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75-642: The Goddard Earth Observing System ( GEOS ) is an integrated Earth system model and data assimilation system developed at the Global Modeling and Assimilation Office (GMAO) at NASA 's Goddard Space Flight Center . The components of the model use the Earth System Modeling Framework (ESMF), enabling them to be connected in a flexible manner and supporting the investigation of many different aspects of Earth science , in particular questions related to coupled processes involving

150-436: A 100% chance of getting pancreatic cancer. Assuming the incidence rate of pancreatic cancer is 1/100000, while 10/99999 healthy individuals have the same symptoms worldwide, the probability of having pancreatic cancer given the symptoms is only 9.1%, and the other 90.9% could be "false positives" (that is, falsely said to have cancer; "positive" is a confusing term when, as here, the test gives bad news). Based on incidence rate,

225-518: A European project in the FP4-ENV program which took place in the Alpilles region, South-East of France (1996–97). The Flow-chart diagram (right), excerpted from the final report of that project, shows how to infer variables of interest such as canopy state, radiative fluxes, environmental budget, production in quantity and quality, from remote sensing data and ancillary information. In that diagram,

300-611: A Problem in the Doctrine of Chances" (1763), which appeared in Philosophical Transactions , and contains Bayes' theorem. Price wrote an introduction to the paper which provides some of the philosophical basis of Bayesian statistics and chose one of the two solutions offered by Bayes. In 1765, Price was elected a Fellow of the Royal Society in recognition of his work on the legacy of Bayes. On 27 April

375-519: A distribution for the probability parameter of a binomial distribution (in modern terminology). On Bayes's death his family transferred his papers to a friend, the minister, philosopher, and mathematician Richard Price . Over two years, Richard Price significantly edited the unpublished manuscript, before sending it to a friend who read it aloud at the Royal Society on 23 December 1763. Price edited Bayes's major work "An Essay Towards Solving

450-582: A fixed receiver, as well as from weather satellites . The World Meteorological Organization acts to standardize the instrumentation, observing practices and timing of these observations worldwide. Stations either report hourly in METAR reports, or every six hours in SYNOP reports. These observations are irregularly spaced, so they are processed by data assimilation and objective analysis methods, which perform quality control and obtain values at locations usable by

525-455: A letter sent to his friend Benjamin Franklin was read out at the Royal Society, and later published, where Price applies this work to population and computing 'life-annuities'. Independently of Bayes, Pierre-Simon Laplace in 1774, and later in his 1812 Théorie analytique des probabilités , used conditional probability to formulate the relation of an updated posterior probability from

600-549: A particular test for whether someone has been using cannabis is 90% sensitive , meaning the true positive rate (TPR) = 0.90. Therefore, it leads to 90% true positive results (correct identification of drug use) for cannabis users. The test is also 80% specific , meaning true negative rate (TNR) = 0.80. Therefore, the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or false positive rate (FPR) = 0.20, for non-users. Assuming 0.05 prevalence , meaning 5% of people use cannabis, what

675-406: A prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work. The Bayesian interpretation of probability was developed mainly by Laplace. About 200 years later, Sir Harold Jeffreys put Bayes's algorithm and Laplace's formulation on an axiomatic basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what

750-470: A randomly selected item is defective, what is the probability it was produced by machine C? Once again, the answer can be reached without using the formula by applying the conditions to a hypothetical number of cases. For example, if the factory produces 1,000 items, 200 will be produced by Machine A, 300 by Machine B, and 500 by Machine C. Machine A will produce 5% × 200 = 10 defective items, Machine B 3% × 300 = 9, and Machine C 1% × 500 = 5, for

825-410: A satisfactory solution. Real world measurements contain errors both due to the quality of the instrument and how accurately the position of the measurement is known. These errors can cause instabilities in the models that eliminate any level of skill in a forecast. Thus, more sophisticated methods were needed in order to initialize a model using all available data while making sure to maintain stability in

