formulasearchengine - User contributions [en]

Alpha beta filter

2014-01-13T17:03:05Z

199.46.200.232: /* The alpha beta gamma extension */

A '''confidence band''' is used in [[statistics|statistical analysis]] to represent the uncertainty in an estimate of a curve or function based on limited or noisy data. Similarly, a '''prediction band''' is used to represent the uncertainty about the value of a new data-point on the curve, but subject to noise. Confidence and prediction bands are often used as part of the graphical presentation of results of a [[regression analysis]].

Confidence bands are closely related to [[confidence intervals]], which represent the uncertainty in an estimate of a single numerical value. "As confidence intervals, by construction, only refer to a single point, they are narrower (at this point) than a confidence band which is supposed to hold simultaneously at many points."<ref>p.65 in W. Härdle, M. Müller, S. Sperlich, A. Werwatz (2004), Nonparametric and Semiparametric Models, Springer, ISBN 3540207228 [http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/ebooks/html/spm/spmhtmlnode17.html],[http://books.google.com/books?id=qPCmAOS-CoMC&lpg=PA65&vq=As%20confidence%20intervals%2C%20by%20construction%2C%20only%20refer%20to%20a%20single%20point%2C%20they%20are%20narrower%20(at%20this%20point)%20than%20a%20confidence%20band%20which%20is%20supposed%20to%20hold%20simultaneously%20at%20many%20points&pg=PA65#v=snippet&q=As%20confidence%20intervals,%20by%20construction,%20only%20refer%20to%20a%20single%20point,%20they%20are%20narrower%20(at%20this%20point)%20than%20a%20confidence%20band%20which%20is%20supposed%20to%20hold%20simultaneously%20at%20many%20points&f=false]</ref>

==Pointwise and simultaneous confidence bands==

Suppose our aim is to estimate a function ''f(x)''. For example, ''f(x)'' might be the proportion of people of a particular age ''x'' who support a given candidate in an election. If ''x'' is measured at the precision of a single year, we can construct a separate 95% confidence interval for each age. Each of these confidence intervals covers the corresponding true value ''f(x)'' with probability 0.95. Taken together, these confidence intervals constitute a ''95% pointwise confidence band'' for ''f(x)''.

In mathematical terms, a pointwise confidence band <math>\hat{f}(x)\pm w(x)</math> with coverage probability 1−α satisfies the following condition separately for each value of ''x'':

:<math>
{\rm Pr}\Big(\hat{f}(x)-w(x) \le f(x) \le \hat{f}(x)+w(x)\Big) = 1-\alpha,
</math>

where <math>\hat{f}(x)</math> is the point estimate of ''f''(''x'').

The ''simultaneous coverage probability'' of a collection of confidence intervals is the probability that all of them cover their corresponding true values simultaneously. In the example above, the simultaneous coverage probability is the probability that the intervals for x=18,19,... all cover their true values (assuming that 18 is the youngest age at which a person can vote). If each interval individually has coverage probability 0.95, the simultaneous coverage probability is generally less than 0.95. A ''95% simultaneous confidence band'' is a collection of confidence intervals for all values ''x'' in the domain of ''f(x)'' that is constructed to have simultaneous coverage probability 0.95.

In mathematical terms, a simultaneous confidence band <math>\hat{f}(x)\pm w(x)</math> with coverage probability 1−α satisfies the following condition:

:<math>
{\rm Pr}\Big(\hat{f}(x)-w(x) \le f(x) \le \hat{f}(x)+w(x) \;\;\;\; \text{ for all } x\Big) = 1-\alpha.
</math>

In nearly all cases, a simultaneous confidence band will be wider than a pointwise confidence band with the same coverage probability.

[[Image:binomial confidence band.svg|thumb|325px|Confidence bands for simulated data depicting the proportion of voters supporting a given candidate in election, as a function of the voters' ages. Pointwise 95% confidence bands, and simultaneous 95% confidence bands constructed using the [[Bonferroni method]] are shown.]]

