Statistical Modeling and Model Selection

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Questions and Answers

Which criterion tends to select models with fewer variables and thus potentially lower test error?

Adjusted R2

BIC (correct)

AIC

Mallow's Cp

What should be minimized to achieve a high Adjusted R2 value?

RSS (correct)

TSS

BIC

Which approach can be used to adjust training error for model selection?

Cross-Validation (CV)

Validation Sets

All of the Above (correct)

Estimating Test Error Indirectly

Which criterion is likely to favor models with a smaller test error due to its penalty formulation?

BIC Signup and view all the answers

What characteristic is desired for values of Mallow's Cp?

Small values Signup and view all the answers

What does a larger Adjusted R2 indicate when comparing two models?

Better fit of the model Signup and view all the answers

Why is the likelihood function important when estimating the best model?

It reflects the goodness of fit for the model. Signup and view all the answers

What is the objective when choosing the best model with AIC or BIC?

To minimize test error Signup and view all the answers

What is the primary characteristic of natural cubic splines?

They extrapolate linearly between boundary knots. Signup and view all the answers

How many parameters are associated with a cubic spline that has k knots?

k + 1 Signup and view all the answers

What does adding more internal knots to a natural cubic spline allow for?

Better control over the spline’s fit. Signup and view all the answers

What is the main advantage of using piecewise polynomial functions?

They allow for different polynomial functions in various regions. Signup and view all the answers

What aspect do cubic splines need to maintain at the knots?

Continuity of the function and its first two derivatives. Signup and view all the answers

What does the term 'control wagging' refer to in spline models?

Manipulating curve shapes through knot placement. Signup and view all the answers

What is a key benefit of enforcing continuity in spline models?

It helps achieve a smoother transition between intervals. Signup and view all the answers

What is the role of knots in spline functions?

Define points where the polynomial changes its degree. Signup and view all the answers

What is the purpose of cross-validation (CV) in model selection?

To determine the tuning parameters for different models. Signup and view all the answers

Which component of Principal Components Regression (PCR) captures the largest variance?

1st Principal Component (PC). Signup and view all the answers

Why is dimension reduction important in regression modeling?

It helps in fitting models using fewer predictors while managing bias and variance. Signup and view all the answers

In the context of Ridge and Lasso regression, what role does cross-validation (CV) play?

It selects the optimal tuning parameter for the models. Signup and view all the answers

What does it mean when a model uses new predictors that are transformations of existing predictors?

New predictors help mitigate the bias-variance tradeoff. Signup and view all the answers

What does the loss function in regression help to achieve?

It assesses the effect of imposing penalties on coefficients. Signup and view all the answers

What is an expected consequence of dimensionality reduction in a regression context?

Improvement in the generalization of the model. Signup and view all the answers

What is the goal when dividing the predictor space in decision trees?

To find high dimensional rectangles with minimal RSS Signup and view all the answers

What does using a loss function that is equivalent to ordinary least squares imply?

The model will not consider regularization techniques. Signup and view all the answers

What is the most common approach used for selecting the best split in decision trees?

Top down greedy approach known as binary splitting Signup and view all the answers

What risk is associated with building a large decision tree?

Overfitting the model Signup and view all the answers

What method can be used to improve the decision tree after it has been built?

Pruning non-significant branches Signup and view all the answers

How is the value of α determined in the context of tree pruning?

Through cross-validation (CV) Signup and view all the answers

What is meant by a class tree in classification tasks?

A tree that assumes a sample belongs to the dominant class in its region Signup and view all the answers

Which outcome is achieved by tuning hyperparameters in decision trees?

Optimization of tree size for better generalization Signup and view all the answers

What does the process of binary splitting in decision trees involve?

Evaluating splits without considering future impact Signup and view all the answers

What is the main purpose of adding more trees in boosting?

To reduce the prediction bias of the model. Signup and view all the answers

What is the purpose of using the Gini index in classification?

To determine the purity of classes Signup and view all the answers

Which parameter is often tuned to change tree depth in a boosting model?

Split number Signup and view all the answers

What does CV help to determine in boosting?

The optimal number of trees to be used. Signup and view all the answers

How does bagging contribute to reducing variance?

Through averaging predictions from multiple trees Signup and view all the answers

What is indicated by a larger drop in RSS during tree construction?

The predictor variable is more important. Signup and view all the answers

What is a key characteristic of the Random Forest algorithm?

It decorrelates trees by using random selections of predictors Signup and view all the answers

In boosting, what do you start with when building a tree?