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900-419: A set of observed data and estimated errors that are present in both the observations and the forecast itself. The difference between the forecast and the observations at that time is called the departure or the innovation (as it provides new information to the data assimilation process). A weighting factor is applied to the innovation to determine how much of a correction should be made to the forecast based on

975-406: A total of 24. Thus, the likelihood that a randomly selected defective item was produced by machine C is 5/24 (~20.83%). This problem can also be solved using Bayes' theorem: Let X i denote the event that a randomly chosen item was made by the i machine (for i  = A,B,C). Let Y denote the event that a randomly chosen item is defective. Then, we are given the following information: If

1050-550: A variant of the Kalman-Bucy filter (a continuous time version of the Kalman filter ) with the gain matrix prescribed rather than obtained from covariances. A major development was achieved by L. Gandin (1963) who introduced the "statistical interpolation" (or "optimal interpolation") method, which developed earlier ideas of Kolmogorov. This is a 3DDA method and is a type of regression analysis which utilizes information about

1125-859: Is a fluid . The idea of numerical weather prediction is to sample the state of the fluid at a given time and use the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future. The process of entering observation data into the model to generate initial conditions is called initialization . On land, terrain maps available at resolutions down to 1 kilometer (0.6 mi) globally are used to help model atmospheric circulations within regions of rugged topography, in order to better depict features such as downslope winds, mountain waves and related cloudiness that affects incoming solar radiation. The main inputs from country-based weather services are observations from devices (called radiosondes ) in weather balloons that measure various atmospheric parameters and transmits them to

1200-485: Is a cannabis user given that they test positive," which is what is meant by PPV. We can write: The denominator P ( Positive ) = P ( Positive | User ) P ( User ) + P ( Positive | Non-user ) P ( Non-user ) {\displaystyle P({\text{Positive}})=P({\text{Positive}}\vert {\text{User}})P({\text{User}})+P({\text{Positive}}\vert {\text{Non-user}})P({\text{Non-user}})}

1275-454: Is a direct application of the Law of Total Probability . In this case, it says that the probability that someone tests positive is the probability that a user tests positive, times the probability of being a user, plus the probability that a non-user tests positive, times the probability of being a non-user. This is true because the classifications user and non-user form a partition of a set , namely

1350-454: Is a linear operator (matrix). Factors driving the rapid development of data assimilation methods for NWP models include: Data assimilation has been used, in the 1980s and 1990s, in several HAPEX (Hydrologic and Atmospheric Pilot Experiment) projects for monitoring energy transfers between the soil, vegetation and atmosphere. For instance: - HAPEX-MobilHy , HAPEX-Sahel, - the "Alpilles-ReSeDA" (Remote Sensing Data Assimilation) experiment,

1425-430: Is a part of the challenge for every forecasting problem. Dealing with biased data is a serious challenge in data assimilation. Further development of methods to deal with biases will be of particular use. If there are several instruments observing the same variable then intercomparing them using probability distribution functions can be instructive. The numerical forecast models are becoming of higher resolution due to

1500-399: Is needed to map the modeled variable to a form that can be directly compared with the observation. One of the common mathematical philosophical perspectives is to view data assimilation as a Bayesian estimation problem. From this perspective, the analysis step is an application of Bayes' theorem and the overall assimilation procedure is an example of recursive Bayesian estimation . However,

1575-464: Is performed many times. P ( A ) is the proportion of outcomes with property A (the prior) and P ( B ) is the proportion with property B . P ( B  |  A ) is the proportion of outcomes with property B out of outcomes with property A , and P ( A  |  B ) is the proportion of those with A out of those with  B (the posterior). The role of Bayes' theorem is best visualized with tree diagrams. The two diagrams partition

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1650-406: Is raised to 100% and specificity remains at 80%, the probability of someone testing positive really being a cannabis user only rises from 19% to 21%, but if the sensitivity is held at 90% and the specificity is increased to 95%, the probability rises to 49%. Even if 100% of patients with pancreatic cancer have a certain symptom, when someone has the same symptom, it does not mean that this person has

