Chapter 17 Introduction to Multivariate Statistics

Question 1

What’s covariance?

P 157

Answer 1

In probability, covariance is the measure of the joint probability for two random variables. It describes how the two variables change together.

cov(X, Y ) = E[(X − E[X]) × (Y − E[Y ])]

Question 2

What does the sign of the covariace indicate?

P 157

Answer 2

The sign of the covariance can be interpreted as whether the two variables increase together (positive) or decrease together (negative).

Question 3

NumPy does not have a function to calculate the covariance between two variables directly. Instead, it has a function for calculating a ____ called ____ that we can use to retrieve the covariance.

P 157

Answer 3

covariance matrix, cov()

Question 4

The magnitude of the covariance is not easily interpreted. A covariance value of zero indicates that both variables are completely ____.

P 157

Answer 4

independent

Question 5

How can we access the covariance, using the covariance matrix?

P 157

Answer 5

We access just the covariance for the two variables as the [0, 1] element of the square covariance matrix returned.

Question 6

How can the covariance (obtained from covariance matrix) be normalized to a score between -1 and 1 to make the magnitude interpretable?

P 158

Answer 6

r =
cov(X, Y )/(SD_X × SD_Y)
Where r is the correlation coefficient of X and Y , cov(X, Y ) is the sample covariance of X and Y and S_X and S_Y are the standard deviations of X and Y respectively.

Question 7

NumPy provides the ____function for calculating the correlation between two variables directly.

P 158

Answer 7

corrcoef()

Note: .corr() is NOT for numpy, it’s for pandas dataframes
Like cov(), it returns a matrix, in this case a correlation matrix. As with the results from cov() we can access just the correlation of interest from the [0,1] value from the returned squared matrix

Question 8

What’s a covariance matrix?

P 159

Answer 8

The covariance matrix is a square and symmetric matrix that describes the covariance between two or more random variables.

Question 9

The diagonal of the covariance matrix are the ____, as such it is often called the variance-covariance matrix.

P 159

Answer 9

variances of each of the random variables

Question 10

How can we find the covarience between a 2D-array’s columns, using the cov() function?

P 159

Answer 10

The cov() function can be called with a single 2D array where each sub-array contains a feature (e.g. column). If this function is called with your data defined in a normal matrix format (rows then columns), then a transpose of the matrix will need to be provided to the function in order to correctly calculate the covariance of the columns.

Question 11

The covariance matrix is not widely used in multivariate analysis. True/False

P 160

Answer 11

False
The covariance matrix is widely used in multivariate analysis.