Linear regression in the context of "Ordinary least squares"

In statistical modeling, regression analysis is a statistical method for estimating the relationship between a dependent variable (often called the outcome or response variable, or a label in machine learning parlance) and one or more independent variables (often called regressors, predictors, covariates, explanatory variables or features).

The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters (e.g., quantile regression or Necessary Condition Analysis) or estimate the conditional expectation across a broader collection of non-linear models (e.g., nonparametric regression).

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 the random errors are assumed to have normal distributions with the same variance.

In machine learning and pattern recognition, a feature is an individual measurable property or characteristic of a data set. Choosing informative, discriminating, and independent features is crucial to producing effective algorithms for pattern recognition, classification, and regression tasks. Features are usually numeric, but other types such as strings and graphs are used in syntactic pattern recognition, after some pre-processing step such as one-hot encoding. The concept of "features" is related to that of explanatory variables used in statistical techniques such as linear regression.

Regression dilution, also known as regression attenuation, is the biasing of the linear regression slope towards zero (the underestimation of its absolute value), caused by errors in the independent variable.

Consider fitting a straight line for the relationship of an outcome variable y to a predictor variable x, and estimating the slope of the line. Statistical variability, measurement error or random noise in the y variable causes uncertainty in the estimated slope, but not bias: on average, the procedure calculates the right slope. However, variability, measurement error or random noise in the x variable causes bias in the estimated slope (as well as imprecision). The greater the variance in the x measurement, the closer the estimated slope must approach zero instead of the true 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 correlation between y and x corrected by the ratio of standard deviations of these variables. The intercept of the fitted line is such that the line passes through the center of mass (x, y) of the data points.

In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and synergetic nonlinear effects. Contrary to traditional modelling methods, such as linear regression and basic statistical methods, nonlinear modelling can be utilized efficiently in a vast number of situations where traditional modelling is impractical or impossible. The newer nonlinear modelling approaches include non-parametric methods, such as feedforward neural networks, kernel regression, multivariate splines, etc., which do not require a prior knowledge of the nonlinearities in the relations. Thus the nonlinear modelling can utilize production data or experimental results while taking into account complex nonlinear behaviours of modelled phenomena which are in most cases practically impossible to be modelled by means of traditional mathematical approaches, such as phenomenological modelling.

Contrary to phenomenological modelling, nonlinear modelling can be utilized in processes and systems where the theory is deficient or there is a lack of fundamental understanding on the root causes of most crucial factors on system. Phenomenological modelling describes a system in terms of laws of nature. Nonlinear modelling can be utilized in situations where the phenomena are not well understood or expressed in mathematical terms. Thus nonlinear modelling can be an efficient way to model new and complex situations where relationships of different variables are not known.