1725-402: Is sometimes known as the butterfly effect – the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. At any update time, data assimilation usually takes a forecast (also known as the first guess , or background information) and applies a correction to the forecast based on

1800-534: Is the probability that a random person who tests positive is really a cannabis user? The Positive predictive value (PPV) of a test is the proportion of persons who are actually positive out of all those testing positive, and can be calculated from a sample as: If sensitivity, specificity, and prevalence are known, PPV can be calculated using Bayes theorem. Let P ( User | Positive ) {\displaystyle P({\text{User}}\vert {\text{Positive}})} mean "the probability that someone

1875-449: Is then P X , Y ( d x , d y ) = P Y x ( d y ) P X ( d x ) {\displaystyle P_{X,Y}(dx,dy)=P_{Y}^{x}(dy)P_{X}(dx)} . The conditional distribution P X y {\displaystyle P_{X}^{y}} of X {\displaystyle X} given Y = y {\displaystyle Y=y}

1950-469: Is then determined by P X y ( A ) = E ( 1 A ( X ) | Y = y ) {\displaystyle P_{X}^{y}(A)=E(1_{A}(X)|Y=y)} Existence and uniqueness of the needed conditional expectation is a consequence of the Radon–Nikodym theorem . This was formulated by Kolmogorov in his famous book from 1933. Kolmogorov underlines

2025-462: Is used operationally at forecast centres such as the Met Office . The process of creating the analysis in data assimilation often involves minimization of a cost function . A typical cost function would be the sum of the squared deviations of the analysis values from the observations weighted by the accuracy of the observations, plus the sum of the squared deviations of the forecast fields and

2100-489: The Apollo program , GPS , and atmospheric chemistry . Examples of how variational assimilation is implemented weather forecasting at: Other examples of assimilation: Bayes%27 theorem Bayes' theorem (alternatively Bayes' law or Bayes' rule , after Thomas Bayes ) gives a mathematical rule for inverting conditional probabilities , allowing us to find the probability of a cause given its effect. For example, if

2175-559: The Ensemble Kalman filter and the Reduced-Rank Kalman filters (RRSQRT). Another significant advance in the development of the 4DDA methods was utilizing the optimal control theory (variational approach) in the works of Le Dimet and Talagrand (1986), based on the previous works of J.-L. Lions and G. Marchuk, the latter being the first to apply that theory in the environmental modeling. The significant advantage of

2250-544: The European Centre for Medium-Range Weather Forecasts (ECMWF) and at the NOAA National Centers for Environmental Prediction (NCEP). Data assimilation can also be achieved within a model update loop, where we will iterate an initial model (or initial guess) in an optimisation loop to constrain the model to the observed data. Many optimisation approaches exist and all of them can be setup to update

2325-486: The Kalman filter . Many methods represent the probability distributions only by the mean and input some pre-calculated covariance. An example of a direct (or sequential ) method to compute this is called optimal statistical interpolation, or simply optimal interpolation ( OI ). An alternative approach is to iteratively solve a cost function that solves an identical problem. These are called variational methods, such as 3D-Var and 4D-Var. Typical minimization algorithms are

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2400-461: The MM5 model. They are based on the simple idea of Newtonian relaxation (the 2nd axiom of Newton). They introduce into the right part of dynamical equations of the model a term that is proportional to the difference of the calculated meteorological variable and the observed value. This term that has a negative sign keeps the calculated state vector closer to the observations. Nudging can be interpreted as

2475-470: The Pythagorean theorem is to geometry". Stephen Stigler used a Bayesian argument to conclude that Bayes' theorem was discovered by Nicholas Saunderson , a blind English mathematician, some time before Bayes; that interpretation, however, has been disputed. Martyn Hooper and Sharon McGrayne have argued that Richard Price 's contribution was substantial: By modern standards, we should refer to

2550-436: The conjugate gradient method or the generalized minimal residual method . The ensemble Kalman filter is sequential method that uses a Monte Carlo approach to estimate both the mean and the covariance of a Gaussian probability distribution by an ensemble of simulations. More recently, hybrid combinations of ensemble approaches and variational methods have become more popular (e.g. they are used for operational forecasts both at

