10 - Multidimensional Scaling Flashcards
What is Multidimensional Scaling about?
On how to map objects when characteristics that determine the objects’ representation are not available or are unknown.
What is the principle of multidimensional scaling?
Find a visual representation of the objects in a space with a minimal number of dimensions that reproduces the subjective distances between the objects as accurately as possible.
What are the 2 goals of multidimensional scaling?
- Map objects relative
- Identify dimensions
- Map the objects relative to each other –> which objects are perceived to be similar to each other and which are dissimilar?
- Identify dimensions that determine subjective representations
How to build an n-dimensional representation from pairwise distances?
Find a spatial mapping of the objects that reproduces these pairwise distances as well as possible.
What does the distance stand for?
dissimilarity
What are the two methods for collecting dissimilarity ratings?
- Rating method
- Ranking method
What is the Rating method about for collecting dissimilarity ratings?
Rate similarity of each pair of objects:
“Please rate each pair on a scale from very dissimilar (1) to very similar (9).
- Cheeseburger vs. Hamburger 9
- Filet-o-Fish vs. BigMac 1
How to determine the number of pairs?
Npairs = Nobjects * (Nobjects -1)/ 2
How does the ranking method work for collecting dissimilarity ratings?
Rank pairs of objects in terms of similarity
“Please order the pairs from least to most similar”
- Cheeseburger vs. Hamburger
- Filet-o-Fish vs. BigMac
What are two types of multidimensional scaling (MDS)?
- Classic (metric) MDS
- Nonmetric (ordinal) MDS
What is the classical (metric) MDS about?
- assumes distance data on a metric level of measurement
- has an analytics solution
What is the nonmetric (ordinal) MDS about?
- is adequate for ordinal distance data (ranks, ratings, etc.; makes less strong assumptions about measurement level)
- finds a solution iteratively
First we have a dissimilarity matrix, then we have an initial configuration map with 2D. What comes after that?
Shephard diagram.
What is done in the Shephard diagram.
We identify violations of monotonicity.
What do we do when we identify a violation of monotonicity?
What is the revision of coordinates by Kruskal about?
The revision of coordinates by Kruskal is a method used in multidimensional scaling (MDS) to improve the fit between the distances between objects in the original high-dimensional space and the distances between the objects in the reduced-dimensional space. It is a post-processing step that involves adjusting the coordinates of the objects in the reduced space to minimize the stress and improve the accuracy of the MDS solution.
How to quantify model fit?
With STRESS
What is STRESS about?
= Summed discrepancy between current and target distances
The stress is calculated as the sum of the squared differences between the distances in the original space and the distances in the reduced space.
Adjustment of configuration is done until:
- STRESS level reaches criterion (convergence)
- Maximum number of iterations has been reached
What is the difference between STRESS 1 AND 2?
STRESS 1 has lower values
What is a very poor fit and a nearly perfect fit according to STRESS 1?
Very poor = 20% STRESS 1
Near perfect = 0% STRESS 1
Procedure of Nonmetric MDS Starting point + 5 steps
Starting point:
- Matrix with the pairwise dissimilarities between objects
Step1:
- Initial configuration: Position objects at random coordinates in an n-dimensional space
Step2:
- Calculate the distance between pairs of objects and identify nonmonotonic pairs.
Step3:
- compute target distance for nonmonotonic pairs to fulfill monotonicity
Step 4:
- Adjust coordinates of the objects to get closer to target distance
- calculate STRESS of revised configuration
Step5:
- repeat step 4 until convergence
How to determine how many dimensions to use?
- usally 2-3 so you can plot visually
- you can use a scree plot
- you can use the Q coefficient
What value should the Q coefficient have?
> = 2
