Lecture 12 – Model Assessment and Selection

1 / 45

# Lecture 12 – Model Assessment and Selection - PowerPoint PPT Presentation

Lecture 12 – Model Assessment and Selection. Rice ECE697 Farinaz Koushanfar Fall 2006. 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

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

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Lecture 12 – Model Assessment and Selection

Rice ECE697

Farinaz Koushanfar

Fall 2006

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 Criteria
• Training Error
• Loss Function
• Generalization Error
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)||22
• 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 X11/2 and apply k-NN
• Right: Y is 1 if j=110Xj is 5 and 0 otherwise

Prediction error

Squared bias

Variance

Example – loss function

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 AICCp
• 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 xp.
• 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 (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