Flat distributions

As discussed before, in the conventional data reconciliation approach, auxiliary gross error detection techniques are required to remove any gross error before applying reconciliation techniques. Furthermore, the reconciled states are only the maximum likelihood states of the plant, if feasible plant states are equally likely. That is, P x = 1 if the constraints are satisfied and P x = 0 otherwise. This is the so-called binary assumption (Johnston and Kramer, 1995) or flat distribution. 

Case 1 Binary assumption on P[x (flat distribution), and sensor errors are assumed to follow a normal distribution... 

A relatively flat distribution can represent a situation of relative uncertainty. For example, when one has only a maximum and minimum, the conventional default distribution is uniform between those values. The main difficulty in determining the flat distribution, to be used in a situation of relative ignorance, is that a distribution that is flat in one scale may be far from flat in another. 

Ordinal data tend not to form normal distributions. For a start, it is often recorded on scales with a very limited number of possible values. Scales of four, five or six points are frequently seen. In such cases, it is impossible for the data to form the sort of smooth, bell-shaped distribution that constitutes a true normal distribution. However, then the problem is further exacerbated. Although there is no necessary reason for it, anybody who has worked with real-world, ordinal data knows that it is frequently hideously non-normal. Offered a scale of possible scores, people will quite frequently do bizarre things like only using the extreme upper and lower values but not the middle ones, or else they will produce a completely flat distribution, with no peak frequencies anywhere. No amount of mathematical transformation is going to convert that sort of mess into anything remotely resembling a normal distribution. 

Maxent is one method for determining the probability of a model. A simple example is that of the toss of a six sided unbiassed die. What is the most likely underlying frequency distribution, and how can each possible distribution be measured Figure 3.31 illustrates a flat distribution and Figure 3.32 a skew distribution (expressed... 

We note that in the case z > 2, a distribution with the form 1 /yz 1 would yield a divergent normalization factor because for z > 2 (p < 2) an invariant measure does not exist. If we set the system in an initial flat distribution, the pdf p(y, t) will keep changing forever, becoming sharper and sharper in the vicinity of y = 0, without ever reaching any equilibrium distribution. 

Here we show that slow modulation does not yield any aging and that consequently superstatistics is not the proper approach to to the BQD complexity. Let us assume that the renewal condition applies and that Eq. (234) can be used. Let us assume that the initial condition is the flat distribution, p(y. 0) = 1 for any value of v from y = 0 to y = 1. We decide to start the observation process at t = ta > 0. The waiting time distribution of the first sojourn times is given by... 

We see that in both case (a), Eq. (288), and case (b), Eq. (291), the flat distribution coincides with the equilibrium distribution. Consequently,... 

The flat distribution implies that a free one-dimensional random walk in the potential energy space is realized in this ensemble. This allows the simulation to escape from any local minimum-energy states and to sample the configurational space much more widely than the conventional canonical MC or MD methods. 

We have numerically diagonalized small spin systems containing up to 5 by 5 spins subjected to a small random field hf j flatly distributed in the interval (—(5/2, (5/2). We see that indeed the gap closes rather fast away from the special Jx = Jy point (Fig. 3) but remains significant near Jz = Jx point where it clearly has a much weaker size dependence. Interestingly, the gap between the lowest 2ra states and the rest of the spectrum expected in the limits Jz 33> Jx or Jz -C Jx appears only at Jx/ Jz > jc with a practically size independent jc 1.2. We also see that the condition l 1 eliminates all low lying states in the Jz lowest excited state in l 1 sector... 

For g(x) to be proportional to 1/x, it requires the relaxation time to be an exponential function of some random variable such that x = xq exp ( ), where itself has a flat distribution. It means that ( ) = constant, and rt(x) = n ). (d /dx) oc x. If a, the polarisability, is also a function of then it can lead to a sub-linear frequency dependence of a (co). The functional form given for the variation of x can arise from two different relaxation mechanisms. The first is a classical barrier hopping, in which two energetically favourable sites like in a double well potential are separated by a barrier fV and = W/kT. The second mechanism is a phonon assisted quantum tunneling through a barrier, which separates two equilibrium positions, in which case = 2aR, where a is the localization length and R is the separation between the sites. In the first case, by treating JV as independent of R, it has been shown (Poliak and Pike, 1972) that... 

FIGURE 4.3 Graphical representation of kurtosis, K, and skewness, S, in comparison to the Gaussian (standard) distribution (upper left). The right-hand side shows leptokurtic (peaked) or platykurtic (flatted) distribution as well as positive skewed distribution (fronting) and negative skewed distribution (tailing). 

The main problem is that such "odd" shape excitation is difficult to implement. For example, a flat distribution of excitation would result from a transmitter output having an envelope of the form (sinu)t)/ujt but you would first have to generate such a time dependence, either by analog or digital means and then have a transmitter which would respond linearly to such an input. A better alternative is to generate such an irradiation pattern by a series of equally spaced pulses whose widths or amplitudes are modulated appropriately (Tomlinson and Hill, 1973). 

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