formulasearchengine - User contributions [en]

Acutance

2015-01-11T10:12:56Z

92.39.193.151: /* Definition */

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Cubic function

2015-01-10T01:26:29Z

92.39.193.151: /* General formula for roots */

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Steinhart–Hart equation

2014-12-08T19:40:44Z

92.39.197.166: /* Inverse of the equation */

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Finite field arithmetic

2013-11-09T02:15:28Z

92.39.203.117: /* Implementation Tricks */

In the field of [[computer science]], a '''binary decision diagram''' ('''BDD''') or '''branching program''', like a [[negation normal form]] (NNF) or a [[propositional directed acyclic graph]] (PDAG), is a [[data structure]] that is used to represent a [[Boolean function]]. On a more abstract level, BDDs can be considered as a [[data compression|compressed]] representation of [[set (mathematics)|sets]] or relations. Unlike other compressed representations, operations are performed directly on the compressed representation, i.e. without decompression.

== Definition ==

A Boolean function can be represented as a rooted, directed, acyclic [[graph theory|graph]], which consists of several decision nodes and terminal nodes. There are two types of terminal nodes called 0-terminal and 1-terminal. Each decision node <math>N</math> is labeled by Boolean variable <math>V_N</math> and has two [[child node]]s called low child and high child. The edge from node <math>V_N</math> to a low (or high) child represents an assignment of <math>V_N</math> to 0 (resp. 1).
Such a '''BDD''' is called 'ordered' if different variables appear in the same order on all paths from the root. A BDD is said to be 'reduced' if the following two rules have been applied to its graph:
* Merge any [[Graph isomorphism|isomorphic]] subgraphs.
* Eliminate any node whose two children are [[Graph isomorphism|isomorphic]].

In popular usage, the term '''BDD''' almost always refers to '''Reduced Ordered Binary Decision Diagram''' ('''ROBDD''' in the literature, used when the ordering and reduction aspects need to be emphasized). The advantage of an ROBDD is that it is canonical (unique) for a particular function and variable order.<ref>Graph-Based Algorithms
for Boolean Function Manipulation, Randal E. Bryant, 1986</ref> This property makes it useful in [[functional equivalence]] checking and other operations like functional technology mapping.

A path from the root node to the 1-terminal represents a (possibly partial) variable assignment for which the represented Boolean function is true. As the path descends to a low (or high) child from a node, then that node's variable is assigned to 0 (resp. 1).

=== Example ===
The left figure below shows a binary [[decision tree|decision ''tree'']] (the reduction rules are not applied), and a [[truth table]], each representing the function f (x1, x2, x3). In the tree on the left, the value of the function can be determined for a given variable assignment by following a path down the graph to a terminal. In the figures below, dotted lines represent edges to a low child, while solid lines represent edges to a high child. Therefore, to find (x1=0, x2=1, x3=1), begin at x1, traverse down the dotted line to x2 (since x1 has an assignment to 0), then down two solid lines (since x2 and x3 each have an assignment to one). This leads to the terminal 1, which is the value of f (x1=0, x2=1, x3=1).

The binary decision ''tree'' of the left figure can be transformed into a binary decision ''diagram'' by maximally reducing it according to the two reduction rules. The resulting '''BDD''' is shown in the right figure.

{| align="center"
|-
| [[File:BDD.png|thumb|546px|Binary decision tree and truth table for the function <math>f(x_1, x_2, x_3)=\bar{x_1} \bar{x_2} \bar{x_3} + x_1 x_2 + x_2 x_3</math>]]
| [[File:BDD simple.svg|thumb|189px|BDD for the function f]]
|}

== History ==
The basic idea from which the data structure was created is the [[Shannon expansion]]. A [[switching function]] is split into two sub-functions (cofactors) by assigning one variable (cf. ''if-then-else normal form''). If such a sub-function is considered as a sub-tree, it can be represented by a ''binary decision tree''. Binary decision diagrams (BDD) were introduced by Lee,<ref name="Lee">C. Y. Lee. "Representation of Switching Circuits by Binary-Decision Programs". Bell Systems Technical Journal, 38:985–999, 1959.</ref> and further studied and made known by Akers<ref name="Akers">Sheldon B. Akers. Binary Decision Diagrams, IEEE Transactions on Computers, C-27(6):509–516, June 1978.</ref> and Boute.<ref>Raymond T. Boute, "The Binary Decision Machine as a programmable controller". [[EUROMICRO]] Newsletter, Vol. 1(2):16–22, January 1976.</ref>

The full potential for efficient algorithms based on the data structure was investigated by [[Randal Bryant]] at [[Carnegie Mellon University]]: his key extensions were to use a fixed variable ordering (for canonical representation) and shared sub-graphs (for compression). Applying these two concepts results in an efficient data structure and algorithms for the representation of sets and relations.<ref name="Bryant-1986">Randal E. Bryant. "[http://www.cs.cmu.edu/~bryant/pubdir/ieeetc86.ps Graph-Based Algorithms for Boolean Function Manipulation]". IEEE Transactions on Computers, C-35(8):677–691, 1986.</ref><ref name="Bryant-1992">R. E. Bryant, "[http://www.cs.cmu.edu/~bryant/pubdir/acmcs92.ps Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams"], ACM Computing Surveys, Vol. 24, No. 3 (September, 1992), pp. 293–318.
</ref> By extending the sharing to several BDDs, i.e. one sub-graph is used by several BDDs, the data structure ''Shared Reduced Ordered Binary Decision Diagram'' is defined.<ref name="Brace">Karl S. Brace, Richard L. Rudell and Randal E. Bryant. "[http://portal.acm.org/citation.cfm?id=123222&coll=portal&dl=ACM Efficient Implementation of a BDD Package"]. In Proceedings of the 27th ACM/IEEE Design Automation Conference (DAC 1990), pages 40–45. IEEE Computer Society Press, 1990.</ref> The notion of a BDD is now generally used to refer to that particular data structure.

In his video lecture ''Fun With Binary Decision Diagrams (BDDs)'',<ref>http://scpd.stanford.edu/knuth/index.jsp</ref> [[Donald Knuth]] calls BDDs "one of the only really fundamental data structures that came out in the last twenty-five years" and mentions that Bryant's 1986 paper was for some time one of the most-cited papers in computer science.

== Applications ==

BDDs are extensively used in [[Computer Aided Design|CAD]] software to synthesize circuits ([[logic synthesis]]) and in [[formal verification]]. There are several lesser known applications of BDD, including [[Fault tree]] analysis, [[Bayesian probability|Bayesian]] Reasoning, Product Configuration, and [[Private information retrieval]] <ref name="Jensen">R.M. Jensen. [http://www.cs.cmu.edu/~runej/data/papers/JSW04.pdf "CLab: A C+ + library for fast backtrack-free interactive product configuration"]. Proceedings of the Tenth International Conference on Principles and Practice of Constraint Programming, 2004.</ref>
<ref name="Lipmaa">H.L. Lipmaa. [http://eprint.iacr.org/2009/395.pdf "First CPIR Protocol with Data-Dependent Computation"]. ICISC 2009.</ref>{{Citation needed|reason=Please provide examples of these applications in the literature.|date=June 2010}}.

Every arbitrary BDD (even if it is not reduced or ordered) can be directly implemented by replacing each node with a 2 to 1 [[Multiplexer#Digital multiplexers|multiplexer]]; each multiplexer can be directly implemented by a 4-LUT in a [[FPGA]]. It is not so simple to convert from an arbitrary network of logic gates to a BDD{{Citation needed|date=March 2008}} (unlike the [[and-inverter graph]]).

