Introduction |
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xv | |
Preface |
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xvii | |
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1 Examples of spatio-temporal data |
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1 | (22) |
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1 | (1) |
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1.2 Spatio-temporal data types |
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2 | (2) |
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1.3 Point referenced data sets used in the book |
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4 | (8) |
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1.3.1 New York air pollution data set |
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4 | (2) |
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1.3.2 Air pollution data from England and Wales |
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6 | (1) |
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1.3.3 Air pollution in the eastern US |
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7 | (1) |
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1.3.4 Hubbard Brook precipitation data |
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7 | (2) |
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1.3.5 Ocean chlorophyll data |
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9 | (1) |
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1.3.6 Atlantic ocean temperature and salinity data set |
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10 | (2) |
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1.4 Areal unit data sets used in the book |
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12 | (8) |
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1.4.1 Covid-19 mortality data from England |
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12 | (2) |
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1.4.2 Childhood vaccination coverage in Kenya |
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14 | (2) |
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1.4.3 Cancer rates in the United States |
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16 | (2) |
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1.4.4 Hospitalization data from England |
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18 | (1) |
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1.4.5 Child poverty in London |
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18 | (2) |
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20 | (1) |
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20 | (3) |
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2 Jargon of spatial and spatio-temporal modeling |
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23 | (26) |
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23 | (1) |
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23 | (2) |
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25 | (1) |
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2.4 Variogram and covariogram |
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26 | (2) |
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28 | (1) |
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2.6 Matern covariance function |
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29 | (3) |
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2.7 Gaussian processes (GP) GP(O, C(Ψ)) |
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32 | (2) |
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2.8 Space-time covariance functions |
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34 | (3) |
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2.9 Kriging or optimal spatial prediction |
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37 | (1) |
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2.10 Autocorrelation and partial autocorrelation |
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38 | (1) |
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2.11 Measures of spatial association for areal data |
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39 | (2) |
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2.12 Internal and external standardization for areal data |
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41 | (1) |
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42 | (2) |
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44 | (2) |
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46 | (1) |
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47 | (1) |
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47 | (2) |
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3 Exploratory data analysis methods |
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49 | (20) |
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49 | (1) |
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3.2 Exploring point reference data |
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50 | (5) |
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3.2.1 Non-spatial graphical exploration |
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50 | (1) |
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3.2.2 Exploring spatial variation |
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51 | (4) |
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3.3 Exploring spatio-temporal point reference data |
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55 | (3) |
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3.4 Exploring areal Covid-19 case and death data |
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58 | (8) |
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3.4.1 Calculating the expected numbers of cases and deaths |
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59 | (2) |
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3.4.2 Graphical displays and covariate information |
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61 | (5) |
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66 | (1) |
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67 | (2) |
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4 Bayesian inference methods |
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69 | (52) |
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69 | (4) |
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4.2 Prior and posterior distributions |
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73 | (1) |
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4.3 The Bayes theorem for probability |
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73 | (1) |
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4.4 Bayes theorem for random variables |
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74 | (2) |
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4.5 Posterior α Likelihood × Prior |
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76 | (1) |
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4.6 Sequential updating of the posterior distribution |
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77 | (1) |
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4.7 Normal-Normal example |
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77 | (3) |
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80 | (4) |
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81 | (1) |
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82 | (1) |
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82 | (2) |
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84 | (1) |
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85 | (2) |
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4.10.1 Conjugate prior distribution |
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85 | (1) |
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4.10.2 Locally uniform prior distribution |
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86 | (1) |
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4.10.3 Non-informative prior distribution |
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86 | (1) |
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4.11 Posterior predictive distribution |
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87 | (4) |
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4.11.1 Normal-Normal example |
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89 | (2) |
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4.12 Prior predictive distribution |
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91 | (1) |
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4.13 Inference for more than one parameter |
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92 | (1) |
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4.14 Normal example with both parameters unknown |
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93 | (5) |
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98 | (5) |
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98 | (2) |
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4.15.2 Posterior probability of a model |
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100 | (1) |
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4.15.3 Hypothesis testing |
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101 | (2) |
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4.16 Criteria-based Bayesian model selection |
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103 | (10) |
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105 | (2) |
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107 | (3) |
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4.16.3 Posterior predictive model choice criteria (PMCC) |
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110 | (3) |
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4.17 Bayesian model checking |
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113 | (2) |
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4.17.1 Nuisance parameters |
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114 | (1) |
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4.18 The pressing need for Bayesian computation |
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115 | (1) |
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116 | (1) |
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116 | (5) |
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5 Bayesian computation methods |
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121 | (32) |
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121 | (1) |
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5.2 Two motivating examples for Bayesian computation |
