Linear Models and Regularisation Techniques
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Questions and Answers

Linear models are an example of selection and tuning methods.

True (A)

The term 'Equa-' refers specifically to nonlinear models.

False (B)

A polynomial degree of 1 is associated with high bias and low variance.

True (A)

The only model being discussed is a linear model.

<p>False (B)</p> Signup and view all the answers

The number '200' appears multiple times in the provided content.

<p>True (A)</p> Signup and view all the answers

Increasing the regularisation strength parameter λ results in a more complex model.

<p>False (B)</p> Signup and view all the answers

Iterative criteria is unrelated to tuning in model selection.

<p>False (B)</p> Signup and view all the answers

LASSO is one of the approaches used for regularisation in linear models.

<p>True (A)</p> Signup and view all the answers

When λ is large, the model parameters are not penalised significantly.

<p>False (B)</p> Signup and view all the answers

More regularisation generally increases the variance of a model.

<p>False (B)</p> Signup and view all the answers

The tradeoff between bias and variance can be managed through the regularisation strength parameter λ.

<p>True (A)</p> Signup and view all the answers

A polynomial degree of 14 is typically associated with low bias and high variance.

<p>True (A)</p> Signup and view all the answers

Regularisation cannot help improve generalisation error if a model is overfitting.

<p>False (B)</p> Signup and view all the answers

Higher model complexity always leads to lower error rates.

<p>False (B)</p> Signup and view all the answers

Regularization techniques like LASSO and Ridge regression help in reducing model overfitting.

<p>True (A)</p> Signup and view all the answers

The bias-variance tradeoff involves balancing the model's tendency to underfit and overfit.

<p>True (A)</p> Signup and view all the answers

Polynomial regression always results in better fitting than linear regression.

<p>False (B)</p> Signup and view all the answers

The degree of a polynomial model has no effect on its complexity.

<p>False (B)</p> Signup and view all the answers

Both training error and cross-validation error can be high in underfitting scenarios.

<p>True (A)</p> Signup and view all the answers

Model complexity can be defined solely based on the number of parameters used.

<p>False (B)</p> Signup and view all the answers

Feature selection is unrelated to regularization in model tuning.

<p>False (B)</p> Signup and view all the answers

L1-regularisation increases the contribution of features by driving coefficients to zero.

<p>False (B)</p> Signup and view all the answers

Regularisation is used to manage the complexity of a model in order to improve its generalisation.

<p>True (A)</p> Signup and view all the answers

Standardized coefficients are not typically associated with regularisation techniques.

<p>False (B)</p> Signup and view all the answers

Higher regularisation strength parameter λ leads to a simpler model by penalising model parameters more heavily.

<p>True (A)</p> Signup and view all the answers

Feature selection through regularisation is irrelevant when using linear models.

<p>False (B)</p> Signup and view all the answers

Ridge regression aims to achieve coefficient estimates that fit the data well by minimizing the R-squared value.

<p>False (B)</p> Signup and view all the answers

The tuning parameter λ in ridge regression must always be greater than 1.

<p>False (B)</p> Signup and view all the answers

The shrinkage penalty in ridge regression serves to reduce the coefficients towards zero.

<p>True (A)</p> Signup and view all the answers

In ridge regression, increasing the value of λ generally leads to lower variance and higher bias.

<p>True (A)</p> Signup and view all the answers

Ridge regression involves selecting the best value for the tuning parameter λ through trial and error methods only.

<p>False (B)</p> Signup and view all the answers

Ridge regression is fundamentally different from least squares regression in terms of the estimation approach used.

<p>True (A)</p> Signup and view all the answers

In ridge regression, if the coefficients β1, ..., βp are large, the shrinkage penalty will have a minimal effect.

<p>True (A)</p> Signup and view all the answers

The tuning parameter λ is not critical when using ridge regression and can be ignored.

<p>False (B)</p> Signup and view all the answers

The tuning parameter $ au$ is used to control the penalty in the regularization process.

<p>False (B)</p> Signup and view all the answers

Regularization helps in feature selection by identifying which features are important to include in the model.

