Description of Multivariate Data

Multivariate Analysis The analysis of many variables

Multivariate Analysis : The analysis of many variables More precisely and also more traditionally this term stands fo the study of a random sample pf n objects (units or cases) such that on each object we measure p variables or characteristics. So that for each object there is a vector: Each with p components:

The variables will be correlated as they are measured on the same object. This may lead to incorrect and inadequate analysis: A common practice is to treat each variable separately by applying methods of univariate analysis.

The challenge of multivariate analysis is to untangle the overlapping information provided by a set of correlated variables and to reveal the underlying structure.

This is done by a variety of methods, some of which are generalizations of univariate methods and some which are multivariate with without univariate counterparts

The purpose of this course is to describe and perhaps justify these methods, and also provide some guidance about how to select an appropriate method for a given multivariate data set.

Example x 1 = age (in years) at entry to university, Randomly select n = 5 students as objects and for each student measure: x 2 = mark out of 100 in an exam at the end of the first year, x 3 = sex (0 = female, 1= male)

The result may look something like this: Objectsx1x1 x2x2 x3x It is of interest to note that the variables in the example are not of the same type: –x 1 is a continuous variable, –x 2 is a discrete variable and –x 3 is a binary variable

variables The Data Matrix

where We can write = the i th row of X.

where We can also write

is the p-vector denoting the p observations on the first object, while In this notation is the n-vector denoting the observations on the first variable The rows form a random sample while the columns do not (this is emphasized in the notation by the use of parentheses)

The objective of multivariate analysis will be a attempt to find some feature of the variables (i.e. the columns of the data matrix) At other times, the objective of multivariate analysis will be a attempt to find some feature of the individuals (i.e. the rows of the data matrix) The feature that we often look for is grouping of the individuals or of the variables. We will give a classification of multivariate methods later

Summarization of the data

Even when n and p are moderately large, the amount of information (np elements of the data matrix) can be overwhelming and it is necessary to find ways of summarizing data. Later on we will discuss way of graphical representation of the data

Definitions: 1.The sample mean for the i th variable 2.The sample variance for the i th variable 3.The sample covariance between the i th variable and the j th variable

Defn: The sample mean vector Putting the definitions together we are led to the following definitions:

Defn: The sample covariance matrix

Expressing the sample mean vector and the sample covariance matrix in terms of the data matrix

The sample mean vector

Note where is the n-vector whose components are all equal to 1.

The sample covariance matrix

We can write

then because

The final step is to realize that that It is easy to check that

So that

In the text book And then

Another Expression for S

Note: and

Thus Hence

Data are frequently scaled as well as centered. The scaling is done by introducing: Defn: the sample correlation coefficient for (between) the i th and the j th variables the sample correlation matrix

Obviously and using the Schwartz’s inequality If R = I then we say the variables are uncorrelated

Note: if we denote Then it can be checked that

Measures of Multivariate Scatter

The sample variance-covariance matrix S is an obvious generalization of the univariate concept of variance, which measures scatter about the mean. Sometimes it is convenient to have a single number to measure the overall multivariate scatter.

There are two common measures of this type: Defn: The generalized sample variance Defn: The total sample variance

In both cases, large values indicate a high degree of scatter about the centroid: low values indicate concentration about the centroid: Using the eigenvalues 1, 2, …, p of the matrix S,it can be shown that

If p = 0 then This says that there is a linear dependence amongst the variables. Normally, S is positive definite and all the eigenvalues are positive.

Linear combinations

Taking linear combinations of variables is one of the most important tools of multivariate analysis. This is for basically two reasons: 1.A few appropriately chosen combinations may provide more of the information than a lot of the original variables. (this is called dimension reduction.) 2.Linear combinations can simplify the structure of the variance-covariance matrix, which can help in the interpretation of the data.

For a given vector of constraints: We consider a linear combination For i = 1, 2, …, n. Then

And the variance of the Y’s is