Linear Discriminant Analysis- Submitted by Partha Pratim Das (Roll no.-12036)

Linear discriminant analysis (LDA) and the related Fisher's linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate two or more classes of objects or events. The resulting combination may be used as a linear classifier or, more commonly, for dimensionality reduction before later classification.

LDA is closely related to ANOVA (analysis of variance) and regression analysis, which also attempt to express one dependent variable as a linear combination of other features or measurements. In the other two methods however, the dependent variable is a numerical quantity, while for LDA it is a categorical variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA, as they also explain a categorical variable. These other methods are preferable in applications where it is not reasonable to assume that the independent variables are normally distributed, which is a fundamental assumption of the LDA method.

LDA is also closely related to principal component analysis (PCA) and factor analysis in that both look for 

linear combinations of variables which best explains the data. LDA explicitly attempts to model the difference between the classes of data. PCA on the other hand does not take into account any difference in class, and factor analysis builds the feature combinations based on differences rather than similarities. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made.

LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is discriminant correspondence analysis

Applications

Bankruptcy prediction

In bankruptcy prediction based on accounting ratios and other financial variables, linear discriminant analysis was the first statistical method applied to systematically explain which firms entered bankruptcy vs. survived. Despite limitations including known non-conformance of accounting ratios to the normal distribution assumptions of LDA, Edward Altman's 1968 model is still a leading model in practical applications.

Face recognition

In computerised face recognition, each face is represented by a large number of pixel values. Linear discriminant analysis is primarily used here to reduce the number of features to a more manageable number before classification. Each of the new dimensions is a linear combination of pixel values, which form a template. The linear combinations obtained using Fisher's linear discriminant are called Fisher faces, while those obtained using the related principal component analysis are called eigenfaces.

Marketing

In marketing, discriminant analysis was once often used to determine the factors which distinguish different types of customers and/or products on the basis of surveys or other forms of collected data. Logistic regression or other methods are now more commonly used. The use of discriminant analysis in marketing can be described by the following steps:

Formulate the problem and gather data - Identify the salient attributes consumers use to evaluate products in this category - Use quantitative marketing research techniques (such as surveys) to collect data from a sample of potential customers concerning their ratings of all the product attributes. The data collection stage is usually done by marketing research professionals. Survey questions ask the respondent to rate a product from one to five (or 1 to 7, or 1 to 10) on a range of attributes chosen by the researcher. Anywhere from five to twenty attributes are chosen. They could include things like: ease of use, weight, accuracy, durability, colourfulness, price, or size. The attributes chosen will vary depending on the product being studied. The same question is asked about all the products in the study. The data for multiple products is codified and input into a statistical program such as R, SPSS or SAS. (This step is the same as in Factor analysis).

Estimate the Discriminant Function Coefficients and determine the statistical significance and validity - Choose the appropriate discriminant analysis method. The direct method involves estimating the discriminant function so that all the predictors are assessed simultaneously. The stepwise method enters the predictors sequentially. The two-group method should be used when the dependent variable has two categories or states. The multiple discriminant method is used when the dependent variable has three or more categorical states. Use Wilks’s Lambda to test for significance in SPSS or F stat in SAS. The most common method used to test validity is to split the sample into estimation or analysis sample, and a validation or holdout sample. The estimation sample is used in constructing the discriminant function. The validation sample is used to construct a classification matrix which contains the number of correctly classified and incorrectly classified cases. The percentage of correctly classified cases is called the hit ratio.

Plot the results on a two dimensional map, define the dimensions, and interpret the results. The statistical program (or a related module) will map the results. The map will plot each product (usually in two dimensional spaces). The distance of products to each other indicate either how different they are. The dimensions must be labelled by the researcher. This requires subjective judgement and is often very challenging. See perceptual mapping.

In addition to the examples given below, LDA is applied in positioning and product management.

Source: www.wikipedia.org

Discriminate Analysis

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. Alternatively, in star-galaxy separation, for instance, using digitised images, the analyst may define group (stars, galaxies) membership visually for a conveniently small training set or design set.

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 n_Ygroups lie in a space of dimension at most n_Y - 1. This will be the number of discriminant axes or factors obtainable in the most common practical case when n > m > n_Y (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 neighbours belongs.

There is no best discrimination method. A few remarks concerning the advantages and disadvantages of the methods studied are as follows.

• Analytical simplicity or computational reasons may lead to initial consideration of linear discriminant analysis or the NN-rule.

