In multivariate analysis, we are concerned with the joint analysis of multiple dependent variables. These variables can be represented using matrices and vectors. This provides simplification of notation and a format for expressing important formulas.

Vector notation • The modern approach to presenting data uses vectors. • Specifically, a vector is an ordered set of n elements enclosed by a pair of brackets.

The scalar quantities used in the univariate setting are replaced by vectors in the multivariate setting: Sample Mean Vector \(\bar{\mathbf{y}}_{i.} = \frac{1}{n_i}\sum_{j=1}^{n_i}\mathbf{Y}_{ij} = \left(\begin{array}{c}\bar{y}_{i.1}\\ \bar{y}_{i.2} \\ \vdots \\ \bar{y}_{i.p}\end{array}\right)\) = sample mean vector for group i .

Definition of a vector and a scalar A column vector is an r × 1 matrix, that is, a matrix with only one column. A vector is almost often denoted by a single lowercase letter in boldface type.

A random sample is used to model unobserved data that are intended to be observed. For example, assume the p-dimensional population is repre-sented by a p-dimensional random vector X~= 2 6 4 X 1... X p 3 7 5˘N( ;~ ); where X j represents the j-th variable. A collection of random vectors X~ 1;:::;X~ n i:i:d:˘N( ;~ ) is referred to as a random ...

multivariate problems, known as multivariate analysis of variance methods (MANOVA). These may be viewed as extensions of usual analysis of variance methods, where now, an \observation" is an entire vector from an unit rather than just a single, scalar response.

multivariable real-valued func-tions. We graph functions of th. iables and the “output” variable. By contrast, for graphing single-variable vector-valued functions we do not given an axis for the “input” variable, for example the var.

Table 1.1 shows some examples of multivariate observations. The reader will notice that in some cases all the variables are measured in the same scale (see 1 and 2 in Table 1.1). In other cases, measurements are in different scales (see 3 in Table 1.1).

Vector Analysis with Mathematica and Java. At the University of Minnesota we have a Multivariable Calculus and Vector Analysis course which makes heavy use of technology. Students spend two hours per week working in a computer lab using Mathematica.

Multivariate analysis requires that we can build models on M(n, p) of n×p matrices. This is a vector space and in many ways it is simply Rnp, but the organization of data in the matrix form does actually mean something. Some prefer to consequently map everything into Rnp and work with the theory for the normal distribution, say, on Rnp.