<p>True (A)</p> Signup and view all the answers

A $eta$ coefficient of zero indicates that a feature has some level of significance in the model.

<p>False (B)</p> Signup and view all the answers

The #1 norm shrinks the penalty effect while the #2 norm has a different impact on the model's coefficients.

<p>True (A)</p> Signup and view all the answers

The notation $β̂$ refers to the estimated coefficients in the context of regularization.

<p>True (A)</p> Signup and view all the answers

A higher value of the tuning parameter $eta$ always results in better model performance.

<p>False (B)</p> Signup and view all the answers

Shrinkage based on the tuning parameter λ effectively reduces the coefficients to minimize model complexity.

<p>True (A)</p> Signup and view all the answers

The regularization process does not impact the selection of features from the dataset.

<p>False (B)</p> Signup and view all the answers

Flashcards

Variance

The degree to which a model's predictions are consistent across different datasets.

Bias

The error that arises from a model being too simplistic and not capturing the underlying patterns in the data.

Bias-Variance Trade-Off

The trade-off between bias and variance in model selection. A model with high bias will have low variance, and vice versa.

Model Complexity

The complexity of a model, often measured by the number of parameters or the flexibility of the function used to fit the data.

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Regularization

A technique used to reduce overfitting by adding a penalty term to the loss function, which discourages overly complex models and encourages weights towards zero.

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LASSO Regression

A type of regularization where the penalty is proportional to the sum of the absolute values of the model's coefficients. This method encourages sparsity in the model, which promotes feature selection.

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Ridge Regression

A type of regularization where the penalty is proportional to the sum of the squared values of the model's coefficients. This method shrinks the magnitude of the model's parameters towards zero but doesn't force them to be exactly zero.

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Feature Selection

The process of selecting a subset of relevant features from a larger set of variables for inclusion in your model.

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Ensemble Learning

A machine learning technique that involves finding the best combination of models to improve prediction accuracy.

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L1 Regularization (LASSO)

A type of regularization technique that penalizes large weights in a model, encouraging simpler and potentially more robust models.

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L2 Regularization (Ridge)

A type of regularization technique that penalizes the sum of squared weights in a model, leading to smaller but non-zero weights.

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Model Selection

Choosing the best model from a set of candidate models based on their performance on a validation dataset.

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What is a linear model?

A linear model is a statistical method that uses a linear function to predict an output value based on one or more input variables. It assumes a straight-line relationship between the inputs and the output.

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What is polynomial degree?

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, a polynomial with degree 1 is a linear function, while a polynomial with degree 2 is a quadratic function.

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What does high bias mean?

High bias means that the model is too simple and cannot capture the complexity of the data, leading to underfitting. It results in poor prediction accuracy on both training and unseen data.

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What does low bias mean?

Low bias means that the model is complex enough to capture the underlying patterns in the data, but it may be too sensitive to the training data and overfit. It results in high accuracy on training data but low accuracy on unseen data.

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What does high variance mean?

High variance means that the model is too sensitive to the training data and does not generalize well to unseen data. This can lead to overfitting, where the model performs well on the training data but poorly on new data.

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What does low variance mean?

Low variance means that the model is less sensitive to the training data and generalizes well to unseen data. This can however lead to underfitting, where the model does not perform well on the training data because it's too simple.

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What is model regularization?

Regularization techniques aim to prevent overfitting in machine learning models by penalizing complex models, effectively encouraging simpler models that generalize better to unseen data. This is achieved by adding a penalty term to the cost function.

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What is Ridge regression?

Ridge regression is a type of regularization that adds a penalty proportional to the square of the magnitude of the coefficients. This shrinks the coefficients toward zero, reducing the impact of less important features and preventing overfitting.

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λ (Lambda)

A tuning parameter used in regularization methods to control the strength of the penalty applied to the model coefficients. Higher values of λ lead to stronger regularization and smaller coefficient magnitudes.

