formulasearchengine - User contributions [en]

Compound Poisson process

2014-11-08T18:01:17Z

157.27.130.133: /* Exponentiation of measures */

54 yrs old Clothing Patternmaker Albert from Cookshire, loves to spend some time creating model cars, diet and poole pottery. Last month just arrived at Sapi and Chewore Safari Areas.

Index of dissimilarity

2013-05-09T14:05:19Z

157.27.190.15:

[[Image:Eccentricity.svg|thumb|right|All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.]]
In [[mathematics]], the '''eccentricity''', denoted ''e'' or <math>\varepsilon</math>, is a [[parameter]] associated with every [[Conic section#Eccentricity|conic section]]. It can be thought of as a measure of how much the conic section deviates from being circular.

In particular,
*The eccentricity of a [[circle]] is zero.[[Image:Cubic surface.gif|thumb|right|Ellipses, hyperbolas with all possible eccentricites from zero to infinity and a parabola on one cubic surface.]]
*The eccentricity of an [[ellipse]] which is not a circle is greater than zero but less than 1.
*The eccentricity of a [[parabola]] is 1.
*The eccentricity of a [[hyperbola]] is greater than 1.

Furthermore, two conic sections are [[similarity (geometry)|similar]] if and only if they have the same eccentricity.

==Definitions==
Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called eccentricity, commonly denoted as ''e''.

The eccentricity can also be defined in terms of the intersection of a plane and a [[Cone (geometry)|double-napped cone]] associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is

<math> e = \frac{\sin \alpha}{\sin \beta} </math>

where α is the angle between the plane and the horizontal and β is the angle between the cone's slant generator and the horizontal.

The '''linear eccentricity''' of a conic section, denoted ''c'' (or sometimes ''f'' or ''e''), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the [[semimajor axis]] ''a'': that is, <math> e = \frac{c}{a} </math>.

==Alternative names==
The eccentricity is sometimes called '''first eccentricity''' to distinguish it from the '''second eccentricity''' and '''third eccentricity''' defined for ellipses (see below). The eccentricity is also sometimes called '''numerical eccentricity'''.

In the case of ellipses and hyperbolas the linear eccentricity is sometimes called '''half-focal separation'''.

==Notation==
Three notational conventions are in common use:
#''e'' for the eccentricity and ''c'' for the linear eccentricity.
#<math>\varepsilon</math> for the eccentricity and ''e'' for the linear eccentricity.
#''e'' or <math>\epsilon</math> for the eccentricity and ''f'' for the linear eccentricity (mnemonic for half-''f''ocal separation).
This article makes use of the first notation.

==Values==
{| class="wikitable"
! conic section !! equation !! eccentricity (''e'') !! linear eccentricity (''c'')
|-
| [[Circle]] || <math>x^2+y^2=r^2</math> || <math>0</math> || <math>0</math>
|-
| [[Ellipse]] || <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math> || <math>\sqrt{1-\frac{b^2}{a^2}}</math> || <math>\sqrt{a^2-b^2}</math>
|-
| [[Parabola]] || <math>y^2=4ax</math> || <math>1</math> || <math>a</math>
|-
| [[Hyperbola]] || <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1</math> || <math>\sqrt{1+\frac{b^2}{a^2}}</math> || <math>\sqrt{a^2+b^2}</math>
|}

where, when applicable, ''a'' is the length of the semi-major axis and ''b'' is the length of the semi-minor axis.

When the conic section is given in the general quadratic form

:<math>Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0,</math>

the following formula gives the eccentricity ''e'' if the conic section is not a parabola (which has eccentricity equal to 1), not a [[Degenerate conic|degenerate hyperbola or degenerate ellipse]], and not an imaginary ellipse:<ref>Ayoub, Ayoub B., "The eccentricity of a conic section", ''[[The College Mathematics Journal]]'' 34(2), March 2003, 116-121.</ref>

:<math>e=\sqrt{\frac{2\sqrt{(A-C)^2 + B^2}}{\eta (A+C) + \sqrt{(A-C)^2 + B^2}}}</math>

where <math>\eta = 1</math> if the determinant of the 3×3 matrix

:<math>\begin{bmatrix}A & B/2 & D/2\\B/2 & C & E/2\\D/2&E/2&F\end{bmatrix}</math>

is negative or <math>\eta = -1</math> if that determinant is positive.