• Linear discrimination is the most widely used in practice. Often the 2-group method is used repeatedly for the analysis of pairs of multigroup data (yielding decision surfaces for k groups).

• To estimate the parameters required in quadratic discrimination more computation and data is required than in the case of linear discrimination. If there is not a great difference in the group covariance matrices, then the latter will perform as well as quadratic discrimination.

• The k-NN rule is simply defined and implemented, especially if there is insufficient data to adequately define sample means and covariance matrices.

• MDA is most appropriately used for feature selection. As in the case of PCA, we may want to focus on the variables used in order to investigate the differences between groups; to create synthetic variables which improve the grouping ability of the data; to arrive at a similar objective by discarding irrelevant variables; or to determine the most parsimonious variables for graphical representational purposes.

Blog by_ Akhilesh Agarwal (Operations)

Factor Analysis in Marketing ( By- Karanbir Singh,12141)

• Overview
In essence, marketing factor analysis is changing one marketing variable to see what affect, if any, the change has on the outcome. The change in sales also affects the bottom line of the company, so factor analysis in marketing helps companies determine which marketing efforts it should pursue, which efforts need work and which marketing efforts may meet the cutting room floor.
• Significance
Ffactor analysis in marketing requires an evaluation of how changing one marketing point, such as price, changes the sales of the product. In order to measure how the factor changes the sales, it requires that only one marketing variable is changed at a time in order to measure the relationship between the variables and the outcome. In marketing, changing one variable can be significant because it may cause an increase or decrease the sales of the product.
• Variables
An infinite number of marketing variables can exist, which is why it’s necessary to alter only one at a time. Marketing variables include the product, the product packaging, the size of the product and the color of the product. The price, distribution channels and marketing strategies may also be variables of the product that can be changed to see how the change makes a difference in the sales of the product.
• What Customers Want
Factor analysis in marketing is important because it reflects the perception of the buyer of the product. By testing variables, it is possible for marketing professionals to determine what is important to the customers of the product. For example, if a product is only available in black and the sales reach $150,000 in one year, but when the company adds color options of red, blue and silver, and sales reach $300,000, then the company can conclude through factor analysis that color options are important to the customers of the product. Ultimately, it is imperative to use factor analysis in marketing to create the ideal product for customers, which in turn, increases the sales of the product.
• Conducting Factor Analysis
Generally, companies test variables with factor analysis in marketing using tools such as focus groups and surveys. Since making changes to the product itself in order to test variables can be expensive, surveys and focus groups allows companies to gather pertinent information without increasing the cost to manufacture the product. Focus groups and surveys allow companies to gather perceptual information from current and potential customers of the product. This information is important because it allows the company to see the product from the vantage point of the customers and determine which factors in marketing are the most important to the customers. For example, a focus group may see four different package versions of a product and then ask the focus group participants to choose which package they like best and explain why. Companies can use this information to alter product packaging to attract more customers and sell more products.
• Misconceptions
Factor analysis in marketing can change the way a company sells its product, but factor analysis in marketing is not a science. Grand scale testing is typically required to come to a fair conclusion on the relationship of the variables. It is wise to have a market research expert help to conduct the factor analysis and evaluate the test results to ensure it is not an incorrectly implied cause/effect relationship.

