Decimal degrees - Revision history

128.46.119.2: /* Precision */ using better unit prefixes

2014-12-18T20:57:39Z

Precision: using better unit prefixes

← Older revision		Revision as of 22:57, 18 December 2014
Line 1:		Line 1:
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164.151.4.18: /* Accuracy */

2014-02-11T04:55:18Z

Accuracy

← Older revision		Revision as of 06:55, 11 February 2014
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	~~The '''Hartley function'''~~ is ~~a measure of uncertainty, introduced by [[Ralph Hartley]] in 1928. If we pick a sample from a finite set~~ '~~'A'' uniformly at random, the information revealed after we know the outcome is given by the Hartley function~~		Nice to satisfy you, my name is Numbers Held though I don't really like being called like that. His wife doesn't like it the way he [http://biztoc.com/mod/groups/topicposts.php?topic=1311479&group_guid=1311468 http://biztoc.com/mod/groups/topicposts.php?topic=1311479&group_guid=1311468] does but what he truly likes performing is to do aerobics and he's been performing it at home std testing for fairly a whilst. My working day occupation is a meter reader. home std test kit South Dakota is exactly where I've usually been [http://chastity.com/chastity-qa/stds/infections/whats-most-common-std/whats-most residing].<br><br>Take a [http://www.dev.adaptationlearning.net/solid-advice-regards-yeast-infection http://www.dev.adaptationlearning.net] look at my weblog ... std testing at home, [http://flashtrue.com/stay-yeast-infection-free-using-these-useful-tips/ Read the Full Report],
	~~:<math> H_0(A) := \mathrm{log}_b \vert A \vert .</math>~~
	If the [[base (exponentiation)\|base]] of the [[logarithm]] is 2, then the uncertainty is measured in bits. If it is the [[natural logarithm]], then the unit is [[Nat (information)\|nats]]. (Hartley himself used a base-ten logarithm, and this unit of information is sometimes called ~~the '''[[Ban (information)\|hartley]]''' in his honor.) It is also known as the Hartley entropy~~.

	~~== Hartley function, Shannon~~'~~s entropy, and Rényi entropy ==~~
	~~The Hartley function coincides with~~ the ~~[[Shannon entropy]] (as well as with the Rényi entropies of all orders) in the case of a uniform probability distribution.~~ ~~It is actually a special case of the [~~[~~Rényi entropy]] since~~:
	~~:<math>H_0(X) = \frac 1 {1-0} \log \sum_{i=1}^{\|X\|} p_i^0 = \log \|X\|~~.</~~math>~~

	But it can also be viewed as a primitive construction, since, as emphasized by Kolmogorov and Rényi (see George, J. Klirr's "Uncertainty and information", p.423), the Hartley function can be defined without introducing any notions of probability.

	==~~Characterization of the Hartley function==~~
	~~The Hartley function only depends on the number of elements in a set, and hence can be viewed as a function on natural numbers~~. ~~Rényi showed that the Hartley function in base 2 is the only function mapping natural numbers to real numbers that satisfies~~

	~~# <math>H(mn) = H(m)+H(n)<~~/~~math> (additivity)~~
	~~# <math>H(m) \leq H(m+1)<~~/~~math> (monotonicity)~~
	~~# <math>H(2)=1<~~/~~math> (normalization)~~

	Condition 1 says that the uncertainty of the Cartesian product of two finite sets ''A'' and ''B'' is the sum of uncertainties of ''A'' and ''B''. Condition 2 says that larger set has larger uncertainty.