[[File:Ellipse and hyperbola.gif|thumb|250px|Ellipse and hyperbola with constant ''a'' and changing eccentricity ''e''.]]

==Ellipses==

The eccentricity of an ellipse is strictly less than 1. When circles are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0.

For any ellipse, let ''a'' be the length of its [[semi-major axis]] and ''b'' be the length of its [[semi-minor axis]].

We define a number of related additional concepts (only for ellipses):

{| class="wikitable"
! name !! symbol !! in terms of ''a'' and ''b'' !! in terms of ''e''
|-
|   '''first eccentricity''' ||   <math>e</math> ||   <math>\sqrt{1-\frac{b^2}{a^2}}</math> ||   <math>e</math>
|-
|   '''second eccentricity''' ||   <math>e'</math> ||   <math>\sqrt{\frac{a^2}{b^2}-1}</math> ||   <math>\frac{e}{\sqrt{1-e^2}}</math>
|-
|   '''third eccentricity''' ||   <math>e''=\sqrt m</math> ||  <math>\frac{\sqrt{a^2-b^2}}{\sqrt{a^2+b^2}}</math> ||   <math> \frac{e}{\sqrt{2-e^2}} </math>
|-
|   '''[[angular eccentricity]]''' ||   <math>\alpha</math> ||  <math>\cos^{-1}\left(\frac{b}{a}\right)</math> ||   <math>\sin^{-1} e</math>
|}

===Other formulas for the eccentricity of an ellipse===

The eccentricity of an ellipse is, most simply, the ratio of half the distance between its two foci, to the length of the semimajor axis.

The eccentricity is also the ratio of the semimajor axis ''a'' to the distance ''d'' from the center to the directrix:

:<math>e = \frac{a}{d}.</math>

The eccentricity can be expressed in terms of the [[flattening factor]] ''g'' (defined as ''g'' = 1 – ''b''/''a'' for semimajor axis ''a'' and semiminor axis ''b''):
:<math>e = \sqrt{g(2-g)}.</math>
Comment: flattening is denoted by ''f'' in some subject areas, particularly geodesy.

Define the maximum and minimum radii <math>r_\text{max}</math> and <math>r_\text{min}</math> as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis ''a'', the eccentricity is given by

:<math>e = \frac{r_\text{max}-r_\text{min}}{r_\text{max}+r_\text{min}} = \frac{r_\text{max}-r_\text{min}}{2a}.</math>

==Hyperbolas==

The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a [[rectangular hyperbola]] is <math>\sqrt{2}</math>.

==Quadrics==
The eccentricity of a three-dimensional [[quadric]] is the eccentricity of a designated [[Cross section (geometry)|section]] of it. For example, on a triaxial ellipsoid, the ''meridional eccentricity'' is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the ''equatorial eccentricity'' is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

==Celestial mechanics==
In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the [[apocenter]] distance is close to the [[pericenter]] distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipses, in Keplerian, i.e., <math>1/r</math> potentials.

