Linear least squares
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Linear least squares is an important computational problem, that arises primarily in applications when it is desired to fit a linear mathematical model to measurements obtained from experiments. The goals of linear least squares are to extract predictions from the measurements and to reduce the effect of measurement errors. Mathematically, it can be stated as the problem of finding an approximate solution to an overdetermined system of linear equations. In statistics, it corresponds to the maximum likelihood estimate for a linear model with normally distributed error.
Linear least square problems admit a closedform solution, in contrast to nonlinear least squares problems, which often have to be solved by an iterative procedure.
[edit] Motivational example
As a result of an experiment, four (x,y) data points were obtained, (1,6), (2,5), (3,7), and (4,10) (shown in red in the picture on the right). It is desired to find a line y = β_{1} + β_{2}x that fits "best" these four points. In other words, we would like to find the numbers β_{1} and β_{2} that approximately solve the overdetermined linear system
of four equations in two unknowns in some "best" sense.
The least squares approach to solving this problem is to try to make as small as possible the sum of squares of "errors" between the right and lefthand sides of these equations, that is, to find the minimum of the function
The minimum is determined by calculating the partial derivatives of S(β_{1},β_{2}) in respect to β_{1} and β_{2} and setting them to zero. This results in a system of two equations in two unknowns, called the normal equations, which, when solved, gives the solution
 β_{1} = 3.5
 β_{2} = 1.4
and the equation y = 3.5 + 1.4x of the line of best fit. The residuals, that is, the discrepancies between the y values from the experiment and the y values calculated using the line of best fit are then found to be 1.1, − 1.3, − 0.7, and 0.9 (see the picture on the right). The minimum value of the sum of squares is S(3.5,1.4) = 1.1^{2} + ( − 1.3)^{2} + ( − 0.7)^{2} + 0.9^{2} = 4.2.
[edit] Computation
The common computational procedure to find a firstdegree polynomial function approximation in a situation like this is as follows.
Use for the number of data points.
Find the four sums: , , , and .
The calculations for the slope, m, and the yintercept, b, are as follows.
and
[edit] The general problem
Consider an overdetermined system
of m linear equations in n unknowns, with m > n, written in matrix form as
Such a system usually has no solution, and the goal is then to find the numbers β_{j} which fit the equations "best", in the sense of solving the quadratic minimization problem
A justification for choosing this criterion is given in properties below. This minimization problem has a unique solution, provided that the n columns of the matrix X are linearly independent, given by solving the normal equations
[edit] Uses in data fitting
The primary application of linear least squares is in data fitting. Given a set of m data points consisting of experimentally measured values taken at m values of an independent variable (x_{i} may be scalar or vector quantities), and given a model function with it is desired to find the parameters β_{j} such that the model function fits "best" the data. In linear least squares, linearity is meant to be with respect to parameters β_{j}, so
Here, the functions φ_{j} may be nonlinear with respect to the variable x.
Ideally, the model function fits the data exactly, so
for all This is usually not possible in practice, as there are more data points than there are parameters to be determined. The approach chosen then is to find the minimal possible value of the sum of squares of the residuals
so to minimize the function
After substituting for r_{i} and then for f, this minimization problem becomes the quadratic minimization problem above with X_{ij} = φ_{j}(x_{i}), and the best fit can be found by solving the normal equations.
[edit] Derivation of the normal equations
S is minimized when its gradient with respect to each parameter is equal to zero. The elements of the gradient vector are the partial derivatives of S with respect to the parameters:
Since , the derivatives are
Substitution of the expressions for the residuals and the derivatives into the gradient equations gives
Upon rearrangement, the normal equations
are obtained. The normal equations are written in matrix notation as
The solution of the normal equations yields the vector of the optimal parameter values.
[edit] Computation
[edit] Inverting the normal equations
Although the algebraic solution of the normal equations can be written as
it is not good practice to invert the normal equations matrix. An exception occurs in numerical smoothing and differentiation where an analytical expression is required.
If the matrix is wellconditioned and positive definite, that is, it has full rank, the normal equations can be solved directly by using the Cholesky decomposition , where R is an upper triangular matrix, giving
The solution is obtained in two stages, a forward substitution, , followed by a backward substitution . Both subtitutions are facilitated by the triangular nature of R.
See example of linear regression for a workedout numerical example with three parameters.
[edit] Orthogonal decomposition methods
Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable.
The extra stability results from not having to form the product . The residuals are written in matrix notation as
The matrix X is subjected to an orthogonal decomposition; the QR decomposition will serve to illustrate the process.
where Q is an orthogonal matrix and R is an matrix which is partitioned into a block, , and a zero block. is upper triangular.
The residual vector is leftmultiplied by .
The sum of squares of the transformed residuals, , is the same as before, because Q is orthogonal.
The minimum value of S is attained when the upper block, U, is zero. Therefore the parameters are found by solving
These equations are easily solved as is upper triangular.
