genPositiveDefMat     package:clusterGeneration     R Documentation

_G_E_N_E_R_A_T_E _A _P_O_S_I_T_I_V_E _D_E_F_I_N_I_T_E _M_A_T_R_I_X/_C_O_V_A_R_I_A_N_C_E _M_A_T_R_I_X

_D_e_s_c_r_i_p_t_i_o_n:

     Generate a positive definite matrix/covariance matrix.

_U_s_a_g_e:

     genPositiveDefMat(dim, covMethod=c("eigen", "onion", "c-vine", "unifcorrmat"), 
                       alphad=1, eta=1, rangeVar=c(1,10), lambdaLow=1, ratioLambda=10)

_A_r_g_u_m_e_n_t_s:

     dim: Dimension of the matrix to be generated. 

covMethod: Method to generate positive definite matrices/covariance
          matrices. Choices are "eigen", "onion", "c-vine", or
          "unifcorrmat"; see details below.  

  alphad: parameter for unifcorrmat method to generate random
          correlation matrix $alphad=1$ for uniform. 'alphad' should be
          positive.

     eta: parameter for "c-vine" and "onion" methods to generate random
          correlation matrix $eta=1$ for uniform. 'eta' should be
          positive.

rangeVar: Range for variances of a covariance matrix (see details). The
          default range is [1, 10] which can generate reasonable
          variability of variances. 

lambdaLow: Lower bound on the eigenvalues of cluster covariance
          matrices.  If the argument 'covMethod="eigen"', eigenvalues
          are generated for cluster covariance matrices. The
          eigenvalues are randomly generated from the interval
          ['lambdaLow', 'lambdaLow'*'ratioLambda'].  In our experience,
          'lambdaLow'=1 and 'ratioLambda'=10  can give reasonable
          variability of the diameters of clusters. 'lambdaLow' should
          be positive. 

ratioLambda: The ratio of the upper bound of the eigenvalues to the
          lower bound of the  eigenvalues of cluster covariance
          matrices.  See 'lambdaLow'. 

_D_e_t_a_i_l_s:

     The current version of the function 'genPositiveDefMat' implements
     four  methods to generate random covariance matrices. The first
     method, denoted by  "eigen", first randomly generates eigenvalues 
     (lambda_1,...,lambda_p) for the covariance matrix 
     (boldsymbol{Sigma}), then uses columns of a randomly generated
     orthogonal matrix 
     (boldsymbol{Q}=(boldsymbol{alpha}_1,...,boldsymbol{alpha}_p))  as
     eigenvectors. The covariance matrix boldsymbol{Sigma} is then 
     contructed as 
     boldsymbol{Q}*diag(lambda_1,...,lambda_p)*boldsymbol{Q}^T.

     The remaining methods, denoted as "onion", "c-vine", and
     "unifcorrmat" respectively, first generates a random  correlation
     matrix (boldsymbol{R}) via the method mentioned and proposed in
     Joe (2006), then randomly generates variances
     (sigma_1^2,...,sigma_p^2) from  an interval specified by the
     argument 'rangeVar'. The covariance matrix  boldsymbol{Sigma} is
     then constructed as 
     diag(sigma_1,...,sigma_p)*boldsymbol{R}*diag(sigma_1,...,sigma_p).

_V_a_l_u_e:

egvalues: eigenvalues of Sigma 

   Sigma: positive definite matrix/covariance matrix 

_A_u_t_h_o_r(_s):

     Weiliang Qiu stwxq@channing.harvard.edu
      Harry Joe harry@stat.ubc.ca

_R_e_f_e_r_e_n_c_e_s:

     Joe, H. (2006) Generating Random Correlation Matrices Based on
     Partial Correlations.  _Journal of Multivariate Analysis_, *97*,
     2177-2189.

     Ghosh, S., Henderson, S. G. (2003).  Behavior of the NORTA method
     for correlated random vector generation  as the dimension
     increases. _ACM Transactions on Modeling and Computer Simulation
     (TOMACS)_, *13(3)*, 276-294.

     Kurowicka and Cooke, 2006. _Uncertainty Analysis with High
     Dimensional Dependence Modelling_, Wiley, 2006.

_E_x_a_m_p_l_e_s:

     genPositiveDefMat("unifcorrmat",dim=4)
     genPositiveDefMat("eigen",dim=5)

