Least squares

Founding

The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on land sightings for navigation.

The method was the culmination of several advances that took place during the course of the eighteenth century: ^[6]

• The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, perhaps first expressed by Roger Cotes in 1722.
• The combination of different observations taken under the same conditions contrary to simply trying one's best to observe and record a single observation accurately. The approach was known as the method of averages. This approach was notably used by Tobias Mayer while studying the librations of the Moon in 1750, and by Pierre-Simon Laplace in his work in explaining the differences in motion of Jupiter and Saturn in 1788.
• The combination of different observations taken under different conditions. The method came to be known as the method of least absolute deviation . It was notably performed by Roger Joseph Boscovich in his work on the shape of the Earth in 1757 and by Pierre-Simon Laplace for the same problem in 1789 and 1799.
• The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved. Laplace tried to specify a mathematical form of the probability density for the errors and define a method of estimation that minimizes the error of estimation. For this purpose, Laplace used a symmetric two-sided exponential distribution we now call Laplace distribution to model the error distribution, and used the sum of absolute deviation as error of estimation. He felt these to be the simplest assumptions he could make, and he had hoped to obtain the arithmetic mean as the best estimate. Instead, his estimator was the posterior median.

The method

The first clear and concise exposition of the method of least squares was published by Legendre in 1805. ^[7] The technique is described as an algebraic procedure for fitting linear equations to data and Legendre demonstrates the new method by analyzing the same data as Laplace for the shape of the Earth. Within ten years after Legendre's publication, the method of least squares had been adopted as a standard tool in astronomy and geodesy in France, Italy, and Prussia, which constitutes an extraordinarily rapid acceptance of a scientific technique. ^[6]

In 1809 Carl Friedrich Gauss published his method of calculating the orbits of celestial bodies. In that work he claimed to have been in possession of the method of least squares since 1795. ^[8] This naturally led to a priority dispute with Legendre. However, to Gauss's credit, he went beyond Legendre and succeeded in connecting the method of least squares with the principles of probability and to the normal distribution. He had managed to complete Laplace's program of specifying a mathematical form of the probability density for the observations, depending on a finite number of unknown parameters, and define a method of estimation that minimizes the error of estimation. Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method of estimation. He then turned the problem around by asking what form the density should have and what method of estimation should be used to get the arithmetic mean as estimate of the location parameter. In this attempt, he invented the normal distribution.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres. On 1 January 1801, the Italian astronomer Giuseppe Piazzi discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the Sun. Based on these data, astronomers desired to determine the location of Ceres after it emerged from behind the Sun without solving Kepler's complicated nonlinear equations of planetary motion. The only predictions that successfully allowed Hungarian astronomer Franz Xaver von Zach to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

In 1810, after reading Gauss's work, Laplace, after proving the central limit theorem, used it to give a large sample justification for the method of least squares and the normal distribution. In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, normally distributed, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. An extended version of this result is known as the Gauss–Markov theorem.

The idea of least-squares analysis was also independently formulated by the American Robert Adrain in 1808. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares. ^[9]

Linear least squares

A regression model is a linear one when the model comprises a linear combination of the parameters, i.e., $${\displaystyle f(x,{\boldsymbol {\beta }})=\sum _{j=1}^{m}\beta _{j}\phi _{j}(x),}$$ where the function ${\displaystyle \phi _{j}}$ is a function of ${\displaystyle x}$. ^[12]

Letting ${\displaystyle X_{ij}=\phi _{j}(x_{i})}$ and putting the independent and dependent variables in matrices ${\displaystyle X}$ and ${\displaystyle Y}$, respectively, we can compute the least squares in the following way. Note that ${\displaystyle D}$ is the set of all data. ^{[12] [13]} $${\displaystyle L(D,{\boldsymbol {\beta }})=\left\|Y-X{\boldsymbol {\beta }}\right\|^{2}=(Y-X{\boldsymbol {\beta }})^{\mathsf {T}}(Y-X{\boldsymbol {\beta }})=Y^{\mathsf {T}}Y-Y^{\mathsf {T}}X{\boldsymbol {\beta }}-{\boldsymbol {\beta }}^{\mathsf {T}}X^{\mathsf {T}}Y+{\boldsymbol {\beta }}^{\mathsf {T}}X^{\mathsf {T}}X{\boldsymbol {\beta }}}$$$${\displaystyle =Y^{\mathsf {T}}Y-X^{\mathsf {T}}Y{\boldsymbol {\beta }}-X^{\mathsf {T}}Y{\boldsymbol {\beta }}+X^{\mathsf {T}}X{\boldsymbol {\beta }}^{2}}$$