== Variable ordering ==
The size of the BDD is determined both by the function being represented and the chosen ordering of the variables. There exist Boolean functions <math>f(x_1,\ldots, x_{n})</math> for which depending upon the ordering of the variables we would end up getting a graph whose number of nodes would be linear (in ''n'') at the best and exponential at the worst case (e.g., a ripple carry adder). Let us consider the Boolean function <math>f(x_1,\ldots, x_{2n}) = x_1x_2 + x_3x_4 + \cdots + x_{2n-1}x_{2n}.</math>
Using the variable ordering <math>x_1 < x_3 < \cdots < x_{2n-1} < x_2 < x_4 < \cdots < x_{2n}</math>, the BDD needs 2''n''+1 nodes to represent the function. Using the ordering <math>x_1 < x_2 < x_3 < x_4 < \cdots < x_{2n-1} < x_{2n}</math>, the BDD consists of 2''n''+2 nodes.

{| align="center"
|-
| [[File:BDD Variable Ordering Bad.svg|thumb|638px|BDD for the function ''&fnof;''(''x''1, ..., ''x''8) = ''x''1''x''2 + ''x''3''x''4 + ''x''5''x''6 + ''x''7''x''8 using bad variable ordering]]
| [[File:BDD Variable Ordering Good.svg|thumb|156px|Good variable ordering]]
|}

It is of crucial importance to care about variable ordering when applying this data structure in practice.
The problem of finding the best variable ordering is [[NP-hard]].<ref name="Bollig">Beate Bollig, Ingo Wegener. {{doi-inline|10.1109/12.537122|Improving the Variable Ordering of OBDDs Is NP-Complete}}, IEEE Transactions on Computers, 45(9):993–1002, September 1996.
</ref> For any constant ''c'' > 1 it is even NP-hard to compute a variable ordering resulting in an OBDD with a size that is at most c times larger than an optimal one.<ref name="Sieling">Detlef Sieling. "The nonapproximability of OBDD minimization." Information and Computation 172, 103–138. 2002.
</ref> However there exist efficient heuristics to tackle the problem.<ref>{{cite web|last=Rice|first=Michael|title=A Survey of Static Variable Ordering Heuristics for Eﬃcient BDD/MDD Construction|url=http://alumni.cs.ucr.edu/~skulhari/StaticHeuristics.pdf}}</ref>

There are functions for which the graph size is always exponential — independent of variable ordering. This holds e. g. for the multiplication function (an indication{{Citation needed|date=March 2007}} as to the apparent complexity of [[factorization]] ).

Researchers have of late suggested refinements on the BDD data structure giving way to a number of related graphs, such as BMD ([[Binary Moment Diagrams]]), ZDD ([[Zero Suppressed Decision Diagram]]), FDD ([[Free Binary Decision Diagrams]]), PDD ([[Parity decision Diagrams]]), and MTBDDs (Multiple terminal BDDs).

== Logical operations on BDDs ==
Many logical operations on BDDs can be implemented by
polynomial-time graph manipulation algorithms.
* [[logical conjunction|conjunction]]
* [[logical disjunction|disjunction]]
* [[negation]]
* existential abstraction
* universal abstraction
However, repeating these operations several times, for example forming the conjunction or disjunction of a set of BDDs, may in the worst case result in an exponentially big BDD. This is because any of the preceding operations for two BDDs may result in a BDD with a size proportional to the product of the BDDs' sizes, and consequently for several BDDs the size may be exponential.

== See also ==
* [[Boolean satisfiability problem]]
* [[L/poly]], a [[complexity class]] that captures the complexity of problems with polynomially sized BDDs
* [[Model checking]]
* [[Radix tree]]
* [[Binary key]] – a method of species identification in biology using binary trees
* [[NC (complexity)#Barrington's theorem|Barrington's theorem]]

== References ==
<references/>
* R. Ubar, "Test Generation for Digital Circuits Using Alternative Graphs (in Russian)", in Proc. Tallinn Technical University, 1976, No.409, Tallinn Technical University, Tallinn, Estonia, pp. 75–81.

== Further reading ==
* D. E. Knuth, "The Art of Computer Programming Volume 4, Fascicle 1: Bitwise tricks & techniques; Binary Decision Diagrams" (Addison–Wesley Professional, March 27, 2009) viii+260pp, ISBN 0-321-58050-8. [http://www-cs-faculty.stanford.edu/~knuth/fasc1b.ps.gz Draft of Fascicle 1b] available for download.
* H. R. Andersen "[http://configit.com/configit_wordpress/wp-content/uploads/2013/07/bdd-eap.pdf An Introduction to Binary Decision Diagrams,]" Lecture Notes, 1999, IT University of Copenhagen.
* Ch. Meinel, T. Theobald, "[http://www.hpi.uni-potsdam.de/fileadmin/hpi/FG_ITS/books/OBDD-Book.pdf Algorithms and Data Structures in VLSI-Design: OBDD – Foundations and Applications"], Springer-Verlag, Berlin, Heidelberg, New York, 1998. Complete textbook available for download.
* {{cite book|author1=Rüdiger Ebendt|author2=Görschwin Fey|author3=Rolf Drechsler|title=Advanced BDD optimization|year=2005|publisher=Springer|isbn=978-0-387-25453-1}}
* {{cite book|author1=Bernd Becker|author2=Rolf Drechsler|title=Binary Decision Diagrams: Theory and Implementation|year=1998|publisher=Springer|isbn=978-1-4419-5047-5}}

== External links ==
{{Commons category|Binary decision diagrams}}
'''Available OBDD Packages'''
* [http://fmv.jku.at/abcd/ ABCD]: The ABCD package by Armin Biere, Johannes Kepler Universität, Linz.
* [http://www-2.cs.cmu.edu/~modelcheck/bdd.html CMU BDD], BDD package, Carnegie Mellon University, Pittsburgh
* [http://buddy.sourceforge.net/manual/ BuDDy]: A BDD package by Jørn Lind-Nielsen
* [http://biddy.meolic.com/ Biddy]: Academic multiplatform BDD package, University of Maribor
* [http://vlsi.colorado.edu/~fabio/CUDD/ CUDD]: BDD package, University of Colorado, Boulder
* [http://javabdd.sourceforge.net JavaBDD], a Java port of BuDDy that also interfaces to CUDD, CAL, and JDD
* [http://javaddlib.sourceforge.net/jdd/ JDD] is a pure java implementation of BDD and ZBDD. [http://javaddlib.sourceforge.net/jbdd/ JBDD] by the same author has a similar API but is a Java interface to BuDDy and CUDD
* The Berkeley [http://embedded.eecs.berkeley.edu/Research/cal_bdd/ CAL] package which does breadth-first manipulation
* [http://ddd.lip6.fr DDD]: A C++ library with support for integer valued and hierarchical decision diagrams.
* [http://www.jossowski.de/projects/jinc/jinc.html JINC]: A C++ library developed at University of Bonn, Germany, supporting several BDD variants and multi-threading.
* [http://myvideos.stanford.edu/player/slplayer.aspx?coll=ea60314a-53b3-4be2-8552-dcf190ca0c0b&co=18bcd3a8-965a-4a63-a516-a1ad74af1119&o=true Fun With Binary Decision Diagrams (BDDs)], lecture by [[Donald Knuth]]

{{Data structures}}

{{DEFAULTSORT:Binary Decision Diagram}}
[[Category:Diagrams]]
[[Category:Graph data structures]]
[[Category:Model checking]]
[[Category:Articles with example code]]
[[Category:Boolean algebra]]

Fontana bridge

2013-10-04T22:16:36Z

92.39.196.62:

{{Gravitational Lensing}}

While the presence of any mass bends the path of light passing near it, this effect rarely produces the giant arcs and multiple images associated with [[gravitational lens|strong gravitational lensing]]. Most lines of sight in the universe are thoroughly in the weak lensing regime, in which the deflection is impossible to detect in a single background source. However, even in these cases, the presence of the foreground mass can be detected, by way of a systematic alignment of background sources around the lensing mass. '''Weak gravitational lensing''' is thus an intrinsically statistical measurement, but it provides a way to measure the masses of astronomical objects without requiring assumptions about their composition or dynamical state.

