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en>Qwertyus: O(n²) worst-case / O(n log n) average is more similar to quicksort's performance bounds than to mergesort's

2014-10-02T12:35:52Z

O(n²) worst-case / O(n log n) average is more similar to quicksort's performance bounds than to mergesort's

← Older revision		Revision as of 14:35, 2 October 2014
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	~~{{Refimprove\|date=July 2012}}~~		I would like to introduce myself to you, I am Andrew and [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ psychic phone readings] my wife doesn't like it at all. Some time ago she selected to live in Alaska and her mothers and fathers reside nearby. Doing ballet is something she would by no means give up. Distributing production is exactly where my main earnings comes from and it's something best psychic readings ([http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going http://www.zavodpm.ru]) I really enjoy.<br><br>Review my web page :: real psychic readings ([http://fashionlinked.com/index.php?do=/profile-13453/info/ http://fashionlinked.com])
	~~{{Context\|date=October 2009}}~~
	~~'''Estimation of a Rasch model''' is used~~ to ~~estimate the parameters of the [[Rasch model]]. Various techniques are employed~~ to ~~estimate the parameters from matrices of response data. The most common approaches are types of [[maximum likelihood]] estimation~~, ~~such as joint~~ and conditional maximum likelihood estimation. Joint maximum likelihood (JML) equations are efficient, but inconsistent for a finite number of items, whereas conditional maximum likelihood (CML) equations give consistent and unbiased item estimates. Person estimates are generally thought to have [[~~bias of an estimator\|bias]] associated with them, although weighted likelihood estimation methods for the estimation of person parameters reduce the bias.~~

	~~==Rasch model==~~
	~~The Rasch model for dichotomous data takes the form:~~

	:~~<math>~~
	~~\Pr \{X_{ni}=1\}=\frac{\exp({\beta_n} - {\delta_i})}{1 + \exp({\beta_n} - {\delta_i})},~~
	</~~math>~~

	~~where <math> \beta_n <~~/~~math> is the ability of person <math> n <~~/~~math> and <math> \delta_i <~~/~~math> is the difficulty of item <math> i </math>.~~

	~~==Joint maximum likelihood==~~
	~~Let <math>x_{ni}</math> denote the observed response for person ''n'' on item ''i''. The probability of the observed data matrix,~~ which ~~is the product of the probabilities~~ of ~~the individual responses, is given by the likelihood function~~

	~~:<math>~~
	~~\Lambda = \frac{\prod_{n} \prod_{i} \exp(x_{ni}(\beta_n~~-~~\delta_i))}{\prod_{n} \prod_{i}(1+\exp(\beta_n~~-~~\delta_i))}.~~
	~~</math>~~

	~~The log~~-~~likelihood function is then~~

	~~:<math>~~
	~~\log \Lambda = \sum_n^N \beta_n r_n~~ - ~~\sum_i^I \delta_i s_i~~ - ~~\sum_n^N \sum_i^I \log(1+\exp(\beta_n~~-~~\delta_i))~~
	~~</math>~~

	~~where <math>r_n=\sum_i^I x_{ni}</math> is~~ the ~~total raw score for person ''n'', <math>s_i=\sum_n^N x_{ni}<~~/~~math> is the total raw score for item~~ '~~'i'', ''N'' is the total number of persons and ''I'' is the total number of items~~.

	~~Solution equations are obtained by taking partial derivatives with respect~~ to ~~<math>\delta_i</math>~~ and ~~<math>\beta_n</math>~~ and ~~setting the result equal to 0~~. ~~The JML solution equations are:~~

	~~:<math>~~
	~~s_i = \sum_n^N p_{ni},\quad i=1,\dots,I~~
	~~</math>~~

	~~:<math>~~
	~~r_n = \sum_i^I p_{ni},\quad n=1,\dots,N~~
	~~</math>~~

	where ~~<math>p_{ni}=\exp~~(~~\beta_n-\delta_i)~~/~~(1+\exp(\beta_n-\delta_i))<~~/~~math>~~. ~~A more accurate estimate of each <math>\delta_i<~~/~~math> is obtained by multiplying the estimates by <math>(I-1)~~/I</~~math>.~~

