Ehrenfeucht–Fraïssé game: Difference between revisions

Revision as of 22:12, 1 July 2013

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.

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 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.

The case of independent, identically normally-distributed errors

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. 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-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, ${\boldsymbol {\beta }}$ 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 unknown variance $\sigma ^{2}$ .

A joint 100(1 - $\alpha$ ) % confidence region for the elements of ${\boldsymbol {\beta }}$ is represented by the set of values of the vector b which satisfy the following inequality:^[1]

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

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 ${\boldsymbol {\beta }},$ and s² is an unbiased estimate of $\sigma ^{2}$ equal to

s^{2}={\frac {\varepsilon ^{\prime }\varepsilon }{n-p}}.

Further, F is the quantile function of the F-distribution, with p and $\nu =n-p$ degrees of freedom, $\alpha$ 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 R^p. 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.

Weighted and generalised least squares

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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.

It is possible to find^[2] 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 ${\boldsymbol {\beta }}$ , is then bounded by the ellipsoid given by:^[3]

(\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).

Here F represents the percentage point of the F distribution and the quantities p and n-p are the 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 $\chi ^{2}$ (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.

Notes

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Template:More footnotes

References

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

↑ Draper and Smith (1981, p. 94)
↑ Draper and Smith (1981, p. 108)
↑ Draper and Smith (1981, p. 109)

[1] Draper and Smith (1981, p. 94)

[2] Draper and Smith (1981, p. 108)

[3] Draper and Smith (1981, p. 109)

[1]

[2]

[3]

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-On the contrary if the particular person is always underneath stress more often than not there are probabilities that the person's blood stress will remain excessive for some period and it's likely to remain in that greater value. Leading a anxious life more often than not results in high blood pressure.<br><br>This paper highlights the problems that exist in treating Anxiousness and Panic disorder in the absence of a unified mechanism for tying collectively the foundation causes and the branching symptoms. This paper argues that focusing on the one or most recent signs will do little to replicate the broader clinical reality of the situation. It's a beta blocker, which must be consumed orally twice a day for treating high blood pressure (hypertension). It is important to observe the directions on the label and speak to the physician earlier than starting the course of this medication. Angina, a form of chest ache and heart attacks are also handled with pindolol, nonetheless,  lose weight fast consult your doctor for the doable risks concerned in using this drug. Anxiousness and Panic Disorder<br><br>Still, there is not any doubt that treatment for depression is usually a big present to the heart. There are indications that depressed people who are otherwise healthy are extra probably than their non-depressed peers to develop coronary heart illness. For instance, a 13-12 months study of 1,500 topics performed at Johns Hopkins University found that an episode of despair increased the risk of a coronary heart attack more than fourfold. And researchers at Duke University Medical Heart recently reported that a stress-management program minimize the chances that a coronary  low carb diet heart patient would undergo a coronary  how to lose belly fat (simpledietburnfat.com) heart attack or need surgery by 74 p.c.<br><br>Points regarding antihypertensive therapy for masked hypertension are to regulate blood strain within the normal range over 24 h, to perform antihypertensive therapy on the idea of early morning blood pressure and to maintain it at <135 / 85 mm Hg. c. Morning hypertension Though there is no consensus on the definition of morning hypertension, a condition by which blood stress is specifically greater within the morning than at other occasions of day may be regarded as morning hypertension in a slim sense. As the criterion of hypertension primarily based on residence blood strain is one hundred thirty five / 85 mm Hg, an average blood strain early within the morning of one hundred thirty five / 85 mm Hg is outlined as morning hypertension in a broad sense. Normalize your High Blood Strain in three Weeks or Much less<br><br>To keep anxiousness away from our life you could keep away from junk foods and drinks. To get rid from anxiousness it is necessary to keep away from carbohydrates and refined sugar, caffeine, artificial sweeteners, smoking and the other things similar to these things. Nervousness could also be outlined a really uncomfortable feeling of nervousness, irritability orforeboding concerning the future. Attimes, it's connected to a particular event or scenario or particular person. At other times, there isn't any obviouscause for it. That is calledfree-floating anxiety. I've also learn somewhere that having a full bladder whereas your blood pressure is taken can give an elevated reading so it is smart that going to the bathroom would help in lowering ranges too. hypoglycaemia (lowered quantity of glucose within the blood)
+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.
+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.
+==The case of independent, identically normally-distributed errors==
+{{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>.
+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>
+:<math> (\boldsymbol{\beta} - \mathbf{b})^\prime \mathbf{X}^\prime\mathbf{X}(\boldsymbol{\beta} - \mathbf{b}) \le  ps^2 F_{1 - \alpha}(p,\nu) ,</math>
+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
+:<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>.
+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.
+==Weighted and generalised least squares==
+{{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]]

Ehrenfeucht–Fraïssé game: Difference between revisions

Revision as of 22:12, 1 July 2013

Contents

The case of independent, identically normally-distributed errors

Weighted and generalised least squares

Nonlinear problems

See also

Notes

References

External links

Navigation menu

Ehrenfeucht–Fraïssé game: Difference between revisions

Revision as of 22:12, 1 July 2013

The case of independent, identically normally-distributed errors

Weighted and generalised least squares

Nonlinear problems

See also

Notes

References

External links

Navigation menu

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