IIT M FOUNDATION AN3 EXAM QPF3 16 JULY 2023 - 16 Jul 2023

MLT Quiz 111 questions46 marks5 MSQ · 4 SA · 2 MCQFoundation / QPF3

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

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.
Key Equation:

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.
Euclidean Distance: Distance metric used to calculate the distance between points.
Centroids: Mean of the points in a cluster.
Key Steps:
1. Initialize centroids.
2. Assign each data point to the nearest centroid.
3. Recalculate the centroids.
4. Repeat until convergence.

PCA: Technique to reduce the dimensionality of a dataset while preserving its variance.
Covariance Matrix: Matrix used to find the principal components.
Eigenvalues and Eigenvectors: Used to determine the principal components.
Key Concepts:
- The first principal component has the largest eigenvalue.
- The variance along the first principal component is maximum.
Key Equations:

\text{Covariance Matrix} = \frac{1}{n-1} X^T X

\text{Principal Components} = \text{Eigenvectors of the Covariance Matrix}

Kernel PCA: Extension of PCA using kernel functions to handle non-linear data.
Key Concepts:
- The number of principal components cannot be more than the dimensionality of the dataset.
- Valid kernel functions can be combined to form new kernel functions.
Key Equations:

k(x_1, x_2) = (x_1^T x_2)^3

ak_1 + bk_2 \text{ is a valid kernel function for } a, b > 0

Convergence: Algorithm terminates when centroids do not change.
Key Concepts:
- Initial centroids can affect the final clusters.
- The algorithm may not converge to the global optimum.

p = \frac{\text{Number of successes}}{\text{Total number of trials}}

\text{MLE for } p = 0.3 \text{ implies } 70 \text{ zeros in a dataset of } 100 \text{ points}

Beta Distribution: Probability distribution for the parameter $p$ of a Bernoulli distribution.
Prior and Posterior Distributions: Used in Bayesian inference to update the probability of a hypothesis.
Key Equations:

\text{Prior: } \text{Beta}(10, 5)

\text{Posterior: } \text{Beta}(30, 45)

\text{Number of zeros} = 40 \text{ in a dataset of } 60 \text{ points}

GMM: Probabilistic model that assumes data is generated from a mixture of Gaussian distributions.
Key Concepts:
- Each component has a mean ( $\mu$ ) and variance ( $\sigma^2$ ).
- The probability of a point belonging to a component is given by the mixture weights ( $\pi$ ).
Key Equations:

P(x \mid \text{Component } i) = \mathcal{N}(x \mid \mu_i, \sigma_i^2)

P(\text{Component } i) = \pi_i

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