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A modification of the Kaiser and Dichman (1962) procedure of generating multivariate random numbers with specified intercorrelation is proposed. The procedure works with positive and non-positive definite population correlation matrix. A SAS module is also provided to run the procedure.
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Sample size influences considerably the variability of results of estimation processes, but this issue has never previously been analyzed for line-based positional assessment methods. The basis of the analysis is a simulation process which extracts homologous road axes from the product and from the control survey and applies the four methods. Sampl...
Additional details on simulations described in this work. Also includes parameter values and run conditions for proof-of-concept analyses in laboratory cross and wild population experiments.
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... Here, · F represents the Frobenius norm of the corresponding matrix. This problem attracted great interest of a number of researchers in past decade [1,4,5,6,9,11]. A general methodology for a valid correlation matrix was firstly presented in [6], which include the hypersphere decomposition method and the spectral decomposition method. These methods were further improved in [9]. ...
... Further, we depict the curve of Frobenius norm between A and computed approximate matrix versus the number of iteration in Figure 1. with A − H MMU F = 0.1107. This further verified that the MMU algorithm is efficient, since the obtained object function is smaller than that of the method in [1]. Moreover, the eigenvalues of these matrices are listed Table 3. Table 3, it is observed that H MMU is symmetric definite (nearly semidefinite), and thus a valid nonnegative correlation matrix. ...
A modified multiplicative update algorithms is presented for computing the nearest correlation matrix. The convergence property is analyzed in details and a sufficient condition is given to guarantee that the proposed approach will not breakdown. A number of numerical experiments show that the modified multiplicative updating algorithm is efficient, and comparable with the existing algorithms.
... Journal of the National Science Foundation of Sri Lanka 39 (1) hypothesis, the mean difference between treatments (ε) ...
In crossover trials, patients receive two or more treatments in a random order in different periods. The sample size determination is often an important step in planning a crossover study. This paper concerns sample size calculations in 2x2 crossover trials, with random patient effects and no interaction between the treatment and the patient under two scenarios, namely the exact and the large sample size approaches. Simulation was carried out for determining the sample size for both scenarios. For varying parameter values, simulation was used for generating samples of the required size and examining whether the significance level and power of the tests are maintained. The results indicate that when the sample size was ≤ 5, neither method maintained error rates and when the sample size was >5 and < 12 only the exact approach maintained error rates. However, when the sample size is approximately > 12 both methods maintained error rates. In addition it was found that a saving in sample size can be achieved depending on the extent of the correlation between the observations on the same patient. The simulation results indicate that crossover studies should not be conducted when the anticipated sample size is ≤ 5 and when a sample size of >5 and < 12 is anticipated, the exact method of determining sample size should be used. When larger sample sizes are anticipated either method can be used but the method based on large sample size approximation is simpler.
... The eighth approach is a method to generate random numbers from a general symmetric pseudocorrelation matrix R (possibly indefinite) [1]. The method is based on a Cholesky factorization of R which is modified to work with indefinite R matrices. ...
... At the first row iteration we find X (1) 1 such that f 1 (X (1) ...
... . At the second row iteration we find X (1) 2 such that f 2 (X (1) 2 ) < f 2 (X (0) 2 ). But at the third row iteration we have to enforce the constraint X 3 X T 2 = 0. X 2 changed in the previous step, so the X 3 that we find may result in f 3 (X ...
We desire to find a correlation matrix R^ of a given rank that is as close as possible to an input matrix R, subject to the constraint that specified elements in R^ must be zero. Our optimality criterion is the weighted Frobenius norm of the approximation error, and we use a constrained majorization algorithm to solve the problem. Although many correlation matrix approximation approaches have been proposed, this specific problem, with the rank specification and the R^ij=0 constraints, has not been studied until now. We discuss solution feasibility, convergence, and computational effort. We also present several examples.
... The estimated matrices are presented in table-2.2. As the third example, we use the invalid correlation matrix (Q) reported in Al-Subaihi (2004). He has obtained the valid matrix, which is grossly inoptimal (Mishra, 2004) and hence we do not feel a necessity to present it here. ...
The Pearsonian coefficient of correlation as a measure of association between two variates is highly prone to the deleterious effects of outlier observations (in data). Statisticians have proposed a number of formulas to obtain robust measures of correlation that are considered to be less affected by errors of observation, perturbation or presence of outliers. Spearman’s rho, Blomqvist’s signum, Bradley’s absolute r and Shevlyakov’s median correlation are some of such robust measures of correlation. However, in many applications, correlation matrices that satisfy the criterion of positive semi-definiteness are required. Our investigation finds that while Spearman’s rho, Blomqvist’s signum and Bradley’s absolute r make positive semi-definite correlation matrices, Shevlyakov’s median correlation very often fails to do that. The use of correlation matrices based on Shevlyakov’s formula, therefore, is problematic.
... Multiple linear regression (MLR), deals with issues related to estimation, or prediction of the expected value of the dependent variable (y), using the known values of (k) predictors (x's). The statistical model of MLR is: 1 ( 1) ( 1) 1 1 n n k k n y X b × × + + × × = + e (1.1) where y is a vector of responses of n (independent) observations, X is the design matrix of rank k+1, b is a vector of parameters to be estimated or predicted, and e is the vector of residuals. ...
