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Topic: Bivariate Gaussian distribution

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  Software for Flexible Bayesian Modeling
This is an index to documenation for software implementing flexible Bayesian models based on neural networks, Gaussian processes, and finite and infinite mixture models, implemented using Markov chain Monte Carlo methods.
The documentation for the demonstration of sampling from a bivariate Gaussian distribution [bvg] is also not indexed above, but is listed below.
Markov chain sampling for a bivariate Gaussian [bvg]:
www.cs.toronto.edu /~radford/fbm.1997-07-22.doc   (146 words)

 The MathWorks - Demos - Simulation of dependent random variables using copulas
The bivariate lognormal distribution is a simple solution in the case, and of course easily generalizes to higher dimensions and cases where the marginal distributions are _different_ lognormals.
The family of bivariate Gaussian copulas is parameterized by Rho = [1 rho; rho 1], the linear correlation matrix.
Compared to the bivariate Gamma/t distribution constructed earlier, which was based on a Gaussian copula, the distribution constructed here, based on a t(1) copula, has the same marginal distributions and the same rank correlation between variables, but a very different dependence structure.
www.mathworks.com /products/demos/statistics/copulademo.html   (2408 words)

 Normal distribution Summary
As a Gaussian function with the denominator of the exponent equal to two, the standard normal distribution is an eigenfunction of the Fourier transform.
For the normal distribution, this accounts for 68% of the set while two standard deviations from the mean (blue and brown) account for 95% and three standard deviations (blue, brown and green) account for 99.7%.
Normally distributed and uncorrelated does not imply independent (an example of two normally distributed uncorrelated random variables that are not independent; this cannot happen in the presence of joint normality)
www.bookrags.com /Normal_distribution   (4551 words)

 Multivariate normal distribution - Wikipedia, the free encyclopedia
In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution).
That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems.
The cumulative distribution function (cdf) F(x) is defined as the probability that all values in a random vector X are less than or equal to the corresponding values in vector x.
en.wikipedia.org /wiki/Multivariate_normal_distribution   (1118 words)

 Gaussian distribution: FAQ. D'Errico.
The normal (or Gaussian) distribution is one which appears in an incredible variety of statistical applications.
Even when the right conditions are not met however, the distributions found for many experimentally generated sets of data still tend to have a bell shaped curve that often looks quite like that of a normal.
Even when a distribution may not be truly normal, it may still be convenient to assume that a normal distribution is a good approximation.
www.pitt.edu /~wpilib/statfaq/gaussfaq.html   (1510 words)

 Normal distribution - Wikipedia, the free encyclopedia
The normal distribution, also called Gaussian distribution (named after Carl Friedrich Gauss, a German mathematician, although Gauss was not the first to work with it), is a probability distribution of great importance in many fields.
The standard normal distribution is the normal distribution with a mean of zero and a variance of one (the green curves in the plots to the right).
The standard normal distribution has been tabulated (usually in the form of value of the cumulative distribution function Φ), and the other normal distributions are the simple transformations, as described above, of the standard one.
en.wikipedia.org /wiki/Gaussian_distribution   (4041 words)

 GNU Scientific Library -- Reference Manual - Random Number Distributions
In the simplest cases a non-uniform distribution can be obtained analytically from the uniform distribution of a random number generator by applying an appropriate transformation.
For \alpha = 2 it is a Gaussian distribution with @c{$\sigma = \sqrt{2} c$} \sigma = \sqrt{2} c.
Each component is generated to have a gaussian distribution, and then the components are normalized.
www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref_19.html   (2925 words)

 [No title]
Probability distributions are presented for the length and orientation of volcanic dikes within the repository footprint and for the number of eruptive centers located within the repository footprint (conditional on the dike intersecting the repository).
The calculation of conditional distributions for the number of eruptive centers within the potential repository footprint requires an assessment of the number of eruptive centers associated with a volcanic event and the spatial distribution for eruptive centers along the length of the dike.
These were defined as symmetric distributions over the range of 0 to 1, typically with higher probability for locations at the midpoint [the dike centered on point (x,y)] than at the ends [the dike extending for its full length in one direction away from point (x,y)].
www.ocrwm.doe.gov /documents/amr/22731/22731.txt   (16425 words)

Gaussian plume models predict concentrations of particles in a downwind plume and are probably the most commonly used dispersion models.
Particle sizes are assumed to be lognormally distributed; mean and variance of the distribution are specified in advance of the simulation run.
Dispersion due to turbulence was assumed to be random normally distributed (mean = 0 m) in the parallel, perpendicular horizontal, and perpendicular vertical directions to prevailing winds.
www.nbiap.vt.edu /brarg/brasym95/knudsen95.htm   (3955 words)