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β̂λ / β̂

A measure of how well a model generalizes to unseen data. It is calculated as the ratio of the model's coefficients with regularization to the coefficients without regularization.

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βˆλ

The coefficients obtained after applying regularization.

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βˆ

The coefficients obtained without applying regularization.

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What is Regularization?

Regularization is a technique that helps reduce overfitting in machine learning models by adding a penalty term to the loss function. This penalty discourages complex models and encourages smaller coefficients, promoting simpler and more generalizable models.

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How Does Regularization Work?

One of the main ways Regularization works is by shrinking the contribution of features towards the model's predictions, eventually driving some coefficients to zero.

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What is L1 Regularization? (LASSO)

L1 Regularization, also known as LASSO, uses a penalty that's proportional to the sum of the absolute values of the coefficients. This promotes sparsity, which is a model with many coefficients set to zero, meaning some features are completely eliminated.

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What is Ridge Regression? (L2 Regularization)

Ridge Regression, also known as L2 Regularization, uses a penalty that's proportional to the sum of the squared values of the coefficients. This shrinks the coefficients towards zero, but won't necessarily eliminate them completely.

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How Does Regularization Relate to Feature Selection?

By shrinking coefficients and eventually driving some to zero, Regularization effectively performs feature selection, focusing on the most relevant features for the model.

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What is the shrinkage penalty in Ridge Regression?

The shrinkage penalty in ridge regression is a term that penalizes large coefficient values, encouraging them to be closer to zero. This helps to prevent overfitting by reducing the influence of individual features with high variability.

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What is the role of the tuning parameter λ in Ridge Regression?

The tuning parameter λ controls the strength of the shrinkage penalty in ridge regression. A higher λ value results in stronger shrinkage, pushing coefficients closer to zero and reducing the model's complexity. Conversely, a lower λ value allows for larger coefficients, leading to a more complex model.

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When is Ridge Regression used?

Ridge regression is often used when dealing with datasets with a high number of features (p) compared to the number of observations (n), which can potentially lead to overfitting. By shrinking the coefficients, ridge regression reduces the variance of the model, making it more robust and less prone to overfitting.

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Why is Ridge Regression beneficial when dealing with correlated features?

Ridge regression is generally preferred when dealing with datasets with correlated features. By shrinking the coefficients, ridge regression helps to stabilize the model and reduce the influence of features that are highly correlated.

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How is the tuning parameter λ selected in Ridge Regression?

Cross-validation is a technique used to find the optimal value for the tuning parameter λ in ridge regression. It involves splitting the data into multiple folds, training the model on a subset of the folds and evaluating its performance on the remaining fold. This process is repeated multiple times, and the value of λ that yields the best overall performance is selected.

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Is Ridge regression used with feature scaling?

Ridge regression is often used in conjunction with feature scaling, which involves transforming the features to have similar scales. This helps to prevent features with larger scales from having a disproportionate influence on the model.

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How is Ridge Regression related to Linear Regression?

Ridge regression can be seen as a regularized version of linear regression. Regularization refers to the process of adding a penalty term to the loss function, which helps to prevent overfitting by reducing the complexity of the model.

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Study Notes

Learning Objectives

  • To understand the trade-off between model complexity and error.
  • To understand bias and variance of a model.
  • To understand regularisation as an approach to overfitting.

Outline of Lecture

  • Introduction to model complexity and bias-variance tradeoff:
    • Defining bias and variance, relating them to model complexity.
    • Discussing different sources of model error and their connection to bias and variance.
    • Clarifying the bias-variance tradeoff, how it relates to complexity, and optimal balance between bias, variance and model complexity vs. error.
  • Introduction to LASSO and Ridge regression:
    • Explaining how to tune a model through regularisation.
    • Analysing the relationship between regularisation and feature selection.

Recap from Lecture 4

  • Information from previous lecture, visualized by using diagram of test data, training data and all data.

Model Complexity vs. Error

  • How error metrics relate to model complexity, illustrated by figures of polynomial degree = 1, polynomial degree = 4 and polynomial degree = 14.