== Methodology ==
[[Image:Shapenoise.svg|thumb|400px|Distortions of the type produced by lensing, acting on circles and a distribution of ellipses similar to that of real galaxies. Note that the distortion shown here is greatly exaggerated relative to real astronomical systems.]]

Gravitational lensing acts as a [[coordinate transformation]] that distorts the images of background objects (usually galaxies) near a foreground mass. The transformation can be split into two terms, the [[Gravitational Lensing Formalism#Lensing Jacobian|convergence and shear]]. The convergence term magnifies the background objects by increasing their size, and the shear term stretches them tangentially around the foreground mass.

To measure this tangential alignment, it is necessary to measure the [[flattening|ellipticities]] of the background galaxies and construct a statistical estimate of their systematic alignment. The fundamental problem is that galaxies are not intrinsically circular, so their measured ellipticity is a combination of their intrinsic ellipticity and the gravitational lensing shear. Typically, the intrinsic ellipticity is much greater than the shear (by a factor of 3-300, depending on the foreground mass). The measurements of many background galaxies must be combined to average down this "shape noise". The orientation of intrinsic ellipticities of galaxies should be almost<ref>{{cite journal | last = Hirata | first = C.M. | coauthors = Mandelbaum, R.; Ishak, M.; Seljak, U.; Nichol, R.; Pimbblet, K.A.; Ross, N.P.; Wake, D. |date=November 2007 | title = Intrinsic galaxy alignments from the 2SLAQ and SDSS surveys: luminosity and redshift scalings and implications for weak lensing surveys| journal = Monthly Notices of the Royal Astronomical Society| issue = 3 | volume = 381 | pages = 1197–1218 | bibcode = 2007MNRAS.381.1197H | accessdate = 2008-06-06 | doi = 10.1111/j.1365-2966.2007.12312.x|arxiv = astro-ph/0701671 }}</ref> entirely random, so any systematic alignment between multiple galaxies can generally be assumed to be caused by lensing.

Another major challenge for weak lensing is correction for the [[point spread function]] (PSF) due to instrumental and atmospheric effects, which causes the observed images to be smeared relative to the "true sky". This smearing tends to make small objects more round, destroying some of the information about their true ellipticity. As a further complication, the PSF typically adds a small level of ellipticity to objects in the image, which is not at all random, and can in fact mimic a true lensing signal. Even for the most modern telescopes, this effect is usually at least the same order of magnitude as the gravitational lensing shear, and is often much larger. Correcting for the PSF requires building for the telescope a model for how it varies across the field. Stars in our own galaxy provide a direct measurement of the PSF, and these can be used to construct such a model, usually by [[interpolating]] between the points where stars appear on the image. This model can then be used to reconstruct the "true" ellipticities from the smeared ones. Ground-based and space-based data typically undergo distinct reduction procedures due to the differences in instruments and observing conditions.

[[Angular diameter distance]]s to the lenses and background sources are important for converting the lensing observables to physically meaningful quantities. These distances are often estimated using [[photometric redshift]]s when [[spectroscopy|spectroscopic redshifts]] are unavailable. Redshift information is also important in separating the background source population from other galaxies in the foreground, or those associated with the mass responsible for the lensing. With no redshift information, the foreground and background populations can be split by an [[apparent magnitude]] or a [[color index|color]] cut, but this is much less accurate.

== Weak lensing by clusters of galaxies ==
[[Image:Gravitational-lensing-3d.png|thumb|left|400px|The effects of foreground galaxy cluster mass on background galaxy shapes. The upper left panel shows (projected onto the plane of the sky) the shapes of cluster members (in yellow) and background galaxies (in white), ignoring the effects of weak lensing. The lower right panel shows this same scenario, but includes the effects of lensing. The middle panel shows a 3-d representation of the positions of cluster and source galaxies, relative to the observer. Note that the background galaxies appear stretched tangentially around the cluster.]]
[[Galaxy clusters]] are among the largest [[gravitation]]ally bound structures in the [[Universe]], surpassed only by [[superclusters]], with approximately 80% of cluster content in the form of [[dark matter]].<ref>{{ cite journal | last = Diaferio | first=A. | coauthors = Schindler, S.; Dolag, K.|date=February 2008| title = Clusters of Galaxies: Setting the Stage| journal = Space Science Reviews| issue = 1–4 | volume = 134| pages = 7–24| bibcode = 2008SSRv..134....7D| accessdate=2008-06-01 | doi = 10.1007/s11214-008-9324-5}}</ref> The gravitational fields of these clusters deflect light-rays traveling near them. As seen from [[Earth]], this effect can cause dramatic distortions of a background source object detectable by eye such as multiple images, arcs, and rings (cluster strong lensing). More generally, the effect causes small, but statistically coherent, distortions of background sources on the order of 10% (cluster weak lensing). [[Abell 1689]], [[CL0024+17]], and the [[Bullet Cluster]] are among the most prominent examples of lensing clusters.

===History===
The effects of cluster strong lensing were first detected by Roger Lynds of the [[National Optical Astronomy Observatory|National Optical Astronomy Observatories]] and Vahe Petrosian of [[Stanford University]] who discovered giant luminous arcs in a survey of galaxy clusters in the late 1970s. Lynds and Petrosian published their findings in 1986 without knowing the origin of the arcs.<ref>{{cite journal | first = R.| last = Lynds | coauthors= Petrosian, V. |date=September 1986| title = Giant Luminous Arcs in Galaxy Clusters| journal=Bulletin of the American Astronomical Society|volume = 18 | page = 1014| bibcode = 1986BAAS...18R1014L| accessdate = 2008-06-01}}</ref> In 1987, Genevieve Soucail of the [[Toulouse Observatory]] and her collaborators presented data of a blue ring-like structure in Abell 370 and proposed a gravitational lensing interpretation.<ref>{{cite journal | last = Soucail | first = G. | coauthors = Mellier, Y.; Fort, B.; Mathez, G.; Hammer, F. |date=October 1987 | title =Further data on the blue ring-like structure in A 370| journal = Astronomy and Astrophysics |ISSN= 0004-6361| issue=1–2| volume = 184 | pages = L7–L9| bibcode = 1987A&A...184L...7S | accessdate=2008-06-01}}</ref> The first cluster weak lensing analysis was conducted in 1990 by J. Anthony Tyson of [[Bell Laboratories]] and collaborators. Tyson et al. detected a coherent alignment of the [[ellipse|ellipticities]] of the [[faint blue galaxy|faint blue galaxies]] behind both [[Abell 1689]] and CL 1409+52.<ref>{{cite journal | last = Tyson | first = J.A. | coauthors = Valdes, F.; Wenk, R.A. |date=January 1990| title = Detection of systematic gravitational lens galaxy image alignments - Mapping dark matter in galaxy clusters| journal = Astrophysical Journal |ISSN= 0004-637X| volume=349 | pages = L1–L4 | bibcode = 1990ApJ...349L...1T| accessdate = 2008-06-01 | doi = 10.1086/185636}}</ref> Lensing has been used as a tool to investigate a tiny fraction of the thousands of [[List of galaxy clusters|known galaxy clusters]].