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122 | (2) |
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5.3 Monte Carlo integration |
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124 | (1) |
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125 | (3) |
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128 | (1) |
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5.6 Notions of Markov chains for understanding MCMC |
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129 | (2) |
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5.7 Metropolis-Hastings algorithm |
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131 | (3) |
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134 | (2) |
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5.9 Hamiltonian Monte Carlo |
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136 | (3) |
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5.10 Integrated nested Laplace approximation (INLA) |
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139 | (2) |
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5.11 MCMC implementation issues and MCMC output processing |
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141 | (5) |
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5.11.1 Diagnostics based on visual plots and autocorrelation |
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142 | (1) |
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143 | (2) |
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5.11.3 Method of batching |
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145 | (1) |
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5.12 Computing Bayesian model choice criteria |
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146 | (3) |
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146 | (1) |
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147 | (1) |
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148 | (1) |
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5.12.4 Computing the model choice criteria for the New York air pollution data |
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149 | (1) |
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149 | (1) |
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150 | (3) |
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6 Bayesian modeling for point referenced spatial data |
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153 | (40) |
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153 | (2) |
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6.2 Model versus procedure based methods |
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155 | (2) |
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6.3 Formulating linear models |
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157 | (7) |
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6.3.1 Data set preparation |
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157 | (1) |
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6.3.2 Writing down the model formula |
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158 | (3) |
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6.3.3 Predictive distributions |
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161 | (3) |
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6.4 Linear model for spatial data |
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164 | (4) |
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6.4.1 Spatial model fitting using bmstdr |
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166 | (2) |
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6.5 A spatial model with nugget effect |
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168 | (3) |
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6.5.1 Marginal model implementation |
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170 | (1) |
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6.6 Model fitting using software packages |
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171 | (12) |
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171 | (3) |
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174 | (5) |
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179 | (4) |
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183 | (1) |
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6.8 Model validation methods |
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184 | (7) |
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6.8.1 Four most important model validation criteria |
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185 | (2) |
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6.8.2 K-fold cross-validation |
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187 | (1) |
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6.8.3 Illustrating the model validation statistics |
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188 | (3) |
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6.9 Posterior predictive checks |
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191 | (1) |
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191 | (1) |
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192 | (1) |
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7 Bayesian modeling for point referenced spatio-temporal data |
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193 | (46) |
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193 | (4) |
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7.2 Models with spatio-temporal error distribution |
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197 | (9) |
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7.2.1 Posterior distributions |
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198 | (1) |
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7.2.2 Predictive distributions |
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199 | (2) |
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7.2.3 Simplifying the expressions: Σ12H-1 and Σ12H-1Σ21 |
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201 | (2) |
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203 | (1) |
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7.2.5 Illustration of a spatio-temporal model fitting |
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203 | (3) |
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7.3 Independent GP model with nugget effect |
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206 | (11) |
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7.3.1 Full model implementation using spTimer |
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207 | (5) |
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7.3.2 Marginal model implementation using Stan |
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212 | (5) |
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7.4 Auto regressive (AR) models |
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217 | (7) |
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7.4.1 Hierarchical AR Models using spTimer |
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217 | (4) |
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7.4.2 AR modeling using INLA |
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221 | (3) |
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7.5 Spatio-temporal dynamic models |
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224 | (5) |
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7.5.1 A spatially varying dynamic model spTDyn |
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224 | (2) |
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7.5.2 A dynamic spatio-temporal model using spBayes |
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226 | (3) |
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7.6 Spatio-temporal models based on Gaussian predictive processes (GPP) |
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229 | (5) |
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7.7 Performance assessment of all the models |
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234 | (2) |
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236 | (1) |
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237 | (2) |
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8 Practical examples of point referenced data modeling |
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239 | (38) |
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239 | (1) |
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8.2 Estimating annual average air pollution in England and Wales |
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239 | (5) |
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8.3 Assessing probability of non-compliance in air pollution |
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244 | (7) |
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8.4 Analyzing precipitation data from the Hubbard Experimental Forest |
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251 | (15) |
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8.4.1 Exploratory data analysis |
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251 | (6) |
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8.4.2 Modeling and validation |
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257 | (4) |
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8.4.3 Predictive inference from model fitting |
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261 | (1) |
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8.4.3.1 Selecting gauges for possible downsizing |
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261 | (1) |
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8.4.3.2 Spatial patterns in 3-year rolling average annual precipitation |
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262 | (2) |
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8.4.3.3 Catchment specific trends in annual precipitation |
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264 | (1) |
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8.4.3.4 A note on model fitting |
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265 | (1) |
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8.5 Assessing annual trends in ocean chlorophyll levels |
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266 | (2) |
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8.6 Modeling temperature data from roaming ocean Argo floats |
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268 | (7) |
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8.6.1 Predicting an annual average temperature map |
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272 | (3) |
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275 | (1) |
