Lecture 11 - MDS

Question 1

What is different about MDS compared to clustering? How are they similar?

Answer 1

• MDS is older
• MDS has an explicit model (clustering has weak/no model)

SIMILAR: both analyse distance (dissimilarities) and both can do variables OR cases

Question 2

How do you represent MDS?

Answer 2

• geometrical picture
• each case/variable is a point in space
• 1D: line
• 2D: chart
  ^ can do more than this, but 1/2 usually preferred because easier to visualise on page

Question 3

What are the 3 types of MDS?

Answer 3

• CLASSICAL: simplest, 1 proxmity, matrix, interval data (often ratio)
• NON-METRIC: most common, 1 distance matrix, assumes ordinal only
• > 1 MATRIX: can create matrix for each subject, unweighted = replicated MDS, weighted = individual diffs MDS

Question 4

What is the MDS problem?

Answer 4

• locate points in space to represent variables, so that distances b/w points is similar to original distances
• need to find: coordinates aj and ak of points j and k on the mth of r dimensions
• so that distance djk is distance b/w j and k, in r-dimensional Euc space

Question 5

What is the goal of MDS?

Answer 5

• dimension reduction
• n variables, you have an n-dimensional space
• want to “shrink” down from many data points to fewer dimensions
• AIM: make data more understandable

Question 6

What are the advantages and issues with dimension reduction?

Answer 6

• issues: always lose something
• good: very useful simplification of the data
• can oversimplify! (can lose too much info)

Question 7

How do we define distance in MDS?

Answer 7

• we let the distance d be some function of r (where r = original distance)
• djk = f (rjk )

Question 8

What is the difference between classic and modern MDS?

Answer 8

• classic: function is linear regression, djk = a + b.rjk

- modern: rank-ordered function (i.e. want rank order of distances to be the same as original)

Question 9

How does MDS work?

Answer 9

• start with points located randomly in space (of a dimensionality chosen by the user)
• math procedures ‘move’ points around to minimise stress
• many iterations > until rank order of their distances ‘best’ matched the rank order of the data (i.e. until stress is the lowest point)

Question 10

What is stress? What situations produce higher stress?

Answer 10

• badness of fit. How well does MDS representation fit data
• Kruskal
• less than 0.15 is good fit
• more variables = higher stress; higher dimensions = low stress
• want to use #dimensions that are required for acceptable stress value

Question 11

What did Kruskal do? What is good about this?

Answer 11

• devised a method of rank-order transformations
• monotone regression
• distance of the proximity matrix converted into rank order
• no need for assumptions of ratio scales

Question 12

What are the 2 MDS methods in SPSS?

Answer 12

• ALSCAL: method of steepest descent

- PROXSCAL: iterative majorisation > use this

Question 13

Why use PROXSCAL?

Answer 13

• doesn’t need a good starting point

- always converges to minimum stress

Question 14

How do you know when to stop?

Answer 14

• stress doesn’t change by larger than a preset criterion
• stress value reaches a preset min value
• program has reached set no. of iterations

Question 15

What are the subjective vs. statistical interpretation of MDS?

Answer 15

• subjective: look at diagram to see patterns and how they group together > MORE COMMON
• statistical: regression of dimensional coordinates of an MDS solution against other variables

Question 16

What are the 2 spatial manifolds?

Answer 16

• simplex: points form chain or sequence

- circumplex: points form a circle

Question 17

What is the assumption of MDS? How do you check this?

Answer 17

• rship b/w proximity data and derived from distances is smooth

CHECK transformation plot for DEGENERACY

• want smooth plot
• degeneracy: points of representation are located in a few tight clusters
• these tight clusters may be small part of structure of data, but can swamp interpretation
• may make stress close to 0

Question 18

Why is it r-dimensional?

Answer 18

r = the no. of dimensions we choose to represent the data

Question 19

How do we actually reduce dimensions?

Answer 19

• TRANSFORMATION of distances!!!

- distances in new space must be different to original data

Question 20

What is important about the axes in MDS?

Answer 20

• they are arbitrary!!

- relative distances are key

Question 21

What can you use to determine how many dimensions to use in MDS?

Answer 21

scree plot > at elbow

- elbow may not be obvious if you have too few variables

Question 22

What is the difference between transformations in classic vs. modern MDS?

Answer 22

• classical: linear
  djk = a + b*rjk
• modern: rank-order, not linear

Question 23

What is the stress from random data thing? How was this done?

Answer 23

• when stress is less than random data, we can accept the solution
• graph: stress (y) v. dimensions (x), line for diff no. of variables
• you use the no. of dimensions and variables you are using to find the stress of the appropriate random data, want yours to be less than this!
• some dudes made random data, replicated many times and found stress for the data in a no. of solutions

Question 24

In an MDS solution, what are other names for “informal groups” and “partitioning space”?

Answer 24

• informal groups: cluster analysis

- partitioning space: regression

Question 25

What is in the transformation plot?

Answer 25

• Y: transformed proximities

- X: (original) proximities

Question 26

What does MDS transform ordinal proximities in to?

Answer 26

distance data