Abstract
Much of this book has been devoted to describing relationships among multiple noisy variables, yet we have until now managed to avoid a general discussion of multivariate co-variation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The term “learning” tends to be used interchangeably with “estimation,” i.e., the process of determining a parameter value from data. Because it may sometimes refer to significance testing, learning is somewhat broader, and it is often associated with techniques used heavily in the field of machine learning. See Hastie et al. (2009).
- 2.
In constructing the \(F\)-statistic, the values of SS \(_{group }\) and SSE are first standardized by dividing by their respective degrees of freedom, but that is for the convenience of judging the ratio relative to the number 1.
- 3.
Here we are using \(T^2\) both as an observed value of a statistic based on data and as a random variable that has a probability distribution. To be consistent with earlier notation, in using \(T^2\) as a random variable we should replace \(\bar{y}_1\) and \(\bar{y}_2\) in (17.13) and (17.11) with \(\bar{Y}_1\) and \(\bar{Y}_2\).
- 4.
This assumes that the variance matrix is well-defined in the sense that every linear combination \(a^TX\) has finite variance. There exist multivariate distributions for which nonzero linear combinations \(a^TX\) have infinite variance. We do not consider these here.
- 5.
A subspace \(N\) of \(R^m\) is a smooth manifold if at every point \(x \in N\) there is a local coordinate representation in which all points near \(x\) in \(N\) have the form \((u,v)\) where \(v=0\). In other words, everywhere in \(N\) there is a local coordinate system that makes \(N\) look like a linear subspace. See Appendix A of Kass and Vos (1997).
- 6.
The most famous example is Spearman’s general intelligence index \(g\), which is obtained from factor analysis. See, e.g., Gould (1996); Devlin et al. (1997).
- 7.
This use of “kernel” is different than that in Section 15.3.1.
- 8.
Each matrix \(\Sigma _k\) has \(m(m+1)/2\) parameters so there are \(Km(m+1)/2\) parameters when the matrices are allowed to be different and only \(m(m+1)/2\) if they are assumed to be equal.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kass, R.E., Eden, U.T., Brown, E.N. (2014). Multivariate Analysis. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_17
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9602-1_17
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9601-4
Online ISBN: 978-1-4614-9602-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)