Define mean, variance, and standard deviation. How do variance and standard deviation differ in terms of units?
What is a z-score, and how is it calculated?
Why are z-scores considered to be dimensionless, and what advantage does this provide when comparing datasets?
Explain the difference between a population and a sample in statistics.
What is an unbiased estimator?
Describe the key conceptual difference between \(s_n\) and \(s_{n-1}\) for estimating population variance.
Define percentile (i. e., quantile), quartiles, and median.
Explain the difference between the sample median calculation for odd and even numbers of observations.
What does the interquartile range represent, and how is it calculated?
Discuss the potential advantage of using the median and IQR over the mean and standard deviation for summarizing a dataset.
Section 2.2: Distributions
Define the cumulative distribution function (CDF) for a population. Explain why it is a nondecreasing function ranging between 0 and 1.
What is the CDF of a uniform distribution? How does it differ for discrete and continuous cases?
What is an empirical cumulative distribution function (ECDF)?
How does the shape of the ECDF change as the sample size increases?
Define the probability density function (PDF) and explain its relationship to the CDF.
Discuss the concept of a histogram and its normalization.
Describe formulas for computing the mean and variance of a continuous distribution from its PDF.
What are the mean and variance for a continuous uniform distribution over \([0, 1]\)?
What are the characteristics of a standard normal distribution?
How do the mean and variance of a normal distribution affect its PDF?
Discuss the 68-95-99.7 (empirical) rule in the context of the normal distribution.
What does kernel density estimation (KDE) do?
Section 2.3: Grouping
Why might it be useful to analyze data in groups defined by categorical values or other criteria?
What is a facet plot?
Describe the information conveyed by a box plot and a violin plot.
Describe how aggregation works with grouping in data analysis. What are some common aggregation functions?
Describe how transformation works with grouping in data analysis. What are some common aggregation functions?
What is the purpose of filtering in grouped data analysis?
Discuss the impact of standardizing data within groups versus across the entire dataset.
Section 2.4: Outliers
What is an informal definition of an outlier?
Why might outliers be of real interest in certain applications?
List some common reasons why outliers might appear in a dataset.
Provide an example where changing a single value in a dataset significantly affects the mean but not the median.
What is the Interquartile Range (IQR), and how is it calculated?
Describe how outliers are identified using the IQR method.
How are outliers represented in a box plot?
What are the criteria for considering values as outliers in a normal distribution based on standard deviation?
Discuss the potential impact of removing outliers on the analysis of a dataset. When might it be appropriate to remove outliers, and when might it be important to investigate them further?
Section 2.5: Correlation
Define correlation in the context of statistical analysis.
Why is it important to visually inspect data before calculating correlation coefficients?
Define covariance and explain its significance in measuring the relationship between two variables.
Why is covariance not always easy to interpret?
Explain the Pearson correlation coefficient and how it is calculated for both populations and samples.
What does a Pearson coefficient of -1, 0, and 1 signify?
How does the Pearson coefficient address the limitations of covariance?
Discuss how outliers can affect the Pearson correlation coefficient.
Provide an example where a single outlier significantly impacts the Pearson coefficient.
What is the Spearman correlation coefficient, and how does it differ from the Pearson coefficient?
How does the Spearman coefficient mitigate the impact of outliers?
Provide an example demonstrating the robustness of the Spearman coefficient against outliers.
Explain how correlation can be assessed when dealing with categorical variables.
Section 2.6: Cautionary tales
What is the main lesson from the Datasaurus Dozen example?
How can relying solely on summary statistics be misleading?
Explain the difference between correlation and dependence.
Describe an example where two variables are dependent but not correlated.
What is Simpson’s paradox, and how does it manifest in the penguin dataset example?
Can you create a hypothetical example where Simpson’s paradox might lead to incorrect conclusions if not properly understood?
Why is it important to consider both linear and nonlinear relationships when analyzing data?
How can the misuse of statistical methods lead to misconceptions or mistakes?
Chapter 3
Section 3.1
Define a feature vector and label in the context of classification problems.
What distinguishes a binary classification problem from a multiclass problem?
Describe the structure of a feature matrix and a label vector.
Discuss the limitations of assigning arbitrary numerical values to qualitative data. Why is one-hot or dummy encoding considered a better strategy?
Outline the basic steps involved in training and applying a machine learning classifier.
What is a query vector? How is it used in the context of a classifier? Evaluating Classifier Performance:
How can the accuracy of a classifier be determined?
Besides accuracy, what other metrics or considerations might be important in evaluating the effectiveness of a classifier?
Section 3.2
Define the concept of generalization in the context of machine learning classifiers.
