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Colpitts oscillator

2014-02-01T15:10:52Z

117.206.170.141:

{{Merge| Observer-expectancy effect |date=October 2013}}
In [[experimental science]], '''experimenter's bias''', also known as '''research bias''', is a [[subjectivity|subjective]] [[bias]] towards a result expected by the human experimenter.<ref name= Sackett>{{cite journal |last=Sackett |first=D. L. |title=Bias in analytic research |journal=Journal of Chronic Diseases |year=1979 |volume=32 |issue=1–2 |pages=51–63 |doi=10.1016/0021-9681(79)90012-2 |pmid=447779 }}</ref> For example, it occurs when scientists unconsciously affect subjects in experiments.<ref name="KantowitzIII2009">{{cite book|author1=Barry H. Kantowitz|author2=Henry L. Roediger, III|author3=David G. Elmes|title=Experimental Psychology|url=http://books.google.com/books?id=2-5VL8PHLsIC&pg=PA371|accessdate=7 September 2013|year=2009|publisher=Cengage Learning|isbn=978-0-495-59533-5|page=371}}</ref>

== Observer-expectancy effect ==
The experimenter may introduce [[cognitive bias]] into a study in several ways. In what is called the [[observer-expectancy effect]], the experimenter may subtly communicate their expectations for the outcome of the study to the participants, causing them to alter their behavior to conform to those expectations.

Clever Hans (in [[German (language)|German]], ''der Kluge Hans'') was an [[Orlov Trotter]] [[horse]] that was claimed to have been able to perform [[Elementary arithmetic|arithmetic]]. As a result of the large amount of public interest in Clever Hans, [[Philosopher]] and [[psychologist]] [[Carl Stumpf]] investigated von Osten's scientific claims.

Using a substantial number of trials, Pfungst found that the horse could get the correct answer even if von Osten himself did not ask the questions, ruling out the possibility of fraud. However, the horse got the right answer only when the questioner knew what the answer was, and the horse could see the questioner. He observed that when von Osten knew the answers to the questions, Hans got 89 percent of the answers correct, but when von Osten did not know the answers to the questions, Hans only answered six percent of the questions correctly.

Pfungst then proceeded to examine the behaviour of the questioner in detail, and showed that as the horse's taps approached the right answer, the questioner's [[Human position|posture]] and facial expression changed in ways that were consistent with an increase in tension, which was released when the horse made the final, correct tap. This provided a cue that the horse could use to tell it to stop tapping.

Evidence for the experimenter's bias has also been found among tests on humans. Two essentially alike groups were given the same task. This consisted of rating portrait pictures of people and estimating how successful they were on a scale of -10 to 10. However, in group a) the experimenters were told they could expect positive ratings. Conversely, in group b) the experimenters were told they could expect negative ratings. Group a) gave a much more optimistic appraisal than group b). According to those who instigated the experiment, it is explained by the fact that the researchers were giving subtle cues in the same manner Orlov Trotter had given his horse.<ref>Rosenthal R. Experimenter effects in behavioral research. New York, NY: Appleton-Century-Crofts, 1966. 464 p.</ref>

== Seven stages ==
In a review of biases in [[clinical trials|clinical studies]], states that biases can occur in any one of seven stages of research:<ref name= Sackett/>

# in reading-up on the field,
# in specifying and selecting the study sample,
# in executing the experimental manoeuvre (or exposure),
# in measuring exposures and outcomes,
# in analyzing the data,
# in interpreting the analysis, and
# in publishing the results.
The inability of a human being to be [[objectivity (philosophy)|objective]] is the ultimate source of this bias. It occurs more often in sociological and medical sciences, where [[double blind]] techniques are often employed to combat the bias. But experimenter's bias can also be found in some physical sciences, for instance, where the experimenter [[rounding|rounds off]] measurements.

