Total mean squares

The mean-square errors due to slope contribution and the total mean-square errors are defined as... 

These observations may be summarized conveniently in an analysis-of-variance table-. Table 26-7 illustrates this type of table for the above case. The overall variance (total mean square) Sj(N — 1) contains contributions due to variances within as well as between classes. The variation between classes contains both variation within classes and a variation associated with the classes themselves and is given by the expected mean square aj + not. Whether not is significant can be determined by the F test. Under the null hypothesis, = 0. Whether the ratio... 

FIGURE 30.14 Relationship between mean total displacement and temperature factor B. At temperature factors of 60 and higher, the displacement becomes larger than 1.5 A and the electron density becomes very poor (see Figure 30.15). The formula used in the figure is derived from the relationship B = ir , where <)F> represents the displacement perpendicular to the diffracting planes. The total mean square displacement = 3 . 

Y represents a quantity from a set of N observations. This gives the total sum of squares N-1 degrees of freedom. The total mean squares, MST, is an estimate of the variance of the dependent variable Y and is ... 

Since the low frequency modes are so weak, phonon wing like combinations ( 5.2.1.2) are irrelevant and the total mean square vibration of the hydrogen atom is dominated by the optic modes. Indeed, many hydrogen-in-metal systems are studied at room temperature with only modest spectral degradation. This single fact sets the INS spectra of hydrogen in metal systems clearly apart from those of molecular crystals. 

The mode specific vibrational root mean squared displacement vector is thus replaced by the mean square displacement tensor of that atom in that specific mode, it has units of A. The tensor is represented in Eq. (A2.85) by a matrix, in more compact notation it is written, for a specific vibrational mode and atom Bi. The colon-operator ( ) or tensor contraction operation is found by taking the trace (Tr) of the product of the two tensors, see Eq. (2.50). Further we write the total mean square displacement tensor of an atom as the sum over all the individual vibrational contributions. 

An exponential power series is recuperated by dividing through by the first term, Eq. (A2.91). Where the total mean square displacement tensor y4 is a spheroid that deviates little from the isotropic, i.e. one that is almost a sphere, then terms linear in A will be adequate [6]. This is the almost isotropic approximation and Eq. (A2.87) becomes Eq. (2,41) for = 1. 

When a thermal detector is at a temperature Tdet different from that of the background, Tback, the total mean square of the radiation power is given by ... 

Use of acoustic flow noise to monitor solid/gas flow rate is probably the simplest method, but it lacks accuracy. The technique has been used to measure mass flow rate of a powder flow line. A linear dependence was observed between the mass flow rate and the total mean square voltage of the flow noise. To measure the acoustic flow noise, only a wide-band transducer (or... 

This value represents the total mean square error of the n fitted values for each k. 

The Universal Approximation Theorem states that a single layer net, with a suitably large number of hidden nodes, can well approximate any suitably smooth function. Hence for a given input, the network output may be compared with the required output. The total mean square error function is then used to measure how close the actual... 

The values of below the critical temperature, derived from the areas of the spectra, are associated with the local thermal motions of the individual iron atoms relative to their neighbours. The. total mean square displacements of the iron atoms above 7, derived from the areas of the narrow subspectra, were resolved into three components associated with different modes of thermal motions ., s and fc corresponding to local, slow collective and fast collective modes, respectively. This is shown in Figure 6.17 for metmyoglobin and ferritin. The value of ,o,. is obtained by a linear extrapolation from the values of below the critical temperature. The difference between and ioc gives the mean-square collective displacement associated with large-scale motions of parts of the surrounding protein and is resolved into the two parts and fc. The values of correspond to slow... 

According to this result, the amplitudes of the Rouse-modes rapidly decrease with increasing mode order m. If we consider the contributions of the different Rouse-modes to the known total mean squared end-to-end distance, B ) = we find that a large part is already provided by the three lowest order Rouse-modes. Contributions to the end-to-end distance in the 2 -direction, R ) = i o/ 5 come from all z-polarized Rouse-modes with odd m s... 

Hence, 90% of the total mean squared end-to-end distance of a chain originates from the lowest order Rouse-modes. In theorical treatments, polymer chains are sometimes substituted by elastic dumbells, set up by two beads connected by a spring. The justification for this simplification follows from the dominant role of the primary Rouse-modes. 

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