Caption: Figure 25: Comparison of the magnitude of positional covariances
in each direction (x, y, and z).
The key problem in nonlinear Kalman filter is to calculate the intractable nonlinear Gaussian weighted integral as [mathematical expression not reproducible], where x [member of] [R.sup.n], g(x) represents the arbitrary nonlinear function, and N(x; [??], [P.sub.x]) denotes the Gaussian distribution with mean x and covariance
Figure 4 schematically illustrates these inversions and their relations with conditional covariances
. Let us note that these conditional covariances
can also be expressed directly in terms of subcovariances by using again the Schur complement:
The vectors [w.sub.k] and [v.sub.k] are two zero-mean white Gaussian noise processes with covariance
matrixes [Q.sub.w] and [Q.sub.v], respectively.
Third, we tested for the invariance of all factor loadings plus all specified error covariances
, with equality constraints being placed on these parameters, Finally, we tested for the invariance of factor loadings, error covariances
, and latent factor correlations in combination.
To calculate the probability of making a type I error in the LRT for the independence between two groups of variables, a computational simulation by the Monte Carlo method was performed, considering the following scenarios: 16 sample sizes--25, 30, 50, 100, 200, 300, 400, 500, 600, 750, 1,000, 1,500, 2,000, 3,000, 4,000 and 5,000; 40 combinations of the number of variables between the two groups--starting with 3+3, 3+4, 3+5, 3+6, 3+7, 3+8, 3+9, 3+10, 4+4, 4+5, 4+6, 4+7, 4+8, 4+9, 4+10, and 5+3 up to 14+10; and a degree of correlation between the variables of the covariance
matrices: [[summation].sub.XY] = 0; totaling 640 scenarios (16 x 40 x 1).
between direct and maternal genetic effects
Since there is no correlation between measurement errors and signals, the total covariance
matrix of the observations in the LSC model is obtained by the sum of the covariance
matrices of the signal and the noise.
Given the expected excess rate of return vector R - r on n risky securities and the non-singular covariance
matrix [OMEGA] between n risky securities rate of returns, the portfolio [omega] in Equation (1) is the unique risky optimal mean-variance efficient within the mean-variance framework if and only if [omega] = [[OMEGA].sup.-1](R-r)/ [[e.sup.T][[OMEGA].sup.-1](R-r)].
[mu] = E[[mu]([[theta].sub.i]) is the vector of collective premiums S = E[[SIGMA]([[theta].sub.i])], where [SIGMA]([[theta].sub.i]) = Cov([X.sub.i], [X.sup.T.sub.i]|[[theta].sub.i]) is the covariance
matrix of the vector [X.sub.i] = [([X.sub.i1], ..., [X.sub.in]).sup.T]
As mentioned in Section 1, the covariances
between growth-related parameters [alpha], [rho], and [delta] are directly influential upon the respective magnitudes of the ranges of variations of the functionally relevant parameters E, D, and K, according to the sign of the dependence of each parameter E, D, and K upon each parameter [alpha], [rho], and [delta] (signs of dependence provided at Table 1).
One particular, unexpected feature emerging from this study is a negative covariance
occurring between the differential growth index [rho] and the apical angle [alpha] (Figure 3).