== Analogous classifications ==
{{Expand section|date=March 2009}}
A number of classifications in mathematics use derived terminology from the classification of conic sections by eccentricity:
*[[SL2(R)#Classification_of_elements|Classification of elements]] of [[SL2(R)|SL2(R)]] as elliptic, parabolic, and hyperbolic – and similarly for [[Möbius transformation#Classification|classification of elements]] of PSL2(R), the real [[Möbius transformation]]s.
*Classification of discrete distributions by [[variance-to-mean ratio]]; see [[Cumulant#Cumulants_of_some_discrete_probability_distributions|cumulants of some discrete probability distributions]] for details.
*Classification of [[partial differential equations]] is by analogy with the conic sections classification; see [[Elliptic partial differential equation|elliptic]], [[Parabolic partial differential equation|parabolic]] and [[Hyperbolic partial differential equation|hyperbolic]] partial differential equations.<ref name="ornl.gov">{{cite web | url=http://www.phy.ornl.gov/csep/pde/node3.html | title=Classification of Linear PDEs in Two Independent Variables | accessdate=2 July 2013}}</ref>

==See also==
*[[Kepler orbit]]s
*[[Eccentricity vector]]
*[[Orbital eccentricity]]
*[[Roundness (object)]]

==References==
{{reflist}}

==External links==
{{commons category|Eccentricity}}
*[http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity]

{{orbits}}

{{DEFAULTSORT:Eccentricity (Mathematics)}}
[[Category:Conic sections]]
[[Category:Analytic geometry]]

Hydroxyacylglutathione hydrolase

2011-06-08T16:50:27Z

157.27.81.108:

'''Bayes linear statistics''' is a subjectivist statistical methodology and framework. Traditional subjective Bayesian analysis is based upon fully specified probability distributions, which are very difficult to specify at the necessary level of detail. Bayes linear analysis attempts to solve this problem by developing theory and practise for using partially specified probability models. Bayes linear in its current form has been primarily developed by Michael Goldstein. Mathematically and philosophically it extends [[Bruno de Finetti]]'s [[Operational Subjective]] approach to probability and statistics.

Consider first a traditional Bayesian Analysis where you expect to shortly know ''D'' and you would like to know more about some other observable ''B''. In the traditional Bayesian approach it is required that every possible outcome is enumerated i.e. every possible outcome is the cross product of the [[partition of a set]] of ''B'' and ''D''. If represented on a computer where ''B'' requires ''n'' bits and ''D'' ''m'' bits then the number of states required is ''2n+m''. The first step to such an analysis is to determine a persons subjective probabilities e.g. by asking about their betting behaviour for each of these outcomes. When we learn ''D'' conditional probabilities for ''B'' are determined by the application of Bayes' rule.

Practitioners of subjective Bayesian statistics routinely analyse datasets where the size of this set is large enough that subjective probabilities cannot be meaningfully determined for every element of ''D × B''. This is normally accomplished by assuming [[exchangeability]] and then the use of parameterized models with prior distributions over parameters and appealing to the [[de Finetti's theorem]] to justify that this produces valid operational subjective probabilities over ''D × B''. The difficulty with such an approach is that the
validity of the statistical analysis requires that the subjective probabilities are a good representation of an individual's beliefs however this method results in a very precise specification over ''D × B'' and it is often difficult to articulate what it would mean to adopt these belief specifications.

In contrast to the traditional Bayesian paradigm Bayes linear statistics following de Finetti uses [[Prevision]] or subjective expectation as a primitive, probability is then defined as the expectation of an indicator variable. Instead of specifying a subjective probability for every element in the partition ''D × B'' the analyst specifies subjective expectations for just a few quantities that they are interested in or feel knowledgeable about. Then instead of conditioning an adjusted expectation is computed by a rule that is a generalization of Bayes' rule that is based upon expectation.

The use of the word linear in the title refers to de Finetti's arguments that probability theory is a linear theory (de Finetti argued against the more common measure theory approach).

==Example==
In Bayes linear statistics, the probability model is only partially specified, and it is not possible to calculate conditional probability by Bayes' rule. Instead Bayes linear suggests the calculation of an Adjusted Expectation.

To conduct a Bayes linear analysis it is necessary to identify some values that you expect to know shortly by making measurements ''D'' and some future value which you would like to know ''B''. Here ''D'' refers to a vector containing data and ''B'' to a vector containing quantities you would like to predict. For the following example ''B'' and ''D'' are taken to be two-dimensional vectors i.e.