	==~~Derivation of the Hartley function==~~
	~~We want to show that the Hartley function, log<sub>2</sub>(''n''),~~ is ~~the only function mapping natural numbers~~ to ~~real numbers that satisfies~~

	~~# <math>H(mn) = H(m)+H(n)\,</math> (additivity)~~
	~~# <math>H(m) \leq H(m+1)\,</math> (monotonicity)~~
	~~# <math>H(2)=1\,</math> (normalization)~~

	~~Let~~ '~~'&fnof;'' be~~ a ~~function on positive integers that satisfies the above three properties~~. ~~From the additive property, we can show that for any integer ''n'' and ''k'',~~

	~~:<math>f(n^k) = kf(n).\,</math>~~

	~~Let ''~~a~~'', ''b'', and ''t'' be any positive integers~~. ~~There~~ is ~~a unique integer~~ '~~'s'' determined by~~

	:~~<math>a^s \leq b^t \leq a^{s+1}. \qquad(1)<~~/~~math>~~

	~~Therefore,~~

	~~:<math>s \log_2 a\leq t \log_2 b \leq (s+1) \log_2 a \, <~~/~~math>~~

	~~and~~

	~~:<math>\frac{s}{t} \leq \frac{\log_2 b}{\log_2 a} \leq \frac{s+1}{t}~~.</~~math>~~

	~~On the other hand, by monotonicity,~~

	~~:<math>f(a^s) \leq f(b^t) \leq f(a^{s+1}). \, <~~/~~math>~~

	~~Using Equation (1), we get~~

	~~:<math>s f(a) \leq t f(b) \leq (s+1) f(a),\,<~~/~~math>~~

	~~and~~

	~~:<math>\frac{s}{t} \leq \frac{f(a)}{f(b)} \leq \frac{s+1}{t}.<~~/~~math>~~

	~~Hence,~~

	~~:<math>\Big\vert \frac{f(a)}{f(b)}~~ - ~~\frac{\log_2(a)}{\log_2(b)} \Big\vert \leq \frac{1}{t}~~.<~~/math~~>

	~~Since ''t'' can be arbitrarily large, the difference on the left hand side of the above inequality must be zero,~~

	:<~~math~~>~~\frac{f(~~a~~)}{f(b)} = \frac{\log_2(a)}{\log_2(b)}~~.</~~math>~~

	~~So,~~

	:~~<math>f(a) = \mu \log_2(a)\,<~~/~~math>~~

	~~for some constant ''μ'', which must be equal to 1 by the normalization property~~.

	~~==See also==~~
	* [[Rényi entropy]]

	~~{{PlanetMath attribution\|id=6070\|title=Hartley function}}~~
	~~{{PlanetMath attribution\|id=6082\|title=Derivation of Hartley function}}~~

	[~~[Category~~:~~Information theory]~~]

216.121.186.40: /* Example */

2013-12-09T11:42:29Z

Example

← Older revision		Revision as of 13:42, 9 December 2013
Line 1:		Line 1:
	~~Oscar~~ is ~~how he~~'~~s called and he completely enjoys this title. He utilized to be unemployed~~ at ~~home std test but now he~~ is ~~a pc operator but his promotion never arrives~~. [~~http://www~~.~~ranker~~.~~com/list/famous~~-~~people-with-herpes/celebrity-lists California~~] is ~~where I~~'~~ve usually been~~ [~~http~~:/~~/www~~.~~hamcass~~.~~org/index~~.~~php?document_srl~~=~~268708&mid~~=~~gido std home test] living~~ and ~~I love every day living right [http:~~//~~www~~.~~alemcheap~~.~~fi/~~show/the~~-way-~~to~~-cure-an-unpleasant-candidiasis over~~ the ~~counter std test] here~~. ~~The factor she adores most is body std home test building~~ and ~~now she home std test kit~~ is ~~attempting to earn money with it~~.<br><br>~~Also visit my weblog~~ ..~~. over the counter std test~~ (~~[http~~://~~www~~.~~january-yjm~~.~~com~~/xe/~~index~~.~~php?mid~~=~~video&document_srl~~=~~182582 relevant resource site~~])		The '''Hartley function''' is a measure of uncertainty, introduced by [[Ralph Hartley]] in 1928. If we pick a sample from a finite set ''A'' uniformly at random, the information revealed after we know the outcome is given by the Hartley function
			:<math> H_0(A) := \mathrm{log}_b \vert A \vert .</math>
			If the [[base (exponentiation)\|base]] of the [[logarithm]] is 2, then the uncertainty is measured in bits. If it is the [[natural logarithm]], then the unit is [[Nat (information)\|nats]]. (Hartley himself used a base-ten logarithm, and this unit of information is sometimes called the '''[[Ban (information)\|hartley]]''' in his honor.) It is also known as the Hartley entropy.

