Confidence region

From Wikipedia, the free encyclopedia

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 any shape can occur.

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, 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.

Contents

[edit] The case of independent, identically normally-distributed errors

Suppose we have found a solution \boldsymbol{\beta} to the following overdetermined problem:

\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}

where Y is an n-dimensaional 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, \boldsymbol{\beta} is a column vector containing the p parameters which are to be estimated, and \boldsymbol{\varepsilon} is an n-dimensional column vector of errors which are assumed to be independently distributed with normal distributions with zero mean and each having the same the same unknown variance σ2.

According to Draper and Smith, a joint (100 - α) % confidence region for the elements of \boldsymbol{\beta} is represented by the set of values of the vector b which satisfy the following inequality:

(\boldsymbol{\beta} - \mathbf{b})^\prime \mathbf{X}^\prime\mathbf{X}(\boldsymbol{\beta} - \mathbf{b}) \le ps^2 F(p,\nu,1 - \alpha)

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 \boldsymbol{\beta}, s2 is an unbiassed estimate of σ2 equal to \frac{\varepsilon^\prime\varepsilon}{n - p}, F is the F-distribution, ν = n − p, and α is the statistical significance level, and the symbol X^\prime means the transpose of X.

The above inequality defines an ellipsoidal region in the p-dimensional Cartesian parameter space Rp. The centre of the ellipsoid is at the solution \boldsymbol\beta. 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.

[edit] Weighted least squares

Now let us consider the more general case where some distinct elements of \boldsymbol{\varepsilon} 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 \boldsymbol{\varepsilon} is \mathbf{V}\sigma^2, where V is an n-by-n nonsingular matrix which was equal to \mathbf{I}\sigma^2 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.

According to Draper and Smith (p. 108), it is possible to find a nonsingular symmetric matrix P such that

\mathbf{P}^\prime\mathbf{P} = \mathbf{P}\mathbf{P} = \mathbf{V}

In effect, P is a square root of the covariance matrix V.

The least-squares problem

\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}

can then be transformed by left-multiplying each term by the inverse of P, forming the new problem formulation

\mathbf{Z} = \mathbf{Q}\boldsymbol{\beta} + \mathbf{f}

where

\mathbf{Z} = \mathbf{P}^{-1}\mathbf{Y}
\mathbf{Q} = \mathbf{P}^{-1}\mathbf{X} and
\mathbf{f} = \mathbf{P}^{-1}\boldsymbol{\varepsilon}

A joint confidence region for the parameters, i.e. for the elements of \boldsymbol{\beta}, according to Draper and Smith p. 109 is then bounded by the ellipsoid given by:

(\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(p,n-p,1-\alpha)

[edit] 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 χ2 (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.

[edit] See also

[edit] References

  • Draper, N.R.; H. Smith [1966] (1981). Applied Regression Analysis, 2nd, USA: John Wiley and Sons Ltd. ISBN 0471029955. 
  • Press, W.H.; S.A. Teukolsky, W.T. Vetterling, B.P. Flannery [1988] (1992). Numerical Recipes in C: The Art of Scientific Computing, 2nd, Cambridge UK: Cambridge University Press.