Factor Analysis

Factor analysis includes both component analysis and common factor analysis. More than other statistical techniques, factor analysis has suffered from confusion concerning its very purpose. This affects my presentation in two ways. First, I devote a long section to describing what factor analysis does before examining in later sections how it does it. Second, I have decided to reverse the usual order of presentation. Component analysis is simpler, and most discussions present it first. However, I believe common factor analysis comes closer to solving the problems most researchers actually want to solve. Thus learning component analysis first may actually interfere with understanding what those problems are. Therefore component analysis is introduced only quite late in this chapter.
What Factor Analysis Can and Can't Do
I assume you have scores on a number of variables-- anywhere from 3 to several hundred variables, but most often between 10 and 100. Actually you need only the correlation or covariance matrix--not the actual scores. The purpose of factor analysis is to discover simple patterns in the pattern of relationships among the variables. In particular, it seeks to discover if the observed variables can be explained largely or entirely in terms of a much smaller number of variables called factors.
Some Examples of Factor-Analysis Problems
1. Factor analysis was invented nearly 100 years ago by psychologist Charles Spearman, who hypothesized that the enormous variety of tests of mental ability--measures of mathematical skill, vocabulary, other verbal skills, artistic skills, logical reasoning ability, etc.--could all be explained by one underlying "factor" of general intelligence that he called g. He hypothesized that if g could be measured and you could select a subpopulation of people with the same score on g, in that subpopulation you would find no correlations among any tests of mental ability. In other words, he hypothesized that g was the only factor common to all those measures.
It was an interesting idea, but it turned out to be wrong. Today the College Board testing service operates a system based on the idea that there are at least three important factors of mental ability--verbal, mathematical, and logical abilities--and most psychologists agree that many other factors could be identified as well.
2. Consider various measures of the activity of the autonomic nervous system--heart rate, blood pressure, etc. Psychologists have wanted to know whether, except for random fluctuation, all those measures move up and down together--the "activation" hypothesis. Or do groups of autonomic measures move up and down together, but separate from other groups? Or are all the measures largely independent? An unpublished analysis of mine found that in one data set, at any rate, the data fitted the activation hypothesis quite well.
3. Suppose many species of animal (rats, mice, birds, frogs, etc.) are trained that food will appear at a certain spot whenever a noise--any kind of noise--comes from that spot. You could then tell whether they could detect a particular sound by seeing whether they turn in that direction when the sound appears. Then if you studied many sounds and many species, you might want to know on how many different dimensions of hearing acuity the species vary. One hypothesis would be that they vary on just three dimensions--the ability to detect high-frequency sounds, ability to detect low-frequency sounds, and ability to detect intermediate sounds. On the other hand, species might differ in their auditory capabilities on more than just these three dimensions. For instance, some species might be better at detecting sharp click-like sounds while others are better at detecting continuous hiss-like sounds.
4. Suppose each of 500 people, who are all familiar with different kinds of automobiles, rates each of 20 automobile models on the question, "How much would you like to own that kind of automobile?" We could usefully ask about the number of dimensions on which the ratings differ. A one-factor theory would posit that people simply give the highest ratings to the most expensive models. A two-factor theory would posit that some people are most attracted to sporty models while others are most attracted to luxurious models. Three-factor and four-factor theories might add safety and reliability. Or instead of automobiles you might choose to study attitudes concerning foods, political policies, political candidates, or many other kinds of objects.
5. Rubenstein (1986) studied the nature of curiosity by analyzing the agreements of junior-high-school students with a large battery of statements such as "I like to figure out how machinery works" or "I like to try new kinds of food." A factor analysis identified seven factors: three measuring enjoyment of problem-solving, learning, and reading; three measuring interests in natural sciences, art and music, and new experiences in general; and one indicating a relatively low interest in money.
The Goal: Understanding of Causes
Many statistical methods are used to study the relation between independent and dependent variables. Factor analysis is different; it is used to study the patterns of relationship among many dependent variables, with the goal of discovering something about the nature of the independent variables that affect them, even though those independent variables were not measured directly. Thus answers obtained by factor analysis are necessarily more hypothetical and tentative than is true when independent variables are observed directly. The inferred independent variables are called factors. A typical factor analysis suggests answers to four major questions:
1. How many different factors are needed to explain the pattern of relationships among these variables?
2. What is the nature of those factors?
3. How well do the hypothesized factors explain the observed data?
4. How much purely random or unique variance does each observed variable include?
I illustrate these questions later.
Absolute Versus Heuristic Uses of Factor Analysis
A heuristic is a way of thinking about a topic which is convenient even if not absolutely true. We use a heuristic when we talk about the sun rising and setting as if the sun moved around the earth, even though we know it doesn't. "Heuristic" is both a noun and an adjective; to use a heuristic is to think in heuristic terms.
The previous examples can be used to illustrate a useful distinction--between absolute and heuristic uses of factor analysis. Spearman's g theory of intelligence, and the activation theory of autonomic functioning, can be thought of as absolute theories which are or were hypothesized to give complete descriptions of the pattern of relationships among variables. On the other hand, Rubenstein never claimed that her list of the seven major factors of curiosity offered a complete description of curiosity. Rather those factors merely appear to be the most important seven factors--the best way of summarizing a body of data. Factor analysis can suggest either absolute or heuristic models; the distinction is in how you interpret the output.
Is Factor Analysis Objective?
The concept of heuristics is useful in understanding a property of factor analysis which confuses many people. Several scientists may apply factor analysis to similar or even identical sets of measures, and one may come up with 3 factors while another comes up with 6 and another comes up with 10. This lack of agreement has tended to discredit all uses of factor analysis. But if three travel writers wrote travel guides to the United States, and one divided the country into 3 regions, another into 6, and another into 10, would we say that they contradicted each other? Of course not; the various writers are just using convenient ways of organizing a topic, not claiming to represent the only correct way of doing so. Factor analysts reaching different conclusions contradict each other only if they all claim absolute theories, not heuristics. The fewer factors the simpler the theory; the more factors the better the theory fits the data. Different workers may make different choices in balancing simplicity against fit.