:<math>B = (Y_1,Y_2),~ D = (X_1,X_2).</math>

In order to specify a Bayes linear model it is necessary to supply expectations for the vectors ''B'' and ''D'', and to also specify the correlation between each component of ''B'' and each component of ''D''.

For example the expectations are specified as:

: <math>E(Y_1)=5,~E(Y_2)=3,~E(X_1)=5,~E(X_2)=3</math>

and the covariance matrix is specified as :

: <math>
\begin{matrix}
& X_1 & X_2 & Y_1 & Y_2 \\
X_1 & 1 & u & \gamma & \gamma \\
X_2 & u & 1 & \gamma & \gamma \\
Y_1 & \gamma & \gamma & 1 & v \\
Y_2 & \gamma & \gamma & v & 1 \\
\end{matrix}.
</math>

The repetition in this matrix, has some interesting implications to be discussed shortly.

An adjusted expectation is a linear estimator of the form

: <math>c_0 + c_1X_1 + c_2X_2</math>

where <math>c_0, c_1</math> and <math>c_2</math> are chosen to minimise the prior expected loss for the observations i.e. <math>Y_1, Y_2</math> in this case. That is for <math>Y_1</math>

: <math>E([Y_1 - c_0 - c_1X_1 - c_2X_2]^2)\,</math>

where

: <math>c_0, c_1, c_2\,</math>

are chosen in order to minimise the prior expected loss in estimating <math>Y_1</math>

In general the adjusted expectation is calculated with

: <math>E_D(X) = \sum^k_{i=0} h_iD_i .</math>

Setting <math>h_0, \dots, h_k</math> to minimise

: <math>E\left(\left[X-\sum^k_{i=0}h_iD_i\right]^2\right). </math>

From a proof provided in (Goldstein and Wooff 2007) it can be shown that:

: <math>E_D(X) = E(X) + Cov(X,D)Var(D)^{-1}(D-E(D)) . \,</math>

For the case where Var(''D'') is not invertible the [[Moore–Penrose pseudoinverse]] should be used instead.

Furthermore, the adjusted variance of the variable <math>X</math> after observing the data <math>D</math> is given by

: <math>Var_D(X) = Var(X) - Cov(X,D)Var(D)^{-1}Cov(D,X). </math>

==See also==
*[[Imprecise probability]]

== External links==
* [http://maths.dur.ac.uk/stats/bayeslin/ Bayes Linear Methods]

== References ==

* Goldstein, M. (1981) ''Revising Previsions: a Geometric Interpretation (with Discussion)''. [[Journal of the Royal Statistical Society]], Series B, 43(2), 105-130
* Goldstein, M. (2006) ''Subjectivism principles and practice''. Bayesian Analysis][http://ba.stat.cmu.edu/journal/2006/vol01/issue03/goldstein.pdf]
* Michael Goldstein, David Wooff (2007) ''Bayes Linear Statistics, Theory & Methods'', Wiley. ISBN 978-0-470-01562-9
* de Finetti, B. (1931) "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science," (translation of 1931 article) in ''Erkenntnis,'' volume 31, September 1989. The entire double issue is devoted to de Finetti's philosophy of probability.
* de Finetti, B. (1937) “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré,
: - "Foresight: its Logical Laws, Its Subjective Sources," (translation of the [http://www.numdam.org/item?id=AIHP_1937__7_1_1_0 1937 article] in French) in H. E. Kyburg and H. E. Smokler (eds), ''Studies in Subjective Probability,'' New York: Wiley, 1964.
* de Finetti, B. (1974) ''Theory of Probability'', (translation by A Machi and [[AFM Smith]] of 1970 book) 2 volumes, New York: Wiley, 1974-5.



{{DEFAULTSORT:Bayes Linear Statistics}}
[[Category:Bayesian statistics|Linear statistics]]
[[Category:Probability interpretations]]