Problems tagged with "covariance"

Problem #38

Tags: covariance

Consider a data set of \(n\) points in \(\mathbb R^d\), \(\nvec{x}{1}, \ldots, \nvec{x}{n}\). Suppose the data are standardized, creating a set of new points \(\nvec{z}{1}, \ldots, \nvec{z}{n}\). That is, if the new points are stacked into an \(n \times d\) matrix, \(Z\), the mean and variance of each column of \(Z\) would be zero and one, respectively.

True or False: the covariance matrix of the standardized data must be the \(d\times d\) identity matrix; that is, the \(d \times d\) matrix with ones along the diagonal and zeros off the diagonal.

True False
Solution

False.

Problem #41

Tags: covariance

Suppose a data set consists of the following three measurements for each Saturday last year: \(X_1\): The day's high temperature \(X_2\): The number of people at Pacific Beach on that day \(X_3\): The number of people wearing coats on that day

Suppose the covariance between these features is calculated and placed into a \(3 \times 3\) sample covariance matrix, \(C\). Which of the below options most likely shows the sign of each entry of the sample covariance matrix?

Solution

The second option.

Problem #42

Tags: covariance

Suppose we have two data sets, \(\mathcal{D}_1\) and \(\mathcal{D}_2\), each containing \(n/2\) points in \(\mathbb R^d\). Let \(\nvec{\mu}{1}\) and \(C^{(1)}\) be the mean and sample covariance matrix of \(\mathcal{D}_1\), and let \(\nvec{\mu}{2}\) and \(C^{(2)}\) be the mean and sample covariance matrix of \(\mathcal{D}_2\).

Suppose the two data sets are combined into a single data set \(\mathcal D\) containing \(n\) points.

Part 1)

True or False: the mean of the combined data \(\mathcal{D}\) is equal to \(\displaystyle\frac{\nvec{\mu}{1} + \nvec{\mu}{2}}{2}\).

True False
Solution

True.

Part 2)

True or False: the sample covariance matrix of the combined data \(\mathcal{D}\) is equal to \(\displaystyle\frac{C^{(1)} + C^{(2)}}{2}\).

True False
Solution

False

Problem #43

Tags: covariance

Suppose a random vector \(\vec X = (X_1, X_2)\) has a multivariate Gaussian distribution. Suppose it is known that known that \(X_1\) and \(X_2\) are independent.

Let \(C\) be the Gaussian distribution's covariance matrix.

Part 1)

True or False: \(C\) must be diagonal.

True False
Solution

True.

Part 2)

True or False: each entry of \(C\) must the same.

True False
Solution

False.

Problem #52

Tags: covariance

Let \(\mathcal D\) be a set of data points in \(\mathbb R^d\), and let \(C\) be the sample covariance matrix of \(\mathcal D\). Suppose each point in the data set is shifted in the same direction and by the same amount. That is, suppose there is a vector \(\vec\delta\) such that if \(\nvec{x}{i}\in\mathcal D\), then \(\nvec{x}{i} + \vec\delta\) is in the new data set.

True or False: the sample covariance matrix of the new data set is equal to \(C\)(the sample covariance matrix of the original data set).

True False
Solution

True.

Problem #55

Tags: covariance

Consider the data set \(\mathcal D\) shown below.

What will be the sign of the \((1,2)\) entry of the data's sample covariance matrix?

Solution

The sign will be negative.