Machine Learning Techniques Quiz 1: 25 February 2024 (January 2024 term)

Suppose a dataset lies in $\mathbb{R}^4$ and undergoes Principal Component Analysis (PCA) after being centered. The resulting first and second principal components are given by:

$\frac{1}{\sqrt{3}} \cdot \begin{bmatrix} 1 \\ -1 \\ 0 \\ 1 \end{bmatrix}, \quad \frac{1}{\sqrt{3}} \cdot \begin{bmatrix} 1 \\ 0 \\ 1 \\ -1 \end{bmatrix}$

Which of the following could be the third principal component?

Consider a dataset consisting of 1000 samples in a 20-dimensional space. You apply Kernel PCA with a polynomial kernel of degree 3 to reduce the dimensionality of the data. Which of the following statements regarding Kernel PCA is true?

Standard PCA has been performed on a centered dataset in $\mathbb{R}^3$. The first two principal components are given below:

$\mathbf{w}_1 = \frac{1}{\sqrt{3}} \cdot \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \frac{1}{\sqrt{2}} \cdot \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$

Consider the data point in the dataset: $\begin{bmatrix} 2 & 1 & -1 \end{bmatrix}^T$. $(a, b)$ is the representation of this point in the coordinate system formed by the two principal components given above. The first and second coordinates correspond to PC-1 and PC-2 respectively.

Based on the above data, answer the given subquestions.

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

Standard PCA has been performed on a centered dataset in $\mathbb{R}^3$. The first two principal components are given below:

$\mathbf{w}_1 = \frac{1}{\sqrt{3}} \cdot \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \frac{1}{\sqrt{2}} \cdot \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$

Consider the data point in the dataset: $\begin{bmatrix} 2 & 1 & -1 \end{bmatrix}^T$. $(a, b)$ is the representation of this point in the coordinate system formed by the two principal components given above. The first and second coordinates correspond to PC-1 and PC-2 respectively.

Based on the above data, answer the given subquestions.

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

Consider the data points shown in the following image:

Based on the above data, answer the given subquestions.

Perform K-means clustering with K = 2 and initial cluster centers at (0, 0) and (6, 0). What are the final means of clusters after convergence?

Consider the data points shown in the following image:

Based on the above data, answer the given subquestions.

After introducing a new data point (7, 0) , the cluster centers were updated. Enter the sum of the updated x-coordinates for both cluster centers accurately, rounding your answer to two decimal places.

Based on the above data, answer the given subquestions.

What is the log-likelihood of this dataset under this distribution? ln represents the natural logarithm or loge.

Based on the above data, answer the given subquestions.

Consider Lloyd’s algorithm used for k-means clustering and choose the correct statements:

Consider a dataset with 100 total data points. Each data point is classified as either type A or type B. We model this using a Bernoulli distribution, where p is the probability of a data point being type A. If the maximum likelihood estimate (MLE) of p based on the dataset is 0.4, how many data points of type B are there in this dataset?