Historically, lensing analyses were conducted on galaxy clusters detected via their [[baryon]] content (e.g. from [[visible-light astronomy|optical]] or [[X-ray astronomy|X-ray]] surveys). The sample of galaxy clusters studied with lensing was thus subject to various selection effects; for example, only the most [[luminous intensity|luminous]] clusters were investigated. In 2006, David Wittman of the [[University of California at Davis]] and collaborators published the first sample of galaxy clusters detected via their lensing signals, completely independent of their baryon content.<ref>{{cite journal | first = D. | last = Wittman | coauthors = Dell'Antonio, I.P.; Hughes, J.P.; Margoniner, V.E.; Tyson, J.A.; Cohen, J.G.; Norman, D. |date=May 2006 | title = First Results on Shear-selected Clusters from the Deep Lens Survey: Optical Imaging, Spectroscopy, and X-Ray Follow-up | journal = The Astrophysical Journal | issue = 1 | volume = 643 | pages = 128–143 | bibcode = 2006ApJ...643..128W | accessdate = 2008-06-01 | doi = 10.1086/502621|arxiv = astro-ph/0507606 }}</ref> Clusters discovered through lensing are subject to mass selection effects because the more massive clusters produce lensing signals with higher [[signal-to-noise]].

===Observational products===
The projected [[density|mass density]] can be recovered from the measurement of the ellipticities of the lensed background galaxies through techniques that can be classified into two types: direct reconstruction<ref>{{cite journal | first = N.| last = Kaiser | coauthors = Squires, G.|date=February 1993 | title = Mapping the dark matter with weak gravitational lensing| journal = Astrophysical Journal |ISSN= 0004-637X| issue = 2 | volume = 404| pages = 441–450 | bibcode = 1993ApJ...404..441K | accessdate = 2008-06-01 | doi = 10.1086/172297}}</ref> and [[inverse problem|inversion]].<ref>{{cite journal| last = Bartelmann|first = M. |coauthors = Narayan, R; Seitz, S.; Schneider, P. |date=June 1996 | title = Maximum-likelihood Cluster Reconstruction|journal = Astrophysical Journal Letters | volume = 464 | pages = L115 | bibcode = 1996ApJ...464L.115B | accessdate = 2008-06-01| doi = 10.1086/310114|arxiv = astro-ph/9601011| issue = 2 }}</ref> However, a [[mass distribution]] reconstructed without knowledge of the [[Gravitational Lensing Formalism#Magnification|magnification]] suffers from a limitation known as the mass sheet [[Degeneracy (mathematics)|degeneracy]], where the cluster surface mass density κ can be determined only up to a [[Transformation (geometry)|transformation]] <math>\kappa \rightarrow \kappa^{\prime} = \lambda \kappa+(1-\lambda)</math> where λ is an arbitrary constant.<ref>{{cite journal| last = Schneider| first = P.|coauthors = Seitz, C.|date=February 1995 | title = Steps towards nonlinear cluster inversion through gravitational distortions. 1: Basic considerations and circular clusters| journal = Astronomy and Astrophysics |ISSN= 0004-6361| issue = 2 | volume = 294 | pages = 411–431 | bibcode = 1995A&A...294..411S| accessdate = 2008-06-01|arxiv = astro-ph/9407032 }}</ref> This degeneracy can be broken if an independent measurement of the magnification is available because the magnification is not [[invariant (mathematics)|invariant]] under the aforementioned degeneracy transformation.

Given a [[centroid]] for the cluster, which can be determined by using a reconstructed mass distribution or optical or X-ray data, a model can be fit to the shear profile as a function of clustrocentric radius. For example, the [[singular isothermal sphere profile|singular isothermal sphere (SIS) profile]] and the [[Navarro-Frenk-White profile|Navarro-Frenk-White (NFW) profile]] are two commonly used [[parametric model]]s. Knowledge of the lensing cluster [[redshift]] and the redshift distribution of the background galaxies is also necessary for estimation of the mass and size from a model fit; these redshifts can be measured precisely using [[spectroscopy]] or [[photometric redshift|estimated using photometry]]. Individual mass estimates from weak lensing can only be derived for the most massive clusters, and the accuracy of these mass estimates are limited by projections along the line of sight.<ref>{{cite journal| last = Metzler| first = C.A.| coauthors = White, M.; Norman, M.; Loken, C.|date=July 1999 | title = Weak Gravitational Lensing and Cluster Mass Estimates| journal = The Astrophysical Journal| issue = 1 | volume = 520 | pages = L9–L12 | bibcode = 1999ApJ...520L...9M|accessdate = 2008-06-01| doi = 10.1086/312144|arxiv = astro-ph/9904156 }}</ref>

===Scientific implications===
[[Image:Bullet cluster lensing.jpg|thumb|300px|Image of the Bullet Cluster from the Hubble Space Telescope with total mass contours (dominated by dark matter) from a lensing analysis overlaid.]]

Cluster mass estimates determined by lensing are valuable because the method requires no assumption about the dynamical state or [[star formation|star formation history]] of the cluster in question. Lensing mass maps can also potentially reveal "dark clusters," clusters containing overdense concentrations of dark matter but relatively insignificant amounts of baryonic matter. Comparison of the dark matter distribution mapped using lensing with the distribution of the baryons using optical and X-ray data reveals the interplay of the dark matter with the [[star|stellar]] and [[intracluster medium|gas]] components. A notable example of such a joint analysis is the so-called [[Bullet Cluster]].<ref>{{cite journal| last = Clowe | first = D. |coauthors = Gonzalez, A.H.; Markevitch, M.|date=April 2004 | title = Weak-Lensing Mass Reconstruction of the Interacting Cluster 1E 0657-558: Direct Evidence for the Existence of Dark Matter| journal = The Astrophysical Journal| issue = 2 | volume = 604 | pages = 596–603 | bibcode = 2004ApJ...604..596C | accessdate = 2008-06-01| doi = 10.1086/381970|arxiv = astro-ph/0312273 }}</ref> The Bullet Cluster data provide constraints on models relating light, gas, and dark matter distributions such as [[MOND|Modified Newtonian dynamics (MOND)]] and [[Lambda-CDM model|Λ-Cold Dark Matter (Λ-CDM)]].

In principle, since the number density of clusters as a function of mass and redshift is sensitive to the underlying [[physical cosmology|cosmology]], cluster counts derived from large weak lensing [[astronomical survey|surveys]] should be able to constrain cosmological parameters. In practice, however, projections along the line of sight cause many [[false positives]].<ref>{{cite journal | last = Hoekstra | first = H. | coauthors = Jain, B.|date=May 2008 | title = Weak Gravitational Lensing and its Cosmological Applications | journal = Eprint arXiv | bibcode = 2008ARNPS..58...99H | accessdate = 2008-06-01 | doi=10.1146/annurev.nucl.58.110707.171151 | arxiv=0805.0139 | volume = 58 | pages = 99–123}}</ref> Weak lensing can also be used to [[calibrate]] the mass-observable relation via a stacked weak lensing signal around an ensemble of clusters, although this relation is expected to have an intrinsic [[variance|scatter]].<ref>{{cite journal| last = Reyes | first = R.|coauthors = Mandelbaum, R.;Hirata, C.;Bahcall, N.;Seljak, U. |date=February 2008 | title = Improved optical mass tracer for galaxy clusters calibrated using weak lensing measurements | journal = Eprint arXiv | bibcode = 2008MNRAS.390.1157R| accessdate = 2008-06-01 | doi=10.1111/j.1365-2966.2008.13818.x | arxiv=0802.2365| volume = 390| issue = 3| pages = 1157–1169}}</ref> In order for lensing clusters to be a precision probe of cosmology in the future, the projection effects and the scatter in the lensing mass-observable relation need to be thoroughly characterized and modeled.

