
- What should we chose for K?
- With K=N, CV is unbiased, but can have a high variance since the K training sets are almost the same
- Computational complexity

CV (cont’d)

- With lower K, CV has a lower variance, but bias could be a problem!
- The most common are 5-fold and 10-fold!

CV (cont’d)

- Generalized leave-one-out cross validation, for linear fitting with square error loss ỷ=Sy
- For linear fits (Sii is the ith on S diagonal)
- The GCV approximation is

GCV maybe sometimes advantageous where the trace is computed

more easily than the individual Sii’s

Bootstrap

- Denote the training set by Z=(z1,…,zN) where zi=(xi,yi)
- Randomly draw a dataset with replacement from training data
- This is done B times (e.g., B=100)
- Refit the model to each of the bootstrap datasets and examine the behavior over the B replications
- From the bootstrap sample, we can estimate any aspect of the distribution of S(Z) – where S(z) can be any quantity computed from the data

Bootstrap - Schematic

For e.g.,

Bootstrap (Cont’d)

- Bootstrap to estimate the prediction error
- E^rrboot does not provide a good estimate
- Bootstrap dataset is acting as both training and testing and these two have common observations
- The overfit predictions will look unrealistically good
- By mimicking CV, better bootstrap estimates
- Only keep track of predictions from bootstrap samples not containing the observations

Bootstrap (Cont’d)

- The leave-one-out bootstrap estimate of prediction error
- C-i is the set of indices of the bootstrap sample b that do not contain observation I
- We either have to choose B large enough to ensure that all of |C-i| is greater than zero, or just leave-out the terms that correspond to |C-i|’s that are zero

Bootstrap (Cont’d)

- The leave-one-out bootstrap solves the overfitting problem, we has a training size bias
- The average number of distinct observations in each bootstrap sample is 0.632.N
- Thus, if the learning curve has a considerable slope at sample size N/2, leave-one-out bootstrap will be biased upward in estimating the error
- There are a number of proposed methods to alleviate this problem, e.g., .632 estimator, information error rate (overfitting rate)

Bootstrap (Example)

- Five-fold CV and .632 estimate for the same problems as before

- Any of the measures could be biased but not affecting, as long as relative performance is the same

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