==Confidence bands in regression analysis==

Confidence bands commonly arise in [[regression analysis]].<ref>{{cite journal |last= Liu|first= W|coauthors=Lin S., Piegorsch W.W.|year=2008| title= Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model|journal= International Statistical Review|volume=76| issue=1|pages=39–57|doi=10.1111/j.1751-5823.2007.00027.x }}</ref> In the case of a simple regression involving a single independent variable, results can be presented in the form of a plot showing the estimated regression line along with either point-wise or simultaneous confidence bands. Commonly used methods for constructing simultaneous confidence bands in regression are the [[Bonferroni method|Bonferroni]] and [[Scheffé's method|Scheffé]] methods; see [[Familywise error rate#Controlling procedures|Familywise error rate controlling procedures]] for more.

[[Image:Regression confidence band.svg|thumb|325px|Confidence bands for a simple linear regression analysis using simulated data. Pointwise 95% confidence bands, and simultaneous 95% confidence bands constructed using [[Scheffé's method]] are shown.]]

==Other applications of confidence bands==

Confidence bands arise whenever a statistical analysis focuses on estimating a function.

For example, confidence bands can be constructed around estimates of the [[empirical distribution function]]. Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the [[Kolmogorov-Smirnov test#Setting confidence limits for the shape of a distribution function|Kolmogorov-Smirnov test]], or by using non-parametric likelihood methods
.<ref>{{cite journal |last=Owen |first= A.B. |year= 1995 |title=Nonparametric likelihood confidence bands for a distribution function |journal= Journal of the American Statistical Association|volume= 90|pages= 516–521| jstor=2291062 |doi=10.2307/2291062 |issue=430 |publisher=American Statistical Association}}</ref>

Confidence bands have also been devised for estimates of [[probability density function|density functions]], [[spectral density]] functions<ref>{{cite journal |last=Neumann | first=M.H.|coauthors=Paparoditis, E.|year=2008 |title=Simultaneous confidence bands in spectral density estimation|journal=Biometrika|volume=95|issue=2| doi=10.1093/biomet/asn005 |pages=381 }}</ref>
, [[quantile]] functions, [[smoothing|scatterplot smooths]], [[survival function]]s, and [[characteristic function (probability theory)|characteristic functions]]{{citation needed|date=April 2013}}.

==Prediction bands==

Prediction bands are related to [[prediction interval]]s in the same way that confidence bands are related to confidence intervals. Prediction bands commonly arise in regression analysis. The goal of a prediction band is to cover with a prescribed probability the values of one or more future observations from the same population from which a given data set was sampled. Just as prediction intervals are wider than confidence intervals, prediction bands will be wider than confidence bands.

In mathematical terms, a prediction band <math>\hat{f}(x)\pm w(x)</math> with coverage probability 1−α satisfies the following condition for each value of ''x'':

:<math>
{\rm Pr}\Big(\hat{f}(x)-w(x) \le y^* \le \hat{f}(x)+w(x)\Big) = 1-\alpha,
</math>

where ''y''* is an observation taken from the data-generating process at the given point ''x'' that is independent of the data used to construct the point estimate <math>\hat{f}(x)</math> and the confidence interval ''w(x)''. This is a pointwise prediction interval. It would be possible to construct a simultaneous interval for a finite number of independent observations using, for example, the Bonferroni method to widen the interval by an appropriate amount.

== References ==
{{reflist}}

{{Statistics}}

[[Category:Statistical inference]]

Ruze's Equation

2013-10-03T01:04:50Z

199.46.200.231: /* Application to phased array */

In [[mathematics]], a '''Bose–Mesner algebra''' is a special set of [[Matrix (mathematics)|matrices]] which arise from a combinatorial structure known as an [[association scheme]], together with the usual set of rules for combining (forming the products of) those matrices, such that they form an [[associative algebra]], or, more precisely, a [[Unital algebra|unitary commutative algebra]]. Among these rules are:
:*the result of a product is also within the set of matrices,
:*there is an identity matrix in the set, and
:*such that taking products is [[Commutativity|commutative]].