== Galaxy-galaxy lensing==
Galaxy-galaxy lensing is a specific type of weak (and occasionally strong) [[gravitational lensing]], in which the foreground object responsible for distorting the shapes of background galaxies is itself an individual [[field galaxy]] (as opposed to a [[galaxy cluster]] or the [[large-scale structure of the cosmos]]). Of the three typical mass regimes in weak lensing, galaxy-galaxy lensing produces a “mid-range” signal (shear correlations of ~1%) that is weaker than the signal due to cluster lensing, but stronger than the signal due to cosmic shear.

===History===
J.A. Tyson and collaborators first postulated the concept of galaxy-galaxy lensing in 1984, though the observational results of their study were inconclusive.<ref>{{ cite journal | last = Tyson| first = J.A.| coauthors = Valdes, F.; Jarvis, J.F.; Mills, A.P., Jr.|date=June 1984| title = Galaxy mass distribution from gravitational light deflection| journal = Astrophysical Journal |ISSN= 0004-637X| volume = 281| pages = L59–L62| bibcode = 1984ApJ...281L..59T| accessdate = 2008-05-26 | doi = 10.1086/184285}}</ref> It was not until 1996 that evidence of such distortion was tentatively discovered,<ref>{{ cite journal | last = Brainerd| first = Tereasa G.| coauthors = Blanford, Roger D.; Smail, Ian;|date=August 1996| title = Weak Gravitational Lensing by Galaxies| journal = The Astrophysical Journal| volume = 466| page = 623| bibcode = 1996ApJ...466..623B| accessdate = 2008-05-26 | doi = 10.1086/177537|arxiv = astro-ph/9503073 }}</ref> with the first statistically significant results not published until the year 2000.<ref>{{ cite journal | last = Fischer| first = Philippe| coauthors = McKay, Timothy A.; Sheldon, Erin; Connolly, Andrew; Stebbins, Albert; Frieman, Joshua A.; Jain, Bhuvnesh; Joffre, Michael; Johnston, David; Bernstein, Gary; Annis, James; Bahcall, Neta A.; Brinkmann, J.; Carr, Michael A.; Csabai, István; Gunn, James E.; Hennessy, G. S.; Hindsley, Robert B.; Hull, Charles; Ivezić, Željko; Knapp, G. R.; Limmongkol, Siriluk; Lupton, Robert H.; Munn, Jeffrey A.; Nash, Thomas; Newberg, Heidi Jo; Owen, Russell; Pier, Jeffrey R.; Rockosi, Constance M.; Schneider, Donald P.; Smith, J. Allyn; Stoughton, Chris; Szalay, Alexander S.; Szokoly, Gyula P.; Thakar, Aniruddha R.; Vogeley, Michael S.; Waddell, Patrick; Weinberg, David H.; York, Donald G.; The SDSS Collaboration|date=September 2000| title = Weak Lensing with Sloan Digital Sky Survey Commissioning Data: The Galaxy-Mass Correlation Function to 1 H-1 Mpc| journal = The Astronomical Journal| issue = 3| volume = 466| pages = 1198–1208| bibcode = 2000AJ....120.1198F| accessdate = 2008-05-26 | doi = 10.1086/301540|arxiv = astro-ph/9912119 }}</ref> Since those initial discoveries, the construction of larger, high resolution telescopes and the advent of dedicated wide field [[redshift survey|galaxy surveys]] have greatly increased the observed number density of both background source and foreground lens galaxies, allowing for a much more robust statistical sample of galaxies, making the lensing signal much easier to detect. Today, measuring the shear signal due to galaxy-galaxy lensing is a widely used technique in [[observational astronomy]] and [[cosmology]], often used in parallel with other measurements in determining physical characteristics of foreground galaxies.

===Stacking===
Much like in [[Weak gravitational lensing#Weak lensing by clusters of galaxies|cluster-scale weak lensing]], detection of a galaxy-galaxy shear signal requires one to measure the shapes of background source galaxies, and then look for statistical shape correlations (specifically, source galaxy shapes should be aligned tangentially, relative to the lens center.) In principle, this signal could be measured around any individual foreground lens. In practice, however, due to the relatively low mass of field lenses and the inherent randomness in intrinsic shape of background sources (the “shape noise”), the signal is impossible to measure on a galaxy by galaxy basis. However, by combining the signals of many individual lens measurements together (a technique known as “stacking”), the [[signal-to-noise ratio]] will improve, allowing one to determine a statistically significant signal, averaged over the entire lens set.

===Scientific applications===
Galaxy-galaxy lensing (like all other types of gravitational lensing) is used to measure several quantities pertaining to [[mass]]:

*Mass density profiles
**Using techniques similar to those in cluster-scale lensing, galaxy-galaxy lensing can provide information about the shape of mass density profiles, though these profiles correspond to galaxy-sized objects instead of larger clusters or groups. Given a high enough number density of background sources, a typical galaxy-galaxy mass density profile can cover a wide range of distances (from ~1 to ~100 [[effective radius|effective radii]]).<ref>{{ cite journal | last = Gavazzi| first = Raphaël| coauthors = Treu, Tommaso; Rhodes, Jason D.; Koopmans, Léon V. E.; Bolton, Adam S.; Burles, Scott; Massey, Richard J.; Moustakas, Leonidas A.|date=September 2007| title = The Sloan Lens ACS Survey. IV. The Mass Density Profile of Early-Type Galaxies out to 100 Effective Radii| journal = The Astrophysical Journal| volume = 667| issue = 1| pages = 176–190| bibcode = 2007ApJ...667..176G| accessdate = 2008-06-06 | doi = 10.1086/519237|arxiv = astro-ph/0701589 }}</ref> Since the effects of lensing are insensitive to the matter type, a galaxy-galaxy mass density profile can be used to probe a wide range of matter environments: from the central cores of galaxies where [[baryon]]s dominate the total mass fraction, to the outer [[dark matter halo|halos]] where [[dark matter]] is more prevalent.

*Mass-to-light ratios
**Comparing the measured mass to the [[luminosity]] (averaged over the entire galaxy stack) in a specific [[Filter (optics)|filter]], galaxy-galaxy lensing can also provide insight into the [[mass to light ratio]]s of field galaxies. Specifically, the quantity measured through lensing is the total (or [[virial]]) mass to light ratio – again due to the insensitivity of lensing to matter type. Assuming that luminous matter can trace dark matter, this quantity is of particular importance, since measuring the ratio of luminous (baryonic) matter to total matter can provide information regarding the overall ratio of baryonic to dark matter in the universe.<ref>{{ cite journal | last = Hoekstra| first = H.| coauthors = Franx, M.; Kuijken, K.; Carlberg, R. G.; Yee, H. K. C.|date=April 2003| title = Lensing by galaxies in CNOC2 fields| journal = Monthly Notice of the Royal Astronomical Society| volume = 340| issue = 2| pages = 609–622| bibcode = 2003MNRAS.340..609H| accessdate = 2008-06-06 | doi = 10.1046/j.1365-8711.2003.06350.x|arxiv = astro-ph/0211633 }}</ref>