A similar balancing problem arises in regression and analysis of variance, but it generally doesn't prevent different workers from reaching nearly or exactly the same conclusions. After all, if two workers apply an analysis of variance to the same data, and both workers drop out the terms not significant at the .05 level, then both will report exactly the same effects. However, the situation in factor analysis is very different. For reasons explained later, there is no significance test in component analysis that will test a hypothesis about the number of factors, as that hypothesis is ordinarily understood. In common factor analysis there is such a test, but its usefulness is limited by the fact that it frequently yields more factors than can be satisfactorily interpreted. Thus a worker who wants to report only interpretable factors is still left without an objective test.
A similar issue arises in identifying the nature of the factors. Two workers may each identify 6 factors, but the two sets of factors may differ--perhaps substantially. The travel-writer analogy is useful here too; two writers might each divide the US into 6 regions, but define the regions very differently.
Another geographical analogy may be more parallel to factor analysis, since it involves computer programs designed to maximize some quantifiable objective. Computer programs are sometimes used to divide a state into congressional districts which are geographically continguous, nearly equal in population, and perhaps homogeneous on dimensions of ethnicity or other factors. Two different district-creating programs might come up with very different answers, though both answers are reasonable. This analogy is in a sense too good; we believe that factor analysis programs usually don't yield answers as different from each other as district-creating programs do.
Factor Analysis Versus Clustering and Multidimensional Scaling
Another challenge to factor analysis has come from the use of competing techniques such as cluster analysis and multidimensional scaling. While factor analysis is typically applied to a correlation matrix, those other methods can be applied to any sort of matrix of similarity measures, such as ratings of the similarity of faces. But unlike factor analysis, those methods cannot cope with certain unique properties of correlation matrices, such as reflections of variables. For instance, if you reflect or reverse the scoring direction of a measure of "introversion", so that high scores indicate "extroversion" instead of introversion, then you reverse the signs of all that variable's correlations: -.36 becomes +.36, +.42 becomes -.42, and so on. Such reflections would completely change the output of a cluster analysis or multidimensional scaling, while factor analysis would recognize the reflections for what they are; the reflections would change the signs of the "factor loadings" of any reflected variables, but would not change anything else in the factor analysis output.
Another advantage of factor analysis over these other methods is that factor analysis can recognize certain properties of correlations. For instance, if variables A and B each correlate .7 with variable C, and correlate .49 with each other, factor analysis can recognize that A and B correlate zero when C is held constant because .72 = .49. Multidimensional scaling and cluster analysis have no ability to recognize such relationships, since the correlations are treated merely as generic "similarity measures" rather than as correlations.
We are not saying these other methods should never be applied to correlation matrices; sometimes they yield insights not available through factor analysis. But they have definitely not made factor analysis obsolete. The next section touches on this point.
Factors "Differentiating" Variables Versus Factors "Underlying" Variables
When someone says casually that a set of variables seems to reflect "just one factor", there are several things they might mean that have nothing to do with factor analysis. If we word statements more carefully, it turns out that the phrase "just one factor differentiates these variables" can mean several different things, none of which corresponds to the factor analytic conclusion that "just one factor underlies these variables".
One possible meaning of the phrase about "differentiating" is that a set of variables all correlate highly with each other but differ in their means. A rather similar meaning can arise in a different case. Consider several tests A, B, C, D which test the same broadly-conceived mental ability, but which increase in difficulty in the order listed. Then the highest correlations among the tests may be between adjacent items in this list (rAB, rBC and rCD) while the lowest correlation is between items at the opposite ends of the list (rAD). Someone who observed this pattern in the correlations among the items might well say the tests "can be put in a simple order" or "differ in just one factor", but that conclusion has nothing to do with factor analysis. This set of tests would not contain just one common factor.
A third case of this sort may arise if variable A affects B, which affects C, which affects D, and those are the only effects linking these variables. Once again, the highest correlations would be rAB, rBC and rCD while the lowest correlation would be rAD. Someone might use the same phrases just quoted to describe this pattern of correlations; again it has nothing to do with factor analysis.
A fourth case is in a way a special case of all the previous cases: a perfect Guttman scale. A set of dichotomous items fits a Guttman scale if the items can be arranged so that a negative response to any item implies a negative response to all subsequent items while a positive response to any item implies a positive response to all previous items. For a trivial example consider the items
• Are you above 5 feet 2 inches in height?
• Are you above 5 feet 4 inches in height?
• Are you above 5 feet 6 inches in height?
• Etc.
To be consistent, a person answering negatively to any of these items must answer negatively to all later items, and a positive answer implies that all previous answers must be positive. For a nontrivial example consider the following questionnaire items:
• Should our nation lower tariff barriers with nation B?
• Should our two central banks issue a single currency?
• Should our armies become one?
• Should we fuse with nation B, becoming one nation?
If it turned out that these items formed a perfect Guttman scale, it would be easier to describe peoples' attitutes about "nation B" than if they didn't. When a set of items does form a Guttman scale, interestingly it does not imply that factor analysis would discover a single common factor. A Guttman scale implies that one factor differentiates a set of items (e.g, "favorableness toward cooperation with nation B"), not that one factor underlies those items.
Applying multidimensional scaling to a correlation matrix could discover all these simple patterns of differences among variables. Thus multidimensional scaling seeks factors which differentiate variables while factor analysis looks for the factors which underlie the variables. Scaling may sometimes find simplicity where factor analysis finds none, and factor analysis may find simplicity where scaling finds none.

