Package: copent
copentEstimating Copula Entropy and Transfer EntropyIntroductionThe nonparametric methods for estimating copula entropy, transfer entropy, and the statistics for multivariate normality test and two-sample test are implemented. The change point detection method based on this two-sample test is also implemented.The method for estimating copula entropy composes of two simple steps: estimating empirical copula by rank statistic and estimating copula entropy with the KSG method. Copula Entropy is a mathematical concept for multivariate statistical independence measuring and testing, and proved to be equivalent to mutual information. Different from Pearson Correlation Coefficient, Copula Entropy is defined for non-linear, high-order and multivariate cases, which makes it universally applicable. Estimating copula entropy can be applied to many cases, including but not limited to variable selection and causal discovery (by estimating transfer entropy). Please refer to Ma and Sun (2011) <doi:10.1016/S1007-0214(11)70008-6> for more information.The nonparametric method for estimating transfer entropy composes of two steps: estimating three copula entropy and calculating transfer entropy from the estimated copula entropies. A function for conditional independence testing is also provided. Please refer to Ma (2019) <arXiv:1910.04375> for more information.The copula entropy based statistics for multivariate normality test and two-sample test are implemented. The change point detection method based on this two-sample test is also implemented. Please refer to Ma (2022) <arXiv:2206.05956>, Ma (2023) <arXiv:2307.07247>, and Ma (2024) <doi:10.13140/RG.2.2.16378.26562> for more details.Functionscopent -- estimating copula entropy;construct_empirical_copula -- the first step of the copent function, which estimates empirical copula for data by rank statistics;entknn -- the second step of the copent function, which estimates copula entropy from empirical copula with kNN method;ci -- conditional independence testing based on copula entropytransent -- estimating transfer entropy via copula entropymvnt -- estimating the copula entropy-based statistic for multivariate normality testtst -- estimating the copula entropy-based statistic for two-sample testcpd -- single change point detectionmcpd -- multiple change point detectionParametersx: N * d data, N samples, d dimensionsk: kth nearest neighbour, parameter for kNN entropy estimation. default = 3dtype: distance type, can be 'euclidean' or 'chebychev' (for Maximum Distance)lag: time lag. default = 1s0,s1: two samples with same dimensionn: repeat time of estimationthd	: threshold for the statistic of two-sample testmaxp	: maximal number of change pointsminseglen : minimal length of binary segmentationInstallationThe package can be installed from PyPI directly:pip install copentThe package can be installed from Github:pip install git+https://github.com/majianthu/pycopent.gitUsage Examplesestimating copula entropyfromnumpy.randomimportmultivariate_normalasmnormimportcopentrho=0.6mean1=[0,0]cov1=[[1,rho],[rho,1]]x=mnorm(mean1,cov1,200)# bivariate gaussiance1=copent.copent(x)# estimated copula entropyestimating transfer entropyfromcopentimporttransentfrompandasimportread_csvimportnumpyasnpurl="https://archive.ics.uci.edu/ml/machine-learning-databases/00381/PRSA_data_2010.1.1-2014.12.31.csv"prsa2010=read_csv(url)# index: 5(PM2.5),6(Dew Point),7(Temperature),8(Pressure),10(Cumulative Wind Speed)data=prsa2010.iloc[2200:2700,[5,8]].valueste=np.zeros(24)forlaginrange(1,25):te[lag-1]=transent(data[:,0],data[:,1],lag)str="TE from pressure to PM2.5 at%dhours lag :%f"%(lag,te[lag-1])print(str)multivariate normality testfromnumpy.randomimportmultivariate_normalasmnormfromcopentimportmvntmean1=[0,0]cov1=[[1,0.65],[0.65,1]]data=mnorm(mean1,cov1,500)stat1=mvnt(data)two-sample testfromcopentimporttstfromnumpyimportzerosfromnumpy.randomimportmultivariate_normalasmnormm0=[0,0]rho1=0.5v0=[[1,rho1],[rho1,1]]s0=mnorm(m0,v0,400)# bivariate gaussianstat1=zeros(9)foriinrange(0,9):m1=[i,i]s1=mnorm(m1,v0,500)stat1[i]=tst(s0,s1)print(stat1[i])change point detectionfromcopentimportmcpdimportnumpyasnpfromnumpy.randomimportmultivariate_normalasmnormn1=50x1=np.random.normal(0,1,n1)x2=np.random.normal(5,1,n1)x3=np.random.normal(10,1,n1)x4=np.random.normal(1,1,n1)x=np.concatenate((x1,x2,x3,x4))pos,maxstat=mcpd(x,thd=0.2)print(pos)x1=mnorm([0,0],[[1,0],[0,1]],n1)x2=mnorm([10,10],[[1,0],[0,1]],n1)x3=mnorm([5,5],[[1,0],[0,1]],n1)x4=mnorm([1,1],[[1,0],[0,1]],n1)x=np.concatenate((x1,x2,x3,x4))pos,maxstat=mcpd(x,thd=0.2)print(pos)ReferencesJian Ma and Zengqi Sun. Mutual information is copula entropy. Tsinghua Science & Technology, 2011, 16(1): 51-54. See also arXiv preprint arXiv:0808.0845, 2008.Jian Ma. Estimating Transfer Entropy via Copula Entropy. arXiv preprint arXiv:1910.04375, 2019.Jian Ma. Multivariate Normality Test with Copula Entropy. arXiv preprint arXiv:2206.05956, 2022.Jian Ma. Two-Sample Test with Copula Entropy. arXiv preprint arXiv:2307.07247, 2023.Jian Ma. Change Point Detection with Copula Entropy based Two-Sample Test. DOI:10.13140/RG.2.2.16378.26562, 2024.