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In [[statistics]], a '''confidence region''' is a multi-dimensional generalization of a [[confidence interval]]. It is a set of points in an ''n''-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, although other shapes can occur. 
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The confidence region is calculated in such a way that if a set of measurements were repeated many times and a confidence region calculated in the same way on each set of measurements, then a certain percentage of the time, on average, (e.g. 95%) the confidence region would include the point representing the "true" values of the set of variables being estimated. However, unless certain assumptions about [[prior probability|prior probabilities]] are made, it does '''not''' mean, when one confidence region has been calculated, that there is a 95% probability that the "true" values lie inside the region, since we do not assume any particular probability distribution of the "true" values and we may or may not have other information about where they are likely to lie.
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</ul>


==The case of independent, identically normally-distributed errors==
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{{see also|Ordinary least squares}}
Suppose we have found a solution <math>\boldsymbol{\beta}</math> to the following overdetermined problem:
:<math>\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}</math>


where '''Y''' is an ''n''-dimensional column vector containing observed values, '''X''' is an ''n''-by-''p'' matrix which can represent a physical model and which is assumed to be known exactly, <math>\boldsymbol{\beta}</math> is a column vector containing the ''p'' parameters which are to be estimated, and <math>\boldsymbol{\varepsilon}</math> is an ''n''-dimensional column vector of errors which are assumed to be [[Statistical independence|independently distributed]] with [[normal distribution]]s with zero mean and each having the same unknown variance <math>\sigma^2</math>.
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A joint 100(1 - <math>\alpha</math>) % confidence region for the elements of <math>\boldsymbol{\beta}</math> is represented by the set of values of the vector '''b''' which satisfy the following inequality:<ref>Draper and Smith (1981, p. 94)</ref>
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:<math> (\boldsymbol{\beta} - \mathbf{b})^\prime \mathbf{X}^\prime\mathbf{X}(\boldsymbol{\beta} - \mathbf{b}) \le  ps^2 F_{1 - \alpha}(p,\nu) ,</math>
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where the variable '''b''' represents any point in the confidence region, ''p'' is the number of parameters, i.e. number of elements of the vector <math>\boldsymbol{\beta},</math> and ''s''<sup>2</sup> is  an [[Variance#Population variance and sample variance|unbiased estimate]] of <math>\sigma^2</math> equal to
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:<math>s^2=\frac{\varepsilon^\prime\varepsilon}{n - p}.</math>
 
Further, ''F'' is the [[quantile function]] of the [[F-distribution]], with ''p'' and <math> \nu = n - p</math> [[Degrees of freedom (statistics)|degrees of freedom]], <math>\alpha</math> is the [[statistical significance]] level, and the symbol <math>X^\prime</math> means the [[transpose]] of <math>X</math>.
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The above inequality defines an [[ellipsoid]]al region in the ''p''-dimensional Cartesian parameter space R<sup>''p''</sup>. The centre of the ellipsoid is at the solution <math>\boldsymbol\beta</math>. According to Press et al., it's easier to plot the ellipsoid after doing [[singular value decomposition]].  The lengths of the axes of the ellipsoid are proportional to the reciprocals of the values on the diagonals of the diagonal matrix, and the directions of these axes are given by the rows of the 3rd matrix of the decomposition.
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==Weighted and generalised least squares==
</ul>
{{Main|Weighted least squares|Generalized least squares}}
Now let us consider the more general case where some distinct elements of <math>\boldsymbol{\varepsilon}</math> have known nonzero [[covariance]] (in other words, the errors in the observations are not independently distributed), and/or the standard deviations of the errors are not all equal.  Suppose the covariance matrix of <math>\boldsymbol{\varepsilon}</math> is <math>\mathbf{V}\sigma^2</math>, where '''V''' is an ''n''-by-''n'' nonsingular matrix which was equal to <math>\mathbf{I}\sigma^2</math> in the more specific case handled in the previous section, (where '''I''' is the [[identity matrix]],) but here is allowed to have nonzero off-diagonal elements representing the covariance of pairs of individual observations, as well as not necessarily having all the diagonal elements equal.
 
It is possible to find<ref>Draper and Smith (1981, p. 108)</ref> a nonsingular symmetric matrix '''P''' such that
 
:<math>\mathbf{P}^\prime\mathbf{P} = \mathbf{P}\mathbf{P} = \mathbf{V}</math>
 
In effect, '''P''' is a square root of the covariance matrix '''V'''.
 
The least-squares problem
 
:<math>\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}</math>
 
can then be transformed by left-multiplying each term by the inverse of '''P''', forming the new problem formulation
:<math>\mathbf{Z} = \mathbf{Q}\boldsymbol{\beta} + \mathbf{f} ,</math>
 
where  
 
:<math>\mathbf{Z} = \mathbf{P}^{-1}\mathbf{Y}</math>
: <math>\mathbf{Q} = \mathbf{P}^{-1}\mathbf{X}</math> and
:<math>\mathbf{f} = \mathbf{P}^{-1}\boldsymbol{\varepsilon}</math>
 
A joint confidence region for the parameters, i.e. for the elements of <math>\boldsymbol{\beta}</math>, is then bounded by the ellipsoid given by:<ref>Draper and Smith (1981, p. 109)</ref>
 
:<math> (\mathbf{b} - \boldsymbol{\beta})^\prime \mathbf{Q}^\prime\mathbf{Q}(\mathbf{b} - \boldsymbol{\beta}) = {\frac{p}{n - p}} (\mathbf{Z}^\prime\mathbf{Z}
- \mathbf{b}^\prime\mathbf{Q}^\prime\mathbf{Z})F_{1 - \alpha}(p,n-p).
</math>
 
Here ''F'' represents the percentage point of the [[F distribution]] and the quantities ''p'' and ''n-p'' are the [[Degrees of freedom (statistics)|degrees of freedom]] which are the parameters of this distribution.
 
==Nonlinear problems==
 
Confidence regions can be defined for any probability distribution.  The experimenter can choose the significance level and the shape of the region, and then the size of the region is determined by the probability distribution. A natural choice is to use as a boundary a set of points with constant <math>\chi^2</math> ([[Chi-squared distribution|chi-squared]]) values.
 
One approach is to use a linear approximation to the nonlinear model, which may be a close approximation in the vicinity of the solution, and then apply the analysis for a linear problem to find an approximate confidence region. This may be a reasonable approach if the confidence region is not very large and the second derivatives of the model are also not very large.
 
See [[Uncertainty Quantification#Methodologies for forward uncertainty propagation]] for related concepts.
 
==See also==
 
*[[Linear regression]]
*[[Confidence band]]
 
==Notes==
{{reflist}}
 
{{More footnotes|date=September 2011}}
 
==References==
*{{cite book |title=Applied Regression Analysis |last=Draper |first=N.R. |authorlink= |coauthors=H. Smith |year=1981 |origyear=1966 |publisher=John Wiley and Sons Ltd |location=USA |isbn=0-471-02995-5 |pages= |edition=2nd}}
*{{cite book |title=Numerical Recipes in C: The Art of Scientific Computing |last=Press |first=W.H. |authorlink= |coauthors=S.A. Teukolsky, W.T. Vetterling, B.P. Flannery |year=1992 |origyear=1988 |publisher=Cambridge University Press |location=Cambridge UK |isbn= |pages= |edition=2nd}}
 
== External links ==
 
 
[[Category:Estimation theory]]
[[Category:Statistical inference]]

Latest revision as of 21:25, 30 August 2014

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