*Galaxy mass evolution
**Since the [[speed of light]] is finite, an observer on the Earth will see distant galaxies not as they look today, but rather as they appeared at some earlier time. By restricting the lens sample of a galaxy-galaxy lensing study to lie at only one particular redshift, it is possible to understand the mass properties of the field galaxies that existed during this earlier time. Comparing the results of several such redshift-restricted lensing studies (with each study encompassing a different redshift), one can begin to observe changes in the mass features of galaxies over a period of several [[timeline of the big bang|epoch]]s, leading towards a better understanding of the evolution of mass on the smallest cosmological scales.<ref>{{ cite journal | last = Parker| first = Laura C.| coauthors = Hoekstra, Henk; Hudson, Michael J.; van Waerbeke, Ludovic; Mellier, Yannick|date=November 2007| title = The Masses and Shapes of Dark Matter Halos from Galaxy-Galaxy Lensing in the CFHT Legacy Survey| journal = The Astrophysical Journal| volume = 669| issue = 1| pages = 21–31| bibcode = 2007ApJ...669...21P| accessdate = 2008-06-06 | doi = 10.1086/521541}}</ref>

*Other mass trends
**Lens redshift is not the only quantity of interest that can be varied when studying mass differences between galaxy populations, and often there are several parameters used when segregating objects into galaxy-galaxy lens stacks.<ref>{{ cite journal | last = Sheldon| first = Erin S.| coauthors = Johnston, David E.; Frieman, Joshua A.; Scranton, Ryan; McKay, Timothy A.; Connolly, A. J.; Budavári, Tamás; Zehavi, Idit; Bahcall, Neta A.; Brinkmann, J.; Fukugita, Masataka|date=May 2004| title = The Galaxy-Mass Correlation Function Measured from Weak Lensing in the Sloan Digital Sky Survey| journal = The Astronomical Journal | volume = 127| issue = 5| pages = 2544–2564| bibcode = 2004AJ....127.2544S| accessdate = 2008-06-06|arxiv = astro-ph/0312036 |doi = 10.1086/383293 }}</ref><ref>{{ cite journal | last = Mandelbaum| first = Rachel| coauthors = Seljak, Uroš; Kauffmann, Guinevere; Hirata, Christopher M.; Brinkmann, Jonathan|date=May 2006| title = Galaxy halo masses and satellite fractions from galaxy-galaxy lensing in the Sloan Digital Sky Survey: stellar mass, luminosity, morphology and environment dependencies| journal = Monthly Notices of the Royal Astronomical Society| volume = 368| issue = 2| pages = 715–731| bibcode = 2006MNRAS.368..715M| accessdate = 2008-06-06 | doi = 10.1111/j.1365-2966.2006.10156.x|arxiv = astro-ph/0511164 }}</ref> Two widely used criteria are galaxy [[color index|color]] and [[morphology (astronomy)|morphology]], which act as tracers of (among other things) stellar population, galaxy age, and local mass environment. By separating lens galaxies based on these properties, and then further segregating samples based on redshift, it is possible to use galaxy-galaxy lensing to see how several different types of galaxies evolve through time.

==Cosmic shear==

The gravitational lensing by [[large-scale structure]] also produces an observable pattern of alignments in background galaxies, but this distortion is only ~0.1%-1% - much more subtle than cluster or galaxy-galaxy lensing. The [[Gravitational Lensing Formalism#Thin lens approximation|thin lens approximation]] usually used in cluster and galaxy lensing does not always work in this regime, because structures can be elongated along the line of sight. Instead, the distortion can be derived by assuming that the deflection angle is always small (see [[Gravitational Lensing Formalism]]). As in the thin lens case, the effect can be written as a mapping from the unlensed angular position <math>\vec{\beta}</math> to the lensed position <math>\vec{\theta}</math>. The [[Jacobian]] of the transform can be written as an integral over the gravitational potential <math>\Phi</math> along the line of sight

<math>
\frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} + \int_0^{r_\infty} dr
g(r) \frac{\partial^2 \Phi(\vec{x}(r))}{\partial x^i
\partial x^j}
</math>

where <math>r</math> is the [[comoving distance]], <math>x^i</math> are the transverse distances, and

<math>
g(r) = 2 r \int^{r_\infty}_r
\left(1-\frac{r^\prime}{r}\right)W(r^\prime)
</math>

is the ''lensing kernel'', which defines the efficiency of lensing for a distribution of sources <math>W(r)</math>.

As in the thin-lens approximation, the Jacobian can be decomposed into [[Gravitational Lensing Formalism#Lensing Jacobian|shear and convergence terms]].

=== Shear correlation functions ===

Because large-scale cosmological structures do not have a well-defined location, detecting cosmological gravitational lensing typically involves the computation of ''shear correlation functions'', which measure the mean product of the shear at two points as a function of the distance between those points. Because there are two components of shear, three different correlation functions can be defined:

<math>
\xi_{++}(\Delta\theta) = \langle \gamma_+(\vec{\theta}) \gamma_+(\vec{\theta}+\vec{\Delta\theta}) \rangle
</math>

<math>
\xi_{\times\times}(\Delta\theta) = \langle \gamma_\times(\vec{\theta}) \gamma_\times(\vec{\theta}+\vec{\Delta\theta}) \rangle
</math>

<math>
\xi_{\times +}(\Delta\theta)=\xi_{+ \times}(\Delta\theta) = \langle \gamma_+(\vec{\theta}) \gamma_\times(\vec{\theta}+\vec{\Delta\theta}) \rangle
</math>

where <math>\gamma_{+~}</math> is the component along or perpendicular to <math>\vec{\Delta\theta}</math>, and <math>\gamma_\times</math> is the component at 45°. These correlation functions are typically computed by averaging over many pairs of galaxies. The last correlation function, <math>\xi_{\times +}</math>, is not affected at all by lensing, so measuring a value for this function that is inconsistent with zero is often interpreted as a sign of [[systematic error]].

The functions <math>\xi_{++~}</math> and <math>\xi_{\times\times}</math> can be related to projections (integrals with certain weight functions) of the dark matter density correlation function, which can be predicted from theory for a cosmological model through its Fourier transform, the [[matter power spectrum]].<ref>{{ cite journal | last = Miralda-Escudé| first = Jordi|date=October 1991| title = The Correlation Function of Galaxy Ellipticities Produced By Gravitational Lensing| journal = Astrophysical Journal| volume = 380| pages = 1–8| bibcode = 1991ApJ...380....1M| accessdate = 2008-06-01 | doi = 10.1086/170555}}</ref>

Because they both depend on a single scalar density field, <math>\xi_{++~}</math> and <math>\xi_{\times\times}</math> are not independent, and they can be decomposed further into ''E-mode'' and ''B-mode'' correlation functions.<ref>{{ cite journal | last = Schneider| first = P. | coauthors = van Waerbekere, L., Kilbinger, M., Mellier, Y.|date=December 2002 | title=Analysis of two-point statistics of cosmic shear| journal = Astronomy and Astrophysics | volume=396 | pages = 1–19 | bibcode = 2002A&A...396....1S | doi = 10.1051/0004-6361:20021341|arxiv = astro-ph/0206182 }}</ref> In analogy with electric and magnetic fields, the E-mode field is curl-free and the B-mode field is divergence-free. Because gravitational lensing can only produce an E-mode field, the B-mode provides yet another test for systematic errors.

The E-mode correlation function is also known as the ''aperture mass variance''

<math>
\langle M_{ap}^2 \rangle (\theta) = \int_0^{2\theta} \frac{\phi d\phi}{\theta^2}
\left[\xi_{++}(\phi)+\xi_{\times\times}(\phi)\right] T_+\left(\frac{\phi}{\theta}\right)
= \int_0^{2\theta} \frac{\phi d\phi}{\theta^2}
\left[\xi_{++}(\phi)-\xi_{\times\times}(\phi)\right] T_-\left(\frac{\phi}{\theta}\right)
</math>

<math>
T_+(x) = 576\int^\infty_0 \frac{dt}{t^3}J_0(xt)[J_4(t)]^2
</math>

<math>
T_-(x) = 576\int^\infty_0 \frac{dt}{t^3}J_4(xt)[J_4(t)]^2
</math>

where <math>J_0~</math> and <math>J_4~</math> are [[Bessel Functions]].