Bose–Mesner algebras have applications in [[physics]] to [[spin model]]s, and in [[statistics]] to the [[design of experiments]]. They are named for [[R. C. Bose]] and Dale Marsh Mesner.<ref>Bose & Mesner (1959)</ref>

==Definition==
Let ''X'' be a set of ''v'' elements. Consider a partition of the 2-element subsets of ''X'' into ''n'' non-empty subsets, ''R''1, ..., ''R''''n'' such that:
* given an <math>x \in X</math>, the number of <math>y \in X</math> such that <math>\{x,y\} \in R_i</math> depends only on i (and not on ''x''). This number will be denoted by vi, and
* given <math>x,y \in X</math> with <math>\{x,y\} \in R_k</math>, the number of <math>z \in X</math> such that <math>\{x,z\} \in R_i</math> and <math>\{z,y\} \in R_j</math> depends only on ''i'',''j'' and ''k'' (and not on ''x'' and ''y''). This number will be denoted by <math>p^k_{ij}</math>.
This structure is enhanced by adding all pairs of repeated elements of ''X'' and collecting them in a subset ''R''0. This enhancement permits the parameters ''i'', ''j'', and ''k'' to take on the value of zero, and lets some of ''x'',''y'' or ''z'' be equal.

A set with such an enhanced partition is called an [[Association scheme]].<ref>{{harvnb|Cameron|van Lint|1991|loc=pp.197–198}}</ref> One may view an association scheme as a partition of the edges of a [[complete graph]] (with vertex set ''X'') into n classes, often thought of as color classes. In this representation, there is a loop at each vertex and all the loops receive the same 0th color.

The association scheme can also be represented algebraically. Consider the [[Matrix (mathematics)|matrices]] ''D''''i'' defined by:
: <math>(D_i)_{x,y} = \begin{cases}
1,& \text{if } \left(x,y\right)\in R_{i},\\
0,& \text{otherwise.} \end{cases} \qquad (1)</math>

Let <math>\mathcal{A}</math> be the [[vector space]] consisting of all [[Matrix (mathematics)|matrices]] <math>\sideset{}{_{i=0}^{n}}\sum a_{i}D_{i}</math>, with <math>a_{i}</math> complex.<ref>{{harvnb|Camion|1998}}</ref><ref>{{harvnb|Delsarte|Levenshtein|1998}}</ref>

The definition of an [[association scheme]] is equivalent to saying that the <math>D_{i}</math> are ''v'' × ''v'' (0,1)-[[Matrix (mathematics)|matrices]] which satisfy

# <math>D_i</math> is symmetric,
# <math>\sum_{i=0}^n D_{i}=J </math> (the all-ones matrix),
# <math>D_0=I,</math>
# <math>D_i D_j = \sum_{k=0}^n p^k_{ij} D_k = D_j D_i,\qquad i,j=0,\ldots,n.</math>

The (''x'',''y'')-th entry of the left side of 4. is the number of two colored paths of length two joining ''x'' and ''y'' (using "colors" ''i'' and ''j'') in the graph. Note that the rows and columns of <math>D_i</math> contain <math>v_i</math> 1s:

: <math>D_i J=J D_i = v_i J. \qquad (2)</math>

From 1., these [[Matrix (mathematics)|matrices]] are [[Symmetric matrix|symmetric]]. From 2., <math>D_{0},\ldots,D_{n}</math> are [[Linear independence|linearly independent]], and the dimension of <math>\mathcal{A}</math> is <math>n+1</math>. From 4., <math>\mathcal{A}</math> is closed under multiplication, and multiplication is always associative. This [[Associative algebra|associative]] [[commutative algebra]] <math>\mathcal{A}</math> is called the '''Bose–Mesner algebra''' of the [[association scheme]]. Since the [[Matrix (mathematics)|matrices]] in <math>\mathcal{A}</math> are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a [[Matrix (mathematics)|matrix]] <math>S</math> such that to each <math>A\in\mathcal{A}</math> there is a [[diagonal matrix]] <math>\Lambda_{A}</math> with <math>S^{-1}A S=\Lambda_{A}</math>. This means that <math>\mathcal{A}</math> is semi-simple and has a unique basis of primitive idempotents <math>J_{0},\ldots,J_{n}</math>. These are complex n × n [[Matrix (mathematics)|matrices]] satisfying

