ECON 471: Lecture 12, 13 - Machine Learning Fundamentals

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Questions and Answers

What is the main concern when estimating statistical models in high-dimensional settings?

Bias

Variance

Underfitting

Overfitting (correct)

A 20th degree polynomial will have low variance across different samples.

False

What is the tradeoff represented in choosing between a simpler and a more complicated model?

Bias and variance

What is a bad approach to model selection mentioned in the content?

Backward stepwise regression Signup and view all the answers

What is forward stepwise regression?

A method starting with a simple model and progressively adding variables. Signup and view all the answers

Regularization methods help reduce the overall ______ of a model.

variance Signup and view all the answers

What is a potential consequence of starting with a complex model and then simplifying it?

You may accidentally remove useful variables. Signup and view all the answers

What is the primary focus when dealing with high-dimensional data?

Achieving superior prediction Signup and view all the answers

In linear regression, adding interaction terms such as $X_1 imes X_{23}$ is done for better interpretability.

False Signup and view all the answers

What does R² measure?

Goodness of fit of a model Signup and view all the answers

What issue arises when a model is too complex and fits the in-sample data poorly?

Overfitting Signup and view all the answers

The process of checking how well a model performs on data it hasn't seen is called _____ validation.

cross Signup and view all the answers

The __________ of an estimator relates to how far it deviates from the true parameter on average.

bias Signup and view all the answers

What is a common value of K in K-fold cross-validation?

5 Signup and view all the answers

What does MSE stand for?

Mean Squared Error Signup and view all the answers

What is the main goal of the bias-variance trade-off?

Minimize mean squared error Signup and view all the answers

What does the James-Stein estimator aim to do?

Shrink sample means towards a preliminary guess Signup and view all the answers

Study Notes

Dealing with Complex Data

Basic modeling techniques like linear and logistic regression excel in low-dimensional data where interpretability is essential.
High-dimensional data presents challenges; examples include predicting consumer behavior for online purchases or forecasting electricity demand.
In high-dimensional scenarios, interpretability may be less important than achieving superior predictions.
Complex models can automatically incorporate varying terms, such as interactions (e.g., X1·X23), when prediction is the goal.
Among model options, practitioners use generalized linear models as well as advanced techniques like Lasso, random forests, or neural networks.
Model evaluation in high-dimensional settings requires careful thinking as traditional criteria like R² may not translate well.

In Sample vs. Out of Sample Fit

R² assesses goodness-of-fit using observed data but isn’t adequate for evaluating predictions on unseen data.
Out-of-sample performance is measured through Mean Squared Error (MSE), defined as E[(Y - ĝ(X))²].
Simple models usually have high R² and generalize better, while complex models can suffer from overfitting, performing poorly out of sample.
Overfitting occurs when a model fits noise in the sample data instead of the underlying data distribution.

Cross-Validation

Cross-validation fights overfitting by testing a model against data withheld during fitting.
K-fold cross-validation divides the dataset into K parts to train the model on K-1 subsets and validate it on the remaining piece.
Each segment is used for testing in turn, generating MSE for unbiased error estimation.
The average out-of-sample MSE, or cross-fit (CV) score, helps in choosing the model with the optimal performance based on generalizability.

Mean Squared Error and the Bias-Variance Trade-off

In low-dimensional settings, unbiased estimators such as sample means maintain a desirable property ensuring estimates center around true values.
When focusing on predictions rather than estimation accuracy, unbiasedness may not be the best target.
MSE can be decomposed into variance and bias components: MSE(ĝ(x0)) = Var(ĝ(x0)) + [Bias(ĝ(x0))]².
Variance reflects how estimates disperse, while bias indicates deviation from true values.
The Bias-Variance Trade-off suggests that adding bias can lower total MSE by significantly reducing variance.

Bias-Variance Trade-off and the James-Stein Mean Estimate

Standard sample means yield low bias but do not minimize MSE, prompting exploration into biased estimators.
The James-Stein estimator introduces bias by shrinking estimates towards a prior guess, yielding lower total MSE when estimating multiple means.
This result specifically applies when estimating three or more means; the benefits derive from a tailored approach of adjusting standard unbiased estimators.

Regularization

Overfitting arises when models become too complex, potentially leading to unreliable predictions.
Simpler models may be biased but have lower variance, while complex models can adjust closely to sample data but suffer from increased variance.
Regularization techniques introduce controlled bias towards simpler models, reducing model variance and enhancing predictive accuracy.
A key goal of regularization is to find a balance between bias and variance to achieve lower overall MSE without excessive reliance on user input.### Model Selection Approaches
Naive model selection may involve starting with linear regression using all available variables, then removing insignificant variables based on p-values.
This process, called backward stepwise regression, may result in the erroneous exclusion of beneficial variables due to multicollinearity among regressors.
When regressors are highly correlated, p-values may indicate insignificance, masking the utility of individual variables.

Issues with Backward Stepwise Regression

P-values may not be reliable when testing multiple coefficients simultaneously, akin to challenges in multiple hypothesis testing.
Backward stepwise regression is discouraged as it begins with a complex model, which can misrepresent relationships.

Forward Stepwise Regression

Forward stepwise regression starts with a simple model and gradually incorporates variables based on their explanatory power.
The process involves fitting univariate models and selecting the one with the highest in-sample R² value.
Next, bivariate models are fitted, including one previously selected variable, and again the model with the highest in-sample R² is chosen for addition.

Iterative Model Building

The selection process continues iteratively, adding variables that enhance explanatory power until a predefined model complexity or stopping rule is achieved.
This approach serves as a regularization technique, aiming to reduce model complexity while preventing overfitting and minimizing estimator variance.

Practical Implementation

In R, the “step” function can be used to automate the forward selection process with specified variables contained in a data frame.
Model selection strategies prioritize constructing parsimonious models that accurately capture the underlying relationships in the data.

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Description

This quiz covers fundamental concepts in machine learning, focusing on techniques such as linear regression and logistic regression. These models are crucial for economists dealing with complex, high-dimensional data. Enhance your understanding of data science applications in economics.