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Catégorie :Category: nCreator TI-Nspire
Auteur Author: SPITZER2001
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 7.35 Ko KB
Mis en ligne Uploaded: 06/05/2025 - 06:31:52
Uploadeur Uploader: SPITZER2001 (Profil)
Téléchargements Downloads: 15
Visibilité Visibility: Archive publique
Shortlink : https://tipla.net/a4622460
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 7.35 Ko KB
Mis en ligne Uploaded: 06/05/2025 - 06:31:52
Uploadeur Uploader: SPITZER2001 (Profil)
Téléchargements Downloads: 15
Visibilité Visibility: Archive publique
Shortlink : https://tipla.net/a4622460
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
<<
Context: You are given a univariate dataset that exhibits a nonlinear relationship between the input and the output variables (see Figure 1.1). To model this relationship, you decide to apply polynomial regression for varying degrees of complexity, with the polynomial degree n ranging from 1 to 25. For each model, you evaluate its performance using the following metrics: Mean Squared Error (MSE) on the training and test sets, and Coefficient of Determination (R²) on the training and test sets. These results are visualized in Figure 1.2, which shows how model performance changes with increasing polynomial degree. a) By analyzing the MSE curves in Figure 1.2, which polynomial degree appears to provide the best model fit? Justify your choice based on the trade-off between training and test error. Answer: The polynomial degree that provides the best fit appears to be around degree 10 to 15, where the test MSE is minimized and the gap between training and test MSE is relatively small. This indicates a good bias-variance trade-off: the model is flexible enough to capture the nonlinear trend (low bias) but not too complex to overfit (low variance). Higher degrees (e.g., >20) lead to overfitting the training error decreases further, but test error increases significantly. b) Why do both training and test MSE begin to increase at higher polynomial degrees (e.g., beyond degree 20), even though the model is more flexible? Answer: While more complex models (higher degree polynomials) can fit the training data more closely, they also tend to fit noise in the data. This results in overfitting, which leads to poor generalization and higher test MSE. Additionally, very high-degree polynomials may suffer from numerical instability and oscillations (e.g., Runges phenomenon), worsening both training and test performance. c) Suggest one or two non-parametric regression models that could be used instead of polynomial regression. What are the advantages and limitations of using non-parametric models for this kind of dataset? Answer: k-Nearest Neighbors (k-NN) Regression: Advantage: Makes no assumption about the functional form; highly flexible. Limitation: Sensitive to noise and scales poorly in high dimensions. Decision Tree Regression / Random Forests: Advantage: Naturally model nonlinear relationships; handle interactions automatically. Limitation: Prone to overfitting without pruning or ensemble techniques; less smooth predictions. Both models adapt to the data structure and are ideal when the true form of the relationship is unknown. Context: You are training a linear regression model on a dataset with 100 features and only 150 training observations. After fitting a standard least squares model, you notice low training error but high test error. You decide to try Lasso and Ridge Regression to regularize the model. a) Why is regularization needed in this case? Answer: The model is overfitting because there are too many parameters relative to the number of observations. Without regularization, the model tries to perfectly fit the training data, capturing noise and resulting in poor generalization to new data. Regularization constrains the model complexity and improves test performance. b) What is the difference between Ridge and Lasso regression in this context? When would you prefer Lasso? Answer: Ridge (L2): Shrinks all coefficients but does not set any to zero. Useful when many features are informative. Lasso (L1): Can shrink some coefficients exactly to zero, performing feature selection. Lasso is preferred when you expect that many features are irrelevant and want a sparse model. c) How does cross-validation help when applying regularization? Answer: Cross-validation helps find the optimal value of the regularization parameter (») that balances bias and variance. It ensures the model performs well on unseen data and prevents underfitting or overfitting by selecting the best complexity level. Context: You are developing a binary classifier to detect fraudulent credit card transactions, where frauds make up only 1% of the dataset. You train a logistic regression model and obtain 99% accuracy, but the model fails to detect most frauds. a) Why is accuracy a poor performance metric in this scenario? Answer: In imbalanced datasets, a model can achieve high accuracy by predicting the majority class only. Here, predicting non-fraud for all cases yields 99% accuracy but 0% recall for fraud detection. More appropriate metrics: Recall, Precision, F1-Score, ROC AUC, or Precision-Recall AUC. b) Suggest two techniques to improve the classifiers fraud detection capability. Answer: Resampling the data: Use oversampling (e.g., SMOTE) for the minority class or undersample the majority. Cost-sensitive learning: Assign higher misclassification cost to frauds, encouraging the model to detect them. c) How do precision-recall curves help in evaluating this model? Answer: Precision-recall curves focus on the positive (minority) class, sho
[...]
