Time Series, or Statistics for Stochastic Processes and Dynamical Systems

31 Oct 2022 00:17

Rates of convergence of estimators; distribution-free learning theory. Large deviation techniques. Prediction schemes. Are there universal schemes which do not demand exponentially growing volumes of data?

If you have an ergodic process, then the sample-path mean for any nice statistic you care to measure will, almost surely, converge to the distributional mean. This is even true of trajectory probabilities (i.e., if you want to know the probability of a certain finite-length trajectory, simply count how often it happens). So "sit and count" is a reliable and consistent statistical procedure. If the process mixes sufficiently quickly, the rate of convergence might even be respectable. But this doesn't say anything about the efficiency of such procedures, which is surely a consideration. And what do you do for non-ergodic processes? (Take multiple runs and hope they're telling you about different ergodic components?) Non-stationary, even?

I need to learn more about frequency-domain approaches; despite being raised as a physicist, I find the time domain much more natural. After all, the frequency domain is effectively just one choice of a function basis, and there are infinitely many others, which might in some sense be more appropriate to the process at hand. But that's at least in part a rationalization against having to learn more math.

LSE econometrics and its "general-to-specific" modeling procedure is very interesting, and I think possibly even related to stuff I've done, but I need to understand it much better than I do.

(This notebook really needs subdivision.)

See also: Bootstrapping; Change-Point Problems; Classifiers and Clustering for Time Series; Control Theory; Dynamical Systems; Equations of Motion from a Time Series; Ergodic Theory; Factor Models for High-Dimensional Time Series and Spatio-Temporal Data; Filtering, State Estimation and Signal Processing; Grammatical Inference; Inference for Stochastic Differential Equations; Information Theory; Koopman Operators for Modeling Dynamical Systems and Time Series; Machine Learning, Statistical Inference and Induction; Markov Models and Hidden Markov Models; Multiple or Vector Time Series; Neural Coding; Point Processes; Power Law Distributions, 1/f Noise and Long-Memory Processes; Prediction Processes; Markovian (and Conceivably Causal) Representations of Stochastic Processes; Recurrence Times of Stochastic Processes (also Hitting, Waiting, and First-Passage Times) Sequential Decisions Under Uncertainty; State-Space Reconstruction; Statistical Learning Theory with Dependent Data; Statistics; Stochastic Processes; Symbolic Dynamics; Universal Prediction Algorithms