Home / Past papers / MLT Quiz 1 / this paper

IIT M FOUNDATION AN2 EXAM QPF2 16 JULY 2023 - 16 Jul 2023

MLT Quiz 111 questions46 marks5 MSQ · 4 SA · 2 MCQFoundation / QPF2

Elapsed 00:00

Q1MCQ4 marks

Q2MCQ4 marks

Q3MSQ4 marksselect all that apply

Select all true statements regarding standard PCA applied on a centered dataset.

The variance of the dataset along the first principal component is maximum.
The variance of the dataset along the first principal component is minimum.
The first principal component is the unit norm eigenvector corresponding tothe largest eigenvalue of the covariance matrix.
The first principal component is the unit norm eigenvector corresponding tothe smallest eigenvalue of the covariance matrix.

Q4MSQ4 marksselect all that apply

The covariance matrix corresponding to this dataset has 30 non-zeroeigenvalues.
The covariance matrix corresponding to this dataset has only four non-zeroeigenvalues.
The residues will become zero after four rounds
By utilizing four principal component vectors and their correspondingcoefficients, we have the ability to reconstruct the complete dataset using representatives.

Q5MSQ5 marksselect all that apply

Which of the following statements about kernel PCA is/are always true?

Q6MSQ5 marksselect all that apply

Q7MSQ3.5 marksselect all that apply

Which of the following statements is/are true?

Q8SA4 marks

Your numeric answer:

Q9SA4 marks

Your numeric answer:

Q10SA4 marks

Your numeric answer:

Q11SA4.5 marks

Your numeric answer:

Answer key

Q1 (MCQ, 4 marks): C
Q2 (MCQ, 4 marks): C
Q3 (MSQ, 4 marks): A, C
Q4 (MSQ, 4 marks): B, C, D
Q5 (MSQ, 5 marks): B, C
Q6 (MSQ, 5 marks): A, B, D
Q7 (MSQ, 3.5 marks): A, C
Q8 (SA, 4 marks): 60
Q9 (SA, 4 marks): 70
Q10 (SA, 4 marks): 40
Q11 (SA, 4.5 marks): 0.15 to 0.35

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. Kernel Functions and Transformations

Kernel Functions: Functions that compute the dot product of the transformed input data in a higher-dimensional space.
Transformation Mapping ( $\phi$ ): A function that maps input data into a higher-dimensional space.
Example: Given a kernel function $k([x_1, x_2]^T, [y_1, y_2]^T)$ , find the appropriate transformation mapping $\phi$ .

2. K-means Clustering

Algorithm: Iterative algorithm that partitions data into $k$ clusters.
Steps:
1. Initialize centroids.
2. Assign each data point to the nearest centroid.
3. Recalculate the centroids.
4. Repeat until convergence.
Euclidean Distance: Metric used to calculate the distance between points.
Example: Clustering problem with four points $P(1, 1)$ , $Q(2, 1)$ , $R(4, 3)$ , and $S(5, 4)$ with $k = 2$ and initial centroids $C_1 = (0, 0)$ and $C_2 = (4, 4)$ .

3. Principal Component Analysis (PCA)

Objective: Reduce the dimensionality of a dataset while preserving as much variability as possible.
Covariance Matrix: Matrix used to understand the variance and covariance of the dataset.
Eigenvalues and Eigenvectors: Used to determine the principal components.
Example: PCA applied on a centered dataset with statements about variance and eigenvalues.

4. Kernel PCA

Concept: Extension of PCA using kernel methods to handle non-linear data.
Valid Kernels: Functions that satisfy Mercer's condition.
Example: Statements about kernel PCA and valid kernel functions.

5. Lloyd's Algorithm (K-means)

Behavior: Understanding the final clusters obtained using Lloyd's algorithm.
Example: Clusters that cannot be obtained as final clusters using Lloyd's algorithm with $k = 2$ .

6. Bernoulli Distribution and Maximum Likelihood Estimate (MLE)

Bernoulli Distribution: Models binary data.
MLE: Method to estimate the parameters of a statistical model.
Example: Dataset with 100 data points modeled using a Bernoulli distribution with MLE estimate for $p = 0.3$ .

7. Beta Distribution

Prior and Posterior Distributions: Used in Bayesian inference.
Example: Dataset with 60 points modeled using a Bernoulli distribution with prior and posterior distributions as Beta(10,5) and Beta(30,45) respectively.

8. Gaussian Mixture Models (GMM)

Components: Mixture of Gaussian distributions.
Probability Calculation: Calculating the probability that a point comes from a specific component.
Example: GMM with two components and calculating the probability that the point $x = 1.2$ comes from the first component.

Equations

Kernel Function

k([x_1, x_2]^T, [y_1, y_2]^T) = 1 + x_1 y_1 + x_2 y_2 + x_1^2 y_1^2 x_2 y_2 + x_1 y_1 x_2^2 y_2^2

Transformation Mapping

\phi([x_1, x_2]^T) = [1, x_1, x_2, x_1^2 x_2, x_1 x_2^2]

Euclidean Distance

d(p, q) = \sqrt{(q_1 - p_1)^2 + (q_2 - p_2)^2 + \dots + (q_n - p_n)^2}

Covariance Matrix

\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})

Eigenvalues and Eigenvectors

\text{Cov}(X) \mathbf{v} = \lambda \mathbf{v}

Bernoulli Distribution and MLE

\hat{p} = \frac{\text{number of ones}}{\text{total number of data points}}

Beta Distribution

\text{Beta}(\alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)}

Gaussian Mixture Models

P(x) = \sum_{i=1}^{k} \pi_i \mathcal{N}(x | \mu_i, \sigma_i^2)