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| | On-line games can give you a meaningful universe of experience, relax and exhilaration. You may learn, get a sound judgment of success or merely enjoy beating down specific bad-guy. No matter form of video online video video games you are into, add the helpful tips in this particular post to give your self more fun whenever any person play your next pc game Website.<br><br> |
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| [[Image:Michelsonmorley-boxplot.svg|thumb|Figure 1. [[Box plot]] of data from the [[Michelson–Morley experiment]] displaying outliers in the middle column.]]
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| In [[statistics]], an '''outlier''' is an observation point that is distant from other observations.<ref>{{Citation |last=Grubbs |first=F. E. |date=February 1969 |title=Procedures for detecting outlying observations in samples |journal=Technometrics |volume=11 |issue=1 |pages=1–21 |doi= |quote=An outlying observation, or "outlier," is one that appears to deviate markedly from other members of the sample in which it occurs.}}</ref> An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the [[data set]].<ref>{{harvnb|Grubbs|1969|p=1}} stating "An outlying observation may be merely an extreme manifestation of the random variability inherent in the data. ... On the other hand, an outlying observation may be the result of gross deviation from prescribed experimental procedure or an error in calculating or recording the numerical value."</ref>
| | Interweaving social styles form an actual strong net in and we are all captured. When The Tygers of Pan Tang sang 'It's lonely at most of the top. Everybody's your to do you in', these people borrowed significant from clash of clans hack tool no feedback. A society owning clash of [http://en.Wiktionary.org/wiki/clans+chop clans chop] tool no survey is ordinarily like a society that includes no knowledge, in which often it is quite really good.<br><br>For those who have little ones who delight in video games, then you are aware how challenging it really in order to use pull them out of this t. v.. Their eye can automatically be stuck towards the watch out for for hours as they play their preferred games. If you want aid regulating your children's clash of clans Hack time, your pursuing article has some suggestions for you.<br><br>Control system game playing is excellent kids. Consoles present you with far better control linked content and safety, merely kids can simply wind turbine by way of parent or guardian regulates on your notebook computer. Using this step might help to protect your young ones between harm.<br><br>Always your game just as much as possible. While [https://Www.google.com/search?hl=en&gl=us&tbm=nws&q=car-preservation car-preservation] is a good characteristic, do not count regarding this. If you have any concerns with regards to in which and how to use [http://prometeu.net clash of clans hack free download], you can get hold of us at our web-page. Particularly, when you initial start playing a game, you may not provide any thought when currently the game saves, which might result in a drop of significant info the day after tomorrow. Until you thoroughly grasp the sport better, systematically save yourself.<br><br>Individual world can be forced by supply and great quality. We shall look during the the Greek-Roman model. Using special care to highlight the role including clash of clans hack tool no survey inside of a the vast framework this also usually this provides.<br><br>These are not really cheats, they are excuses. The odds are really good that unless you can be found dating a certain pro golfer or a are insane star along the possibility this is not probably gonna happen to you. In John 4:23 not to mention 24 Jesus tells usa we are to worship God "in spirit additionally truth. Once entered, the Ruzzle cheat can show a list dried up possible words that can be done. Using a PSP Casino game Emulator is a straightforward way to hack a new PSP and open to the peak new worlds of fun. s these university students played Internet poker and other casino pastimes simply for fun. |
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| Outliers can occur by chance in any distribution, but they are often indicative either of [[measurement error]] or that the population has a [[heavy-tailed distribution]]. In the former case one wishes to discard them or use statistics that are [[robust statistic|robust]] to outliers, while in the latter case they indicate that the distribution has high [[kurtosis]] and that one should be very cautious in using tools or intuitions that assume a [[normal distribution]]. A frequent cause of outliers is a mixture of two distributions, which may be two distinct sub-populations, or may indicate 'correct trial' versus 'measurement error'; this is modeled by a [[mixture model]].
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| In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable. This can be due to incidental [[systematic error]] or flaws in the [[theory]] that generated an assumed family of [[probability distribution]]s, or it may be that some observations are far from the center of the data. Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected (and not due to any anomalous condition).
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| Outliers, being the most extreme observations, may include the [[sample maximum]] or [[sample minimum]], or both, depending on whether they are extremely high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations.