==Classification of experimenter's biases==

Modern electronic or computerized data acquisition techniques have greatly reduced the likelihood of such bias, but it can still be introduced by a poorly designed analysis technique. Experimenter's bias was not well recognized until the 1950s and 60's, and then it was primarily in medical experiments and studies. Sackett (1979) catalogued 56 biases that can arise in sampling and measurement in clinical research, among the above-stated first six stages of research. These are as follows:
# '''''In reading-up the field'''''
## the biases of rhetoric
## the all's well literature bias
## one-sided reference bias
## positive results bias
## hot stuff bias
# '''''In specifying and selecting the study sample'''''
## popularity bias
## centripetal bias
## referral filter bias
## diagnostic access bias
## diagnostic suspicion bias
## unmasking (detection signal) bias
## mimicry bias
## previous opinion bias
## wrong sample size bias
## admission rate (Berkson) bias
## prevalence-incidence (Neyman) bias
## diagnostic vogue bias
## diagnostic purity bias
## procedure selection bias
## missing clinical data bias
## non-contemporaneous control bias
## starting time bias
## unacceptable disease bias
## migrator bias
## membership bias
## non-respondent bias
## volunteer bias
# '''''In executing the experimental manoeuvre (or exposure)'''''
## contamination bias
## withdrawal bias
## compliance bias
## therapeutic personality bias
## bogus control bias
# '''''In measuring exposures and outcomes'''''
## insensitive measure bias
## underlying cause bias (rumination bias)
## end-digit preference bias
## apprehension bias
## unacceptability bias
## obsequiousness bias
## expectation bias
## substitution game
## family information bias
## exposure suspicion bias
## recall bias
## attention bias
## instrument bias
# '''''In analyzing the data'''''
## post-hoc significance bias
## data dredging bias (looking for the pony)
## scale degradation bias
## tidying-up bias
## repeated peeks bias
# '''''In interpreting the analysis'''''
## mistaken identity bias
## cognitive dissonance bias
## magnitude bias
## significance bias
## correlation bias
## under-exhaustion bias

The effects of bias on experiments in the physical sciences have not always been fully recognized.

==Prevention==

In principle, if a measurement has a resolution of <math>R</math>, then if the experimenter averages <math>N</math> independent measurements the average will have a resolution of <math>R/\sqrt{N}</math> (this is the [[central limit theorem]] of statistics). This is an important experimental technique used to reduce the impact of randomness on an experiment's outcome. This requires that the measurements be statistically independent; there are several reasons why they may not be. If independence is not satisfied, then the average may not actually be a better statistic but may merely reflect the correlations among the individual measurements and their non-independent nature.

The most common cause of non-independence is [[systematic error]]s (errors affecting all measurements equally, causing the different measurements to be highly correlated, so the average is no better than any single measurement). Experimenter bias is another potential cause of non-independence.

===In medical sciences===

The complexity of living systems and the [[ethical]] impossibility of performing fully controlled experiments with certain species of animals and humans provide a rich, and difficult to control, source of experimental bias. The scientific knowledge about the phenomenon under study, and the systematic elimination of probable causes of bias, by detecting [[confounding factor]]s, is the only way to isolate true [[cause-effect relationship]]s. It is also in [[epidemiology]] that experimenter bias has been better studied than in other sciences.

A number of studies into Spiritual Healing illustrate how the design of the study can introduce experimenter bias into the results. A comparison of two studies illustrates that subtle differences in the design of the tests can adversely affect the results of one. The difference was due to the intended result: a positive or negative outcome rather than positive or neutral.

A 1995 paper<ref name=Hodges>{{Cite journal|volume=88|journal=Journal of the Royal Society of Medicine|issue=4 | year=1995 | pages=203–207 | title=Is spiritual healing a valid and effective therapy? | author=Hodges, RD and Scofield, AM |pmid=7745566|pmc=1295164}}</ref> by Hodges & Scofield of '''spiritual healing''' used the growth rate of [[cress]] seeds as their [[independent variable]] in order to eliminate a placebo response or participant bias. The study reported positive results as the test results for each sample were '''consistent''' with the healers intention that healing ''should'' or ''should not'' occur. However the healer involved in the experiment was a personal acquaintance of the study authors raising the distinct possibility of experimenter bias. A randomized clinical trial,<ref>{{Cite journal|doi=10.1016/S0304-3959(00)00421-8|title=Spiritual healing as a therapy for chronic pain: a randomized, clinical trial|author=Abbot, NC, Harkness, EF, Stevinson, C, Marshall, FP, Conn, DA and Ernst, E. | journal=Pain | volume=91 | issue=1–2 | year=2001 | pages=79–89|pmid=11240080}}</ref> published in 2001, investigated the efficacy of spiritual healing (both at a distance and face-to-face) on the treatment of chronic pain in 120 patients. Healers were observed by "simulated healers" who then mimicked the healers movements on a control group while silently counting backwards in fives - a ''neutral'' rather than ''should not heal'' intention. The study found a decrease in pain in all patient groups but "no statistically significant '''differences''' between healing and control groups ... it was concluded that a specific effect of face-to-face or distant healing on chronic pain could not be demonstrated."

