Random vector

Essentially the same argument used above enables one to prove an important multidimensional version of the central limit theorem that applies to sums of independent random vectors. A -dimensional random vector is simply a group of k random variables,... 

Finally, an infinite set of random vectors is defined to be statistically independent if all finite subfamilies are statistically independent. Given an infinite family of identically distributed, statistically independent random vectors having finite means and covariances, we define their normalized sum to be the vector sfn, , sj where... 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

We have considered in some detail in Section 4.2 the case where the random vector Y of n ancillary or dependent variables relates linearly to those of a vector X of n principal or independent variables (e.g., raw data) with covariance matrix L through the matrix equality... 

The vectors pm(t) and pm co) lie in the plane of the mth collision. Accordingly, fim(a>) is a random vector uniformly distributed in its azimuth about vm, the direction of the atom of type A as it approaches an atom B during an early phase of the mth collision. In order to compute the... 

Suppose we change the assumptions of the model in Section 5.3 to AS5 (x ) are an independent and identically distributed sequence of random vectors such that x, has a finite mean vector, finite positive definite covariance matrix Zxx and finite fourth moments E[xjxj xixm] = for all variables. How does the proof of consistency and asymptotic normality of b change Are these assumptions weaker or stronger than the ones made in Section 5.2 ... 

If we assume that the magnetic parameters M and K, (the first-order anisotropy constant) are not distributed (have fixed value for all the magnetic particles and throughout the volume of each particle), the components of the reduced random vector in expression (16) will include only the geometrical... 

For simplicity, we take into consideration only axial distortions of the particle shapes, in which case nx = ny = — fnz, so that the random vector f... 

Let n0 be an m-dimensional deterministic vector representing the number of particles contained in the drug amount qo initially given in each compartment. Also, let (t) be an m-dimensional random vector that takes on zero and positive integer values. (t) represents, at time t, the random distribution among the m compartments of the number of molecules starting in i. Since all of the molecules are independent by assumption, (f) follows a multinomial distribution ... 

Statistical characteristics of the random vector N (t) can be directly obtained from cumulants nSl.Sm (t) with all Sj = 0 except ... 

Theorem 3 (Decomposability). The average entropy of the random vector 9 = 0i, 02 can be decomposed as... 

Once we fix the initial state ip,) and the final state ipj), the optimal field E(t) is obtained by some numerical procedures for appropriate values of the target time T and the penalty factor a. Though there should be many situations corresponding to the choice of ip,) and ipj), we only consider the case where they are Gaussian random vectors. It is defined by... 

We show two numerical examples for a 64 x 64 random matrix Hamiltonian One is the relatively short-time case with T = 20 and a = 1 shown in Fig. 1, and the other is the case with T = 200 and ot = 10 shown in Fig. 2. In both cases, we obtain the optimal field s(f) after 100 iterations using the Zhu-Botina-Rabitz (ZBR) scheme [13] with s(f) = 0 as an initial guess of the field. The initial and the target state is chosen as Gaussian random vectors as mentioned above. The final overlaps are Jo = 0.971 and 0.982, respectively. 

Figure 1. Optimal control between Gaussian random vectors in a 64 x 64 random matrix system by the Zhu-Botina-Rabitz scheme with T — 20 and a = 1. a) The optimal field after 100 iterations b) its power spectrum (c) the optimal evolution of the squared overlap with the target (([)(r) (py) as well as its magnified values near the target time in the inset (d) the convergence behavior of the overlap Jq (solid curve) and the functional J (dashed curve) versus the number of iteration steps.

We next examine when and how the analytic optimal field works for a random matrix system (256 x 256 GOE random matrix). Figure 9 demonstrates the coarse-grained Rabi oscillation induced by the analytic field, Eq. (45), with k = 3, where smooth oscillations of ((j)o(f) (t)(f))p and (Xo(0l4 (0)P observed. The initial and the target states are both Gaussian random vectors with 256 elements. This result shows that the field actually produces the CG Rabi oscillation in the random matrix system. 

Figure 10. The target-time dependence of the final overlap 7o by the analytic optimal field with k — 1 is shown. The residual probability 1 — 7o from perfect control = 1 is depicted for various matrix sizes N of GOE random matrices. The initial and the final states are Gaussian random vectors.

This sequence shows the instant values of the exit of the process conditioned by the vector parameter P = P(pi,p2,. ..Pl). Indeed,Yi.j/P is the exit random vector conditioned by the vector parameter P. In this case p(Yn/P), which is the probability density of this variable, must be a maximum when the parameter vector P is quite near or superposed on the exact or theoretical vector P. Therefore, the maximum likelihood method (MLM) estimates the unknown parameter vector P as P, which maximizes the likelihood function given by ... 

Mean and covariance for conditional gaussian random vector. The minimum mean square estimate of a gaussian random vector when we only have observations of some of its elements is the conditional mean of the remaining elements. The error covariance of this estimate is the conditional covariance. Consequently, if Z is a random gaussian vector composed of sub-vectors x and y, then we may write ... 

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