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El. knyga: Flexible Regression and Smoothing: Using GAMLSS in R [Taylor & Francis e-book]

, , (Department of Statistics, Faculty of Science and Enginerring, Macquarie University, Australia), ,
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This book is about learning from data using the Generalized Additive Models for Location, Scale and Shape (GAMLSS). GAMLSS extends the Generalized Linear Models (GLMs) and Generalized Additive Models (GAMs) to accommodate large complex datasets, which are increasingly prevalent.





In particular, the GAMLSS statistical framework enables flexible regression and smoothing models to be fitted to the data. The GAMLSS model assumes that the response variable has any parametric (continuous, discrete or mixed) distribution which might be heavy- or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution (location, scale, shape) can be modelled as linear or smooth functions of explanatory variables.





Key Features:



















Provides a broad overview of flexible regression and smoothing techniques to learn from data whilst also focusing on the practical application of methodology using GAMLSS software in R.













Includes a comprehensive collection of real data examples, which reflect the range of problems addressed by GAMLSS models and provide a practical illustration of the process of using flexible GAMLSS models for statistical learning.













R code integrated into the text for ease of understanding and replication.













Supplemented by a website with code, data and extra materials.











This book aims to help readers understand how to learn from data encountered in many fields. It will be useful for practitioners and researchers who wish to understand and use the GAMLSS models to learn from data and also for students who wish to learn GAMLSS through practical examples.
Preface xvii
I Introduction to models and packages
1(56)
1 Why GAMLSS?
3(28)
1.1 Introduction
3(1)
1.2 The 1980s Munich rent data
4(2)
1.3 The linear regression model (LM)
6(4)
1.4 The generalized linear model (GLM)
10(6)
1.5 The generalized additive model (GAM)
16(4)
1.6 Modelling the scale parameter
20(3)
1.7 The generalized additive model for location, scale and shape (GAMLSS)
23(4)
1.8 Bibliographic notes
27(1)
1.9 Exercises
28(3)
2 Introduction to the gamlss packages
31(26)
2.1 Introduction
31(1)
2.2 The gamlss packages
32(1)
2.3 A simple example using the gamlss packages
33(21)
2.3.1 Fitting a parametric model
34(6)
2.3.2 Fitting a nonparametric smoothing model
40(1)
2.3.2.1 P-splines
40(3)
2.3.2.2 Cubic splines
43(1)
2.3.2.3 Loess
44(1)
2.3.2.4 Neural networks
44(2)
2.3.3 Extracting fitted values
46(1)
2.3.4 Modelling both fi and a
46(2)
2.3.5 Diagnostic plots
48(1)
2.3.6 Fitting different distributions
49(1)
2.3.7 Selection between models
50(4)
2.4 Bibliographic notes
54(1)
2.5 Exercises
55(2)
II Algorithms, functions and inference
57(94)
3 The algorithms
59(28)
3.1 Introduction
59(3)
3.2 Estimating β and γ for fixed α
62(14)
3.2.1 The RS algorithm
63(1)
3.2.1.1 The outer iteration (GAMLSS iteration)
63(1)
3.2.1.2 The inner iteration (GLM or GLIM iteration)
64(4)
3.2.1.3 The modified backfitting algorithm
68(2)
3.2.2 The CG algorithm
70(1)
3.2.2.1 The outer iteration
70(1)
3.2.2.2 The inner iteration
70(2)
3.2.2.3 The modified backfitting algorithm
72(1)
3.2.3 Fish species example
72(3)
3.2.4 Remarks on the GAMLSS algorithms
75(1)
3.3 MAP estimators of ft and 7 for fixed A
76(1)
3.4 Estimating the hyperparameters A
