Poisson statistics analysis

While radioactive decay is itself a random process, the Gaussian distribution function fails to account for probability relationships describing rates of radioactive decay Instead, appropriate statistical analysis of scintillation counting data relies on the use of the Poisson probability distribution function ... 

After ruling out slow modulation as a possible approach to complexity, we are left with the search for a more satisfactory approach to complexity that accounts for the renewal BQD properties. Is it possible to propose a more exhaustive approach to complexity, which explains both non-Poisson statistics and renewal at the same time We attempt at realizing this ambitious task in Section XVII. In Section XVII.A we show that a non-Ohmic bath can regarded as a source of memory and cooperation. It can be used for a dynamic approach to Fractional Brownian Motion, which, is, however, a theory without critical events. In Section XVIII.B we show, however, that the recursion process is renewal and fits the requests emerging from the statistical analysis of real data afforded by the researchers in the BQD held. In Section XVII.C we explain why this model might afford an exhaustive approach to complexity. 

The variation that is observed in experimental results can take many different forms or distributions. We consider here three of the best known that can be expressed in relatively straightforward mathematical terms the binomial distribution, the Poisson distribution and the Gaussian, or normal, distribution. These are all forms of parametric statistics which are based on the idea that the data are spread in a specific manner. Ideally, this should be demonstrated before a statistical analysis is carried out, but this is not often done. 

Lotte A, Wasz-Hockert O, Poisson N, Dumitrescu N, Verron M, Couvet E. BCG complications. Estimates of the risks among vaccinated subjects and statistical analysis of their main characteristics. Adv Tuberc Res 1984 21 107-93. 

Since radioactive decay follows Poisson statistics, a lower limit to the precision of an analysis can be obtained by a single measurement. In practice, counting statistics generally is the limiting uncertainty, since chi-squared tests often show that the single-measurement precision is an excellent predictor of sample-to-sample repeatability. 

A study is considered valid if the results obtained with positive and negative controls are consistent with the laboratory s historical data and with the literature. Statistical analysis is usually applied to compare treated and negative control groups. Both pairwise and linear trend tests can be used. Because of the low background and Poisson distribution, data transformation (e.g., log transformation) is sometimes needed before using tests applicable to normally distributed data. Otherwise, nonparametric analyses should be preferred. 

Criteria for evaluating the degree of fit between measured fluorescence decay curves and trial decay functions have been discussed.In some instances plots of weighted residuals were found to be sufficient, but a generalized statistical test was proposed for all other cases. An analysis of the statistical distribution of noise in fluorescence decay measurements by SPC has shown, as expected, that Poisson statistics dominate. A method for obtaining decay information from pulse fluorimetry without the need for consideration of the excitation pulse, has been described. ... 

The standard deviation of the null signal in this expression is given in terms of counting statistics if Poisson statistics are not likely to account for most of the random counting error, then it would be prudent to deduce Op from a moderate number of replicates -- le, replace the second term in the numerator of the second factor by 2t Sg Jri, where t is Student s-t and Sp is the estimated standard deviation for the blank (counts). Bounds for systematic error should be based on sound experience or analysis of the measurement process default values that reflect much low-level radionuclide measurement experience are set at 1% [baseline], 5% [blank], and 10% [calibration], respectively. Poisson deviations from normality are adequately accounted for by this expression down to B - 5 background... 

This variation in peak shape between different injections represents a fundamental limitation of chemical analysis at the single molecule level. The distribution in the number of plolecules sampled in any experiment is ultimately limited by Poisson statistics, where the standard deviation of the number of molecules found in any subpopulation is equal to the square root of the average number of molecules taken. 

One of the principal advantages of the use of photon-counting methods over analog methods for detection and analysis of CPL is the statistical nature of these measurements. In fig. 5 we plot the distribution of lum results obtained for a sample measurement. Schippers has shown that the observed spread in values is what one expects fi om Poisson statistics (Schippers, 1982). The standard deviation, aj, of a particular measurement of gium in which a total of N photons were counted is as follows... 

The upper limit for ion beam currents that can be measured using ion counting detection (for trace quantitative analysis) is determined by the dead time of the ion detector, since ions arriving during a detector dead time do not give rise to an output pulse and are thus not counted. The time distribution of individual ions arriving at the detector is essentially random but is well described by Poisson statistics this can be exploited in application of a correction factor applied to ion counts to extend the linear range by up to a factor of 10 above the limit imposed by the detector dead time. 

A model which considers circular crystallites on a circular substrate at low nucleation rate would lead to the growth of a complete planar layer following the formation of a single nucleus, the growth of which would be limited only by the boundaries of the substrate. When the current-time transients were calculated, a statistical analysis of the transients alone was compared with experiments at low overpotentials. The moments of the transients alone consolidated the model and showed that nucleation was uniform over the substrate. The power spectral density of the whole experiment provided the steady-state nucleation rate and showed that, in a stationary state, nucleation could be adequately described as a Poisson process. 

All spectrum analysis programs will make the assumption that count uncertainties are described by Poisson statistics. If the actual count situation is one of those described above, an extra uncertainty allowance would have to be made externally from the program. 

The method of statistical analysis in many bioassays focuses on analyzing the number and pattern of choices made by subjects. In general, these assays will not involve truly continuous variables, but will involve counts, e.g., the number of times that each branch of an olfactometer was chosen, the number of times that upwind flight was observed, the number of eggs deposited on test or control substrates, or the number of times that test or control feeding substrates were selected. Such data often are distributed following a Poisson distribution and can... 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

One common characteristic of many advanced scientific techniques, as indicated in Table 2, is that they are applied at the measurement frontier, where the net signal (S) is comparable to the residual background or blank (B) effect. The problem is compounded because (a) one or a few measurements are generally relied upon to estimate the blank—especially when samples are costly or difficult to obtain, and (b) the uncertainty associated with the observed blank is assumed normal and random and calculated either from counting statistics or replication with just a few degrees of freedom. (The disastrous consequences which may follow such naive faith in the stability of the blank are nowhere better illustrated than in trace chemical analysis, where S B is often the rule [10].) For radioactivity (or mass spectrometric) counting techniques it can be shown that the smallest detectable non-Poisson random error component is approximately 6, where ... 

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