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| Naive interpretation of statistics derived from data sets that include outliers may be misleading. For example, if one is calculating the [[average]] temperature of 10 objects in a room, and nine of them are between 20 and 25 [[degrees Celsius]], but an oven is at 175 °C, the [[median]] of the data will be between 20 and 25 °C but the [[mean]] temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object than the mean; naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect. As illustrated in this case, outliers may be indicative of data points that belong to a different [[Statistical population|population]] than the rest of the [[Sample (statistics)|sample]] set.
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| [[Estimator]]s capable of coping with outliers are said to be [[Robust statistics|robust]]: the median is a robust statistic, while the mean is not.<ref>Ripley, Brian D. 2004. [http://www.stats.ox.ac.uk/pub/StatMeth/Robust.pdf Robust statistics]</ref>
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| == Occurrence and causes ==
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| In the case of [[normal distribution|normally distributed]] data, the [[three sigma rule]] means that roughly 1 in 22 observations will differ by twice the standard deviation or more from the mean, and 1 in 370 will deviate by three times the standard deviation<ref>Ruan, D., Chen, G., Kerre, E.E. and Wets G.(Eds.): Intelligent Data Mining: Techniques and Applications. Studies in Computational Intelligence Vol. 5 Springer 2005, ISBN 978-3-540-26256-5 (page 318)</ref> for details. In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see [[Poisson distribution]], and not indicative of an anomaly. If the sample size is only 100, however, just three such outliers are already reason for concern, being more than 11 times the expected number.
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| In general, if the nature of the population distribution is known a priori, it is possible to test if the number of outliers deviate [[Statistical significance|significant]]ly from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability ''p'') of a given distribution, the number of outliers will follow a [[binomial distribution]] with parameter ''p'', which can generally be well-approximated by the [[Poisson distribution]] with λ = ''pn''. Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, ''p'' is approximately .3%, and thus for 1,000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3.
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| === Causes ===
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| Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered a transient malfunction. There may have been an error in data transmission or transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error or simply through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end ([[King effect]]).
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| == Identifying outliers ==
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| There is no rigid mathematical definition of what constitutes an outlier; determining whether or not an observation is an outlier is ultimately a subjective exercise.
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| Outlier detection<ref>[[Peter Rousseeuw|Rousseeuw, P.]] and Leroy, A.: 1996, Robust Regression and Outlier Detection. John
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| Wiley & Sons., 3rd edition.</ref><ref>Victoria J. Hodge and Jim Austin, A Survey of Outlier Detection Methodologies, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.109.1943&rep=rep1&type=pdf</ref>
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| Model-based methods which are commonly used for identification assume that the data are from a normal distribution, and identify observations which are deemed "unlikely" based on [[mean]] and [[standard deviation]]:
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| * [[Chauvenet's criterion]]
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| * [[Grubbs' test for outliers]]
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| * [[Peirce's criterion]]<ref>[[Benjamin Peirce]], [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1852AJ......2..161P;data_type=PDF_HIGH "Criterion for the Rejection of Doubtful Observations"], ''Astronomical Journal'' II 45 (1852) and [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1852AJ......2..176P;data_type=PDF_HIGH Errata to the original paper].</ref><ref>{{cite journal
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| |title=On Peirce's criterion
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| |authorlink=Benjamin Peirce
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| |first=Benjamin
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| |last=Peirce
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| |journal=Proceedings of the [[American Academy of Arts and Sciences]]
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| |volume=13
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| |month=May
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| |year=1877–1878
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| |pages=348–351
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| |jstor=25138498
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| |doi=10.2307/25138498
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| }}</ref><ref>{{cite journal
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| |first=Charles Sanders
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| |last=Peirce
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| |authorlink=Charles Sanders Peirce
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| |title=Appendix No. 21. On the Theory of Errors of Observation
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| |journal=Report of the Superintendent of the United States [[Coast Survey]] Showing the Progress of the Survey During the Year 1870
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| |year=1870 [published 1873]
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| |pages=200–224
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| }}. NOAA [http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0019.PDF#page=215 PDF Eprint] (goes to Report p. 200, PDF's p. 215).</ref><ref>{{cite book
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| |first=Charles Sanders
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| |last=Peirce
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| |authorlink=Charles Sanders Peirce
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| |contribution=On the Theory of Errors of Observation [Appendix 21, according to the editorial note on page 515]
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| |title=Writings of Charles S. Peirce: A Chronological Edition
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| |volume=Volume 3, 1872-1878
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| |editor=Kloesel, Christian J. W., ''et alia''
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| |publisher=Indiana University Press
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| |location=Bloomington, Indiana <!-- |copyright=1986 -->
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| |year=1982 [1986 copyright]<!-- copyright=1986, but publication is listed as 1982 -->
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| |pages=140–160
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| |isbn=0-253-37201-1
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| }}</ref>
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| <blockquote>
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| It is proposed to determine in a series of <math>m</math> observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as <math>n</math> such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations. (Quoted in the editorial note on page 516 to Peirce (1982 edition) from ''A Manual of Astronomy'' 2:558 by Chauvenet.)
