PERCEPTUAL
MAPPING USING
DISCRIMINANT
ANALYSIS
Presented By
Ritika Gupta
Mani kant roy
Atharva deshpandey
Shweta parashar
What is Perceptual Map
• A diagrammatic technique used by
marketers to visualize (potential) customers’
perceptions and opinions about products,
product lines or brands.
• A perceptual map can also be used to
represent a company’s situation relevant to
the competition.
• Perceptual mapping is also called
positioning mapping which helps you to
develop a market positioning strategy for
your product or service
How perceptual mapping can help
01
02
03
04
Understand the
thought and
behavior of the
customer.
Understand
competitor
Track market
trend
Identify the
white space for
company to
operate
Perceptual
mapping of
car brand
How to Create
Perceptual
Mapping
Step 1: Select
Attributes.
Step 2: Set
Dimensions.
Step 3: Decide
Products/Brands To
Map.
Step 4: Conduct a
Market research.
Step 5: Identify the
attribute which
effects more using
discriminate
analysis.
Step 6 : Plot the
map depending on
attribute and
dimensions.
Types of perceptual mapping
1. Standard Perceptual
Map
2. Multi-Dimensional
Scaled Perceptual Map
Standard
Perceptual Map
The standard perceptual map has
two-axis and visualizes collected
data to communicate key findings. In
other words, a standard perceptual
map highlights the key findings from
a survey quickly with minimum
hassle.
Multi-dimensional
perceptual map
Multi-dimensional maps have more
than two axes with various attributes.
The items are plotted on the map
based on their closeness to the vector
representing a certain product or brand
feature.
Case problem
A marketing researcher wishes to determine the present position of four leading brands of wrist
watches (Casio, Citizen, HMT and Titan) in the minds of the consumer and how these brands measure up on different attributes.
Input Data
The brand of the wristwatch is taken as a dependent variable for analysis and has been coded as a brand. The four brands of wristwatches
chosen for study and their codes are as shown below.
• Casio
• Citizen
• HMT
• Titan
The attributes chosen for analysis and their codes are as follows:
•
•
•
•
•
•
Digital Display- digidisp
Waterproff Feature- wetproof
Provision for alarm
Provision for date- date
Use of jewels in the watch- jewels
Provision of a metal band- metband
Twenty respondents were asked to rate the four brands listed above (each brand assessed by five respondents) on the above-mentioned
attributes on a five-point scale (1 Very bad to 5 Very good). The data collected is as shown in Table 1.1. This data was entered into the SPSS
package and a Discriminant Analysis carried out.
Brand
1.
Casio
2.
Citizen
3.
HMT
4.
Titan
• Total Datapoints(N) = 20
• Total I.V. = 6
Digidisp
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
4
3
4
4
3
3
2
3
3
2
2
2
3
2
2
3
3
3
4
3
Watproof Alarm
4
3
4
4
3
3
3
3
4
2
3
3
3
3
2
3
3
3
2
4
Date
2
3
3
3
2
3
3
2
3
3
2
2
3
2
3
3
3
3
2
3
Jewels
3
3
4
4
3
2
4
3
3
2
2
2
2
2
3
3
3
2
4
3
2
2
3
2
3
4
4
4
3
4
4
3
4
4
4
4
4
4
3
4
Metband
3
2
2
2
3
4
3
4
4
3
4
4
3
3
4
4
4
4
3
3
Group Statistics
Brands
Digidisp Watproof
Casio
3.6
3.6
Citizen
2.6
3
HMT
2.2
2.8
Titan
3.2
3
Total AVG
2.9
3.1
Casio
Citizen
HMT
Titan
Total STD
0.547723
0.547723
0.447214
0.447214
0.718185
0.547723
0.707107
0.447214
0.707107
0.640723
Mean
Alarm
Date
Jewels
Metband
2.6
3.4
2.4
2.4
2.8
2.8
3.8
3.6
2.4
2.2
3.8
3.6
2.8
3
3.8
3.6
2.65
2.85
3.45
3.3
0.547723 0.547723 0.547723 0.547723
0.447214 0.83666 0.447214 0.547723
0.547723 0.447214 0.447214 0.547723
0.447214 0.707107 0.447214 0.547723
0.48936 0.74516 0.759155 0.732695
ANOVA
Sum of
Squares
Digidisp
Watproof
Tests of
equality of
group means
Alarm
Date
Jewels
Metband
Mean
Square
df
Between
Groups
Within
Groups
Total
5.800
3
1.933
4.000
16
0.250
9.800
19
Between
Groups
Within
Groups
Total
1.800
3
0.600
6.000
16
0.375
7.800
19
Between
Groups
Within
Groups
Total
0.550
3
0.183
4.000
16
0.250
4.550
19
Between
Groups
Within
Groups
Total
3.750
3
1.250
6.800
16
0.425
10.550
19
7.350
3
2.450
3.600
16
0.225
10.950
19
5.400
3
1.800
4.800
16
0.300
10.200
19
Between
Groups
Within
Groups
Total
Between
Groups
Within
Groups
Total
Wilks
Lambda
F
Sig.
