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E-grāmata: Predicting the Future: Completing Models of Observed Complex Systems

  • Formāts: PDF+DRM
  • Sērija : Understanding Complex Systems
  • Izdošanas datums: 12-Jun-2013
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781461472186
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Predicting the Future: Completing Models of Observed Complex SystemsPalielināt
  • Formāts: PDF+DRM
  • Sērija : Understanding Complex Systems
  • Izdošanas datums: 12-Jun-2013
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781461472186
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This book discusses model building and evaluation across disciplines, by means of an exact path integral for transferring information from observations to a model of the observed system. Offers examples in geosciences, nonlinear electrical circuits and more.

Through the development of an exact path integral for use in transferring information from observations to a model of the observed system, the author provides a general framework for the discussion of model building and evaluation across disciplines. Through many illustrative examples drawn from models in neuroscience, geosciences, and nonlinear electrical circuits, the concepts are exemplified in detail. Practical numerical methods for approximate evaluations of the path integral are explored, and their use in designing experiments and determining a model’s consistency with observations is explored.
1 An Overview: The Challenge of Complex Systems
1(6)
1.1 Introduction
1(6)
1.1.1 Data Assimilation as a Communications Problem
3(2)
1.1.2 Outline of this Book
5(2)
2 Examples as a Guide to the Issues
7(44)
2.1 The Malkus Waterwheel
8(9)
2.1.1 A Physics Question About the Waterwheel
10(7)
2.2 The Colpitts Oscillator
17(19)
2.2.1 Colpitts Circuit Equations
19(1)
2.2.2 Estimation with Chaotic Signals
20(5)
2.2.3 Instability of the Synchronization Manifold
25(2)
2.2.4 Regularized Cost Function
27(1)
2.2.5 Experimental Colpitts Oscillator Redux
28(5)
2.2.6 Numerical Optimization Methods
33(3)
2.3 A Hodgkin-Huxley Neuron Model
36(10)
2.3.1 Biophysics of the Hodgkin-Huxley Model
36(4)
2.3.2 Estimating Parameters and Unobserved States of the HH Model
40(2)
2.3.3 Predicting the Response of the HH Model
42(3)
2.3.4 Consequences of the Wrong Model
45(1)
2.4 Synopsis and Perspectives: "Slightly Complex" Examples
46(5)
3 General Formulation of Statistical Data Assimilation
51(34)
3.1 Data Assimilation Without Data
52(6)
3.1.1 Deterministic Dynamics: Path Integral
52(4)
3.1.2 Relation to the Quantum Mechanical Path Integral
56(1)
3.1.3 Noisy Dynamics
57(1)
3.2 Data Assimilation with a Little Bit of Data
58(9)
3.2.1 Mutual Information
58(3)
3.2.2 One Measurement
61(3)
3.2.3 Two Measurements
64(3)
3.3 The General Data Assimilation Problem
67(8)
3.3.1 Differential Equations to Discrete Time Maps
67(2)
3.3.2 Errors and Noise: Stochastic Data Assimilation
69(6)
3.4 Approximating the Action
75(2)
3.5 The Value of a Measurement
77(1)
3.6 Predicting
78(1)
3.7 The Scientific Value of the Path Integral
79(1)
3.8 Data Assimilation Path Integrals in Continuous Time
80(3)
3.9 Earlier Work on Path Integrals in Statistical Data Assimilation
83(1)
3.10 Synopsis and Perspectives: Statistical Data Assimilation
84(1)
4 Evaluating the Path Integral
85(40)
4.1 Guide to Methods for Estimating the Path Integral
85(2)
4.2 Stationary Path Methods
87(5)
4.2.1 No Model Error
89(2)
4.2.2 Model Errors
91(1)
4.3 Beyond the Stationary Path: Loop Expansions