77(5)
3.4.1 Global estimation
79(1)
3.4.1.1 Maximum likelihood
79(1)
3.4.1.2 Generalized Akaike information criterion
79(1)
3.4.1.3 Validation
80(1)
3.4.2 Local estimation
81(1)
3.4.2.1 Maximum likelihood
81(1)
3.4.2.2 Generalized Akaike information criterion
82(1)
3.4.2.3 Generalized cross validation
82(1)
3.5 Bibliographic notes
82(2)
3.6 Exercises
84(3)
4 The gamlss () function
87(26)
4.1 Introduction to the gamlss () function
87(1)
4.2 The arguments of the gamlss () function
88(10)
4.2.1 The algorithmic control functions
91(3)
4.2.2 Weighting out observations: the weights and data=subset () arguments
94(4)
4.3 The refit and update functions
98(4)
4.3.1 Refit ()
98(1)
4.3.2 Update ()
99(3)
4.4 The gamlss object
102(6)
4.5 Methods and functions for gamlss objects
108(1)
4.6 Bibliographic notes
109(1)
4.7 Exercises
110(3)
5 Inference and prediction
113(38)
5.1 Introduction
113(5)
5.1.1 Asymptotic behaviour of a parametric GAMLSS model
114(1)
5.1.2 Types of inference in a GAMLSS model
114(2)
5.1.3 Likelihood-based inference
116(1)
5.1.4 Bootstrapping
117(1)
5.2 Functions to obtain standard errors
118(8)
5.2.1 The gen.likelihood() function
118(2)
5.2.2 The vcov() and rvcovO functions
120(3)
5.2.3 The summary () function
123(3)
5.3 Functions to obtain confidence intervals
126(9)
5.3.1 The conf int() function
126(1)
5.3.2 The prof dev() function
127(3)
5.3.3 The prof term() function
130(5)
5.4 Functions to obtain predictions
135(9)
5.4.1 The predict () function
135(8)
5.4.2 The predictAll() function
143(1)
5.5 Appendix: Some theoretical properties of GLM and GAMLSS
144(1)
5.6 Bibliographic notes
145(1)
5.7 Exercises
146(5)
III Distributions
151(70)
6 The GAMLSS family of distributions
153(38)
6.1 Introduction
153(3)
6.2 Types of distribution within the GAMLSS family
156(12)
6.2.1 Explicit GAMLSS family distributions
156(5)
6.2.2 Extending GAMLSS family distributions
161(7)
6.3 Displaying GAMLSS family distributions
168(4)
6.3.1 Using the distribution demos
168(1)
6.3.2 Using the pdf plot() function
169(3)
6.4 Amending an existing distribution and constructing a new distribution
172(7)
6.4.1 Definition of the link functions
173(1)
6.4.2 The fitting information
174(2)
6.4.3 The S3 class definition
176(1)
6.4.4 Definition of the d, p, q and r functions
176(1)
6.4.5 Example: reparameterizing the NO distribution
177(2)
6.5 The link functions
179(3)
6.5.1 How to display the available link functions
179(1)
6.5.2 Changing the default link function
180(1)
6.5.3 Defining a link function
180(1)
6.5.4 Creating a link function
181(1)
6.5.5 Using the own link function
181(1)
6.6 Bibliographic notes
182(1)
6.7 Exercises
183(8)
7 Finite mixture distributions
191(30)
7.1 Introduction to finite mixtures
191(2)
7.2 Finite mixtures with no parameters in common
193(4)
7.2.1 The likelihood function
193(1)
7.2.2 Maximizing the likelihood function using the EM algorithm
194(2)
7.2.3 Modelling the mixing probabilities
196(1)
7.2.4 Estimating the total number of components
197(1)
7.2.5 Zero components
197(1)
7.3 The gamlssMXO function
197(3)
7.4 Example using gamlssMXO: Reading glasses data
200(7)
7.5 Finite mixtures with parameters in common
207(2)
7.6 The gamlssNPO function
209(2)
7.7 Example using gamlssNPO: Animal brain data
211(6)
7.8 Bibliographic notes
217(1)
7.9 Exercises
218(3)
IV Model terms
221(154)
8 Linear parametric additive terms
223(32)
8.1 Introduction to linear and additive terms
223(2)
8.2 Linear additive terms
225(6)
8.2.1 Linear main effects
227(1)
8.2.2 Linear interactions
227(4)
8.3 Polynomials
231(2)
8.4 Fractional polynomials
