Ian T. Jolliffe

Forecast verification : a practitioner's guide in atmospheric science

List of Contributors. Preface. 1. Introduction (I. Jolliffe & D. Stephenson). 2. Basic Concepts (J. Potts). 3. Binary Events (I. Mason). 4. Categorical Events (R. Livezey). 5. Continuous Variables (M. Deque). 6. Verification of Spatial Fields (W. Drosdowsky & H. Zhang). 7. Probability and Ensemble Forecasts (Z. Toth, et al.). 8. Economic Value and Skill (D. Richardson). 9. Forecast Verification: Past, Present and Future (D. Stephenson & I. Jolliffe). Glossary. References. Author Index. Subject Index.

Principal Component Analysis

Introduction * Properties of Population Principal Components * Properties of Sample Principal Components * Interpreting Principal Components: Examples * Graphical Representation of Data Using Principal Components * Choosing a Subset of Principal Components or Variables * Principal Component Analysis and Factor Analysis * Principal Components in Regression Analysis * Principal Components Used with Other Multivariate Techniques * Outlier Detection, Influential Observations and Robust Estimation * Rotation and Interpretation of Principal Components * Principal Component Analysis for Time Series and Other Non-Independent Data * Principal Component Analysis for Special Types of Data * Generalizations and Adaptations of Principal Component Analysis

Discarding Variables in a Principal Component Analysis. I: Artificial Data

Often, results obtained from the use of principal component analysis are little changed if some of the variables involved are discarded beforehand. This paper examines some of the possible methods for deciding which variables to reject and these rejection methods are tested on artificial data containing variables known to be “redundant”. It is shown that several of the rejection methods, of differing types, each discard precisely those variables known to be redundant, for all but a few sets of data.

A Note on the Use of Principal Components in Regression

The use of principal components in regression has received a lot of attention in the literature in the past few years, and the topic is now beginning to appear in textbooks. Along with the use of principal component regression there appears to have been a growth in the misconception that the principal components with small eigenvalues will very rarely be of any use in regression. The purpose of this note is to demonstrate that these components can be as important as those with large variance. This is illustrated with four examples, three of which have already appeared in the literature.

A Modified Principal Component Technique Based on the LASSO

In many multivariate statistical techniques, a set of linear functions of the original p variables is produced. One of the more difficult aspects of these techniques is the interpretation of the linear functions, as these functions usually have nonzero coefficients on all p variables. A common approach is to effectively ignore (treat as zero) any coefficients less than some threshold value, so that the function becomes simple and the interpretation becomes easier for the users. Such a procedure can be misleading. There are alternatives to principal component analysis which restrict the coefficients to a smaller number of possible values in the derivation of the linear functions, or replace the principal components by “principal variables.” This article introduces a new technique, borrowing an idea proposed by Tibshirani in the context of multiple regression where similar problems arise in interpreting regression equations. This approach is the so-called LASSO, the “least absolute shrinkage and selection operator,” in which a bound is introduced on the sum of the absolute values of the coefficients, and in which some coefficients consequently become zero. We explore some of the properties of the new technique, both theoretically and using simulation studies, and apply it to an example.

Discarding Variables in a Principal Component Analysis. II: Real Data

In this paper it is shown for four sets of real data, all published examples of principal component analysis, that the number of variables used can be greatly reduced with little effect on the results obtained. Five methods for discarding variables, which have previously been successfully tested on artificial data (Jolliffe, 1972), are used. The methods are compared and all are shown to be satisfactory for real, as well as artificial, data, although none is shown to be overwhelmingly superior to the others.

Principal Component Analysis and Factor Analysis

Principal component analysis has often been dealt with in textbooks as a special case of factor analysis, and this tendency has been continued by many computer packages which treat PCA as one option in a program for factor analysis—see Appendix A2. This view is misguided since PCA and factor analysis, as usually defined, are really quite distinct techniques. The confusion may have arisen, in part, because of Hotelling’s (1933) original paper, in which principal components were introduced in the context of providing a small number of ‘more fundamental’ variables which determine the values of the p original variables. This is very much in the spirit of the factor model introduced in Section 7.1, although Girschick (1936) indicates that there were soon criticisms of Hotelling’s method of PCs, as being inappropriate for factor analysis. Further confusion results from the fact that practitioners of ‘factor analysis’ do not always have the same definition of the technique (see Jackson, 1981). The definition adopted in this chapter is, however, fairly standard.

Principal component analysis: a review and recent developments.

Large datasets are increasingly common and are often difficult to interpret. Principal component analysis (PCA) is a technique for reducing the dimensionality of such datasets, increasing interpretability but at the same time minimizing information loss. It does so by creating new uncorrelated variables that successively maximize variance. Finding such new variables, the principal components, reduces to solving an eigenvalue/eigenvector problem, and the new variables are defined by the dataset at hand, not a priori, hence making PCA an adaptive data analysis technique. It is adaptive in another sense too, since variants of the technique have been developed that are tailored to various different data types and structures. This article will begin by introducing the basic ideas of PCA, discussing what it can and cannot do. It will then describe some variants of PCA and their application.

Revised “LEPS” Scores for Assessing Climate Model Simulations and Long-Range Forecasts

Abstract The most commonly used measures for verifying forecasts or simulators of continuous variables are root-mean-squared error (rmse) and anomaly correlation. Some disadvantages of these measures are demonstrated. Existing assessment systems for categorical forecasts are discussed briefly. An alternative unbiased verification measure is developed, known as the linear error in probability space (LEPS) score. The LEPS scare may be used to assess forecasts of both continuous and categorical variables and has some advantages over rmse and anomaly correlation. The properties of the version of LEPS discussed here are reviewed and compared with an earlier form of LEPS. A skill-score version of LEPS may be used to obtain an overall measure of the skill of a number of forecasts. This skill score is biased, but the bias is negligible if the number of effectively independent forecasts or simulations is large. Some examples are given in which the LEPS skill score is compared with rmse and anomaly correlation.

Rotation of principal components: choice of normalization constraints

Following a principal component analysis, it is fairly common practice to rotate some of the components, often using orthogonal rotation. It is a frequent misconception that orthogonal rotation will produce rotated components which are pairwise uncorrelated, and/or whose loadings are orthogonal In fact, it is not possible, using the standard definition of rotation, to preserve both these properties. Which of the two properties is preserved depends on the normalization chosen for the loadings, prior to rotation. The usual ‘default’ normalization leads to rotated components which possess neither property.