# Linear least squares

**normal equationslinear least-squaresLinear least squares (mathematics)least squaresLinear least squares (disambiguation)Weighted linear least squaresconstrained linear equationsLinearmultiple regression modelordinary least squares regression formula**

Linear least squares (LLS) is the least squares approximation of linear functions to data.wikipedia

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### Ordinary least squares

**OLSleast squaresOrdinary least squares regression**

ordinary (unweighted),

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model.

### Numerical methods for linear least squares

**least squareslinear least squaresnormal equations**

Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

Numerical methods for linear least squares entails the numerical analysis of linear least squares problems.

### Constrained least squares

In constrained least squares, one is interested in solving a linear least squares problem with an additional constraint on the solution.

In constrained least squares one solves a linear least squares problem with an additional constraint on the solution.

### Linear regression

**regression coefficientmultiple linear regressionregression**

It is a set of formulations for solving statistical problems involved in linear regression, including variants for In statistics, linear least squares problems correspond to a particularly important type of statistical model called linear regression which arises as a particular form of regression analysis.

(See also Weighted linear least squares, and Generalized least squares.) Heteroscedasticity-consistent standard errors is an improved method for use with uncorrelated but potentially heteroscedastic errors.

### Outline of regression analysis

**outline**

See outline of regression analysis for an outline of the topic.

### Regression analysis

**regressionmultiple regressionregression model**

In statistics, linear least squares problems correspond to a particularly important type of statistical model called linear regression which arises as a particular form of regression analysis.

Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators,.

### Total least squares

**orthogonal regressiontotal least squares algorithmstotal least squares problem**

When this is not the case, total least squares or more generally errors-in-variables models, or rigorous least squares, should be used.

In linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector, so the residuals are given by

### Savitzky–Golay filter

**Numerical smoothing and differentiationSavitzky–Golay smoothing filterSavitzky-Golay filter**

This is achieved, in a process known as convolution, by fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares.

### Overdetermined system

**overdeterminedover-determined system overdetermined equation system**

Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations, where the best approximation is defined as that which minimizes the sum of squared differences between the data values and their corresponding modeled values.

the solution of which can be written with the normal equations,

### Non-linear least squares

**nonlinear least squaresNLLSnon-linear least-squares estimation**

In contrast, non-linear least squares problems generally must be solved by an iterative procedure, and the problems can be non-convex with multiple optima for the objective function.

There are many similarities to linear least squares, but also some significant differences.

### Least squares

**least-squaresmethod of least squaresleast squares method**

Linear least squares (LLS) is the least squares approximation of linear functions to data.

See linear least squares for a fully worked out example of this model.

### Beer–Lambert law

**Beer-Lambert lawBeer's lawLambert–Beer law**

In practice it is better to use linear least squares to determine the two amount concentrations from measurements made at more than two wavelengths.

### Line fitting

**line of best fitfitting a line**

### Linear function

**linearlinear factorlinear functions**

Linear least squares (LLS) is the least squares approximation of linear functions to data.

### Weighted least squares

**Batch Least Squareslinear least squaresweighted**

weighted, and

### Generalized least squares

**feasible generalized least squaresgeneralizedgeneralized (correlated)**

generalized (correlated) residuals.

### Mathematical optimization

**optimizationmathematical programmingoptimal**

In OLS (i.e., assuming unweighted observations), the optimal value of the objective function is found by substituting the optimal expression for the coefficient vector:

### Loss function

**objective functioncost functionrisk function**

In OLS (i.e., assuming unweighted observations), the optimal value of the objective function is found by substituting the optimal expression for the coefficient vector:

### Expected value

**expectationexpectedmean**

It can be shown from this that under an appropriate assignment of weights the expected value of S is m − n.

### Chi-squared distribution

**chi-squaredchi-square distributionchi square distribution**

If it is assumed that the residuals belong to a normal distribution, the objective function, being a sum of weighted squared residuals, will belong to a chi-squared (\chi ^2) distribution with m − n degrees of freedom.

### Degrees of freedom (statistics)

**degrees of freedomdegree of freedomEffective degrees of freedom**

If it is assumed that the residuals belong to a normal distribution, the objective function, being a sum of weighted squared residuals, will belong to a chi-squared (\chi ^2) distribution with m − n degrees of freedom.

### Goodness of fit

**goodness-of-fitfitgoodness-of-fit test**

:These values can be used for a statistical criterion as to the goodness of fit.

### Statistics

**statisticalstatistical analysisstatistician**

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.

### Mathematics

**mathematicalmathmathematician**

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.

### Mathematical model

**mathematical modelingmodelmathematical models**

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.