233(2)
8.5 Piecewise polynomials and regression splines
235(4)
8.6 B-splines
239(3)
8.7 Free knot models
242(1)
8.8 Example: the CD4 data
243(10)
8.8.1 Orthogonal polynomials
245(2)
8.8.2 Fractional polynomials
247(2)
8.8.3 Piecewise polynomials
249(1)
8.8.4 Free knot models
250(3)
8.9 Bibliographic notes
253(1)
8.10 Exercises
254(1)
9 Additive smoothing terms
255(66)
9.1 Introduction
256(2)
9.2 What is a scatterplot smoother?
258(3)
9.3 Local regression smoothers
261(4)
9.4 Penalized smoothers: Univariate
265(31)
9.4.1 Demos on penalized smoothers
269(1)
9.4.2 The pb(), pbo() and ps() functions for fitting a P-splines smoother
270(4)
9.4.3 The pbz() function for fitting smooth curves which can shrink to a constant
274(1)
9.4.4 The pbm() function for fitting monotonic smooth functions
275(2)
9.4.5 The pbc() and cy() functions for fitting cyclic smooth functions
277(1)
9.4.6 The cs() and scs() functions for fitting cubic splines
278(4)
9.4.7 The ri() function for fitting ridge and lasso regression terms
282(5)
9.4.8 The pcat() function for reducing levels of a factor
287(6)
9.4.9 The gmrf () function for fitting Gaussian Markov random fields
293(3)
9.5 Penalized smoothers: Multivariate
296(9)
9.5.1 The pvc() function for fitting varying coefficient models
296(1)
9.5.1.1 Continuous z
297(1)
9.5.1.2 Categorical z
298(3)
9.5.2 Interfacing with gam(): The ga() function
301(1)
9.5.2.1 Additive terms
301(1)
9.5.2.2 Smooth surface fitting
302(3)
9.6 Other smoothers
305(10)
9.6.1 Interfacing with nnet(): nn()
305(3)
9.6.2 Interfacing with rpart(): tr()
308(2)
9.6.3 Interfacing with loess(): lo()
310(4)
9.6.4 Interfacing with earth(): ma()
314(1)
9.7 Bibliographic notes
315(2)
9.8 Exercises
317(4)
10 Random effects
321(54)
10.1 Introduction
322(5)
10.1.1 Random effects at the observational and at the factor level
323(1)
10.1.2 Marginal and joint likelihood
324(1)
10.1.3 Functions available for fitting random effects
324(3)
10.2 Nonparametric random effect models
327(7)
10.2.1 Nonparametric random intercept model for μ at the factor level
327(1)
10.2.2 Fitting the nonparametric random intercept model for μ at the factor level
328(3)
10.2.3 Nonparametric random intercept and slopes model for μ
331(3)
10.3 Normal random effect models
334(2)
10.3.1 Summary of the (r + 1)st iteration of the EM algorithm
335(1)
10.4 The function gamlssNP() for random effects
336(3)
10.4.1 Fitting a normal random intercept for μ
337(1)
10.4.2 Fitting nonparametric random effects
337(1)
10.4.2.1 Fitting a nonparametric random intercept in the predictor for μ
337(1)
10.4.2.2 Fitting nonparametric random intercept and slopes in the predictor for μ
337(1)
10.4.2.3 Fitting nonparametric random coefficients in the predictor for other distribution parameters
338(1)
10.5 Examples using gamlssNP()
339(7)
10.5.1 Example: Binary response with normal random intercept
339(2)
10.5.2 Example: Binomial response with nonparametric random intercept and slope
341(5)
10.6 The function random()
346(1)
10.7 Examples using random()
347(7)
10.7.1 The Hodges data
347(5)
10.7.2 Revisiting the respiratory infection in children
352(2)
10.8 The function re(), interfacing with lme()
354(4)
10.9 Examples using re()
358(8)
10.9.1 Refitting Hodges data using re ()
358(1)
10.9.2 Fitting a P-spline smoother using re ()
359(2)
10.9.3 Leukemia data
361(5)
10.10 Bibliographic notes
366(1)
10.11 Exercises
367(8)
V Model selection and diagnostics
375(72)
11 Model selection techniques
377(40)
11.1 Introduction: Statistical model selection
377(3)
11.2 GAMLSS model selection
380(5)