0.408
7.733
0.002
0.769
1.600
0.229
0.879
0.733
0.547
0.645
2.941
0.065
0.329
10.889
0.000
0.471
6.000
0.006
Steps to calculate One-Way Anova
• SSW (Sum of Squares within Groups) : The is the sum of squares within each group. We can
calculate it using std deviation of each and summing them across all groups as below
Summation over all groups [std.dev(s)^2 * (n-1)]
• SSB (Sum of Squares between Groups) : This is the sum of squares with the groups taken as
single elements.
• SST(Sum of squares Total) = SSB + SSW
• Degree of freedom Within (dfw) = N – k (N = Total obs , k = no.of groups)
• Degree of freedom between (dfb) = k-1
• F = (SSB/dfb) / ( SSW/dfw)
Interpretations from One way Anova
• Only Digidisp , Metband and Jewels Independent Variables are significant having significance less
than 0.05 . Also note they have high F value and low Wilks Lambda (less than 0.5)
Eigen Values
V= 3.5371
0
0
0
0.9880
0
0
0
0
0
0
0
0
0
0.0783
0
0
0
0
0
0
0.0000
0
0
0
0
0
0
0
0
0
0
0.0000 0
0
0.0000
Command
W=w1’*w1+ w2’*w2 +w3’*w3+ w4’*w4
B=T-W
X=inv(W)*B
[u,v]=eig(X)
Linear Discriminant Coefficients
Command
X=inv(W)*B
X*a=Lambda*a
Get three linear eqn-1
Use cubic formula
a’*Spooled*a=1-2
Using 1 and 2 we get the solution
LD1
LD2
Digidisp
0.446259 1.687925
Watproof
0.348486
Alarm
-0.08683 0.930529
Date
Jewels
-0.04719 0.638264
-1.39837 0.73948
Metband
Constant
-1.07499 0.752047
6.362
-13.125
-0.35087
Brand 1
2.900819 -0.14247
Brand 2
Brand 3
-0.99129 0.121001
-1.17644 -1.23916
Brand 4
-0.73297 1.261409
Substituting Group means in
unstandardized discriminant
functions we get the following
centroid values which we plot
on 2-D scale
Discriminant Loadings
S = Pooled Variance Matrix
We find discriminant loadings
which gives us the correlation
between discriminant
function and independent
variables
0.2500 0.0875 -0.0250 0.0750 -0.0750 -0.0125
0.0875 0.3750 0.0375 -0.0000 -0.0250 0.0250
-0.0250 0.0375 0.2500 0.0125 0.0500 -0.0750
0.0750 -0.0000 0.0125 0.4250 -0.0500 -0.1125
-0.0750 -0.0250 0.0500 -0.0500 0.2250
0
-0.0125 0.0250 -0.0750 -0.1125
0 0.3000
loadings=S_d*b
Taking sqrt of diagonals of S = S_d
loadings =
0.5 0.0875
-0.025
0.075
0.0875 0.6124 0.0375
0
-0.025
0.025
0.5 0.0125
0.05
-0.075
-0.025 0.0375
0.075
0 0.0125 0.6519
-0.075
-0.025
-0.0125
0.025
0.05
-0.075 -0.0125
-0.05 -0.1125
-0.05 0.4743
-0.075 -0.1125
0
0 0.5477
0.3706
0.2573
-0.0314
0.1925
-0.7074
-0.5738
0.773
-0.032
0.3985
0.4327
0.2475
0.2404
Perceptual Mapping
Loadings
Digidisp
LD1
Attribute
LD2
F
DIGIDISP
7.733
0.3671
0.7765
0.2196
-0.0514
-0.0384
0.4055
ALARM
0.733
0.189
0.4362
DATE
2.941
Jewels
-0.7039
0.244
Metband
-0.5668
0.2334
Attribute vectors are obtained by multiplying each attribute
discriminant loadings with corresponding F value for that variable Watproof
Alarm
obtained earlier
Date
Attribute
WATPROOF
JEWELS
METBAND
Function1
1.6
10.898
6
Function2
DIGIDISP
2.838784
2.3295
WATPROOF
ALARM
0.35136
-0.02815
-0.1542
1.2165
DATE
JEWELS
METBAND
0.555849
-7.6711
-3.4008
1.3086
0.732
0.7002
Varimax- Worked Out Manually
Below are the rotated
Attribute
Function1
Function2
DIGIDISP
2.838784
2.3295
WATPROOF
ALARM
0.35136
-0.02815
-0.1542
1.2165
DATE
JEWELS
METBAND
0.555849
-7.6711
-3.4008
1.3086
0.732
0.7002
Case Study 2 Televisions
• Problem
A consumer durables firm is interested in evaluating
the consumer perceptions about television sets
of three different brands - Samsung, Thomson and
LG. The consumers were interviewed on
eight different parameters and were asked to rank
them on an 8-point scale.