: <math>
J_i^2 =J_i, i=0,\ldots,n, \qquad (3)
</math>

: <math>
J_i J_k=0, i\neq k, \qquad (4)
</math>

: <math>
\sum_{i=0}^n J_i = I. \qquad (5)
</math>

The '''Bose–Mesner algebra''' has two distinguished bases: the basis consisting of the [[Adjacency matrix|adjacency matrices]] <math>D_i</math>, and the basis consisting of the irreducible [[Idempotent matrix|idempotent matrices]] <math>E_k</math>. By definition, there exist well-defined [[complex number]]s such that

: <math>
D_{i}=\sum_{k=0}^n p_i (k) E_k, \qquad (6)
</math>

and

: <math>
|X|E_{k}=\sum_{i=0}^n q_k\left(i\right)D_i. \qquad (7)
</math>

The p-numbers <math>p_i (k)</math>, and the q-numbers <math>q_k(i)</math>, play a prominent role in the theory.<ref>{{harvnb|Camion|1998}}</ref> They satisfy well-defined orthogonality relations. The p-numbers are the [[eigenvalues]] of the [[adjacency matrix]] <math>D_i</math>.

==Theorem==

The [[eigenvalues]] of <math>p_{i}\left(k\right)</math> and <math>q_{k}\left(i\right)</math>, satisfy the orthogonality conditions:

: <math>
\sum_{k=0}^n \mu_i p_i (k)p_\ell (k)=v v_i \delta_{i \ell}, \quad(8)
</math>

: <math>
\sum_{k=0}^n \mu_i q_k (i) q_\ell (i)=v \mu_k \delta_{k \ell}. \quad(9)
</math>

Also

: <math>
\mu_j p_i (j) = v_i q_ j (i),\quad i,j=0,\ldots,n. \quad(10)
</math>

In [[Matrix (mathematics)|matrix]] notation, these are

: <math>
P^T \Delta_\mu P=v\Delta_v, \quad(11)
</math>

: <math>
Q^T \Delta_v Q=v\Delta_\mu, \quad(12)
</math>

where <math>\Delta_v = \operatorname{diag} \{v_0,v_1,\ldots,v_n\},\qquad \Delta_\mu = \operatorname{diag} \{\mu_0,\mu_1,\ldots,\mu_n\}.</math>

==Proof of theorem==

The [[eigenvalue]]s of <math>D_i D_\ell</math> are <math>p_i (k)p_\ell (k)</math> with multiplicities <math>\mu_k</math>. This implies that

: <math>
v v_i \delta_{i\ell} = \operatorname{trace}D_i D_\ell = \sum_{k=0}^n \mu_i p_i(k) p_\ell (k), \quad(13)
</math>

which proves Equation <math>\left(8\right)</math> and Equation <math>\left(11\right)</math>,

: <math>
Q = v P^{-1} = \Delta_v^{-1} P^T \Delta_\mu, \quad(14)
</math>

which gives Equations <math>(9)</math>, <math>(10)</math> and <math>(12)</math>.<math>\Box</math>

There is an analogy between extensions of [[association scheme]]s and [[Kronecker's theorem|extensions]] of [[finite field]]s. The cases we are most interested in are those where the extended schemes are defined on the <math>n</math>-th [[Cartesian power]] <math>X=\mathcal{F}^{n}</math> of a set <math>\mathcal{F}</math> on which a basic [[association scheme]] <math>\left(\mathcal{F},K\right)</math> is defined. A first [[association scheme]] defined on <math>X=\mathcal{F}^{n}</math> is called the <math>n</math>-th [[Kronecker product|Kronecker power]] <math>\left(\mathcal{F},K\right)_{\otimes}^{n}</math> of <math>\left(\mathcal{F},K\right)</math>. Next the extension is defined on the same set <math>X=\mathcal{F}^{n}</math> by gathering classes of <math>\left(\mathcal{F},K\right)_{\otimes}^{n}</math>. The [[Kronecker product|Kronecker power]] corresponds to the [[polynomial ring]] <math>F\left[X\right]</math> first defined on a [[Finite field|field]] <math>\mathbb{F}</math>, while the extension scheme corresponds to the [[extension field]] obtained as a quotient. An example of such an extended scheme is the [[Hamming scheme]].

