This is the second edition of *Linear Models for Multivariate,
Time Series and Spatial Data*. It has a new title to indicate that
it contains much new material. The primary changes are the
addition of two new chapters: one on nonparametric regression and
one on response surface maximization. As before, the presentations
focus on the linear model aspects of the subject. For example, in
the nonparametric regression chapter there is very little about
kernal regression estimation but quite a bit about series
approximations, splines, and regression trees, all of which can be
viewed as linear modeling.

The new edition also includes various smaller changes. Of particular note are a subsection in Chapter 1 on modeling longitudinal (repeated measures) data and a section in Chapter 6 on covariance structures for spatial lattice data. I would like to thank Dale Zimmerman for the suggestion of incorporating material on spatial lattices. Another change is that the subject index is now entirely alphabetical.

This is a companion volume to *Plane Answers to Complex
Questions: The Theory of Linear Models*. It consists of six
additional chapters written in the same spirit as the last six
chapters of the earlier book. Brief introductions are given to
topics related to linear model theory. No attempt is made to give
a comprehensive treatment of the topics. Such an effort would be
futile. Each chapter is on a topic so broad that an in depth
discussion would require a book length treatment.

People need to impose structure on the world in order to understand it. There is a limit to the number of unrelated facts that anyone can remember. If ideas can be put within a broad, sophisticatedly simple structure, not only are they easier to remember but often new insights become available. In fact, sophisticatedly simple models of the world may be the only ones that work. I have often heard Arnold Zellner say that, to the best of his knowledge, this is true in econometrics. The process of modeling is fundamental to understanding the world.

In Statistics, the most widely used models revolve around linear
structures. Often the linear structure is exploited in ways that
are peculiar to the subject matter. Certainly this is true of
frequency domain times series and geostatistics. The purpose of
this volume is to take three fundamental ideas from standard linear
model theory and exploit their properties in examining
multivariate, time series and spatial data. In decreasing order
of importance to the presentation, the three ideas are: best
linear prediction, projections and Mahalanobis's distance.
(Actually, Mahalanobis's distance is a fundamentally multivariate
idea that has been appropriated for use in linear models.)
Numerous references to results in *Plane Answers* are made.
Nevertheless, I have tried to make this book as independent as
possible. Typically, when a result from Plane Answers is needed
not only is the reference given but also the result itself. Of
course, for proofs of these results the reader will have to refer
to the original source.

I want to re-emphasize that this is a book about linear models. It is not traditional multivariate analysis, time series, or geostatistics. Multivariate linear models are viewed as linear models with a nondiagonal covariance matrix. Discriminant analysis is related to the Mahalanobis distance and multivariate analysis of variance. Principle components are best linear predictors. Frequency domain time series involves linear models with a peculiar design matrix. Time domain analysis involves models that are linear in the parameters but have random design matrices. Best linear predictors are used for forecasting time series; they are also fundamental to the estimation techniques used in time domain analysis. Spatial data analysis involves linear models in which the covariance matrix is modeled from the data; a primary objective in analyzing spatial data is making best linear unbiased predictions of future observables. While other approaches to these problems may yield different insights, there is value in having a unified approach to looking at these problems. Developing such a unified approach is the purpose of this book.

There are two well known models with linear structure that are conspicuous by their absence in my two volumes on linear models. One is Cox's (1972) proportional hazards model. The other is the generalized linear model of Nelder and Wedderburn (1972). The proportional hazards methodology is a fundamentally nonparametric technique for dealing with censored data having linear structure. The emphasis on nonparametrics and censored data would make its inclusion here awkward. The interested reader can see Kalbfleisch and Prentice (1980). Generalized linear models allow the extension of linear model ideas to many situations that involve independent nonnormally distributed observations. Beyond the presentation of basic linear model theory, these volumes focus on methods for analyzing correlated observations. While it is true that generalized linear models can be used for some types of correlated data, such applications do not flow from the essential theory. McCullagh and Nelder (1989) give a detailed exposition of generalized linear models and Christensen (1990) contains a short introduction.

**Acknowledgements**

I would like to thank MINITAB for providing me with a copy of release 6.1.1, BMDP for providing me with copies of their programs 4M, 1T, 2T, and 4V and Dick Lund for providing me with a copy of MSUSTAT. Nearly all of the computations were performed with one of these programs. Many were performed with more than one.

I would not have tackled this project but for Larry Blackwood and Bob Shumway. Together Larry and I reconfirmed, in my mind anyway, that multivariate analysis is just the same old stuff. Bob's book put an end to a specter that has long haunted me: a career full of half-hearted attempts at figuring out basic time series analysis.

At my request, Ed Bedrick, Bert Koopmans, Wes Johnson, Bob Shumway and Dale Zimmerman tried to turn me from the errors of my ways. I sincerely thank them for their valuable efforts. The reader must judge how successful they were with a recalcitrant subject. As always, I must thank my editors Steve Fienberg and Ingram Olkin for their suggestions. Jackie Damrau did an exceptional job in typing the first draft of the manuscript.