An alternative decomposition of X is the singular value decomposition (SVD)^{[1]}
This is effectively another kind of orthogonal decomposition as both U and V are orthogonal. This method is the most computationally intensive, but is particularly useful if the normal equations matrix, , is very illconditioned (i.e. if its condition number multiplied by the machine's relative roundoff error is appreciably large). In that case, including the smallest singular values in the inversion merely adds numerical noise to the solution. This can be cured using the truncated SVD approach, giving a more stable and exact answer, by explicitly setting to zero all singular values below a certain threshold and so ignoring them, a process closely related to factor analysis.
[edit] Properties of the leastsquares estimators
The gradient equations at the minimum can be written as
A geometrical interpretation of these equations is that the vector of residuals, is orthogonal to the column space of , since the dot product is equal to zero for any conformal vector, . This means that is the shortest of all possible vectors , that is, the variance of the residuals is the minimum possible. This is illustrated at the right.
If the experimental errors, , are uncorrelated, have a mean of zero and a constant variance, σ, the GaussMarkov theorem states that the leastsquares estimator, , has the minimum variance of all estimators that are linear combinations of the observations. In this sense it is the best, or optimal, estimator of the parameters. Note particularly that this property is independent of the statistical distribution function of the errors. In other words, the distribution function of the errors need not be a normal distribution. However, for some probability distributions, there is no guarantee that the leastsquares solution is even possible given the observations; still, in such cases it is the best estimator that is both linear and unbiased.
For example, it is easy to show that the arithmetic mean of a set of measurements of a quantity is the leastsquares estimator of the value of that quantity. If the conditions of the GaussMarkov theorem apply, the arithmetic mean is optimal, whatever the distribution of errors of the measurements might be.
However, in the case that the experimental errors do belong to a Normal distribution, the leastsquares estimator is also a maximum likelihood estimator.^{[2]}
These properties underpin the use of the method of least squares for all types of data fitting, even when the assumptions are not strictly valid.
[edit] Limitations
An assumption underlying the treatment given above is that the independent variable, x, is free of error. In practice, the errors on the measurements of the independent variable are usually much smaller than the errors on the dependent variable and can therefore be ignored. When this is not the case, total least squares also known as Errorsinvariables model, or Rigorous least squares, should be used. This can be done by adjusting the weighting scheme to take into account errors on both the dependent and independent variables and then following the standard procedure.^{[3]}^{[4]}
In some cases the (weighted) normal equations matrix is illconditioned; this occurs when the measurements have only a marginal effect on one or more of the estimated parameters.^{[5]} In these cases, the least squares estimate amplifies the measurement noise and may be grossly inaccurate. Various regularization techniques can be applied in such cases, the most common of which is called Tikhonov regularization. If further information about the parameters is known, for example, a range of possible values of x, then minimax techniques can also be used to increase the stability of the solution.
Another drawback of the least squares estimator is the fact that the norm of the residuals, is minimized, whereas in some cases one is truly interested in obtaining small error in the parameter , e.g., a small value of . However, since is unknown, this quantity cannot be directly minimized. If a prior probability on is known, then a Bayes estimator can be used to minimize the mean squared error, . The least squares method is often applied when no prior is known. Surprisingly, however, better estimators can be constructed, an effect known as Stein's phenomenon. For example, if the measurement error is Gaussian, several estimators are known which dominate, or outperform, the least squares technique; the best known of these is the JamesStein estimator.
[edit] Weighted linear least squares
When the observations are not equally reliable, a weighted sum of squares
may be minimized.
Each element of the diagonal weight matrix, W should,ideally, be equal to the reciprocal of the variance of the measurement.^{[6]} The normal equations are then
[edit] Parameter errors, correlation and confidence limits
The parameter values are linear combinations of the observed values
Therefore an expression for the errors on the parameter can be obtained by error propagation from the errors on the observations. Let the variancecovariance matrix for the observations be denoted by M and that of the parameters by M^{β}. Then,
When , this simplifies to
When unit weights are used () it is implied that the experimental errors are uncorrelated and all equal: , where is known as the variance of an observation of unit weight, and is an identity matrix. In this case is approximated by , where S is the minimum value of the objective function
In all cases, the variance of the parameter β_{i} is given by and the covariance between parameters β_{i} and β_{j} is given by . Standard deviation is the square root of variance and the correlation coefficient is given by . These error estimates reflect only random errors in the measurements. The true uncertainty in the parameters is larger due to the presence of systematic errors which, by definition, cannot be quantified. Note that even though the observations may be uncorrelated, the parameters are always correlated.
It is often assumed, for want of any concrete evidence, that the error on a parameter belongs to a Normal distribution with a mean of zero and standard deviation σ. Under that assumption the following confidence limits can be derived.
