**Quantification Analysis**- Recurrence Quantification Analysis (Springer, Understanding Complex Systems. (springer.com)
- In contrast to the heuristic approach of the recurrence quantification analysis, which depends on the choice of the embedding parameters, some dynamical invariants as correlation dimension, K2 entropy or mutual information, which are independent on the embedding, can also be derived from recurrence plots. (wikipedia.org)
- Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. (wikipedia.org)
- The recurrence quantification analysis was developed in order to quantify differently appearing recurrence plots (RPs) based on the small-scale structures therein. (wikipedia.org)
- Recurrence quantification analysis, another approach to quantify recurrence properties. (wikipedia.org)

**Eternal Recurrence**- Hypothetically, in the extreme, the concept of historic recurrence assumes the form of the Doctrine of Eternal Recurrence, which has been written about in various forms since antiquity and was described in the 19th century by Heinrich Heine and Friedrich Nietzsche. (wikipedia.org)
- Eternal Recurrence is an EP by Deradoorian, a former member of Dirty Projectors. (wikipedia.org)
- Paul Simpson of AllMusic gave Eternal Recurrence 3.5 stars out of 5, calling it "a fine, thoughtful piece of ambient drone-folk that is as challenging as it is assuring. (wikipedia.org)
- gave Eternal Recurrence a 7 out of 10, stating that "[Eternal Recurrence] pulls back on the rhythmic side of her intercontinental folkloric psychedelia in favour of something more meditative, untethered and texturally focused. (wikipedia.org)
- Xak III: The Eternal Recurrence is the final game in a fantasy role-playing video game series called Xak developed and published by the Japanese software developers MicroCabin and NEC. (wikipedia.org)

**relation**- The recurrence relation can be written as. (mathhelpforum.com)
- So H(2) is divisable by 3 too, and since we have a recurrence relation, this applies to all positive integers following the relation. (mathhelpforum.com)
- How can I Obtain a recurrence relation for the delay suffered by a packet arriving at the input buffer? (mathhelpforum.com)
- Is it possible to find a closed form to a single variable sequence with a multivariable recurrence relation? (mathhelpforum.com)
- In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms. (wikipedia.org)
- The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. (wikipedia.org)
- However, "difference equation" is frequently used to refer to any recurrence relation. (wikipedia.org)
- Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n. (wikipedia.org)
- The recurrence satisfied by the Fibonacci numbers is the archetype of a homogeneous linear recurrence relation with constant coefficients (see below). (wikipedia.org)
- A simple example of a multidimensional recurrence relation is given by the binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} , which count the number of ways of selecting k elements out of a set of n elements. (wikipedia.org)
- However, the Ackermann numbers are an example of a recurrence relation that do not map to a difference equation, much less points on the solution to a differential equation. (wikipedia.org)

**introduced recurrence plots**- 1987) introduced recurrence plots, which provide a way to visualize the periodic nature of a trajectory through a phase space. (wikipedia.org)

**risk of recurrence**- Patients with PSA doubling times of one to 10 years are more likely to have a local rather than systematic recurrence, and patients with a PSA doubling time of greater than 10 years are at a low risk of recurrence. (medindia.net)

**Nonlinear**- Some simply defined recurrence relations can have very complex (chaotic) behaviours, and they are a part of the field of mathematics known as nonlinear analysis. (wikipedia.org)
- Moreover, the recurrence of states, in the meaning that states are again arbitrarily close after some time of divergence, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems (cf. (wikipedia.org)
- The recurrence period density is a sparse representation for nonlinear, non-Gaussian and nondeterministic signals, whereas the Fourier transform is only sparse for purely periodic signals. (wikipedia.org)
- M.A. Little, P.E. McSharry, S.J. Roberts, D.A.E. Costello, I.M. Moroz (2007) Exploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder Detection, BioMedical Engineering OnLine, 6:23 J. Stark, D. S. Broomhead, M. E. Davies and J. Huke (2003) Delay Embeddings for Forced Systems. (wikipedia.org)

**cross recurrence**- Multivariate extensions of recurrence plots were developed as cross recurrence plots and joint recurrence plots. (wikipedia.org)
- Cross-recurrence quantification (CRQ) is a non-linear method that quantifies how similarly two observed data series unfold over time. (wikipedia.org)

**Plots**- Close returns plots are similar to recurrence plots. (wikipedia.org)
- The main advantage of recurrence plots is that they provide useful information even for short and non-stationary data, where other methods fail. (wikipedia.org)
- Recurrence plots are tools which visualise the recurrence behaviour of the phase space trajectory of dynamical systems. (wikipedia.org)
- The RQA quantifies the small-scale structures of recurrence plots, which present the number and duration of the recurrences of a dynamical system. (wikipedia.org)