Explain the purpose of splitting a dataset into training and testing sets. How does this practice help in evaluating the generalization of a classifier?
Describe the significance of true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN) in assessing classifier performance.
Define accuracy, recall (sensitivity), specificity, precision, and negative predictive value (NPV).
What information does a confusion matrix convey about a classifier’s performance?
Explain the F₁ score and balanced accuracy. How do they provide a more nuanced view of classifier performance compared to using accuracy alone?
How do the concepts of binary classification metrics extend to multiclass classification problems?
Discuss the difference between macro averaging and other methods of averaging precision scores across multiple classes. Why might one averaging method be preferred over another?
Describe a hypothetical situation where a high recall is more important than high precision.
Describe a hypothetical situation where a high precision is more important than high recall.
Section 3.3
Describe the process of building a decision tree for classification.
Define Gini impurity and explain its significance in the context of decision trees.
Describe the criteria for partitioning samples in a decision tree.
What is an indicator function, and how is it utilized in expressing Gini impurity?
Explain the concept of a decision boundary in relation to decision trees.
In general terms, how does the depth of a decision tree affect its decision boundary?
What is a greedy algorithm, and how does it apply to the context of finding a decision tree?
What does it mean to say a classifier is interpretable?
Besides Gini impurity, are there other metrics or considerations that might be important in evaluating the partitioning in a decision tree?
Section 3.4
Describe the basic principle behind the k-nearest neighbors algorithm. How does it determine the class of a query point?
What role do distance metrics play in the kNN algorithm? Explain the properties that a distance metric must satisfy.
Define the 2-norm, 1-norm, and infinity-norm. How are these norms used to calculate distances between feature vectors?
How does the choice of \(k\) in the kNN algorithm influence the algorithm’s decision boundary and predictions?
Why is standardization or scaling of features important in kNN?
How is kNN applied to multiclass classification problems?
Section 3.5
Explain how a probability vector arises from vote-based classification methods. How does it provide more information than a winner-takes-all approach?
Define the ROC curve and explain its significance in evaluating binary classifiers.
What is a decision threshold in the context of probabilistic classifiers? How does changing the threshold affect classification outcomes?
What is the AUC metric?
How can ROC curves and AUC scores be used to compare different classifiers or different settings of the same classifier?
Explain how the concepts of ROC curves and AUC scores extend to multiclass classification problems.
Chapter 4
Section 4.1
How is the bias of a classifier defined in the context of expected prediction error?
Explain the concept of variance in the context of machine learning models. How does variance affect a model’s performance on unseen data?
Explain how bias and variance contribute to the total prediction error of a model. Why do we refer to a “tradeoff” between bias and variance?
How does the size of the training set impact the bias and variance of a machine learning model? Answer using sketches of learning curves.
Discuss how a learner’s capacity to capture complex behavior influences both its bias and variance.
What might a large gap between training and test errors suggest about a model’s ability to generalize? What steps could be taken to reduce the gap?
Section 4.2
Define overfitting in the context of machine learning and explain why it is problematic for model generalization.
Why does a model that perfectly fits the training data not necessarily perform well on new, unseen data?
How does the choice of \(k\) in kNN affect the likelihood of overfitting, particularly with noisy data?
Explain how the depth of a decision tree can contribute to overfitting.
Compare and contrast overfitting and underfitting. How can one identify these issues based on the model’s performance?
How is overfitting related to the concept of variance in a model’s predictions? Discuss the relationship between overfitting and the gap between training and testing performance.
What strategies can be employed to reduce the risk of overfitting in machine learning models?
How does overfitting fit into the broader discussion of the bias-variance tradeoff?
Describe how learning curves can be used to diagnose overfitting.
Section 4.3
Define ensemble methods in machine learning and explain why they are used to improve model performance.
What is bootstrap aggregation (bagging), and how does it help in reducing the variance of a machine learning model?
Precisely how does bagging affect the bias and variance of a collection of \(n\) learners?
Is bagging better with constituent classifiers that have small bias and large variance, or that have large bias and small variance?
What is a random forest?
Which is more likely to improve bagging results: smaller individual training sets, or larger ones? Why?
How does selecting random subsets of features for each tree in n ensemble improve the ensemble performance?
What are the two chief disadvantages of using ensemble methods?
Section 4.4
Why is validation important in the process of selecting optimal hyperparameters and models?
Describe the steps involved in \(k\)-fold cross-validation.
Explain stratified \(k\)-fold cross-validation and when it might be preferred over standard \(k\)-fold cross-validation.
How can cross-validation be used to tune hyperparameters? Describe the process of creating a validation curve and interpreting its results.