Blog by - Amardeep Singh Randhawa (Operations)

Discriminate Analysis

Descriptive Analysis

Descriptive statistics are used throughout data analysis in a number of different ways. Simply stated, they refer to means, ranges, and numbers of valid cases of one variable.

First, desc

riptive statisti

cs are important in data cleaning. They are regularly used -- generated or reviewed from hard copy -- during analysis to keep an eye on the variables being used, especially when a considerable number are being studied. It is very important to monitor the ‘N’ (number of valid cases) for each variable. If the 'N' differs greatly between variables, consider this an early warning of problems that may arise when the variables

are examined together late

r on. You would then want to track down why cases are being lost between variables. At the same time, out of range values with unexpectedly high (or low) means and standard deviations, and other simple parameters, are looked for. This is illustrated in the examples later in this section.

Second, a typical use of descrip

tive analysis is to produce a situation analysis which usually consists of national or sub-national level information such as land size, population, income, health expenditure, illness, malnutrition, mortality, access to safe water, sanitation, and agricultural production levels, to name a few. This data provides a snap shot of the situation under study. Often the information from the situation analysis comes from a variety of sources such as Multi-Indicator Cluster Surveys, national statistics, and compilations done by organizations such as UNICEF and WHO

1. Producing Descriptive Summary Data Using SPSS

It is fairly simple to produce these types of information from the data sets available in this learning package. Remember that the descriptive analysis can often be presented more accurately for the continuous variables than for categorical variables because of lost information from collapsing it into categories. The linked exercises present continuous variables only and categorical analysis will follow. This outcome shows a possible presentation of a Descriptive Statistics exercise. The descriptive statistics chosen include: N, Minimum, Maximum, Mean, and Standard Deviation. There are numerous other optional descriptives to choose from in the SPSS program.

. Producing Histograms and Examining Frequency Distributions

Histograms are used to simply show the distribution of a quantitative variable by its relative frequency of data points in an interval (in this case the intervals are standard deviations on a z-score distribution). The histogram function in SPSS gives the option of overlaying a normal curve,which helps determine how well the HAZ, WAZ, and WHZ fit the normal distribution, or a bell curve. If there is a strong deviation from the normal distribution or if the distribution mean is off-set far from 0, then this might tell you something about possible problems with the data (e.g. measurement or calculation errors might lead to a non-normal distribution) or also show severe nutrition problems in the population (e.g. a normal distribution with a mean at -1.8 SD). Histograms are most useful for continuous data to see if the distribution is normal and whether there was adequate data cleaning (are there outliers?). Also, histograms are useful for determining arbitrary cut-points for creating categories from continuous data (e.g. creating three levels of income by visually dividing income distribution into three equal parts using the histogram)

INTPRETATION:

This histogram does approximate a normal or Gaussian distribution, as expected for a z-score. The WAZ variable has also evidently been cleaned (all values above and below 5SD have been eliminated in the data cleaning exercise CH 2) and shows a shift in the mean down the scale to -1.29 SD. This would indicate that the average child under 5 years (for which the measurements were taken) is at least mildly underweight in comparison with a normal population (and likely the population has a malnutrition problem).

