| Preface |
|
ix | |
|
1 A Data Assimilation Reminder |
|
|
1 | (4) |
|
1.1 Recalling the Basic Idea of Statistical Data Assimilation |
|
|
1 | (3) |
|
1.2 What Is in the Following
Chapters? |
|
|
4 | (1) |
|
2 Remembrance of Things Path |
|
|
5 | (9) |
|
2.1 Recursion Relation along the Path X |
|
|
6 | (2) |
|
2.2 The `Action' A(X) = --log[ P(X|Y)] |
|
|
8 | (1) |
|
2.3 Multiple Measurement Windows in Time |
|
|
9 | (1) |
|
2.4 The Standard Model for SDA |
|
|
9 | (2) |
|
2.5 The Standard Model Action for the Hodgkin-Huxley NaKL Model |
|
|
11 | (2) |
|
|
|
13 | (1) |
|
3 SDA Variational Principles |
|
|
14 | (12) |
|
3.1 Estimating Expected Value Integrals |
|
|
14 | (1) |
|
3.2 Laplace's Method for Estimating Expected Value Integrals |
|
|
15 | (3) |
|
3.3 The Euler-Lagrange Equations for the Standard Model: Continuous Time |
|
|
18 | (8) |
|
4 Using Waveform Information |
|
|
26 | (21) |
|
4.1 Inconsistency in the Standard Model Action |
|
|
26 | (1) |
|
4.2 Time Delay State Vectors and Data |
|
|
27 | (3) |
|
4.3 "Old Nudging" for Proxy Vectors |
|
|
30 | (1) |
|
4.4 L = 1; x1(t) Is Observed |
|
|
31 | (2) |
|
4.5 Regularizing the Local Inverse of ∂S/∂x |
|
|
33 | (1) |
|
4.6 Computing the Pseudoinverse with Singular Value Decomposition |
|
|
33 | (1) |
|
|
|
34 | (1) |
|
4.8 Synchronization Errors in Time |
|
|
35 | (5) |
|
|
|
40 | (4) |
|
4.10 The Euler-Lagrange Equations for Measurement Terms in Proxy Vectors |
|
|
44 | (1) |
|
4.11 Simplified Use of Waveforms |
|
|
45 | (2) |
|
5 Annealing in the Model Precision Rf |
|
|
47 | (19) |
|
5.1 Varying the Hyperparameter Rf |
|
|
49 | (4) |
|
5.2 Lorenz96 Model with D = 5 |
|
|
53 | (5) |
|
5.3 Hodgkin-Huxley NaKL Neuron |
|
|
58 | (5) |
|
5.4 Qualitative Commentary about Precision Annealing |
|
|
63 | (3) |
|
6 Discrete Time Integration in Data Assimilation Variational Principles: Lagrangian and Hamiltonian Formulations |
|
|
66 | (29) |
|
6.1 Symplecticity in Variational Data Assimilation |
|
|
69 | (8) |
|
6.2 A Symplectic Annealing Method |
|
|
77 | (2) |
|
6.3 Three Integration Methods |
|
|
79 | (2) |
|
6.4 Numerical Twin Experiments |
|
|
81 | (11) |
|
6.5 Summary of Symplectic Annealing Methods |
|
|
92 | (3) |
|
|
|
95 | (24) |
|
7.1 Metropolis-Hastings -- Random Proposals |
|
|
98 | (2) |
|
7.2 Precision Annealing MHR Sampling |
|
|
100 | (4) |
|
7.3 Hamiltonian Monte Carlo Methods -- Structured Proposals |
|
|
104 | (4) |
|
7.4 Underappreciating HMC |
|
|
108 | (1) |
|
7.5 Using PAHMC on Two Model Dynamics |
|
|
108 | (1) |
|
|
|
108 | (3) |
|
7.7 Lorenz96 Model; D = 20 |
|
|
111 | (2) |
|
7.8 Computational Considerations for PAHMR and PAHMC Procedures |
|
|
113 | (6) |
|
8 Machine Learning and Its Equivalence to Statistical Data Assimilation |
|
|
119 | (21) |
|
8.1 (A(X)) = (-- log P(X|Y)); Action Is Information |
|
|
119 | (1) |
|
8.2 General Discussion of ML |
|
|
120 | (3) |
|
8.3 Using ML to Predict Subsequent Terms in a Time Series {s(n)} |
|
|
123 | (3) |
|
8.4 Action Levels for the Given Time Series {s(n)} |
|
|
126 | (1) |
|
8.5 Errors in Training and Validation |
|
|
126 | (5) |
|
8.6 "Twin Experiment" with a Multi-layer Perceptron |
|
|
131 | (6) |
|
8.7 Continuous Layers: Deepest Learning |
|
|
137 | (1) |
|
8.8 Comments on a Set of Curated Retinal Images |
|
|
138 | (2) |
|
9 Two Examples of the Practical Use of Data Assimilation |
|
|
140 | (32) |
|
9.1 Data Assimilation in Action |
|
|
140 | (1) |
|
9.2 Experimental Data on Neurons in the Avian Brain |
|
|
141 | (8) |
|
9.3 Shallow Water Equations; Lagrangian Drifters |
|
|
149 | (6) |
|
9.4 Time-Delayed Nudging Used in the Shallow Water Equations |
|
|
155 | (4) |
|
|
|
159 | (1) |
|
9.6 Nonlinear Shallow Water Equations |
|
|
160 | (1) |
|
9.7 Results with Time Delay Nudging for the Shallow Water Equations |
|
|
161 | (10) |
|
|
|
171 | (1) |
|
|
|
172 | (2) |
| Bibliography |
|
174 | (9) |
| Index |
|
183 | |
Lisainfo e-raamatute kohta