|
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] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), 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 statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Norm
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 zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can 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". It can also refer to a norm that can take infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squared Deviations
A square is a regular quadrilateral with four equal sides and four right angles. Square or Squares may also refer to: Mathematics and science *Square (algebra), multiplying a number or expression by itself *Square (cipher), a cryptographic block cipher *Global square, a principle in infinitary combinatorics *Square number, an integer that is the square of another integer *Square of a graph *Square wave (waveform), a non-sinusoidal periodic waveform Construction *Square (area), an Imperial unit of floor area and other construction materials *Square, a public meeting place: **Garden square, an open space with buildings surrounding a garden **Market square, an open area where market stalls are traditionally set out for trading **Town square, an open area commonly found in the heart of a traditional town used for community gatherings *Square (tool), an L- or T-shaped tool: **Combination square, a tool with a ruled blade and one or more interchangeable heads **Machinist square, a meta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Chi-squared Statistic
In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares. Its square root is called regression standard error, standard error of the regression, or standard error of the equation (see ) Definition It is defined as chi-square per degree of freedom: \chi^2_\nu = \frac \nu, where the chi-squared is a weighted sum of squared deviations: \chi^2 = \sum_ with inputs: variance \sigma_i^2, observations ''O'', and calculated data ''C''. The degree of freedom, \nu = n - m, equals the number of observations ''n'' minus the number of fitted parameters ''m''. In weighted least squares, the definition is often written in matrix notation as \chi^2_\nu = \frac, where ''r'' is the vector of residuals, and ''W'' is the weight matrix, the inverse of the input (diagonal) covariance matrix of observations. If ''W' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Squared Error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the erro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lack-of-fit Sum Of Squares
In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable value(s). These are errors that could never be avoided by any predictive equation that assigned a predicted value for the dependent variable as a function of the value(s) of the independent variable(s). The remainder of the residual sum of squares is attributed to lack of fit of the model since it would be mathematically possible to eliminate these errors entirely. Principle In order for the lack-of-fit sum of squares to differ from the sum of squares ... [...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] |
Degrees Of Freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself. For example, if the variance is to be estimated from a random sample of N independent scores, then the degrees of freedom is equal to the number of independent scores (''N'') minus the number of parameters estimated as intermediate steps (one, namely, the sample mean) and is therefore equal to N-1. Mathematically, degrees of freedom is the number of dimensions of the domain of a random vector, or e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chi-squared Distribution
In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random variables. The chi-squared distribution \chi^2_k is a special case of the gamma distribution and the univariate Wishart distribution. Specifically if X \sim \chi^2_k then X \sim \text(\alpha=\frac, \theta=2) (where \alpha is the shape parameter and \theta the scale parameter of the gamma distribution) and X \sim \text_1(1,k) . The scaled chi-squared distribution s^2 \chi^2_k is a reparametrization of the gamma distribution and the univariate Wishart distribution. Specifically if X \sim s^2 \chi^2_k then X \sim \text(\alpha=\frac, \theta=2 s^2) and X \sim \text_1(s^2,k) . The chi-squared distribution is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akaike Information Criterion
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection. AIC is founded on information theory. When a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. AIC estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model. In estimating the amount of information lost by a model, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model. In other words, AIC deals with both the risk of overfitting and the risk of underfitting. The Akaike information crite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pearson Correlation Coefficient
In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations. As a simple example, one would expect the age and height of a sample of children from a school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation). Naming and history It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s, and for which the mathematical formula was derived and published by Auguste Bravais in 1844. The nami ... [...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] |