Now try running an exercise to create a HISTOGRAM for the continuous outcome variables HAZ and WHZ in Eastern Kenya.

If you started with continuous outcome data, in many cases it is useful to convert that data into categorical and report the percentages of the categories representing underweight, stunted and/or wasted children under 5 years of age. As you remember, the WHO cut-point for being classified as at least moderately malnourished is -2.00 standard deviations.

3. Scatterplotting

Although scatterplotting is normally used to examine the association of two continuous variables, it has a restricted practical use here to examine descriptively the scatter of values when the sample size is quite small -- for example prevalences by a few divisions in a national dataset. This gives a first idea as to differences between areas, and whether they are due to a few extreme values. For the whole national dataset (or one area at a time) this is better done with the histogram of frequency distributions we saw in the previous section. But scatterplotting is so easy and quick that it’s convenient even when the classifying variable is categorical and unordered, like administrative divisions. An example from the Bangladesh district data looks like this (the point labels are the district numbers from the data set).

INTERPRETATION: The scatterplot shows that there is quite a lot of variability within the divisions, for the range of prevalences for the districts has a broad range. For instance, Division 1 has 14 districts that range from approximately 8% low arm circumference in <5>

Blog By: Vivek Sarmal (12114_Operations)

Perceptual Mapping

Perceptual Mapping is a data analysis tool that is used to represent the varying relationships that exist between one or many marketplace competitors. The mapping attempts to measure the factors that consumers use in making their purchase decisions. Using a perceptual map has become a popular tool for data analysts to use when briefing upper management on complex issues because it can represent complex relationships graphically.

The map has the following characteristics:

1. Pair-wise distances between product alternatives directly indicate how close or far apart the  

    products are in the minds of customers

2. A vector on the map indicates both magnitude and direction in the Euclidean space. Vectors 

    are usually used to geometrically denote attributes of the perceptual maps

3. The axes of the map are a special set of vectors suggesting the underlying dimensions that 

     best characterize how customers differentiate between alternatives

Submitted By:

Vaibhav Khaparde

12054

Marketing

What is Cluster ? Types of Clustering ? Process of Clustering?

Cluster Analysis 

There are two major methods of clustering

n  hierarchical clustering

n  k-means clustering.

Hierarchical clustering

In hierarchical clustering the data are not partitioned into a particular cluster in a single step. Instead, a series of partitions takes place, which may run from a single cluster containing all objects to n clusters each containing a single object.

Hierarchical clustering may be represented by a two dimensional diagram known as dendogram which illustrates the fusions or divisions made at each successive stage of analysis.

Divisive Clustering(hierarchical): All the objects which we want to cluster are submerged into group and then dividing them into particular blocks.

Agglomerative methods 

An agglomerative hierarchical clustering procedure produces a series of partitions of the data, say from 1 to n. The first number say ‘An’ consists of n single object 'clusters', the last A1, consists of single group. While clustering the numbers joins together the two clusters at each iteration which are closest together (most similar). Then the other group which are similar they then the other group and form another cluster.

Euclidean distance: Short distance from two points.

Higher number = more proximity=similarity=1

Lower number=more proximity=dissimilarity=0

An example of how Cluster forms of different mobile services and offers? Funused0-funused9 determine the offers and facilities of mobile services.

Posted By

Tushar Sonkusare

Roll no 12171 (marketing)

Cluster analysis for Marketing ( Saurabh Trivedi ,12046 )

Cluster analysis' is a class of statistical techniques that can be applied to data that exhibit “natural” groupings. Cluster analysis sorts through the raw data and groups them into clusters. A cluster is a group of relatively homogeneous cases or observations. Objects in a cluster are similar to each other. They are also dissimilar to objects outside the cluster, particularly objects in other clusters.

In Marketing, Cluster analysis can be used for:-
 Segmenting the market and determining target markets
 Product positioning and New Product Development
 Selecting test markets

Clustering Methods:
 objects are organized into an hierarchical structure as part of the procedure
 examples:
 Divisive clustering - start by treating all objects as if they are part of a single large cluster, then divide the cluster into smaller and smaller clusters
 Agglomerative clustering - start by treating each object as a separate cluster, then group them into bigger and bigger clusters

Difference in Two clustering methods:Agglomerative vs. Divisive Clustering

Once the measure of association as well as the method for determining the distances between clusters have been considered, out last choice for cluster analysis follows. There are two methods for proceeding...

