IIT M FOUNDATION AN4 EXAM QPF4 16 JULY 2023 - 16 Jul 2023

MLT Quiz 111 questions46 marks5 MSQ · 4 SA · 2 MCQFoundation / QPF4

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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

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

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 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

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

Algorithm: Iterative algorithm that partitions data into $k$ clusters.
Steps:
1. Initialize centroids.
2. Assign each data point to the nearest centroid.
3. Recalculate centroids as the mean of the assigned points.
4. Repeat until convergence.
Example:
- Points: $P(1, 1)$ , $Q(2, 1)$ , $R(4, 3)$ , $S(5, 4)$
- Initial centroids: $C_1 = (0, 0)$ , $C_2 = (4, 4)$
- Final centroids after convergence: $\langle 1.5, 1 \rangle$ , $\langle 4.5, 3.5 \rangle$

Objective: Reduce the dimensionality of a dataset while preserving its variance.
Steps:
1. Center the data.
2. Compute the covariance matrix.
3. Compute the eigenvectors and eigenvalues of the covariance matrix.
4. Select the top $k$ eigenvectors (principal components).
Key Points:
- The first principal component has the maximum variance.
- The first principal component is the unit norm eigenvector corresponding to the largest eigenvalue of the covariance matrix.
Example:
- Dataset of 200 points in $\mathbb{R}^{30}$ lying in a four-dimensional subspace.
- After PCA, the covariance matrix has only four non-zero eigenvalues.

Objective: Extend PCA to non-linear datasets using kernel functions.
Key Points:
- The number of principal components can exceed the dimensionality of the original dataset.
- Valid kernel functions can be combined linearly with positive coefficients.
- Not all kernel functions are positive definite.
Example:
- Kernel function: $k(x_1, x_2) = (x_1^T x_2)^3$
- Valid kernel function: $ak_1 + bk_2$ where $a, b > 0$

Behavior: Minimizes within-cluster variance.
Limitations: Cannot obtain clusters that do not minimize within-cluster variance, such as concentric circles or highly imbalanced clusters.
Example:
- Dataset: $\{-3, -1, 0, 1, 4, 299, 300, 301\}$
- Initial centroids: $-10$ , $310$
- Final centroids: $a$ , $b$
- Product of final centroids: $ab = 60$

Bernoulli Distribution: Models binary outcomes with parameter $p$ .
MLE for $p$ : Estimates the parameter $p$ based on observed data.
Example:
- Dataset with 100 data points, MLE estimate for $p$ is 0.3.
- Number of zeros: $100 \times (1 - 0.3) = 70$

Beta Distribution: Prior and posterior distributions for the parameter $p$ of a Bernoulli distribution.
Example:
- Prior: $\text{Beta}(10,5)$
- Posterior: $\text{Beta}(30,45)$
- Number of zeros in the dataset: 40

Objective: Model data as a mixture of Gaussian distributions.
Probability Calculation: Use the Gaussian density function to compute the probability of a data point belonging to a component.
Example:
- GMM with two components:
  - $\pi_1 = 0.3$ , $\mu_1 = 0$ , $\sigma_1^2 = 1$
  - $\pi_2 = 0.7$ , $\mu_2 = 1$ , $\sigma_2^2 = 2$
- Probability that $x = 1.2$ comes from the first component: between 0.15 and 0.35