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The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined …

The result of fitting a set of data points with a quadratic function

Conic fitting a set of points using least-squares approximation

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1. Least squares – The method of least squares is a standard approach in regression analysis to the approximate solution of overdetermined systems, i. e. sets of equations in which there are more equations than unknowns. Least squares means that the overall solution minimizes the sum of the squares of the made in the results of every single equation. The most important application is in data fitting, the best fit in the least-squares sense minimizes the sum of squared residuals. Least squares problems fall into two categories, linear or ordinary least squares and non-linear least squares, depending on whether or not the residuals are linear in all unknowns, the linear least-squares problem occurs in statistical regression analysis, it has a closed-form solution. The non-linear problem is solved by iterative refinement, at each iteration the system is approximated by a linear one. Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable, when the observations come from an exponential family and mild conditions are satisfied, least-squares estimates and maximum-likelihood estimates are identical. The method of least squares can also be derived as a method of moments estimator, the following discussion is mostly presented in terms of linear functions but the use of least-squares is valid and practical for more general families of functions. Also, by iteratively applying local quadratic approximation to the likelihood, for the topic of approximating a function by a sum of others using an objective function based on squared distances, see least squares. The least-squares method is credited to Carl Friedrich Gauss. The accurate description of the behavior of bodies was the key to enabling ships to sail in open seas. The combination of different observations taken under the same conditions contrary to simply trying ones best to observe, the approach was known as the method of averages. The combination of different observations taken under different conditions, the method came to be known as the method of least absolute deviation. It was notably performed by Roger Joseph Boscovich in his work on the shape of the earth in 1757, the development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved. Laplace tried to specify a mathematical form of the probability density for the errors and he felt these to be the simplest assumptions he could make, and he had hoped to obtain the arithmetic mean as the best estimate. Instead, his estimator was the posterior median, the first clear and concise exposition of the method of least squares was published by Legendre in 1805. The technique is described as a procedure for fitting linear equations to data. The value of Legendres method of least squares was immediately recognized by leading astronomers, in 1809 Carl Friedrich Gauss published his method of calculating the orbits of celestial bodies. In that work he claimed to have been in possession of the method of least squares since 1795 and this naturally led to a priority dispute with Legendre

2. Overdetermined system – In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent when constructed with random coefficients, however, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology can be described in terms of the concept of constraint counting, each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that one degree of freedom. Therefore, the critical case occurs when the number of equations, for every variable giving a degree of freedom, there exists a corresponding constraint. The overdetermined case occurs when the system has been overconstrained — that is, in contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns. Such systems usually have an amount of solutions. Consider the system of 3 equations and 2 unknowns, which is overdetermined because 3>2, there is one solution for each pair of linear equations, for the first and second equations, for the first and third, and for the second and third. However, there is no solution that all three simultaneously. Diagrams #2 and 3 show other configurations that are inconsistent because no point is on all of the lines, systems of this variety are deemed inconsistent. The only cases where the system does in fact have a solution are demonstrated in Diagrams #4,5. These exceptions can occur only when the system contains enough linearly dependent equations that the number of independent equations does not exceed the number of unknowns. Linear dependence means that some equations can be obtained from linearly combining other equations, for example, Y = X +1 and 2Y = 2X +2 are linearly dependent equations because the second one can be obtained by taking twice the first one. Any system of equations can be written as a matrix equation. The previous system of equations can be written as follows, = Notice that the rows of the coefficient matrix outnumber the columns, the rank of this matrix is 2, which corresponds to the number of dependent variables in the system. A linear system is consistent if and only if the coefficient matrix has the rank as its augmented matrix. The augmented matrix has rank 3, so the system is inconsistent, the nullity is 0, which means that the null space contains only the zero vector and thus has no basis. In linear algebra the concepts of row space, column space, the informal discussion of constraints and degrees of freedom above relates directly to these more formal concepts

3. Statistics – Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes

4. Robust regression – In robust statistics, robust regression is a form of regression analysis designed to circumvent some limitations of traditional parametric and non-parametric methods. Regression analysis seeks to find the relationship between one or more independent variables and a dependent variable, Robust regression methods are designed to be not overly affected by violations of assumptions by the underlying data-generating process. In particular, least squares estimates for regression models are sensitive to outliers. While there is no definition of an outlier, outliers are observations which do not follow the pattern of the other observations. One instance in which robust estimation should be considered is when there is a suspicion of heteroscedasticity. In the homoscedastic model, it is assumed that the variance of the term is constant for all values of x. Heteroscedasticity allows the variance to be dependent on x, which is accurate for many real scenarios. For example, the variance of expenditure is often larger for individuals with higher income than for individuals with lower incomes, software packages usually default to a homoscedastic model, even though such a model may be less accurate than a heteroscedastic model. One simple approach is to apply least squares to percentage errors as this reduces the influence of the values of the dependent variable compared to ordinary least squares. Another common situation in which robust estimation is used occurs when the data contain outliers, in the presence of outliers that do not come from the same data-generating process as the rest of the data, least squares estimation is inefficient and can be biased. Because the least squares predictions are dragged towards the outliers, and because the variance of the estimates is artificially inflated, the result is that outliers can be masked. Although it is claimed that least squares are robust, they are only robust in the sense that the type I error rate does not increase under violations of the model. In fact, the type I error rate tends to be lower than the level when outliers are present. The reduction of the type I error rate has been labelled as the conservatism of classical methods, despite their superior performance over least squares estimation in many situations, robust methods for regression are still not widely used. Several reasons may explain their unpopularity. One possible reason is there are several competing methods and the field got off to many false starts. Another reason may be that some popular software packages failed to implement the methods. The belief of many statisticians that classical methods are robust may be another reason, although uptake of robust methods has been slow, modern mainstream statistics text books often include discussion of these methods