Model Complexity vs. Error (Underfitting & Overfitting)

  • Graph showing how training error and cross-validation error both increase as the model is made more complex (underfitting).
  • Graph showing how the training error decreases to a minimum and then rises again as the model complexity increases (overfitting).
  • Defining under-fitting and over-fitting.

Model Complexity vs. Error (Just Right)

  • Graph showing a "just right" model where the training error and cross-validation error are low.

Choosing the Level of Complexity

  • Importance of achieving a balance between model complexity and prediction accuracy
  • Visual representation of polynomial degrees (1, 4, 14) with their corresponding fitting, to illustrate the bias-variance tradeoff through the example of polynomial regression.

Intuition: Bias and Variance

  • Intuitive explanation of bias and variance using an analogy of a dartboard. Low bias means the darts are close to the bullseye, high bias means the darts are far from the bullseye, also low variance means darts are close together, and high variance means darts are widely dispersed.
  • Definition of bias and variance.
  • Ideally, a model has low bias and low variance.

Three Sources of Model Error

  • Bias: Failing to capture the relationship between features and outcome variables, leading to consistently poor predictions.
  • Variance: Overly sensitive to small changes in input data, resulting in inconsistent predictions.
  • Irreducible error: Inherent unpredictability in any real-world dataset.

Bias-variance Tradeoff

  • Summarising the bias-variance trade-off analogy.
  • Finding the right level of model complexity is crucial for achieving a balance between bias and variance.
  • Balancing model complexity to avoid underfitting and overfitting
  • Explaining the bias-variance tradeoff from the complexity point of view.

Bias-variance Tradeoff: Our Example

  • Illustrations of polynomial degrees (1, 4, 14) representing high-bias, just right, and high-variance models.
  • Explaining the relationships between model complexity, bias and variance.

Linear model regularisation (or shrinkage)

  • Regularization method to control model complexity, adding a penalty to the cost function that scales with the size of parameters.
  • This approach, termed shrinkage, penalises complex models.
  • Showing the cost function and defining the tuning parameter, λ.

Linear model regularisation (or shrinkage)

  • Discussing how the regularisation strength parameter λ controls the complexity trade-off. Describing how lambda influences the model complexity and bias/variance.
  • The two regularization methods presented are ridge and lasso.

Ridge regression

  • Ridge regression's penalty is proportional to the squared coefficient values.
  • The shrinkage penalty associated with ridge regression.

Ridge regression (Example)

  • Examples of ridge regression for polynomial regression models of order 9.

LASSO regression

  • LASSO regression penalises coefficients proportionally to their absolute values.
  • Description of the shrinkage penalty of LASSO regression.

L1 and L2 regularisation

  • Explaining L1 and L2 regularisation in terms of norms of a vector.
  • L1-norm is the sum of absolute values of elements in the vector; L2-norm is the square root of the sum of squared values.
  • Connecting regularisation to L1 and L2 norms.

LASSO and Ridge Regression

  • Visual comparison of how coefficients approach zero in ridge vs. lasso regression as λ changes.
  • Showing the coefficients plotted for different levels of λ.

Regularisation and Feature Selection

  • How regularization can be used for feature selection.
  • Regularization shrinks the contribution of features, often driving some coefficients to zero, reducing the number of important features.
  • Visualising the effect of L1 regularization on coefficient values.

Feature selection

  • Discussing the importance of feature selection for preventing overfitting and improving model performance/interpretation.
  • Explaining how feature selection can be performed by removing features.

Regularization: Python syntax

  • Showing Python syntax for implementing Ridge and Lasso regression in a machine learning context.

Lessons learned

  • Summarising the key concepts covered in the lecture.
  • Summarising the use of regularisation and feature selection, and their relationship.

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Learning from Data Lecture 5 PDF

Description

This quiz explores the concepts of linear models, regularisation methods like LASSO, and the bias-variance tradeoff in machine learning. Test your understanding of model selection, tuning parameters, and the impact of polynomial degrees on model performance. Gain insights into how regularisation influences model complexity and generalisation error.

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