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Ordinary Least Squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable. Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation. The OLS estimator is c ... [...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 squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German: '' Statistik'', "description of a 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 surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Error
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 are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (it has no solution) 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 restricts one degree of freedom. Therefore, the critical case occurs when the number of equations and the number of free variables are equal. 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, when the equations outnumber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Design Matrix
In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables (ones and zeros) that indicate group membership in an ANOVA, or it can contain values of continuous variables. The design matrix contains data on the independent variables (also called explanatory variables) in statistical models which attempt to explain observed data on a response variable (often called a dependent variable) in terms of the explanatory variables. The theory relating to such models makes substantial use of matrix manipulations involving the design matrix: see for example linear regression. A n ... [...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 err ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row And Column Vectors
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, consisting of a single row of n entries, \boldsymbol a = \begin a_1 & a_2 & \dots & a_n \end. (Throughout this article, boldface is used for both row and column vectors.) The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: \begin x_1 \; x_2 \; \dots \; x_m \end^ = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end and \begin x_1 \\ x_2 \\ \vdots \\ x_m \end^ = \begin x_1 \; x_2 \; \dots \; x_m \end. The set of all row vectors with ''n'' entries in a given field (such as the real numbers) forms an ''n''-dimensional vector space; similarly, the set of all column vectors with ''m'' entries forms an ''m''-dimensional vector space. The space of row vectors with ''n'' entries can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Regression Model
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called ''simple linear regression''; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Unit
In statistics, a unit is one member of a set of entities being studied. It is the main source for the mathematical abstraction of a "random variable". Common examples of a unit would be a single person, animal, plant, manufactured item, or country that belongs to a larger collection of such entities being studied. Experimental and sampling units Units are often referred to as being either experimental units, sampling units or units of observation: * An "experimental unit" is typically thought of as one member of a set of objects that are initially equal, with each object then subjected to one of several experimental treatments. Put simply, it is the smallest entity to which a treatment is applied. * A "sampling unit" is typically thought of as an object that has been sampled from a statistical population. This term is commonly used in opinion polling and survey sampling. For example, in an experiment on educational methods, methods may be applied to classrooms of students. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Likelihood Estimator
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is differentiable, the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when all observed outcomes are assumed to have Normal distributions with the same variance. From the perspective of Bayesian inference, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
