Weighted matrices

US model can be combined with the Monte Carlo simulation approach to calculate a r range of properties them is available from the simple matrix multiplication method. 2 RIS Monte Carlo method the statistical weight matrices are used to generate chain irmadons with a probability distribution that is implied in their statistical weights. 

Q and R are the state and control weighting matrices and are always square and symmetric. J is always a scalar quantity. 

In the studies described above, the samples were supported in low atomic weight matrices, melted in situ, and measured in transmission mode. Similarly, second... 

In this case we minimize a weighted SSE with non-constant weights. The user-supplied weighting matrices differ from experiment to experiment. 

Of course, it is not at all clear how one should select the weighting matrices Q i=l,...,N, even for cases where a constant weighting matrix Q is used. Practical guidelines for the selection of Q can be derived from Maximum Likelihood (ML) considerations. 

However, the requirement of exact knowledge of all covariance matrices (E i=l,2,...,N) is rather unrealistic. Fortunately, in many situations of practical importance, we can make certain quite reasonable assumptions about the structure of E, that allow us to obtain the ML estimates using Equation 2.21. This approach can actually aid us in establishing guidelines for the selection of the weighting matrices Q, in least squares estimation. 

Upon substitution of E, into Equation 2.21 it becomes apparent that the ML parameter estimates are the same as the weighted LS estimates when the following weighting matrices are used,... 

The converged parameter values represent the Least Squares (LS), Weighted LS or Generalized LS estimates depending on the choice of the weighting matrices Q,. Furthermore, if certain assumptions regarding the statistical distribution of the residuals hold, these parameter values could also be the Maximum Likelihood (ML) estimates. 

Experimental data are available as measurements of the output vector as a function of time, i.e., [yj, t ], i=l,...,N where withyj we denote the measurement of the output vector at time t,. These are to be matched to the values calculated by the model at the same time, y(t,), in some optimal fashion. Based on the statistical properties of the experimental error involved in the measurement of the output vector, we determine the weighting matrices Qj (i=l,...,N) that should be used in the objective function to be minimized as mentioned earlier in Chapter 2. The objective function is of the form,... 

The values of the elements of the weighting matrices R, depend on the type of estimation method being used. When the residuals in the above equations can be assumed to be independent, normally distributed with zero mean and the same constant variance, Least Squares (LS) estimation should be performed. In this case, the weighting matrices in Equation 14.35 are replaced by the identity matrix I. Maximum likelihood (ML) estimation should be applied when the EoS is capable of calculating the correct phase behavior of the system within the experimental error. Its application requires the knowledge of the measurement... 

Since no correlation is assumed for the corresponding noises, the weighting matrices can be partitioned as well, that is,... 

Structure Crystal structures, Point defects, Dislocations Crystal structures, Defect reactions, The glassy state Configuration, Conformation, Molecular Weight Matrices, Reinforce- ments Biochemistry, Tissue stracture... 

You may also wish to use different weighting matrices for different observations. For this purpose the weighting option WI = 3 is provided. To use this option you must supply a second subroutine starting at line 800, where you have access to the index M of the current sample point. The task of the second routine is to compute the NYtfvIY weighting matrix for the current sample point and to place it into the array W. 

The observed concentrations have been listed in Table 1.3. Let us first fit the above response function to the data by the least sqares method with the weighting matrices = I, i.e., without weighting. Module M45 results in the estimates shown in the first row of of Table 3.5. 

The pitfall of iterative reweighting stems from the fact that parameter-dependent matrices [JjV J ]- cannot simply be considered as weighting matrices. We can give, however, a true sum-of-squares structure... 

While the study of diarylpentanes is helpful in understanding the conformational behavior of aryl vinyl polymers, a simple weighting of the properties of the model compounds by the tacticity of the polymer does not yield the properties of the polymer. For example, the presence of dl dyads surrounding a meso dyad will suppress the tt conformer in the meso dyad 14fl). Thus, in order to obtain the fraction of tt meso conformers within an atactic P2VN sample, it is necessary to resort to a Monte Carlo calculation utilizing an extended product of statistical weight matrices, 26). 

Collect the statistical weights into statistical weight matrices (of dimension Vj.j X v , where v,- is the number of rotational isomeric states of bond i), one per skeletal bond subject to conformational change, indexing rows and columns of the matrices with the RIS of the bond. The generic statistical weight matrix for bond i is termed U,-. 

Once the statistical weight matrices and the local geometry of the individual conformers is determined, the RIS approach allows for the extremely efficient estimation of global characteristics. The essential simplifying assumption of the... 

A three-state RIS model is derived for POE, based upon ab initio electronic structure analyses of model molecules DME and DEE. It is demonstrated that the low energy of the tg gI conformation of DME, resulting from strong O -H attractions, as indicated by the ab initio studies, necessitates the inclusion of third-order interactions in the RIS model. This is realized by adopting 9x9 statistical weight matrices. 

Conformational energy calculations indicate the g state to be at least 30 kJ mol-1 higher than the t and the g states. With the exclusion of the former conformation, all interactions of long range are eliminated, and the statistical weight matrices for the respective bond pairs reduce to 2x2 order. 

All ester groups in the PET chain are assigned confidently to be planar trans. The restriction of bonds 1 and 3 to the trans states and bond 2 to a choice between cis and trans leads to the simple statistical weight matrices. The length of the span of the terephthaloyl residue in PET guarantees independence of the conformations of successive repeating units of the chain. 

Geometrical parameters employed for polyesters II are those used in the analysis of aromatic polyesters by Erman, Flory, and Hummel Macromolecules 1980, 13, 4841. Statistical weight matrices may be formulated for any given residue by the usual procedure. 

Experimental values are presented of the molar Kerr constants /x and dipole moments squared, lx, for the copolymers poly(styrene-co-p-bromostyrene), where x is the degree of polymerization. Some results are also presented for poly(styrene-co-p-chlorostyrene) and related polymers. The RIS model of Yoon etal. (Yoon, D. Y. Sundararajan, P. R. Flory, P. J. Macromolecules 1975, 8, 776) is used to calculate mK/x and /x values as a function of tacticity and composition. The statistical weight matrices are identical with those used by Saiz etal. (Saiz, E. Mark, J. E. Flory, P. J. Macromolecules 1977, 10, 967), with the following parameters h = 0.8 exp 397/RT), co = o = 1.3 exp - 1987/RT) and m,= 1.B exp -(2186/RT), where T = 298 K is the temperature. 

Near a trifunctional branch point the following statistical weight matrices are required (the term o is factored out of 3Un(- j. 

When a trifunctional branch point is present, the configuration partition function is written as Z = U1 Statistical weight matrices used are ... 

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