**displaystyle**- The Fibonacci sequence is defined using the recurrence F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} with initial conditions (seed values) F 0 = 0 {\displaystyle F_{0}=0} F 1 = 1. (wikipedia.org)
- Explicitly, the recurrence yields the equations F 2 = F 1 + F 0 {\displaystyle F_{2}=F_{1}+F_{0}} F 3 = F 2 + F 1 {\displaystyle F_{3}=F_{2}+F_{1}} F 4 = F 3 + F 2 {\displaystyle F_{4}=F_{3}+F_{2}} etc. (wikipedia.org)
- An order-d homogeneous linear recurrence with constant coefficients is an equation of the form a n = c 1 a n − 1 + c 2 a n − 2 + ⋯ + c d a n − d , {\displaystyle a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+\cdots +c_{d}a_{n-d},} where the d coefficients ci (for all i) are constants. (wikipedia.org)
- To make the plot, continuous time and continuous phase space are discretized, taking e.g. x → ( i ) {\displaystyle {\vec {x}}(i)} as the location of the trajectory at time i τ {\displaystyle i\tau } and counting as a recurrence any time the trajectory gets sufficiently close (say, within ε) to a point it has been previously. (wikipedia.org)
- The difference is that the relative time between recurrences is used for the y {\displaystyle y} -axis (instead of absolute time). (wikipedia.org)
- The simplest measure is the recurrence rate, which is the density of recurrence points in a recurrence plot: RR = 1 N 2 ∑ i , j = 1 N R ( i , j ) . {\displaystyle {\text{RR}}={\frac {1}{N^{2}}}\sum _{i,j=1}^{N}\mathbf {R} (i,j). (wikipedia.org)

**Dynamical Systems**- Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal. (wikipedia.org)

**periodic**- For a given phase in a volume, the recurrence is not necessarily a periodic recurrence. (wikipedia.org)

**plot**- In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space. (wikipedia.org)
- However, making a recurrence plot enables us to investigate certain aspects of the m-dimensional phase space trajectory through a two-dimensional representation. (wikipedia.org)
- Caused by characteristic behaviour of the phase space trajectory, a recurrence plot contains typical small-scale structures, as single dots, diagonal lines and vertical/horizontal lines (or a mixture of the latter, which combines to extended clusters). (wikipedia.org)
- 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. (wikipedia.org)
- Recurrence plot, a powerful visualisation tool of recurrences in dynamical (and other) systems. (wikipedia.org)

**2002**- Relentless Recurrence is the third studio album by the Taiwanese black metal band Chthonic, released in 2002. (wikipedia.org)

**Webber**- We would like to thank C.L. Webber, Jr. for the invitation to contribute a chapter to this book on Recurrence Analysis after one of us (H.L.) gave a presentation at the Recurrence Symposium in Grenoble, France. (springer.com)

**yields**- It is also a consequence of Krylov-Bogolyubov theorem, or existence of invariant probability measures theorem, which asserts that every continuous transformation in a compact metric space admits an invariant probability measure, and an application of Poincaré Recurrence theorem to that invariant probability measure yields Birkhoff Recurrence theorem. (planetmath.org)

**sequence**- A constant-recursive sequence is a sequence satisfying a recurrence of this form. (wikipedia.org)
- Recurrence period density entropy is useful for characterising the extent to which a time series repeats the same sequence, and is therefore similar to linear autocorrelation and time delayed mutual information, except that it measures repetitiveness in the phase space of the system, and is thus a more reliable measure based upon the dynamics of the underlying system that generated the signal. (wikipedia.org)

**sequences**- Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). (wikipedia.org)
- Recurrence sequences. (wikipedia.org)

**diagonal lines**- The base for these dynamical invariants are the recurrence rate and the distribution of the lengths of the diagonal lines. (wikipedia.org)

**tumor**- Many factors can cause a rise in tumor markers other than a recurrence. (medhelp.org)
- The current standards are: If there is no internal disease and recurrence is confined to the area of the original tumor, surgery and/or radiation therapy is used, plus tamoxifen in selected cases. (medhelp.org)

**probability**- The recurrence rate corresponds with the probability that a specific state will recur. (wikipedia.org)

**percentage**- CRQ produces measures reflecting coordination, such as how often two data series have similar values or reflect similar system states (called percentage recurrence, or %REC), among other measures. (wikipedia.org)

**form**- Since difference equations are a very common form of recurrence, some authors use the two terms interchangeably. (wikipedia.org)

**Theorem**- There is also a generalization of Birkhoff recurrence theorem for multiple commuting transformations, known as Birkhoff Multiple Recurrence theorem. (planetmath.org)
- In physics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. (wikipedia.org)
- Q.E.D.. The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee: There may be some special phases that never return to the starting phase volume, or that only return to the starting volume a finite number of times then never return again. (wikipedia.org)

**depends**- The recurrence time depends on the required degree of closeness (the size of the phase volume). (wikipedia.org)

