Day 5 - Group B

The 2 sessions on Day 5 were focused on two important parameters.

1.       Understanding of Multi-dimensional Scaling (MDS)

2.       Usage of Permap which is the software used for Perceptual mapping

Multi-dimensional Scaling (MDS)

Multidimensional scaling (MDS) is a set of related statistical techniques often used in information visualization for exploring similarities or dissimilarities in data. MDS is a special case of ordination. An MDS algorithm starts with a matrix of item–item similarities and then assigns a location to each item in N-dimensional space, where N is specified a priori. For sufficiently small N, the resulting locations may be displayed in a graph or 3D visualisation.

The different types of MDS are:

1.       Classical multidimensional scaling: Also known as Principal Coordinates Analysis, Torgerson Scaling or Torgerson–Gower scaling. Takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain.

2.       Metric multidimensional scaling: A superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization.

3.       Non-metric multidimensional scaling: In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression.

4.       Generalized multidimensional scaling: An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In case when the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another.

The Multi-dimensional scaling is done to understand the following points:

1.       Overall similarity between the parameters: this gives us the brief idea as how similar/dissimilar are the pair of objects that are compared. If the distance between two objects in a matrix is 1 then the matrix is called as the similarity matrix. If the distance between two objects in a matrix is 0 then the matrix is called as the dissimilarity matrix. All the overall attributes are considered.

2.       Attribute based: Here, the ranking on basis of attributes to objects is done. For e.g. if we need to understand the similarity between Coke and Pepsi, then the parameters that we might cover are taste, packaging, colour and so on. Any type of distances can be used like the probability, Euclidean, etc.

However, the disadvantage here is that we might miss a few attributes and this might not give us the accurate data.

               

Permap

It is the map that uses MDS to reduce multiple pairwise relationships to 2-D pictures commonly called perceptual map. The purpose of Permap is to provide a perfectly convenient way of producing perceptual maps and to do so in a way that helps researcher avoid a number of common mistakes.

A major advantage with Permap is that it deals with the problems associated with substantiating and communicating results based on data evolving in more than two dimensions. It is also forgiving of missing of imprecise data points. The basic look of the software is mentioned below.

We can understand the working of Permap with an example. We will find out the perceptual mapping of soft drinks.

Here, the respondents have replied on a scale of 0-9 (high similarity will get high marks and vice versa) for comparison of Coke, Pepsi, Slice, Maaza, Mountain Dew and Sprite.

After the responses, these were put in the table and average score of each combination was calculated. The number of combinations will be n*(n-1)/2. That is 6*5/2 which is 15.

 The matrix format is mentioned below:

title = perception of soft drinks

nobjects = 6

similaritylist

Coke      1                                                                             

Pepsi     0.82        1                                                             

Slice       0.08        0.08        1                                             

Maaza   0.16        0.18        0.86        1                             

Mt Dew                0.4          0.48        0.26        0.2          1             

Sprite    0.46        0.36        0.2          0.18        0.74        1

When this is fed to the Permap software, the result is as shown below:

   

According to the responses, following conclusions can be recorded:

1.       Pepsi and Coke have similar attributes.

2.       Mountain Dew and Slice have similar attributes.

3.       Maaza and Slice have similar attributes.

Attributes that observed were:

1.       Taste/flavour

2.       Colour

3.       Fizz

4.       Content of the drink

5.       Packaging

6.       Smell

7.       consistency