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Nonlinear Model
In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and synergetic nonlinear effects. Contrary to traditional modelling methods, such as linear regression and basic statistical methods, nonlinear modelling can be utilized efficiently in a vast number of situations where traditional modelling is impractical or impossible. The newer nonlinear modelling approaches include non-parametric methods, such as feedforward neural network A feedforward neural network (FNN) is an artificial neural network wherein connections between the nodes do ''not'' form a cycle. As such, it is different from its descendant: recurrent neural networks. The feedforward neural network was the ...s, kernel regression, multivariate splines, etc., which do not require a ''priori'' knowled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used 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). 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, and to make predictions about behavior. Elements of a mathematical model Mathematical models can take many forms, including dynamical systems, statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Regression
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called '' simple linear regression''; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical
Statistics (from German: ''Statistik'', "description of a 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 surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feedforward Neural Network
A feedforward neural network (FNN) is an artificial neural network wherein connections between the nodes do ''not'' form a cycle. As such, it is different from its descendant: recurrent neural networks. The feedforward neural network was the first and simplest type of artificial neural network devised. In this network, the information moves in only one direction—forward—from the input nodes, through the hidden nodes (if any) and to the output nodes. There are no cycles or loops in the network. Single-layer perceptron The simplest kind of neural network is a ''single-layer perceptron'' network, which consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. The sum of the products of the weights and the inputs is calculated in each node, and if the value is above some threshold (typically 0) the neuron fires and takes the activated value (typically 1); otherwise it takes the deactivated value (typically -1). Ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Regression
In statistics, kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables ''X'' and ''Y''. In any nonparametric regression, the conditional expectation of a variable Y relative to a variable X may be written: : \operatorname(Y \mid X) = m(X) where m is an unknown function. Nadaraya–Watson kernel regression Nadaraya and Watson, both in 1964, proposed to estimate m as a locally weighted average, using a kernel as a weighting function. The Nadaraya–Watson estimator is: : \widehat_h(x)=\frac where K_h is a kernel with a bandwidth h. Derivation : \operatorname(Y \mid X=x) = \int y f(y\mid x) \, dy = \int y \frac \, dy Using the kernel density estimation for the joint distribution ''f''(''x'',''y'') and ''f''(''x'') with a kernel ''K'', : \hat(x,y) = \frac\sum_^n K_h(x-x_i) K_h(y-y_i), : \hat(x) = \frac \sum_^n K_h(x-x_i), we get ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spline (mathematics)
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term ''spline'' more frequently refers to a piecewise polynomial ( parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. Introduction The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