			== Hartley function, Shannon's entropy, and Rényi entropy ==
			The Hartley function coincides with the [[Shannon entropy]] (as well as with the Rényi entropies of all orders) in the case of a uniform probability distribution. It is actually a special case of the [[Rényi entropy]] since:
			:<math>H_0(X) = \frac 1 {1-0} \log \sum_{i=1}^{\|X\|} p_i^0 = \log \|X\|.</math>

			But it can also be viewed as a primitive construction, since, as emphasized by Kolmogorov and Rényi (see George, J. Klirr's "Uncertainty and information", p.423), the Hartley function can be defined without introducing any notions of probability.

			==Characterization of the Hartley function==
			The Hartley function only depends on the number of elements in a set, and hence can be viewed as a function on natural numbers. Rényi showed that the Hartley function in base 2 is the only function mapping natural numbers to real numbers that satisfies

			# <math>H(mn) = H(m)+H(n)</math> (additivity)
			# <math>H(m) \leq H(m+1)</math> (monotonicity)
			# <math>H(2)=1</math> (normalization)

			Condition 1 says that the uncertainty of the Cartesian product of two finite sets ''A'' and ''B'' is the sum of uncertainties of ''A'' and ''B''. Condition 2 says that larger set has larger uncertainty.

			==Derivation of the Hartley function==
			We want to show that the Hartley function, log<sub>2</sub>(''n''), is the only function mapping natural numbers to real numbers that satisfies

			# <math>H(mn) = H(m)+H(n)\,</math> (additivity)
			# <math>H(m) \leq H(m+1)\,</math> (monotonicity)
			# <math>H(2)=1\,</math> (normalization)

			Let ''&fnof;'' be a function on positive integers that satisfies the above three properties. From the additive property, we can show that for any integer ''n'' and ''k'',

			:<math>f(n^k) = kf(n).\,</math>

			Let ''a'', ''b'', and ''t'' be any positive integers. There is a unique integer ''s'' determined by

			:<math>a^s \leq b^t \leq a^{s+1}. \qquad(1)</math>

			Therefore,

			:<math>s \log_2 a\leq t \log_2 b \leq (s+1) \log_2 a \, </math>

			and

			:<math>\frac{s}{t} \leq \frac{\log_2 b}{\log_2 a} \leq \frac{s+1}{t}.</math>

			On the other hand, by monotonicity,

			:<math>f(a^s) \leq f(b^t) \leq f(a^{s+1}). \, </math>

			Using Equation (1), we get

			:<math>s f(a) \leq t f(b) \leq (s+1) f(a),\,</math>

			and

			:<math>\frac{s}{t} \leq \frac{f(a)}{f(b)} \leq \frac{s+1}{t}.</math>

			Hence,

			:<math>\Big\vert \frac{f(a)}{f(b)} - \frac{\log_2(a)}{\log_2(b)} \Big\vert \leq \frac{1}{t}.</math>

			Since ''t'' can be arbitrarily large, the difference on the left hand side of the above inequality must be zero,

			:<math>\frac{f(a)}{f(b)} = \frac{\log_2(a)}{\log_2(b)}.</math>

			So,

			:<math>f(a) = \mu \log_2(a)\,</math>

			for some constant ''μ'', which must be equal to 1 by the normalization property.

			==See also==
			* [[Rényi entropy]]

			{{PlanetMath attribution\|id=6070\|title=Hartley function}}
			{{PlanetMath attribution\|id=6082\|title=Derivation of Hartley function}}

			[[Category:Information theory]]

en>Mike Rosoft: Reverted edits by 82.7.95.228 (talk) to last revision by Loew Galitz (HG)

2012-06-07T16:03:40Z

Reverted edits by 82.7.95.228 (talk) to last revision by Loew Galitz (HG)

New page

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