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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 regressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Simple Linear Regression
In statistics, simple linear regression 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. Other regression methods that can be used in place of ordinary least sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Statistics
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 ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Regression Analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Least Squares Example2
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, 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 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. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lasso (statistics)
In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term. Lasso was originally formulated for linear regression models. This simple case reveals a substantial amount about the estimator. These include its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so-called soft thresholding. It also reveals that (like standard linear regression) the coefficient estimates do not need to be unique if covariates are collinear. Though originally defined for linear regression, lasso regularization is easily extended to other statistical models including generalized linear models, g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ridge Regression
Ridge regression is a method of estimating the coefficients of multiple- regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff). The theory was first introduced by Hoerl and Kennard in 1970 in their '' Technometrics'' papers “RIDGE regressions: biased estimation of nonorthogonal problems” and “RIDGE regressions: applications in nonorthogonal problems”. This was the result of ten years of research into the field of ridge an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Loss Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Least Absolute Deviations
Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based minimizing the '' sum of absolute deviations'' (sum of absolute residuals or sum of absolute errors) or the ''L''1 norm of such values. It is analogous to the least squares technique, except that it is based on '' absolute values'' instead of squared values. It attempts to find a function which closely approximates a set of data by minimizing residuals between points generated by the function and corresponding data points. The LAD estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution. It was introduced in 1757 by Roger Joseph Boscovich. Formulation Suppose that the data set consists of the points (''x''''i'', ''y''''i'') with ''i'' = 1, 2, ..., ''n''. We want to find a function ''f'' such that f(x_i)\approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 regr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Data Set
A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the data set in question. The data set lists values for each of the variables, such as for example height and weight of an object, for each member of the data set. Data sets can also consist of a collection of documents or files. In the open data discipline, data set is the unit to measure the information released in a public open data repository. The European data.europa.eu portal aggregates more than a million data sets. Some other issues ( real-time data sources, non-relational data sets, etc.) increases the difficulty to reach a consensus about it. Properties Several characteristics define a data set's structure and properties. These include the number and types of the attributes or variables, and various statistical measures applica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Forecasting
Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual results. Prediction is a similar but more general term. Forecasting might refer to specific formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgmental methods or the process of prediction and resolution itself. Usage can vary between areas of application: for example, in hydrology the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period. Risk and uncertainty are central to forecasting and prediction; it is generally considered a good practice to indicate the degree of uncertai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |