Tweedie distributions: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>AppliedMathematics
m Further detail on multifractal systems
en>BG19bot
m WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)
 
Line 1: Line 1:
The '''Wigner distribution (WD)''' was first proposed for corrections to classical statistical mechanics in 1932 by [[Eugene Wigner]]. The [[Wigner distribution function|Wigner distribution]], or Wigner–Ville distribution (WVD) for analytic signals, also has applications in time frequency analysis. The Wigner distribution gives better auto term localisation compared to the smeared out [[spectrogram]] (SP). However when applied to a signal with multi frequency components cross terms appear due to its quadratic nature. Several methods have been proposed to reduce the cross terms. For example in 1994 L. Stankovic proposed a novel technique, now mostly referred to as S-method, resulting in the reduction or removal of cross terms. The concept of the S-method is a combination between the spectrogram and the Pseudo Wigner Distribution (PWD), the windowed version of the WD.
CMS provides the best platform to create websites that fulfill all the specifications of SEO. Medical word press themes give you the latest medical designs. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. When you liked this information and you want to acquire details with regards to [http://lapgamer.com/profile/paeastman wordpress backup plugin] kindly stop by the web-site. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. After sending these details, your Word - Press blog will be setup within a few days. Use this section to change many formatting elements. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. After that the developer adds the unordered list for navigations. <br><br>It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. But the Joomla was created as the CMS over years of hard work. Normally, the Word - Press developers make a thorough research on your website goals and then ingrain the most suitable graphical design elements to your website. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. Word - Press is an open source content management system which is easy to use and offers many user friendly features. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and  full user registration plug-ins.
 
The original WD, the spectrogram, and the modified WDs all belong to the [[Cohen's class]] of bilinear time-frequency representations :
:<math>C_x(t, f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_x(\theta,\nu) \Pi(t - \theta,f - \nu)\, d\theta\, d\nu \quad = [W_x\,\ast\,\Pi] (t,f)</math>
where <math>\Pi \left(t, f\right)</math> is Cohen's '''kernel function''', which is often a low-pass function, and normally serves to mask out the interference in the original Wigner representation.
 
== Mathematical definition ==
 
*'''Wigner distribution'''
:<math> W_x(t,f) = \int_{-\infty}^\infty x(t+\tau/2) x^*(t-\tau/2) e^{-j2\pi\tau\,f} \, d\tau</math>
Cohen's kernel function : <math>\Pi (t,f) = \delta_{(0,0)} (t,f) </math>
 
*'''Spectrogram'''
:<math>SP_x (t,f) = |ST_x (t,f)|^2 = ST_x (t,f)\,ST_x^* (t,f)</math>
where <math>ST_x</math> is the [[short-time Fourier transform]] of <math>x</math>.  
:<math> ST_x(t,f) = \int_{-\infty}^\infty x(\tau) w^*(t-\tau) e^{-j2\pi f\tau} \, d\tau</math>
 
Cohen's kernel function : <math>\Pi (t,f) = W_h(t,f) </math> which is the WD of the window function itself. This can be verified by applying the convolution property of the [[Wigner distribution function]].
 
The spectrogram cannot produce interference since it is a positive-valued quadratic distribution.
 
*'''Pseudo Wigner distribution'''
:<math> PW_x(t,f) = \int_{-\infty}^\infty w(\tau/2) w^*(-\tau/2) x(t+\tau/2) x^*(t-\tau/2) e^{-j2\pi\tau\,f} \, d\tau</math>
Cohen's kernel function : <math>\Pi (t,f) = \delta_0 (t)\,W_h(t,f) </math> which is concentred on the frequency axis.
 