Sample Datapoints
Fuzzy
Weight Brand
Logic
Price Remote Colour Screen Audio Internet
Below is part of datapoints that were collected in
survey.
4
4
4
5
6
3
3
4
1
4
5
3
3
4
5
3
5
1
20 participants are asked to rate each of the 3 brands.
So we have :
3
6
4
3
5
5
4
4
1
3
7
2
4
4
4
3
5
1
Total Datapoints (N) = 60
3
6
3
4
3
6
5
5
1
4
6
2
4
4
2
6
4
1
4
6
3
4
5
4
6
4
1
Datapoints in each Group (n)= 20
Total Independent Variable (p) = 8
No. of groups (g) = 3
Group Statistics
This table presents the distribution of observations into
the three groups within "brand".
Here there are equal number of datapoints in each
group and default weight of each datapoint is taken
as 1.
We can see screen and color variables have high
standard deviation as compared to other observed
variables.
One-Way Anova for each Independent Variable
The tests of equality of group means measure each
independent variable's potential before the model is
created.
•
Higher Values of F-statistic means the i.v.
discriminates well , here "Color" variable has highest
F value and "Remote" variable has lowest F- value
•
Significance value is greater than 0.10, the variable
probably does not contribute to the model - "Remote"
and "Audio" non-contributing
•
Wilks' lambda is another measure of a variable's
potential. Smaller values indicate the variable is better
at discriminating between groups.
Summary of Canonical Discriminant Functions
• These eigenvalues are related to the canonical
correlations and describe how much discriminating
ability a function possesses. The magnitudes of the
eigenvalues are indicative of the functions’
discriminating abilities.
• Wilks’ Lambda is one of the multivariate statistic
calculated by SPSS. It is the product of the values of
(1-canonical correlation2). In this example, our
canonical correlations are .870 and 0.658, so the Wilks’
Lambda testing both canonical correlations is (10.8702)*(1-0.6582) =0.138 and the Wilks’ Lambda
testing the second canonical correlation is (1-0.6582) =
0.567.
Summary of Canonical Discriminant Functions
•
The magnitudes of Standardized Canonical Discriminant Function coefficients indicate how strongly the
discriminating variables effect the score. In 1st function "Color" has highest value(0.964) and thus, will
have the greatest impact of the 8 on the first discriminant score.
Plot of Brands on Discriminant Functions
• We find each discriminant function points for each group centroids by replacing group variable means in
functions.
• We then plot each datapoint function values on the map with dimensions of discriminant function as well
as Group centroids for each brand .
Perceptual Map of Television Brands & Attributes
• To plot the attributes on the map we can
use the standardised coefficients of the
original variables in the discriminant
function. For "Price" the co-ordinates are (0.402,-.436) . X-axis –Function 1, Y-axis –
Function 2.
• These vectors represent the effect of
discriminating on each dimension.
• Longer arrows pointing more closely
towards a given group centroid, represent
variables most strongly associated with
that particular group (or Brand, in this
case).
• Vectors pointing in the opposite
direction from a given group centroid
represent lower association with the
concerned group
Perceptual Map of Television Brands & Attributes
• Samsung, Thomson and LG have their
unique positions on the map. In addition,
on the same map,
• Dimension 1 seems to be a combination
of Colour and Weight (closest to the xaxis)
• Dimension 2 seems to comprise Screen
(size), Internet the vector that is closest
to the vertical axis.