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| </blockquote>
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| * [[Dixon's Q test]]
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| * [[ASTM]] E178 Standard Practice for Dealing With Outlying Observations
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| Other methods flag observations based on measures such as the [[interquartile range]]. For example, if <math>Q_1</math> and <math>Q_3</math> are the lower and upper [[quartile]]s respectively, then one could define an outlier to be any observation outside the range:
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| :<math> \big[ Q_1 - k (Q_3 - Q_1 ) , Q_3 + k (Q_3 - Q_1 ) \big]</math>
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| for some nonnegative constant <math>k</math>.
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| Other approaches are distance-based<ref>Knorr, E. M. and Ng, R. T.: 1998, Algorithms for Mining Distance-Based Outliers
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| in Large Datasets. In: Proceedings of the VLDB Conference. New York, USA, pp. 392–403</ref><ref>Ramaswamy, S., Rastogi, R., and Shim, K.: 2000, ‘Efficient Algorithms for Mining Outliers from Large Data Sets’. In: Proceedings of the ACM SIGMOD Conference on Management of Data. Dallas, TX, pp.427–438.</ref> and density-based,<ref>Markus Breunig and Hans-Peter Kriegel and Raymond T. Ng and Jörg Sander: 2000, LOF: Identifying Density-Based Local Outliers. In: Proceedings of the ACM SIGMOD Conference. pp. 93-104</ref> and all of them frequently use the distance to the [[k-nearest neighbor]]s to label observations as outliers or non-outliers.
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| * Modified Thompson Tau test<ref>{{cite web | url=http://www.mne.psu.edu/me345/Lectures/Outliers.pdf | title=Outliers | date=September 12, 2011 | author=John M. Cimbala }}</ref>
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| The modified Thompson Tau test is a method used to determine if outlier exists in a data set. The strength of this method lies in the fact that it takes into account a data set’s standard deviation, average and provides a statistically determined rejection zone; thus providing an objective method to determine if a data point is an outlier.
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| How it works:
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| First, a data set's average is determined. Next the absolute deviation between each data point and the average are determined. Thirdly, a rejection region is determined using the formula: <math>Rejection Region{{=}} \frac{{t_{\alpha/2}}{n-1}}{\sqrt{n}\sqrt{n-2+{t_{\alpha/2}^2}}}
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| </math>; where <math>{t_{\alpha/2}}</math> is the critical value from the Student t distribution, n is the sample size, and s is the sample standard deviation. | |
| To determine if a value is an outlier:
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| If δ > Rejection Region, the data point is an outlier.
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| If δ ≤ Rejection Region, the data point is not an outlier.
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| The modified Thompson Tau test is used to find one outlier at a time (largest value of δ is removed if it is an outlier). Meaning, if a data point is found to be an outlier, it is removed from the data set and the test is applied again with a new average and rejection region. This process is continued until no outliers remain in a data set.
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| == Working with outliers ==
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| The choice of how to deal with an outlier should depend on the cause. | |
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| === Retention ===
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| Even when a normal distribution model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that is the case. The application should use a classification algorithm that is robust to outliers to model data with naturally occurring outlier points.
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| === Exclusion ===
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| Deletion of outlier data is a controversial practice frowned on by many scientists and science instructors; while mathematical criteria provide an objective and quantitative method for data rejection, they do not make the practice more scientifically or methodologically sound, especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. An outlier resulting from an instrument reading error may be excluded but it is desirable that the reading is at least verified.
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| In [[Regression analysis|regression]] problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the parameters, using a measure such as [[Cook's distance]].<ref>Cook, R. Dennis (Feb 1977). "Detection of Influential Observations in Linear Regression". Technometrics (American Statistical Association) 19 (1): 15–18.</ref> | |
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| If a data point (or points) is excluded from the [[data analysis]], this should be clearly stated on any subsequent report.