Discuss how the variance of cross-validation scores across folds can inform us about a model’s reliability.
What is a grid search for hyperparameter optimization? When does it become impractical?
Chapter 5
Section 5.1
What is the mathematical form of a linear regression model? Use vector notation.
Using vector notation, define the loss function used in linear regression. How is it related to the selection of model coefficients?
What are the mathematical criteria that determine the coefficients of a linear regression model?
What is the inner product between two vectors, and why is it important in the context of linear regression?
What does the coefficient of determination measure, and what do its values indicate about a model’s performance?
Define residuals in the context of regression and discuss their significance in evaluating model fit.
Compare and contrast MSE and MAE as measures of regression model performance. In what situations might one be preferred over the other?
Section 5.2
Define multilinear regression and explain how it extends the concept of (single) linear regression.
What is the role of the weight vector in multilinear regression?
Describe the loss function used in multilinear regression and explain how it defines the optimal weight vector.
Use a matrix-vector product to express the loss function of multilinear regression.
Explain how feature scaling affects the interpretation of regression coefficients in multilinear regression.
Explain the polynomial regression method. How does it relate to linear regression?
Which is more likely to produce overfitting, and why: increasing the degree of a polynomial regression model, or decreasing it?
Section 5.3
What is regularization in the context of machine learning? How does it help in addressing overfitting?
Explain what ill-posed and ill-conditioned problems are. How does regularization address them in machine learning models?
Explain Ridge and LASSO regression in terms of their effects on the loss function.
How does the regularization parameter influence the behavior of Ridge and LASSO regression models?
How does one determine the optimal value of the regularization parameter?
Discuss the particular advantages of LASSO regression in terms of feature selection.
Section 5.4
Describe the two mathematical properties that define linearity.
Why are nonlinear regression methods necessary/useful?
Describe how the k-nearest neighbors classification algorithm can be adapted for regression.
Describe how decision trees can be used for regression.
Discuss the difference between using mean square error and mean absolute error as quality measures.
Describe the concept of a random forest regressor. How does it improve upon decision tree regression?
Discuss the bias-variance tradeoff in the context of kNN and decision tree regression.
Section 5.5
Define the logistic function. Why is it particularly suitable for modeling probabilities?
Explain mathematically how logistic regression models the probability of a binary outcome.
What is cross-entropy loss, and why is it preferred over mean squared error in logistic regression?
Discuss the advantages and disadvantages of logistic regression compared to linear regression.
Explain mathematically how LASSO and ridge regularization modify logistic regression.
Describe how logistic regression can be extended to handle multiclass classification problems.
Explain the concept of a decision threshold in the context of logistic regression. How does the ROC curve help to evaluate the effect of the decision threshold?
What is the AUC metric? What would be considered poor and excellent values for it?
Chapter 6
Section 6.1
Explain the relationship between similarity and distance in the context of clustering. How is similarity typically quantified when a distance metric is available?
Define a distance matrix and describe its properties.
Define angular distance and its significance in comparing the similarity between data points.
Explain the concept of cosine pseudodistance and how it differs from true distance metrics. Why might one choose to use cosine pseudodistance over angular distance?
Discuss some counterintuitive properties of high-dimensional spaces that affect the use of distance-based methods in data analysis.
What is the curse of dimensionality, and how does it impact the effectiveness of clustering and nearest-neighbor algorithms in high-dimensional data?
Discuss factors to consider when choosing a distance metric for clustering analysis. How do the characteristics of the data influence this choice?
Describe how distance and similarity measures can be applied in text analysis, such as sentiment analysis or document clustering. What challenges arise in this context, and how are they addressed?
Section 6.2
Define the Rand index and explain how it is used to compare two clusterings of the same dataset.
What is the adjusted Rand index, and how does it improve upon the Rand index for evaluating clustering performance?
Describe the silhouette value for a sample within a clustering. What is its maximum possible value, and what does that value indicate about the sample’s placement?
What do negative silhouette values signify?
Identify the limitations of silhouette scores in evaluating clusterings. Why might they favor certain types of cluster shapes?
Section 6.3
What is the k-means algorithm, and what is its primary goal?
Define the terms “centroid” and “inertia” as they relate to the k-means clustering algorithm. How do these concepts contribute to the formation of clusters?
Describe Lloyd’s algorithm. What does convergence mean in this context?
Discuss the significance of initialization in the k-means algorithm.
Explain how the number of clusters, \(k\), is determined in k-means clustering. What are some strategies or metrics used to select an optimal \(k\)?
Identify and explain some significant limitations of the k-means clustering algorithm.
