en>Monkbot: /* References */Task 5: Fix CS1 deprecated coauthor parameter errors

2014-07-24T03:54:24Z

References: Task 5: Fix CS1 deprecated coauthor parameter errors

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-Since the arithmetic mean is not always appropriate for angles, the following method can be used to obtain both a mean value and measure for the [[variance]] of the angles:
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-Convert all angles to corresponding points on the [[unit circle]], e.g., <math>\alpha</math> to <math>(\cos\alpha,\sin\alpha)</math>. That is, convert [[polar coordinates]] to [[Cartesian coordinates]]. Then compute the [[arithmetic mean]] of these points. The resulting point will lie on the unit disk. Convert that point back to polar coordinates. The angle is a reasonable mean of the input angles. The resulting radius will be 1 if all angles are equal. If the angles are uniformly distributed on the circle, then the resulting radius will be 0, and there is no circular mean. In other words, the radius measures the concentration of the angles.
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-Given the angles <math>\alpha_1,\dots,\alpha_n</math> the mean is computed by
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206.49.107.56 at 03:35, 7 November 2013

2013-11-07T03:35:03Z

← Older revision		Revision as of 05:35, 7 November 2013
Line 1:		Line 1:
	~~Oscar~~ is ~~how he'~~s ~~known~~ as and ~~he totally enjoys this title~~. ~~His spouse doesn't like it~~ the ~~way he does but what he truly likes performing~~ is to ~~do aerobics~~ and ~~he's been performing it~~ for ~~fairly~~ a ~~whilst~~. ~~Years ago he moved to North Dakota~~ and ~~his family members enjoys it~~. ~~My working day occupation~~ is ~~a meter reader~~.<br><br>~~my website~~ :: [http://www.~~fuguporn~~.com/~~user~~/~~RMitchell www~~.~~fuguporn.com~~]		In [[mathematics]], a '''mean of circular quantities''' is a [[mean]] which is sometimes better-suited for quantities like [[angle]]s, [[hour\|daytime]]s, and [[fractional part]]s of [[real number]]s. This is necessary since most of the usual means may not be appropriate on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because for most purposes 360° is the same thing as 0°.<ref> Christopher M. Bishop: ''Pattern Recognition and Machine Learning (Information Science and Statistics)'', ISBN 0-387-31073-8 </ref> As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. This is one of the simplest examples of [[statistics of non-Euclidean spaces]].

			== Mean of angles ==

			Since the arithmetic mean is not always appropriate for angles, the following method can be used to obtain both a mean value and measure for the [[variance]] of the angles:

			Convert all angles to corresponding points on the [[unit circle]], e.g., <math>\alpha</math> to <math>(\cos\alpha,\sin\alpha)</math>. That is, convert [[polar coordinates]] to [[Cartesian coordinates]]. Then compute the [[arithmetic mean]] of these points. The resulting point will lie on the unit disk. Convert that point back to polar coordinates. The angle is a reasonable mean of the input angles. The resulting radius will be 1 if all angles are equal. If the angles are uniformly distributed on the circle, then the resulting radius will be 0, and there is no circular mean. In other words, the radius measures the concentration of the angles.

			Given the angles <math>\alpha_1,\dots,\alpha_n</math> the mean is computed by

			:<math>\bar{\alpha} = \operatorname{atan2}\left(\frac{1}{n}\cdot\sum_{j=1}^n \sin\alpha_j, \frac{1}{n}\cdot\sum_{j=1}^n \cos\alpha_j\right) </math>

			using the [[atan2]] variant of the [[arctangent]] function, or

			:<math>\bar{\alpha} = \arg\left(\frac{1}{n}\cdot\sum_{j=1}^n \exp(i\cdot\alpha_j)\right) </math>

			using [[complex number]]s.

			== Properties ==

			The mean <math>\bar{\alpha}</math>
			* maximizes the [[likelihood]] of the mean parameter of the [[von Mises distribution]] and
			* minimizes the sum of a certain distance on the circle, more precisely
			::<math>\bar{\alpha} = \underset{\beta}{\operatorname{argmin}} \sum_{j=1}^n d(\alpha_j,\beta)</math>, where <math>d(\varphi,\beta) = 1-\cos(\varphi-\beta).</math>
			:The distance <math>d(\varphi,\beta)</math> is equal to half the squared [[Euclidean distance]] between the two points on the unit circle associated with <math>\varphi</math> and <math>\beta</math>.

			== See also ==

			* [[Directional statistics]]
			* [[Circular distribution]]

			== References ==

			<references />

			==External links==
			* [http://www.codeproject.com/Articles/190833/Circular-Values-Math-and-Statistics-with-Cplusplus Circular Values Math and Statistics with C++11], A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics


			[[Category:Means]]
			[[Category:Directional statistics]]

en>ZéroBot: r2.7.1) (Robot: Adding ja:ロビン境界条件

2012-06-28T13:16:51Z

r2.7.1) (Robot: Adding ja:ロビン境界条件

New page

Oscar is how he's known as and he totally enjoys this title. His spouse doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been performing it for fairly a whilst. Years ago he moved to North Dakota and his family members enjoys it. My working day occupation is a meter reader.<br><br>my website :: [http://www.fuguporn.com/user/RMitchell www.fuguporn.com]

Robin boundary condition - Revision history

en>Monkbot: /* References */Task 5: Fix CS1 deprecated coauthor parameter errors

206.49.107.56 at 03:35, 7 November 2013

en>ZéroBot: r2.7.1) (Robot: Adding ja:ロビン境界条件