An exact decomposition thus requires knowledge of the shear correlation functions at zero separation, but an approximate decomposition is fairly insensitive to these values because the filters <math>T_+~</math> and <math>T_-~</math> are small near <math>\theta=0~</math>.

=== Weak lensing and cosmology ===

The ability of weak lensing to constrain the [[matter power spectrum]] makes it a potentially powerful probe of cosmological parameters, especially when combined with other observations such as the [[cosmic microwave background]], [[Type Ia supernova|supernovae]], and [[Redshift survey|galaxy surveys]]. Detecting the extremely faint cosmic shear signal requires averaging over many background galaxies, so surveys must be both deep and wide, and because these background galaxies are small, the image quality must be very good. Measuring the shear correlations at small scales also requires a high density of background objects (again requiring deep, high quality data), while measurements at large scales push for wider surveys.

While weak lensing of large-scale structure was discussed as early as 1967,<ref>{{ cite journal | last = Gunn| first = James E.|date=December 1967| title = On the Propagation of Light in Inhomogeneous Cosmologies. I. Mean Effects| journal = Astrophysical Journal| volume = 150| pages = 737G| bibcode = 1967ApJ...150..737G| accessdate = 2008-06-01 | doi = 10.1086/149378}}</ref> due to the challenges mentioned above, it was not detected until more than 30 years later when large [[Charge-coupled device|CCD]] cameras enabled surveys of the necessary size and quality. In 2000, four independent groups<ref>{{ cite journal | last = Wittman| first = David| coauthors = Tyson, J.A.; Kirkman, David; Dell'Antonio, Ian; Bernstein, Gary |date=May 2000| title = Detection of weak gravitational lensing distortions of distant galaxies by cosmic dark matter at large scales| journal = Nature| volume = 405| pages = 143–148| bibcode = 2000Natur.405..143W| accessdate = 2008-06-01 | doi = 10.1038/35012001 | pmid = 10821262 | issue = 6783|arxiv = astro-ph/0003014 }}</ref><ref>{{ cite journal | last = Bacon| first = David| coauthors = Refregier, Alexandre; Ellis, Richard |date=October 2000| title = Detection of weak gravitational lensing by large-scale structure| journal = MNRAS| volume = 318| pages = 625–640| bibcode = 2000MNRAS.318..625B| accessdate = 2008-06-01 | doi = 10.1046/j.1365-8711.2000.03851.x|arxiv = astro-ph/0003008 | issue = 2 }}</ref><ref>{{ cite journal | last = Kaiser| first = Nick| coauthors = Wilson, Gillian; Luppino, Gerard|date=March 2000| title = Large-Scale Cosmic Shear Measurements| journal = | bibcode = 2000astro.ph..3338K| accessdate = 2008-06-01|arxiv = astro-ph/0003338 | page = 3338 }}</ref><ref>{{ cite journal | last = Van Waerbeke| first = L.| coauthors = Mellier, Y.; Erben, T.; Cuillandre, J.C.; Bernardeau, F.; Maoli, R.; Bertin, E.; McCracken, H.J.; Le Fèvre, O.; Fort, B.; Dantel-Fort, M.; Jain, B.; Schneider, P.|date=June 2000| title = Detection of correlated galaxy ellipticities from CFHT data: first evidence for gravitational lensing by large-scale structures| journal = Astronomy and Astrophysics |bibcode = 2000A&A...358...30V| accessdate = 2008-06-01|arxiv = astro-ph/0002500 }}</ref> published the first detections of cosmic shear, and subsequent observations have started to put constraints on cosmological parameters (particularly the [[Lambda-CDM model|dark matter density <math>\Omega_m~</math> and power spectrum amplitude <math>\sigma_8~</math>]]) that are competitive with other cosmological probes.

For current and future surveys, one goal is to use the redshifts of the background galaxies (often approximated using [[photometric redshift]]s) to divide the survey into multiple redshift bins. The low-redshift bins will only be lensed by structures very near to us, while the high-redshift bins will be lensed by structures over a wide range of redshift. This technique, dubbed "cosmic [[tomography]]", makes it possible to map out the 3D distribution of mass. Because the third dimension involves not only distance but cosmic time, tomographic weak lensing is sensitive not only to the matter power spectrum today, but also to its evolution over the history of the universe, and the expansion history of the universe during that time. This is a much more valuable cosmological probe, and many proposed experiments to measure the properties of [[dark energy]] and [[dark matter]] have focused on weak lensing, such as the [[The Dark Energy Survey|Dark Energy Survey]], [[Pan-STARRS]], and [[LSST]].

Weak lensing also has an important effect on the [[Cosmic Microwave Background]] and diffuse [[Hydrogen line|21cm line radiation]]. Even though there are no distinct resolved sources, perturbations on the origining surface are sheared in a similar way to galaxy weak lensing, resulting in changes to the power spectrum and statistics of the observed signal. Since the source plane for the CMB and high-redshift diffuse 21 cm are at higher redshift that resolved galaxies, the lensing effect probes cosmology at higher redshifts than galaxy lensing.

==See also==
*[[Large Synoptic Survey Telescope]]

== References ==
{{reflist|2}}

==External links==
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=-17503 Weak gravitational lensing on arxiv.org]
*[http://www.kaggle.com/c/DarkWorlds Observing Dark Worlds]
[[Category:Gravitational lensing]]

Template:DoseConcentrationClearance

2013-07-04T21:42:00Z

92.39.192.184:

{{Use dmy dates|date=October 2012}}
The '''smoothing spline''' is a method of [[smoothing]] (fitting a [[smooth curve]] to a set of noisy [[observation]]s) using a [[Spline (mathematics)|spline]] function.

==Definition==
Let <math>(x_i,Y_i);x_1<x_2<\dots<x_n, i \in \mathbb{Z} </math> be a sequence of observations, modeled by the relation <math>Y_i = \mu(x_i)</math>. The smoothing spline estimate <math>\hat\mu</math> of the function <math>\mu</math> is defined to be the minimizer (over the class of twice differentiable functions) of<ref>{{Cite book|title=Generalized Additive Models|last=Hastie|first=T. J.|coauthors=Tibshirani, R. J.|year=1990|publisher=Chapman and Hall|isbn=0-412-34390-8}}</ref>
:<math>
\sum_{i=1}^n (Y_i - \hat\mu(x_i))^2 + \lambda \int_{x_1}^{x_n} \hat\mu''(x)^2 \,dx.
</math>

Remarks:
# <math>\lambda \ge 0</math> is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate.
# The integral is evaluated over the range of the <math>x_i</math>.
# As <math>\lambda\to 0</math> (no smoothing), the smoothing spline converges to the interpolating spline.
# As <math>\lambda\to\infty</math> (infinite smoothing), the roughness penalty becomes paramount and the estimate converges to a [[Ordinary least squares|linear least squares]] estimate.
# The roughness penalty based on the [[second derivative]] is the most common in modern statistics literature, although the method can easily be adapted to penalties based on other derivatives.
# In early literature, with equally-spaced <math>x_i</math>, second or third-order differences were used in the penalty, rather than derivatives.
# When the sum-of-squares term is replaced by a log-likelihood, the resulting estimate is termed ''penalized likelihood''. The smoothing spline is the special case of penalized likelihood resulting from a Gaussian likelihood.

