Abstract
The atmospheric fields are three-dimensional fields by nature, the variation in longitude, latitude and altitude of winds, temperatures and the other quantities are normal. A major jump forward in the development of climate science was reached when it was realized that the analysis of simultaneous values of the variables contained significant information. Indeed, to advance scientific understanding it is essential to have a view of the relation that links together various climate variables, for instance the temperature and the pressure in various places.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
This matrix notation is very common in meteorological data, whereas in many other fields, data are stored as an n ×m matrix, namely as X ∗. This difference affects the whole notation in later chapters, when defining the covariance matrix and other statistical quantities.
- 2.
More precisely, using \(\bar{\mathbf{x}} = \frac{1} {n}\mathbf{X}\mathbf{1}\), we have
$$\mathbf{X} -\bar{\mathbf{x}}{\mathbf{1}}^{{_\ast}} = \mathbf{X}({\mathbf{I}}_{n} - \frac{1} {n}\mathbf{1}{\mathbf{1}}^{{_\ast}}).$$Since the matrix \({\mathbf{I}}_{n} - \frac{1} {n}\mathbf{1}{\mathbf{1}}^{{_\ast}}\) has rank n − 1, the relation above shows that \((\mathbf{X} -\bar{\mathbf{x}}{\mathbf{1}}^{{_\ast}})\) has rank not greater than min{n − 1, m}. Therefore, the scaled covariance matrix \((\mathbf{X} -\bar{\mathbf{x}}{\mathbf{1}}^{{_\ast}}){(\mathbf{X} -\bar{\mathbf{x}}{\mathbf{1}}^{{_\ast}})}^{{_\ast}}\) has rank at most n − 1, if m ≥ n.
Reference
Jolliffe IT (2002) Principal component analysis, 2nd edn. Springer Series in Statistics, Springer, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Navarra, A., Simoncini, V. (2010). Empirical Orthogonal Functions. In: A Guide to Empirical Orthogonal Functions for Climate Data Analysis. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3702-2_4
Download citation
DOI: https://doi.org/10.1007/978-90-481-3702-2_4
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3701-5
Online ISBN: 978-90-481-3702-2
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)