Radial basis function network - Revision history

en>Krishnavedala: /* Network architecture */ image thumbnail set to static 250px in sync with other images on this page

2015-01-09T23:11:42Z

Network architecture: image thumbnail set to static 250px in sync with other images on this page

← Older revision		Revision as of 01:11, 10 January 2015
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	In the ~~[[stochastic calculus]], '''Tanaka's formula''' states that~~		The name of the writer is Numbers. For many years I've been working as a payroll clerk. Minnesota is exactly where he's been living for years. He is really fond of doing ceramics but he is struggling to find time for it.<br><br>Feel free to surf to my web page :: [http://www.blaze16.com/blog/257588 std testing at home]

	~~:<math>\|B_t\| = \int_0^t \sgn(B_s)\, dB_s + L_t</math>~~

	~~where ''B''<sub>''t''</sub>~~ is ~~the standard [[Brownian motion]], sgn denotes the [[sign function]]~~

	~~:<math>\sgn (x) = \begin{cases} +1, & x \geq 0; \\ -1, & x < 0~~. ~~\end{cases}</math>~~

	~~and~~ '~~'L''<sub>''t''</sub>~~ is ~~its [[Local time (mathematics)\|local time]] at 0 (the local time spent by ''B'' at 0 before time ''t'') given by the [[Lp space\|''L'~~'~~<sup>2</sup>-limit]]~~

	~~:<math>L_{t} = \lim_{\varepsilon \downarrow 0} \frac1{2 \varepsilon} \| \{~~ s ~~\in [0, t] \| B_{s} \in (- \varepsilon, + \varepsilon) \} \|~~.~~</math>~~

	~~Tanaka's formula~~ is ~~the explicit [[Doob–Meyer decomposition theorem\|Doob–Meyer decomposition]]~~ of ~~the submartingale \|''B''<sub>''t''</sub>\| into the [[martingale (probability theory)\|martingale]] part (the [[integral]] on the right-hand side), and a continuous increasing process (local~~ time). It can also be seen as the analogue of [[Itō's lemma]] for the (nonsmooth) absolute value function <math>f(x)=\|x\|</math>, with <math> f'(x) = \sgn(x)</math> and <math> f''(x) = 2\delta(x) </math>; see [[Local time (mathematics)\|local time]] for ~~a formal explanation of the Itō term.~~

	~~== Outline of proof ==~~
	~~The [[Function (mathematics)\|function]] \|''x''\| is not [[smooth function\|''C''<sup>2</sup>]] in ''x'' at ''x'' = 0, so we cannot apply [[Itō's formula]] directly. But if we approximate~~ it ~~near zero (i.e~~. ~~in [−''ε'', ''ε'']) by [[parabola]]s~~

	:<~~math~~>~~\frac{x^2}{2\|\varepsilon\|}+\frac{\|\varepsilon\|}{2}.~~<~~/math~~>

	~~And using [[Itō's formula]] we can then take the [[Limit (mathematics)\|limit]] as ''ε'' → 0, leading~~ to ~~Tanaka's formula.~~

	~~== References ==~~

	* {{cite book
	~~\| last = Øksendal~~
	~~\| first = Bernt K.~~
	~~\| authorlink = Bernt Øksendal~~
	~~\| title = Stochastic Differential Equations~~: ~~An Introduction with Applications~~
	~~\| edition = Sixth edition~~
	~~\| publisher=Springer~~
	~~\| location = Berlin~~
	~~\| year = 2003~~
	~~\| isbn = 3-540-04758-1~~
	~~}} (Example 5.3.2)~~
	* {{cite book
	~~\| last = Shiryaev~~
	~~\| first = Albert N.~~
	~~\|authorlink= Albert Shiryaev~~
	~~\| title = Essentials of stochastic finance~~: ~~Facts, models, theory~~
	~~\| series = Advanced Series on Statistical Science & Applied Probability No~~. 3
	~~\|coauthors = trans~~. ~~N. Kruzhilin~~
	~~\|publisher = World Scientific Publishing Co. Inc.~~
	~~\| location = River Edge, NJ~~
	~~\| year = 1999~~
	~~\| isbn = 981-02-3605-0~~
	}}

