Machine Learning Techniques Quiz 1: 7 July 2024 (May 2024 term)

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.

Let $\mathbf{C}$ be the covariance matrix of a mean-centered dataset:

$\mathbf{C} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$

Standard PCA is performed on this dataset. The first two PCs are given below:

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

Based on the above data, answer the given subquestions.

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

Let $\mathbf{C}$ be the covariance matrix of a mean-centered dataset:

$\mathbf{C} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$

Standard PCA is performed on this dataset. The first two PCs are given below:

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

Based on the above data, answer the given subquestions.

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

Let $\mathbf{C}$ be the covariance matrix of a mean-centered dataset:

$\mathbf{C} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$

Standard PCA is performed on this dataset. The first two PCs are given below:

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

Based on the above data, answer the given subquestions.

Consider the following dataset of six points in $\mathbb{R}^2$:

K-means clustering is run on this dataset with $k = 2$. In this version, the means are initialized first. The mean of the first cluster is initialized to $(-4, -4)$ and the mean of the second cluster is initialized to $(6, 6)$.

Based on the above data, answer the given subquestions.

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

Consider the following dataset of six points in $\mathbb{R}^2$:

K-means clustering is run on this dataset with $k = 2$. In this version, the means are initialized first. The mean of the first cluster is initialized to $(-4, -4)$ and the mean of the second cluster is initialized to $(6, 6)$.

Based on the above data, answer the given subquestions.

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

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.

What is the posterior?

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.

Consider a dataset of $n$ data-points all of which are non-negative integers. These data-points are sampled from a Poisson distribution, whose probability mass function is given below:

$f(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}, \ \lambda > 0$

Here $\lambda$ is a parameter.

Based on the above data, answer the given subquestions.

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

Consider a dataset of $n$ data-points all of which are non-negative integers. These data-points are sampled from a Poisson distribution, whose probability mass function is given below:

$f(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}, \ \lambda > 0$

Here $\lambda$ is a parameter.

Based on the above data, answer the given subquestions.

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.