=== In physical sciences===
If the signal being measured is actually smaller than the rounding error and the data are over-averaged, a positive result for the measurement can be found in the data where none exists (i.e. a more precise experimental apparatus would conclusively show no such signal). If an experiment is searching for a sidereal variation of some measurement, and if the measurement is rounded-off by a human who knows the [[sidereal time]] of the measurement, and if hundreds of measurements are averaged to extract a "signal" which is smaller than the apparatus' actual resolution, then it should be clear that this "signal" can come from the non-random round-off, and not from the apparatus itself. In such cases a single-blind experimental protocol is required; if the human observer does not know the sidereal time of the measurements, then even though the round-off is non-random it cannot introduce a spurious sidereal variation.

=== In forensic sciences ===
Observer effects are rooted in the universal human tendency to interpret data in a manner consistent with one’s expectations.<ref>{{cite book |first=R. |last=Rosenthal |title=Experimenter Effects in Behavioral Research |location=NY |publisher=Appleton-Century-Crofts |year=1966 |isbn= }}</ref> This tendency is particularly likely to distort the results of a scientific test when the underlying data are ambiguous and the scientist is exposed to domain-irrelevant information that engages emotions or desires.<ref>{{cite journal |first=D. M. |last=Risinger |first2=M. J. |last2=Saks |first3=W. C. |last3=Thompson |first4=R. |last4=Rosenthal |title=The ''Daubert''/''Kumho'' Implications of Observer Effects in Forensic Science: Hidden Problems of Expectation and Suggestion |journal=[[California Law Review|Calif. L. Rev.]] |year=2002 |volume=90 |issue=1 |pages=1–56 |doi= 10.2307/3481305|jstor=3481305 }}</ref> Despite impressions to the contrary, forensic DNA analysts often must resolve ambiguities, particularly when interpreting difficult evidence samples such as those that contain mixtures of DNA from two or more individuals, degraded or inhibited DNA, or limited quantities of DNA template. The full potential of forensic DNA testing can only be realized if observer effects are minimized.<ref>{{Cite journal
| url = http://www.bioforensics.com/articles/sequential_unmasking.html
| author = D. Krane, S. Ford, J. Gilder, K. Inman, A. Jamieson, R. Koppl, I. Kornfield, D. Risinger, N. Rudin, M. Taylor, W.C. Thompson
| title = Sequential unmasking: A means of minimizing observer effects in forensic DNA interpretation
| journal = [[Journal of Forensic Sciences]]
| year = 2008
| volume = 53
| issue = 4
| pages = 1006–1007
| doi = 10.1111/j.1556-4029.2008.00787.x
| pmid = 18638252
}}</ref>

==In social science ==
After the data are collected, bias may be introduced during data interpretation and analysis. For example, in deciding which variables to control in analysis, social scientists often face a trade-off between [[omitted-variable bias]] and post-treatment bias.<ref>King, Gary. [http://gking.harvard.edu/gking/talks/bigprobP.pdf "Post-Treatment Bias in Big Social Science Questions"], accessed February 7, 2011.</ref>

==See also==
* [[Bias]]
* [[Publication bias]]
* [[Confirmation bias]]
* [[List of cognitive biases]]
* [[Systematic bias]]
* [[Funding bias]]

==References==
{{Reflist}}

[[Category:Cognitive biases]]
[[Category:Design of experiments]]
[[Category:Cognitive inertia]]

Polynomial-time algorithm for approximating the volume of convex bodies

2013-12-05T13:11:15Z

117.206.127.232:

{{ref improve|date=March 2013}}
A '''microwave cavity''' or ''[[radio frequency]] (RF) cavity'' is a special type of [[resonator]], consisting of a closed (or largely closed) metal structure that confines [[electromagnetic fields]] in the [[microwave]] region of the spectrum. The structure is either hollow or filled with [[dielectric]] material.