	~~==Conditional maximum likelihood==~~
	~~The conditional likelihood function is defined as~~

	~~:<math>~~
	~~\Lambda = \prod_{n} \Pr\{(x_{ni})\mid r_n\} =\frac{\exp(\sum_i~~ -~~s_i\delta_i)}{\prod_{n} \gamma_r}~~
	~~</math>~~

	~~in which~~

	~~: <math>~~
	~~\gamma_r = \sum_{(x) \mid r}\exp(~~-~~\sum_i x_{ni}\delta_i)~~
	~~</math>~~

	~~is the [[elementary symmetric function]] of order ''r'', which represents the sum over all combinations of ''r'' items. For example, in the case of three items,~~

	~~: <math>\gamma_2 = \exp(~~-~~\delta_1~~-~~\delta_2)+\exp(~~-~~\delta_1~~-~~\delta_3)+\exp(~~-~~\delta_2~~-~~\delta_3).<~~/~~math>~~

	~~==Estimation algorithms==~~
	~~Some kind of [[expectation-maximization algorithm]] is used in the estimation of the parameters of Rasch models~~. Algorithms for implementing Maximum Likelihood estimation commonly employ [[Newton-Raphson]] iterations to solve for solution equations obtained from setting the partial derivatives of the log-likelihood functions equal to 0. Convergence criteria are used to determine when the iterations cease. For example, the criterion might be that the mean item estimate changes by less than a certain value, such as 0.001, between one iteration and another for all items.

	~~==See also==~~
	* [[Expectation-maximization algorithm]]
	* [[Rasch model]]

	~~{{inline\|date=July 2012}}~~
	~~==References==~~
	* Linacre, J.M. (2004). ~~''Estimation methods for Rasch measures''. Chapter 2 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN~~: ~~JAM Press.~~
	* Linacre, J.M. (~~2004). ''Rasch model estimation~~: ~~further topics''~~. ~~Chapter 24 in E~~.~~V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN~~: ~~JAM Press~~.

	~~{{DEFAULTSORT:Rasch Model Estimation}}~~
	~~[[Category:Psychometrics]]~~
	~~[[Category:Statistical models]~~]

en>Qwertyus: /* Pseudocode */

2013-12-30T14:04:30Z

Pseudocode

← Older revision		Revision as of 16:04, 30 December 2013
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	~~Hi there~~. ~~Allow me begin by introducing~~ the ~~writer~~, ~~her title~~ is ~~Myrtle Cleary~~. ~~Managing people~~ is ~~his occupation~~. The ~~preferred hobby~~ for ~~my kids and me~~ is ~~to perform baseball~~ and I'~~m attempting~~ to ~~make it a occupation. Years in~~ the ~~past he moved~~ to ~~North Dakota and his family members enjoys it~~.<br><br>~~my web blog; [http~~://~~www~~.~~alemcheap~~.fi/~~people~~/~~anhipkiss at home std testing~~]		{{Refimprove\|date=July 2012}}
			{{Context\|date=October 2009}}
			'''Estimation of a Rasch model''' is used to estimate the parameters of the [[Rasch model]]. Various techniques are employed to estimate the parameters from matrices of response data. The most common approaches are types of [[maximum likelihood]] estimation, such as joint and conditional maximum likelihood estimation. Joint maximum likelihood (JML) equations are efficient, but inconsistent for a finite number of items, whereas conditional maximum likelihood (CML) equations give consistent and unbiased item estimates. Person estimates are generally thought to have [[bias of an estimator\|bias]] associated with them, although weighted likelihood estimation methods for the estimation of person parameters reduce the bias.

			==Rasch model==
			The Rasch model for dichotomous data takes the form:

			:<math>
			\Pr \{X_{ni}=1\}=\frac{\exp({\beta_n} - {\delta_i})}{1 + \exp({\beta_n} - {\delta_i})},
			</math>

			where <math> \beta_n </math> is the ability of person <math> n </math> and <math> \delta_i </math> is the difficulty of item <math> i </math>.

