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?

    <p>Backward stepwise regression</p> Signup and view all the answers

    What is forward stepwise regression?

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

    Regularization methods help reduce the overall ______ of a model.

    <p>variance</p> Signup and view all the answers

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

    <p>You may accidentally remove useful variables.</p> Signup and view all the answers

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

    <p>Achieving superior prediction</p> Signup and view all the answers

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

    <p>False</p> Signup and view all the answers

    What does R² measure?

    <p>Goodness of fit of a model</p> Signup and view all the answers

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

    <p>Overfitting</p> Signup and view all the answers

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

    <p>cross</p> Signup and view all the answers

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

    <p>bias</p> Signup and view all the answers

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

    <p>5</p> Signup and view all the answers

    What does MSE stand for?

    <p>Mean Squared Error</p> Signup and view all the answers

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

    <p>Minimize mean squared error</p> Signup and view all the answers

    What does the James-Stein estimator aim to do?

    <p>Shrink sample means towards a preliminary guess</p> 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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    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.

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