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Predicted Value
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 correlat ... [...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] |
Natural Numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Distribution Function
In statistics, an empirical distribution function ( an empirical cumulative distribution function, eCDF) is the Cumulative distribution function, distribution function associated with the empirical measure of a Sampling (statistics), sample. This cumulative distribution function is a step function that jumps up by at each of the data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. The empirical distribution function is an Estimator, estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of Convergence of random variables#Convergence in distribution, convergence of the empirical distribution function to the underlying cumulative distribution function. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proofs Involving Ordinary Least Squares
The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at the same time retaining the completeness of exposition. Derivation of the normal equations Define the ith residual to be :r_i= y_i - \sum_^ X_\beta_j. Then the objective S can be rewritten :S = \sum_^m r_i^2. Given that ''S'' is convex, it is minimized when its gradient vector is zero (This follows by definition: if the gradient vector is not zero, there is a direction in which we can move to minimize it further – see maxima and minima.) The elements of the gradient vector are the partial derivatives of ''S'' with respect to the parameters: :\frac=2\sum_^m r_i\frac \qquad (j=1,2,\dots, n). The derivatives are :\frac=-X_. Substitution of the expressions for the residuals and the derivatives into the gradient equations gives :\frac = 2\sum_^ \left( y_i-\sum_^ X_\beta_k \right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