2625-486: The interpretation of probability ascribed to the terms. The two predominant interpretations are described below. In the Bayesian (or epistemological) interpretation , probability measures a "degree of belief". Bayes' theorem links the degree of belief in a proposition before and after accounting for evidence. For example, suppose it is believed with 50% certainty that a coin is twice as likely to land heads than tails. If

2700-494: The posterior probability ). Bayes' theorem is named after the Reverend Thomas Bayes ( / b eɪ z / ), also a statistician and philosopher. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. His work was published in 1763 as An Essay Towards Solving a Problem in the Doctrine of Chances . Bayes studied how to compute

2775-420: The troposphere and well into the stratosphere . Information from weather satellites is used where traditional data sources are not available. Commerce provides pilot reports along aircraft routes and ship reports along shipping routes. Research projects use reconnaissance aircraft to fly in and around weather systems of interest, such as tropical cyclones . Reconnaissance aircraft are also flown over

2850-843: The Bayes–Price rule. Price discovered Bayes's work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes's name alone is unfair but so entrenched that anything else makes little sense. Bayes' theorem is stated mathematically as the following equation: P ( A | B ) = P ( B | A ) P ( A ) P ( B ) {\displaystyle P(A\vert B)={\frac {P(B\vert A)P(A)}{P(B)}}} where A {\displaystyle A} and B {\displaystyle B} are events and P ( B ) ≠ 0 {\displaystyle P(B)\neq 0} . Bayes' theorem may be derived from

2925-478: The KF algorithms as a 4DDA tool for NWP models came later. However, this was (and remains) a difficult task because the full version requires solution of the enormous number of additional equations (~N*N~10**12, where N=Nx*Ny*Nz is the size of the state vector, Nx~100, Ny~100, Nz~100 – the dimensions of the computational grid). To overcome this difficulty, approximate or suboptimal Kalman filters were developed. These include

3000-603: The Mars Climate Sounder onboard NASA's Mars Reconnaissance Orbiter . Two methods of data assimilation have been applied to these datasets: an Analysis Correction scheme and two Ensemble Kalman Filter schemes, both using a global circulation model of the martian atmosphere as forward model. The Mars Analysis Correction Data Assimilation (MACDA) dataset is publicly available from the British Atmospheric Data Centre. Data assimilation

3075-573: The Pacific. In 1922, Lewis Fry Richardson published the first attempt at forecasting the weather numerically. Using a hydrostatic variation of Bjerknes's primitive equations , Richardson produced by hand a 6-hour forecast for the state of the atmosphere over two points in central Europe, taking at least six weeks to do so. His forecast calculated that the change in surface pressure would be 145 millibars (4.3  inHg ), an unrealistic value incorrect by two orders of magnitude. The large error

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3150-941: The analyzed fields weighted by the accuracy of the forecast. This has the effect of making sure that the analysis does not drift too far away from observations and forecasts that are known to usually be reliable. J ( x ) = ( x − x b ) T B − 1 ( x − x b ) + ( y − H [ x ] ) T R − 1 ( y − H [ x ] ) , {\displaystyle J(\mathbf {x} )=(\mathbf {x} -\mathbf {x} _{b})^{\mathrm {T} }\mathbf {B} ^{-1}(\mathbf {x} -\mathbf {x} _{b})+(\mathbf {y} -{\mathit {H}}[\mathbf {x} ])^{\mathrm {T} }\mathbf {R} ^{-1}(\mathbf {y} -{\mathit {H}}[\mathbf {x} ]),} where B {\displaystyle \mathbf {B} } denotes

3225-551: The atmosphere, ocean, and/or land. Uses of GEOS span a range of spatiotemporal scales and include the representation of dynamical, physical, chemical and biological processes. This article related to the National Aeronautics and Space Administration is a stub . You can help Misplaced Pages by expanding it . This climatology -related article is a stub . You can help Misplaced Pages by expanding it . Data assimilation Data assimilation initially developed in

