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PubMed2006

Statistical distribution of the measure of coherence.

The recent papers by Miranda de Sa et al. (2001) and Miranda de Sa (2004) proposed a new measure of coherence between two signals. In this paper, we derive the exact probability density function and moments of the measure. We also provide simple programs for computing the associated cumulative distribution function and percentile points as well as for generating random samples of the measure.

Adolescent↗
PubMed2005

Analytical results for the statistical distribution related to a memoryless deterministic walk: dimensionality effect and mean-field models.

Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding mu steps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial nonperiodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parametrized by the normalized incomplete beta function Id= I1/4 [1/2, (d+1) /2] . The joint distribution S(N) (mu,d) (t,p) is relevant, and the marginal distributions previously studied are particular cases. We show that, for the memory-less deterministic tourist walk in the euclidean space, this distribution is Sinfinity(1,d) (t,p) = [Gamma (1+ I(-1)(d)) (t+ I(-1)(d) ) /Gamma(t+p+ I(-1)(d)) ] delta(p,2), where t=0, 1,2, ... infinity, Gamma(z) is the gamma function and delta(i,j) is the Kronecker delta. The mean-field models are the random link models, which correspond to d-->infinity, and the random map model which, even for mu=0 , presents nontrivial cycle distribution [ S(N)(0,rm) (p) proportional to p(-1) ] : S(N)(0,rm) (t,p) =Gamma(N)/ {Gamma[N+1- (t+p) ] N( t+p)}. The fundamental quantities are the number of explored points n(e)=t+p and Id. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.

Algorithms↗
PubMed1994

The evolution of proteins from random amino acid sequences: II. Evidence from the statistical distributions of the lengths of modern protein sequences.

This paper continues an examination of the hypothesis that modern proteins evolved from random heteropeptide sequences. In support of the hypothesis, White and Jacobs (1993, J Mol Evol 36:79-95) have shown that any sequence chosen randomly from a large collection of nonhomologous proteins has a 90% or better chance of having a lengthwise distribution of amino acids that is indistinguishable from the random expectation regardless of amino acid type. The goal of the present study was to investigate the possibility that the random-origin hypothesis could explain the lengths of modern protein sequences without invoking specific mechanisms such as gene duplication or exon splicing. The sets of sequences examined were taken from the 1989 PIR database and consisted of 1,792 "super-family" proteins selected to have little sequence identity, 623 E. coli sequences, and 398 human sequences. The length distributions of the proteins could be described with high significance by either of two closely related probability density functions: The gamma distribution with parameter 2 or the distribution for the sum of two exponential random independent variables. A simple theory for the distributions was developed which assumes that (1) protoprotein sequences had exponentially distributed random independent lengths, (2) the length dependence of protein stability determined which of these protoproteins could fold into compact primitive proteins and thereby attain the potential for biochemical activity, (3) the useful protein sequences were preserved by the primitive genome, and (4) the resulting distribution of sequence lengths is reflected by modern proteins. The theory successfully predicts the two observed distributions which can be distinguished by the functional form of the dependence of protein stability on length. The theory leads to three interesting conclusions. First, it predicts that a tetra-nucleotide was the signal for primitive translation termination. This prediction is entirely consistent with the observations of Brown et al. (1990a,b, Nucleic Acids Res 18:2079-2086 and 18: 6339-6345) which show that tetra-nucleotides (stop codon plus following nucleotide) are the actual signals for termination of translation in both prokaryotes and eukaryotes. Second, the strong dependence of statistical length distributions on sequence-termination signaling codes implies that the evolution of stop codons and translation-termination processes was as important as gene splicing in early evolution. Third, because the theory is based upon a simple no-exon stochastic model, it provides a plausible alternative to a limited universe of exons from which all proteins evolved by gene duplication and exon splicing (Dorit et al. 1990, Science 250:1377-1382).

Amino Acid Sequence↗
PubMedDate not supplied

Statistical distributions of uncertainty and variability in activated sludge model parameters.

All models used in activated sludge design and analysis use parameters to characterize process performance. The values of these parameters are often assumed based on default values recommended in the literature, but to date, no quantitative estimates of the parameter uncertainties have been published. Similarly, little attention has been given to quantifying site-specific parameter variability, even though its occurrence has been observed several times in the literature. In this paper, universal uncertainty distributions of the model parameters from Activated Sludge Model No. 1 are developed from a database of parameter values reported in the literature using Bayesian statistics. Site-specific distributions of parameter variability were developed using the same techniques. All parameter distributions developed demonstrated that significant uncertainty and variability exist, which could lead to overdesign or plant failure if not considered during the design process.