Finally, I have to recognize the contribution of Magic Johnson.
I was so upset when the 1987-88 Lakers won a second consecutive NBA
title that I began writing this book in order to block the mental
anguish. I am reminded of Woody Allen's dilemma: is the
importance of life more accurately reflected in watching *The
Sorrow and the Pity* or in watching the Knicks? (In my case, the
Jazz and the Celtics.) It's a tough call. Perhaps life is about
actually making movies and doing statistics.

- Preface to the Second Edition
- Preface to the First Edition
- Multivariate Linear Models
- Estimation
- BLUEs
- Maximum Likelihood
- Unbiased Estimation of Sigma

- Testing Hypotheses
- Test Statistics
- Prediciton Regions
- Multiple Comparison Methods

- One Sample Problems
- Two Sample Problems
- One-way Analysis of Variance and Profile Analysis
- Profile Analysis

- Growth Curves
- Longitudinal Data

- Testing for Additional Information
- Additional Exercises

- Estimation
- Discrimination and Allocation
- The General Allocation Problem
- Mahalanobis's Distance
- Maximum Likelihood
- Bayesian Methods
- Estimated Allocation

- Equal Covariance Matrices
- Cross-Validation

- Linear Discrimination Coordinates
- Additional Exercises

- The General Allocation Problem
- Principal Components and Factor Analysis
- Properties of Best Linear Predictors
- The Theory of Principal Components
- Sequential Prediction
- Joint Prediction
- Other Derivations of Principal Components
- Principal Components Based on the Correlation Matrix

- Sample Principal Components
- The Sample Prediction Error
- Using Principal Components

- Factor Analysis
- Terminology and Applications
- Maximum Likelihood Theory
- Principal Factor Estimation
- Discussion

- Additional Exercises

- Frequency Analysis of Time Series
- Stationary Processes
- Basic Data Analysis
- Spectral Approximation of Stationary Time Series
- The Random Effects Model
- The Measurement Error Model
- Linear Filtering
- Recursive Filters

- The Coherence of Two Time Series
- Fourier Analysis
- Additional Exercises

- Time Domain Analysis
- Correlations
- Partial Correlations and Best Linear Prediction

- Time Domain Models
- Autoregressive Models: AR(p)'s
- Moving Average Modesl: MA(q)'s
- Autoregressive Moving Average Models: ARMA(p,q)'s
- Autoregressive Integrated Moving Average Models: ARIMA(p,d,q)'s

- Time Domain Prediction
- Nonlinear Least Squares
- The Gauss-Newton Algorithm
- Nonlinear Regression

- Estimation
- Correlations
- Conditional Estimation for AR(p) Models
- Conditional Least Squares for ARMA(p,q)'s
- Conditional MLEs for ARMA(p,q)'s
- Unconditional Estimation for ARMA(p,q) Models
- Estimation for ARIMA(p,d,q) Models

- Model Selection
- Box-Jenkins
- Model Selection Criteria
- An Example

- Seasonal Adjustment
- The Multivariate State-Space Model and the Kalman Filter
- The Kalman Filter
- Parameter Estimation
- Missing Values

- Additional Exercises

- Correlations
- Linear Models for Spatial Data: Kriging
- Modeling Spatial Data
- Stationarity

- Best Linear Unbiased Prediction of Spatial Data: Kriging
- Block Kriging

- Prediction Based on the Semivariogram: Geostatistical Kriging
- Measurement Error and the Nugget Effect
- The Effect of Estimated Covariances on Prediction
- Spatial Data
- Mathematical Results

- Models for Covariance Functions and Semivariograms
- The Linear Covariance Model
- Nonlinear Isotropic Covariance Models
- Modeling Anisotropic Covariance Functions
- Nonlinear Semivariograms

- Covariance Models for Lattice Data
- Spatial Covariance Selection Models
- Spatial Autoregression Models
- Spatial Autoregressive Moving Average Models

- Estimation of Covariance Functions and Semivariograms
- Estimation for Linear Covariance Functions
- Maximum Likelihood Estimation
- Residual Maximum Likelihood Estimation
- Traditional Geostatistical Estimation

- Modeling Spatial Data
- Nonparametric Regression
- Orthogonal Series Approximations
- Simple Nonparametric Regression
- Estimation
- Variable Selection
- Heteroscedastic Simple Nonparametric Regression
- Other Methods: Cubic Splines and Kernal Estimates
- Nonparametric Multiple Regression
- Testing Lack of Fit
- Other Methods: Regression Trees
- Density Estimation
- Exercises

- Response Surface Maximization
- Approximating Response Functions
- First-Order Models and Steepest Ascent
- Fitting Quadratic Models
- Interpreting Quadratic Response Functions

- References
- Author Index
- Subject Index

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