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Hat Notation
A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses. Estimated value In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value. For example, in the context of errors and residuals, the "hat" over the letter \hat indicates an observable estimate (the residuals) of an unobservable quantity called \varepsilon (the statistical errors). Another example of the hat denoting an estimator occurs in simple linear regression. Assuming a model of y_i = \beta_0+\beta_1 x_i+\varepsilon_i, with observations of independent variable data x_i and dependent variable data y_i, the estimated model is of the form \hat_i = \hat_0+\hat_1 x_i where \sum_i (y_i-\hat_i)^2 is commonly minimized via least squares by finding optimal values of \hat_0 and \hat_1 for the observed data. Hat matrix In statistics, the hat matrix ''H'' projects the observed values y of response variable to the predicted values ŷ: :\hat = H \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumflex
The circumflex () is a diacritic in the Latin and Greek scripts that is also used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from "bent around"a translation of the (). The circumflex in the Latin script is chevron-shaped (), while the Greek circumflex may be displayed either like a tilde () or like an inverted breve (). For the most commonly encountered uses of the accent in the Latin alphabet, precomposed characters are available. In English, the circumflex, like other diacritics, is sometimes retained on loanwords that used it in the original language (for example '' entrepôt, crème brûlée''). In mathematics and statistics, the circumflex diacritic is sometimes used to denote a function and is called a '' hat operator''. A free-standing version of the circumflex symbol, , is encoded in ASCII and Unicode and has become known as '' caret'' and has acquired special uses, particularly i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Notation
Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling them into expression (mathematics), expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and property (philosophy), properties in a concise, unambiguous, and accurate way. For example, the physicist Albert Einstein's formula E=mc^2 is the quantitative representation in mathematical notation of mass–energy equivalence. Mathematical notation was first introduced by François Viète at the end of the 16th century and largely expanded during the 17th and 18th centuries by René Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler. Symbols and typeface The use of many symbols is the basis of mathematical notation. They play a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "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. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), 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 estimator, 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. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Errors And Residuals In Statistics
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the '' estimated'' value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances. Introduction Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Linear Regression
In statistics, simple linear regression (SLR) is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x'' and ''y'' coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective ''simple'' refers to the fact that the outcome variable is related to a single predictor. It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared '' residual'' (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible. In this case, the slope of the fitted line is equal to the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The method is widely used in areas such as regression analysis, curve fitting and data modeling. The least squares method can be categorized into linear and nonlinear forms, depending on the relationship between the model parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. History Founding The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Screw Theory
Screw theory is the algebraic calculation of pairs of Vector (mathematics and physics), vectors, also known as ''dual vectors'' – such as Angular velocity, angular and linear velocity, or forces and Moment (physics), moments – that arise in the kinematics and Dynamics (mechanics), dynamics of Rigid body, rigid bodies. Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axis, screw axes of spatial movement and the Line of action, lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and Vector addition, addition of vectors. Important theorems of screw theory include: the ''transfer principle'' proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws; Chasles' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations. The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. Definition Let ''V'' be a vector space of dimension ''n'' over a field ''F'' and let : B = \ be an ordered basis for ''V''. Then for every v \in V there is a unique linear combination of the basis vectors that equals '' v '': : v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