 [No title]   (Site not responding. Last check: 2007-10-20)
A recent universal construction of bivariate Gaussian distributions, leading to unforeseen kaleidoscopic decompositions of circular bells in terms of a host of elegant patterns having arbitrary n-fold symmetries, is reviewed.
Also, it has been discovered that the new construction of the bivariate Gaussian distribution leads, as the iterations are performed, to a host of exotic kaleidoscopic patterns of a transient nature having arbitrary n-fold symmetries, that decompose the circular bell in unforeseen and remarkable ways.
The remarkable kaleidoscopic decompositions of the bivariate Gaussian distribution.
www.mi.sanu.ac.yu /vismath/puente   (2787 words)

 RSS Matters
Then assuming that the data is truly Gaussian, a Gaussian probability distribution is fit with the estimated mean and standard deviation as parameters.
For example, the difference between the assumed Gaussian model and the nonparametric kernel density estimate curves can be used to define a test of the goodness of fit for the Gaussian distribution.
For example, the bootstrap is a methodology that is improved by substituting the empirical distribution function by a smoothed version of it (see RSS Matters Oct. 2001).
www.unt.edu /benchmarks/archives/2003/february03/rss.htm   (1970 words)

 [No title]
Nonhomogeneous parametric spatial distributions of future volcano occurrences were also modeled, for example, that the location of future volcanoes will follow a bivariate Gaussian distribution based on the location of volcanoes in Crater Flat.
Specifically, a probability distribution of the annual frequency of intersection of a basaltic dike with the proposed repository footprint was defined.
From these individual probability distributions, an aggregate probability distribution for the annual frequency of intersection of the proposed repository footprint by a dike was computed that reflected the uncertainty across the entire expert panel (CRWMS M&O 1996 [100116], Figure 4-32).
www.ocrwm.doe.gov /documents/amr/34758/34758.txt   (8088 words)

 Tales of Statisticians | Siméon-Denis Poisson
This was the Poisson distribution, which predicts the pattern in which random events of very low probability occur in the course of a very large number of trials.
Another type of data which is amenable to Poisson description is the distribution of yeast cells in a suspension.
This was the application of Gosset, who worked for the Guinness brewery firm, and who had arrived at the Poisson distribution without knowledge of Poisson or von Bortkiewicz.
www.umass.edu /wsp/statistics/tales/poisson.html   (790 words)

 Data Assimilation with sparse GPs   (Site not responding. Last check: 2007-10-20)
The second problem is the inference with the inherently non-tractable likelihood model, governed by the underlying physics governing the wave formation and light reflection from water surfaces.
Here the forward wind-field to scatterometer mapping was learned using a truncated Fourier series for the scatterometer values and the coefficients of the Fourier series were the outputs of RBF networks whose inputs are the relative wind direction (to the satellite) and the incidence angle.
A gaussian noise model was assumed and a Taylor expansion of the RBF networks has been made, obtaining an approximation to the log-average.
www.ncrg.aston.ac.uk /Projects/SSGP/examples/data_assimilation.html   (973 words)

 Bundling Information Goods:
Under reasonable assumptions about the distribution of valuations, the law of large numbers guarantees that the distribution of valuations for the bundle has proportionately more mass near the mean.
However, the distribution for the bundle converges more slowly to a Gaussian distribution, and the number of goods required to achieve a given level of profits and efficiency gains generally increases.
In this case, the distribution of consumer valuations for the bundle does not converge to a Gaussian distribution as more goods are added.
www.gsm.uci.edu /~bakos/big/big96-12.html   (10916 words)

 EPA - Distribution-Free Methods
Before computers, the mathematically "nice" features of the Gaussian, or normal, distribution meant that parametric methods were used most often.
Many of the classical tests that require an assumption of normality have nonparametric counterparts where the actual observed values are replaced by their ranks.
Nonparametric methods should be used when the data are known to not be normally distributed, for example, when the distribution of the data has a long tail or ordinal values (e.g., low, medium, or high) rather than continuous values (e.g., 0-100) with a unimodal distribution.
www.epa.gov /bioindicators/statprimer/distribution.html   (249 words)

 The Skew-Normal Distribution
The SN distribution is an extension of the normal (Gaussian) probability distribution, allowing for the presence of skewness.
The distribution is obtained by introducing a skewness parameter to the usual t density.
The `package sn' (or 'library' in Splus terminology) is a suite of functions for handling skew-normal and skew-t distributions, both in the univariate and the multivariate cases.
azzalini.stat.unipd.it /SN/index.html   (1158 words)

 Using Rank-Order Geostatistics for Spatial Interpolation of Highly Skewed Data in a Heavy-Metal Contaminated Site -- ...
The spatial distribution of a pollutant in contaminated soils
multi-normal distribution), the kriging estimation in the normal-scored
Gaussian check were used to assess whether the kriging estimation
jeq.scijournals.org /cgi/content/full/30/3/894   (3304 words)