11.2.1 Component D: Selection of the distribution
381(1)
11.2.2 Component G: Selection of the link functions
381(1)
11.2.3 Component T: Selection of the additive terms in the model
382(1)
11.2.4 Component L: Selection of the smoothing parameters
383(1)
11.2.5 Selection of all components using a validation data set
384(1)
11.2.6 Summary of the GAMLSS functions for model selection
384(1)
11.3 The addterm() and dropterm() functions
385(7)
11.4 The stepGAIC() function
392(5)
11.4.1 Selecting a model for μ
393(3)
11.4.2 Selecting a model for σ
396(1)
11.5 Strategy A: The stepGAICAll.A() function
397(2)
11.6 Strategy B: The stepGAICAll.B() function
399(2)
11.7 K-fold cross validation
401(1)
11.8 Validation and test data
402(6)
11.8.1 The gamlssVGD() and VGD() functions
402(2)
11.8.2 The getTGD() and TGD() functions
404(1)
11.8.3 The stepTGD() function
404(2)
11.8.4 The stepTGDAll.A() function
406(2)
11.9 The find.hyper() function
408(3)
11.10 Bibliographic notes
411(1)
11.11 Exercises
411(6)
12 Diagnostics
417(30)
12.1 Introduction
417(1)
12.2 Normalized (randomized) quantile residuals
418(4)
12.3 The plot () function
422(4)
12.4 The wp() function
426(7)
12.4.1 Single worm plot
426(2)
12.4.2 Multiple worm plot
428(4)
12.4.3 Arguments of the wp function
432(1)
12.5 The dtop() function
433(2)
12.5.1 Arguments of the dtop function
434(1)
12.6 The () stats() function
435(4)
12.6.1 Examples
436(3)
12.6.2 Arguments of the Q. stats function
439(1)
12.7 The rqres.plot() function
439(2)
12.7.1 Example
439(1)
12.7.2 Arguments of the rqres.plot () function
440(1)
12.8 Appendix
441(1)
12.8.1 Proof of probability integral transform: Continuous case
441(1)
12.8.2 Proof of calibration: Calibrating the pdf
441(1)
12.9 Bibliographic notes
442(1)
12.10 Exercises
443(4)
VI Applications
447(76)
13 Centile estimation
449(48)
13.1 Introduction
449(6)
13.1.1 Quantile regression
451(1)
13.1.2 The LMS method and extensions
452(3)
13.1.3 Example: The Dutch boys BMI data
455(1)
13.2 Fitting centile curves
455(10)
13.2.1 The lms () function
456(2)
13.2.2 Estimating the smoothing degrees of freedom using a local GAIC
458(1)
13.2.3 The find.hyper() function
459(1)
13.2.4 Residual diagnostics
460(2)
13.2.5 The fittedPlot) function
462(3)
13.3 Plotting centile curves
465(10)
13.3.1 centiles()
465(3)
13.3.2 calibration()
468(2)
13.3.3 centiles.fan()
470(1)
13.3.4 centiles.split()
471(2)
13.3.5 Comparing centile curves: centiles. com()
473(1)
13.3.6 Plot of distribution of y for specific values of x
474(1)
13.4 Predictive centile curves: centiles.predO, z.scores()
475(5)
13.4.1 Case 1: Centile for y given x and centile percentage
476(1)
13.4.2 Case 2: Centile for y given x and centile z-score
477(1)
13.4.3 Case 3: z-score given y and x
478(2)
13.5 Quantile sheets: quantSheets()
480(7)
13.5.1 Smoothing parameters
481(1)
13.5.2 Residuals
481(1)
13.5.3 Fitting the model
482(5)
13.6 Bibliographic notes
487(1)
13.7 Exercises
488(9)
14 Further applications
497(26)
14.1 Introduction
497(1)
14.2 Count data: The fish species data
498(10)
14.3 Binomial data: The hospital stay data
508(5)
14.4 Continuous data: Revisiting the 1990s film data
513(6)
14.4.1 Preliminary analysis
513(1)
14.4.2 Modelling the data using the normal distribution
514(4)
14.4.3 Modelling the data using the BCPE distribution
518(1)
14.5 Epilogue
519(4)
Bibliography 523(20)
Index 543
Mikis D. Stasinopoulos, Robert A. Rigby, Gillian Z. Heller, Vlasios Voudouris, Fernanda De Bastiani
Pastovi nuoroda: https://www.kriso.lt/db/9781315269870_pe.html
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