**historic**- Historic recurrence is the repetition of similar events in history. (wikipedia.org)
- The concept of historic recurrence has variously been applied to the overall history of the world (e.g., to the rises and falls of empires), to repetitive patterns in the history of a given polity, and to any two specific events which bear a striking similarity. (wikipedia.org)
- Historic recurrences can sometimes induce a sense of "convergence", "resonance" or déjà vu. (wikipedia.org)
- Prior to the theory of historic recurrence that was offered by Polybius, a Greek Hellenistic historian (ca 200 - ca 118 BCE), ancient western thinkers who had thought about recurrence had largely been concerned with cosmological rather than historic recurrence. (wikipedia.org)
- Western philosophers and historians who have discussed various concepts of historic recurrence include Polybius, the Greek historian and rhetorician Dionysius of Halicarnassus (c. 60 BCE - after 7 BCE), Luke the Evangelist, Niccolò Machiavelli (1469-1527), Giambattista Vico (1668-1744), Arnold J. Toynbee (1889-1975). (wikipedia.org)
- An eastern concept that bears a kinship to western concepts of historic recurrence is the Chinese concept of the Mandate of Heaven, by which an unjust ruler will lose the support of Heaven and be overthrown. (wikipedia.org)
- G.W. Trompf notes that most western concepts of historic recurrence imply that "the past teaches lessons for. (wikipedia.org)

**investigate**- In this chapter, the potential of Recurrence Analysis (RA) to investigate the dynamics of this atmosphere-vegetation boundary system is elucidated. (springer.com)

**quantifies**- It quantifies the number and duration of recurrences of a dynamical system presented by its phase space trajectory. (wikipedia.org)

**cancer**- Learn more about cancer recurrence and living with this possibility here. (cancer.org)
- What Is Cancer Recurrence? (cancer.org)
- If cancer is found after treatment, and after a period of time when the cancer couldn't be detected, it's called a cancer recurrence. (cancer.org)
- Can I Do Anything to Prevent Cancer Recurrence? (cancer.org)
- Many issues and questions come with cancer recurrence. (cancer.org)
- Easy reading information about cancer recurrence. (cancer.org)
- Do they suspect recurrence of breast cancer? (medhelp.org)
- In a new Mayo Clinic study, the concept of PSA doubling time (DT) is found to be a reliable tool to distinguish which patients have prolonged innocuous PSA levels after therapy from those who are at great risk for disease recurrence and death from prostate cancer. (medindia.net)
- UCA President Lu Hardin today told college Board members of a recurrence of his eye cancer, which will require surgery in August. (arktimes.com)
- Biochemical recurrence is a rise in the blood level of prostate-specific antigen (PSA) in prostate cancer patients after treatment with surgery or radiation. (wikipedia.org)
- Biochemical recurrence Defining Biochemical Recurrence of Prostate Cancer After Radical Prostatectomy: A Proposal for a Standardized Definition. (wikipedia.org)
- 2005 Guidelines:Prostate cancer/Management/Locally advanced and metastatic/Biochemical relapse alternative causes, and implications for treatment Defining Biochemical Recurrence of Prostate Cancer After Radical Prostatectomy: A Proposal for a Standardized Definition 10 definitions compared. (wikipedia.org)
- Electronic Saviors Volume 2: Recurrence is the second Electronic Saviors cancer charity compilation compiled by Jim Semonik of Rein[Forced], a cancer survivor. (wikipedia.org)

**packet**- A recurrence tracking microscope (RTM) is a microscope that is based on the quantum recurrence phenomenon of an atomic wave packet. (wikipedia.org)
- In 2006, Farhan Saif used quantum recurrence phenomena of wave packet as a probe to study nano-structures on a surface, naming it Recurrence Tracking Microscope (RTM). (wikipedia.org)

**time**- The recurrence of states in nature has been known for a long time and has also been discussed in early work (e.g. (wikipedia.org)
- A recurrence is a time the trajectory returns to a location it has visited before. (wikipedia.org)
- The Poincaré recurrence time is the length of time elapsed until the recurrence (this time may vary greatly depending on the exact initial state and required degree of closeness). (wikipedia.org)
- To achieve greater accuracy of recurrence, we need to take smaller initial volume, which means longer recurrence time. (wikipedia.org)
- The second recurrence time does not need to be double the first recurrence time. (wikipedia.org)

**Given**- Recurrence Index (RI) in genomics denotes the number of mutations per kilobase of a given genomic locus of a patient carrying particular mutations. (wikipedia.org)

**similar**- I need(ed) an outlook-similar recurrence setup for my application. (sencha.com)

**relations**- Multi-variable or n-dimensional recurrence relations are about n-dimensional grids. (wikipedia.org)

**cases**- Other minor cases of recurrence thinking," he writes, "include the isolation of any two specific events which bear a very striking similarity [his emphasis], and the preoccupation with parallelism [his emphasis], that is, with resemblances, both general and precise, between separate sets of historical phenomena. (wikipedia.org)

**take**- Recurrences take place due to ascertainable circumstances and chains of causality. (wikipedia.org)

**treatment**- After having a lumpectomy, chemo and radiation my wife is 2 1/2 years out from treatment and has a recurrence. (medhelp.org)

**book**- G.W. Trompf, in his book The Idea of Historical Recurrence in Western Thought, traces historically recurring patterns of political thought and behavior in the west since antiquity. (wikipedia.org)

**level**- RI represents a patient level recurrence frequency estimated for somatic mutations. (wikipedia.org)