 Spatial-temporal models
Additionally, spatial clustering is incorporated using a Neyman-Scott-type mechanism in which the displacements of the cell origins from the storm centre follow a bivariate distribution in space.
An important modification to the model of [Cox and Isham1988] is to have the storm centre moving with the same velocity as the cells so that cells are born within the existing structure of the storm.
Expressions for the mean and second order properties of the model can be derived [Northrop1996] although the second order properties require the numerical evaluation of integrals or the use of approximations.
www.ucl.ac.uk /stats/research/Resrprts/rr176/node8.html   (679 words)

 S-WoBA: Some new bivariate IG and NIG-distributions for modelling covariate nancial returns
We present two approaches for constructing bivariate NIG distribution that take advantage of the correlation between the univariate latent IG-variables that characterizes the marginal NIG-distribution.
A context for implementation in finance is given.
Keywords: Financial returns; bivariate distribution; NIG distribution; mixture representation; inverse Gaussian distribution; bivariate simulation; (follow links to similar papers)
swoba.hhs.se /nhhfms/abs/nhhfms2007_001.htm   (250 words)

 Vincent Zoonekynd's Blog
The inverted chi squared distribution is often used as a prior for variances, becausse it is amenable to computations: it is said to be conjugate -- but in the computer era, there is no reason to limit ourselves to these.
In the bivariate gaussian example, since X1 and X2 are correlated, successive values of (X1, X2) will be correlated: the samples do not contain as much information as an independant sample of the same size.
Extreme Value Theory (EVT) also studies the distribution of the maximum of iid random variables: there is a limit theorem, similar to the central limit theorem (with "max" instead of "mean") that identifies the limit distribution as one of the GEV (Generalized Extreme Values) distribution (of which the Gumbel, Frechet, Weibut are special cases).
zoonek.free.fr /blosxom/R/2006-06-22_useR2006.html   (5021 words)

 Messages (STK4050 - høst 2005)   (Site not responding. Last check: 2007-10-20)
The estimate of sigma²_theta was wrong, the prior distribution of one of the variances had a typo, and the mixture weights in exercise 2 did not add to one.
Then from this bivariate distribution, find the conditional (Gaussian) distribution for x(t) given y(t).
This gives the conditional distribution for x(t) given x(t-1) and y(t) (since we already have conditioned on x(t-1)).
www.uio.no /studier/emner/matnat/math/STK4050/h05/beskjeder.xml   (478 words)

 GNU Scientific Library -- Reference Manual: The Bivariate Gaussian Distribution   (Site not responding. Last check: 2007-10-20)
This function generates a pair of correlated gaussian variates, with mean zero, correlation coefficient
The probability distribution for bivariate gaussian random variates is,
) for a bivariate gaussian distribution with standard deviations
www.linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_270.html   (94 words)

 Proteome Science | Full text | Estimating probabilities of peptide database identifications to LC-FTICR-MS observations
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The probability of matching is estimated for each peptide in the reference database using statistical classification methods assuming bivariate Gaussian probability distributions on the uncertainties in the masses and the normalized elution times.
This gave the following distribution of amino acid residues: 0.27% from bovine peptides, 0.54% from standard proteins and known contaminants, and 99.2% from Shewanella and other unexpected proteins.
www.proteomesci.com /content/4/1/1   (3797 words)

 This weeks seminar
Also reviewed is a universal connection between arbitrary diffuse measures and the univariate and bivariate Gaussian distributions, as found when the fractal interpolating functions become plane- and space filling, respectively.
It is explained how the latter yields an infinite class of two-dimensional symmetric crystalline sets (Borges’ alephs) making up exotic kaleidoscopes of arbitrary symmetries, that decompose the bivariate Gaussian distribution in nontrivial ways.
It is shown that such mathematical designs include the structure of natural ice crystals and the rosette structure of relevant biochemical units, including even life’s own DNA.
www.jhu.edu /ceafm/Seminars-Spring-04/C-Puente.htm   (133 words)

 Interactive Statistical Calculation Pages
Lilliefors Test for Exponential Distribution -- tests whether a set of observed values are consistent with an exponential distribution.
Bivariate Sampling Statistics -- calculates means, variances, and covariance for up to 42 [x,y] measurements.
Comparison of Two Survival Distributions, using data from a data file in your computer (many different file types are supported).
statpages.org   (9672 words)

 The Bivariate Gaussian Distribution - GNU Scientific Library -- Reference Manual   (Site not responding. Last check: 2007-10-20)
Next: The Exponential Distribution, Previous: The Gaussian Tail Distribution, Up: Random Number Distributions
This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient
The probability distribution for bivariate Gaussian random variates is,
www.gnu.org /software/gsl/manual/html_node/The-Bivariate-Gaussian-Distribution.html   (135 words)

 [No title]   (Site not responding. Last check: 2007-10-20)
Expected value of uniform, exponential and Gaussian random variables.
Special Case: Erlang distribution and its mean, variance and Laplace transform.
We skipped section 7.6 (on the moment generating function since we already covered the z-transform and the Laplace transform) and section 7.8.
www-ee.eng.hawaii.edu /~jyee/342.00/classcov.txt   (859 words)

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