Ehrenfeucht–Fraïssé game

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

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

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 $\mathit{\beta }$ to the following overdetermined problem:

$\mathbf{Y}=\mathbf{X}\mathit{\beta }+\mathit{\epsilon }$

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, $\mathit{\beta }$ is a column vector containing the p parameters which are to be estimated, and $\mathit{\epsilon }$ 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 $\mathit{\beta }$ is represented by the set of values of the vector b which satisfy the following inequality:

$(\mathit{\beta }-\mathbf{b}{)}^{\prime }{\mathbf{X}}^{\prime }\mathbf{X}(\mathit{\beta }-\mathbf{b})\le p{s}^2{F}_{1-\alpha }(p,\nu ),$

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

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

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 $\mathit{\beta }$. 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 $\mathit{\epsilon }$ 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 $\mathit{\epsilon }$ 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 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

$\mathbf{Y}=\mathbf{X}\mathit{\beta }+\mathit{\epsilon }$

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

$\mathbf{Z}=\mathbf{Q}\mathit{\beta }+\mathbf{f},$

where

$\mathbf{Z}={\mathbf{P}}^{-1}\mathbf{Y}$

$\mathbf{Q}={\mathbf{P}}^{-1}\mathbf{X}$ and

$\mathbf{f}={\mathbf{P}}^{-1}\mathit{\epsilon }$

A joint confidence region for the parameters, i.e. for the elements of $\mathit{\beta }$, is then bounded by the ellipsoid given by:

$(\mathbf{b}-\mathit{\beta }{)}^{\prime }{\mathbf{Q}}^{\prime }\mathbf{Q}(\mathbf{b}-\mathit{\beta })=\frac{p}{n-p}({\mathbf{Z}}^{\prime }\mathbf{Z}-{\mathbf{b}}^{\prime }{\mathbf{Q}}^{\prime }\mathbf{Z})F_{1-\alpha }(p,n-p).$

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.

See also

• Linear regression
• Confidence band

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• 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
• 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