Agglomerative Clustering:

(Leaves to trunk)
• We start out with all sample units in n clusters of size 1.
• Then, at each step of the algorithm, the pair of clusters with the shortest distance are combined into a single cluster.
• The algorithm stops when all sample units are combined into a single cluster of size n.
Divisive Clustering:
(Trunk to leaves)
• We start out with all sample units in a single cluster of size n.
• Then, at each step of the algorithm, clusters are partitioned into a pair of daughter clusters, selected to maximize the distance between each daughter.
• The algorithm stops when sample units are partitioned into n clusters of size 1.


Agglomerative clustering methods Methods for grouping multivariate data into clusters. Suppose there are n data items. The agglomerative clustering methods start by regarding these as n separate clusters of size 1. The two clusters judged closest together (on some criterion) are then merged to reduce the number of clusters to (n − 1). This procedure continues until all the items are collected into a single cluster.
Agglomerative algorithms begin with each element as a separate cluster and merge them into successively larger clusters. Divisive algorithms begin with the whole set and proceed to divide it into successively smaller clusters.


This method builds the hierarchy from the individual elements by progressively merging clusters. In our example, we have six elements {a} {b} {c} {d} {e} and {f}. The first step is to determine which elements to merge in a cluster. Usually, we want to take the two closest elements, according to the chosen distance.
Optionally, one can also construct a distance matrix at this stage, where the number in the i-th row j-th column is the distance between the i-th and j-th elements. Then, as clustering progresses, rows and columns are merged as the clusters are merged and the distances updated. This is a common way to implement this type of clustering, and has the benefit of caching distances between clusters. A simple agglomerative clustering algorithm is described in thesingle-linkage clustering page; it can easily be adapted to different types of linkage.
Suppose we have merged the two closest elements b and c, we now have the following clusters {a}, {b, c}, {d}, {e} and {f}, and want to merge them further. To do that, we need to take the distance between {a} and {b c}, and therefore define the distance between two clusters. Usually the distance between two clusters and is one of the following:
 The maximum distance between elements of each cluster (also called complete linkage clustering):

 The minimum distance between elements of each cluster (also called single-linkage clustering):

 The mean distance between elements of each cluster (also called average linkage clustering):

 The sum of all intra-cluster variance.
 The increase in variance for the cluster being merged.
 The probability that candidate clusters spawn from the same distribution function (V-linkage).
Each agglomeration occurs at a greater distance between clusters than the previous agglomeration, and one can decide to stop clustering either when the clusters are too far apart to be merged (distance criterion) or when there is a sufficiently small number of clusters (number criterion).


Algorithm for Agglomerative clustering (simplified):
1. Initially, put each article in its own cluster.
2. Among all current clusters, pick the two clusters with the smallest distance.
3. Replace these two clusters with a new cluster, formed by merging the two original ones.
4. Repeat the above two steps until there is only one remaining cluster in the pool.
Thus, the agglomerative clustering algorithm will result in a binary cluster tree with single article clusters as its leaf nodes and a root node containing all the articles.


Source:http://www.stat.psu.edu/online/courses/stat505/18_cluster/06_cluster_versus.html
http://www.norusis.com/pdf/SPC_v13.pdf
http://www.highbeam.com/doc/1O106-agglomerativeclstrngmthds.html
http://en.wikipedia.org/wiki/Cluster_analysis

Perceptual Mapping - A Technique

The market’s image of a product is based on its features and benefits, whether real or perceived. We illustrate these perceptions to give you a powerful “at-a-glance” look at where your brand lives on the competitive landscape. Based on how customers rate the functional and/or emotional attributes of your brand and competitive brands, we create visual representations that depict your position relative to the competition and to customers’ ideals. 

The result is a clear picture that reveals opportunities for positioning your brand more effectively.

Shedding Light

Perceptual mapping provides answers to these vital questions:

·         What is your market position relative to your competitors?

·         What are your strengths and weaknesses?

·         Where are there gaps in the market?

·         How do you measure up to the “ideal?”

·         What strategies will improve your competitive position so that you are both relevant to the market and differentiated from competitors?

The process includes developing the attributes—either functional, emotional, or both.  Survey respondents rate the client brand and competitive brands on each of the attributes, and attribute importance and ideal attribute ratings are obtained. These perceptions are used to create the maps, and analysis of gaps and ideal points is performed.

An Example of Perceptual Mapping

This hypothetical example of the beverage market illustrates one of the primary results of a perceptual mapping study—the map itself. For illustrative purposes, this example uses types of beverages instead of actual beverage brands.

