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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

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lecture 12 model assessment and selection

Lecture 12 – Model Assessment and Selection

Rice ECE697

Farinaz Koushanfar

Fall 2006

summary
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
Model Selection Criteria
  • Training Error
  • Loss Function
  • Generalization Error
model selection and assessment
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
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
Bias-Variance Decomposition (cont’d)
  • For K-nearest neighbor
  • For linear regression
bias variance decomposition cont d1
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
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
Example – loss function

Prediction error

Squared bias

Variance

optimism of training error
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
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
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
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
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
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 d1
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
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 parameters1
The effective number of parameters
  • Thus, for a regularized model:
  • Hence
  • and
the bayesian approach and bic
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 (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 d1
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 d2
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 d3
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
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 d1
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 d2
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 d3
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 d4
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 d5
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
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
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 d1
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 d3
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 d4
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
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 (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 d1
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 d2
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
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