	~~[[Category:Equations]]~~
	~~[[Category:Martingale theory]]~~
	~~[[Category:Probability theorems]~~]

en>Monkbot: /* Network architecture */Fix CS1 deprecated date parameter errors

2014-01-29T01:56:41Z

Network architecture: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 03:56, 29 January 2014
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	~~Greetings.~~ [~~http~~://~~www~~.~~teenvalley.net~~/~~user~~/~~RLay over~~ the ~~counter std test~~] ~~writer~~'s ~~name~~ is ~~Phebe and she feels comfy when individuals use~~ the ~~complete title. His family lives in South Dakota but his wife desires them to transfer. I utilized to be unemployed home home std test kit std test but now std~~ [~~http:~~/~~/kpchurch.org/?document_srl=1359143 at home std test~~] ~~test I am~~ a [~~http:~~//~~Www.huffingtonpost.ca~~/~~colette-kenney/dating-with-herpes_b_1609226.html librarian~~] ~~and~~ the ~~wage has been truly satisfying~~. ~~To gather coins~~ is ~~what his family members and him appreciate.~~<br><br>~~my page~~; [~~http~~:/~~/indianrajneet~~.~~com/groups/best~~-~~strategies~~-~~for~~-~~keeping~~-~~candida~~-~~albicans~~-~~from-increasing/ http~~:~~//indianrajneet.com/~~]		In the [[stochastic calculus]], '''Tanaka's formula''' states that

			:<math>\|B_t\| = \int_0^t \sgn(B_s)\, dB_s + L_t</math>

			where ''B''<sub>''t''</sub> is the standard [[Brownian motion]], sgn denotes the [[sign function]]

			:<math>\sgn (x) = \begin{cases} +1, & x \geq 0; \\ -1, & x < 0. \end{cases}</math>

			and ''L''<sub>''t''</sub> is its [[Local time (mathematics)\|local time]] at 0 (the local time spent by ''B'' at 0 before time ''t'') given by the [[Lp space\|''L''<sup>2</sup>-limit]]

			:<math>L_{t} = \lim_{\varepsilon \downarrow 0} \frac1{2 \varepsilon} \| \{ s \in [0, t] \| B_{s} \in (- \varepsilon, + \varepsilon) \} \|.</math>

			Tanaka's formula is the explicit [[Doob–Meyer decomposition theorem\|Doob–Meyer decomposition]] of the submartingale \|''B''<sub>''t''</sub>\| into the [[martingale (probability theory)\|martingale]] part (the [[integral]] on the right-hand side), and a continuous increasing process (local time). It can also be seen as the analogue of [[Itō's lemma]] for the (nonsmooth) absolute value function <math>f(x)=\|x\|</math>, with <math> f'(x) = \sgn(x)</math> and <math> f''(x) = 2\delta(x) </math>; see [[Local time (mathematics)\|local time]] for a formal explanation of the Itō term.

			== Outline of proof ==
			The [[Function (mathematics)\|function]] \|''x''\| is not [[smooth function\|''C''<sup>2</sup>]] in ''x'' at ''x'' = 0, so we cannot apply [[Itō's formula]] directly. But if we approximate it near zero (i.e. in [−''ε'', ''ε'']) by [[parabola]]s

			:<math>\frac{x^2}{2\|\varepsilon\|}+\frac{\|\varepsilon\|}{2}.</math>

			And using [[Itō's formula]] we can then take the [[Limit (mathematics)\|limit]] as ''ε'' → 0, leading to Tanaka's formula.

			== References ==

			* {{cite book
			\| last = Øksendal
			\| first = Bernt K.
			\| authorlink = Bernt Øksendal
			\| title = Stochastic Differential Equations: An Introduction with Applications
			\| edition = Sixth edition
			\| publisher=Springer
			\| location = Berlin
			\| year = 2003
			\| isbn = 3-540-04758-1
			}} (Example 5.3.2)
			* {{cite book
			\| last = Shiryaev
			\| first = Albert N.
			\|authorlink= Albert Shiryaev
			\| title = Essentials of stochastic finance: Facts, models, theory
			\| series = Advanced Series on Statistical Science & Applied Probability No. 3
			\|coauthors = trans. N. Kruzhilin
			\|publisher = World Scientific Publishing Co. Inc.
			\| location = River Edge, NJ
			\| year = 1999
			\| isbn = 981-02-3605-0
			}}

			[[Category:Equations]]
			[[Category:Martingale theory]]
			[[Category:Probability theorems]]

