# Won 1x2 fixed ratio 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.

1, this term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

2, 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.

3, 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 the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.

Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.

4 This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.

Linear regression has many practical uses.

Most applications fall into one of the following two broad categories: If the goal is prediction, forecasting, or error reduction, clarification needed linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables.

After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.

If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory.

Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression.

Conversely, the least squares approach can be used to fit models that are not linear models.

Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.

Contents Formulation edit In linear regression, the observations ( red ) are assumed to be the result of random deviations ( green ) from an underlying relationship ( blue ) between a dependent variable ( y ) and an independent variable ( x ).

Given a data set yi, xi1,xipi1ndisplaystyle y_i,x_i1,ldots,x_ip_i1n of n statistical units, a linear regression model assumes that the relationship between the dependent variable y and the p -vector of regressors x is linear.

This relationship is modeled through a disturbance term or error variable an unobserved random variable that adds "noise" to the linear relationship between the dependent variable and regressors.

Thus the model takes the form y_ibeta _0beta _1x_i1cdots beta _px_ipvarepsilon _imathbf x _imathsf Tboldsymbol beta varepsilon _i,qquad i1,ldots,n, where T denotes the transpose, so that x i T is the inner product between vectors x i and.

Often these n equations are stacked together and written in matrix notation as yX, displaystyle mathbf y Xboldsymbol beta boldsymbol varepsilon, where y(y1y2yn displaystyle mathbf y beginpmatrixy_1y_2vdots y_nendpmatrix, quad Xbeginpmatrixmathbf x _1mathsf Tmathbf x _2mathsf Tvdots mathbf x _nmathsf.

Notation and terminology edit ydisplaystyle mathbf y is a vector of observed values yi (i1,n)displaystyle y_i (i1,ldots,n) of the variable called the regressand, endogenous variable, response variable, measured variable, criterion variable, or dependent variable.

This variable is also sometimes known as the predicted variable, but this should not be confused with predicted values, which are denoted ydisplaystyle hat.

The decision as to which variable in a data set is modeled as the dependent variable and which are modeled as the independent variables may be based on a presumption that the value of one of the variables.

Alternatively, there may be an operational reason to model one of the variables in terms of the others, in which case there need be no presumption of causality.

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