On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal

The general expressions for the moments of the stochastic shrinkage parameters of the Liu type estimator

One of the problems with the Liu estimator is the appropriate value for the unknown biasing parameter d . In this article we consider the optimum value for d and give upper bound for the expected value of the estimator of this biasing parameter. We also derive the general expressions for the moments of the stochastic shrinkage parameters of the Liu estimator and the generalized Liu estimator. Numerical calculations are carried out to illustrate the behavior of the mean and variance of the biasing parameter. Also, a numerical example is given to illustrate the effect of the biasing parameter d , on the mean square error of the Liu estimator

Convergence of Estimates of Unique Variances in Factor Analysis, Based on the Inverse Sample Covariance Matrix

If the ratio m/p tends to zero, where m is the number of factors m and p the number of observable variables, then the inverse diagonal element of the inverted observable covariance matrix (ςpjj)-1 tends to the corresponding unique variance ψjj for almost all of these (Guttman, 1956). If the smallest singular value of the loadings matrix from Common Factor Analysis tends to infinity as p increases, then m/p tends to zero. The same condition is necessary and sufficient for (ςpjj)-1 to tend to ψjj for all of these. Several related conditions are discussed. (PsycINFO Database Record (c) 2009 APA, all rights reserved) (journal abstract

Constrained Candecomp/Parafac via the Lasso

The Candecomp/Parafac (CP) model is a well-known tool for summarizing a three-way array by extracting a limited number of components. Unfortunately, in some cases, the model suffers from the socalled degeneracy, that is a solution with diverging and uninterpretable components. To avoid degeneracy, orthogonality constraints are usually applied to one of the component matrices. This solves the problem only from a technical point of view because the existence of orthogonal components underlying the data is not guaranteed. For this purpose, we consider some variants of the CP model where the orthogonality constraints are relaxed either by constraining only a pair, or a subset, of components or by stimulating the CP solution to be possibly orthogonal. We theoretically clarify that only the latter approach, based on the least absolute shrinkage and selection operator and named the CP-Lasso, is helpful in solving the degeneracy problem. The results of the application of CP-Lasso on simulated and real life data show its effectiveness

Exponential smoothing weighted correlations

In many practical applications, correlation matrices might be affected by the “curse of dimensionality” and by an excessive sensitiveness to outliers and remote observations. These shortcomings can cause problems of statistical robustness especially accentuated when a system of dynamic correlations over a running window is concerned. These drawbacks can be partially mitigated by assigning a structure of weights to observational events. In this paper, we discuss Pearson’s ρ and Kendall’s τ correlation matrices, weighted with an exponential smoothing, computed on moving windows using a data-set of daily returns for 300 NYSE highly capitalized companies in the period between 2001 and 2003. Criteria for jointly determining optimal weights together with the optimal length of the running window are proposed. We find that the exponential smoothing can provide more robust and reliable dynamic measures and we discuss that a careful choice of the parameters can reduce the autocorrelation of dynamic correlations whilst keeping significance and robustness of the measure. Weighted correlations are found to be smoother and recovering faster from market turbulence than their unweighted counterparts, helping also to discriminate more effectively genuine from spurious correlations