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

MLT Quiz 113 questions50 marks6 SA · 5 MSQ · 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

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Q10SA4 marks

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Q11SA4.5 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

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

Q12SA2 marks

What is the value of a? Enter your answer correct to three decimal places.

Your numeric answer:

Q13SA2 marks

What is the value of b? Enter your answer correct to two decimal places.

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 ( $\phi$ ): Maps input data to a higher-dimensional space.
Example: For a given kernel $k([x_1, x_2]^T, [y_1, y_2]^T) = 1 + x_1 y_1 + x_2 y_2 + x_1^2 y_1^2 x_2 y_2 + x_1 y_1 x_2^2 y_2^2$ , the appropriate transformation mapping is:

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

Standard PCA: Applied on a centered dataset to find the directions (principal components) of maximum variance.
Variance: The variance of the dataset along the first principal component is maximum.
Eigenvalues and Eigenvectors: The first principal component is the unit norm eigenvector corresponding to the largest eigenvalue of the covariance matrix.
Low-Dimensional Subspace: If all points lie in a four-dimensional low subspace, the covariance matrix has only four non-zero eigenvalues.

Kernel PCA: Extension of PCA using kernel functions to handle non-linear data.
Valid Kernel Functions: For every transformation mapping $\phi : \mathbb{R}^d \rightarrow \mathbb{R}^D$ , there exists a valid kernel function.
Combination of Kernels: For two valid kernels $k_1, k_2 : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ , $ak_1 + bk_2$ is a valid kernel function where $a$ and $b$ are positive real numbers.

Lloyd's Algorithm: Iterative algorithm to partition data into $k$ clusters.
Initialization: Initial cluster centers are chosen, and the algorithm iteratively updates the centers to minimize within-cluster variance.
Example: For a dataset $\{-3, -1, 0, 1, 4, 299, 300, 301\}$ with initial centers $-10$ and $310$ , the final cluster centers $a$ and $b$ can be found, and their product $ab$ can be computed.

GMM: Probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions.
Probability Calculation: The probability that a point $x$ comes from a specific component can be calculated using the Gaussian density function.
Example: For a GMM with two components, the probability that the point $x = 1.2$ comes from the first component can be calculated using the given table of Gaussian densities.

Beta Distribution: Used as a prior and posterior distribution for the parameter $p$ in a Bernoulli distribution.
Example: For a dataset with 60 points, if the prior and posterior distributions for $p$ are $\text{Beta}(10,5)$ and $\text{Beta}(30,45)$ respectively, the number of data points with the value 0 can be determined.