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This 8 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 797A at North Carolina State University taught by Arkady Kheyfets in Fall. Since its upload, it has received 10 views. For similar materials see /class/223693/ma-797a-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
Stochastic Models Estimators and Emulators The best model of a cat is another cat or better yet the cat itself Norbert Wiener Stochastic Models Motivation No mathematical system model is perfect Even physical laws are often approximations eg Newton versus Einstein Physical laws provide framework but parameters are unknown 0 Dynamic systems often driven by disturbances that we can neither control nor model deterministically Sensors do not provide perfect measurements Emulators Definition A mathematical model and the computer program used to implement it are often referred to as a simulator Definition An emulator is a statistical approximation of a simulator It is ideally designed to be simpler and more efficient than the original model Hence it is much quicker to run If the approximation is sufficiently accurate it can be used to provide uncertainty and sensitivity measures Because the emulator is a statistical approximation it can be used to construct a probability function for the original model Bayesian approaches often considered to incorporate prior knowledge References A O Hagan Bayesian analysis of computer code outputs A tutorial 1 MC Kennedy and A O Hagan Bayesian calibration of computer models Frequentist Versus Bayesian Analysis Frequentist Analysis Based on the interpretation of the probability of an event as the long term limiting frequency with which the event occurs if the experiment is repeated an infinite number of times This is limited if event can be repeated only a few times eg oncel eg the permeability of a certain oil region may only be tested once Aleatory Uncertainty uncertainty in repeated events due to intrinsic randomness and unpredictability Epistemic Uncertainty Uncertainty in nonrepeatable events due simply to lack of knowledge Bayesian Inference Statistical inference in which evidence or observations are used to update or infer the probability that a hyposthesis is true As evidence is gained the degree of belief in the hypothesis changes and is updated Bayesian Inference Bayesian Inference Example For billions of years the sun has risen after it has set The sun has set tonight V th very high probability the sun will rise tomorrow V th very low probability the sun will not rise tomorrow Bayes Theorem PElH0PH0 Here H0 Null hypothesis inferred before new evidence E became available PHU prior probability of H0 PEH0 Conditional probability of observing evidence E given that H0 is true Often termed the likelihood function when expressed as a function of E given H0 PE Marginal probability of E Probability of observing new evidence E under all mutually exclusive hypotheses Computed as EPEHPHi PH0E posterior probability of H0 given E Note PEH0PE is impact that evidence has on belief in hypothesis Kalman Filter Rudolf Kalman References PS Maybeck Stochastic Models Estimation and Control Academic Press New York 1979 Material on the website httpwwwcsunceduwelchkalman High Level Definition An optimal recursive data processing algorithm Optimal with regard to most sensible criteria Recursive implies that previous data does not have to be stored and reprocessed every time a new measurement is taken Kalman Filter Basic Assumptions System can be described by a linear model For many applications linear models are adequate and if not we can often linearize about a nominal point or trajectory For certain nonlinear applications the theory can be extended to obtain nonlinear filters System and measurement noise are white and Gaussian White noise implies that the noise value is not correlated in time This also implies noise has equal power at all frequencies which cannot be true since it would result in infinite noise However physical systems have a natural bandpass above which effects are basically negligible Assumption of Gaussian influences amplitude and shape at certain time Motivated by Central Limit Theorem Requires knowledge only of first and secondorder statistics mean and variance Hence highly tractable Kalman Filter Motivating Example