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Fitting A Line
Line fitting is the process of constructing a straight line that has the best fit to a series of data points. Several methods exist, considering: *Vertical distance: Simple linear regression *Resistance to outliers: Robust simple linear regression *Perpendicular distance: Orthogonal regression (this is not scale-invariant i.e. changing the measurement units leads to a different line.) *Weighted geometric distance: Deming regression *Scale invariant approach: Major axis regression This allows for measurement error in both variables, and gives an equivalent equation if the measurement units are altered. See also *Linear least squares * Linear segmented regression * Linear trend estimation *Polynomial regression *Regression dilution Regression dilution, also known as regression attenuation, is the biasing of the linear regression slope towards zero (the underestimation of its absolute value), caused by errors in the independent variable. Consider fitting a straight line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straight Line
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its ''endpoints''). Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. Properties In the Greek deductive geometry of Euclid's ''Elements'', a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Linear Regression
In statistics, simple linear regression (SLR) 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. In this case, the slope of the fitted line is equal to the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outlier (statistics)
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 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, which may be two dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Simple Linear Regression
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system's functional body. In the same line ''robustness'' can be defined as "the ability of a system to resist change without adapting its initial stable configuration". "Robustness in the small" refers to situations wherein perturbations are small in magnitude, which considers that the "small" magnitude hypothesis can be difficult to verify because "small" or "large" depends on the specific problem. Conversely, "Robustness in the large problem" refers to situations wherein no assumptions can be made about the magnitude of perturbations, which can either be small or large.C.Alippi: "Robustness Analysis" chapter in ''Intelligence for Embedded Systems.'' Springer, 2014, 283pp, . It has been discussed that robustness has two dimensions: resistance and avoidance.Durach, C.F. et al. (2015)Antecedents and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular Distance
In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line. Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve. The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point. Other instances include: *'' Point on plane closest to origin'', for the perpendicular distance from the origin to a plane in three-dimensional space *'' Nearest distance between skew lines'', for the perpendicular distance b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Regression
In statistics, Deming regression, named after W. Edwards Deming, is an errors-in-variables model that tries to find the line of best fit for a two-dimensional data set. It differs from the simple linear regression in that it accounts for errors in observations on both the ''x''- and the ''y''- axis. It is a special case of total least squares, which allows for any number of predictors and a more complicated error structure. Deming regression is equivalent to the maximum likelihood estimation of an errors-in-variables model in which the errors for the two variables are assumed to be independent and normally distributed, and the ratio of their variances, denoted ''δ'', is known. In practice, this ratio might be estimated from related data-sources; however the regression procedure takes no account for possible errors in estimating this ratio. The Deming regression is only slightly more difficult to compute than the simple linear regression. Most statistical software packages used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Major Axis Regression
In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models. The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix. Linear model Background In the least squares method of data modeling, the objective function to be minimized, ''S'', is a Quadratic form (statistics), quadratic form: :S=\mathbf, where ''r'' is the vector of errors and residuals in statistics, residuals and ''W'' is a weighting matrix. In linear least squares (mathematics), linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector \boldsymbol\beta, so the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Least Squares
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. Basic formulation Consider the linear equation where A \in \mathbb^ and b \in \mathbb^m are given and x \in \mathbb^n is variable to be computed. When m > n, it is generally the case that () has no solution. For example, there is no value of x that satisfies \begin 1 & 0 \\ 0 & 1 \\ 1 & 1 \end x = \begin 1 \\ 1 \\ 0 \end, because the first two rows require that x = (1, 1), but then the third row is not satisfied. Thus, for m > n, the goal of solving () exactly is typically replaced by finding the value of x that minimizes some error. There are many ways th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Segmented Regression
Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented regression analysis can also be performed on multivariate data by partitioning the various independent variables. Segmented regression is useful when the independent variables, clustered into different groups, exhibit different relationships between the variables in these regions. The boundaries between the segments are ''breakpoints''. Segmented linear regression is segmented regression whereby the relations in the intervals are obtained by linear regression. Segmented linear regression, two segments Segmented linear regression with two segments separated by a ''breakpoint'' can be useful to quantify an abrupt change of the response function (Yr) of a varying influential factor (x). The breakpoint can be interpreted as a ''critica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Trend Estimation
Linear trend estimation is a statistical technique used to analyze data patterns. Data patterns, or trends, occur when the information gathered tends to increase or decrease over time or is influenced by changes in an external factor. Linear trend estimation essentially creates a straight line on a graph of data that models the general direction that the data is heading. Fitting a trend: Least-squares Given a set of data, there are a variety of functions that can be chosen to fit the data. The simplest function is a straight line with the dependent variable (typically the measured data) on the vertical axis and the independent variable (often time) on the horizontal axis. The least-squares fit is a common method to fit a straight line through the data. This method minimizes the sum of the squared errors in the data series y. Given a set of points in time t and data values y_t observed for those points in time, values of \hat a and \hat b are chosen to minimize the sum of squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Regression
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable ''x'' and the dependent variable ''y'' is modeled as a polynomial in ''x''. Polynomial regression fits a nonlinear relationship between the value of ''x'' and the corresponding conditional mean of ''y'', denoted E(''y'' , ''x''). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(''y'' , ''x'') is linear in the unknown parameters that are estimated from the data. Thus, polynomial regression is a special case of linear regression. The explanatory (independent) variables resulting from the polynomial expansion of the "baseline" variables are known as higher-degree terms. Such variables are also used in classification settings. History Polynomial regression models are usually fit using the method of least squares. The leas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
