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Least Trimmed Squares
Least trimmed squares (LTS), or least trimmed sum of squares, is a robust statistical method that fits a function to a set of data whilst not being unduly affected by the presence of outliers . It is one of a number of methods for robust regression. Description of method Instead of the standard least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ... method, which minimises the sum of squared residuals over ''n'' points, the LTS method attempts to minimise the sum of squared residuals over a subset, k, of those points. The unused n - k points do not influence the fit. In a standard least squares problem, the estimated parameter values β are defined to be those values that minimise the objective function ''S''(β) of squared residuals: :S = \sum_^n r_i(\beta)^2, where th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics that maintain their properties even if the underlying distributional assumptions are incorrect. Robust Statistics, statistical methods have been developed for many common problems, such as estimating location parameter, location, scale parameter, scale, and regression coefficient, regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a Parametric statistics, parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set, measurement error, or that the population has a heavy-tailed distribution. In the case of measurement error, one wishes to discard them or use statistics that are robust statistics, robust to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution. A frequent cause of outliers is a mixture of two distributions, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Regression
In robust statistics, robust regression seeks to overcome some limitations of traditional regression analysis. A regression analysis models the relationship between one or more independent variables and a dependent variable. Standard types of regression, such as ordinary least squares, have favourable properties if their underlying assumptions are true, but can give misleading results otherwise (i.e. are not robust to assumption violations). Robust regression methods are designed to limit the effect that violations of assumptions by the underlying data-generating process have on regression estimates. For example, least squares estimates for regression models are highly sensitive to outliers: an outlier with twice the error magnitude of a typical observation contributes four (two squared) times as much to the squared error loss, and therefore has more leverage over the regression estimates. The Huber loss function is a robust alternative to standard square error loss that r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The method is widely used in areas such as regression analysis, curve fitting and data modeling. The least squares method can be categorized into linear and nonlinear forms, depending on the relationship between the model parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. History Founding The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residual Sum Of Squares
In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection. In general, total sum of squares = explained sum of squares + residual sum of squares. For a proof of this in the multivariate ordinary least squares (OLS) case, see partitioning in the general OLS model. One explanatory variable In a model with a single explanatory variable, RSS is given by: :\operatorname = \sum_^n (y_i - f(x_i))^2 where ''y''''i'' is the ''i''th value of the variable to be predicted, ''x''''i'' is the ''i''th value of the explanatory variable, and f(x_i) is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Errors And Residuals In Statistics
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the '' estimated'' value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances. Introduction Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dependent And Independent Variables
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function), on the values of other variables. Independent variables, on the other hand, are not seen as depending on any other variable in the scope of the experiment in question. Rather, they are controlled by the experimenter. In pure mathematics In mathematics, a function (mathematics), function is a rule for taking an input (in the simplest case, a number or set of numbers)Carlson, Robert. A concrete introduction to real analysis. CRC Press, 2006. p.183 and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is , and the most common symbol for the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Statistical Association
The ''Journal of the American Statistical Association'' is a quarterly peer-reviewed scientific journal published by Taylor & Francis on behalf of the American Statistical Association. It covers work primarily focused on the application of statistics, statistical theory and methods in economic, social, physical, engineering, and health sciences. The journal also includes reviews of books which are relevant to the field. The journal was established in 1888 as the ''Publications of the American Statistical Association''. It was renamed ''Quarterly Publications of the American Statistical Association'' in 1912, obtaining its current title in 1922. Reception According to the ''Journal Citation Reports ''Journal Citation Reports'' (''JCR'') is an annual publication by Clarivate. It has been integrated with the Web of Science and is accessed from the Web of Science Core Collection. It provides information about academic journals in the natur ...'', the journal has a 2023 impac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Applied Statistics
The ''Journal of Applied Statistics'' or ''J.Appl.Stat.'' is a peer-reviewed scientific journal covering applied statistics that is published by Taylor & Francis. Its ''Journal Citation Reports'' impact factor was 1.2 in 2023. Creation The journal was founded by Gopal Kanji in 1974 as the ''Bulletin in Applied Statistics (BIAS)''. Kanji remained as editor until the end of 2007. Quality rankings In 2020, the Australian Mathematical Society listed ''J.Appl.Stat.'' with a ranking of ''B'' on a scale including ''A*'', ''A'', ''B'' and ''C''. ''J.Appl.Stat.'' states that its ''Journal Citation Reports'' impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.699 in 2017 and 1.2 in 2023. References External links * {{Statistics journals Statistics journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics that maintain their properties even if the underlying distributional assumptions are incorrect. Robust Statistics, statistical methods have been developed for many common problems, such as estimating location parameter, location, scale parameter, scale, and regression coefficient, regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a Parametric statistics, parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