3300-1492: The background error covariance, R {\displaystyle \mathbf {R} } the observational error covariance. ∇ J ( x ) = 2 B − 1 ( x − x b ) − 2 H T R − 1 ( y − H [ x ] ) {\displaystyle \nabla J(\mathbf {x} )=2\mathbf {B} ^{-1}(\mathbf {x} -\mathbf {x} _{b})-2{\mathit {H}}^{T}\mathbf {R} ^{-1}(\mathbf {y} -{\mathit {H}}[\mathbf {x} ])} J ( x ) = ( x − x b ) T B − 1 ( x − x b ) + ∑ i = 0 n ( y i − H i [ x i ] ) T R i − 1 ( y i − H i [ x i ] ) {\displaystyle J(\mathbf {x} )=(\mathbf {x} -\mathbf {x} _{b})^{\mathrm {T} }\mathbf {B} ^{-1}(\mathbf {x} -\mathbf {x} _{b})+\sum _{i=0}^{n}(\mathbf {y} _{i}-{\mathit {H}}_{i}[\mathbf {x} _{i}])^{\mathrm {T} }\mathbf {R} _{i}^{-1}(\mathbf {y} _{i}-{\mathit {H}}_{i}[\mathbf {x} _{i}])} provided that H {\displaystyle {\mathit {H}}}

3375-514: The coin is flipped a number of times and the outcomes observed, that degree of belief will probably rise or fall, but might even remain the same, depending on the results. For proposition A and evidence B , For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see Bayesian inference . In the frequentist interpretation , probability measures a "proportion of outcomes". For example, suppose an experiment

3450-444: The definition of conditional density : Therefore, Let P Y x {\displaystyle P_{Y}^{x}} be the conditional distribution of Y {\displaystyle Y} given X = x {\displaystyle X=x} and let P X {\displaystyle P_{X}} be the distribution of X {\displaystyle X} . The joint distribution

3525-551: The definition of conditional probability : where P ( A ∩ B ) {\displaystyle P(A\cap B)} is the probability of both A and B being true. Similarly, Solving for P ( A ∩ B ) {\displaystyle P(A\cap B)} and substituting into the above expression for P ( A | B ) {\displaystyle P(A\vert B)} yields Bayes' theorem: For two continuous random variables X and Y , Bayes' theorem may be analogously derived from

3600-420: The field of numerical weather prediction . Numerical weather prediction models are equations describing the dynamic behavior of the atmosphere, typically coded into a computer program. In order to use these models to make forecasts, initial conditions are needed for the model that closely resemble the current state of the atmosphere. Simply inserting point-wise measurements into the numerical models did not provide

3675-427: The following table presents the corresponding numbers per 100,000 people. Which can then be used to calculate the probability of having cancer when you have the symptoms: A factory produces items using three machines—A, B, and C—which account for 20%, 30%, and 50% of its output respectively. Of the items produced by machine A, 5% are defective; similarly, 3% of machine B's items and 1% of machine C's are defective. If

3750-560: The grid scale, or clouds, in the atmospheric models. This increasing non-linearity in the models and observation operators poses a new problem in the data assimilation. The existing data assimilation methods such as many variants of ensemble Kalman filters and variational methods, well established with linear or near-linear models, are being assessed on non-linear models. Many new methods are being developed, e.g. particle filters for high-dimensional problems, and hybrid data assimilation methods. Other uses include trajectory estimation for

3825-459: The importance of conditional probability by writing "I wish to call attention to ... and especially the theory of conditional probabilities and conditional expectations ..." in the Preface. The Bayes theorem determines the posterior distribution from the prior distribution. Uniqueness requires continuity assumptions. Bayes' theorem can be generalized to include improper prior distributions such as

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3900-627: The increase of computational power , with operational atmospheric models now running with horizontal resolutions of order of 1 km (e.g. at the German National Meteorological Service, the Deutscher Wetterdienst ( DWD ) and Met Office in the UK). This increase in horizontal resolutions is starting to allow to resolve more chaotic features of the non-linear models, e.g. to resolve convection on

3975-444: The item is defective, the probability that it was made by machine C is 5/24. Although machine C produces half of the total output, it produces a much smaller fraction of the defective items. Hence the knowledge that the item selected was defective enables us to replace the prior probability P ( X C ) = 1/2 by the smaller posterior probability P (X C  |  Y ) = 5/24. The interpretation of Bayes' rule depends on

4050-436: The item was made by the first machine, then the probability that it is defective is 0.05; that is, P ( Y  |  X A ) = 0.05. Overall, we have To answer the original question, we first find P (Y). That can be done in the following way: Hence, 2.4% of the total output is defective. We are given that Y has occurred, and we want to calculate the conditional probability of X C . By Bayes' theorem, Given that

4125-414: The meaning of a positive test result correctly and avoid the base-rate fallacy . One of the many applications of Bayes' theorem is Bayesian inference , a particular approach to statistical inference , where it is used to invert the probability of observations given a model configuration (i.e., the likelihood function ) to obtain the probability of the model configuration given the observations (i.e.,

4200-495: The model's mathematical algorithms. Some global models use finite differences , in which the world is represented as discrete points on a regularly spaced grid of latitude and longitude; other models use spectral methods that solve for a range of wavelengths. The data are then used in the model as the starting point for a forecast. A variety of methods are used to gather observational data for use in numerical models. Sites launch radiosondes in weather balloons which rise through

4275-424: The model, for instance, evolutionary algorithm have proven to be efficient as free of hypothesis, but computationally expensive. In numerical weather prediction applications, data assimilation is most widely known as a method for combining observations of meteorological variables such as temperature and atmospheric pressure with prior forecasts in order to initialize numerical forecast models. The atmosphere

4350-424: The most cited publication in all of the geosciences is an application of data assimilation to reconstruct the observed history of the atmosphere. Classically, data assimilation has been applied to chaotic dynamical systems that are too difficult to predict using simple extrapolation methods. The cause of this difficulty is that small changes in initial conditions can lead to large changes in prediction accuracy. This

4425-400: The new information from the observations. The best estimate of the state of the system based on the correction to the forecast determined by a weighting factor times the innovation is called the analysis . In one dimension, computing the analysis could be as simple as forming a weighted average of a forecasted and observed value. In multiple dimensions the problem becomes more difficult. Much of

4500-431: The numerical model. Such data typically includes the measurements as well as a previous forecast valid at the same time the measurements are made. If applied iteratively, this process begins to accumulate information from past observations into all subsequent forecasts. Because data assimilation developed out of the field of numerical weather prediction, it initially gained popularity amongst the geosciences. In fact, one of

4575-426: The open oceans during the cold season into systems which cause significant uncertainty in forecast guidance, or are expected to be of high impact from three to seven days into the future over the downstream continent. Sea ice began to be initialized in forecast models in 1971. Efforts to involve sea surface temperature in model initialization began in 1972 due to its role in modulating weather in higher latitudes of

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4650-400: The pattern. The rare subspecies is 0.1% of the total population. How likely is the beetle having the pattern to be rare: what is P (Rare | Pattern)? From the extended form of Bayes' theorem (since any beetle is either rare or common), For events A and B , provided that P ( B ) ≠ 0, In many applications, for instance in Bayesian inference , the event B is fixed in

4725-577: The probabilistic analysis is usually simplified to a computationally feasible form. Advancing the probability distribution in time would be done exactly in the general case by the Fokker–Planck equation , but that is not feasible for high-dimensional systems; so, various approximations operating on simplified representations of the probability distributions are used instead. Often the probability distributions are assumed Gaussian so that they can be represented by their mean and covariance, which gives rise to

4800-452: The risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the individual is typical of the population as a whole. Based on Bayes law both the prevalence of a disease in a given population and the error rate of an infectious disease test have to be taken into account to evaluate

