Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.įormulation 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). Conversely, the least squares approach can be used to fit models that are not linear models. So, cost functions that are robust to outliers should be used if the dataset has many large outliers. Use of the Mean Squared Error(MSE) as the cost on a dataset that has many large outliers, can result in a model that fits the outliers more than the true data due to the higher importance assigned by MSE to large errors. 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), or by minimizing a penalized version of the least squares cost function as in ridge regression ( L 2-norm penalty) and lasso ( L 1-norm penalty). 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 variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.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 error i.e variance reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables.Most applications fall into one of the following two broad categories: Linear regression has many practical uses. 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 was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. 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. 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. In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. The case of one explanatory variable is called simple linear regression for more than one, the process is called multiple linear regression. In statistics, linear regression is a statistical model which estimates the linear relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). There is no meaningful interpretation for the correlation coefficient as there is for the \(R^2\) value.Statistical modeling method Part of a series on Know what various correlation coefficient values mean.Know how to calculate the correlation coefficient r from the \(R^2\) value.Understand the cautions necessary in using the \(R^2\) value as a way of assessing the strength of the linear association.Know how to interpret the \(R^2\) value.That is, they can be 0 even if there is a perfect nonlinear association. Know that the coefficient of determination (\(R^2\)) and the correlation coefficient (r) are measures of linear association.Interpret the intercept \(b_\) from Minitab's fitted line plot and regression analysis output.Understand the concept of the least squares criterion.Distinguish between a deterministic relationship and a statistical relationship.Upon completion of this lesson, you should be able to:
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