136.242.131.53: /* Interpolation */

2012-02-27T22:35:50Z

Interpolation

New page

Greetings. [http://www.teenvalley.net/user/RLay over the counter std test] writer's name is Phebe and she feels comfy when individuals use the complete title. His family lives in South Dakota but his wife desires them to transfer. I utilized to be unemployed home home std test kit std test but now std [http://kpchurch.org/?document_srl=1359143 at home std test] test I am a [http://Www.huffingtonpost.ca/colette-kenney/dating-with-herpes_b_1609226.html librarian] and the wage has been truly satisfying. To gather coins is what his family members and him appreciate.<br><br>my page; [http://indianrajneet.com/groups/best-strategies-for-keeping-candida-albicans-from-increasing/ http://indianrajneet.com/]

← Older revision		Revision as of 01:11, 10 January 2015
Line 1:		Line 1:
	In the ~~[[stochastic calculus]], '''Tanaka's formula''' states that~~		The name of the writer is Numbers. For many years I've been working as a payroll clerk. Minnesota is exactly where he's been living for years. He is really fond of doing ceramics but he is struggling to find time for it.<br><br>Feel free to surf to my web page :: [http://www.blaze16.com/blog/257588 std testing at home]

	~~:<math>\|B_t\| = \int_0^t \sgn(B_s)\, dB_s + L_t</math>~~

	~~where ''B''<sub>''t''</sub>~~ is ~~the standard [[Brownian motion]], sgn denotes the [[sign function]]~~

	~~:<math>\sgn (x) = \begin{cases} +1, & x \geq 0; \\ -1, & x < 0~~. ~~\end{cases}</math>~~

	~~and~~ '~~'L''<sub>''t''</sub>~~ is ~~its [[Local time (mathematics)\|local time]] at 0 (the local time spent by ''B'' at 0 before time ''t'') given by the [[Lp space\|''L'~~'~~<sup>2</sup>-limit]]~~

	~~:<math>L_{t} = \lim_{\varepsilon \downarrow 0} \frac1{2 \varepsilon} \| \{~~ s ~~\in [0, t] \| B_{s} \in (- \varepsilon, + \varepsilon) \} \|~~.~~</math>~~

	~~Tanaka's formula~~ is ~~the explicit [[Doob–Meyer decomposition theorem\|Doob–Meyer decomposition]]~~ of ~~the submartingale \|''B''<sub>''t''</sub>\| into the [[martingale (probability theory)\|martingale]] part (the [[integral]] on the right-hand side), and a continuous increasing process (local~~ time). It can also be seen as the analogue of [[Itō's lemma]] for the (nonsmooth) absolute value function <math>f(x)=\|x\|</math>, with <math> f'(x) = \sgn(x)</math> and <math> f''(x) = 2\delta(x) </math>; see [[Local time (mathematics)\|local time]] for ~~a formal explanation of the Itō term.~~

	~~== Outline of proof ==~~
	~~The [[Function (mathematics)\|function]] \|''x''\| is not [[smooth function\|''C''<sup>2</sup>]] in ''x'' at ''x'' = 0, so we cannot apply [[Itō's formula]] directly. But if we approximate~~ it ~~near zero (i.e~~. ~~in [−''ε'', ''ε'']) by [[parabola]]s~~

	:<~~math~~>~~\frac{x^2}{2\|\varepsilon\|}+\frac{\|\varepsilon\|}{2}.~~<~~/math~~>

	~~And using [[Itō's formula]] we can then take the [[Limit (mathematics)\|limit]] as ''ε'' → 0, leading~~ to ~~Tanaka's formula.~~

	~~== References ==~~

	* {{cite book
	~~\| last = Øksendal~~
	~~\| first = Bernt K.~~
	~~\| authorlink = Bernt Øksendal~~
	~~\| title = Stochastic Differential Equations~~: ~~An Introduction with Applications~~
	~~\| edition = Sixth edition~~
	~~\| publisher=Springer~~
	~~\| location = Berlin~~
	~~\| year = 2003~~
	~~\| isbn = 3-540-04758-1~~
	~~}} (Example 5.3.2)~~
	* {{cite book
	~~\| last = Shiryaev~~
	~~\| first = Albert N.~~
	~~\|authorlink= Albert Shiryaev~~
	~~\| title = Essentials of stochastic finance~~: ~~Facts, models, theory~~
	~~\| series = Advanced Series on Statistical Science & Applied Probability No~~. 3
	~~\|coauthors = trans~~. ~~N. Kruzhilin~~
	~~\|publisher = World Scientific Publishing Co. Inc.~~
	~~\| location = River Edge, NJ~~
	~~\| year = 1999~~
	~~\| isbn = 981-02-3605-0~~
	}}

	~~[[Category:Equations]]~~
	~~[[Category:Martingale theory]]~~
	~~[[Category:Probability theorems]~~]