The gradient of the loss is: $${\displaystyle {\frac {\partial L(D,{\boldsymbol {\beta }})}{\partial {\boldsymbol {\beta }}}}={\frac {\partial \left(Y^{\mathsf {T}}Y-X^{\mathsf {T}}Y{\boldsymbol {\beta }}-X^{\mathsf {T}}Y{\boldsymbol {\beta }}+X^{\mathsf {T}}X{\boldsymbol {\beta }}^{2}\right)}{\partial {\boldsymbol {\beta }}}}=-2X^{\mathsf {T}}Y+2X^{\mathsf {T}}X{\boldsymbol {\beta }}}$$

Setting the gradient of the loss to zero and solving for ${\displaystyle {\boldsymbol {\beta }}}$, we get: ^{[13] [12]} $${\displaystyle -2X^{\mathsf {T}}Y+2X^{\mathsf {T}}X{\boldsymbol {\beta }}=0\Rightarrow X^{\mathsf {T}}Y=X^{\mathsf {T}}X{\boldsymbol {\beta }}}$$$${\displaystyle {\boldsymbol {\hat {\beta }}}=\left(X^{\mathsf {T}}X\right)^{-1}X^{\mathsf {T}}Y}$$

Non-linear least squares

There is, in some cases, a closed-form solution to a non-linear least squares problem – but in general there is not. In the case of no closed-form solution, numerical algorithms are used to find the value of the parameters ${\displaystyle \beta }$ that minimizes the objective. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation: $${\displaystyle {\beta _{j}}^{k+1}={\beta _{j}}^{k}+\Delta \beta _{j},}$$ where a superscript k is an iteration number, and the vector of increments ${\displaystyle \Delta \beta _{j}}$ is called the shift vector. In some commonly used algorithms, at each iteration the model may be linearized by approximation to a first-order Taylor series expansion about ${\displaystyle {\boldsymbol {\beta }}^{k}}$: $${\displaystyle {\begin{aligned}f(x_{i},{\boldsymbol {\beta }})&=f^{k}(x_{i},{\boldsymbol {\beta }})+\sum _{j}{\frac {\partial f(x_{i},{\boldsymbol {\beta }})}{\partial \beta _{j}}}\left(\beta _{j}-{\beta _{j}}^{k}\right)\\[1ex]&=f^{k}(x_{i},{\boldsymbol {\beta }})+\sum _{j}J_{ij}\,\Delta \beta _{j}.\end{aligned}}}$$

The Jacobian J is a function of constants, the independent variable and the parameters, so it changes from one iteration to the next. The residuals are given by $${\displaystyle r_{i}=y_{i}-f^{k}(x_{i},{\boldsymbol {\beta }})-\sum _{k=1}^{m}J_{ik}\,\Delta \beta _{k}=\Delta y_{i}-\sum _{j=1}^{m}J_{ij}\,\Delta \beta _{j}.}$$

To minimize the sum of squares of ${\displaystyle r_{i}}$, the gradient equation is set to zero and solved for ${\displaystyle \Delta \beta _{j}}$: $${\displaystyle -2\sum _{i=1}^{n}J_{ij}\left(\Delta y_{i}-\sum _{k=1}^{m}J_{ik}\,\Delta \beta _{k}\right)=0,}$$ which, on rearrangement, become m simultaneous linear equations, the normal equations: $${\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{m}J_{ij}J_{ik}\,\Delta \beta _{k}=\sum _{i=1}^{n}J_{ij}\,\Delta y_{i}\qquad (j=1,\ldots ,m).}$$

The normal equations are written in matrix notation as $${\displaystyle \left(\mathbf {J} ^{\mathsf {T}}\mathbf {J} \right)\Delta {\boldsymbol {\beta }}=\mathbf {J} ^{\mathsf {T}}\Delta \mathbf {y} .}$$

These are the defining equations of the Gauss–Newton algorithm.