A stump, or tree with a single split. Signup and view all the answers

When building trees using bootstrap samples in bagging, what portion of data is typically used?

Approximately 67% of the dataset Signup and view all the answers

Why might Random Forest not overfit despite the number of trees used?

It decorrelates individual trees Signup and view all the answers

Which statistical measure is used in classification trees to assess variable importance?

Total Gini index Signup and view all the answers

What is a characteristic of boosting compared to random forests?

Boosting trees capture signals missed by previous trees. Signup and view all the answers

What does the term 'majority rules' refer to in a bagging context?

The final prediction based on the majority of tree outputs Signup and view all the answers

What is the role of predictors in Random Forest when making splits?

A random selection of predictors is used for each split Signup and view all the answers

What does updating residuals in a boosting model achieve?

It adjusts the output of each tree based on previous predictions. Signup and view all the answers

What does a small Gini index indicate about the classes?

The classes are mostly pure Signup and view all the answers

Study Notes

Cross Validation + Bootstrapping

Resampling methods are used to get more information about model fit
Bootstrapping is best when a large dataset is not available, to get estimates for test set prediction error
Cross validation is a method of estimating test set prediction error when a large dataset is not available
Cross validation is typically used to estimate prediction error when creating a model
A subsample of training observations is randomly selected
This subset of data is used to train and build the model
Randomly divide data into training and validation subsets
Model is then fit using training subset
Predictions are made using the model on the validation subset.
Validation set error is then calculated and used as an estimate of the test error
Validation error is used as an estimate of test error, usually to avoid overfitting
Cross-validation provides an estimate of the test error
Bootstrap can be used to estimate error
Cross-validation is often preferable over a simple train/test split
A lack of independence amongst data can make simple train/test splits less accurate

Drawbacks of Validation

Validation error depends on the split of data into training and validation
Validation error varies widely based on the random selection of data for the validation part.
The validation error can be highly variable based on which data points are included in the validation / training subsets

Summary

Cross-validation can be used to estimate prediction error
Bootstrap can be used to estimate prediction error
Cross-validation is often preferred for prediction error calculations versus simple train/test splits
Variability of the validation set error depends on the random selection of data for the validation part.
The validation error can be highly variable based on which data points are included in the validation / training subsets

K-Fold Validation

Divide data into K equal parts
Leave one part out to create the validation set
Fit the model on the remaining K-1 parts of the data
Use the model on the remaining part to calculate prediction error for validation
Repeat process k times, using a different part of the data for each iteration
Combine all test errors and divide by k to get an average validation error

Leave-One-Out Cross Validation

A special type of k-fold cross validation
A single observation is used for the validation dataset at each iteration in the process.
Number of folds = Number of observations

Cross-Validation for Classification

Estimate the test error for classification type models
Used to estimate model fit on independent data

Issues in Cross Validation

Training set is only a subset of the original data, can lead to biased estimates
Need enough data for all the folds and iterations to be representative and provide trustworthy estimates
Large dataset may lead to high computing cost

More Advanced Study Material

If k=N, where N is the number of observations, this is Leave-One-Out Cross Validation

Best Subset Selection

Starts with a null model (no predictors)
Examines all possible models containing 1, 2, 3 ... up to all predictors
Select the model with the minimal prediction error (e.g. lowest RSS)

Stepwise Selection

Starts with the null model and adds predictors one at a time
At each step, add a predictor that reduces the prediction error the most
Alternative: start with a full model and remove one predictor at a time, choosing the model with minimal prediction error

Other Methods

Best subset and step-wise selection are computationally expensive for a very large number of predictors

Shrinkage Methods

Penalize model complexity using a penalty term, often to avoid overfitting
Common example of shrinkage method = ridge regression
Penalizing model complexity helps prevent overfitting
If using a shrinkage method, tuning parameter lambda can be selected using cross validation.

Principle Component Regression

Dimensionality reduction technique
Reduces the number of predictors in the model
Tries to identify combinations of predictors that explain most of the variance
Useful for high-dimensional data

Partial Least Squares Regression

Dimensionality reduction technique
Combines elements of PCR and regression
Can be used when predictors are correlated, can overcome issues with simple PCA
Similar to principle component analysis

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Description

This quiz explores various criteria and approaches for model selection in statistical modeling, focusing on aspects like Adjusted R2, AIC, BIC, and Mallow's Cp. It delves into the characteristics of natural cubic splines and the advantages of piecewise polynomial functions. Test your understanding of these concepts and their implications in achieving accurate model predictions.