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Aliasing (factorial Experiments)
In the statistical theory of factorial experiments, aliasing is the property of fractional factorial designs that makes some effects "aliased" with each other – that is, indistinguishable from each other. A primary goal of the theory of such designs is the control of aliasing so that important effects are not aliased with each other. In a "full" factorial experiment, the number of ''treatment combinations'' or ''cells'' (see #Overview, below) can be very large. This necessitates limiting observations to a ''fraction'' (subset) of the treatment combinations. Aliasing is an automatic and unavoidable result of observing such a fraction. The aliasing properties of a design are often summarized by giving its #Resolution, ''resolution''. This measures the degree to which the design avoids aliasing between main effects and important interactions. Fractional factorial experiments have long been a basic tool in agriculture, food technology, industry, medicine and public health, and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Experiment
In statistics, a factorial experiment (also known as full factorial experiment) investigates how multiple factors influence a specific outcome, called the response variable. Each factor is tested at distinct values, or levels, and the Experimental unit, experiment includes every possible combination of these levels across all factors. This comprehensive approach lets researchers see not only how each factor individually affects the response, but also how the factors interaction (statistics), interact and influence each other. Often, factorial experiments simplify things by using just two levels for each factor. A 2x2 factorial design, for instance, has two factors, each with two levels, leading to four unique combinations to test. The interaction between these factors is often the most crucial finding, even when the individual factors also have an effect. If a full factorial design becomes too complex due to the sheer number of combinations, researchers can use a fractional fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality (mathematics)
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity'' to linear algebra of bilinear forms. Two elements and of a vector space with bilinear form B are orthogonal when B(\mathbf,\mathbf)= 0. Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are used to form an orthogonal basis (linear algebra), basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics. Definitions * In geometry, two Euclidean vectors are orthogonal if they are perpendicular, ''i.e.'' they form a right angle. * Two vector space, vectors and in an inner product space V are ''orthogonal'' if their inner product \langle \mathbf, \mathbf \rangle is zero. This relationship is denoted \mathbf \perp \mathbf. * A set of vectors in an inner produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A ''Solution (mathematics), solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the Tuple, ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A Nonlinear system, system of non-linear equations can often be Approximation, approximated by a linear system (see linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Experiment
In statistics, a factorial experiment (also known as full factorial experiment) investigates how multiple factors influence a specific outcome, called the response variable. Each factor is tested at distinct values, or levels, and the Experimental unit, experiment includes every possible combination of these levels across all factors. This comprehensive approach lets researchers see not only how each factor individually affects the response, but also how the factors interaction (statistics), interact and influence each other. Often, factorial experiments simplify things by using just two levels for each factor. A 2x2 factorial design, for instance, has two factors, each with two levels, leading to four unique combinations to test. The interaction between these factors is often the most crucial finding, even when the individual factors also have an effect. If a full factorial design becomes too complex due to the sheer number of combinations, researchers can use a fractional fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Products
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see ''Inner product space'' for more). It should not be confused with the cross product. Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Triple
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is only one 0-tuple, called the ''empty tuple''. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term ''"infinite tuple"'' is occasionally used for ''"infinite sequences"''. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning. An -tuple can be formally defined as the image of a function that has the set of the first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from an ordered pair; indeed, an -tuple can be identified with the ordered pair of its first elements and its th e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mod 2
Mod, MOD or mods may refer to: Places * Modesto City–County Airport, Stanislaus County, California, US Arts, entertainment, and media Music * Mods (band), a Norwegian rock band * M.O.D. (Method of Destruction), a band from New York City, US * The Mods (band), a punk rock band from Toronto, Canada Other uses in arts, entertainment, and media * Manufactured on demand for CD, DVD distribution * ''Mod'' (film), 2011 * ''The Mods'' (film), 2014 * Video game modding * , a Scottish Gaelic festival * Media-on-demand * '' MuchOnDemand'', a Canadian TV program Brands and enterprises * Mod Club Theatre, Toronto, Canada * MOD Pizza, US Organizations * MoD, Ministry of defence of some countries ** MoD (UK), Ministry of Defence * Masters of Deception, a US hacker group * Ministry of Development (Brunei) Science and technology Computing and Internet * Mod, a module for Apache HTTP Server * Case modding of a computer * Forum moderator, of an online forum * Module file, a music fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Prime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
