Generalized Additive Models

An Introduction with R, Second Edition

Simon N. Wood

• Publisher: Chapman and Hall/CRC
• Year: 2017
• ISBN:
  9781498728379
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Edition: 2

The first edition of this book has established itself as one of the leading references on generalized additive models (GAMs), and the only book on the topic to be introductory in nature with a wealth of practical examples and software implementation. It is self-contained, providing the necessary background in linear models, linear mixed models, and generalized linear models (GLMs), before presenting a balanced treatment of the theory and applications of GAMs and related models.

The author bases his approach on a framework of penalized regression splines, and while firmly focused on the practical aspects of GAMs, discussions include fairly full explanations of the theory underlying the methods. Use of R software helps explain the theory and illustrates the practical application of the methodology. Each chapter contains an extensive set of exercises, with solutions in an appendix or in the book's R data package gamair, to enable use as a course text or for self-study.

Chapter 1

Linear Models

How old is the universe? The standard Big Bang model of the origin of the universe says that it expands uniformly, and locally, according to Hubble’s law,

$y=\beta x,$

where y is the relative velocity of any two galaxies separated by distance x, and β is ‘Hubble’s constant’ (in standard astrophysical notation y ≡ v, x ≡ d and β ≡ H₀). β⁻¹ gives the approximate age of the universe, but β is unknown and must somehow be estimated from observations of y and x, made for a variety of galaxies at different distances from us.

Figure 1.1 plots velocity against distance for 24 galaxies, according to measurements made using the Hubble Space Telescope. Velocities are assessed by measuring the Doppler effect red shift in the spectrum of light observed from the galaxies concerned, although some correction for ‘local’ velocity components is required. Distance measurement is much less direct, and is based on the 1912 discovery, by Henrietta Leavitt, of a relationship between the period of a certain class of variable stars, known as the Cepheids, and their luminosity. The intensity of Cepheids varies regularly with a period of between 1.5 and something over 50 days, and the mean intensity increases predictably with period. This means that, if you can find a Cepheid, you can tell how far away it is, by comparing its apparent brightness to its period predicted intensity.

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Figure 1.1 A Hubble diagram showing the relationship between distance, x, and velocity, y, for 24 galaxies containing Cepheid stars. The data are from the Hubble Space Telescope key project to measure the Hubble constant as reported in Freedman et al. (2001).

It is clear, from the figure, that the observed data do not follow Hubble’s law exactly, but given the measurement process, it would be surprising if they did. Given the apparent variability, what can be inferred from these data? In particular: (i) what value of β is most consistent with the data? (ii) what range of β values is consistent with the data and (iii) are some particular, theoretically derived, values of β consistent with the data? Statistics is about trying to answer these three sorts of questions.

One way to proceed is to formulate a linear statistical model of the way that the data were generated, and to use this as the basis for inference. Specifically, suppose that, rather than being governed directly by Hubble’s law, the observed velocity is given by Hubble’s constant multiplied by the o...