==Derivation of the smoothing spline==

It is useful to think of fitting a smoothing spline in two steps:
# First, derive the values <math>\hat\mu(x_i);i=1,\ldots,n</math>.
# From these values, derive <math>\hat\mu(x)</math> for all ''x''.

Now, treat the second step first.

Given the vector <math>\hat{m} = (\hat\mu(x_1),\ldots,\hat\mu(x_n))^T</math> of fitted values, the sum-of-squares part of the spline criterion is fixed. It remains only to minimize <math>\int \hat\mu''(x)^2 \, dx</math>, and the minimizer is a natural cubic [[Spline (mathematics)|spline]] that interpolates the points <math>(x_i,\hat\mu(x_i))</math>. This interpolating spline is a linear operator, and can be written in the form
:<math>
\hat\mu(x) = \sum_{i=1}^n \hat\mu(x_i) f_i(x)
</math>
where <math>f_i(x)</math> are a set of spline basis functions. As a result, the roughness penalty has the form
:<math>
\int \hat\mu''(x)^2 dx = \hat{m}^T A \hat{m}.
</math>
where the elements of ''A'' are <math>\int f_i''(x) f_j''(x)dx</math>. The basis functions, and hence the matrix ''A'', depend on the configuration of the predictor variables <math>x_i</math>, but not on the responses <math>Y_i</math> or <math>\hat m</math>.

Now back to the first step. The penalized sum-of-squares can be written as
:<math>
\|Y - \hat m\|^2 + \lambda \hat{m}^T A \hat m,
</math>
where <math>Y=(Y_1,\ldots,Y_n)^T</math>.
Minimizing over <math>\hat m</math> gives
:<math>
\hat m = (I + \lambda A)^{-1} Y.
</math>

==De Boor's approach==

De Boor's approach exploits the same idea, of finding a balance between having a smooth curve and being close to the given data.<ref name="DeBoor2001">{{Cite book|title=A Practical Guide to Splines (Revised Edition)|last=De Boor|first=C.|year=2001|publisher=Springer|pages=207–214|isbn=0-387-90356-9}}</ref>

<math>p\sum_{i=1}^n \left ( \frac{Y_i - \hat\mu \left (x_i \right )}{\delta_i} \right )^2+\left ( 1-p \right )\int \left ( \hat\mu^{\left (m \right )}\left ( x \right ) \right )^2 \, dx</math>

where <math>p</math> is a parameter called smooth factor and belongs to the interval <math>[0,1]</math>, and <math>\delta_i;i=1,\dots,n</math> are the quantities controlling the extent of smoothing (they represent the weight <math>\delta_i^{-2}</math> of each point <math>Y_i</math>). In practice, since [[cubic splines]] are mostly used, <math>m</math> is usually <math>2</math>. The solution for <math>m=2</math> was proposed by Reinsch in 1967.<ref name="Reinsch1967" /> For <math>m=2</math>, when <math>p</math> approaches <math>1</math>, <math>\hat\mu</math> converges to the "natural" spline interpolant to the given data.<ref name="DeBoor2001" /> As <math>p</math> approaches <math>0</math>, <math>\hat\mu</math> converges to a straight line (the smoothest curve). Since finding a suitable value of <math>p</math> is a task of trial and error, a redundant constant <math>S</math> was introduced for convenience.<ref name="Reinsch1967">{{Cite web|author=Reinsch, Christian H|title=Smoothing by Spline Functions|url=http://www.cise.ufl.edu/class/cap5416fa10/resources/Reinsch_1967.pdf |accessdate=11 March 2011}}</ref>
<math>S</math> is used to numerically determine the value of <math>p</math> so that the function <math>\hat\mu</math> meets the following condition:

<math>\sum_{i=1}^n \left ( \frac{Y_i - \hat\mu \left (x_i \right )}{\delta_i} \right )^2 \le S</math>

The algorithm described by de Boor starts with <math>p=0</math> and increases <math>p</math> until the condition is met.<ref name="DeBoor2001" /> If <math>\delta_i</math> is an estimation of the standard deviation for <math>Y_i</math>, the constant <math>S</math> is recommended to be chosen in the interval <math>\left [ n-\sqrt{2n},n+\sqrt{2n} \right ]</math>. Having <math>S=0</math> means the solution is the "natural" spline interpolant.<ref name="Reinsch1967" /> Increasing <math>S</math> means we obtain a smoother curve by getting farther from the given data.

==Creating a multidimensional spline==

Given the constraint from the definition formula <math>x_1<x_2< \dots <x_n</math> we can conclude that the algorithm doesn't work for any sets of data. If we plan to use this algorithm for random points in a multidimensional space we need to find a solution to give as input to the algorithm sets of data where these constraints are met. A solution for this is to introduce a parameter so that the input data would be represented as single-valued functions depending on that parameter; after this the smoothing will be performed for each function. In a bidimensional space a solution would be to parametrize <math>x</math> and <math>y</math> so that they would become <math>x(t)</math> and <math>y(t)</math> where <math>t_1<t_2< \dots <t_n</math>. A convenient solution for <math>t</math> is the cumulating distance <math>t_{i+1}=t_{i}+\sqrt{(x_{i+1}-x_{i})^2+(y_{i+1}-y_{i})^2}</math> where <math>t_1=0</math>.<ref>{{Cite web|author=Robert E. Smith Jr., Joseph M Price and Lona M. Howser|title=A Smoothing Algorithm Using Cubic Spline Functions|url=http://www.pdas.com/refs/tnd7397.pdf |accessdate=31 May 2011}}</ref><ref>{{Cite web|author=N. Y. Graham|title=Smoothing With Periodic Cubic Splines|url=http://www.alcatel-lucent.com/bstj/vol62-1983/articles/bstj62-1-101.pdf |accessdate=31 May 2011}}</ref>

A more detailed analysis on parametrization is done by E.T.Y Lee.<ref>{{Cite web|author=E.T.Y. Lee|title=Choosing nodes in parametric curve interpolation|url=http://www.cs.bgu.ac.il/~leonid/na105/Splines/Lee.pdf |accessdate=28 June 2011}}</ref>

==Related methods==

Smoothing splines are related to, but distinct from:
* Regression splines. In this method, the data is fitted to a set of spline basis functions with a reduced set of knots, typically by least squares. No roughness penalty is used.
* Penalized Splines. This combines the reduced knots of regression splines, with the roughness penalty of smoothing splines.<ref>{{Cite book|title=Semiparametric Regression|last=Ruppert|first=David|coauthors=Wand, M. P. and Carroll, R. J.|publisher=Cambridge University Press|year=2003|isbn=0-521-78050-0}}</ref>
* [[Elastic map]]s method for [[manifold learning]]. This method combines the [[least squares]] penalty for approximation error with the bending and stretching penalty of the approximating manifold and uses the coarse discretization of the optimization problem.

==Source code==

Source code for [[Spline (mathematics)|spline]] smoothing can be found in the examples from [[Carl R. de Boor|Carl de Boor's]] book ''A Practical Guide to Splines''. The examples are in [[Fortran]] [[programming language]]. The updated sources are available also on Carl de Boor's official site [http://pages.cs.wisc.edu/~deboor/].

==Further reading==

* Wahba, G. (1990). ''Spline Models for Observational Data''. SIAM, Philadelphia.
* Green, P. J. and Silverman, B. W. (1994). ''Nonparametric Regression and Generalized Linear Models''. CRC Press.
* De Boor, C. (2001). ''A Practical Guide to Splines (Revised Edition)''. Springer.

==References==
{{Reflist}}

[[Category:Regression analysis]]
[[Category:Splines]]
[[Category:Statistical methods]]

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