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Nearest Neighbor Conditional Estimation for Harris Recurrent Markov Chains
This paper is concerned with consistent nearest neighbor time series estimation for data generated by a Harris recurrent Markov chain. The goal is to validate nearest neighbor estimation in this general time series context, using simple and weak conditions. The framework considered covers, in a unified manner, a wide variety of statistical quantities, e.g. autoregression function, conditional quantiles, conditional tail estimators and, more generally, extremum estimators. The focus is theoretical, but examples are given to highlight applications
Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
We introduce a data-driven forecasting method for high-dimensional chaotic
systems using long short-term memory (LSTM) recurrent neural networks. The
proposed LSTM neural networks perform inference of high-dimensional dynamical
systems in their reduced order space and are shown to be an effective set of
nonlinear approximators of their attractor. We demonstrate the forecasting
performance of the LSTM and compare it with Gaussian processes (GPs) in time
series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation
and a prototype climate model. The LSTM networks outperform the GPs in
short-term forecasting accuracy in all applications considered. A hybrid
architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is
proposed to ensure convergence to the invariant measure. This novel hybrid
method is fully data-driven and extends the forecasting capabilities of LSTM
networks.Comment: 31 page
Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data
Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used
in the machine learning and dynamical systems literature to represent complex
dynamical or sequential relationships between variables. More recently, as deep
learning models have become more common, RNNs have been used to forecast
increasingly complicated systems. Dynamical spatio-temporal processes represent
a class of complex systems that can potentially benefit from these types of
models. Although the RNN literature is expansive and highly developed,
uncertainty quantification is often ignored. Even when considered, the
uncertainty is generally quantified without the use of a rigorous framework,
such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a
more formal framework while maintaining the forecast accuracy that makes these
models appealing, by presenting a Bayesian RNN model for nonlinear
spatio-temporal forecasting. Additionally, we make simple modifications to the
basic RNN to help accommodate the unique nature of nonlinear spatio-temporal
data. The proposed model is applied to a Lorenz simulation and two real-world
nonlinear spatio-temporal forecasting applications
Optimal model-free prediction from multivariate time series
Forecasting a time series from multivariate predictors constitutes a
challenging problem, especially using model-free approaches. Most techniques,
such as nearest-neighbor prediction, quickly suffer from the curse of
dimensionality and overfitting for more than a few predictors which has limited
their application mostly to the univariate case. Therefore, selection
strategies are needed that harness the available information as efficiently as
possible. Since often the right combination of predictors matters, ideally all
subsets of possible predictors should be tested for their predictive power, but
the exponentially growing number of combinations makes such an approach
computationally prohibitive. Here a prediction scheme that overcomes this
strong limitation is introduced utilizing a causal pre-selection step which
drastically reduces the number of possible predictors to the most predictive
set of causal drivers making a globally optimal search scheme tractable. The
information-theoretic optimality is derived and practical selection criteria
are discussed. As demonstrated for multivariate nonlinear stochastic delay
processes, the optimal scheme can even be less computationally expensive than
commonly used sub-optimal schemes like forward selection. The method suggests a
general framework to apply the optimal model-free approach to select variables
and subsequently fit a model to further improve a prediction or learn
statistical dependencies. The performance of this framework is illustrated on a
climatological index of El Ni\~no Southern Oscillation.Comment: 14 pages, 9 figure
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