4875-448: The same cost function . However, in practical applications these assumptions are never fulfilled, the different methods perform differently and generally it is not clear what approach (Kalman filtering or variational) is better. The fundamental questions also arise in application of the advanced DA techniques such as convergence of the computational method to the global minimum of the functional to be minimised. For instance, cost function or

4950-403: The same outcomes by A and B in opposite orders, to obtain the inverse probabilities. Bayes' theorem links the different partitionings. An entomologist spots what might, due to the pattern on its back, be a rare subspecies of beetle . A full 98% of the members of the rare subspecies have the pattern, so P (Pattern | Rare) = 98%. Only 5% of members of the common subspecies have

5025-479: The set in which the solution is sought can be not convex. The 4DDA method which is currently most successful is hybrid incremental 4D-Var, where an ensemble is used to augment the climatological background error covariances at the start of the data assimilation time window, but the background error covariances are evolved during the time window by a simplified version of the NWP forecast model. This data assimilation method

5100-590: The set of people who take the drug test. This combined with the definition of conditional probability results in the above statement. In other words, even if someone tests positive, the probability that they are a cannabis user is only 19%—this is because in this group, only 5% of people are users, and most positives are false positives coming from the remaining 95%. If 1,000 people were tested: The 1,000 people thus yields 235 positive tests, of which only 45 are genuine drug users, about 19%. The importance of specificity can be seen by showing that even if sensitivity

5175-408: The small blue-green arrows indicate the direct way the models actually run. Data assimilation methods are currently also used in other environmental forecasting problems, e.g. in hydrological and hydrogeological forecasting. Bayesian networks may also be used in a data assimilation approach to assess natural hazards such as landslides. Given the abundance of spacecraft data for other planets in

5250-574: The solar system, data assimilation is now also applied beyond the Earth to obtain re-analyses of the atmospheric state of extraterrestrial planets. Mars is the only extraterrestrial planet to which data assimilation has been applied so far. Available spacecraft data include, in particular, retrievals of temperature and dust/water/ice optical thicknesses from the Thermal Emission Spectrometer onboard NASA's Mars Global Surveyor and

5325-462: The spatial distributions of covariance functions of the errors of the "first guess" field (previous forecast) and "true field". These functions are never known. However, the different approximations were assumed. The optimal interpolation algorithm is the reduced version of the Kalman filtering (KF) algorithm and in which the covariance matrices are not calculated from the dynamical equations but are pre-determined in advance. Attempts to introduce

5400-718: The uniform distribution on the real line. Modern Markov chain Monte Carlo methods have boosted the importance of Bayes' theorem including cases with improper priors. Bayes' rule and computing conditional probabilities provide a solution method for a number of popular puzzles, such as the Three Prisoners problem , the Monty Hall problem , the Two Child problem and the Two Envelopes problem . Suppose,

5475-491: The variational approaches is that the meteorological fields satisfy the dynamical equations of the NWP model and at the same time they minimize the functional, characterizing their difference from observations. Thus, the problem of constrained minimization is solved. The 3DDA variational methods were developed for the first time by Sasaki (1958). As was shown by Lorenc (1986), all the above-mentioned 4DDA methods are in some limit equivalent, i.e. under some assumptions they minimize

5550-427: The work in data assimilation is focused on adequately estimating the appropriate weighting factor based on intricate knowledge of the errors in the system. The measurements are usually made of a real-world system, rather than of the model's incomplete representation of that system, and so a special function called the observation operator (usually depicted by h() for a nonlinear operator or H for its linearization)

5625-676: Was caused by an imbalance in the pressure and wind velocity fields used as the initial conditions in his analysis, indicating the need for a data assimilation scheme. Originally "subjective analysis" had been used in which numerical weather prediction (NWP) forecasts had been adjusted by meteorologists using their operational expertise. Then "objective analysis" (e.g. Cressman algorithm) was introduced for automated data assimilation. These objective methods used simple interpolation approaches, and thus were 3DDA (three-dimensional data assimilation) methods. Later, 4DDA (four-dimensional data assimilation) methods, called "nudging", were developed, such as in

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