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| '''Recurrence quantification analysis (RQA)''' is a method of [[nonlinear]] [[data analysis]] (cf. [[chaos theory]]) for the investigation of [[dynamical systems]]. It quantifies the number and duration of recurrences of a dynamical system presented by its [[phase space]] trajectory.
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>When you have virtually any inquiries with regards to wherever as well as how you can employ [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC], you are able to contact us from our own web-site. |
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| ==Background==
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| The recurrence quantification analysis was developed in order to quantify differently appearing [[recurrence plot]]s (RPs) based on the small-scale structures therein. [[Recurrence plot]]s are tools which visualise the recurrence behaviour of the [[phase space]] trajectory of [[dynamical systems]]. They mostly contain single dots and lines which are parallel to the mean diagonal (''line of identity'', LOI) or which are vertical/horizontal. Lines parallel to the LOI are referred to as ''diagonal lines'' and the vertical structures as ''vertical lines''. Because an RP is usually symmetric, horizontal and vertical lines correspond to each other, and, hence, only vertical lines are considered. The lines correspond to a typical behaviour of the phase space trajectory: whereas the diagonal lines represent such segments of the phase space trajectory which run parallel for some time, the vertical lines represent segments which remain in the same [[phase space]] region for some time.
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| If only a [[time series]] is available, the phase space can be reconstructed by using a time delay embedding (see [[Takens' theorem]]):
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| :<math>\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)),</math>
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| where <math>u(i)</math> is the time series, <math>m</math> the embedding dimension and <math>\tau</math> the time delay.
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| The RQA quantifies the small-scale structures of recurrence plots, which present the number and duration of the recurrences of a dynamical system. The measures introduced for the RQA were developed heuristically between 1992 and 2002 (Zbilut & Webber 1992; Webber & Zbilut 1994; Marwan et al. 2002). They are actually [[measure of complexity|measures of complexity]]. The main advantage of the recurrence quantification analysis is that it can provide useful information even for short and non-stationary data, where other methods fail.
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| RQA can be applied to almost every kind of data. It is widely used in [[physiology]], but was also successfully applied on problems from [[engineering]], [[chemistry]], [[Earth sciences]] etc.
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| ==RQA measures==
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| The simplest measure is the '''recurrence rate''', which is the density of recurrence points in a recurrence plot:
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| :<math>\text{RR} = \frac{1}{N^2} \sum_{i,j=1}^N \mathbf{R}(i,j).</math>
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| The recurrence rate corresponds with the probability that a specific state will recur. It is almost equal with the definition of the [[correlation sum]], where the LOI is excluded from the computation.
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| The next measure is the percentage of recurrence points which form diagonal lines in the recurrence plot of minimal length <math>\ell_\min</math>:
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| :<math>\text{DET} = \frac{\sum_{\ell=\ell_\min}^N \ell\, P(\ell)}{\sum_{i,j=1}^N R(i,j)},</math>
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| where <math>P(\ell)</math> is the [[frequency distribution]] of the lengths <math>\ell</math> of the diagonal lines. This measure is called '''determinism''' and is related with the [[predictability]] of the [[dynamical system]], because [[white noise]] has a recurrence plot with almost only single dots and very few diagonal lines, whereas a [[deterministic process]] has a recurrence plot with very few single dots but many long diagonal lines.
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| The amount of recurrence points which form vertical lines can be quantified in the same way:
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| :<math> \text{LAM} = \frac{\sum_{v=v_\min}^{N}vP(v)}{\sum_{v=1}^{N}vP(v)},</math>
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| where <math>P(v)</math> is the frequency distribution of the lengths <math>v</math> of the vertical lines, which have at least a length of <math>v_\min</math>. This measure is called '''laminarity''' and is related with the amount of [[laminar phase]]s in the system ([[intermittency]]).
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| The lengths of the diagonal and vertical lines can be measured as well. The '''averaged diagonal line length'''
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| :<math>\text{L} = \frac{\sum_{\ell=\ell_\min}^N \ell\, P(\ell)}{\sum_{\ell=\ell_\min}^N P(\ell)}</math>
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| is related with the ''predictability time'' of the dynamical system
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| and the '''trapping time''', measuring the average length | |
| of the vertical lines, | |
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| :<math>TT = \frac{\sum_{v=v_\min}^{N} v P(v)} {\sum_{v=v_\min}^{N} P(v)}</math>
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| is related with the ''laminarity time'' of the dynamical system, i.e. how long the system remains in a specific state.
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| Because the length of the diagonal lines is related on the time how long segments of the [[phase space]] trajectory run parallel, i.e. on the [[divergence]] behaviour of the trajectories, it was sometimes stated that the [[Multiplicative inverse|reciprocal]] of the maximal length of the diagonal lines (without LOI) would be an estimator for the positive maximal [[Lyapunov exponent]] of the dynamical system. Therefore, the '''maximal diagonal line length''' <math>L_\max</math> or the '''divergence'''
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| :<math>DIV = \frac{1}{L_\max}</math>
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| are also measures of the RQA. However, the relationship between these measures with the positive maximal Lyapunov exponent is not as easy as stated, but even more complex (to calculate the Lyapunov exponent from an RP, the whole frequency distribution of the diagonal lines has to be considered). The divergence can have the trend of the positive maximal Lyapunov exponent, but not more. Moreover, also RPs of white noise processes can have a really long diagonal line, although very seldom, just by a finite probability. It is obvious that therefore the divergence cannot reflect the maximal Lyapunov exponent.
