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Partition Of The Sum Of Squares
The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion (also called variability). When scaled for the number of degrees of freedom, it estimates the variance, or spread of the observations about their mean value. Partitioning of the sum of squared deviations into various components allows the overall variability in a dataset to be ascribed to different types or sources of variability, with the relative importance of each being quantified by the size of each component of the overall sum of squares. Background The distance from any point in a collection of data, to the mean of the data, is the deviation. This can be written as y_i - \overline, where y_i is the ith data point, and \overline is the estimate of the mean. If all such deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inferential Statistics
Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner-product Space
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite Dimension (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squared Deviations
Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average value (for actual experimental data). Computations for ''analysis of variance'' involve the partitioning of a sum of SDM. Background An understanding of the computations involved is greatly enhanced by a study of the statistical value : \operatorname( X ^ 2 ), where \operatorname is the expected value operator. For a random variable X with mean \mu and variance \sigma^2, : \sigma^2 = \operatorname( X ^ 2 ) - \mu^2.Mood & Graybill: ''An introduction to the Theory of Statistics'' (McGraw Hill) Therefore, : \operatorname( X ^ 2 ) = \sigma^2 + \mu^2. From the above, the following can be derived: : \operatorname\left( \sum\left( X ^ 2\right) \right) = n\sigma^2 + n\mu^2, : \operatorname\left( \left(\sum X \right)^ 2 \right) = n\sigma^2 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Squared Error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', often called the Pythagorean equation: :a^2 + b^2 = c^2 , The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P : V \to V such that P^2 = P. When V has an inner product and is complete (i.e. when V is a Hilbert space) the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a Hilber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthomodular Lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b'' = 1 and ''a'' ∧ ''b'' = 0. Complements need not be unique. A relatively complemented lattice is a lattice such that every interval 'c'', ''d'' viewed as a bounded lattice in its own right, is a complemented lattice. An orthocomplementation on a complemented lattice is an involution that is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law is called an orthomodular lattice. In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. Definition and basic properties A complemented lattice is a bounded lattice (with least element 0 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every vector in ''W''. Informally, it is called the perp, short for perpendicular complement. It is a subspace of ''V''. Example Let V = (\R^5, \langle \cdot, \cdot \rangle) be the vector space equipped with the usual dot product \langle \cdot, \cdot \rangle (thus making it an inner product space), and let W = \, with A = \begin 1 & 0\\ 0 & 1\\ 2 & 6\\ 3 & 9\\ 5 & 3\\ \end. then its orthogonal complement W^\perp = \ can also be defined as W^\perp = \, being \tilde = \begin -2 & -3 & -5 \\ -6 & -9 & -3 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end. The fact that every column vector in A is orthogonal to every column vector in \tilde can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of \R^n under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Mean Squares
In statistics, expected mean squares (EMS) are the expected values of certain statistics arising in partitions of sums of squares in the analysis of variance (ANOVA). They can be used for ascertaining which statistic should appear in the denominator in an F-test for testing a null hypothesis that a particular effect is absent. Definition When the total corrected sum of squares in an ANOVA is partitioned into several components, each attributed to the effect of a particular predictor variable, each of the sums of squares in that partition is a random variable that has an expected value. That expected value divided by the corresponding number of degrees of freedom is the expected mean square for that predictor variable. Example The following example is from ''Longitudinal Data Analysis'' by Donald Hedeker and Robert D. Gibbons. Donald Hedeker, Robert D. Gibbons. ''Longitudinal Data Analysis.'' Wiley Interscience. 2006. pp. 21–24 Each of ''s'' treatments (one of which ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