>>
Compatible OS 3.0 et ultérieurs.
<<
Context: You are given a univariate dataset that exhibits a nonlinear relationship between the input and the output variables (see Figure 1.1). To model this relationship, you decide to apply polynomial regression for varying degrees of complexity, with the polynomial degree n ranging from 1 to 25. For each model, you evaluate its performance using the following metrics: Mean Squared Error (MSE) on the training and test sets, and Coefficient of Determination (R²) on the training and test sets. These results are visualized in Figure 1.2, which shows how model performance changes with increasing polynomial degree. a) By analyzing the MSE curves in Figure 1.2, which polynomial degree appears to provide the best model fit? Justify your choice based on the trade-off between training and test error. Answer: The polynomial degree that provides the best fit appears to be around degree 10 to 15, where the test MSE is minimized and the gap between training and test MSE is relatively small. This indicates a good bias-variance trade-off: the model is flexible enough to capture the nonlinear trend (low bias) but not too complex to overfit (low variance). Higher degrees (e.g., >20) lead to overfitting the training error decreases further, but test error increases significantly. b) Why do both training and test MSE begin to increase at higher polynomial degrees (e.g., beyond degree 20), even though the model is more flexible? Answer: While more complex models (higher degree polynomials) can fit the training data more closely, they also tend to fit noise in the data. This results in overfitting, which leads to poor generalization and higher test MSE. Additionally, very high-degree polynomials may suffer from numerical instability and oscillations (e.g., Runges phenomenon), worsening both training and test performance. c) Suggest one or two non-parametric regression models that could be used instead of polynomial regression. What are the advantages and limitations of using non-parametric models for this kind of dataset? Answer: k-Nearest Neighbors (k-NN) Regression: Advantage: Makes no assumption about the functional form; highly flexible. Limitation: Sensitive to noise and scales poorly in high dimensions. Decision Tree Regression / Random Forests: Advantage: Naturally model nonlinear relationships; handle interactions automatically. Limitation: Prone to overfitting without pruning or ensemble techniques; less smooth predictions. Both models adapt to the data structure and are ideal when the true form of the relationship is unknown. Context: You are training a linear regression model on a dataset with 100 features and only 150 training observations. After fitting a standard least squares model, you notice low training error but high test error. You decide to try Lasso and Ridge Regression to regularize the model. a) Why is regularization needed in this case? Answer: The model is overfitting because there are too many parameters relative to the number of observations. Without regularization, the model tries to perfectly fit the training data, capturing noise and resulting in poor generalization to new data. Regularization constrains the model complexity and improves test performance. b) What is the difference between Ridge and Lasso regression in this context? When would you prefer Lasso? Answer: Ridge (L2): Shrinks all coefficients but does not set any to zero. Useful when many features are informative. Lasso (L1): Can shrink some coefficients exactly to zero, performing feature selection. Lasso is preferred when you expect that many features are irrelevant and want a sparse model. c) How does cross-validation help when applying regularization? Answer: Cross-validation helps find the optimal value of the regularization parameter (») that balances bias and variance. It ensures the model performs well on unseen data and prevents underfitting or overfitting by selecting the best complexity level. Context: You are developing a binary classifier to detect fraudulent credit card transactions, where frauds make up only 1% of the dataset. You train a logistic regression model and obtain 99% accuracy, but the model fails to detect most frauds. a) Why is accuracy a poor performance metric in this scenario? Answer: In imbalanced datasets, a model can achieve high accuracy by predicting the majority class only. Here, predicting non-fraud for all cases yields 99% accuracy but 0% recall for fraud detection. More appropriate metrics: Recall, Precision, F1-Score, ROC AUC, or Precision-Recall AUC. b) Suggest two techniques to improve the classifiers fraud detection capability. Answer: Resampling the data: Use oversampling (e.g., SMOTE) for the minority class or undersample the majority. Cost-sensitive learning: Assign higher misclassification cost to frauds, encouraging the model to detect them. c) How do precision-recall curves help in evaluating this model? Answer: Precision-recall curves focus on the positive (minority) class, sho
[...]
>>
