Processing multivariate data

Question 1

What is multivariate data

Answer 1

Data that is described by more than 1 variable, the data is multidimensional

Question 2

How is a multivariate observation represented

Answer 2

Using a vector, x. (column vector)

Question 3

If there are multiple samples for each variable how would the column vectors be combined and represented

Answer 3

In a matrix, having N samples (rows) and L features (columns)

The row contains all the variables
The column contains each sample for that specific variable

Question 4

What is covariance and give the formula

Answer 4

Extension of the variance to multi-dimensions

Measure of association/correlation between two variables

This describes how the two variables relate or change togther

Question 5

If two variables increase/decrease together what will the covariance be

Answer 5

positive

Question 6

If one variable increases and the other decreases, what will the covariance be

Answer 6

negative

Question 7

If two variables are independent, what will the covariance be

Answer 7

close to 0

Question 8

Explain the covariance matrix

Answer 8

The diagonal elements are just the variance values of that feature.

The off-diagonals are the covariances

Element ij is the covariance between fetaure i and feature j

Question 9

Difference between Population and sample statistics

Answer 9

Population refers to the entire group of data points e.g. adults in the UK. Population parameters are true but usually unknown values that describe the population.

A sample is a subset of the population selected for analysis. Sample statistics are the values calculated from the sample, used to estimate population parameters.

Question 10

If X1, X2, … , Xn are random samples drawn from a distribution with the population mean vector u and the population covariance matrix sigma.

Describe the sample mean

Answer 10

The sample mean x bar is an unbiased estimate of the population mean u

Question 11

If X1, X2, … , Xn are random samples drawn from a distribution with the population mean vector u and the population covariance matrix sigma.

Describe the sample covariance

Answer 11

The sample covariance S is a biased estimate of the population covariance sigma

Question 12

If X1, X2, … , XN are random samples drawn from a distribution with the population mean vector u and the population covariance matrix sigma.

Describe the unbiased estimate of the population covariance

Answer 12

The unbiased estimate of the population covariance is (N)/(N-1)S

Question 13

what is meant if an estimator is unbiased

Answer 13

An estimator is unbiased if, on average over many samples, it equals the true population value.

Question 14

How do we visualise 2d data

Answer 14

Scatter plot