92(9)
4.3.1 The Effective Action: Numerical Optimization Reappears
92(1)
4.3.2 The Effective Action
93(1)
4.3.3 Dyson-Schwinger Equations for Statistical Data Assimilation
94(4)
4.3.4 The Effective Action: Loop Expansion
98(3)
4.4 Estimating the Path Distribution exp[ -A0(X)]
101(2)
4.4.1 Langevin Equations: Fokker-Planck
101(2)
4.5 Monte Carlo Methods
103(9)
4.5.1 Metropolis-Hastings (Rosenbluth) Methods
103(1)
4.5.2 Using GPU Parallel Processing
104(2)
4.5.3 Example Monte Carlo Problem: NaKL Neuron Model
106(6)
4.6 Consistency of Model Errors
112(11)
4.6.1 An Example from the Lorenz96 Model with D = 100
115(3)
4.6.2 An Example from the Lorenz96 Model with D = 20
118(4)
4.6.3 Comments on Consistent Model Errors
122(1)
4.7 Synopsis and Perspectives: Evaluating the Path Integral
123(2)
5 Twin Experiments
125(74)
5.1 The Roles of Twin Experiments
126(1)
5.2 Neuron Models
126(19)
5.2.1 NaKL Hodgkin-Huxley Model
127(2)
5.2.2 The Importance of the Regularizing Variable u(t)
129(2)
5.2.3 Frequency Content of the Stimulus Iapp(t)
131(1)
5.2.4 Additive Noise in the Observations
132(2)
5.2.5 An Additional Current: Ih
134(1)
5.2.6 NaKLh Twin Experiments
135(3)
5.2.7 Model Testing Through Prediction
138(1)
5.2.8 Robustness to Model Errors; NaKL Model ↔ NaKLh Model
139(1)
5.2.9 NaKLh Data Presented to an NaKL Model
140(3)
5.2.10 NaKL Data Driving an NaKLh Model; Pruning a Big Model
143(2)
5.3 The Lorenz96 Model
145(14)
5.3.1 Synchronization
146(3)
5.3.2 Lyapunov Exponents
149(5)
5.3.3 CLEs for Lorenz96 Models
154(3)
5.3.4 Lorenz96 Model: Variational Principle; No Model Errors
157(2)
5.4 Monte Carlo Estimation of the Path Integral for the Lorenz96 Model
159(15)
5.4.1 How Many Observations Are Required?
162(4)
5.4.2 Results of Monte Carlo Estimation of the Path Integral for Moments of X and Model Parameters
166(1)
5.4.3 Prediction by Model Equations for t > tm
167(5)
5.4.4 Non-Gaussian Measurement Error
172(1)
5.4.5 How Often Should One Make Measurements?
172(2)
5.5 Shallow Water Equations
174(18)
5.5.1 One-Layer Shallow Water Flow
175(3)
5.5.2 Statistical Data Analysis for the Shallow Water Equations
178(2)
5.5.3 Generating the Data
180(2)
5.5.4 Synchronization of the Data with the Model Output
182(5)
5.5.5 Results for the Shallow Water Equations: Synchronization Implies Predictability
187(5)
5.6 Synopsis and Perspectives: Twin Experiments
192(7)
6 Analysis of Experimental Data
199(22)
6.1 The Avian Song System: Individual Neurons
200(2)
6.2 Experimental Procedures for HVC Neurons
202(1)
6.2.1 HVC Slice Preparation: Experimental Procedure
202(1)
6.3 Experimental Results and Analysis
203(1)
6.4 Model Details
203(4)
6.5 Details of the Numerical Evaluation
207(1)
6.5.1 Estimation and Prediction of HVC Neuron Responses to Injected Current
208(1)
6.6 Results from Data Acquisition and Analysis
208(11)
6.6.1 Neuron 20110413_4_1 Epoch 22
208(2)
6.6.2 Neuron 20120517_1_1 Epochs 11 and 12
210(4)
6.6.3 Neuron 20120406_1_3 Epochs 19 and 15
214(2)
6.6.4 Estimated Currents and Channel Kinetics
216(3)
6.7 Comments on the Analysis of These Data
219(1)
6.8 Synopsis and Perspectives: Analysis of Experimental Data
219(2)
7 Unfinished Business
221(6)
7.1 "More Measurements": The Use of Time Delay Phase Space Reconstruction
223(4)
Bibliography 227(6)
Index 233
Henry Abarbanel is a new member of the Springer Complexity Board. He is a Professor of Physics at UCSD in La Jolla, CA. http://neurograd.ucsd.edu/faculty/detail.php?id=19
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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