CART Decision Trees and Derivatives Quiz

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Questions and Answers

What type of decision trees does CART stand for?

Classification And Regression Trees (correct)

Cumulative And Regression Trees

Classification Algorithm for Regression Trees

Categorical And Regression Trees

What does the directional derivative of f at θ, denoted Du f (θ), primarily measure?

The change in f when moving from θ to θ + hu. (correct)

The curvature of f at θ.

The maximum value of f at θ.

The overall function growth rate of f.

CART trees can only be binary trees and cannot be converted from m-trees.

False

The partial derivative of f with respect to θk represents the directional derivative in the direction of the unit vector ek.

True Signup and view all the answers

What is the primary output variable y in the context of predicting the creditworthiness of individuals?

0 or 1 Signup and view all the answers

In a decision tree, internal nodes correspond to tests of the form IF (xj < _____), where tr is the threshold value.

tr Signup and view all the answers

What does the notation ∇f (θ) represent?

The gradient of the function f at point θ. Signup and view all the answers

The example partial derivative of f with respect to θ1 is ___?

2θ1 + 5θ2 Signup and view all the answers

Which algorithm is used by CART to determine the decision variables and threshold values?

Greedy Algorithm Signup and view all the answers

Which is NOT a characteristic of the directional derivative?

It can be defined for points that are minimum points. Signup and view all the answers

The process of classification using a CART tree begins at the leaves of the tree.

False Signup and view all the answers

Match the following components of a decision tree with their functions:

Internal Nodes = Correspond to tests of attributes Leaves = Indicate predictions for each region Root Node = First decision point in the tree Branches = Show the outcome of tests Signup and view all the answers

Higher-order derivatives are beneficial in solving minimization problems in machine learning.

False Signup and view all the answers

What does the algorithm do to minimize the cost function G in CART?

Partitions the dataset into subsets Signup and view all the answers

Calculate the components of the gradient for the function f (θ1, θ2) = θ1^2 - 5θ1 + θ2^3 + 10.

∇f = [2θ1 - 5, 3θ2^2] Signup and view all the answers

Match the following terms with their definitions:

Directional Derivative = Change of f when moving from θ to θ + hu Partial Derivative = Derivative with respect to one variable, others constant Gradient = Vector of all partial derivatives of f Minimum Point = A point where f does not decrease in any direction Signup and view all the answers

What approach is considered when dealing with categorical features with more than two values?

Sending all examples with one value down the left branch Signup and view all the answers

Decision trees are inherently stable and do not show high variance.

False Signup and view all the answers

What process is employed to reduce the error of a classifier after a decision tree has been created?

Pruning Signup and view all the answers

Examples of algorithms that are considered stable include and .

Logistic regression, k-NN Signup and view all the answers

Match the following terms with their meanings:

Pruning = Reducing a decision tree by merging nodes One-hot encoding = Converting categorical variables into binary values High variance = Sensitivity to changes in input Dynamic programming = A method to optimize decision tree construction Signup and view all the answers

What happens when a decision tree is allowed to grow until only a few examples are in the same leaf?

It leads to overfitting. Signup and view all the answers

Generalized Optimal Sparse Decision Trees (GOSDT) aim to build less interpretable models compared to traditional decision trees.

False Signup and view all the answers

What is the significance of using threshold values in continuous variables?

To calculate impurity at split points Signup and view all the answers

What is the primary risk of choosing too large a number of weak models M in AdaBoost?

Overfitting the data Signup and view all the answers

AdaBoost is inherently stochastic, similar to Random Forests.

False Signup and view all the answers

What is the ideal range of leaves (J) for small decision trees to perform well in AdaBoost?

4 to 10 Signup and view all the answers

The loss function used in AdaBoost is known as the __________ loss.

exponential Signup and view all the answers

Match the following characteristics with their descriptions:

Deterministic method = Does not randomize during training Overfitting = Model performs poorly on unseen data Base function = A component like a tree used in learning LogitBoost = Uses logistic regression loss instead of exponential loss Signup and view all the answers

What weight method is used in AdaBoost for weak classifiers that are CART decision trees?

