Home / Past papers / MLT Quiz 1 / this paper

2024 Oct27: - 27 Oct 2024

MLT Quiz 113 questions50 marks6 SA · 5 MCQ · 2 MSQ

Elapsed 00:00

Q1MCQ4 marks

Q2MCQ4 marks

Q3MCQ3 marks

Q4MSQ4 marksselect all that apply

Q5MSQ5 marksselect all that apply

Q6SA3 marks

Your numeric answer:

Q7SA5 marks

Your numeric answer:

Q8SA4 marks

Your numeric answer:

Q9SA4 marks

Consider a dataset with 100 total data points. Each data point is classified as either type A or type B. We model this using a Bernoulli distribution, where p is the probability of a data point being type A. If the maximum likelihood estimate (MLE) of p based on the dataset is 0.4, how many data points of type B are there in this dataset?

Your numeric answer:

Comprehension passage - the questions below it refer to this

Q10SA3 marks

Your numeric answer:

Q11SA4 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q12MCQ3 marks

(0,1)
(2,3)
(3,2)
(2,4)

Q13MCQ4 marks

Which point has the lowest probability of being chosen as the 3rd cluster mean? Use squared distance to calculate the scores.

(1,0)
(2,3)
(3,2)
(1,2)

Answer key

Q1 (MCQ, 4 marks): B
Q2 (MCQ, 4 marks): A
Q3 (MCQ, 3 marks): A
Q4 (MSQ, 4 marks): A, B
Q5 (MSQ, 5 marks): A, D
Q6 (SA, 3 marks): 80
Q7 (SA, 5 marks): 6
Q8 (SA, 4 marks): 1.50 to 1.60
Q9 (SA, 4 marks): 60
Q10 (SA, 3 marks): 4.8 to 5.1
Q11 (SA, 4 marks): 0.5 to 0.9
Q12 (MCQ, 3 marks): D
Q13 (MCQ, 4 marks): B

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Summary of Core Topics, Concepts, Principles, and Equations

1. Kernels and Kernel Functions

Concepts: Valid kernels, kernel transformations, kernel PCA.
Equations:
- Kernel functions $k(\mathbf{x}, \mathbf{y})$ .
- Example kernels:

k_1(\mathbf{x}, \mathbf{y}) = x_1 y_1 + x_2 y_2 + (x_1 + x_2)(y_1 + y_2)

k_2(\mathbf{x}, \mathbf{y}) = x_1 y_1 + x_2 y_2 + (x_1^2 + y_2^2) + 3

Kernel PCA and dimensionality reduction.

2. Principal Component Analysis (PCA)

Concepts: Eigenvalues, eigenvectors, variance, dimensionality reduction.
Equations:
- Reconstruction error:

\frac{1}{n} \sum_{i=1}^{n} \| x_i - (x_i^T w) w \|^2

Variance captured by principal components:

\text{Percentage of total variance} = \frac{\text{Sum of top eigenvalues}}{\text{Total sum of eigenvalues}} \times 100\%

3. Maximum Likelihood Estimation (MLE)

Concepts: Likelihood functions, probability density functions (PDFs), MLE for parameters.
Equations:
- Likelihood function for different PDFs:

f(x \mid \theta) = \begin{cases} 5x^4, & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}

f(x \mid \theta) = \begin{cases} 1, & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}

MLE condition:

\prod_{i=1}^{n} 5x_i^4 < 1 \text{ or } > 1

4. K-Means Clustering

Concepts: Initialization, cluster assignment, objective function.
Equations:
- Objective function:

F(z_1, z_2, \ldots, z_n) = \sum_{i=1}^{n} \| x_i - \mu_{z_i} \|_2^2

Distance calculations and cluster mean updates.

5. Gaussian Mixture Models (GMM)

Concepts: EM algorithm, mixture components, mean estimation.
Equations:
- Mean estimation in M-step:

\mu_k = \frac{\sum_{i=1}^{n} \lambda_k^i x_i}{\sum_{i=1}^{n} \lambda_k^i}

6. Bernoulli Distribution

Concepts: Probability, maximum likelihood estimation.
Equations:
- MLE for probability $p$ :

\hat{p} = \frac{\text{Number of type A data points}}{\text{Total number of data points}}

7. Projections and Reconstruction Error

Concepts: Projection onto a line, reconstruction error.
Equations:
- Projection length:

\text{Projection length} = \frac{\mathbf{x} \cdot \mathbf{w}}{\|\mathbf{w}\|}

Reconstruction error:

\| \mathbf{x} - (\mathbf{x} \cdot \mathbf{w}) \mathbf{w} \|