Non linear and linear EEG analysis Flashcards
What are the characteristics of a Dynamical System?
It is a model that determines the evolution of a system given only its initial state
Systems have memory - the current state is a function of the previous state
A dynamic system has two properties - state and dynamics
Dynamics of a system refer to the set of laws that describe how the state of the system changes over time
What is trajectory?
It is the sequence of points that solve the equation for dynamic system. It can run away to infinity or be confined to a certain space
What is an attractor?
It is a geometrical object to which dynamical system evolves after a long enough time.
It attracts trajectories from variety of initial conditions.
Name the 4 types of attractors and briefly describe them
Fixed point - the system settles on a point and after which no further changes can occur without external force
Limit cycle - closed loop in the state space
Torus - complex and donut shaped
Strange attractor - a complex object with fractal geometry, usually associated with chaotic dynamics
What is the top-down approach to reconstructing the original state space by only a scalar time serious?
Top down approach allows to reconstruct the state space from the data without knowledge of undelaying dynamics Reconstruct the state space and characterise its validity though surrogate data analysis Use embedding (for state space reconstruction)..there are two types of embedding (time delay and special)
How are embedding parametres chosen?
Data is ideally noise free and measured with infinite precision, however that is not often possible. So search for:
optimal time delay
minimum embedding dimension
optimal time window
There is however no unique method that can solve all issues to set embedding parametres appropriate for all purposes
What are the time delay embedding characteristics ?
If time delay is too small = reconstructed vector will be almost equal and collapse into a diagonal
If time delay is too large=reconstructed vectors will consist of irrelevant components and will fill in the entire space
How to choose time delay?
Based on:
Autocorrelation
Time delayed mutual information BUT:
- the reconstructed attractor should expand from the diagonal but not too much to fold back
-the components of any state vector must be as uncorrelated as possible
How to choose embedding dimension (m)?
Though the methods of:
False nearest neighbour
start with a low “d” and then increase the embedding dimension gradually, while keeping track of the neighbourhood geometry
d0 is the minimum embedding dimension for which no additional false neighbours are found
How is state space characterised?
Correlation dimension
lyapunov exponent
Define the correlation dimension and its characteristics
It is based on the concept of how densely the points of an attractor aggregate around one another. Its estimation is related to the relative frequency with which the attractor visits each covering element
Correlation dimension is:
-computationally different
- biased by autocorrelation
-influenced by noise
- less reliable for high dimensional system
-certain types of noise can lead to finite correlation dimension
What is Lyapunov exponent?
It describes the rate at which two neighbouring trajectories converge or diverge. It also provides an estimate of inherent predictability
- For a system to be chaotic at least one Lyapunov exponent should be positive
- Number of Lyapunov exponents is equal to the number of dimensions in the state space
What is a chaotic system?
It is low-dimensional nonlinear system. Therefore, non linearity is a condition for chaos.
What is surrogate analysis?
It is an analysis that allows to detect nonlinearity in the data
What are Fourier surrogates?
The surrogate signal has the same mean, variance, power spectral density of the original signal
The surrogate signal has no dynamical nonlinearities