[[Association scheme]]s may be merged, but merging them leads to non-symmetric [[association scheme]]s, whereas all usual [[code]]s are [[subgroup]]s in symmetric [[Abelian variety|Abelian schemes]].<ref>{{harvnb|Delsarte|Levenshtein|1998}}</ref><ref>{{harvnb|Camion|1998}}</ref><ref>{{harvnb|MacWilliams|Sloane|1978}}</ref>

==See also==
* [[Association scheme]]

{{More footnotes|date=September 2010}}

==Notes==

{{Reflist}}

==References==

* {{Cite book|first=Rosemary A.|last=Bailey|authorlink=Rosemary A. Bailey|url=http://www.maths.qmul.ac.uk/~rab/Asbook |title=Association schemes: Designed experiments, algebra and combinatorics|series=Cambridge Studies in Advanced Mathematics|volume=84|publisher=Cambridge University Press|year=2004|pages=387|isbn=978-0-521-82446-0| mr=2047311|ref=harv}}

* {{cite book | last1=Bannai | first1=Eiichi | last2=Ito | first2=Tatsuro | title=Algebraic combinatorics I: Association schemes | publisher=The Benjamin/Cummings Publishing Co., Inc. | location=Menlo Park, CA | year=1984 | pages=xxiv+425 | isbn=0-8053-0490-8 | mr=0882540 | unused_data= }}

* Bannai, Etsuko (2001) "Bose–Mesner algebras associated with four-weight spin models", ''Graphs and Combinatorics'', 17 (4),589–598. {{doi|10.1007/PL00007251}}

* {{citation| last1=Bose|first1=R. C.| authorlink1=R. C. Bose| last2=Mesner|first2=D. M.|year=1959|title=On linear associative algebras corresponding to association schemes of partially balanced designs|journal=[[Annals of Mathematical Statistics]]|volume=30|issue=1|pages=21–38| url=http://projecteuclid.org/euclid.aoms/1177706356 | doi=10.1214/aoms/1177706356 | mr = 102157 | jstor = 2237117}}

* {{citation|last=Cameron|first=P. J.|last2=van Lint|first2=J. H.|title=Designs, Graphs, Codes and their Links|year=1991|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-42385-6}}

* {{citation|last1= Camion|first1=P.|authorlink1=Paul Camion|chapter=Codes and association schemes: Basic properties of association schemes relevant to coding| title=Handbook of coding theory|editor1-last= Pless|editor1-first=V. S.|editor1-link=Vera Pless|editor2-last=Huffman|editor2-first=W. C.|publisher=Elsevier|place= The Netherlands|year= 1998}}

* {{citation|last1=Delsarte|first1=P.|last2=Levenshtein|first2=V. I.|authorlink2=Vladimir Levenshtein|title=Association schemes and coding theory|journal=IEEE Transactions in Information Theory| volume= 44| issue= 6|pages= 2477–2504|year= 1998}}

* {{citation| first1=F. J.|last1= MacWilliams|first2=N. J. A.|last2= Sloane|authorlink2=Neil J. A. Sloane|title=The theory of error-correcting codes|publisher= Elsevier|place= New York|year= 1978}}

* Nomura, K. (1997) "An algebra associated with a spin model", ''Journal of Algebraic Combinatorics'', 6 (1), 53–58. {{DOI|10.1023/A:1008644201287}}

{{Experimental design|state=expanded}}

{{DEFAULTSORT:Bose-Mesner Algebra}}
[[Category:Algebraic combinatorics]]
[[Category:Design of experiments]]
[[Category:Analysis of variance]]
[[Category:Representation theory]]
[[Category:Algebra]]