Scaling calculations of Gini impurity Signup and view all the answers

In the iterative process of AdaBoost, each model is dependent on the previous model, denoted as g_m = g_{m-1}(x) + β_m b(x; γ_m). This dependency portrays __________ modeling.

sequential Signup and view all the answers

Predictions made by AdaBoost can be easily parallelized.

True Signup and view all the answers

What is an eigenvector of matrix C associated with?

A corresponding eigenvalue Signup and view all the answers

The largest eigenvalue should be chosen to maximize variance.

True Signup and view all the answers

What happens to the coordinates of points after projecting into the space spanned by the largest eigenvectors?

The coordinates become a combination of the projections onto the selected eigenvectors. Signup and view all the answers

To maximize variance for the dimension of the subspace q, we project X into the space spanned by the ____ eigenvectors.

largest Signup and view all the answers

When performing noise filtering and lossy compression, which part of the eigenvectors are omitted?

The contribution from eigenvectors q + 1,..., p Signup and view all the answers

To select a suitable value for q, one should consider the ratio of eigenvalues.

True Signup and view all the answers

What visual indicator might suggest an optimal choice for q in eigenvalue selection?

A 'kink' or change in slope in the graph of eigenvalues. Signup and view all the answers

What does W(C) measure in clustering?

Dispersion of points within clusters Signup and view all the answers

Maximizing B(C) is equivalent to minimizing W(C).

True Signup and view all the answers

What is the primary distance metric used in the k-means algorithm?

Euclidean distance Signup and view all the answers

The number of points in group k is denoted as _____ .

Nk Signup and view all the answers

Which step is NOT part of the K-means algorithm?

Calculate the total distance from all points to centroids Signup and view all the answers

In K-means, a local minimum of W(C) is reached through iterative _____ .

assignment Signup and view all the answers

There are approximately K^n different partitions in the clustering problem.

False Signup and view all the answers

Study Notes

Linear Algebra Review

A p-vector is a vector with p elements (x₁, x₂, ..., xₚ). Column vectors (p x 1 matrices) are assumed unless otherwise specified.
eₖ is a p-vector with all elements zero except the k-th element, which is one.
1 is a p-vector where all elements are one.
The inner product (dot product) of p-vectors a and b is a ⋅ b = a₁b₁ + a₂b₂ + ... + aₚbₚ = Σⱼaⱼbⱼ.
The notation aᵀ indicates that vector a has been transposed, which results in a row vector.
The norm of vector x is a measure of its length. The 2-norm (Euclidean norm) is ||x||₂ = √(x₁² + x₂² + ... + xₚ²) = √Σⱼxⱼ².
The 1-norm is ||x||₁ = |x₁| + |x₂| + ... + |xₚ| = Σⱼ|xⱼ|.
Vectors a and b are perpendicular if a ⋅ b = 0.
The projection of vector a onto vector b is a ⋅ b / ||b||².
Vectors x₁, ..., xₖ are linearly independent if a₁x₁ + ... + aₖxₖ = 0 if and only if a₁ = ... = aₖ = 0.
A matrix is a table of numbers with n rows and m columns.
Aij represents the element in row i, column j of a matrix A.
The transpose of matrix A (denoted as Aᵀ) is the matrix where rows and columns are swapped.
A matrix A is symmetric if A = Aᵀ.
I is the identity matrix, with 1s on the diagonal and 0s elsewhere.
The rank of a matrix is the number of linearly independent columns in the matrix.

Supervised Learning

In supervised learning, we have a dataset X = {(x⁽¹⁾, y⁽¹⁾), ..., (x⁽ⁿ⁾, y⁽ⁿ⁾)}. x⁽ⁱ⁾ are input variables (features) and y⁽ⁱ⁾ are output variables.
Input variables are often vectors (x⁽ⁱ⁾ ∈ Rᵖ) and output variables are often scalars (y⁽ⁱ⁾ ∈ R).
Example: Predicting apartment prices in Reykjavík—x is the size in m² and y is the price in million ISK.
Example: Predicting heart attacks—x is patient data (age, maximum pulse, blood pressure) and y is whether patient had a heart attack (0 or 1).
Example: Image classification—x is a bitmap image and y is a class from a set of possible categories (dog, cat, car, etc.).