A microwave cavity acts similarly to a [[resonant circuit]] with extremely low loss at its [[frequency]] of operation, resulting in [[quality factor]]s up to the order of 106, compared to 102 for circuits made with separate [[inductors]] and [[capacitors]] at the same frequency. They are used in [[electronic oscillator|oscillator]]s and [[transmitter]]s to create microwave signals, and as [[electronic filter|filter]]s to separate a signal at a given frequency from other signals, in equipment such as [[radar]] equipment, [[microwave relay]] stations, satellite communications, and [[microwave oven]]s.

In addition to their use in electrical networks, RF cavities can manipulate [[charged particle]]s passing through them by application of [[acceleration voltage]] and are thus used in [[particle accelerator]]s.

== Theory of operation ==

Most resonant cavities are made from closed (or short-circuited) sections of [[Waveguide (electromagnetism)|waveguide]] or high-[[permittivity]] [[dielectric]] material (see [[dielectric resonator]]). Electric and magnetic energy is stored in the cavity and the only losses are due to finite [[Electrical resistivity and conductivity|conductivity]] of cavity walls and [[Loss tangent|dielectric losses]] of material filling the cavity. Every cavity has numerous resonant frequencies that correspond to electromagnetic field modes satisfying necessary boundary conditions on the walls of the cavity. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), it follows that cavity length must be an integer multiple of half-wavelength at resonance.<ref name="pozar">David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998.</ref> Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength [[transmission line]] resonator.<ref name="pozar" /> [[Q factor]] of a resonant cavity can be calculated using [[Cavity Perturbation Theory|cavity perturbation theory]] and expressions for stored electric and magnetic energy.

The electromagnetic fields in the cavity are excited via external coupling. An external power source is usually coupled to the cavity by a small [[aperture]], a small wire probe or a loop.<ref name="collin">R. E. Collin, Foundations for Microwave Engineering, 2nd edition, IEEE Press, New York, NY, 2001.</ref> External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.<ref name="montgomery">Montgomery, C. G. & Dicke, Robert H. & Edward M. Purcell, Principles of microwave circuits / edited by C.G. Montgomery, R.H. Dicke, E.M. Purcell, Peter Peregrinus on behalf of the Institution of Electrical Engineers, London, U.K., 1987.</ref>

=== Quality factor ===

The [[quality factor]] <math>\scriptstyle Q</math> of a cavity can be decomposed into three parts, representing different power loss mechanisms.

*<math>\scriptstyle Q_c</math>, resulting from the power loss in the walls which have finite conductivity{{clarify|What is l?|date=October 2012}}
:{{NumBlk|:|<math>Q_c = \frac{(kad)^3b\eta}{2\pi^2R_s} \cdot \frac{1}{l^2a^3\left(2b + d\right) + \left(2b + a\right)d^3}\,</math>|{{EquationRef|3}}}}
*<math>\scriptstyle Q_d</math>, resulting from the power loss in the lossy [[dielectric]] material filling the cavity.
:{{NumBlk|:|<math>Q_d = \frac{1}{\tan \delta}\,</math>|{{EquationRef|4}}}}
*<math>\scriptstyle Q_{ext}</math>, resulting from power loss through unclosed surfaces (holes) of the cavity geometry.

Total Q factor of the cavity can be found as<ref name="pozar" />
{{NumBlk|:|<math>Q = \left( \frac{1}{Q_c}+\frac{1}{Q_d}\right) ^{-1}\,</math>|{{EquationRef|2}}}}

where k is the [[wavenumber]], <math>\scriptstyle \eta</math> is the [[Wave impedance|intrinsic impedance]] of the dielectric, <math>\scriptstyle R_s</math> is the [[Electrical resistivity and conductivity|surface resistivity]] of the cavity walls, <math>\scriptstyle \mu_r</math> and <math>\scriptstyle \epsilon_r</math> are relative [[Permeability (electromagnetism)|permeability]] and [[permittivity]] respectively and <math>\scriptstyle \tan \delta</math> is the [[loss tangent]] of the dielectric.

