Linear least squares
Linear least squares is an important
computational problem, that arises primarily in applications when it is desired to fit a linear mathematical modelto 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 systemof linear equations.
As a result of an experiment, four data points were obtained, and (shown in red in the picture on the right). It is desired to find a line that fits "best" these four points. In other words, we would like to find the numbers and that approximately solve the overdetermined linear system:of four equations in two unknowns in some "best" sense.
least squaresapproach to solving this problem is to try to make as small as possible the sum of squares of "errors" between the right- and left-hand sides of these equations, that is, to find the minimum of the function
The minimum is determined by calculating the
partial derivatives of in respect to and and setting them to zero. This results in a system of two equations in two unknowns, which, when solved, gives the solution
and the equation of the line of best fit. The residuals, that is, the discrepancies between the values from the experiment and the values calculated using the line of best fit are then found to be and (see the picture on the right). The minimum value of the sum of squares is
The general problem
linear equations in unknowns, with written in matrix form as
Such a system usually has no solution, and the goal is then to find the numbers which fit the equations "best", in the sense of minimizing the sum of squares of differences between the right- and left-hand sides of the equations. The justification for choosing this criterion is given in properties, below.
The linear least squares problem has a unique solution, provided that the columns of the matrix are
linearly independent. The solution is obtained by solving the normal equations
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 ( may be scalar or vector quantities), and given a model function with it is desired to find the parameters such that the model function fits "best" the data. In linear least squares the model function is assumed to be linear in the parameters so
Here, the functions may be nonlinear in 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
The problem then reduces to the overdetermined linear system mentioned earlier, with
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.
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 differentiationwhere an analytical expression is required.
If the matrix is well-conditioned 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 worked-out numerical example with three parameters.
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 decompositionwill serve to illustrate the process. :where Q is an orthogonalmatrix and R is an matrix which is partitioned into a block, , and a zero block. is upper triangular.:The residual vector is left-multiplied 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) [C.L. Lawson and R.J. Hanson, Solving Least Squares Problems, Prentice-Hall,1974] :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 ill-conditioned (i.e. if its condition numbermultiplied by the machine's relative round-off erroris 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.
Properties of the least-squares 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 spaceof , 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, , the
Gauss-Markov theoremstates that the least-squares 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 functionof the errors. In other words, "the distribution function of the errors need not be a normal distribution".
For example, it is easy to show that the
arithmetic meanof a set of measurements of a quantity is the least-squares estimator of the value of that quantity. If the conditions of the Gauss-Markov 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 distribuition, the least-squares estimator is also a
maximum likelihoodestimator. [H. Margenau and G.M. Murphy, The Mathematics of Physics and Chemistry, Van Nostrand, 1943, 1956]
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.
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 squaresalso known as "Errors-in-variables 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.P. Gans, Data fitting in the Chemical Sciences, Wiley, 1992] [W.E. Deming, Statistical adjustment of Data, Wiley, 1943]
In some cases the (weighted) normal equations matrix is
ill-conditioned; this occurs when the measurements have only a marginal effect on one or more of the estimated parameters.When fitting polynomials the normal equations matrix is a Vandermonde matrix. Vandermode matrices become increasingly ill-conditioned as the order of the matrix increases.] 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 minimaxtechniques 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 probabilityon is known, then a Bayes estimatorcan 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 James-Stein estimator.
Weighted linear least squares
When the observations are not equally reliable, a weighted sum of squares:may be minimized.
Each element of the
diagonalweight matrix, W should,ideally, be equal to the reciprocal of the varianceof the measurement. [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 variance-covariance matrixof the observations.] The normal equations are then:
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 propagationfrom the errors on the observations. Let the variance-covariance matrixfor the observations be denoted by M and that of the parameters by M. 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 varianceof the parameter is given by and the covariancebetween parameters and is given by . Standard deviationis the square root of variance and the correlation coefficient is given by . These error estimates reflect only random errorsin the measurements. The true uncertainty in the parameters is larger due to the presence of systematic errorswhich, by definition, cannot be quantified.Note that even though the observations may be un-correlated, 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 distributionwith a mean of zero and standard deviation . Under that assumption the following confidence limitscan 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 t-distributionwith "m-n" degrees of freedom. When "m>>n" Student's t-distribution 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. [J. Mandel, The Statistical Analysis of Experimental Data, Interscience, 1964]
When the number of observations is relatively small,
Chebychev's inequalitycan 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.
Residual values and correlation
The residuals are related to the observations by:The symmetric,
idempotentmatrix is known in the statistics literature as the hat matrix, . Thus,:where I is an identity matrix. The variance-covariance matrice of the residuals, Mr is given by:This shows that even though the observations may be uncorrelated, the residuals are alwayscorrelated.
The sum of residual values is equal to zero whenever the model function contains a constant term. Left-multiply the expression for the residuals by .:Say, for example, that the first term of the model is a constant, so that 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, [K.V. Mardia, J.T. Kent and J.M. Bibby, Multivariate analysis, Academic Press, 1979] but since the observations are only a sample of the population of all possible observations, the residuals should belong to a Student's t-distribution. Studentized residuals are useful in making a statistical test for an outlierwhen a particular residual appears to be excessively large.
The objective function can be written as:since is also symmetric and idempotent. It can be shown from this, [W. C. Hamilton, Statistics in Physical Science, The Ronald Press, New York, 1964] that the
expected valueof "S" is "m-n". 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 , where 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 Chi-square () distribution with "m-n" degrees of freedom. Some illustrative percentile values of are given in the following table. [M.R. Spiegel, Probability and Statistics, Schaum's Outline Series, McGraw-Hill 1982] :These values can be used for a statistical criterion as to the
goodness-of-fit. When unit weights are used, the numbers should be divided by the variance of an observation.
Typical uses and applications
* Polynomial fitting: models are
polynomials in an independent variable, "x":
** Straight line: . [F.S. Acton, Analysis of Straight-Line Data, Wiley, 1959]
** Quadratic: .
** Cubic, quartic and higher polynomials. For high-order polynomials the use of
orthogonal polynomialsis recommended. [P.G. Guest, Numerical Methods of Curve Fitting, Cambridge University Press, 1961.]
Numerical smoothing and differentiation— this is an application of polynomial fitting.
*Multinomials in more than one independent variable, including surface fitting
*Curve fitting with
Chemometrics, Calibration curve, Standard addition, Gran plot, analysis of mixtures
*Cite book | author=Björck, Åke | authorlink= | coauthors= | title=Numerical methods for least squares problems | date=1996 | publisher=SIAM | location=Philadelphia | isbn=0-89871-360-9 | pages=
*Cite book | author=Bevington, Philip R | coauthors=Robinson, Keith D | title=Data Reduction and Error Analysis for the Physical Sciences | date=2003 | publisher=McGraw Hill | location= | isbn=0072472278 | pages=
* [http://mathworld.wolfram.com/LeastSquaresFitting.html Least Squares Fitting – From MathWorld]
* [http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html Least Squares Fitting-Polynomial – From MathWorld]
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