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276 | (1) |
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9 Bayesian forecasting for point referenced data |
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277 | (24) |
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277 | (3) |
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9.2 Exact forecasting method for GP |
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280 | (4) |
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9.2.1 Example: Hourly ozone levels in the Eastern US |
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281 | (3) |
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9.3 Forecasting using the models implemented in spTimer |
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284 | (4) |
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9.3.1 Forecasting using GP models |
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285 | (1) |
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9.3.2 Forecasting using AR models |
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286 | (1) |
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9.3.3 Forecasting using the GPP models |
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287 | (1) |
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9.4 Forecast calibration methods |
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288 | (5) |
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288 | (2) |
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9.4.2 Illustrating the calibration plots |
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290 | (3) |
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9.5 Example comparing GP, AR and GPP models |
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293 | (2) |
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9.6 Example: Forecasting ozone levels in the Eastern US |
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295 | (5) |
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300 | (1) |
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300 | (1) |
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10 Bayesian modeling for areal unit data |
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301 | (32) |
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301 | (1) |
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10.2 Generalized linear models |
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302 | (6) |
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10.2.1 Exponential family of distributions |
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302 | (2) |
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304 | (2) |
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306 | (1) |
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10.2.4 The implied likelihood function |
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307 | (1) |
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10.2.5 Model specification using a GLM |
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307 | (1) |
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10.3 Example: Bayesian generalized linear model |
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308 | (4) |
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10.3.1 GLM fitting with binomial distribution |
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309 | (1) |
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10.3.2 GLM fitting with Poisson distribution |
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310 | (1) |
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10.3.3 GLM fitting with normal distribution |
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311 | (1) |
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10.4 Spatial random effects for areal unit data |
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312 | (2) |
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10.5 Revisited example: Bayesian spatial generalized linear model |
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314 | (4) |
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10.5.1 Spatial GLM fitting with binomial distribution |
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315 | (1) |
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10.5.2 Spatial GLM fitting with Poisson distribution |
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316 | (1) |
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10.5.3 Spatial GLM fitting with normal distribution |
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317 | (1) |
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10.6 Spatio-temporal random effects for areal unit data |
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318 | (2) |
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10.6.1 Linear model of trend |
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318 | (1) |
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319 | (1) |
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319 | (1) |
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10.6.4 Temporal autoregressive model |
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320 | (1) |
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10.7 Example: Bayesian spatio-temporal generalized linear model |
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320 | (7) |
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10.7.1 Spatio-temporal GLM fitting with binomial distribution |
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321 | (1) |
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10.7.2 Spatio-temporal GLM fitting with Poisson distribution |
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322 | (1) |
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10.7.3 Examining the model fit |
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323 | (2) |
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10.7.4 Spatio-temporal GLM fitting with normal distribution |
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325 | (2) |
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10.8 Using INLA for model fitting and validation |
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327 | (3) |
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330 | (1) |
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331 | (2) |
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11 Further examples of areal data modeling |
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333 | (24) |
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333 | (1) |
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11.2 Assessing childhood vaccination coverage in Kenya |
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333 | (5) |
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11.3 Assessing trend in cancer rates in the US |
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338 | (4) |
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11.4 Localized modeling of hospitalization data from England |
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342 | (6) |
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344 | (1) |
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11.4.2 Model fitting results |
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345 | (3) |
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11.5 Assessing trend in child poverty in London |
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348 | (6) |
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11.5.1 Adaptive CAR-AR model |
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350 | (1) |
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11.5.2 Model fitting results |
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351 | (3) |
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354 | (1) |
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354 | (3) |
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12 Gaussian processes for data science and other applications |
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357 | (20) |
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357 | (3) |
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12.2 Learning methods and their Bayesian interpretations |
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360 | (9) |
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12.2.1 Learning with empirical risk minimization |
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362 | (2) |
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12.2.2 Learning by complexity penalization |
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364 | (1) |
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12.2.3 Supervised learning and generalized linear models |
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365 | (1) |
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12.2.4 Ridge regression, LASSO and elastic net |
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365 | (3) |
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12.2.5 Regression trees and random forests |
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368 | (1) |
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12.3 Gaussian Process (GP) prior-based machine learning |
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369 | (4) |
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12.3.1 Example: predicting house prices |
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371 | (2) |
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12.4 Use of GP in Bayesian calibration of computer codes |
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373 | (2) |
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375 | (1) |
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375 | (2) |
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Appendix A Statistical densities used in the book |
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377 | (6) |
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377 | (4) |
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381 | (2) |
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Appendix B Answers to selected exercises |
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383 | (12) |
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B.1 Solutions to Exercises in Chapter 4 |
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383 | (7) |
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B.2 Solutions to Exercises in Chapter 5 |
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390 | (5) |
Bibliography |
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395 | (12) |
Glossary |
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407 | (2) |
Index |
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409 | |