Facies are the type of rocks with similar characteristics given by geologists and facies classification is of very significance in geological tasks, such as formation evaluation, reservoir characterization. As the geological data accumulates, there are growing interests in facies classification with machine learning methods [1, 2, 3, 4, 5, 6, 7, 8, 9]. There are two issues with the existing works on facies classification. First, the machine learning models are built without variable selection or with only very primary method, such as cross-validation, which makes the classifiers with useless variable as inputs and therefore with low performance. Second, most of the models for facies classification are block-box, such as deep learning [5, 10, 11], Boostings or SVMs[7], which are un-interpretable to geologists. Variable selection is a common task that selects a subset from all the available variables for machine learning models. By this, the accuracy of the predictive models built with the selected variables can be improved compared with those built without selection. The traditional method for variable selection are mainly based on likelihoods, such as AIC, BIC, or accuracy, such as LASSO [12], or correlation, such as HSIC [13], distance correlation [14], and copula entropy [15]. Copula Entropy (CE) is a recently proposed rigorous mathematical concept for measuring multivariate statistical independence and is proved to be equivalent to mutual information in information theory [16].

LiNGAM determines the variable order from cause to effect using additive noise models, but it faces challenges with confounding. Previous methods maintained LiNGAM's fundamental structure while trying to identify and address variables affected by confounding. As a result, these methods required significant computational resources regardless of the presence of confounding, and they did not ensure the detection of all confounding types. In contrast, this paper enhances LiNGAM by introducing LiNGAM-MMI, a method that quantifies the magnitude of confounding using KL divergence and arranges the variables to minimize its impact. This method efficiently achieves a globally optimal variable order through the shortest path problem formulation. LiNGAM-MMI processes data as efficiently as traditional LiNGAM in scenarios without confounding while effectively addressing confounding situations. Our experimental results suggest that LiNGAM-MMI more accurately determines the correct variable order, both in the presence and absence of confounding.

Change point detection is a typical task that aim to find changes in time series and can be tackled with two-sample test. Copula Entropy is a mathematical concept for measuring statistical independence and a two-sample test based on it was introduced recently. In this paper we propose a nonparametric multivariate method for multiple change point detection with the copula entropy-based two-sample test. The single change point detection is first proposed as a group of two-sample tests on every points of time series data and the change point is considered as with the maximum of the test statistics. The multiple change point detection is then proposed by combining the single change point detection method with binary segmentation strategy. We verified the effectiveness of our method and compared it with the other similar methods on the simulated univariate and multivariate data and the Nile data.

Identifying differential equation governing dynamical system is an important problem with wide applications. Copula Entropy (CE) is a mathematical concept for measuring statistical independence in information theory. In this paper we propose a method for identifying differential equation of dynamical systems with CE. The problem is considered as a variable selection problem and solved with the previously proposed CE-based method for variable selection. The proposed method composed of two components: the difference operator and the CE estimator. Since both components can be done non-parametrically, the proposed method is therefore model-free and hyperparameter-free. The simulation experiment with the 3D Lorenz system verified the effectiveness of the proposed method.

Time lag between variables is a key characteristics of dynamical systems in different fields and identifying such time lag is an important problem in complex systems with many applications. Transfer Entropy (TE) was proposed as a tool for time lag identification recently. Unfortunately, estimating TE has been a notoriously difficult problem. Copula Entropy (CE) is a measure of statistical independence and it was proved that TE can be represented with only CE. Therefore, a non-parametric estimator of TE based on CE was proposed according to such representation recently. In this paper we propose to use the CE-based estimator of TE to identify time lag in dynamical systems. Both simulated and real data are used to verify the effectiveness of the proposed method in the experiments. Experimental results show that the proposed method can identify the time lags in the four simulated systems. The real data experiment with the data on power consumption of the Tetouan city also demonstrates that our method can identify the pattern of time lags through the estimated TE from the weather factors to the power consumption of the city.

Variable selection is an important problem in statistics and machine learning. Copula Entropy (CE) is a mathematical concept for measuring statistical independence and has been applied to variable selection recently. In this paper we propose to apply the CE-based method for variable selection to survival analysis. The idea is to measure the correlation between variables and time-to-event with CE and then select variables according to their CE value. Experiments on simulated data and two real cancer data were conducted to compare the proposed method with two related methods: random survival forest and Lasso-Cox. Experimental results showed that the proposed method can select the 'right' variables out that are more interpretable and lead to better prediction performance.

We propose a novel estimator of the mutual information between two ordinal vectors $x$ and $y$. Our approach is inductive (as opposed to deductive) in that it depends on the data generating distribution solely through some nonparametric properties revealing associations in the data, and does not require having enough data to fully characterize the true joint distributions $P_{x, y}$. Specifically, our approach consists of (i) noting that $I\left(y; x\right) = I\left(u_y; u_x\right)$ where $u_y$ and $u_x$ are the copula-uniform dual representations of $y$ and $x$ (i.e. their images under the probability integral transform), and (ii) estimating the copula entropies $h\left(u_y\right)$, $h\left(u_x\right)$ and $h\left(u_y, u_x\right)$ by solving a maximum-entropy problem over the space of copula densities under a constraint of the type $\alpha_m = E\left[\phi_m(u_y, u_x)\right]$. We prove that, so long as the constraint is feasible, this problem admits a unique solution, it is in the exponential family, and it can be learned by solving a convex optimization problem. The resulting estimator, which we denote MIND, is marginal-invariant, always non-negative, unbounded for any sample size $n$, consistent, has MSE rate $O(1/n)$, and is more data-efficient than competing approaches. Beyond mutual information estimation, we illustrate that our approach may be used to mitigate mode collapse in GANs by maximizing the entropy of the copula of fake samples, a model we refer to as Copula Entropy Regularized GAN (CER-GAN).

Fall is a leading cause of death which suffers the elderly and society. Timed Up and Go (TUG) test is a common tool for fall risk assessment. In this paper, we propose a method for predicting TUG score from gait characteristics extracted from video based on computer vision and machine learning technologies. First, 3D pose is estimated from video captured with 2D and 3D cameras during human motion and then a group of gait characteristics are computed from 3D pose series. After that, copula entropy is used to select those characteristics which are mostly associated with TUG score. Finally, the selected characteristics are fed into the predictive models to predict TUG score. Experiments on real world data demonstrated the effectiveness of the proposed method. As a byproduct, the associations between TUG score and several gait characteristics are discovered, which laid the scientific foundation of the proposed method and make the predictive models such built interpretable to clinical users.

Variable selection is of significant importance for classification and regression tasks in machine learning and statistical applications where both predictability and explainability are needed. In this paper, a Copula Entropy (CE) based method for variable selection which use CE based ranks to select variables is proposed. The method is both model-free and tuning-free. Comparison experiments between the proposed method and traditional variable selection methods, such as Stepwise Selection, regularized generalized linear models and Adaptive LASSO, were conducted on the UCI heart disease data. Experimental results show that CE based method can select the `right' variables out effectively and derive better interpretable results than traditional methods do without sacrificing accuracy performance. It is believed that CE based variable selection can help to build more explainable models.

Causal inference is a fundemental problem in statistics and has wide applications in different fields. Transfer Entropy (TE) is a important notion defined for measuring causality, which is essentially conditional Mutual Information (MI). Copula Entropy (CE) is a theory on measurement of statistical independence and is equivalent to MI. In this paper, we prove that TE can be represented with only CE and then propose a non-parametric method for estimating TE via CE. The proposed method was applied to analyze the Beijing PM2.5 data in the experiments. Experimental results show that the proposed method can infer causality relationships from data effectively and hence help to understand the data better.