Section 6.4
What is hierarchical clustering?
Describe the process of agglomerative clustering.
Describe the different types of linkage criteria: single, complete, average, and Ward.
How is a dendrogram interpreted in the context of hierarchical clustering? How can it be used to determine the optimal number of clusters?
Discuss the advantages of using hierarchical clustering over k-means.
Identify and explain some significant limitations of hierarchical clustering.
How does the choice of linkage type affect the clustering result?
Explain why hierarchical clustering can be performed using only a distance matrix, without access to the original feature vectors.
Chapter 7
Section 7.1
Define a graph. What is the difference between an undirected graph and a directed graph?
Explain how a graph can be represented by its adjacency matrix.
Explain the differences between star graphs, cycle graphs, wheel graphs, complete graphs, and lattice graphs. Provide examples of each.
What does it mean for two nodes to be neighbors in a graph? Define the degree of a node.
Define an ego graph. How can ego graphs be useful in analyzing networks?
How can the degree of nodes and the average degree of a graph be used to analyze the structure of a network?
Provide examples of practical applications where graphs are effectively used.
Section 7.2
What is an Erdős-Rényi graph, and how is it constructed?
How does the probability parameter \(p\) affect the structure of an Erdős-Rényi graph?
Quote the formula for the expected number of edges in an Erdős-Rényi graph with \(n\) nodes and edge probability \(p\).
Describe the process of constructing a Watts-Strogatz graph. What are its key parameters?
Explain how a Watts-Strogatz graph models human social relationships.
Compare and contrast Erdős-Rényi and Watts-Strogatz graphs in terms of their structure.
Discuss potential applications of random graph models in understanding real-world networks.
Section 7.3
What does the local clustering coefficient of a node in a network represent? How is it calculated?
How can the clustering coefficient be used to analyze real-world networks, such as social networks or biological networks?
Explain why the expected value of the average clustering coefficient in Erdős-Rényi graphs is equal to the probability \(p\). How does this property compare to most real-world networks?
Discuss how the structure of a network (e.g., the number of nodes, average degree, presence of community structures) can impact its clustering coefficient. Provide examples to illustrate your points.
What are some practical applications of analyzing clustering in networks?
Section 7.4
How is the distance between two nodes in a graph determined?
Explain the concept of the diameter of a graph. How does it relate to the average distance between nodes?
What does it mean for a graph to be connected?
Explain how the Watts-Strogatz model illustrates the small-world phenomenon.
In the Watts-Strogatz model, how does the rewiring probability \(q\) affect the average distance and clustering coefficient of the graph?
Section 7.5
What does the degree distribution of a network tell us about its structure?
Describe the typical characteristics of degree distributions observed in real-world networks, such as the Twitch network. What are “hubs,” and why are they significant?
Compare and contrast the degree distributions of Erdős-Rényi (ER), Watts-Strogatz (WS), and Barabási-Albert (BA) graphs. Which model is likely to produce a degree distribution similar to that observed in many real-world networks?
Explain the concept of a power-law distribution. How can you identify if a network’s degree distribution follows a power-law?
What is preferential attachment, and how does it relate to the Barabási-Albert model of network growth?
Discuss the clustering characteristics of Barabási-Albert graphs. How well do BA graphs replicate the clustering observed in real-world networks like Twitch?
Considering the degree distributions and clustering coefficients of real-world networks, what challenges do researchers face in accurately modeling these networks? How do the ER, WS, and BA models address these challenges differently?
Section 7.6
What is centrality in the context of network analysis?
Define degree centrality.
Define betweenness centrality. Describe how it differs from degree centrality and what it reveals about a node’s role in a network.
What is eigenvector centrality, and how does it relate to the concept of nodes linking to other important nodes?
Discuss the differences between degree centrality, betweenness centrality, and eigenvector centrality.
Given a small-world network, how might centrality measures help identify key nodes?
Discuss practical applications of centrality measures in real-world scenarios.
Section 7.7
In plain terms, what is the friendship paradox?
Describe the mathematical inequality that represents the friendship paradox.
How does the friendship paradox extend to eigenvector centrality?
Discuss the implications of the friendship paradox for understanding social networks. How might this paradox affect individuals’ perceptions of their social standing?
Section 7.8
What is the purpose of identifying communities within a network?
Describe the process of a random walk on a network. How does the probability distribution of the walker’s location change with each hop?
Explain how the random-walk matrix \(W\) is constructed from the adjacency matrix \(A\). What does each element of \(W\) represent?
Describe the label propagation algorithm and its purpose in community detection. How does the damping parameter \(\lambda\) affect the identification of communities?