16th September
Discriminant Analysis
Discriminant Analysis may be used
for two objectives: either we want to assess the adequacy of classification, given
the group memberships of the objects under study; or we wish to assign objects to one of a number of (known)
groups of objects. Discriminant Analysis may thus have a descriptive or a
predictive objective.
In both cases, some group assignments must be known
before carrying out the Discriminant Analysis. Such group assignments, or
labelling, may be arrived at in any way. Hence Discriminant Analysis can be
employed as a useful complement to Cluster Analysis (in order to judge the
results of the latter) or Principal Components Analysis.
Methods implemented in this area
are Multiple Discriminant Analysis, Fisher's Linear Discriminant Analysis, and
K-Nearest Neighbours Discriminant Analysis.
Multiple Discriminant Analysis
(MDA) is also
termed Discriminant Factor Analysis and Canonical
Discriminant Analysis. It adopts a similar perspective to PCA: the rows of the
data matrix to be examined constitute points in a multidimensional space, as
also do the group mean vectors. Discriminating axes are determined in this
space, in such a way that optimal separation of the predefined groups is
attained. As with PCA, the problem becomes mathematically the eigenreduction of
a real, symmetric matrix. The eigenvalues represent the discriminating power of
the associated eigenvectors. The nYgroups
lie in a space of dimension at most nY - 1. This will be the number of
discriminant axes or factors obtainable in the most common practical case when n > m > nY (where n is the number of rows, and m the number of columns of the input
data matrix).
Linear Discriminant Analysis
is the 2-group
case of MDA. It optimally separates two groups,
using the Mahalanobis metric or generalized
distance. It also gives the same linear
separating decision surface as Bayesian maximum likelihood discrimination in
the case of equal class covariance matrices.
K-NNs Discriminant Analysis
Non-parametric (distribution-free)
methods dispense with the need for assumptions regarding the probability
density function. They have become very popular especially in the image
processing area. The K-NNs method assigns an object of unknown affiliation to
the group to which the majority of its K nearest neighbour’s belongs.
The major underlying assumptions of
DA are:
- the observations are a random sample;
- each predictor variable is normally distributed;
- each of the allocations for the dependent categories in the initial classifi cation are correctly classified;
- there must be at least two groups or categories, with each case belonging to only one group so that the groups are mutually exclusive and collectively exhaustive (all cases can be placed in a group)
Conjoint Analysis
Conjoint analysis is a popular
marketing research technique that marketers use to determine what features a
new product should have and how it should be priced. Conjoint analysis became popular because it
was a far less expensive and more flexible way to address these issues than
concept testing.
Conjoint analysis has been successfully applied in many
industries, such as Air Travel, Smart Phones, Computers, Financial Services,
Health Care, Real Estate, and Electronics. If a job includes configuring a
defined set of features for a product or service and the consumer’s purchase
decision will be “rational,” conjoint analysis can help. If, on the other hand,
one has to decide if a consumer’s purchase decision will be “impulse” or
“image,” conjoint is not the right tool.
Because conjoint analysis helps one understand market’s
preferences, it can be applied to a variety of difficult aspects of the job,
including product development, competitive positioning, pricing, product line
analysis, segmentation and resource allocation. “How should we price our new
product to maximize adoption?” “What features should we include in our next
release to take market share from our competition?” “If we expand our product
line, will overall revenue grow, or will we suffer too much cannibalization?”
“For which value-added features is the market willing to pay?”
Conjoint (trade-off) analysis has become one of the most
widely-used quantitative methods in Marketing Research. It is used to measure
the perceived values of specific product features, to learn how demand for a
particular product or service is related to price, and to forecast what the
likely acceptance of a product would be if brought to market.
Rather than directly ask survey respondents what they
prefer in a product, or what attributes they find most important, conjoint
analysis employs the more realistic context of respondents evaluating potential
product profiles by giving them a variety of product details and arriving at a
score.
Kanya Patil (14021)
Urvashi Singh (14057)
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