			==Joint maximum likelihood==
			Let <math>x_{ni}</math> denote the observed response for person ''n'' on item ''i''. The probability of the observed data matrix, which is the product of the probabilities of the individual responses, is given by the likelihood function

			:<math>
			\Lambda = \frac{\prod_{n} \prod_{i} \exp(x_{ni}(\beta_n-\delta_i))}{\prod_{n} \prod_{i}(1+\exp(\beta_n-\delta_i))}.
			</math>

			The log-likelihood function is then

			:<math>
			\log \Lambda = \sum_n^N \beta_n r_n - \sum_i^I \delta_i s_i - \sum_n^N \sum_i^I \log(1+\exp(\beta_n-\delta_i))
			</math>

			where <math>r_n=\sum_i^I x_{ni}</math> is the total raw score for person ''n'', <math>s_i=\sum_n^N x_{ni}</math> is the total raw score for item ''i'', ''N'' is the total number of persons and ''I'' is the total number of items.

			Solution equations are obtained by taking partial derivatives with respect to <math>\delta_i</math> and <math>\beta_n</math> and setting the result equal to 0. The JML solution equations are:

			:<math>
			s_i = \sum_n^N p_{ni},\quad i=1,\dots,I
			</math>

			:<math>
			r_n = \sum_i^I p_{ni},\quad n=1,\dots,N
			</math>

			where <math>p_{ni}=\exp(\beta_n-\delta_i)/(1+\exp(\beta_n-\delta_i))</math>. A more accurate estimate of each <math>\delta_i</math> is obtained by multiplying the estimates by <math>(I-1)/I</math>.

			==Conditional maximum likelihood==
			The conditional likelihood function is defined as

			:<math>
			\Lambda = \prod_{n} \Pr\{(x_{ni})\mid r_n\} =\frac{\exp(\sum_i -s_i\delta_i)}{\prod_{n} \gamma_r}
			</math>

			in which

			: <math>
			\gamma_r = \sum_{(x) \mid r}\exp(-\sum_i x_{ni}\delta_i)
			</math>

			is the [[elementary symmetric function]] of order ''r'', which represents the sum over all combinations of ''r'' items. For example, in the case of three items,

			: <math>\gamma_2 = \exp(-\delta_1-\delta_2)+\exp(-\delta_1-\delta_3)+\exp(-\delta_2-\delta_3).</math>

			==Estimation algorithms==
			Some kind of [[expectation-maximization algorithm]] is used in the estimation of the parameters of Rasch models. Algorithms for implementing Maximum Likelihood estimation commonly employ [[Newton-Raphson]] iterations to solve for solution equations obtained from setting the partial derivatives of the log-likelihood functions equal to 0. Convergence criteria are used to determine when the iterations cease. For example, the criterion might be that the mean item estimate changes by less than a certain value, such as 0.001, between one iteration and another for all items.

			==See also==
			* [[Expectation-maximization algorithm]]
			* [[Rasch model]]

			{{inline\|date=July 2012}}
			==References==
			* Linacre, J.M. (2004). ''Estimation methods for Rasch measures''. Chapter 2 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN: JAM Press.
			* Linacre, J.M. (2004). ''Rasch model estimation: further topics''. Chapter 24 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN: JAM Press.

			{{DEFAULTSORT:Rasch Model Estimation}}
			[[Category:Psychometrics]]
			[[Category:Statistical models]]

en>EmausBot: r2.7.2+) (Robot: Modifying fa:مرتب‌سازی کتابخانه‌ای

2012-07-16T20:19:28Z

r2.7.2+) (Robot: Modifying fa:مرتب‌سازی کتابخانه‌ای

New page

Hi there. Allow me begin by introducing the writer, her title is Myrtle Cleary. Managing people is his occupation. The preferred hobby for my kids and me is to perform baseball and I'm attempting to make it a occupation. Years in the past he moved to North Dakota and his family members enjoys it.<br><br>my web blog; [http://www.alemcheap.fi/people/anhipkiss at home std testing]