Bayes Theorem↗
PubMed1986

Origin of eukaryotic introns: a hypothesis, based on codon distribution statistics in genes, and its implications.

A hypothesis for the origin of introns in eukaryotic genes is developed. By computer simulation it was found that the reading-frame lengths in a random nucleotide sequence are distributed in a negative exponential manner and that there exists an upper limit of about 200 codons in the length of the reading frames (RFs). These characteristics suggest that, if primordial DNA contained a random nucleotide sequence, the most primitive cells would have been under selective pressure to eliminate interfering stop codons in order to increase the length of RFs. Further, they indicate that the only possible way that a coding sequence that is considerably longer than 600 nucleotides could be derived from the short coding sequences occurring in a random sequence would be to splice the short coding sequences and to eliminate the stretches of sequences containing clusters of inframe stop codons. Thus, introns are suggested to be those stretches of sequences containing interfering stop codons that were originally earmarked in the first primitive cells to be eliminated in order to enable the coding for long polypeptides. Because the statistical characteristics of codon distributions in today's eukaryotic DNA sequences resemble closely those of a random sequence and because the upper limit in the length of RFs (200 codons) in a random sequence corresponds precisely to the observed maximum length of exons in today's eukaryotic genes (600 nucleotides), it is suggested that introns originated in the most primitive unicellular eukaryotes when they evolved from primordial sequences. The data from the prokaryotic gene sequences indicate that prokaryotic genes may have been derived originally from primitive unicellular eukaryotic genes by losing introns from them.

Animals↗
PubMed1990

Truth and consequences of ordinal differences in statistical distributions: toward a theory of hierarchical inference.

A theory is presented that establishes a dominance hierarchy of potential distinctions (order relations) between two distributions. It is proposed that it is worthwhile for researchers to ascertain the strongest possible distinction, because all weaker distinctions are logically implied. Implications of the theory for hypothesis testing, theory construction, and scales of measurement are considered. Open problems for future research are outlined.

Humans↗
PubMed2003

The statistical distribution of the intensity of pixels within spots of DNA microarrays: what is the appropriate single-value representative?

This paper opens a discussion about an important issue in the analysis of data from spotted DNA microarrays: how to summarise into a single value the distribution for the intensity values of the pixels within a spot. Although the most popular statistic used is the median, there is no clear study demonstrating why it is more appropriate than other measures of central tendency such as the mean or the mode. Here, we argue that the median intensity is not the most appropriate measure for many common cases and discuss a frequently encountered case of a 'doughnut'-shaped spot for which the mode is closest to the 'expected' spot intensity. For an 'ideal' spot with a clear boundary and uniformly hybridised, the intensity of its pixels should approximately be normally distributed. In practical situations, these two requirements are often not met due to the physical properties of pins and the particularities of the printing and hybridisation processes. As a consequence, the distribution of the intensity of the pixels is usually negatively skewed. This asymmetry results in a larger displacement for the mean and median than for the mode from the ideal situation mentioned above.

Algorithms↗
PubMed2007

Statistical distribution of the difference of two proportions.

Differences of two proportions or rates occur most frequently in biomedical research. However, as far as published work in the Statistics in Medicine are concerned, only approximations have been used to study the distribution of such differences. In this Letter, we derive the exact probability distribution of the difference of two proportions. The expression involves a well-known special function. It is expected that this result could be widely useful for medical practitioners.

Algorithms↗
PubMed1985

The statistical distribution of nucleic acid similarities.

All pairs of a large set of known vertebrate DNA sequences were searched by computer for most similar segments. Analysis of this data shows that the computed similarity scores are distributed proportionally to the logarithm of the product of the lengths of the sequences involved. This distribution is closely related to recent results of Erdos and others on the longest run of heads in coin tossing. A simple rule is derived for determination of statistical significance of the similarity scores and to assist in relating statistical and biological significance.

Base Sequence↗
PubMed1977

The statistical distribution of macrophage migration distance and its application to MIF test.

The measurement of the distance of macrophage migration is very simple and reasonable as a technique in MIF assay. By the use of this technique, a large scale model experiment was performed and its data were analysed statistically. As the result of analysis, Student's t-test for significance of the difference between control and test readings of migration distance at low activity level in the MIF test was justified, and it was proved that it is applicable to the routine MIF test. In addition to this, the measurement of migration distance was compared with that of migration area.

Animals↗