Detection of Non‐Gaussian Behavior Using Machine Learning Techniques: A Case Study on the Lorenz 63 Model

Abstract

An important assumption made in most variational, ensemble, and hybrid‐based data assimilation systems is that all minimized errors are Gaussian random variables. A theory developed at the Cooperative Institute for Research in the Atmosphere enables for the Gaussian assumption for the different types of errors to be relaxed to a lognormally distributed random variable. While this is a first step toward using more consistent distributions to model the errors involved in numerical weather/ocean prediction, we still need to be able to identify when we need to assign a lognormal distribution in a mixed Gaussian‐lognormal approach. In this paper, we present some machine learning techniques and experiments with the Lorenz 63 model. Using these machine learning techniques, we show detection of non‐Gaussian distributions can be done using two methods: a support vector machine and a neural network. This is done by training past data to classify (1) differences with the distribution statistics (means and modes) and (2) the skewness of the probability density function.

Non‐Gaussian Detection Using Machine Learning With Data Assimilation Applications

“…The training of the machine learning method is performed over 50,000 time steps to obtain a somewhat robust fit for the system. This approach is based on the method in Goodliff et al (2020).…”

“…In this experiment, we predict the probability density function of the z component of the Lorenz 63 model based on the values of x and y components of the model. Using x and y as the training data and the z ‐score of the target data, we have shown (Goodliff et al., 2020) that we can be highly precise in our predictions of the distribution of z . The z ‐score (skewness statistic $$\sqrt{{\beta }_{1}}$$) is calculated and estimated by methods shown in D'agostino et al.…”

“…In the second part of Goodliff et al. (2020), a support vector machine and a neural network were tested with the Lorenz 63 model to detect non‐Gaussian behavior. It was shown that these techniques were very capable of detecting skewness, and differences in descriptive statistics, in order to estimate, and predict, non‐Gaussianity.…”

“…As shown in the first part of Goodliff et al (2020), the trajectory of the z component of the Lorenz 63 model changes distributions on different parts of the underlying attractor, and as such if the data assimilation is to be optimized then these changes need to be used to switch from the Gaussian to the mixed distribution-based cost functions. In the second part of Goodliff et al (2020), a support vector machine and a neural network were tested with the Lorenz 63 model to detect non-Gaussian behavior. It was shown that these techniques were very capable of detecting skewness, and differences in descriptive statistics, in order to estimate, and predict, non-Gaussianity.…”

A New Technique to Quantify the Local Predictability of Extreme Events: The Backward Nonlinear Local Lyapunov Exponent Method

“…The Lorenz63 model can be used as a weather model, with only warm and cold regimes (Evans et al, 2004). It has some dynamical properties consistent with the real atmospheric system (Palmer, 1993;Goodliff et al, 2020). In addition, the Lorenz63 model is simple, which is beneficial for performing a large number of numerical simulations at relatively low computational cost.…”