Note that the pseudo Wigner can also be written as the Fourier transform of the “spectral-correlation” of the STFT
:<math> PW_x(t,f) = \int_{-\infty}^\infty ST_x(t, f+\nu/2) ST_x^*(t, f-\nu/2) e^{j2\pi\nu\,t} \, d\nu</math>
 
*'''Smoothed pseudo Wigner distribution''' :
In the pseudo Wigner the time windowing acts as a frequency direction smoothing. Therefore it suppresses the Wigner distribution interference components that oscillate in the frequency direction. Time direction smoothing can be implemented by a time-convolution of the PWD with a lowpass function <math>q</math> :
:<math> SPW_x(t,f) = [ q\,\ast\, PW_x (.,f)] (t) =  \int_{-\infty}^\infty q(t-u) \int_{-\infty}^\infty w(\tau/2) w^*(-\tau/2) x(u+\tau/2) x^*(u-\tau/2) e^{-j2\pi\tau\,f} \, d\tau\, du</math>
Cohen's kernel function : <math>\Pi (t,f) = q(t)\, W(f) </math> where <math>W</math> is the Fourier transform of the window <math>w</math>.
 
Thus the kernel corresponding to the smoothed pseudo Wigner distribution has a separable form. Note that even if the SPWD and the S-Method both smoothes the WD in the time domain, they are not equivalent in general.
 
*'''S-method'''
:<math> SM(t,f) = \int_{-\infty}^\infty ST_x(t, f+\nu/2) ST_x^*(t, f-\nu/2) G(\nu) e^{j2\pi\nu\,t} \, d\nu</math>
 
Cohen's kernel function : <math>\Pi (t,f) = g(t)\, W_h(t,f) </math>
 
The S-method limits the range of the integral of the PWD with a low-pass windowing function <math>g(t)</math> of Fourier transform <math>G(f)</math>. This results in the cross-term removal, without blurring the auto-terms that are well-concentred along the frequency axis.
The S-method strikes a balance in smoothing between the pseudo-Wigner distribution <math>PW_x</math> [<math>g(t) = 1</math>] and the power spectrogram <math>SP_x</math> [<math>g(t) = \delta_0 (t)</math>].
 
Note that in the original 1994 paper, Stankovic defines the S-methode with a modulated version of the short-time Fourier transform :
:<math> SM(t,f) = \int_{-\infty}^\infty \tilde{ST}_x(t,f+\nu) \tilde{ST}_x^*(t,f-\nu) P(\nu)\, d\nu </math>
 
where
 
:<math> \tilde{ST}_x(t,f) = \int_{-\infty}^\infty x(t+\tau) w^*(\tau) e^{-j2\pi f\tau} \, d\tau \quad = ST_x(t,f)\,e^{j2\pi f t}</math>
 
Even in this case we still have
 
: <math>\Pi (t,f) = p(2t)\, W_h(t,f) </math>
 
== See also ==
* [[Time-frequency representation]]
* [[Wigner distribution function]]
* [[Bilinear time–frequency distribution]]
* [[Short-time Fourier transform]]
* [[Gabor transform]]
 
==References==
*P. Gonçalves and R. Baraniuk, “Pseudo Affine Wigner Distributions : Definition and Kernel Formulation”, IEEE Trans. on Signal Processing, vol. 46, no. 6, Jun. 1998
*L. Stankovic, “A Method for Time-Frequency Signal Analysis”, IEEE Trans. on Signal Processing, vol. 42, no. 1, Jan. 1994
 
[[Category:Signal processing]]
[[Category:Transforms]]

Latest revision as of 02:31, 7 February 2014

CMS provides the best platform to create websites that fulfill all the specifications of SEO. Medical word press themes give you the latest medical designs. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. When you liked this information and you want to acquire details with regards to wordpress backup plugin kindly stop by the web-site. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.

These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.

Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. After sending these details, your Word - Press blog will be setup within a few days. Use this section to change many formatting elements. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. After that the developer adds the unordered list for navigations.

It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. But the Joomla was created as the CMS over years of hard work. Normally, the Word - Press developers make a thorough research on your website goals and then ingrain the most suitable graphical design elements to your website. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Word - Press offers constant updated services and products, that too, absolutely free of cost.

Instead, you can easily just include it with our bodies integration field in e - Panel. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. Word - Press is an open source content management system which is easy to use and offers many user friendly features. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Tweedie_distributions&oldid=259627"