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| === Non-normal distributions ===
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| The possibility should be considered that the underlying distribution of the data is not approximately normal, having "[[fat tails]]". For instance, when sampling from a [[Cauchy distribution]],<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/CauchyDistribution.html Cauchy Distribution. From MathWorld--A Wolfram Web Resource]</ref> the sample variance increases with the sample size, the sample mean fails to converge as the sample size increases, and outliers are expected at far larger rates than for a normal distribution.
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| === Set-membership uncertainties ===
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| A [[set estimation|set membership approach]] considers that the uncertainty corresponding to the ''i''th measurement
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| of an unknown random vector ''x'' is represented by a set ''X''<sub>i</sub> (instead of a probability density function).
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| If no outliers occur, ''x'' should belong to the intersection of all ''X''<sub>i</sub>'s.
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| When outliers occur, this intersection could be empty, and we should relax a small number
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| of the sets ''X''<sub>i</sub> (as small as possible) in order to avoid any inconsistency.
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| <ref>
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| {{cite journal|last1=Jaulin|first1=L.|
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| title=Probabilistic set-membership approach for robust regression|
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| journal=Journal of Statistical Theory and Practice|
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| year=2010|
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| url=http://www.ensta-bretagne.fr/jaulin/paper_probint_0.pdf}}
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| </ref>
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| This can be done using the notion of ''q''-relaxed intersection.
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| As illustrated by the figure, the ''q''-relaxed intersection corresponds to the set of all ''x'' which
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| belong to all sets except ''q'' of them.
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| Sets ''X''<sub>i</sub> that do not intersect the ''q''-relaxed intersection
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| could be suspected to be outliers.
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| [[File:Wiki q inter def.jpg|thumb| Figure 1. ''q''-relaxed intersection of 6 sets for ''q''=2 (red), ''q''=3 (green), ''q''= 4 (blue), ''q''= 5 (yellow).]]
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| === Alternative models ===
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| In cases where the cause of the outliers is known, it may be possible to incorporate this effect into the model structure, for example by using a [[hierarchical Bayes model]] or a [[mixture model]].<ref>Roberts, S. and Tarassenko, L.: 1995, A probabilistic resource allocating network for novelty detection. Neural Computation 6, 270–284.</ref><ref>{{Cite journal |last= Bishop |first=C. M. |date= August 1994 |title= Novelty detection and Neural Network validation |journal= Proceedings of the IEE Conference on Vision, Image and Signal Processing |volume=141 |issue=4 |pages= 217–222 |doi=10.1049/ip-vis:19941330 |postscript= <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| == See also ==
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| * [[Anomaly detection]]
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| * [[Anomaly time series]]
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| * [[Robust statistics]]
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| * [[Robust regression]]
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| * [[Box plot]]
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| * [[King effect]]
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| * [[Studentized residual]]
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| * [[Winsorising]]
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| * [[Data transformation (statistics)]]
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| * [[Local Outlier Factor]]
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| == References ==
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| {{Reflist|30em}}
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| * ISO 16269-4, Statistical interpretation of data — Part 4: Detection and treatment of outliers
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| * {{Cite book| last = Strutz| first = Tilo | title = Data Fitting and Uncertainty - A practical introduction to weighted least squares and beyond | publisher = Vieweg+Teubner| year = 2010| isbn= 978-3-8348-1022-9}}
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| ==External links==
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| * {{MathWorld|Outlier|author=Renze, John}}
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| * {{SpringerEOM|id=O/o110080|title=Outlier|first1=N. |last1=Balakrishnan |first2=A. |last2=Childs}}
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| * [http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h.htm Grubbs test] described by NIST manual
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| * [http://www.psychwiki.com/wiki/Detecting_Outliers_-_Univariate how to detect univariate outliers], [http://www.psychwiki.com/wiki/Detecting_Outliers_-_Multivariate how to detect multivariate outliers] and [http://www.psychwiki.com/wiki/Dealing_with_Outliers how to deal with outliers]
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| [[Category:Statistical charts and diagrams]]
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| [[Category:Statistical terminology]]
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| [[Category:Data analysis]]
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| [[Category:Robust statistics]]
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| [[Category:Statistical outliers| ]]
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