Mode, most probable value

A third measure of location is the mode, which is defined as that value of the measured variable for which there are the most observations. Mode is the most probable value of a discrete random variable, while for a continual random variable it is the random variable value where the probability density function reaches its maximum. Practically speaking, it is the value of the measured response, i.e. the property that is the most frequent in the sample. The mean is the most widely used, particularly in statistical analysis. The median is occasionally more appropriate than the mean as a measure of location. The mode is rarely used. For symmetrical distributions, such as the Normal distribution, the mentioned values are identical. 

Thus the mean and variance of a gamma distribution are sufficient to determine its two parameters, a and /3. Note that the coefficient of variation (standard deviation divided by the mean) is equal to the square root of 1/or. The most probable value of k (the mode) occurs at (a - l)//3 if oc > 1, and as a T oo the gamma distribution itself becomes the gaussian or normal distribution for the variable, /8k.13... 

The mode is defined as the most probable value of x. Therefore, the mode Xj is that X for which fix) is maximum and is obtained from... 

By solving dp(p D, C)jdp = 0, the most probable value (mode) for the mean parameter p is equal to the sample average ... 

The prior PDF p(k, 0, C) implies that, given a class of dynamical models and before using the dynamic test data, the most probable values of X and 0 are those that minimize the Euclidean norm (2-norm) of the error in the eigen equation for the dynamical model. This implies that the prior most probable values of X and 0 are the squared modal frequencies and mode shapes of a dynamical model, but these values are never explicitly required. This prior PDF will have multiple peaks because there is no implied ordering of the modes here. 

Consider next the incomplete mode shape measurements where only six sensors on the first, fourth, fifth, seventh, tenth and top floors are available. The results presented in Table 5.2 are based on five measured modes and show the initial values, final most probable values, standard deviations and COVs of the stiffness parameters, which are comparable to the COV of the modal data. Figure 5.1 shows the iterative history for the most probable values of the stiffness parameters, with convergence occurring in about 120 iterations. Again, the same set of nominal stiffness values is used so the nominal model overestimated the interstory stiffnesses by 100 to 200%. The parameters converge very quickly even for such an unsatisfactory set of initial values. The CPU time for 200 iterations is about 0.8 s with a conventional dual CPU 3.0 GHz personal computer under the MATLAB environment [171]. Figure 5.2 shows the comparison between the identified system mode shapes (solid lines) and the actual mode shapes (dashed lines) for the first five modes but the two sets of curves are on top of each other. Of course, it is no wonder that the mode shape components of the observed degrees of freedom are estimated better than the unobserved ones. 

Figure 5.1 Iteration history for the most probable values of the stiffness parameters with incomplete measurement of mode shapes...

This distribution has the most probable value (mode) 0.005, mean 0.01 and standard deviation /2 X 0.005. After N days of survival, the likelihood function was given by ... 

Mode. The mode m represents the local maximum (or one of the local maxima) of the density/mass function. It is usually referred to as the most probable value in physics and chemistry. (In the case of discrete distributions, only the spectrum points are considered when looking for maxima.) If there is only one maximum, the distrihution is called unimodah if there are two maxima, the distribution is called bimodal and so on. 

It is clear that, for large values of p, the mode(s) is(are) practically equal to the expected value, i.e., the expected value doubles as the most probable value in the case of the Poisson distribution. On the other hand, the relative deviation rapidly decreases as p increases. Thus, the distribution is gradually shrinking on the expected value. 

Note that when the degree of freedom (fc) is large, then the expected value is practically equal to the mode, i.e., the expected value doubles as the most probable value. (The reason for the quotation marks is that with continuous distributions any x value has 0 for probability. What this statement really means is that the probability dP = f x)dx is at its maximum if dxis considered constant.) When evaluating nuclear spectra, distributions with about 2,000 degrees of freedom quite often occur (2,048-channel spectra). In such cases the relative deviation is about 3%, i.e., the value of the y (2,000) random variable is most probably 2,000 3% (2,000 60). For more detail, see the next remark. (The p-quantiles for distributions y (fc) with degree of freedom 1 < fc < 30 can be found in Table 9.4.)... 

Au SK (2011) Fast Bayesian FFT method for ambient modal identification with separated modes. J Eng Mech-ASCE 137(3) 214-226 Au SK (2012a) Fast Bayesian ambient modal identification in the frequency domain, Part I posterior most probable value. Mech Syst Signal Process 26(l) 60-75 Au SK (2012b) Connecting Bayesian and frequentist quantification of parameter uncertainty in system Identification. Mech Syst Signal Process 29 328-342 Au SK (2014a) Uncertainty law in ambient modal identification. Part I theory. Mech Syst Signal Process 48(1-2) 15-33... 

Here the C—H bond is weakened and the frequencies lowered by delocalization of electron density toward the positively charged center (hyperconjugation see Section 10.2 for further discussion). A bending mode is probably again the most important one 58 A H/A D is greater than 1, values ranging up to about 1.4 for favorably situated hydrogens,59 but more typically on the order of 1.1. 

We can also find the statistical mode, o, the most probable strength value for a given distribution. Consider the mean fiber strength (Eq. 10.8) again... 

For /3>1, we shall have 0.881.0. This means that we can regard the quantity as the reference level strength. To find the statistical mode, a, the most probable strength value, we proceed as follows. The Weibull distribution is... 

The characteristic chemisorption data obtained for the various supported catalyst samples are summarized in Table 2. It is evident that, the 02-net adsorption (3n) has in general lower magnitude for catalyst samples supported on pure silica and silica-rich support and higher magnitudes for catalyst samples supported on alumina and alumina-rich supports. However, these adsorption values decrease markedly as the CoPc content increases on one and the same support (97.1 SA) being related most probably to the mode of surface complex dispersion. 

The most probable velocity, up, is the velocity most likely to be possessed by any molecule in the system and corresponds to the maximum value or mode of... 

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