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John Tukey
John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, and the Teichmüller–Tukey lemma all bear his name. He is also credited with coining the term ' bit' and the first published use of the word 'software'. Biography Tukey was born in New Bedford, Massachusetts in 1915, to a Latin teacher father and a private tutor. He was mainly taught by his mother and attended regular classes only for certain subjects like French. Tukey obtained a BA in 1936 and MSc in 1937 in chemistry, from Brown University, before moving to Princeton University, where in 1939 he received a PhD in mathematics after completing a doctoral dissertation titled "On denumerability in topology". During World War II, Tukey worked at the Fire Control Research Office and collaborated with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Bedford, Massachusetts
New Bedford (Massachusett: ) is a city in Bristol County, Massachusetts. It is located on the Acushnet River in what is known as the South Coast region. Up through the 17th century, the area was the territory of the Wampanoag Native American people. English colonists bought the land on which New Bedford would later be built from the Wampanoag in 1652, and the original colonial settlement that would later become the city was founded by English Quakers in the late 17th century. The town of New Bedford itself was officially incorporated in 1787. During the first half of the 19th century, New Bedford was one of the world's most important whaling ports. At its economic height during this period, New Bedford was the wealthiest city in the world per capita. New Bedford was also a center of abolitionism at this time. The city attracted many freed or escaped African-American slaves, including Frederick Douglass, who lived there from 1838 until 1841. The city also served as the primary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cooley–Tukey FFT Algorithm
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N_1N_2 in terms of ''N''1 smaller DFTs of sizes ''N''2, recursively, to reduce the computation time to O(''N'' log ''N'') for highly composite ''N'' ( smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below. Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors. The algorithm, along with its recursive application, was invented by Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but which are not unduly affected by outliers or other small departures from model assumptions. In statistics, classical estimation methods rely heavily on assumptions which are often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tukey Depth
In computational geometry, the Tukey depth is a measure of the depth of a point in a fixed set of points. The concept is named after its inventor, John Tukey. Given a set of points P in ''d''-dimensional space, a point ''p'' has Tukey depth ''k'' where ''k'' is the smallest number of points in any closed halfspace that contains ''p''. For example, for any extreme point of the convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ... there is always a (closed) halfspace that contains only that point, and hence its Tukey depth is 1. Tukey mean and relation to centerpoint A centerpoint ''c'' of a point set of size ''n'' is nothing else but a point of Tukey depth of at least ''n''/(''d'' + 1). See also * Centerpoint (geometry) Computational geometry< ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centerpoint (geometry)
In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in ''d''-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(''d'' + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint. Related concepts Closely related concepts are the Tukey depth of a point (the minimum number of sample points on one side of a hyperplane through the point) and a Tukey median of a point set (a point maximizing the Tukey depth). A centerpoint is a point of depth at least ''n''/(''d'' + 1), and a Tukey median must be a centerpoint, but not every centerpoint is a Tukey median. Both terms are named ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bland–Altman Plot
A Bland–Altman plot (difference plot) in analytical chemistry or biomedicine is a method of data plotting used in analyzing the agreement between two different assays. It is identical to a Tukey mean-difference plot, the name by which it is known in other fields, but was popularised in medical statistics by J. Martin Bland and Douglas G. Altman. Agreement versus correlation Bland and Altman drive the point that any two methods that are designed to measure the same parameter (or property) should have good correlation when a set of samples are chosen such that the property to be determined varies considerably. A high correlation for any two methods designed to measure the same property could thus in itself just be a sign that one has chosen a widespread sample. A high correlation does not necessarily imply that there is good agreement between the two methods. Construction Consider a sample consisting of n observations (for example, objects of unknown volume). Both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller–Tukey Lemma
In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equivalent to the axiom of choice, and therefore to the well-ordering theorem, Zorn's lemma, and the Hausdorff maximal principle. Definitions A family of sets \mathcal is of finite character provided it has the following properties: #For each A\in \mathcal, every finite subset of A belongs to \mathcal. #If every finite subset of a given set A belongs to \mathcal, then A belongs to \mathcal. Statement of the lemma Let Z be a set and let \mathcal\subseteq\mathcal(Z). If \mathcal is of finite character and X\in\mathcal, then there is a maximal Y\in\mathcal (according to the inclusion relation) such that X\subseteq Y. Applications In linear algebra, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tukey's Test Of Additivity
In statistics, Tukey's test of additivity, named for John Tukey, is an approach used in two-way ANOVA (regression analysis involving two qualitative factors) to assess whether the factor variables ( categorical variables) are additively related to the expected value of the response variable. It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance. The test statistic proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test." Introduction The most common setting for Tukey's test of additivity is a two-way factorial analysis of variance (ANOVA) with one observation per cell. The response variable ''Y''''ij'' is observed in a table of cells with the rows indexed by ''i'' = 1,..., ''m'' and the columns indexed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trimean
In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles: : TM= \frac This is equivalent to the average of the median and the midhinge: : TM= \frac\left(Q_2 + \frac\right) The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book which has given its name to a set of techniques called exploratory data analysis. Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%. This beneficial property has been described as follows: Efficiency Despite its simplicity, the trimean is a remarkably efficient estimator of population mean. More precisely, for a large data set (over 100 points) from a symmetric population, the average of the 20th, 50th, and 80th percentile is the most efficient 3 point L-estim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tukey Lambda Distribution
Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly. The Tukey lambda distribution has a single shape parameter, λ, and as with other probability distributions, it can be transformed with a location parameter, μ, and a scale parameter, σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function. Quantile function For the standard form of the Tukey lambda distribution, the quantile function, ~Q(p)~, (i.e. the inverse function to the cumulative distribution function) and the quantile density function (~ q = \operatornameQ / \operatornamep ~ are : Q\left(p;\lambda\right) ~=~ \begin \frac \left ^\lambda - (1 - p)^\lambda\right, & \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tukey's Range Test
Tukey's range test, also known as Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, Also occasionally as "honestly," see e.g. is a single-step multiple comparison procedure and statistical test. It can be used to find means that are significantly different from each other. Named after John Tukey, it compares all possible pairs of means, and is based on a studentized range distribution (''q'') (this distribution is similar to the distribution of ''t'' from the ''t''-test. See below).Linton, L.R., Harder, L.D. (2007) Biology 315 – Quantitative Biology Lecture Notes. University of Calgary, Calgary, AB Tukey's test compares the means of every treatment to the means of every other treatment; that is, it applies simultaneously to the set of all pairwise comparisons :\mu_i-\mu_j \, and identifies any difference between two means that is greater than the expected standard error. The confidence coefficient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tukey–Duckworth Test
In statistics, the Tukey–Duckworth test is a two-sample location test – a statistical test of whether one of two samples was significantly greater than the other. It was introduced by John Tukey, who aimed to answer a request by W. E. Duckworth for a test simple enough to be remembered and applied in the field without recourse to tables, let alone computers. Given two groups of measurements of roughly the same size, where one group contains the highest value and the other the lowest value, then (i) count the number of values in the one group exceeding all values in the other, (ii) count the number of values in the other group falling below all those in the one, and (iii) sum these two counts (we require that neither count be zero). The critical values of the total count are, roughly, 7, 10, and 13, i.e. 7 for a two sided 5% level, 10 for a two sided 1% level, and 13 for a two sided 0.1% level. The test loses some accuracy if the samples are quite large (greater tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