=== Cavity geometry ===

==== Rectangular cavity ====

[[File:Rectangular cavity.JPG|thumb|Rectangular cavity]]

Resonance frequencies of a rectangular microwave cavity for any [[Transverse mode|<math>\scriptstyle TE_{mnl}</math>]] or [[Transverse mode|<math>\scriptstyle TM_{mnl}</math>]] resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by<ref name="pozar" />

{{NumBlk|:|<math>\begin{align}
f_{mnl} &= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\cdot k_{mnl}\\
&= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 + \left(\frac{l\pi}{d}\right)^2}\\
&= \frac{c}{2\sqrt{\mu_r\epsilon_r}}\sqrt{\left( \frac{m}{a}\right) ^2+\left(\frac{n}{b}\right) ^2 + \left(\frac{l}{d}\right) ^2}
\end{align}</math>|{{EquationRef|1}}}}

where <math>\scriptstyle k_{mnl}</math> is the [[wavenumber]], with <math>\scriptstyle m</math>, <math>\scriptstyle n</math>, <math>\scriptstyle l</math> being the mode numbers and <math>\scriptstyle a</math>, <math>\scriptstyle b</math>, <math>\scriptstyle d</math> being the corresponding dimensions; c is the speed of light in vacuum; and <math>\scriptstyle \mu_r</math> and <math>\scriptstyle \epsilon_r</math> are relative [[Permeability (electromagnetism)|permeability]] and [[permittivity]] respectively.

==== Cylindrical cavity ====

[[File:Cylindrical cavity.svg|thumb|Cylindrical cavity]]

The field solutions of a cylindrical cavity of length <math>\scriptstyle L</math> and radius <math>\scriptstyle R</math> follow from the solutions of a cylindrical [[waveguide]] with additional electric boundary conditions at the position of the enclosing plates. The resonance frequencies are different for TE and TM modes.

;TM modes:<ref name="wangler">T. Wangler, ''RF linear accelerators'', Wiley (2008)</ref> <math>f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2}</math>

;TE modes:<ref name="wangler" /> <math>f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X'_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2}</math>

Here, <math>\scriptstyle X_{mn}</math> denotes the <math>\scriptstyle n</math>-th zero of the <math>\scriptstyle m</math>-th [[Bessel function]], and <math>\scriptstyle X'_{mn}</math> denotes the <math>\scriptstyle n</math>-th zero of the ''derivative'' of the <math>\scriptstyle m</math>-th Bessel function.

== Comparison to LC circuits ==

[[File:LC cavity.JPG|thumb|LC cavity|LC circuit equivalent for microwave resonant cavity]]

Microwave resonant cavities can be represented and thought of as simple [[LC circuit]]s.<ref name="montgomery" /> For a microwave cavity, the stored electric energy is equal to the stored magnetic energy at resonance as is the case for a resonant [[LC circuit]]. In terms of inductance and capacitance, the resonant frequency for a given <math>\scriptstyle mnl</math> mode can be written as<ref name="montgomery" />

{{NumBlk|:|<math>L_{mnl} = \mu k_{mnl}^2V\,</math>|{{EquationRef|6}}}}

{{NumBlk|:|<math>C_{mnl} = \frac{\epsilon}{k_{mnl}^4V}\,</math>|{{EquationRef|7}}}}

{{NumBlk|:|<math>\begin{align}
f_{mnl} &= \frac{1}{2\pi\sqrt{L_{mnl}C_{mnl}}}\\
&= \frac{1}{2\pi\sqrt{\frac{1}{k_{mnl}^2} \mu\epsilon}}
\end{align}</math>|{{EquationRef|5}}}}

where V is the cavity volume, <math>\scriptstyle k_{mnl}</math> is the mode wavenumber and <math>\scriptstyle \epsilon</math> and <math>\scriptstyle \mu</math> are permittivity and permeability respectively.

To better understand the utility of resonant cavities at microwave frequencies, it is useful to note that the losses of conventional inductors and capacitors start to increase with frequency in the [[VHF]] range. Similarly, for frequencies above one [[gigahertz]], Q factor values for transmission-line resonators start to decrease with frequency.<ref name="collin" /> Because of their low losses and high Q factors, cavity resonators are preferred over conventional LC and transmission-line resonators at high frequencies.

=== Losses in LC resonant circuits ===

Conventional inductors are usually wound from wire in the shape of a [[helix]] with no core. [[Skin effect]] causes the high frequency resistance of inductors to be many times their [[direct current]] resistance. In addition, capacitance between turns causes [[dielectric]] losses in the [[Insulator (electrical)|insulation]] which coats the wires. These effects make the high frequency resistance greater and decrease the Q factor.

