The data is often found to contain considerable variability, or noise, and thus Hidden Markov model and change-point analysis methods are being developed to infer real copy number changes.

These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis.

Ordinary differential equations appear in the movement of heavenly bodies ( planets, stars and galaxies ); optimization occurs in portfolio management ; numerical linear algebra is important for data analysis ; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

The analysis and processing of various types of corpora are also the subject of much work in computational linguistics, speech recognition and machine translation, where they are often used to create hidden Markov models for part of speech tagging and other purposes.

Alternatives to FTA include dependence diagram ( DD ), also known as reliability block diagram ( RBD ) and Markov analysis.

It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev ( Markov's teacher ), and many sources, especially in analysis, refer to it as Chebyshev's inequality or Bienaymé's inequality.

As used in describing simple linear regression analysis, one assumption of the fitted model ( to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the Gauss – Markov theorem ) is that the standard deviations of the error terms are constant and do not depend on the x-value.

It is still used widely in cellular phones for error correcting codes, as well as for speech recognition, DNA analysis, and many other applications of Hidden Markov models.

The OpenBUGS software ( Bayesian inference Using Gibbs Sampling ) does a Bayesian analysis of complex statistical models using Markov chain Monte Carlo.

JAGS ( Just another Gibbs sampler ) is a GPL program for analysis of Bayesian hierarchical models using Markov Chain Monte Carlo.

The Numerical Recipes books cover a range of topics that include both classical numerical analysis ( interpolation, integration, linear algebra, differential equations, and so on ), signal processing ( Fourier methods, filtering ), statistical treatment of data, and a few topics in machine learning ( hidden Markov models, support vector machines ).

* L. E. Baum, T. Petrie, G. Soules, and N. Weiss, " A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains ", Ann.

In Bayesian analysis, a generally applicable approach creates random samples from the joint posterior distribution of all the parameters: see Markov chain Monte Carlo.

Bayesian analysis uses the likelihood of trees in a Markov chain Monte Carlo ( MCMC ) simulation to sample trees in proportion to their likelihood, thereby producing a credible sample of trees.

This talk was titled " Markov processes and problems in analysis ".

The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery.

If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.

Assuming that the strength of the cable is proportional to its density per unit length, the weight, w, per unit length of the chain can be written T / c, where c is constant, and the analysis for nonuniform chains can be applied.

With no assumptions have been made regarding the force G acting on the chain, the following analysis can be made.

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force.

In 1923, Danish and Dutch scientists Christian Christiansen and Hendrik Anthony Kramers, in an analysis of formation of polymers, pointed out that such a chain reaction need not start with a molecule excited by light, but could also start with two molecules colliding violently in the traditional way classically previously proposed for initiation of chemical reactions, by van't Hoff.

Some of the protactinium-related dating variations rely on the analysis of the relative concentrations for several long-living members of the uranium decay chain – uranium, thorium and protactinium, for example.

Event chain methodology is an uncertainty modeling and schedule network analysis technique that is focused on identifying and managing events and event chains that affect project schedules.

The analysis of VNTR alleles continues, but is now usually performed by polymerase chain reaction ( PCR ) methods.

Around the same time the development of effective polymerase chain reaction techniques made it possible to apply cladistic methods of analysis to biochemical features as well.

However, the unit of analysis of most of these theories is not the system “ supply chain ”, but another system such as the “ firm ” or the “ supplier / buyer relationship ”.

However Daniel P. Whitmire's 1975 analysis indicates that a ramjet may achieve net power via the CNO cycle, which produces fusion at a much higher rate (~ 10 < sup > 16 </ sup > times higher ) than the proton-proton chain.

De la Beche had been inspired to create the painting by a vivid description of the food chain of the Lias by William Buckland that was based on analysis of coprolites.

The analysis is performed by extracting nuclear DNA from the cells of a forensic sample of interest, then amplifying specific polymorphic regions of the extracted DNA by means of the polymerase chain reaction.

Common letters and symbols would appear more often on the chain, according to the frequency analysis of the likely input.

The fragile X abnormality is now directly determined by analysis of the number of CGG repeats using polymerase chain reaction ( PCR ) and methylation status using Southern blot analysis.

In March 2011, Subway was named " the most loved fast food chain in the US " by Amplicate. com, based on their analysis of opinions expressed on social media websites.

At the tactical level briefings are delivered to patrols on current threats and collection priorities, these patrols are then debriefed to elicit information for analysis and communication through the reporting chain.

A hydropathy analysis uses an algorithm that quantifies the hydrophobic character at each position along the polypeptide chain.

Using hydrophobicity analysis to predict transmembrane helices enables a prediction in turn of the " transmembrane topology " of a protein ; i. e. prediction of what parts of it protrude into the cell, what parts protrude out, and how many times the protein chain crosses the membrane.

Critical chain is used as an alternative to critical path analysis.

Various aspects of optimizing the supply chain include liaising with suppliers to eliminate bottlenecks ; sourcing strategically to strike a balance between lowest material cost and transportation, implementing JIT ( Just In Time ) techniques to optimize manufacturing flow ; maintaining the right mix and location of factories and warehouses to serve customer markets, and using location / allocation, vehicle routing analysis, dynamic programming and, of course, traditional logistics optimization to maximize the efficiency of the distribution side.

In the late 1990s, microsatellite analysis was used to identify hybrids of Indian and Siberian tigers through polymerase chain reaction amplification of hair samples.

* University of Oxford – developed much of the experiment's Monte Carlo analysis program ( SNOMAN ), and maintained the program

There is also an ever growing connection between Bayesian methods and simulation-based Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a graphical model structure may allow for efficient simulation algorithms like the Gibbs sampling and other Metropolis – Hastings algorithm schemes.

* Herman Kahn, Stochastic ( Monte Carlo ) attenuation analysis, Santa, Monica, Calif., Rand Corp., 1949

However, in 2010, analysis of retrotransposon insertion sites in the nuclear DNA of a variety of marsupials, while confirming the placement of the Monito del Monte in Australidelphia, showed that its lineage is the most basal of that superorder.

Alternatively a more formal risk analysis can be undertaken using Monte Carlo simulations.

The test should instead be tailored to the purpose of the analysis, e. g. one uses Monte Carlo filtering if one is interested in which factors are most responsible for generating high / low values of the output.

* In Corporate Finance, project finance and real options analysis, Monte Carlo Methods are used by financial analysts who wish to construct " stochastic " or probabilistic financial models as opposed to the traditional static and deterministic models.

He was the president of the 1903-1904 Monte Carlo international tournaments and according to the common, though unreliable beliefs, invited the Russian chess master Mikhail Chigorin to play but later paid him 1, 500 francs ( greater than 3rd prize money ) not to play because Chigorin had published analysis of one of the Prince's games, pointing out he had made gross errors.

This method of generating random data is called " Monte Carlo " analysis, after the famous casino, and it is widely used in statistical analysis to test procedures.

It is also possible to perform Monte Carlo sensitivity analysis.

* Monte Carlo based operations analysis package & air taxi business planning software developed during the SATS program

* Guest JE, Duncan AM & Chester DK ( 1988 ) Monte Vulture Volcano ( Basilicata, Italy ): an analysis of morphology and volcaniclastic facies.

Ensemble forecasting is a form of Monte Carlo analysis: multiple numerical predictions are conducted using slightly different initial conditions that are all plausible given the past and current set of observations, or measurements.