Linear Regression

If there are two input variables, x = (x₁, x₂), and a linear relationship is assumed between input and output variables, then the function f is of the form f(x) = θ₀ + θ₁x₁ + θ₂x₂.
When the number of input variables is p, the model is f(x) = θ₀x₀ + θ₁x₁ + ... + θₚxₚ = Σⱼθⱼxⱼ, where x₀ = 1 for simplicity.
The cost function J(θ) = (½)Σᵢ(f(x⁽ⁱ⁾) - y⁽ⁱ⁾)² is minimized to find suitable parameters θ.

Function Minimization

Learning algorithms often find parameters θ that maximize or minimize an objective function.
The objective function measures how well the model fits the data.

Functions of a Single Variable

The derivative of a function f(θ) at θ indicates how quickly f changes near θ.
A necessary condition for a local minimum is that the derivative is zero.

Functions of Multiple Variables

A directional derivative of a function f(θ₁, θ₂,... θₚ) at point θ indicates how f changes when moving along a line from θ.
A partial derivative of f at θ = (θ₁, θ₂... θₚ) with respect to θₖ is the directional derivative of f in the direction of the unit vector eₖ (in the k-th coordinate axis.)

Nearest Neighbor Classifiers

A nearest neighbor classifier assigns a point x to the class that is most frequent among its k nearest neighbors in the dataset.
Euclidean distance is often used to calculate the distance between points.
This method is simple but can be computationally expensive for large datasets.

Logistic Regression Classifier

The output value is in the interval [0, 1] using a sigmoid function.
Optimization is used to derive suitable parameters θ.
Error rate in logistic regression are different than in linear regression.

Evaluating Model Quality and Choosing a Model

Mean Squared Error (MSE): Measures average squared error between predicted values f(x⁽ⁱ⁾) and actual values y⁽ⁱ⁾. A low MSE on the training data does not guarantee a good model.
Independent dataset error estimation: Split the data into training and testing sets to evaluate a model's performance on unseen data.

Error Estimation from Independent Dataset

Randomly split the dataset X into training (Xtrain) and testing (Xtest) sets.
Compute the MSEtest on the test data.

Model Selection

Evaluate the model's error by iterating over different hyperparameter values and choosing the one that yields the lowest average error on a test dataset.

Performance Metrics

Root Mean Squared Error (RMSE) is the square root of MSE. It has the same units as the output variable (y).

Classification

Accuracy in classification: Percentage of correctly classified examples.
True Positive (TP), True Negative (TN), False Positive (FP), False Negative (FN) are categories for assessing classification errors.
Other metrics might be required if types of errors have different costs or class distributions are uneven.

Class Imbalance

Class imbalance occurs when the number of examples differs significantly between categories.
Modifications of the classifier or the objective function might be necessary to address this.

Clustering

The task in clustering is to partition the examples into groups in X.
The distance function d between examples is important.
K-means is a common clustering algorithm.

Principal Component Analysis (PCA)

PCA finds a subspace where the data lies approximately.
The variances are maximized by choosing appropriate directions.

Non-negative Matrix Factorization (NMF)

Approximates a given (non-negative) data matrix with the product of two non-negative matrices (W, H).
Can be useful for analyzing structure present in the data.

Recommender Systems

Recommending items users might be interested in using user reviews.
Use of a matrix or user/item interaction pattern.

Reinforcement Learning

The agent interacts with an environment (e.g. game, trading algorithms, robotics).
The goal is to maximize the total reward obtained over a period of time, e.g from playing many rounds of a game.
A (Q-) function quantifies how good it would be to perform a given action in a state.

Neural Networks

Multilayer perceptrons (MLP)
Neural networks with multiple layers
Output layer with specific activation functions (logistic regression)

Gradient Descent, Variants, and Regularization

Batch gradient descent
Stochastic gradient descent
Mini-batch gradient descent
Momentum: Technique to accelerate convergence
Gradient clipping: Technique to limit gradient values to prevent exploding gradients.
Regularization functions: To avoid overfitting (e.g., L₁, L₂).