Conventional capacitors use [[air]], [[mica]], [[ceramic]] or perhaps [[teflon]] for a dielectric. Even with a low loss dielectric, capacitors are also subject to skin effect losses in their [[lead (electronics)|leads]] and [[plate (electronics)|plates]]. Both effects increase their [[equivalent series resistance]] and reduce their Q.

Even if the Q factor of VHF inductors and capacitors is high enough to be useful, their [[Parasitic element (electrical networks)|parasitic]] properties can significantly affect their performance in this frequency range. The shunt capacitance of an inductor may be more significant than its desirable series inductance. The series inductance of a capacitor may be more significant than its desirable shunt capacitance. As a result, in the VHF or microwave regions, a capacitor may appear to be an inductor and an inductor may appear to be a capacitor. These phenomena are better known as [[Parasitic element (electrical networks)|parasitic inductance]] and [[parasitic capacitance]].

=== Losses in cavity resonators ===

Dielectric loss of air is extremely low for high frequency electric or magnetic fields. Air-filled microwave cavities confine electric and magnetic fields to the air spaces between their walls. Electric losses in such cavities are almost exclusively due to currents flowing in cavity walls. While losses from wall currents are small, cavities are frequently [[plated]] with [[silver]] to increase their [[electrical conductivity]] and reduce these losses even further. [[Copper]] cavities frequently [[oxidize]], which increases their loss. Silver or [[gold]] plating prevents oxidation and reduces electrical losses in cavity walls. Even though gold is not quite as good a conductor as copper, it still prevents oxidation and the resulting deterioration of Q factor over time. However, because of its high cost, it is used only in the most demanding applications.

Some satellite resonators are silver plated and covered with a gold flash layer. The current then mostly flows in the high-conductivity silver layer, while the gold flash layer protects the silver layer from oxidizing.

== References ==
{{Reflist}}

[[Category:Microwave technology]]
[[Category:Accelerator physics]]

Thermal effusivity

2013-04-28T09:43:13Z

117.206.40.122:

In [[information retrieval]], '''Okapi BM25''' is a [[ranking function]] used by [[search engine]]s to rank matching documents according to their [[Relevance (information retrieval)|relevance]] to a given search query. It is based on the [[Probabilistic relevance model|probabilistic retrieval framework]] developed in the 1970s and 1980s by [[Stephen E. Robertson]], [[Karen Spärck Jones]], and others.

The name of the actual ranking function is BM25. To set the right context, however, it usually referred to as "Okapi BM25", since the Okapi information retrieval system, implemented at [[London]]'s [[City University, London|City University]] in the 1980s and 1990s, was the first system to implement this function.

BM25, and its newer variants, e.g. BM25F (a version of BM25 that can take document structure and anchor text into account), represent state-of-the-art [[TF-IDF]]-like retrieval functions used in document retrieval, such as Web search.

== The ranking function ==

BM25 is a [[Bag of words model|bag-of-words]] retrieval function that ranks a set of documents based on the query terms appearing in each document, regardless of the inter-relationship between the query terms within a document (e.g., their relative proximity). It is not a single function, but actually a whole family of scoring functions, with slightly different components and parameters. One of the most prominent instantiations of the function is as follows.

Given a query <math>Q</math>, containing keywords <math>q_1, ..., q_n</math>, the BM25 score of a document <math>D</math> is:

:<math> \text{score}(D,Q) = \sum_{i=1}^{n} \text{IDF}(q_i) \cdot \frac{f(q_i, D) \cdot (k_1 + 1)}{f(q_i, D) + k_1 \cdot (1 - b + b \cdot \frac{|D|}{\text{avgdl}})},</math>

where <math>f(q_i, D)</math> is <math>q_i</math>'s [[term frequency]] in the document <math>D</math>, <math>|D|</math> is the length of the document <math>D</math> in words, and <math>avgdl</math> is the average document length in the text collection from which documents are drawn. <math>k_1</math> and <math>b</math> are free parameters, usually chosen, in absence of an advanced optimization, as <math>k_1 \in [1.2,2.0]</math> and <math>b = 0.75</math>.<ref>Christopher D. Manning, Prabhakar Raghavan, Hinrich Schütze. ''An Introduction to Information Retrieval'', Cambridge University Press, 2009, p. 233.</ref> <math>\text{IDF}(q_i)</math> is the IDF ([[inverse document frequency]]) weight of the query term <math>q_i</math>. It is usually computed as:

:<math>\text{IDF}(q_i) = \log \frac{N - n(q_i) + 0.5}{n(q_i) + 0.5},</math>

where <math>N</math> is the total number of documents in the collection, and <math>n(q_i)</math> is the number of documents containing <math>q_i</math>.