 68% confidence limits,
 95% confidence limits,
 99% confidence limits,
The assumption is not unreasonable when m>>n. If the experimental errors are normally distributed the parameters will belong to a Student's tdistribution with mn degrees of freedom. When m>>n Student's tdistribution approximates to a Normal distribution. Note, however, that these confidence limits cannot take systematic error into account. Also, parameter errors should be quoted to one significant figure only, as they are subject to sampling error.^{[7]}
When the number of observations is relatively small, Chebychev's inequality can be used for an upper bound on probabilities, regardless of any assumptions about the distribution of experimental errors: the maximum probabilities that a parameter will be more than 1, 2 or 3 standard deviations away from its expectation value are 100%, 25% and 11% respectively.
[edit] Residual values and correlation
The residuals are related to the observations by
The symmetric, idempotent matrix is known in the statistics literature as the hat matrix, . ( is idempotent because it is a projection.) Thus,
where I is an identity matrix. The variancecovariance matrice of the residuals, M^{r} is given by
This shows that even though the observations may be uncorrelated, the residuals are always correlated.
The sum of residual values is equal to zero whenever the model function contains a constant term. Leftmultiply the expression for the residuals by .
Say, for example, that the first term of the model is a constant, so that X_{i1} = 1 for all i. In that case it follows that
Thus, in the motivational example, above, the fact that the sum of residual values is equal to zero it is not accidental but is a consequence of the presence of the constant term, α, in the model.
If experimental error follows a normal distribution, then, because of the linear relationship between residuals and observations, so should residuals,^{[8]} but since the observations are only a sample of the population of all possible observations, the residuals should belong to a Student's tdistribution. Studentized residuals are useful in making a statistical test for an outlier when a particular residual appears to be excessively large.
[edit] Objective function
The objective function can be written as
since is also symmetric and idempotent. It can be shown from this,^{[9]} that the expected value of S is mn. Note, however, that this is true only if the weights have been assigned correctly. If unit weights are assumed, the expected value of S is (m − n)σ^{2}, where σ^{2} is the variance of an observation.
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 Chisquare (χ^{2}) distribution with mn degrees of freedom. Some illustrative percentile values of χ^{2} are given in the following table.^{[10]}

mn 10 9.34 18.3 23.2 25 24.3 37.7 44.3 100 99.3 124 136
These values can be used for a statistical criterion as to the goodnessoffit. When unit weights are used, the numbers should be divided by the variance of an observation.
[edit] Typical uses and applications
 Polynomial fitting: models are polynomials in an independent variable, x:
 Straight line: .^{[11]}
 Quadratic: .
 Cubic, quartic and higher polynomials. For highorder polynomials the use of orthogonal polynomials is recommended.^{[5]}^{[12]}
 Numerical smoothing and differentiation — this is an application of polynomial fitting.
 Multinomials in more than one independent variable, including surface fitting
 Curve fitting with Bsplines ^{[3]}
 Chemometrics, Calibration curve, Standard addition, Gran plot, analysis of mixtures
[edit] Software for solving LLSP
1. Free and opensource, with OSIApproved licenses
Name  License  Brief info 

bvls  BSD  Fortran code by Robert L. Parker & Philip B. Stark 
lapack dgelss  BSD  made by Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University 
OpenOpt  BSD  universal crossplatform Pythonwritten numerical optimization framework; see its LLSP page and full list of problems 
2. Commercial
 MATLAB lsqlin
[edit] Notes
 ^ Lawson, C. L.; Hanson, R. J. (1974). Solving Least Squares Problems. Englewood Cliffs, NJ: PrenticeHall. ISBN 0138225850.
 ^ Margenau, Henry; Murphy, George Moseley (1956). The Mathematics of Physics and Chemistry. Princeton: Van Nostrand.
 ^ ^{a} ^{b} Gans, Peter (1992). Data fitting in the Chemical Sciences. New York: Wiley. ISBN 0471934127.
 ^ Deming, W. E. (1943). Statistical adjustment of Data. New York: Wiley.
 ^ ^{a} ^{b} When fitting polynomials the normal equations matrix is a Vandermonde matrix. Vandermode matrices become increasingly illconditioned as the order of the matrix increases.
 ^ This implies that the observations are uncorrelated. If the observations are correlated, the expression applies. In this case the weight matrix should ideally be equal to the inverse of the variancecovariance matrix of the observations.
 ^ Mandel, John (1964). The Statistical Analysis of Experimental Data. New York: Interscience.
 ^ Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate analysis. New York: Academic Press. ISBN 0124712509.
 ^ Hamilton, W. C. (1964). Statistics in Physical Science. New York: Ronald Press.
 ^ Spiegel, Murray R. (1975). Schaum's outline of theory and problems of probability and statistics. New York: McGrawHill. ISBN 0585267391.
 ^ Acton, F. S. (1959). Analysis of StraightLine Data. New York: Wiley.
 ^ Guest, P. G. (1961). Numerical Methods of Curve Fitting. Cambridge: Cambridge University Press.
[edit] References
 Björck, Åke (1996). Numerical methods for least squares problems. Philadelphia: SIAM. ISBN 0898713609.
 Bevington, Philip R; Robinson, Keith D (2003). Data Reduction and Error Analysis for the Physical Sciences. McGraw Hill. ISBN 0072472278.
[edit] External links
Theory
Online utilities