Convolutional Neural Networks

CNNs are feedforward neural networks specific to image data.
Use convolution, pooling (subsampling) and fully connected layers.
Weight sharing is one main advantage of this type of neural network.
They are useful for tasks that involve image classification and other pattern recognition tasks in which spatial properties are important.

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Test your knowledge on CART decision trees and the concepts of directional derivatives. This quiz covers key terms, functions, and algorithms essential for understanding decision tree classification and gradient measures. Perfect for students in data science or machine learning courses.

CART Decision Trees and Derivatives Quiz

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Questions and Answers

What type of decision trees does CART stand for?

What does the directional derivative of f at θ, denoted Du f (θ), primarily measure?

CART trees can only be binary trees and cannot be converted from m-trees.

The partial derivative of f with respect to θk represents the directional derivative in the direction of the unit vector ek.

What is the primary output variable y in the context of predicting the creditworthiness of individuals?

In a decision tree, internal nodes correspond to tests of the form IF (xj < _____), where tr is the threshold value.

What does the notation ∇f (θ) represent?

The example partial derivative of f with respect to θ1 is ___?

Which algorithm is used by CART to determine the decision variables and threshold values?

Which is NOT a characteristic of the directional derivative?

The process of classification using a CART tree begins at the leaves of the tree.

Match the following components of a decision tree with their functions:

Higher-order derivatives are beneficial in solving minimization problems in machine learning.

What does the algorithm do to minimize the cost function G in CART?

Calculate the components of the gradient for the function f (θ1, θ2) = θ1^2 - 5θ1 + θ2^3 + 10.

Match the following terms with their definitions:

What approach is considered when dealing with categorical features with more than two values?

Decision trees are inherently stable and do not show high variance.

What process is employed to reduce the error of a classifier after a decision tree has been created?

Examples of algorithms that are considered stable include __________ and __________.

Match the following terms with their meanings:

What happens when a decision tree is allowed to grow until only a few examples are in the same leaf?

Generalized Optimal Sparse Decision Trees (GOSDT) aim to build less interpretable models compared to traditional decision trees.

What is the significance of using threshold values in continuous variables?

What is the primary risk of choosing too large a number of weak models M in AdaBoost?

AdaBoost is inherently stochastic, similar to Random Forests.

What is the ideal range of leaves (J) for small decision trees to perform well in AdaBoost?

The loss function used in AdaBoost is known as the __________ loss.

Match the following characteristics with their descriptions:

What weight method is used in AdaBoost for weak classifiers that are CART decision trees?

In the iterative process of AdaBoost, each model is dependent on the previous model, denoted as g_m = g_{m-1}(x) + β_m b(x; γ_m). This dependency portrays __________ modeling.

Predictions made by AdaBoost can be easily parallelized.

What is an eigenvector of matrix C associated with?

The largest eigenvalue should be chosen to maximize variance.

What happens to the coordinates of points after projecting into the space spanned by the largest eigenvectors?

To maximize variance for the dimension of the subspace q, we project X into the space spanned by the ____ eigenvectors.

When performing noise filtering and lossy compression, which part of the eigenvectors are omitted?

To select a suitable value for q, one should consider the ratio of eigenvalues.

What visual indicator might suggest an optimal choice for q in eigenvalue selection?

What does W(C) measure in clustering?

Maximizing B(C) is equivalent to minimizing W(C).

What is the primary distance metric used in the k-means algorithm?

The number of points in group k is denoted as _____ .

Which step is NOT part of the K-means algorithm?

In K-means, a local minimum of W(C) is reached through iterative _____ .

There are approximately K^n different partitions in the clustering problem.

Study Notes

Linear Algebra Review

Supervised Learning

Linear Regression

Function Minimization

Functions of a Single Variable

Functions of Multiple Variables

Nearest Neighbor Classifiers

Logistic Regression Classifier

Evaluating Model Quality and Choosing a Model

Error Estimation from Independent Dataset

Model Selection

Performance Metrics

Classification

Class Imbalance

Clustering

Principal Component Analysis (PCA)

Non-negative Matrix Factorization (NMF)

Recommender Systems

Reinforcement Learning

Neural Networks

Gradient Descent, Variants, and Regularization

Convolutional Neural Networks

Studying That Suits You

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