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Line Of Best Fit
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 **Weighted geometric distance: Deming regression *Scale invariance: Major axis regression See also *Linear least squares *Linear segmented regression *Linear trend estimation *Polynomial regression *Regression dilution Regression dilution, also known as regression attenuation, is the Bias (statistics), biasing of the linear regression regression slope, slope towards zero (the underestimation of its absolute value), caused by errors in the independent variable. ... Further reading *"Fitting lines", chap.1 in LN. Chernov (2010), ''Circular and linear regression: Fitting circles and lines by least squares'', Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straight Line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two Point (geometry), points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as Non-Euclidean geometry, non-Euclidean, Projective geometry, projective and affine geometry). In modern mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 square ... [...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 a ... [...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 between t ... [...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 which tries to find the line of best fit for a two-dimensional dataset. 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] |
Deming Regression
In statistics, Deming regression, named after W. Edwards Deming, is an errors-in-variables model which tries to find the line of best fit for a two-dimensional dataset. 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] |
Scale Invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry. *In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity. *In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale. *In quantum field theory, ... [...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, ''S'', :S=\mathbf, is minimized, where ''r'' is the vector of residuals and ''W'' is a weighting matrix. In linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector \boldsymbol\beta, so the residuals are given by :\mathbf. There are ''m'' observations in y and ''n'' parameters in β with ''m''>''n''. X is a ''m''� ... [...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. Main formulations The three main linear least squares formulations are: * Ordinary least squares (OLS) is the most common estimator. OLS estimates are commonly used to analyze both experimental and observational data. The OLS method minimizes the sum of squared residuals, and leads to a closed-form expression for the estimated value of the unknown parameter vector ''β'': \hat = (\mathbf^\mathsf\mathbf)^ \mathbf^\mathsf \mathbf, where \mathbf is a vector whose ''i''th element is the ''i''th observation of the dependent variable, and \mathbf is a matrix whose ' ... [...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 to aid interpretation of data. When a series of measurements of a process are treated as, for example, a sequences or time series, trend estimation can be used to make and justify statements about tendencies in the data, by relating the measurements to the times at which they occurred. This model can then be used to describe the behaviour of the observed data, without explaining it. In particular, it may be useful to determine if measurements exhibit an increasing or decreasing trend which is statistically distinguished from random behaviour. Some examples are determining the trend of the daily average temperatures at a given location from winter to summer, and determining the trend in a global temperature series over the last 100 years. In the latter case, issues of homogeneity are important (for example, about whether the series is equally reliable throughout its length). Fitting a trend: least-squares Given a set of data an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
