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Guess Value
In mathematical modeling, a guess value is more commonly called a starting value or initial value. These are necessary for most optimization problems which use search algorithms, because those algorithms are mainly deterministic and iterative, and they need to start somewhere. One common type of application is nonlinear regression. Use The quality of the initial values can have a considerable impact on the success or lack of such of the search algorithm. This is because the fitness function or objective function (in many cases a sum of squared errors ( SSE)) can have difficult shapes. In some parts of the search region, the function may increase exponentially, in others quadratically, and there may be regions where the function asymptotes to a plateau. Starting values that fall in an exponential region can lead to algorithm failure because of arithmetic overflow. Starting values that fall in the asymptotic plateau region can lead to algorithm failure because of "dithering". Determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Modeling
A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical modeling''. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). It can also be taught as a subject in its own right. The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in music, linguistics, and philosophy (for example, intensively in analytic philosophy). A model may help to explain a system and to study the effects of different components, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearization
In mathematics, linearization (British English: linearisation) is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization of a function Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that f(x) is differentiable on , b/math> (or , a/math>) and that a is close to b. In short, linearization approximates the output of a function near x = a. For example, \sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genetic Algorithms
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems via biologically inspired operators such as selection, crossover, and mutation. Some examples of GA applications include optimizing decision trees for better performance, solving sudoku puzzles, hyperparameter optimization, and causal inference. Methodology Optimization problems In a genetic algorithm, a population of candidate solutions (called individuals, creatures, organisms, or phenotypes) to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties (its chromosomes or genotype) which can be mutated and altered; traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evolutionary Algorithms
Evolutionary algorithms (EA) reproduce essential elements of the biological evolution in a computer algorithm in order to solve "difficult" problems, at least Approximation, approximately, for which no exact or satisfactory solution methods are known. They belong to the class of Metaheuristic, metaheuristics and are a subset of Population Based Bio-Inspired Algorithms, population based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, genetic recombination, recombination and natural selection, selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic
Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used interchangeably. In probability theory, the formal concept of a '' stochastic process'' is also referred to as a ''random process''. Stochasticity is used in many different fields, including image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance (e.g., stochastic oscillator), due to seemingly random changes in the different markets within the financial sector and in medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. Etymology The word ''stochastic'' in English was originally used as an adjective with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose. The ''arithmetic mean'', also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the '' sample mean'' (\bar) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted \mu or \mu_x. Outside probability and statistics, a wide rang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d. Linearity of a mapping is closely related to '' proportionality''. Examples in physics include the linear relationship of voltage and current in an electrical conductor ( Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are '' nonlinear''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. Linearity of a polynomial means that its de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series ''analysis'' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensions
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: * System of linear equations, * System of nonlinear equations, * System of bilinear equations, * System of polynomial equations, * System of differential equations, or a * System of difference equations See also * Simultaneous equations model, a statistical model in the form of simultaneous linear equations * Elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ..., for elementary methods {{set index article Equations Broad-concept articles de:Gleichung#Gleichungssysteme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