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| The [[probability]] <math>p(\ell)</math> that a diagonal line has exactly length <math>\ell</math> can be estimated from the frequency distribution <math>P(\ell)</math> with <math>p(\ell) = \frac{P(\ell)}{\sum_{\ell=l_\min}^N P(\ell)}</math>. The [[Shannon entropy]] of this probability,
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| :<math>\text{ENTR} = - \sum_{\ell=\ell_\min}^N p(\ell) \ln p(\ell),</math>
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| reflects the complexity of the deterministic structure in the system. However, this entropy depends sensitively on the bin number and, thus, may differ for different realisations of the same process, as well as for different data preparations.
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| The last measure of the RQA quantifies the thinning-out of the recurrence plot. The '''trend''' is the regression coefficient of a linear relationship between the density of recurrence points in a line parallel to the LOI and its distance to the LOI. More exactly, let us consider the recurrence rate in a diagonal line parallel to LOI of distance ''k'' (''diagonal-wise recurrence rate''): | |
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| :<math>\text{RR}_k = \frac{1}{N-k} \sum_{j-i=k}^{N-k} \mathbf{R}(i,j),</math>
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| then the trend is defined by
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| :<math>\text{TREND} = \frac{\sum_{i=1}^\tilde{N} (i-\tilde{N}/2)(RR_i - \langle RR_i \rangle)}{\sum_{i=1}^\tilde{N} (i-\tilde{N}/2)^2},</math>
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| with <math>\langle \cdot \rangle</math> as the average value and <math>\tilde{N} < N</math>. This latter relation should ensure to avoid the edge effects of too low recurrence point densities in the edges of the recurrence plot. The measure ''trend'' provides information about the stationarity of the system.
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| Similar to the diagonal-wise defined recurrence rate, the other measures based on the diagonal lines (DET, L, ENTR) can be defined diagonal-wise. These definitions are useful to study interrelations or [[synchronisation]] between different systems (using [[recurrence plot]]s or [[recurrence plot#Extensions|cross recurrence plots]]).
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| ==Time-dependent RQA==
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| Instead of computing the RQA measures of the entire recurrence plot, they can be computed in small windows moving over the recurrence plot along the LOI. This provides time-dependent RQA measures which allow detecting, e.g., chaos-chaos transitions (Marwan et al. 2002). '''Note:''' the choice of the size of the window can strongly influence the measure ''trend''.
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| ==Example==
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| [[Image:LogisticMap BifurcationDiagram.png|thumb|left|512px|Bifurcation diagram for the Logistic map.]]
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| [[Image:Logistic map rqa.svg|thumb|left|820px|RQA measures of the logistic map for various setting of the control parameter a. The measures RR and DET exhibit maxima at chaos-order/ order-chaos transitions. The measure DIV has a similar trend as the maximal [[Lyapunov exponent]] (but it is not the same!). The measure LAM has maxima at chaos-chaos transitions ([[laminar phase]]s, [[intermittency]]).]]
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| {{Clear}}
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| ==See also==
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| * [[Recurrence plot]], a powerful visualisation tool of recurrences in dynamical (and other) systems.
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| *[[Approximate entropy]]
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| ==References==
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| * {{Cite journal| author=Marwan, N. | year=2008 | title=A Historical Review of Recurrence Plots | journal=European Physical Journal ST | volume=164 | pages=3–12 | doi=10.1140/epjst/e2008-00829-1|bibcode = 2008EPJST.164....3M }}
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| * {{Cite journal| author=Marwan, N., Romano, M. C. ,Thiel, M. ,Kurths, J. | year=2007 | title=Recurrence Plots for the Analysis of Complex Systems | journal=Physics Reports | volume=438 | pages=237–329 | doi=10.1016/j.physrep.2006.11.001|bibcode = 2007PhR...438..237M| issue=5–6 }}
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| * {{Cite journal| author=Marwan, N., Wessel, N., Meyerfeldt, U., Schirdewan, A., Kurths, J. | year=2002 | title=Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data | journal=Physical Review E | volume=66 | pages=026702 | doi=10.1103/PhysRevE.66.026702|arxiv = physics/0201064 |bibcode = 2002PhRvE..66b6702M| issue=2 }}
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| * {{Cite journal| author=Marwan, N., Kurths, J. | year=2002 | title=Nonlinear analysis of bivariate data with cross recurrence plots | journal=Physics Letters A | volume=302 | pages=299–307 | doi=10.1016/S0375-9601(02)01170-2|arxiv = physics/0201061 |bibcode = 2002PhLA..302..299M| issue=5–6 }}
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| * {{Cite journal| author=Webber Jr., C. L., Zbilut, J. P. | year=1994 | title=Dynamical assessment of physiological systems and states using recurrence plot strategies | journal=Journal of Applied Physiology | volume=76 | pages=965–973 | pmid=8175612 | issue=2}}
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| * {{Cite journal| author=Zbilut, J.P., Webber Jr., C.L. | year=1992 | title=Embeddings and delays as derived from quantification of recurrence plots | journal=Physics Letters A | volume=171 | pages=199–203 | doi=10.1016/0375-9601(92)90426-M|bibcode = 1992PhLA..171..199Z| issue=3–4 }}
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| ==External links==
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| * http://www.recurrence-plot.tk/
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| * http://www.scitopics.com/Recurrence_Quantification_Analysis.html
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| {{DEFAULTSORT:Recurrence Quantification Analysis}}
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| [[Category:Signal processing]]
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| [[Category:Dynamical systems]]
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| [[Category:Chaos theory]]
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| [[Category:Nonlinear time series analysis]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
When you have virtually any inquiries with regards to wherever as well as how you can employ dentist DC, you are able to contact us from our own web-site.