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Sieve Estimator
In statistics, sieve estimators are a class of non-parametric estimators which use progressively more complex models to estimate an unknown high-dimensional function as more data becomes available, with the aim of asymptotically reducing error towards zero as the amount of data increases. This method is generally attributed to Ulf Grenander. Method of sieves in Positron emission tomography Sieve estimators have been used extensively for estimating density functions in high-dimensional spaces such as in Positron emission tomography(PET). The first exploitation of Sieves in PET for solving the maximum-likelihood Positron emission tomography#Image reconstruction problem was by Donald Snyder and Michael Miller, where they stabilized the time-of-flight PET problem originally solved by Shepp and Vardi. Shepp and Vardi's introduction of Maximum-likelihood estimators in emission tomography exploited the use of the Expectation-Maximization algorithm, which as it ascended towards the maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-parametric Statistics
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both descriptive statistics and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated. Definitions The term "nonparametric statistics" has been imprecisely defined in the following two ways, among others: Applications and purpose Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimators
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean. There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators. ''Estimation theory'' is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given circumstances. However, in robust statistics, statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulf Grenander
Ulf Grenander (23 July 1923 – 12 May 2016) was a Swedish statistician and professor of applied mathematics at Brown University. His early research was in probability theory, stochastic processes, time series analysis, and statistical theory (particularly the order-constrained estimation of cumulative distribution functions using his sieve estimator). In recent decades, Grenander contributed to computational statistics, image processing, pattern recognition, and artificial intelligence. He coined the term pattern theory to distinguish from pattern recognition. Honors In 1966 Grenander was elected to the Royal Academy of Sciences of Sweden, and in 1996 to the US National Academy of Sciences. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. He received an honorary doctorate in 1994 from the University of Chicago, and in 2005 from the Royal Institute of Technology of Stockholm, Sweden. Schooling Grenander earned his undergraduate degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positron Emission Tomography
Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in Metabolism, metabolic processes, and in other physiological activities including blood flow, regional chemical composition, and absorption. Different tracers are used for various imaging purposes, depending on the target process within the body. For example, 18F-FDG, -FDG is commonly used to detect cancer, Sodium fluoride#Medical imaging, NaF is widely used for detecting bone formation, and Isotopes of oxygen#Oxygen-15, oxygen-15 is sometimes used to measure blood flow. PET is a common medical imaging, imaging technique, a Scintigraphy#Process, medical scintillography technique used in nuclear medicine. A radiopharmaceutical, radiopharmaceutical — a radioisotope attached to a drug — is injected into the body as a radioactive tracer, tracer. When the radiopharmaceutical undergoes beta plus decay, a positron is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Regression
Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship between predictors and dependent variable. Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates. Definition In nonparametric regression, we have random variables X and Y and assume the following relationship: : \mathbb \mid X=x= m(x), where m(x) is some deterministic function. Linear regression is a restricted case of nonparametric regression where m(x) is assumed to be affine. Some authors use a slightly stronger assumption of additive noise: : Y = m(X) + U, where the random variable U is the `noise term', with mean 0. Without the assumption that m belongs to a specific parametric family of functions it is impo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stuart Geman
Stuart Alan Geman (born March 23, 1949) is an American mathematician, known for influential contributions to computer vision, statistics, probability theory, machine learning, and the neurosciences. ikipediaList of important publications in computer science. He and his brother, Donald Geman, are well known for proposing the Gibbs sampler, and for the first proof of convergence of the simulated annealing algorithm. Biography Geman was born and raised in Chicago. He was educated at the University of Michigan (B.S., Physics, 1971), Dartmouth Medical College (MS, Neurophysiology, 1973), and the Massachusetts Institute of Technology (Ph.D, Applied Mathematics, 1977). Since 1977, he has been a member of the faculty at Brown University, where he has worked in the Pattern Theory group, and is currently the James Manning Professor of Applied Mathematics. He has received many honors and awards, including selection as a Presidential Young Investigator and as an ISI Highly Cited research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
