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2024 July07: IIT M AN EXAM QDD2 - 07 Jul 2024

MLT Quiz 116 questions40 marks10 SA · 5 MCQ · 1 MSQafternoon / QDD2

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Q1MCQ2 marks

Q2MCQ3 marks

Both k1 and k2 are valid kernels.
k1 is a valid kernel, but k2 is not a valid kernel.
k2 is a valid kernel, but k1 is not a valid kernel.
Neither k1 nor k2 is a valid kernel.

Q3MSQ3 marksselect all that apply

Q4SA3 marks

Kernel PCA with a polynomial kernel of degree 3 is run on a dataset of 100 data-points. Let ϕ be the transformation corresponding to this kernel. If the largest eigenvalue of the centered kernel matrix is 10, find the largest eigenvalue of the covariance matrix of the transformed dataset, where ϕ is used to effect the transformation.

Your numeric answer:

Q5SA3 marks

Your numeric answer:

Q6SA4 marks

Your numeric answer:

Q7SA4 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

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

Q8SA2 marks

Find the variance of the dataset along the first principal component.

Your numeric answer:

Q9SA1 mark

Find the variance of the dataset along the x-axis.

Your numeric answer:

Q10MCQ2 marks

Comprehension passage - the questions below it refer to this

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

Q11SA2 marks

If (x1, y1) is the mean of the first cluster after convergence, find the value of x1 + y1.

Your numeric answer:

Q12SA2 marks

If (x2, y2) is the mean of the second cluster after convergence, find the value of x2 + y2.

Your numeric answer:

Comprehension passage - the questions below it refer to this

In the context of Bayesian estimation, consider a Beta prior for the parameter p of a Bernoulli distribution: question figure

In the context of Bayesian estimation, consider a Beta prior for the parameter p of a Bernoulli distribution: The dataset has 8 ones and 5 zeros. Based on the above data, answer the given subquestions.

Q13MCQ2 marks

What is the posterior?

Beta(12, 8)
Beta(11, 7)
Br(0.5)
Beta(8, 5)

Q14SA2 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

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

Q15MCQ2.5 marks

Find the log-likelihood of the dataset under this distribution.

Q16SA2.5 marks

Consider a dataset that has 25 data-points. The data-point xi and its frequency is given in the following table: question figure

Consider a dataset that has 25 data-points. The data-point xi and its frequency is given in the following table: In case the table is not clear: the value 0 appears once in the dataset, the value 1 appears four times in the dataset, and so on. Find the maximum likelihood estimate for the parameter λ of the Poisson distribution given this dataset.

Your numeric answer:

Answer key

Q1 (MCQ, 2 marks): A
Q2 (MCQ, 3 marks): A
Q3 (MSQ, 3 marks): A, B
Q4 (SA, 3 marks): 0.1
Q5 (SA, 3 marks): 0.8
Q6 (SA, 4 marks): 16
Q7 (SA, 4 marks): 0.25
Q8 (SA, 2 marks): 3
Q9 (SA, 1 marks): 2
Q10 (MCQ, 2 marks): A
Q11 (SA, 2 marks): -5
Q12 (SA, 2 marks): 9
Q13 (MCQ, 2 marks): A
Q14 (SA, 2 marks): 0.6
Q15 (MCQ, 2.5 marks): A
Q16 (SA, 2.5 marks): 2.52

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. Principal Component Analysis (PCA)

Concepts: Projection of data onto principal components, reconstruction error, covariance matrix, eigenvalues, and eigenvectors.
Equations:
- Projection of a data point $\mathbf{x}$ onto principal components $\mathbf{w}_1$ and $\mathbf{w}_2$ :

\mathbf{x}_{\text{projected}} = (\mathbf{x} \cdot \mathbf{w}_1) \mathbf{w}_1 + (\mathbf{x} \cdot \mathbf{w}_2) \mathbf{w}_2

Reconstruction error:

\text{Reconstruction Error} = \| \mathbf{x} - \mathbf{x}_{\text{projected}} \|

Variance along a principal component:

\text{Variance} = \lambda_i

where $\lambda_i$ is the eigenvalue corresponding to the principal component $\mathbf{w}_i$ .

2. Kernel Methods

Concepts: Kernel functions, polynomial kernels, kernel PCA, valid kernels.
Equations:
- Polynomial kernel of degree $d$ :

k(\mathbf{x}, \mathbf{y}) = (\mathbf{x} \cdot \mathbf{y} + c)^d

Kernel PCA transformation and eigenvalue calculation.

3. Gaussian Mixture Models (GMM) and EM Algorithm

Concepts: Mixture models, expectation-maximization (EM) algorithm, posterior probabilities.
Equations:
- Posterior probability in the E-step:

\lambda_i^k = \frac{\pi_k \mathcal{N}(x_i | \mu_k, \Sigma_k)}{\sum_{j=1}^K \pi_j \mathcal{N}(x_i | \mu_j, \Sigma_j)}

Probability of a data point belonging to a component given the data:

P(z_k | x_i) = \frac{\pi_k \mathcal{N}(x_i | \mu_k, \Sigma_k)}{\sum_{j=1}^K \pi_j \mathcal{N}(x_i | \mu_j, \Sigma_j)}

4. K-Means Clustering

Concepts: Initialization methods (e.g., k-means++), cluster means, distance metrics.
Equations:
- Probability of choosing a point as the next mean in k-means++:

P(x_i) = \frac{d(x_i, \mu)^2}{\sum_{j=1}^n d(x_j, \mu)^2}

5. Bayesian Estimation

Concepts: Prior and posterior distributions, Beta distribution, Bernoulli distribution, maximum likelihood estimation (MLE).
Equations:
- Beta distribution:

p \sim \text{Beta}(\alpha, \beta)

Posterior distribution for a Beta prior and Bernoulli likelihood:

p \sim \text{Beta}(\alpha + \text{number of successes}, \beta + \text{number of failures})

Expected value of the posterior:

\hat{p} = \frac{\alpha + \text{number of successes}}{\alpha + \beta + \text{number of trials}}

Poisson distribution:

f(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}

Log-likelihood for Poisson distribution:

\log \mathcal{L}(\lambda) = \sum_{i=1}^n \left( -\lambda + x_i \log \lambda - \log (x_i!) \right)