PCA

Question 1

What is the meaning of PCA

Answer 1

Principal components analysis is concerned with explaining the covariance structure of a set of variables through a few linear combinations of the original variables. PCA re expresses large amounts of data to account for most information int he data

Question 2

What is the use of PCA

Answer 2

Its a dimension reduction technique or si used as a method for identifying associations among variables

Question 3

Explain the construction and structure of the new principal components

Answer 3

Aim of PCA is to describe variation in a set of correlation variables xi in terms of a new set of uncorrelated prinicpal compnonetns yi where the number of yis is substantially less than xis. Each yi is a linear combination of the xi variables.

Question 4

What is meant by the principal components being in decreasing order of importance

Answer 4

The first principal component yi accounts for most variation in the original data out of all of the linear combinations of xis - Usually would aim to explain 80% to 90% of variation in data using Principal components and PC1 will explain a large part of that.

Question 5

How do you find the eigenvalues and eigenvectors of a matrix

Answer 5

Solve det(A-lamdaI)=0 for eignvalues
Solve (A-lamdaI)v=0 for eigenvectors

Question 6

What is the meaning of the eigenvalues and eigenvectors of covariance matrix S for set of data in terms of PCs

Answer 6

Eigenvalue j quantify how much of the variance is accounted for within each PCj. The eignevalue is the variance of each new PC.
Eigenvector j detail the linear combination of xi’s which form PCj

Question 7

How much of the total variance does PC1 explain?

Answer 7

Lamda1/(sum of all lamdas)= % of total variation explained

Question 8

Define the first principal component

Answer 8

First PC of a data set is the linear combination of the variables which has greatest variance

Question 9

What is the total variance of data

Answer 9

Sum of lamda i’s

Question 10

What is a key assumption of the set of PCs

Answer 10

They are uncorrelated with each other

Question 11

In words : what is the proportion of variation explained by each PC and how do we use this to decide how many PCs to use to describe the data

Answer 11

Each eigenvalue divided by the sum of all eigenvalues gives proportion of the variation explained by the associated principal component. This cumulative proportion of variation helps to decide how many PCs to use.

Question 12

What is a disadvantage to PCA

Answer 12

Interpretation of the new PCs can be difficult
It gives large weight to variables who have a large range of values

Question 13

In the coefficients of linear combinations of the variables that construct PCs - What matters in terms of signs, comparison, magnitude

Answer 13

Signs on the coefficients are aribtray but it matters if they are opposite to another element. The magnitudes also matter

Question 14

Why might standardisation be needed in PCA

Answer 14

To prevent variables with bigger variances perhaps in smaller units being weighted more heavily than other more important variables

Question 15

What does standardisation mean and how does one do it

Answer 15

Standardisation means ensuring the data is expressed as comparable units - We divide each variable by the sample stdev for that variable which forces all variances to be 1. Hence we are now working with a correlation matrix instead of a covariance matrix

Question 16

Relationship between correlation and covariance matrix

Answer 16

Correlation matrix = standardised covariance matrix.

Question 17

Why might you want to avoid standardisation?

Answer 17

Is variance of a variable is an accurate representation of its importance relative to other variables variances then PCA should be performed on this unstandardized data.

Question 18

What is something to be cautious of when clustering

Answer 18

Clustering is very common analysis but not correct! : Uses PCs and even though we only lose a small amount of variability in the data using PCs its not theoretically correct. just to be aware of.

Question 19

What is the function of principal components analysis

Answer 19

prcomp(iris[,1:4])

Question 20

data(iris)
> fit<-prcomp(iris[,1:4])
> fit

What does this r code mean?

Answer 20

Its reading int he iris data set and performing principal components analysis on the data.

Question 21

summary(fit)
> round(fit$rotation,2)

What does this code mean?

Answer 21

It cna be easier to examine a summary of the output of prcomp (in the fit variable)
The ‘summary’ function provides a summary of the PCA output and the ‘round’ function simply rounds the eigenvector values to 2 decimal places.

Question 22

> plot(fit)

What would this r code do? fit is as such:
fit<-prcomp(iris[,1:4])

Answer 22

Would plot the proportion of variance explained by each PC

Question 23

> newiris<-predict(fit)
newiris

What does this code do? Fit is as such:
fit<-prcomp(iris[,1:4])

Answer 23

The ‘predict’ function is a generic function which predicts results of various model fitting functions — in this case it recognizes ‘fit’ as the result of a principal components analysis and calculates the values of the new PCs for each observation

Question 24

What would you expect the following code to output
data(iris)
> iris[1:10,]

Answer 24

Reading in data: would print the first 10 rows of the data set

Question 25

What would you expect the output to be of this code:
summary(iris)

Answer 25

Provides a summary of data set: for each field would give:
min, max, quartiles, mean, median or a count if not numeric

Question 26

What does princomp() do

Answer 26

princomp(obtains the principal components via an eigen-decomposition of the covariance
matrix of the data)

Question 27

What does prcomp() do

Answer 27

prcomp(obtains the principal components via singular value decompositions the data matrix)

Question 28

What would you expect output to be of
fit<-prcomp(iris[,1:4])
fit

Answer 28

Give standard deviations of all 4 components and the linear combinations of each variable that make up the PCs

Question 29

What would you expect output to be of
fit<-prcomp(iris[,1:4])
summary(fit)

Answer 29

Gives importance of components detailing the SD, Proportion of variance and cumulative proportion for each PCi