There are several interpretations for IDF and slight variations on its formula. In the original BM25 derivation, the IDF component is derived from the [[Binary Independence Model]].

Please note that the above formula for IDF shows potentially major drawbacks when using it for terms appearing in more than half of the corpus documents. These terms' IDF is negative, so for any two almost-identical documents, one which contains the term and one which does not contain it, the latter will possibly get a larger score.
This means that terms appearing in more than half of the corpus will provide negative contributions to the final document score. This is often an undesirable behavior, so many real-world applications would deal with this IDF formula in a different way:

* Each summand can be given a floor of 0, to trim out common terms;
* The IDF function can be given a floor of a constant <math>\epsilon</math>, to avoid common terms being ignored at all;
* The IDF function can be replaced with a similarly shaped one which is non-negative, or strictly positive to avoid terms being ignored at all.

== IDF Information Theoretic Interpretation ==
Here is an interpretation from information theory. Suppose a query term <math>q</math> appears in <math>n(q)</math> documents. Then a randomly picked document <math>D</math> will contain the term with probability <math>\frac{n(q)}{N}</math> (where <math>N</math> is again the cardinality of the set of documents in the collection). Therefore, the [[information]] content of the message "<math>D</math> contains <math>q</math>" is:

:<math>-\log \frac{n(q)}{N} = \log \frac{N}{n(q)}.</math>

Now suppose we have two query terms <math>q_1</math> and <math>q_2</math>. If the two terms occur in documents entirely independently of each other, then the probability of seeing both <math>q_1</math> and <math>q_2</math> in a randomly picked document <math>D</math> is:

:<math>\frac{n(q_1)}{N} \cdot \frac{n(q_2)}{N},</math>

and the information content of such an event is:

:<math>\sum_{i=1}^{2} \log \frac{N}{n(q_i)}.</math>

With a small variation, this is exactly what is expressed by the IDF component of BM25.

== Modifications ==
* At the extreme values of the coefficient <math>b</math> BM25 turns into ranking functions known as '''BM11''' (for <math>b=1</math>) and '''BM15''' (for <math>b=0</math>).<ref>http://xapian.org/docs/bm25.html</ref>
* '''BM25F'''<ref>Hugo Zaragoza, Nick Craswell, Michael Taylor, Suchi Saria, and Stephen Robertson. [http://trec.nist.gov/pubs/trec13/papers/microsoft-cambridge.web.hard.pdf ''Microsoft Cambridge at TREC-13: Web and HARD tracks.''] In Proceedings of TREC-2004.</ref> is a modification of BM25 in which the document is considered to be composed from several fields (such as headlines, main text, anchor text) with possibly different degrees of importance.

== Footnotes ==
{{Reflist}}

== References ==
* Stephen E. Robertson, Steve Walker, Susan Jones, Micheline Hancock-Beaulieu, and Mike Gatford. Okapi at TREC-3. In Proceedings of the Third Text REtrieval Conference (TREC 1994). Gaithersburg, USA, November 1994.

* Stephen E. Robertson, Steve Walker, and Micheline Hancock-Beaulieu. Okapi at TREC-7. In Proceedings of the Seventh Text REtrieval Conference. Gaithersburg, USA, November 1998.

* Karen Spärck Jones, Steve Walker, and Stephen E. Robertson. A Probabilistic Model of Information Retrieval: Development and Comparative Experiments (parts 1 and 2). Information Processing and Management, 36(6):779-840. 2000.

== External links ==
* [http://www.soi.city.ac.uk/~ser/papers/foundations_bm25_review.pdf The Probabilistic Relevance Framework: BM25 and Beyond]
* [http://nlp.uned.es/~jperezi/Lucene-BM25/ BM25 implementation for Lucene]

[[Category:Ranking functions]]
[[Category:Information retrieval]]

Theory of computation

2012-09-01T06:02:59Z

117.206.192.131: /* Computability theory */

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117.206.21.38: /* Applications in mechanics */

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