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Related Concept Videos

Absolute and Local Extreme Values01:22

Absolute and Local Extreme Values

The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...
Parametric Survival Analysis: Weibull and Exponential Methods01:14

Parametric Survival Analysis: Weibull and Exponential Methods

Parametric survival analysis models survival data by assuming a specific probability distribution for the time until an event occurs. The Weibull and exponential distributions are two of the most commonly used methods in this context, due to their versatility and relatively straightforward application.
Weibull Distribution
The Weibull distribution is a flexible model used in parametric survival analysis. It can handle both increasing and decreasing hazard rates, depending on its shape parameter...
BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
Pole and System Stability01:24

Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.
Unusual Results01:16

Unusual Results

Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes.
According to the range rule of thumb, any value above or below two standard deviations, 2σ  from the mean, μ  is considered unusual.
Maximum unusual value = μ + 2σ
Minimum unusual value...
Critical Numbers and the Closed Interval Method01:21

Critical Numbers and the Closed Interval Method

Understanding the maximum and minimum values of a function is essential for analyzing its overall behavior. These values, often referred to as extrema, provide insight into how a function behaves across its domain. In mathematical terms, extrema can be either local—representing peaks and valleys within a limited region—or absolute, indicating the highest or lowest points over an entire interval.A function’s extrema occur at critical numbers, which are values in the domain where the derivative...

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How to Create and Use Binocular Rivalry
14:34

How to Create and Use Binocular Rivalry

Published on: November 10, 2010

Extreme events in bimodal systems.

S C Nicolis1, C Nicolis

  • 1Mathematics Department, Uppsala University, P.O. Box 480 SE-751 06 Uppsala, Sweden.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2008
PubMed
Summary

Systems with a pronounced left peak in their probability density exhibit anomalous fluctuations. These non-trivial transient behaviors occur within specific observational time windows, impacting extreme value statistics.

Area of Science:

  • Statistical physics
  • Complex systems analysis

Background:

  • Understanding extreme value statistics is crucial for analyzing complex systems.
  • Systems with multi-modal probability densities present unique statistical challenges.

Purpose of the Study:

  • To investigate the extreme value statistics of systems with a two-hump probability density.
  • To characterize anomalous transient behaviors and fluctuations in such systems.

Main Methods:

  • Analysis of probability density functions with asymmetric peaks.
  • Study of systems with two locally stable states under additive white noise.
  • Examination of dynamical systems in the deterministic chaos regime.

Main Results:

  • Identified non-trivial transient behavior characterized by anomalous fluctuations around the mean.

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  • Demonstrated these effects occur within finite observational time windows.
  • Illustrated findings across independent random variables, bistable systems with noise, and chaotic dynamics.

    • Conclusions:

    • The specific shape of the probability density function significantly influences system dynamics and extreme value statistics.
    • Anomalous fluctuations are a key feature of systems with asymmetric two-hump densities.
    • These findings have implications for diverse fields modeling complex phenomena.