A Simulation study was conducted, to compare a number of univariate variable selection criteria that are available in either SAS or SPSS, in terms of their ability to select the "true" regression model for different sample sizes, intercorrelations, and interacorrelations. The results suggest that the ability of all procedures to identify the "true" model is less than 19%, and that sample size, intercorrelations, and interacorrrelations have no significant effect. The study also shows that all criteria are more likely to overfit by at least two variables, than to select the "true" model or underfit.
... ly convergent and computationally efficient. Additionally, it is straightforward to implement and can handle arbitrary weights on the entries of the correlation matrix. A simulation study by the authors suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. Al-Subaihi (2004) proposed a modification of Kaiser-Dichman procedure (see Kaiser and Dichman, 1962) to generate normally distributed (correlated) variates from a given negative semidefinite Q , which, in the process, is approximated by a positive definite * R matrix. The resulting variates satisfy the * R matrix. It appears that Al-Subaihi's method does ...
... A simulation study by the authors suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. Al-Subaihi (2004) proposed a modification of Kaiser-Dichman procedure (see Kaiser and Dichman, 1962) to generate normally distributed (correlated) variates from a given negative semidefinite Q , which, in the process, is approximated by a positive definite * R matrix. The resulting variates satisfy the * R matrix. ...
The nearest correlation matrix problem is to find a valid (positive semidefinite) correlation matrix, R(m,m), that is nearest to a given invalid (negative semidefinite) or pseudo-correlation matrix, Q(m,m); m larger than 2. In the literature on this problem, 'nearest' is invariably defined in the sense of the least Frobenius norm. Research works of Rebonato and Jaeckel (1999), Higham (2002), Anjos et al. (2003), Grubisic and Pietersz (2004), Pietersz, and Groenen (2004), etc. use Frobenius norm explicitly or implicitly. However, it is not necessary to define 'nearest' in this conventional sense. The thrust of this paper is to define 'nearest' in the sense of the least maximum norm (LMN) of the deviation matrix (R-Q), and to obtain R nearest to Q. The LMN provides the overall minimum range of deviation of the elements of R from those of Q. We also append a computer program (source codes in FORTRAN) to find the LMN R from a given Q. Presently we use the random walk search method for optimization. However, we suggest that more efficient methods based on the Genetic algorithms may replace the random walk algorithm of optimization.
... Once R (the nearmost R to Q ) is obtained, one may go in for obtaining the variates that satisfy the R matrix. Al-Subaihi (2004) proposed a modification of Kaiser-Dichman procedure to generate normally distributed (correlated) variates from a given non-positive definite Q , which, in the process, is approximated by a positive definite * R matrix. The resulting variates satisfy the * R matrix. ...
In simulation we often have to generate correlated random variables by giving a reference intercorrelation matrix, R or Q. The matrix R is positive definite and a valid correlation matrix. The matrix Q may appear to be a correlation matrix but it may be invalid (negative definite). With R(m,m) it is easy to generate X(n,m), but Q(m,m) cannot give real X(n,m). So, Q has to be converted into the near-most R matrix by some procedure. NJ Higham (2002) provides a method to generate R from Q that satisfies the minimum Frobenius norm condition for (Q-R). Ali Al-Subaihi (2004) gives another method, but his method does not produce an optimal R from Q. In this paper we propose an algorithm to produce an optimal R from Q by minimizing the maximum norm of (Q-R). A Computer program (in FORTRAN) also has been provided. Having obtained R from Q, the paper gives an algorithm to obtain X(n,m) from R(m,m). The proposed algorithm is based on factorization of R, yet it is different from the Kaiser Dichman (1962) procedure. A computer program also has been given.
... Note that the correlation matrix should be positive semi-definite and that the multivariate distribution will be the same as In [17] the matrix is decomposed into its eigenvalues and eigenvectors, the highest eigenvalues are chosen and the matrix is reconstruct using them. In [18] a review of existing approaches is given and a new approach is proposed. This consists of decomposing the correlation matrix as a block matrix such that the matrices in the main diagonals are positive definite while the matrices in the other diagonals are transpose of each other. ...
... Multiple linear regression (MLR), deals with issues related to estimation, or prediction of the expected value of the dependent variable (y), using the known values of (k) predictors (x's). The statistical model of MLR is: 1 ( 1) ( 1) 1 1 n n k k n y X b × × + + × × = + e (1.1) where y is a vector of responses of n (independent) observations, X is the design matrix of rank k+1, b is a vector of parameters to be estimated or predicted, and e is the vector of residuals. ...
The correlation matrix has a wide range of applications in finance and risk management. However, due to the constraints of practical operations, the correlation matrix cannot satisfy the positive semidefinite property in most cases. In this paper, an elementwisely alternative gradient algorithm and a columnwisely alternative gradient algorithm are presented to compute the nearest correlation matrix that satisfies the semidefinite property for a given set of constraints. The convergence properties and the implementation of these two algorithms are discussed. Numerical experiments show that the proposed methods are efficient. Furthermore, the columnwisely alternative gradient algorithm outperforms other algorithms in terms of the number of iterations and the objective value of the cost function.