Differences between linear and nonlinear least squares

• The model function, f, in LLSQ (linear least squares) is a linear combination of parameters of the form ${\displaystyle f=X_{i1}\beta _{1}+X_{i2}\beta _{2}+\cdots }$ The model may represent a straight line, a parabola or any other linear combination of functions. In NLLSQ (nonlinear least squares) the parameters appear as functions, such as ${\displaystyle \beta ^{2},e^{\beta x}}$ and so forth. If the derivatives ${\displaystyle \partial f/\partial \beta _{j}}$ are either constant or depend only on the values of the independent variable, the model is linear in the parameters. Otherwise, the model is nonlinear.
• Need initial values for the parameters to find the solution to a NLLSQ problem; LLSQ does not require them.
• Solution algorithms for NLLSQ often require that the Jacobian can be calculated similar to LLSQ. Analytical expressions for the partial derivatives can be complicated. If analytical expressions are impossible to obtain either the partial derivatives must be calculated by numerical approximation or an estimate must be made of the Jacobian, often via finite differences.
• Non-convergence (failure of the algorithm to find a minimum) is a common phenomenon in NLLSQ.
• LLSQ is globally concave so non-convergence is not an issue.
• Solving NLLSQ is usually an iterative process which has to be terminated when a convergence criterion is satisfied. LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the Gauss–Seidel method.
• In LLSQ the solution is unique, but in NLLSQ there may be multiple minima in the sum of squares.
• Under the condition that the errors are uncorrelated with the predictor variables, LLSQ yields unbiased estimates, but even under that condition NLLSQ estimates are generally biased.

These differences must be considered whenever the solution to a nonlinear least squares problem is being sought. ^[12]

Tikhonov regularization

In some contexts, a regularized version of the least squares solution may be preferable. Tikhonov regularization (or ridge regression) adds a constraint that ${\displaystyle \left\|\beta \right\|_{2}^{2}}$, the squared ${\displaystyle \ell _{2}}$-norm of the parameter vector, is not greater than a given value to the least squares formulation, leading to a constrained minimization problem. This is equivalent to the unconstrained minimization problem where the objective function is the residual sum of squares plus a penalty term ${\displaystyle \alpha \left\|\beta \right\|_{2}^{2}}$ and ${\displaystyle \alpha }$ is a tuning parameter (this is the Lagrangian form of the constrained minimization problem). ^[17]

In a Bayesian context, this is equivalent to placing a zero-mean normally distributed prior on the parameter vector.

Lasso method

An alternative regularized version of least squares is Lasso (least absolute shrinkage and selection operator), which uses the constraint that ${\displaystyle \|\beta \|_{1}}$, the L₁-norm of the parameter vector, is no greater than a given value. ^{[18] [19] [20]} (One can show like above using Lagrange multipliers that this is equivalent to an unconstrained minimization of the least-squares penalty with ${\displaystyle \alpha \|\beta \|_{1}}$ added.) In a Bayesian context, this is equivalent to placing a zero-mean Laplace prior distribution on the parameter vector. ^[21] The optimization problem may be solved using quadratic programming or more general convex optimization methods, as well as by specific algorithms such as the least angle regression algorithm.

One of the prime differences between Lasso and ridge regression is that in ridge regression, as the penalty is increased, all parameters are reduced while still remaining non-zero, while in Lasso, increasing the penalty will cause more and more of the parameters to be driven to zero. This is an advantage of Lasso over ridge regression, as driving parameters to zero deselects the features from the regression. Thus, Lasso automatically selects more relevant features and discards the others, whereas Ridge regression never fully discards any features. Some feature selection techniques are developed based on the LASSO including Bolasso which bootstraps samples, ^[22] and FeaLect which analyzes the regression coefficients corresponding to different values of ${\displaystyle \alpha }$ to score all the features. ^[23]

The L¹-regularized formulation is useful in some contexts due to its tendency to prefer solutions where more parameters are zero, which gives solutions that depend on fewer variables. ^[18] For this reason, the Lasso and its variants are fundamental to the field of compressed sensing. An extension of this approach is elastic net regularization.