In the perceptual map below, both functional attributes (“sparkling”) and emotional attributes (“youthful”) are included. Each attribute is represented by an arrow. Attributes that are close to one another are related, and the types of beverages are positioned near the attributes that best describe them.

By understanding the position of the client brand and competitive brands, the following questions may be addressed:

·         What attributes does the brand own? What attributes do competitors own?

·         Are there gaps in the market that may be filled by the client brand?

·         How should the brand be positioned to be both relevant to the market and differentiated from the competition?

In addition to the map itself, analysis of the raw brand ratings, importance ratings, gaps, and ideal market position adds further insight into brand positioning opportunities.

Perceptual Mapping Training

We also offer customized on-site training seminars on perceptual mapping so that your staff members can learn how to perform these types of studies on their own.

Backroom Services

We offer backroom services for perceptual mapping studies, including designing the study, programming and hosting the survey, and developing the perceptual maps.

Beyond Perceptual Mapping

Combining perceptual mapping with conjoint analysis creates an extremely powerful analytical and modeling tool. By linking the two techniques, potential changes in brand perceptions can be tested in the conjoint model in order to predict the market impact of those changes. 

Submitted By:

Ganesh Singh

12021

Marketing

Importance of Business Analytics and Using SPSS

Workshop - Class 1, 2

The class started by installing all the required software for using SPSS 15 tool. I was very excited about it as I had heard about it during Marketing Research Sessions and its important in Marketing was significant.


Initially we were explained the importance of SPSS and how Business Intelligence helped Marketeers to make critical decision accessing right information. The information available to us using initial Market Research Conducted using various techniques like Survey, Interviews, Observation, Questionnaires etc.

The data available can be actually categorized and combined in various ways to evaluate a hypothesis.

We learned about two different technique or Data analytics technique :

1. Frequency
2. Cross Tabs

FREQUENCIES can produce

• Tables that show
• all values that occur for a specified variable
• how frequently the values occur
• the percentage of times they occur, based on
• the total number of cases
• only cases with valid data for the variable
• Any statistic from this list
• MEAN
• SEMEAN
• MEDIAN
• MINIMUM
• MAXIMUM
• RANGE
• Graphs
• bar charts -- for discrete variables
• histograms -- for continuous variables

CROSSTABS

Crosstabs is an SPSS procedure that cross-tabulates two variables, thus displaying their relationship in tabular form. In contrast to Frequencies, which summarizes information about one variable, Crosstabs generates information about bivariate relationships.

Crosstabs creates a table that contains a cell for every combination of categories in the two variables.

• Inside each cell is the number of cases that fit that particular combination of responses.
• SPSS can also report the row, column, and total percentages for each cell of the table.

Because Crosstabs creates a row for each value in one variable and a column for each value in the other, the procedure is not suitable for continuous variables that assume many values. Crosstabs is designed for discretevariables--usually those measured on nominal or ordinal scales.
Crosstabs are usually presented with the independent variable across the top and the dependent along the side.


We worked on Sample file called Retail.sav which was based on data collected for a retail store chain covering 4 different stores and consumers satisfaction levels. We grouped the available data using cross tabs to evaluate a null hypothesis and then determing Chi square value.


Testing statistical significance using chi-square

• Chi-square tests for statistical independence between two variables
• The variables may be nominal level or higher, but it is best suited for discrete variables with limited categories.
• Chi-square is sensitive to any departure from chance relationship
• Strictly speaking, chi-square is only a test of significance, not a measure of a relationship between variables

• Consider the relationship between two discrete variables from the 2000 National Election Study
• Feelings about the Bible is the dependent variable
• Region of the country is the independent variable

Interpretation of chi-square

• The magnitude of the chi-square value must be judged against a table of values of the chi-square distribution:
• One must enter this table using the appropriate degrees of freedom: calculated according to the size of the table:
• X2 degrees of freedom = df
• = (rows - 1) (columns - 1)
• = (3 - 1) (6 - 1)
• = 2 x 5 = 10.
• Given the same degrees of freedom, the larger the chi-square value, the more "significant" it is.
• The entries in the chi-square table for a given chi-square are matched with "alpha" levels at specified levels of significance, e.g., .050 or .010 or .001
• A chi-square as large as 75 for 10 degrees of freedom would be expected by chance fewer than 1 time in 1000.

Submitted by:-
Ankit Agarwal
12070
Marketing