The Geometry of Multivariate Statistics

The Geometry of Multivariate Statistics

The Geometry of Multivariate Statistics

The Geometry of Multivariate Statistics

Synopsis

A traditional approach to developing multivariate statistical theory is algebraic. Sets of observations are represented by matrices, linear combinations are formed from these matrices by multiplying them by coefficient matrices, and useful statistics are found by imposing various criteria of optimization on these combinations. Matrix algebra is the vehicle for these calculations. A second approach is computational. Since many users find that they do not need to know the mathematical basis of the techniques as long as they have a way to transform data into results, the computation can be done by a package of computer programs that somebody else has written. An approach from this perspective emphasizes how the computer packages are used, and is usually coupled with rules that allow one to extract the most important numbers from the output and interpret them. Useful as both approaches are--particularly when combined--they can overlook an important aspect of multivariate analysis. To apply it correctly, one needs a way to conceptualize the multivariate relationships that exist among variables. This book is designed to help the reader develop a way of thinking about multivariate statistics, as well as to understand in a broader and more intuitive sense what the procedures do and how their results are interpreted. Presenting important procedures of multivariate statistical theory geometrically, the author hopes that this emphasis on the geometry will give the reader a coherent picture into which all the multivariate techniques fit.

Excerpt

In simple terms, this little book is designed to help its reader think about multivariate statistics. I say "think" here because I have not written about how one programs the computer or calculates the test statistics. Instead I hope to help the reader understand in a broad and intuitive sense what the multivariate procedures do and how their results are interpreted.

There are many ways to develop multivariate statistical theory. the traditional approach is algebraic. Sets of observations are represented by matrices, linear combinations are formed from these matrices by multiplying them by coefficient matrices, and useful statistics are found by imposing various criteria of optimization on these combinations. Matrix algebra is the vehicle for these calculations. a second approach is computational. Many users of multivariate statistics find that they do not need to know the mathematical basis of the techniques as long as they can transform data into results. the computation can be done by a package of computer programs that somebody else has written. An approach to multivariate statistics from this perspective emphasizes how the computer packages are used, and is usually coupled with rules that allow one to extract the most important numbers from the output and interpret them.

Useful as both approaches are, particularly when combined, they overlook an important aspect of multivariate analysis. To apply it correctly, one needs a way to conceptualize the multivariate relationships among the variables. To some extent, the equations help. a linear combination explicitly defines a new variable, and a correlation matrix accurately expresses the pattern of association among the members of a set of variables. However, I have never found these descriptions sufficient, either for myself or when teaching others. Problems that involve many variables require a deeper understanding than is typically provided by the formal equations or the computer programs. Although knowing the algebra is helpful and a powerful computer program is almost essential, neither is sufficient without a good way to picture the variables.

Fortunately, a tool to develop this understanding is available. Multi-

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